Model independent analysis of B →  (1430)μ+μ−decay

36
JHEP02(2012)045 Published for SISSA by Springer Received: October 16, 2011 Revised: December 21, 2011 Accepted: January 23, 2012 Published: February 15, 2012 Model independent analysis of B K * 2 (1430)μ + μ - decay Ishtiaq Ahmed, a,b M. Jamil Aslam, a,b,c Muhammad Junaid a,b and Saba Shafaq b a National Centre for Physics, Quaid-i-Azam University Campus, Islamabad 45320, Pakistan b Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan c Physics Department, University of Alberta, Edmonton, T6G 2G7, Alberta, Canada E-mail: [email protected], [email protected], [email protected], [email protected] Abstract: A detailed study of the impact of New Physics (NP) operators with differ- ent Lorentz structures, which are absent in the Standard Model Hamiltonian, on the B K 2 (1430)µ + µ decay is performed. In this context, the various observables such as branching ratio, forward-backward asymmetry of leptons, lepton polarization asymme- tries and the helicity fractions of the final state K 2 (1430) meson have been studied. We have examined the effects of new vector-axial vector, scalar-pseudoscalar and tensor type interactions for this decay B K 2 (1430)µ + µ by using the constraints on different NP couplings which come from the B s µ + µ , B X s µ + µ and ¯ B ¯ K µ + µ decays. It is found that the effects of VA, SP and T operators are significant on the zero position of A FB (q 2 ) as well as on its magnitude. In addition to this these NP operators also give sig- nificant effects on the differential decay rate, lepton polarization asymmetries and helicities fractions of final state K 2 (1430) meson. Keywords: Rare Decays, Beyond Standard Model, B-Physics c SISSA 2012 doi:10.1007/JHEP02(2012)045

Transcript of Model independent analysis of B →  (1430)μ+μ−decay

Page 1: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

Published for SISSA by Springer

Received October 16 2011

Revised December 21 2011

Accepted January 23 2012

Published February 15 2012

Model independent analysis of B rarr Klowast

2(1430)micro+microminus

decay

Ishtiaq Ahmedab M Jamil Aslamabc Muhammad Junaidab and Saba Shafaqb

aNational Centre for Physics Quaid-i-Azam University Campus

Islamabad 45320 PakistanbDepartment of Physics Quaid-i-Azam University

Islamabad 45320 PakistancPhysics Department University of Alberta

Edmonton T6G 2G7 Alberta Canada

E-mail ishtiaqncpedupk jamilncpedupk mjunaidncpedupk

sabaqauedupk

Abstract A detailed study of the impact of New Physics (NP) operators with differ-

ent Lorentz structures which are absent in the Standard Model Hamiltonian on the

B rarr Klowast2 (1430)micro

+microminus decay is performed In this context the various observables such

as branching ratio forward-backward asymmetry of leptons lepton polarization asymme-

tries and the helicity fractions of the final state Klowast2 (1430) meson have been studied We

have examined the effects of new vector-axial vector scalar-pseudoscalar and tensor type

interactions for this decay B rarr Klowast2 (1430)micro

+microminus by using the constraints on different NP

couplings which come from the Bs rarr micro+microminus B rarr Xsmicro+microminus and B rarr Klowastmicro+microminus decays It

is found that the effects of V A SP and T operators are significant on the zero position of

AFB(q2) as well as on its magnitude In addition to this these NP operators also give sig-

nificant effects on the differential decay rate lepton polarization asymmetries and helicities

fractions of final state Klowast2 (1430) meson

Keywords Rare Decays Beyond Standard Model B-Physics

ccopy SISSA 2012 doi101007JHEP02(2012)045

JHEP02(2012)045

Contents

1 Introduction 1

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements 3

21 Standard model 3

22 New physics operators 4

23 Parametrization of matrix elements and form factors 5

24 Numerical constraints on NP parameters 6

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay 8

31 Differential decay rate 9

32 Forward-backward asymmetry 10

33 Lepton polarization asymmetries 10

34 Helicity fractions 12

4 Numerical Analysis 13

41 V A new-physics operators 15

42 SP new-physics operators 20

43 T operators and interference of SP and T operators 22

44 All NP operators together 26

5 Conclusions 30

1 Introduction

The wealthy data taken from the Tevatron experiments on Bs and the other B-factories

have reached to the luminosity where billions of B mesons will be produced which will help

us to enter into the precision era of B physics The Standard Model (SM) is enormously

successful in explaining most of the data to date However some discrepancies in the

predication of the SM have been observed in B decays [1ndash4] The pertinent examples are

(i) in B rarr Kπ decays it is difficult to account for all the experimental measurements of

CP asymmetry in neutral and charged modes within the SM [7] (ii) the B0s minus B0

s mixing

phase by the CDF and DO collaboration deviates from the SM prediction [5 6] (iii) the

isospin asymmetry between neutral and the charged decay modes of the B rarr Klowastℓ+ℓminus

decay also deviate from the SM [8 9] The Large Hadron Collider (LHC) is already up

and running where CMS ATLAS and LHCb have already started taking data while the

Super-B factories are on their way Therefore it is an ideal time to test the predictions of

SM and try to identify the nature of physics that is beyond SM

ndash 1 ndash

JHEP02(2012)045

In SM the zero crossing of the AFB(q2) in B rarr Klowastl+lminus is at a well determined

position which is free from the hadronic certainties at the leading order (LO) in strong

coupling αs [11ndash13] Recently the LHCb has announced the results of AFB for the decay

B rarr Klowastmicro+microminus that are close to the SM predictions They have collected over 300 events

for this decay with signal to background ratio above 3 which is the largest sample of such

decays in the world and is even cleaner than the sample used by the B factories These

results of AFB(B rarr Klowastmicro+microminus) are close to the SM predictions with slight error bars thus

they have overwritten the earlier measurements by Belle with better statistics [15] The

collaboration plans to continue to study the channel B rarr Klowastmicro+microminus in finer detail with

measurements that include more angular variables and expected to achieve high sensitivity

to any small deviation from the SM [15]

In order to incorporate the experimental predictions of different physical observables

in B rarr Klowastmicro+microminus this decay has been studied in SM and in number of different NP scenar-

ios [10 12 16ndash36] where NP effects display themselves through modification in the Wilson

coefficients as well as through the new operators Beside these models the general analysis

of B rarr Klowastmicro+microminus decay has also been performed which allow us to include all possible

NP effects like vector-axial vector (V A) scalar-pseudoscalar (SP ) and tensor-axial tensor

(TE) [37] This study has shown that the interference of V A operators among themselves

and with the SM operators is the cause of the larger magnitude of AFB(q2) as well as the

shift in the zero position of the AFB(q2) In addition to this it is also pointed out that the

SP and T operators not only influence the AFB(q2) but also give viable effects in other

physical observables

In general there are two ways to search NP which are considered to resolve anomalies

in the experimental measurements One is the direct search which can be achieved by

increasing the energies of colliders and find the predicted new particles but these particles

are quite massive and therefore hard to produced The other one is indirect search which

is comparatively easier ie to see how much different observables are affected by these new

particles The indirect searches require precise measurement of the observables for the

channel under consideration as well as need rigorous complementary studies

It is therefore indispensable to study the impact of NP on the different observables for

the decay channels other than the B rarr Klowastmicro+microminus With this motivation the B rarr Klowast2ℓ

+ℓminus

channel where Klowast2 is the excited state of Klowast with parity 2+ provide the same testing

ground to study the effects of the NP as B rarr Klowastmicro+microminus decay The exclusive B rarrKlowast

2 (1430)ℓ+ℓminus decay has been studied in the SM and some approaches beyond SM [38ndash46]

For example in refs [38 39] the form factors for B rarr Klowast2ℓ

+ℓminus are calculated using pertur-

bative QCD and LCSR approaches The analysis of different physical observable for this

decay is studied in various beyond SM scenarios like SM4 non-universal Z prime model Uni-

versal Extra dimension model and a model with the flipped sign of Ceff7 has been performed

in [40ndash42 45] The analysis of lepton polarization asymmetries of B rarr Klowast2ℓ

+ℓminus in the pres-

ence of new right-handed currents is done in [43 44] The analysis of polarized decay width

and forward-backward asymmetry by considering beyond SM operator basis is given in [46]

These studies show that like B rarr Klowastℓ+ℓminus decay this channel is also quite promising in

finding NP and will provide us some independent tests of the physics of the SM and beyond

ndash 2 ndash

JHEP02(2012)045

In this paper we will perform the study of B rarr Klowast2 (1430)micro

+microminus on the lines of ref [37]

and study the NP effects without restricting ourselves to any specific model Contrary to

B rarr Klowastmicro+microminus the decay B rarr Klowast2 (1430)micro

+microminus is not yet experimentally observed We have

taken into account the constraints from the different B decay mode such as Bs rarr micro+microminus

B rarr Xsmicro+microminus and B rarr Klowastmicro+microminus on different NP operator couplings allowed by the

most general lorentz structure of transition matrix elements In this paper we will also

incorporate bounds on the input parameters coming from the recent AFB measurements

at LHCb experiment for the decay B rarr Klowast(892)micro+microminus [15] With these new bounds we

will obtain better understanding of how these new physics operators affect branching ratio

zero-position of forward-backward asymmetry different lepton polarization asymmetries

and the helicity fractions of final state meson

The effects of these new operators couplings (VASPT) are significant to the zero

position of AFB(q2) as well as on its magnitude In addition to AFB these NP operators

also give significant effects on decay rate lepton polarization asymmetries and longitudinal

and transverse helicities of final state Klowast2 (1430) meson We hope that in the coming years

the LHCb experiment will collect around 64k events in the full range of q2 for an integrated

luminosity of 2fbminus1 [37] This would allow us to observe the B rarr Klowast2 (1430)micro

+microminus in the

SM and would also permit many of the additional tests of NP

The paper is organized as follows in section 2 the necessary hadronic inputs namely

basic Hamiltonian and form factors and constraints on the NP parameters are collected

Section 3 contains the analytic formulas for differential decay rates forward backward

asymmetry different polarizations (longitudinal normal and transverse) and the helicity

fractions of the final state Klowast2 (1430) meson Section 4 deals with the numerical analysis

and then conclusions are summarized in section 5

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements

21 Standard model

The semileptonic decay mode B rarr Klowast2 (1430)l

+lminus where l = micro for our case is governed

by the quark level transitions b rarr sl+lminus In SM the effective Hamiltonian corresponding

to this transition is given by

HSMeff = minus4GFradic

2V lowasttbVts

10sum

i=1

Ci(micro)Oi(micro) (21)

where Oi(micro)(i = 1 10) are four quark operators and Ci(micro) are the corresponding Wilson

coefficients Out of these 10 operators the O7 O9 and O10 are the relevant for B rarrKlowast

2 (1430)l+lminus decay and these can be written as

O7 =e

16π2[sσmicroν (msPL +mbPR) b]F

microν

O9 =e

16π2[sγmicroPLb]

(

lγmicrol)

O10 =e

16π2[sγmicroPLb]

(

lγmicroγ5l)

(22)

with PLR = (1plusmn γ5) 2

ndash 3 ndash

JHEP02(2012)045

In term of these operators the effective Hamiltonian for B rarr Klowast2 (1430)l

+lminus takes the

form

HSMeff = minusGFαradic

2πVtbV

lowastts

CSM9 (sγmicroPLb)(lγ

microl) + CSM10 (sγmicroPLb)(lγ

microγ5l)

minus2mbCSM7 (siσmicroν

q2PRb)(lγ

microl)

(23)

where q2 is square of the sum of the momentum of final state leptons At the scale micro =

48GeV the SM Wilson coefficients takes the following values which are calculated at next-

to-next-to-leading order

Ceff7 = minus034 Ceff

9 = 4211 + Y (q2) C10 = minus4103 (24)

where Ceff7 = C7 minus C33minus 4C49minus 20C53minus 80C69 The function Y (q2) is given by

Y (z q2) = h(z q2)(3C1(micro) + C2(micro) + 3C3(micro) + C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(1 sprime)(4C3(micro) + 4C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(0 q2)(C3(micro) + 3C4(micro)) +

2

9(3C3(micro) + C4(micro) + 3C5(micro) + C6(micro)) (25)

with

h(z q2) = minus8

9lnz +

8

27+

4

9xminus 2

9(2 + x)|1minus x|12

ln∣

radic1minusx+1radic1minusxminus1

∣minus iπ for x equiv 4z2q2 lt 1

2 arctan 1radicxminus1

for x equiv 4z2q2 gt 1

h(0 q2) =8

27minus 8

9lnmb

microminus 4

9lnq2 +

4

9iπ (26)

and z = mcmb

22 New physics operators

The NP effects in semileptonic B meson decays can be established in two ways in one case

we can change the Wilson Coefficients but leave the operator basis same as that of the SM

and in other case we can add new operators too Here we do not restrict ourself to any

model and discuss all possible operators that corresponds to B rarr Klowast2 (1430)l

+lminus decays

In this scenario the total effective Hamiltonian for brarr sl+lminus can be written as

Heff = HSMeff +HV A

eff +HSPeff +HT

eff (27)

where HSMeff is given in eq (23) while [37]

HV Aeff =minus 4GFradic

2

α

4πV lowasttsVtb RV (sγmicroPLb) microγmicromicro+RA (sγmicroPLb) microγmicroγ5micro

+RprimeV (sγmicroPRb) microγmicromicro+Rprime

A (sγmicroPRb) microγmicroγ5micro

(28)

HSPeff =minus 4GFradic

2

α

4πV lowasttsVtb RS (sPRb) micromicro+RP (sPRb) microγ5micro

+RprimeS (sPLb) micromicro+Rprime

P (sPLb) microγ5micro

(29)

HTeff =minus 4GFradic

2

α

4πV lowasttsVtb

CT (sσmicroνb) microσmicroνmicro+ iCTE (sσmicroνb) microσαβmicroǫmicroναβ

(210)

ndash 4 ndash

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 2: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

Contents

1 Introduction 1

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements 3

21 Standard model 3

22 New physics operators 4

23 Parametrization of matrix elements and form factors 5

24 Numerical constraints on NP parameters 6

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay 8

31 Differential decay rate 9

32 Forward-backward asymmetry 10

33 Lepton polarization asymmetries 10

34 Helicity fractions 12

4 Numerical Analysis 13

41 V A new-physics operators 15

42 SP new-physics operators 20

43 T operators and interference of SP and T operators 22

44 All NP operators together 26

5 Conclusions 30

1 Introduction

The wealthy data taken from the Tevatron experiments on Bs and the other B-factories

have reached to the luminosity where billions of B mesons will be produced which will help

us to enter into the precision era of B physics The Standard Model (SM) is enormously

successful in explaining most of the data to date However some discrepancies in the

predication of the SM have been observed in B decays [1ndash4] The pertinent examples are

(i) in B rarr Kπ decays it is difficult to account for all the experimental measurements of

CP asymmetry in neutral and charged modes within the SM [7] (ii) the B0s minus B0

s mixing

phase by the CDF and DO collaboration deviates from the SM prediction [5 6] (iii) the

isospin asymmetry between neutral and the charged decay modes of the B rarr Klowastℓ+ℓminus

decay also deviate from the SM [8 9] The Large Hadron Collider (LHC) is already up

and running where CMS ATLAS and LHCb have already started taking data while the

Super-B factories are on their way Therefore it is an ideal time to test the predictions of

SM and try to identify the nature of physics that is beyond SM

ndash 1 ndash

JHEP02(2012)045

In SM the zero crossing of the AFB(q2) in B rarr Klowastl+lminus is at a well determined

position which is free from the hadronic certainties at the leading order (LO) in strong

coupling αs [11ndash13] Recently the LHCb has announced the results of AFB for the decay

B rarr Klowastmicro+microminus that are close to the SM predictions They have collected over 300 events

for this decay with signal to background ratio above 3 which is the largest sample of such

decays in the world and is even cleaner than the sample used by the B factories These

results of AFB(B rarr Klowastmicro+microminus) are close to the SM predictions with slight error bars thus

they have overwritten the earlier measurements by Belle with better statistics [15] The

collaboration plans to continue to study the channel B rarr Klowastmicro+microminus in finer detail with

measurements that include more angular variables and expected to achieve high sensitivity

to any small deviation from the SM [15]

In order to incorporate the experimental predictions of different physical observables

in B rarr Klowastmicro+microminus this decay has been studied in SM and in number of different NP scenar-

ios [10 12 16ndash36] where NP effects display themselves through modification in the Wilson

coefficients as well as through the new operators Beside these models the general analysis

of B rarr Klowastmicro+microminus decay has also been performed which allow us to include all possible

NP effects like vector-axial vector (V A) scalar-pseudoscalar (SP ) and tensor-axial tensor

(TE) [37] This study has shown that the interference of V A operators among themselves

and with the SM operators is the cause of the larger magnitude of AFB(q2) as well as the

shift in the zero position of the AFB(q2) In addition to this it is also pointed out that the

SP and T operators not only influence the AFB(q2) but also give viable effects in other

physical observables

In general there are two ways to search NP which are considered to resolve anomalies

in the experimental measurements One is the direct search which can be achieved by

increasing the energies of colliders and find the predicted new particles but these particles

are quite massive and therefore hard to produced The other one is indirect search which

is comparatively easier ie to see how much different observables are affected by these new

particles The indirect searches require precise measurement of the observables for the

channel under consideration as well as need rigorous complementary studies

It is therefore indispensable to study the impact of NP on the different observables for

the decay channels other than the B rarr Klowastmicro+microminus With this motivation the B rarr Klowast2ℓ

+ℓminus

channel where Klowast2 is the excited state of Klowast with parity 2+ provide the same testing

ground to study the effects of the NP as B rarr Klowastmicro+microminus decay The exclusive B rarrKlowast

2 (1430)ℓ+ℓminus decay has been studied in the SM and some approaches beyond SM [38ndash46]

For example in refs [38 39] the form factors for B rarr Klowast2ℓ

+ℓminus are calculated using pertur-

bative QCD and LCSR approaches The analysis of different physical observable for this

decay is studied in various beyond SM scenarios like SM4 non-universal Z prime model Uni-

versal Extra dimension model and a model with the flipped sign of Ceff7 has been performed

in [40ndash42 45] The analysis of lepton polarization asymmetries of B rarr Klowast2ℓ

+ℓminus in the pres-

ence of new right-handed currents is done in [43 44] The analysis of polarized decay width

and forward-backward asymmetry by considering beyond SM operator basis is given in [46]

These studies show that like B rarr Klowastℓ+ℓminus decay this channel is also quite promising in

finding NP and will provide us some independent tests of the physics of the SM and beyond

ndash 2 ndash

JHEP02(2012)045

In this paper we will perform the study of B rarr Klowast2 (1430)micro

+microminus on the lines of ref [37]

and study the NP effects without restricting ourselves to any specific model Contrary to

B rarr Klowastmicro+microminus the decay B rarr Klowast2 (1430)micro

+microminus is not yet experimentally observed We have

taken into account the constraints from the different B decay mode such as Bs rarr micro+microminus

B rarr Xsmicro+microminus and B rarr Klowastmicro+microminus on different NP operator couplings allowed by the

most general lorentz structure of transition matrix elements In this paper we will also

incorporate bounds on the input parameters coming from the recent AFB measurements

at LHCb experiment for the decay B rarr Klowast(892)micro+microminus [15] With these new bounds we

will obtain better understanding of how these new physics operators affect branching ratio

zero-position of forward-backward asymmetry different lepton polarization asymmetries

and the helicity fractions of final state meson

The effects of these new operators couplings (VASPT) are significant to the zero

position of AFB(q2) as well as on its magnitude In addition to AFB these NP operators

also give significant effects on decay rate lepton polarization asymmetries and longitudinal

and transverse helicities of final state Klowast2 (1430) meson We hope that in the coming years

the LHCb experiment will collect around 64k events in the full range of q2 for an integrated

luminosity of 2fbminus1 [37] This would allow us to observe the B rarr Klowast2 (1430)micro

+microminus in the

SM and would also permit many of the additional tests of NP

The paper is organized as follows in section 2 the necessary hadronic inputs namely

basic Hamiltonian and form factors and constraints on the NP parameters are collected

Section 3 contains the analytic formulas for differential decay rates forward backward

asymmetry different polarizations (longitudinal normal and transverse) and the helicity

fractions of the final state Klowast2 (1430) meson Section 4 deals with the numerical analysis

and then conclusions are summarized in section 5

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements

21 Standard model

The semileptonic decay mode B rarr Klowast2 (1430)l

+lminus where l = micro for our case is governed

by the quark level transitions b rarr sl+lminus In SM the effective Hamiltonian corresponding

to this transition is given by

HSMeff = minus4GFradic

2V lowasttbVts

10sum

i=1

Ci(micro)Oi(micro) (21)

where Oi(micro)(i = 1 10) are four quark operators and Ci(micro) are the corresponding Wilson

coefficients Out of these 10 operators the O7 O9 and O10 are the relevant for B rarrKlowast

2 (1430)l+lminus decay and these can be written as

O7 =e

16π2[sσmicroν (msPL +mbPR) b]F

microν

O9 =e

16π2[sγmicroPLb]

(

lγmicrol)

O10 =e

16π2[sγmicroPLb]

(

lγmicroγ5l)

(22)

with PLR = (1plusmn γ5) 2

ndash 3 ndash

JHEP02(2012)045

In term of these operators the effective Hamiltonian for B rarr Klowast2 (1430)l

+lminus takes the

form

HSMeff = minusGFαradic

2πVtbV

lowastts

CSM9 (sγmicroPLb)(lγ

microl) + CSM10 (sγmicroPLb)(lγ

microγ5l)

minus2mbCSM7 (siσmicroν

q2PRb)(lγ

microl)

(23)

where q2 is square of the sum of the momentum of final state leptons At the scale micro =

48GeV the SM Wilson coefficients takes the following values which are calculated at next-

to-next-to-leading order

Ceff7 = minus034 Ceff

9 = 4211 + Y (q2) C10 = minus4103 (24)

where Ceff7 = C7 minus C33minus 4C49minus 20C53minus 80C69 The function Y (q2) is given by

Y (z q2) = h(z q2)(3C1(micro) + C2(micro) + 3C3(micro) + C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(1 sprime)(4C3(micro) + 4C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(0 q2)(C3(micro) + 3C4(micro)) +

2

9(3C3(micro) + C4(micro) + 3C5(micro) + C6(micro)) (25)

with

h(z q2) = minus8

9lnz +

8

27+

4

9xminus 2

9(2 + x)|1minus x|12

ln∣

radic1minusx+1radic1minusxminus1

∣minus iπ for x equiv 4z2q2 lt 1

2 arctan 1radicxminus1

for x equiv 4z2q2 gt 1

h(0 q2) =8

27minus 8

9lnmb

microminus 4

9lnq2 +

4

9iπ (26)

and z = mcmb

22 New physics operators

The NP effects in semileptonic B meson decays can be established in two ways in one case

we can change the Wilson Coefficients but leave the operator basis same as that of the SM

and in other case we can add new operators too Here we do not restrict ourself to any

model and discuss all possible operators that corresponds to B rarr Klowast2 (1430)l

+lminus decays

In this scenario the total effective Hamiltonian for brarr sl+lminus can be written as

Heff = HSMeff +HV A

eff +HSPeff +HT

eff (27)

where HSMeff is given in eq (23) while [37]

HV Aeff =minus 4GFradic

2

α

4πV lowasttsVtb RV (sγmicroPLb) microγmicromicro+RA (sγmicroPLb) microγmicroγ5micro

+RprimeV (sγmicroPRb) microγmicromicro+Rprime

A (sγmicroPRb) microγmicroγ5micro

(28)

HSPeff =minus 4GFradic

2

α

4πV lowasttsVtb RS (sPRb) micromicro+RP (sPRb) microγ5micro

+RprimeS (sPLb) micromicro+Rprime

P (sPLb) microγ5micro

(29)

HTeff =minus 4GFradic

2

α

4πV lowasttsVtb

CT (sσmicroνb) microσmicroνmicro+ iCTE (sσmicroνb) microσαβmicroǫmicroναβ

(210)

ndash 4 ndash

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

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L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 32 ndash

JHEP02(2012)045

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[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

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[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

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[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

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[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

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[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

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[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 3: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

In SM the zero crossing of the AFB(q2) in B rarr Klowastl+lminus is at a well determined

position which is free from the hadronic certainties at the leading order (LO) in strong

coupling αs [11ndash13] Recently the LHCb has announced the results of AFB for the decay

B rarr Klowastmicro+microminus that are close to the SM predictions They have collected over 300 events

for this decay with signal to background ratio above 3 which is the largest sample of such

decays in the world and is even cleaner than the sample used by the B factories These

results of AFB(B rarr Klowastmicro+microminus) are close to the SM predictions with slight error bars thus

they have overwritten the earlier measurements by Belle with better statistics [15] The

collaboration plans to continue to study the channel B rarr Klowastmicro+microminus in finer detail with

measurements that include more angular variables and expected to achieve high sensitivity

to any small deviation from the SM [15]

In order to incorporate the experimental predictions of different physical observables

in B rarr Klowastmicro+microminus this decay has been studied in SM and in number of different NP scenar-

ios [10 12 16ndash36] where NP effects display themselves through modification in the Wilson

coefficients as well as through the new operators Beside these models the general analysis

of B rarr Klowastmicro+microminus decay has also been performed which allow us to include all possible

NP effects like vector-axial vector (V A) scalar-pseudoscalar (SP ) and tensor-axial tensor

(TE) [37] This study has shown that the interference of V A operators among themselves

and with the SM operators is the cause of the larger magnitude of AFB(q2) as well as the

shift in the zero position of the AFB(q2) In addition to this it is also pointed out that the

SP and T operators not only influence the AFB(q2) but also give viable effects in other

physical observables

In general there are two ways to search NP which are considered to resolve anomalies

in the experimental measurements One is the direct search which can be achieved by

increasing the energies of colliders and find the predicted new particles but these particles

are quite massive and therefore hard to produced The other one is indirect search which

is comparatively easier ie to see how much different observables are affected by these new

particles The indirect searches require precise measurement of the observables for the

channel under consideration as well as need rigorous complementary studies

It is therefore indispensable to study the impact of NP on the different observables for

the decay channels other than the B rarr Klowastmicro+microminus With this motivation the B rarr Klowast2ℓ

+ℓminus

channel where Klowast2 is the excited state of Klowast with parity 2+ provide the same testing

ground to study the effects of the NP as B rarr Klowastmicro+microminus decay The exclusive B rarrKlowast

2 (1430)ℓ+ℓminus decay has been studied in the SM and some approaches beyond SM [38ndash46]

For example in refs [38 39] the form factors for B rarr Klowast2ℓ

+ℓminus are calculated using pertur-

bative QCD and LCSR approaches The analysis of different physical observable for this

decay is studied in various beyond SM scenarios like SM4 non-universal Z prime model Uni-

versal Extra dimension model and a model with the flipped sign of Ceff7 has been performed

in [40ndash42 45] The analysis of lepton polarization asymmetries of B rarr Klowast2ℓ

+ℓminus in the pres-

ence of new right-handed currents is done in [43 44] The analysis of polarized decay width

and forward-backward asymmetry by considering beyond SM operator basis is given in [46]

These studies show that like B rarr Klowastℓ+ℓminus decay this channel is also quite promising in

finding NP and will provide us some independent tests of the physics of the SM and beyond

ndash 2 ndash

JHEP02(2012)045

In this paper we will perform the study of B rarr Klowast2 (1430)micro

+microminus on the lines of ref [37]

and study the NP effects without restricting ourselves to any specific model Contrary to

B rarr Klowastmicro+microminus the decay B rarr Klowast2 (1430)micro

+microminus is not yet experimentally observed We have

taken into account the constraints from the different B decay mode such as Bs rarr micro+microminus

B rarr Xsmicro+microminus and B rarr Klowastmicro+microminus on different NP operator couplings allowed by the

most general lorentz structure of transition matrix elements In this paper we will also

incorporate bounds on the input parameters coming from the recent AFB measurements

at LHCb experiment for the decay B rarr Klowast(892)micro+microminus [15] With these new bounds we

will obtain better understanding of how these new physics operators affect branching ratio

zero-position of forward-backward asymmetry different lepton polarization asymmetries

and the helicity fractions of final state meson

The effects of these new operators couplings (VASPT) are significant to the zero

position of AFB(q2) as well as on its magnitude In addition to AFB these NP operators

also give significant effects on decay rate lepton polarization asymmetries and longitudinal

and transverse helicities of final state Klowast2 (1430) meson We hope that in the coming years

the LHCb experiment will collect around 64k events in the full range of q2 for an integrated

luminosity of 2fbminus1 [37] This would allow us to observe the B rarr Klowast2 (1430)micro

+microminus in the

SM and would also permit many of the additional tests of NP

The paper is organized as follows in section 2 the necessary hadronic inputs namely

basic Hamiltonian and form factors and constraints on the NP parameters are collected

Section 3 contains the analytic formulas for differential decay rates forward backward

asymmetry different polarizations (longitudinal normal and transverse) and the helicity

fractions of the final state Klowast2 (1430) meson Section 4 deals with the numerical analysis

and then conclusions are summarized in section 5

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements

21 Standard model

The semileptonic decay mode B rarr Klowast2 (1430)l

+lminus where l = micro for our case is governed

by the quark level transitions b rarr sl+lminus In SM the effective Hamiltonian corresponding

to this transition is given by

HSMeff = minus4GFradic

2V lowasttbVts

10sum

i=1

Ci(micro)Oi(micro) (21)

where Oi(micro)(i = 1 10) are four quark operators and Ci(micro) are the corresponding Wilson

coefficients Out of these 10 operators the O7 O9 and O10 are the relevant for B rarrKlowast

2 (1430)l+lminus decay and these can be written as

O7 =e

16π2[sσmicroν (msPL +mbPR) b]F

microν

O9 =e

16π2[sγmicroPLb]

(

lγmicrol)

O10 =e

16π2[sγmicroPLb]

(

lγmicroγ5l)

(22)

with PLR = (1plusmn γ5) 2

ndash 3 ndash

JHEP02(2012)045

In term of these operators the effective Hamiltonian for B rarr Klowast2 (1430)l

+lminus takes the

form

HSMeff = minusGFαradic

2πVtbV

lowastts

CSM9 (sγmicroPLb)(lγ

microl) + CSM10 (sγmicroPLb)(lγ

microγ5l)

minus2mbCSM7 (siσmicroν

q2PRb)(lγ

microl)

(23)

where q2 is square of the sum of the momentum of final state leptons At the scale micro =

48GeV the SM Wilson coefficients takes the following values which are calculated at next-

to-next-to-leading order

Ceff7 = minus034 Ceff

9 = 4211 + Y (q2) C10 = minus4103 (24)

where Ceff7 = C7 minus C33minus 4C49minus 20C53minus 80C69 The function Y (q2) is given by

Y (z q2) = h(z q2)(3C1(micro) + C2(micro) + 3C3(micro) + C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(1 sprime)(4C3(micro) + 4C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(0 q2)(C3(micro) + 3C4(micro)) +

2

9(3C3(micro) + C4(micro) + 3C5(micro) + C6(micro)) (25)

with

h(z q2) = minus8

9lnz +

8

27+

4

9xminus 2

9(2 + x)|1minus x|12

ln∣

radic1minusx+1radic1minusxminus1

∣minus iπ for x equiv 4z2q2 lt 1

2 arctan 1radicxminus1

for x equiv 4z2q2 gt 1

h(0 q2) =8

27minus 8

9lnmb

microminus 4

9lnq2 +

4

9iπ (26)

and z = mcmb

22 New physics operators

The NP effects in semileptonic B meson decays can be established in two ways in one case

we can change the Wilson Coefficients but leave the operator basis same as that of the SM

and in other case we can add new operators too Here we do not restrict ourself to any

model and discuss all possible operators that corresponds to B rarr Klowast2 (1430)l

+lminus decays

In this scenario the total effective Hamiltonian for brarr sl+lminus can be written as

Heff = HSMeff +HV A

eff +HSPeff +HT

eff (27)

where HSMeff is given in eq (23) while [37]

HV Aeff =minus 4GFradic

2

α

4πV lowasttsVtb RV (sγmicroPLb) microγmicromicro+RA (sγmicroPLb) microγmicroγ5micro

+RprimeV (sγmicroPRb) microγmicromicro+Rprime

A (sγmicroPRb) microγmicroγ5micro

(28)

HSPeff =minus 4GFradic

2

α

4πV lowasttsVtb RS (sPRb) micromicro+RP (sPRb) microγ5micro

+RprimeS (sPLb) micromicro+Rprime

P (sPLb) microγ5micro

(29)

HTeff =minus 4GFradic

2

α

4πV lowasttsVtb

CT (sσmicroνb) microσmicroνmicro+ iCTE (sσmicroνb) microσαβmicroǫmicroναβ

(210)

ndash 4 ndash

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

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[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

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Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

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[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

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[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

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[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

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[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

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[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

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[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

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[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

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[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

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the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

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[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

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[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

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[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

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JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

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J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 4: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

In this paper we will perform the study of B rarr Klowast2 (1430)micro

+microminus on the lines of ref [37]

and study the NP effects without restricting ourselves to any specific model Contrary to

B rarr Klowastmicro+microminus the decay B rarr Klowast2 (1430)micro

+microminus is not yet experimentally observed We have

taken into account the constraints from the different B decay mode such as Bs rarr micro+microminus

B rarr Xsmicro+microminus and B rarr Klowastmicro+microminus on different NP operator couplings allowed by the

most general lorentz structure of transition matrix elements In this paper we will also

incorporate bounds on the input parameters coming from the recent AFB measurements

at LHCb experiment for the decay B rarr Klowast(892)micro+microminus [15] With these new bounds we

will obtain better understanding of how these new physics operators affect branching ratio

zero-position of forward-backward asymmetry different lepton polarization asymmetries

and the helicity fractions of final state meson

The effects of these new operators couplings (VASPT) are significant to the zero

position of AFB(q2) as well as on its magnitude In addition to AFB these NP operators

also give significant effects on decay rate lepton polarization asymmetries and longitudinal

and transverse helicities of final state Klowast2 (1430) meson We hope that in the coming years

the LHCb experiment will collect around 64k events in the full range of q2 for an integrated

luminosity of 2fbminus1 [37] This would allow us to observe the B rarr Klowast2 (1430)micro

+microminus in the

SM and would also permit many of the additional tests of NP

The paper is organized as follows in section 2 the necessary hadronic inputs namely

basic Hamiltonian and form factors and constraints on the NP parameters are collected

Section 3 contains the analytic formulas for differential decay rates forward backward

asymmetry different polarizations (longitudinal normal and transverse) and the helicity

fractions of the final state Klowast2 (1430) meson Section 4 deals with the numerical analysis

and then conclusions are summarized in section 5

2 B rarr Klowast

2(1430)micro+microminus operators and matrix elements

21 Standard model

The semileptonic decay mode B rarr Klowast2 (1430)l

+lminus where l = micro for our case is governed

by the quark level transitions b rarr sl+lminus In SM the effective Hamiltonian corresponding

to this transition is given by

HSMeff = minus4GFradic

2V lowasttbVts

10sum

i=1

Ci(micro)Oi(micro) (21)

where Oi(micro)(i = 1 10) are four quark operators and Ci(micro) are the corresponding Wilson

coefficients Out of these 10 operators the O7 O9 and O10 are the relevant for B rarrKlowast

2 (1430)l+lminus decay and these can be written as

O7 =e

16π2[sσmicroν (msPL +mbPR) b]F

microν

O9 =e

16π2[sγmicroPLb]

(

lγmicrol)

O10 =e

16π2[sγmicroPLb]

(

lγmicroγ5l)

(22)

with PLR = (1plusmn γ5) 2

ndash 3 ndash

JHEP02(2012)045

In term of these operators the effective Hamiltonian for B rarr Klowast2 (1430)l

+lminus takes the

form

HSMeff = minusGFαradic

2πVtbV

lowastts

CSM9 (sγmicroPLb)(lγ

microl) + CSM10 (sγmicroPLb)(lγ

microγ5l)

minus2mbCSM7 (siσmicroν

q2PRb)(lγ

microl)

(23)

where q2 is square of the sum of the momentum of final state leptons At the scale micro =

48GeV the SM Wilson coefficients takes the following values which are calculated at next-

to-next-to-leading order

Ceff7 = minus034 Ceff

9 = 4211 + Y (q2) C10 = minus4103 (24)

where Ceff7 = C7 minus C33minus 4C49minus 20C53minus 80C69 The function Y (q2) is given by

Y (z q2) = h(z q2)(3C1(micro) + C2(micro) + 3C3(micro) + C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(1 sprime)(4C3(micro) + 4C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(0 q2)(C3(micro) + 3C4(micro)) +

2

9(3C3(micro) + C4(micro) + 3C5(micro) + C6(micro)) (25)

with

h(z q2) = minus8

9lnz +

8

27+

4

9xminus 2

9(2 + x)|1minus x|12

ln∣

radic1minusx+1radic1minusxminus1

∣minus iπ for x equiv 4z2q2 lt 1

2 arctan 1radicxminus1

for x equiv 4z2q2 gt 1

h(0 q2) =8

27minus 8

9lnmb

microminus 4

9lnq2 +

4

9iπ (26)

and z = mcmb

22 New physics operators

The NP effects in semileptonic B meson decays can be established in two ways in one case

we can change the Wilson Coefficients but leave the operator basis same as that of the SM

and in other case we can add new operators too Here we do not restrict ourself to any

model and discuss all possible operators that corresponds to B rarr Klowast2 (1430)l

+lminus decays

In this scenario the total effective Hamiltonian for brarr sl+lminus can be written as

Heff = HSMeff +HV A

eff +HSPeff +HT

eff (27)

where HSMeff is given in eq (23) while [37]

HV Aeff =minus 4GFradic

2

α

4πV lowasttsVtb RV (sγmicroPLb) microγmicromicro+RA (sγmicroPLb) microγmicroγ5micro

+RprimeV (sγmicroPRb) microγmicromicro+Rprime

A (sγmicroPRb) microγmicroγ5micro

(28)

HSPeff =minus 4GFradic

2

α

4πV lowasttsVtb RS (sPRb) micromicro+RP (sPRb) microγ5micro

+RprimeS (sPLb) micromicro+Rprime

P (sPLb) microγ5micro

(29)

HTeff =minus 4GFradic

2

α

4πV lowasttsVtb

CT (sσmicroνb) microσmicroνmicro+ iCTE (sσmicroνb) microσαβmicroǫmicroναβ

(210)

ndash 4 ndash

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

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[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

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[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

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[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

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[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

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[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

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J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 5: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

In term of these operators the effective Hamiltonian for B rarr Klowast2 (1430)l

+lminus takes the

form

HSMeff = minusGFαradic

2πVtbV

lowastts

CSM9 (sγmicroPLb)(lγ

microl) + CSM10 (sγmicroPLb)(lγ

microγ5l)

minus2mbCSM7 (siσmicroν

q2PRb)(lγ

microl)

(23)

where q2 is square of the sum of the momentum of final state leptons At the scale micro =

48GeV the SM Wilson coefficients takes the following values which are calculated at next-

to-next-to-leading order

Ceff7 = minus034 Ceff

9 = 4211 + Y (q2) C10 = minus4103 (24)

where Ceff7 = C7 minus C33minus 4C49minus 20C53minus 80C69 The function Y (q2) is given by

Y (z q2) = h(z q2)(3C1(micro) + C2(micro) + 3C3(micro) + C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(1 sprime)(4C3(micro) + 4C4(micro) + 3C5(micro) + C6(micro))

minus1

2h(0 q2)(C3(micro) + 3C4(micro)) +

2

9(3C3(micro) + C4(micro) + 3C5(micro) + C6(micro)) (25)

with

h(z q2) = minus8

9lnz +

8

27+

4

9xminus 2

9(2 + x)|1minus x|12

ln∣

radic1minusx+1radic1minusxminus1

∣minus iπ for x equiv 4z2q2 lt 1

2 arctan 1radicxminus1

for x equiv 4z2q2 gt 1

h(0 q2) =8

27minus 8

9lnmb

microminus 4

9lnq2 +

4

9iπ (26)

and z = mcmb

22 New physics operators

The NP effects in semileptonic B meson decays can be established in two ways in one case

we can change the Wilson Coefficients but leave the operator basis same as that of the SM

and in other case we can add new operators too Here we do not restrict ourself to any

model and discuss all possible operators that corresponds to B rarr Klowast2 (1430)l

+lminus decays

In this scenario the total effective Hamiltonian for brarr sl+lminus can be written as

Heff = HSMeff +HV A

eff +HSPeff +HT

eff (27)

where HSMeff is given in eq (23) while [37]

HV Aeff =minus 4GFradic

2

α

4πV lowasttsVtb RV (sγmicroPLb) microγmicromicro+RA (sγmicroPLb) microγmicroγ5micro

+RprimeV (sγmicroPRb) microγmicromicro+Rprime

A (sγmicroPRb) microγmicroγ5micro

(28)

HSPeff =minus 4GFradic

2

α

4πV lowasttsVtb RS (sPRb) micromicro+RP (sPRb) microγ5micro

+RprimeS (sPLb) micromicro+Rprime

P (sPLb) microγ5micro

(29)

HTeff =minus 4GFradic

2

α

4πV lowasttsVtb

CT (sσmicroνb) microσmicroνmicro+ iCTE (sσmicroνb) microσαβmicroǫmicroναβ

(210)

ndash 4 ndash

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

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L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

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0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

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L

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005

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015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

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12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

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L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 6: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

Here RV RA RV RprimeA RS RP R

primeS R

primeP CT and CTE are the NP effective couplings The

constraints on the numerical values of these NP coupling from different B meson decays

will be given later in this section

23 Parametrization of matrix elements and form factors

The exclusive B rarr Klowast2 (1430)l

+lminus decay involves the hadronic matrix elements of quark

operators which can be parameterized in terms of form factors These form factors are

written as

lang

Klowast2 (k ǫ) |sγmicrob|B(p)

rang

= minus 2V (q2)

mB +mKlowast

2

ǫmicroνρσεlowastναp

α

mBpρkσ (211)

lang

Klowast2 (k ǫ)

∣sγmicroγ5b∣

∣B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mBqνqmicro

+i(mB +mKlowast

2)A1(q

2)

mB

[

gmicroνεlowastναpα minus 1

q2εlowastναp

αqνqmicro]

minusiA2(q2)

εlowastναpαqν

mB(mB +mKlowast

2)

[

(pmicro + kmicro)minusm2

B minusm2Klowast

2

q2qmicro

]

(212)lang

Klowast2 (k ǫ) |sσmicroνqνb|B(p)

rang

= minus2iT1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (213)

lang

Klowast2 (k ǫ)

∣sσmicroνγ5qνb∣

∣B(p)rang

= T2(q2)[

(m2B minusm2

Klowast

2)gmicroνεlowastναp

α minus εlowastναpαqν(pmicro + kmicro)

] 1

mB

+T3(q2)εlowastναp

α

mBqν

[

qmicro minus q2

m2B minusm2

Klowast

2

(pmicro + kmicro)

]

(214)

lang

Klowast2 (k ǫ) |sb|B(p)

rang

=2V (q2)

(mB +mKlowast

2)(mb minusms)

ǫmicroνρσεlowastναp

α

mBpρkσ (215)

lang

Klowast2 (k ǫ)

∣sγ5b∣

∣ B(p)rang

=2imKlowast

2A0(q

2)

q2εlowastναp

α

mB(mb +ms)qνqmicro (216)

lang

Klowast2 (k ǫ) |sσmicroνb|B(p)

rang

= minus2T1(q2)ǫmicroνρσ

εlowastναpα

mBpρkσ (217)

where p(k) is the momentum of the B(Klowast2 ) meson and εlowastνα is the polarization of the final

state Klowast2 (1430) meson In case of the tensor meson the polarization sum is given by [38]

Pmicroναβ =sum

εmicroν(p)εlowastαβ(p) =

1

2(θmicroαθνβ + θmicroβθνα)minus

1

3(θmicroνθαβ) (218)

with

θmicroν = minusgmicroν +kmicrokνm2

Klowast

2

(219)

Defining

εlowastν = εlowastναpαmB (220)

the resulting matrix elements for B rarr Klowast2 (1430) will look just like the B rarr V (eg Klowast

meson) transitions The form factors for B rarr Klowast2 (1430) transition are the non-perturbative

ndash 5 ndash

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 7: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

F (q2) F (0) a b

V (q2) 016+002minus002 208 15

A0(q2) 025+004

minus004 157 01

A1(q2) 014+002

minus002 123 049

A2(q2) 005+002

minus002 132 149

T1(q2) 014+002

minus002 207 15

T2(q2) 014+002

minus002 122 034

T3(q2) 001+001

minus002 991 276

Table 1 B rarr Klowast

2 form factors in the light cone sum rules approach F (0) denotes the value

of form factors at q2 = 0 while a and b are the parameters in the parameterizations shown in

eq (221) [39]

quantities and are needed to be calculated using different approaches (both perturbative

and non-perturbative) like Lattice QCD QCD sum rules Light Cone sum rules etc Here

we will consider the form factors calculated in the LCSR approach [39] and their evolution

with q2 is given by

F (q2) =F (0)

1minus a(q2m2B) + b(q2m2

B)2

(221)

where the value of different parameters is given in table 1

The errors in the values of the form factors arise from number of input parameters

involved in their calculation in LCSR such as variations in the Boral parameters fluctuation

of threshold parameters errors in the b quark mass corrections from the decay constants

of involved mesons and from the Gengenbauer moments in the distribution amplitudes

It is worth mentioning here that it is possible to study this mode with the help of

QCD factorization which might discover additional sources of theoretical errors which are

overlooked in the simple form factor analysis As besides the factorizable contribution of

form factors which has been used here there are in principle non-factorizable contributions

to matrix elements as well [13] QCD factorization not only reduce the number of inde-

pendent parameters in heavy quark limit but also help us to compute the non-factorizable

corrections to B meson decays with the help of light-cone distribution amplitudes and

hard scattering kernel [53 54] The details of QCD factorization and its implications in

semileptonic and radiative B meson decays can be found in [13]

24 Numerical constraints on NP parameters

The constraints on NP parameters can be obtained from the branching ratios of B0s rarr

micro+microminus B0s rarr Xsmicro

+microminus and B0d rarr Kmicro+microminus [5 47ndash50] Due to the large hadronic uncertain-

ties the constraints obtained from the exclusive decay modes are somewhat weaker com-

pared to the inclusive ones The branching ratio of inclusive decay mode B rarr Xsmicro+microminus has

been measured by Babar [49] and Belle [50] and their measurements in low-q2 (1GeV2 le

ndash 6 ndash

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

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[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

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contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

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transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

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and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

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the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

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2 l+lminus in the standard model and new physics scenarios

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2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

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Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

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[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

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[48] Particle Data Group C Amsler et al Review of Particle Physics

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JHEP02(2012)045

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micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 8: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

q2 le 6GeV2) and high-q2 (144GeV2 le q2 le 25GeV2) regions can be summarized as

B(B rarr Xsmicro+microminus)low q2 = (149plusmn 050+041

minus032)times 10minus6 (Belle)

(18plusmn 07plusmn 050)times 10minus6 (BaBar) (222)

(160plusmn 050)times 10minus6 (Average)

B(B rarr Xsmicro+microminus)high q2 = (042plusmn 012+007

minus007)times 10minus6 (Belle)

(050plusmn 025+007minus008)times 10minus6 (BaBar) (223)

(044plusmn 012)times 10minus6 (Average)

where q2 is the invariant mass squared of two muons and low-q2 region corresponds to

1GeV2 le q2 le 6GeV2 and high-q2 region corresponds to q2 ge 144GeV2

The constraints on the new V A couplings come mainly from B0d rarr Xsmicro

+microminus and B0d rarr

Kmicro+microminus [51] It has been shown in ref [51] that if RVA couplings are present only the

constraints on these couplings from above said decays came to be

10 le |RV + 36|2(47)2

+|RA minus 40|2

(48)2|RV + 28|2

(65)2+|RA minus 41|2

(66)2le 1 (224)

Similarly when we have only RprimeVA couplings the constraints on them from the same decay

will become [51]

22 le |RprimeV + 36|2 + |Rprime

A minus 40|2 le 566 |RprimeV |2 + |Rprime

A|2 le 17 (225)

These constraints are weekend when both RVA and RprimeVA couplings are present at the same

time In this case these new physics couplings are restricted to be

|RV + 28|2(65)2

+|RA minus 41|2

(66)2le 1 (226)

and

|RprimeV |2 + |Rprime

A|2 le 40 (227)

The limits on couplings corresponding to scalar-pseudoscalar (SP) operators comes

from the upper bound on B0s rarr micro+microminus which provides the limit

|RS minusRprimeS |2 + |RP minusRprime

P |2 le 044 (228)

This puts a sever constraint on the NP couplings if only RSP or RprimeSP are present However

under the condition where both types of operators are present RSP or RprimeSP these bounds

can be relaxed due to cancelation between the RSP and RprimeSP In this case B0

d rarr Xsmicro+microminus

and B0d rarr Kmicro+microminus can still bound these couplings and the stronger bound is obtained from

the measurement of later quantity which yields

|RS |2 + |RP |2 le 9 RS asymp RprimeS RS asymp Rprime

P (229)

Now the constraint on the NP tensor type couplings comes entirely from the inclusive

B0d rarr Xsmicro

+microminus decay In case when only tensor type operators are present the limits on

tensor type couplings are [51]

|CT |2 + |CTE |2 le 10 (230)

ndash 7 ndash

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

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L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

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06

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1

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Breg

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Breg

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Breg

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(c) (d)

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06

08

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Breg

K2 H

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L

0

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Breg

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0341L+Μ++Μ-L

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L

0

02

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1

f LH

Breg

K2 H

1430L+Μ++Μ-L

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L

0

02

04

06

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1

f LH

Breg

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1430L+Μ++Μ-L

(e) (f)

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L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

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L

0

02

04

06

08

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Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

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Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

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L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

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L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

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L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

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L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

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4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

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L

0

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6acute 10-8

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1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

(c)

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L

0

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4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

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L

-01

-005

0

005

01

015

02

AB

FH

Breg

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0341L+Μ++Μ-L

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L

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0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

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005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

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L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

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L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 9: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

In the present study we will examine B rarr Klowast2 (1430)micro

+microminus decays by using these NP

constraints

3 Decay rate forward-backward asymmetry and lepton polarization a-

symmetries for B rarr Klowast

2(1430)micro+microminus decay

In this section we are going to perform the calculations of some very interesting quantities

like the decay rate the forward-backward asymmetry the polarization asymmetries of

the final state lepton and the helicity fractions of the final state Klowast2 (1430) meson From

eq (27) it is straightforward to obtain the decay amplitude for B rarr Klowast2 (1430)micro

+microminus as

M(B rarr Klowast2 (1430)micro

+microminus) = minus GFα

2radic2π

VtbVlowastts

[

TmicroV lγmicrol + Tmicro

A lγmicroγ5l

+TS ll + TP lγ5l + Tmicroν

T lσmicroν l + iTmicroνE ǫmicroναβ lσαβl

]

(31)

with

TmicroV = minusF1ǫ

microναβεlowastνpKlowastαqβ + iF2εlowastmicro + iF3ε

lowast middot q(pB + pKlowast)micro + iF4εlowast middot qqmicro

TmicroA = minusF5ǫ

microναβεlowastνpKlowastαqβ + iF6εlowastmicro + iF7ε

lowast middot q(pB + pKlowast)micro + iF8εlowast middot qqmicro

TS = iF9εlowast middot q

TP = iF10εlowast middot pB

TmicroνT = CT

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

TmicroνE = CTE

(

iF11ǫmicroναβεlowastα(pB + pKlowast)β + iF12ǫ

microναβεlowastαqβ minus iF13ǫmicroναβεlowast middot qpαKlowastqβ

)

(32)

The auxiliary functions F1 middot middot middot F13 appearing in eq (32) are defined as follows

F1 =

[

2V (q2)

mB +mKlowast

(Ceff9 +RV +Rprime

V ) +4mb

q2Ceff7 T1(q

2)

]

F2 = minus[

(mB +mKlowast)A1(q2)(Ceff

9 +RV minusRprimeV ) +

2mb

q2Ceff7 T1(q

2)(m2B minusmKlowast)

]

F3 =

[

A(q2)

mB +mKlowast

(Ceff9 +RV minusRprime

V ) +2mb

q2Ceff7

(

T1(q2) +

q2T3(q2)

m2B minusmKlowast

)]

F4 =

[

2mKlowast

q2(Ceff

9 +RV minusRprimeV )(A3(q

2)minusA0(q2))minus 2mb

q2Ceff7 T3(q

2)

]

F5 =

[

2V (q2)

mB +mKlowast

(Ceff10 +RA +Rprime

A)

]

F6 = minus[

(mB +mKlowast)A1(q2)(Ceff

10 +RA minusRprimeA)

]

F7 =

[

A(q2)

mB +mKlowast

(Ceff10 +RA minusRprime

A)

]

F8 =

[

2mKlowast

q2(Ceff

10 +RA minusRprimeA)(A3(q

2)minusA0(q2))

]

F9 =

[

minus2(RS minusRprimeS)

mKlowast

mbA0(q

2)

]

ndash 8 ndash

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

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12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

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6acute 10-8

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12acute 10-7

dG qd2H

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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0

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12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

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4acute 10-8

6acute 10-8

8acute 10-8

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12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

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6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

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[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

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[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

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[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

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[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

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[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

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process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

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micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 10: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

F10 =

[

2(RP minusRprimeP )

mKlowast

mbA0(q

2)

]

F11 = minus2T1(q2)

F12 =

[

2(m2B minusm2

Klowast)

q2(T1(q

2)minus T2(q2))

]

F13 =

[

4

q2

(

T1(q2)minus T2(q

2)minus q2T3(q2)

m2B minusm2

Klowast

)]

(33)

31 Differential decay rate

The differential decay width of B rarr Klowast2 (1430)micro

+microminus in dilepton center of mass frame can

be written as

dΓ(B rarr Klowast2 (1430)micro

+microminus)

dq2=

1

(2π)31

32m3B

int u(q2)

minusu(q2)|MBrarrKlowast

2(1430)micro+microminus |2du (34)

where u = (p + pmicrominus)2 minus (p + pmicro+)2 = u(q2) cos θ and q2 = (pmicro+ + pmicrominus)2 p k pmicro+ and

pmicrominus are the four-momenta vectors of B Klowast2 (1430) micro

+ and microminus respectively and u(q2) are

given by

u(q2) =radic

m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2

radic

1minus 4m2l

q2(35)

Collecting everything together one can write the general expression of the differential decay

rate for B rarr Klowast2 (1430)micro

+microminus as

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

18m2lm

4T q

2

3F21m

2Klowast

2q2λ+ 2F2

2m4B(λ+ 10m2

T q2) (36)

+2F23λ

2] (

2m2l + q2

)

minus[

3F25m

2Klowast

2minus 3F2

9m2B

]

K2q2λ+ 12F2

8m2l q

+2F26m

4B

[

2(

λminus 20m2T q

2)

m2l + q2

(

λ+ 10m2T q

2)]

+ 3F210m

2Bq

+2F27λ

[

2(

λminus 3q4 + 6m2T q

2)

m2Klowast

2+ q2λ

]

minus 4F2F3m4B

(

2m2l + q2

)

K1λ2

minus4F6F7m2Bλ

[

2(

K1 + 3q2)

m2l +K1q

2]

minus 24F6F8m2lm

2Bq

2λminus 12F10F6mlm3Bq

+24F7F8m2lK0q

2λ+ 12F10F7mlmBK0q2λ+ 8F11F12q

2(λ+ 10m2Tm

2B minus 10m4

T )K3

minus4F11F13(K1 + 4m2T )λq

2K3 minus 72F1F11mlm2Klowast

2q2λCT

+48F11F2mlm2Bq

2(λ+ 10m2Tm

2B minus 10m4

T )CTE minus 4F12F13K1λq2K3

+48F12F2mlm2Bq

2(

λ+ 10m2T q

2)

CTE minus 48F12F3mlq2K1CTEλ

minus24F13F2mlm2BK1λq

2CTE + 24F13F3mlq2CTEλ

2

where ml is the mass of final state muon and

λ = λ(

m2Bm

2Klowast

2 q2

)

equiv m4B +m4

Klowast

2+ q4 minus 2m2

Klowast

2m2

B minus 2q2m2B minus 2m2

Klowast

2q2 (37)

K0 = (m2B minusm2

Klowast

2) K1 =

(

m2B minusm2

Klowast

2minus q2

)

K2 = q2 minus 4m2l

K3 = q2C2T minus 4m2

l

(

C2T minus 36C2

TE

)

(38)

and the auxiliary functions are the same as defined in eq (33)

ndash 9 ndash

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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2 (1430) and Klowast(892)

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JHEP02(2012)045

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 11: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

32 Forward-backward asymmetry

Now we are in a position to explore the forward-backward asymmetry of final state leptons

in the B rarr Klowast2 (1430)micro

+microminus which is an essential observable sensitive to the new physics

effects To calculate the forward-backward asymmetry we consider the following double

differential decay rate formula for the process B rarr Klowast2 (1430)micro

+microminus

d2Γ(q2 cos θ)

dq2d cos θ=

1

(2π)31

64m3B

u(q2)|MBrarrKlowast

2(1430)micro+microminus |2 (39)

where θ is the angle between the momentum of B baryon and lminus in the dilepton rest frame

The differential and normalized FBAs for the semi-leptonic decay B rarr Klowast2 (1430)l

+lminus are

defined as

dAFB(q2)

dq2=

int 1

0d cos θ

d2Γ(q2 cos θ)

dq2d cos θminusint 0

minus1d cos θ

d2Γ(q2 cos θ)

dq2d cos θ(310)

and

AFB(q2) =

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θminusint 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

int 10 d cos θ d2Γ(q2cos θ)

dq2d cos θ+int 0minus1 d cos θ

d2Γ(q2cos θ)dq2d cos θ

(311)

Following the same procedure as we did for the differential decay rate one can easily get

the expression for the forward-backward asymmetry as follows

dAFB(q2)

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2

u(q2)λ

6mBm4T

[

AV AFB +ASP

FB +ATSPFB +AT

FB

]

(312a)

AV AFB = minus3(F2F5 + F1F6)q

2m2Bm

2K2

(312b)

ASPFB = 2F9mlmB

(

F3λ+ F2m2B

[

q2 minusm2B +m2

K2

])

(312c)

ATSPFB = minusq2 (F10CT + 2F9CTE)mB

(

2F12

(

m2B minusm2

K2minus q2

)

(312d)

minusF13λ+ 2F11

(

m2B + 3m2

K2minus q2

))

ATFB = minus2ml

(

12F12F5q2CTEm

2K2

+ 12F11F5CTEm2K2

(

m2B minusm2

K2

)

(312e)

+F13λCT

(

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F12CT

(

m2B minusm2

K2minus q2

) (

(F6 minusF7)m2B minusF8q

2 + F7m2K2

)

minus2F11CT

(

F6m2B

(

m2B + 9m2

K2minus q2

)

minus(

m2B + 3m2

K2minus q2

) (

F8q2 + F7

(

m2B minusm2

K2

)))

and the expression of the differential decay rate is given in eq (36)

33 Lepton polarization asymmetries

In the rest frame of the lepton microminus the unit vectors along longitudinal normal and transver-

sal component of the microminus can be defined as [55]

sminusmicroL = (0 ~eL) =

(

0~pminus|~pminus|

)

sminusmicroN = (0 ~eN ) =

0~k times ~pminus∣

~k times ~pminus∣

(313)

sminusmicroT = (0 ~eT ) = (0 ~eN times ~eL)

ndash 10 ndash

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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2 (1430) and Klowast(892)

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fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

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process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

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micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 12: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

where ~pminus and ~k are the three-momenta of the microminus and Klowast2 (1430) meson respectively in the

center mass (CM) frame of micro+microminus system Lorentz transformation is used to boost the

longitudinal component of the lepton polarization to the CM frame of the lepton pair as

(

sminusmicroL

)

CM=

( |~pminus|ml

El~pminusml |~pminus|

)

(314)

where El and ml are the energy and mass of the final state muon The normal and

transverse components remain unchanged under the Lorentz boost The longitudinal (PL)

normal (PN ) and transverse (PT ) polarizations of lepton can be defined as

P(∓)i (q2) =

dΓdq2

(~ξ∓ = ~e∓)minus dΓdq2

(~ξ∓ = minus~e∓)dΓdq2

(~ξ∓ = ~e∓) + dΓdq2

(~ξ∓ = minus~e∓)(315)

where i = L N T and ~ξ∓ is the spin direction along the leptons micro∓ The differential

decay rate for polarized lepton l∓ in B rarr Klowast2 (1430)micro

+microminus decay along any spin direction~ξ∓ is related to the unpolarized decay rate (36) with the following relation

dΓ(~ξ∓)

dq2=

1

2

(

dq2

)

[1 + (P∓L ~e∓L + P∓

N~e∓N + P∓T ~e∓T ) middot ~ξ∓] (316)

We can achieve the expressions of longitudinal normal and transverse polarizations for

B rarr Klowast2 (1430)micro

+microminus decays as collected below The longitudinal lepton polarization can

be written as

PL prop1

36m2Bm

4T

4u(q2)radicλ[

P V AL + PSP

L + P TL

]

(317a)

P V AL = ml

(

2λF2 (F6 minusF7)m4B +

(

3λF1F5q2 + 2F2m

2B

(

10q2F6m2B + λF7

))

m2Klowast

2

+2λF3

(

F6

(

q2 minusm2B +m2

Klowast

2

)

m2B + λF7

))

(317b)

PSPL = 3mlλF9mB

(

minus2mlF8q2 minusF10mBq

2 + 2ml

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

(317c)

P TL = m2

lF11

(

24 (CTEF1 minus CTF5)m2Klowast

2λ+ 4CTλF3

(

m2B + 3m2

Klowast

2minus q2

)

+4m2B (4CTEF6 minus CTF2)

(

10(

m2B minusm2

Klowast

2

)

m2Klowast

2+ λ

))

+m2lF12

(

4m2B (4CTEF6 minus CTF2)

(

10q2m2Klowast

2+ λ

)

minus 4CTF3

(

q2 minusm2B +m2

Klowast

2

))

+m2lF13

(

2CTλminus 4CTF2m2B

(

m2B minusm2

Klowast

2minus q2

)

minus 2CTλ2F3

)

(317d)

ndash 11 ndash

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

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[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

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[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

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[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

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2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 13: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

Similarly the normal lepton polarization is

PN prop minusπu(q2)t

(12m2Bm

4T

t)32

[

P V AN + PSP

N + P TN

]

(318a)

P V AN = 2ml

(

F2

(

F8q4 minus

(

(F6 minusF7 + F8)m2B + (minus3F1 + F7 minusF8)m

2Klowast

2

)

q2 (318b)

+(

m2B minusm2

Klowast

2

)

m2B

(

(F6 minusF7)m2B + F7m

2Klowast

2

))

+ λF3

(

F8q2 minusF6m

2B + F7

(

m2B minusm2

Klowast

2

)))

PSPN = mBλ

(

F3F10q2 +

(

q2 minus 4m2l

)

F7F9

)

(318c)

minus(

F2F10q2 +

(

q2 minus 4m2l

)

F6F9

)

m3B

(

m2B minusm2

Klowast

2minus q2

)

P TN = F13

(

8m2l λCTE

(

F8q2 minusF6m

2B minusF7

(

m2Klowast

2minusm2

B

))

+ 4mlmBλCTEF10q2)

+F12

(

16m2lCTE

(

F6m2BminusF7

(

m2Bminusm2

Klowast

2

)

minusF8q2minusmB

2mlF10q

2

)

(

m2Bminusm2

Klowast

2minusq2

)

+ 6(

4m2l + q2

)

CTEF1q2m2

Klowast

2+ 3

(

q2 minus 4m2)

CTF5q2m2

Klowast

2

)

+F11

(

minus16m2lCTE

(

F8q2 + F7

(

m2B minusm2

Klowast

2

)

+mB

2mlF10q

2

)

(

m2B + 3m2

Klowast

2minus q2

)

+6m2Klowast

2

(

4m2l + q2

)

(

CTEF1

(

m2B minusm2

Klowast

2

)

minus CTF2m2B

)

+3(

4m2l minus q2

)

CTF5m2Klowast

2

(

m2Klowast

2minusm2

B

)

+4CTEF6m2B

(

4m2l

(

m2B + 6m2

Klowast

2minus q2

)

minus 3q2m2Klowast

2

)

)

(318d)

and the transverse one is given by

PT propiπλq2u(q2)

12mBm2Klowast

2

λCTEF13

(

4F2m2B minus 3F5m

2Klowast

2+ F3

(

q2 minus 5m2B +m2

Klowast

2

))

(319)

minus2F12

(

2CTE

((

8F3m2B minus (2F3 + 3F5)m

2Klowast

2

)

q2 + 4F2m2B

(

2m2B +m2

Klowast

2minus 2q2

)

minus(

m2Bminusm2

Klowast

2

)(

8F3m2B + (4F3 + 3F5)m

2Klowast

2

))

minus3CTF1m2Klowast

2

(

q2minus3m2B + 3m2

Klowast

2

))

+2F11

(

2CTE

(

3F5m2Klowast

2

(

3m2B +m2

Klowast

2minus q2

)

minus 4F2

(

2m2B + 3m2

Klowast

2minus 2q2

)

m2B

+2F3

(

7m2Klowast

2q2minus4q4+6λ+4m2

B

(

q2+5m2Klowast

2

)))

minus3CTF1m2Klowast

2

(

3m2B+m2

Klowast

2minusq2

)

34 Helicity fractions

Helicity fraction is an observable associated with polarization of the out going meson that

is almost free of hadronic uncertainties The spin-2 polarization tensor which satisfies

ǫmicroνkν = 0 with k being the momentum is symmetric and traceless It can be constructed

by the vector polarization ǫmicro as

ǫmicroν(plusmn2) = ǫmicro(plusmn)ǫν(plusmn) ǫmicroν(plusmn1) =1radic2[ǫmicro(plusmn)ǫν(0) + ǫmicro(0)ǫν(plusmn)]

ǫmicroν(0) =1radic6[ǫmicro(+)ǫν(minus) + ǫmicro(minus)ǫν(+)] +

radic

2

3ǫmicro(0)ǫν(0) (320)

ndash 12 ndash

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

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0

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6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

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L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 14: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

using the definition

εTν(n) =ενα(n)p

α

mB

the above relations simplify to

εTν(plusmn2) = 0 εTν(plusmn1) =ǫ(0)pradic2mB

ǫmicro(plusmn) εTν(0) =

radic

2

3

ǫ(0)p

mBǫmicro(0)

The physical expressions for helicity fractions are given by

fi(q2) =

dΓidq2

dΓdq2 i = L T (321)

Here L and T refers to longitudinal and transverse helicity fractions The explicit expression

of the longitudinal helicity fraction for the decay B rarr Klowast2 (1430)l

+lminus is

dΓL

dq2=

G2Fα

2

211π5m3B

|VtbVlowastts|2 u(q2)

36m4Klowast

2m2

Bq2

(

4F 22m

4B

(

2m2 + q2)

K21 + 4F 2

3 (2m2 + t)λ2

+6F 29 q

2λm2B

(

q2 minus 4m)

+ 6F 210q

4λm2B + 4F 2

6m4B

(

q2K21 + 2m2

(

λminus 8q2m2Klowast

2

))

+4F 27 λ

(

2m2(

4q2(

m2B +m2

Klowast

2

)

+K20 minus 2q4

)

+ q2λ)

+ 24F 28m

2q4λ

minus8F6F7λm2B

(

2m2(

K0 + 2q2)

+ q2K1

)

minus 8F2F3λm2B

(

2m2 + q2)

K1 minus 8F12F13q2λK1K3

+48F7F8m2q2λK0 + 8F 2

11q2K2

4K3 + 8F 212q

2K21K3 + 2F 2

13q2λ2K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+ 48F3F13mq2λ2CTE

minus48F2F13mq2λm2BCTEK1 minus 96F3F12mq2λCTEK1 + 96F2F12mq2m2

BCTEK21

+96F2F11mq2m2BCTEK1K4 minus 48F2F13mq2λm2

BCTEK1 minus 96F3F12mq2λCTEK1

+8F 211q

2K24K3 + 8F 2

12q2K2

1K3 + 2F 213q

2λ2K3 minus 8F12F13q2λK1K3 + 16F11F12q

2K1K4K3

minus8F11F13λ(

4C2Tm

2 (λminusK0K1) + 144m2q2C2TEK4 + 3m2C2

Tm2Klowast

2

)

+96F2F12mq2m2BCTEK

21 +48F3F13mq2λ2CTE + 96F2F11mq2m2

BCTEK1K4

)

(322)

As the sum of the longitudinal and transverse helicity amplitudes is equal to unity ie

fL(q2) + fT (q

2) = 1 for each value of q2 therefore the expression of transverse helicity

fraction of final state meson is quite obvious

4 Numerical Analysis

In this section we will examine the above derived physical observables and analyze the

effects of MI parameters on these observables As B rarr Klowast2 (1430)micro

+microminus is not yet observed

therefore the purpose of this study is not to get constraints on NP parameters but is

to use the constraints on MI parameters defined in section 24 which are coming from

different B meson decays We consider different Lorentz structures of NP as well as

their combinations and examine their implications on differential decay rate the forward-

backward asymmetry the lepton polarization asymmetries and the helicity fractions of the

ndash 13 ndash

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

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L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

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L

0

02

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08

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Breg

K2 H

0341L+Μ++Μ-L

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L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

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L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

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4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

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4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

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L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

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L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

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0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

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L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 15: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

mB = 528GeV mb = 428GeV ms = 013GeV

mmicro = 0105GeV mτ = 177GeV fB = 025GeV

|VtbVlowastts| = 45times 10minus3 αminus1 = 137 GF = 117times 10minus5GeVminus2

τB = 154times 10minus12 sec mKlowast

2= 143GeV

Table 2 Default values of input parameters used in the calculations [56]

C1 C2 C3 C4 C5 C6 C7 C9 C10

1107 -0248 -0011 -0026 -0007 -0031 -0313 4344 -4669

Table 3 The Wilson coefficients Cmicroi at the scale micro sim mb in the SM [56]

final state Klowast2 (1430) meson Here we varied the strength of individual NP couplings such

that they lies inside the bounds given by different flavor decays (cf section 24) In all the

figures the band correspond to the uncertainties in different input parameters where form

factors (cf table 1) are the major contributors The same type of lines correspond to the

same set of NP parameters and all the NP curves are plotted for the central values of the

form factors and other input parameters To make the quantitative analysis complete the

numerical values of input parameters that we have used are given in table 2 and 3

SemileptonicB decays are ideal probes to study the physics in and beyond the Standard

Model There are a large number of observables which are accessible in these decays The

branching ratio in general for semi-leptonic decays like B rarr Klowast2 (1430)micro

+microminus is prune

to many sources of uncertainties The major source of uncertainty originate from the

B rarr Klowast2 (1430) transition form factors calculated using the LCSR approach as shown in

table 1 can bring about 20 minus 30 uncertainty to the differential branching ratio This

goes to show that differential branching ratio is not a suitable observable to look for the

NP effects unless these effects are very drastic It is therefore valuable to look for the

observables where hadronic uncertainties almost have no effect In this regard the most

important observables are the zero position of the forward-backward asymmetry different

lepton polarization asymmetries and the helicity fractions of the final state meson These

observables are almost free from the hadronic uncertainties and serve as an important

probe to look for NP

The SM predicts the zero crossing of AFB(q2) at a well determined position which is

free from the hadronic uncertainties at the leading order (LO) in strong coupling αs [11ndash13]

This zero position of AFB which is independent of form factors is an important observable

in the search of new physics The short distance relation between the Wilson coefficients

and the zero position of AFB is given by [12 13]

real[Ceff9 (q20)] =

2mbmB

q20Ceff7 (41)

where q20 is the zero position (ZP) of the AFB Recently LHCb has published its results on

AFB(B rarr Klowastmicro+microminus) that are quite close to SM predictions [15] The LHCb result shows

with small error bars that the zero position of AFB(B rarr Klowastmicro+microminus) is close to the SMrsquos

zero position Like B rarr Klowastmicro+microminus decay the semileptonic decay B rarr Klowast2 (1430)micro

+microminus also

ndash 14 ndash

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

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L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

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L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

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[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

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Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

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[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

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[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

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[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

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[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

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[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

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[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

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[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

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[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

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[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

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[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

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[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

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[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

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[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

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Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

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[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

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[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

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[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

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[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 16: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

occurs through the quark level transition brarr smicro+microminus Therefore the future measurements

of the AFB(B rarr Klowast2 (1430)micro

+microminus) will shed more light on NP in the flavor sector

Now following the lines of B rarr Klowastmicro+microminus decay the semileptonic decay B rarrKlowast

2 (1430)micro+microminus also occurs through the quark level transition b rarr smicro+microminus therefore the

future measurements of the AFB will shed some light on the searches of physics in and

beyond the SM in the flavor sector

The polarization of the final state leptons can shed light on the helicity structure of

the electroweak interactions The longitudinal and normal lepton polarization asymmetry

are also very good observables to look for NP effects As in the case of forward-backward

asymmetry the long distance effects are also canceled out in these asymmetries Hence

they provide a good tool to test the various new physics extensions of the SM In the

present numerical analysis we have taken the NP couplings to be real The transverse

lepton polarization asymmetry as given in eq (319) constitutes of the imaginary part

of the auxiliary functions hence for real couplings its value comes out to be negligibly

small In this numerical study we will discuss the results of transverse lepton polarization

asymmetry in the presence of all NP operators at the very end

Another interesting observable is the study of the spin effects of the final state meson

which for our case is the Klowast2 (1430) meson A detailed discussion about the effects of NP

operators on the longitudinal and transverse helicity fractions of final state meson will also

be done in the forthcoming numerical analysis

A model independent numerical analysis of this decay channel B rarr Klowast2 (1430)micro

+microminus

is presented in this section where we will analyze the effects of different NP operator

couplings (VASPT) on different physical observables A similar NP study was presented

in [46] where the observables studied were polarized branching ratios as function of NP

couplings and forward backward asymmetry The observables we have presented in this

manuscript are different than the ones presented in [46] For example we have studied

differential observables(that are functions of q2) like branching ratio forward backward

asymmetry lepton polarization asymmetries and helicity fractions of the final state meson

We have also incorporated the old bounds [51 52] as well as the new LHCb bound [15]

on different NP couplings We will study the effects of these NP couplings one by one as

follows

41 V A new-physics operators

In this section we shall consider the scenarios when (a) only RVA couplings are present

(b) only RprimeVA couplings are present and (c) both types of couplings are present

Figures 1ndash5 display the results when the only vector-axial vector V A NP couplings

(RVA and RprimeVA) are present In the case of differential branching ratio it can be seen

that its values increase for certain range of parameters while decrease for other range of

parameters The increase in differential branching ratio is up to 4 time its SM value for

the case when both type of NP couplings are present simultaneously (cf figure 1(cd))

As can be seen in figure 2 that the AFB(q2) is very sensitive to the NP effects The zero

crossing is also disturbed significantly by the VA couplings The short distance relation

ndash 15 ndash

JHEP02(2012)045

(a) (b)

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L

0

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dG qd2H

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(c) (d)

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dG qd2H

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

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6acute 10-8

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dG qd2H

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

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FH

Breg

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-005

0

005

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015

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025

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FH

Breg

K2 H

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L

-005

0

005

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015

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025

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FH

Breg

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(c) (d)

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-01

0

01

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03

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FH

Breg

K2 H

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-01

0

01

02

03

AB

FH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

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0

02

04

06

08

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Breg

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L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

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L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

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L

-02

0

02

04

06

08

P LH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

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02

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Breg

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0 2 4 6 8 10 12 14q2HGeV2

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0

02

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f LH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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0

02

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f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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0

02

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08

f LH

Breg

K2 H

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(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

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02

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f TH

Breg

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0

02

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f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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06

08

f TH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

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f TH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

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1

f TH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

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dG qd2H

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

0

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dG qd2H

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 17: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

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dq2H

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5acute 10-8

1acute 10-7

15acute 10-7

dG

dq2H

Breg

K2 H

1430L+Μ++Μ-L

Figure 1 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of V A couplings inside their allowed region In figure

(a) only RV and RA are present figure (b) when only Rprime

V and Rprime

A are present figure (c) figure (d)

when both RV Rprime

V and RA Rprime

A are present and in all figures the solid line represent the SM with

light green shaded region as uncertainty in SM

eq (41) is modified with the inclusion of new VA couplings which now takes the form

real(Ceff9 (q20)) +RV minus

RprimeV R

primeA

C10=

2mbmB

q20Ceff7 (42)

The experimental results that came in from measurement of AFB(B rarr Klowastmicro+microminus) at

LHCb can put stringent bound on the NP coupling constants In the measurement of the

zero position of AFB(B rarr Klowastmicro+microminus) at LHCb the experimental error bar in this zero

position is roughly 30 percent When this error bar uncertainty is plugged in eq (42) we

get a new bound on the VA coupling constants

minus 1 lt RV minusRprime

V RprimeA

C10lt 32 (43)

When this bound is taken into consideration then the zero position (ZP) stays in the

vicinity of Standard Modelrsquos ZP (see figures 2(abc)) But if we ignore this new bound

then the effects on forward backward asymmetry become very drastic (see figure 2d) These

effects increase when all the couplings RVA and RprimeVA are present (figures 2(cd)) The zero

crossing shifts quite significantly and it even disappears for certain values of RVA and RprimeVA

(figure 2d)

ndash 16 ndash

JHEP02(2012)045

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BH

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015

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AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

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(c) (d)

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0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

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Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

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15acute 10-7

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dG qd2H

Breg

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Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

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Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

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025

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075

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Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

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Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

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0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

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L

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Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 18: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

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K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AF

BH

Breg

K2 H

1430L+Μ++Μ-L

Figure 2 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of V A couplings The caption for the above figures are the same as figure 1

Figures 3 and 4 depict the effects on lepton polarization asymmetries coming from

RVA and RprimeVA couplings The effects are quite obvious in the case of longitudinal lepton

polarization asymmetry The magnitude of PL decrease in most cases and when RprimeVA is

present the effects get more amplified This asymmetry even vanishes (figure 3b) in some

values of RprimeVA couplings When both RVA and Rprime

VA are present PL even changes sign for

some values of couplings(see figures 3(cd)) Seen in eq (317b) the V A effects are only

ml suppressed compared to other NP contributions that are m2l suppressed Qualitatively

these effects are large because of large numerical values and different signs of the new V A

couplings The normal lepton polarization asymmetry is also sensitive to NP effects as it

develops a zero crossing which is not present in SM Still the effects are suppressed by ml

(see eq (318b)) hence in figure 4 all curves show nearly the same behavior

The longitudinal and transverse helicity fractions are depicted in figures 5 The de-

viation from the SM results are fairly large that arise from the effects of different V A

couplings When both RVA and RprimeVA couplings are present (see figures 5(cdgh)) the

traits become even more conspicuous

ndash 17 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-02

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-02

0

02

04

06

08

1

P LH

Breg

K2 H

1430L+Μ++Μ-L

Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f LH

Breg

K2 H

1430L+Μ++Μ-L

(e) (f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(g) (h)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1

f TH

Breg

K2 H

1430L+Μ++Μ-L

Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 19: Model independent analysis of B →  (1430)μ+μ−decay

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Figure 3 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of V A couplings The caption for the above figures are the same as figure 1

(a) (b)

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Figure 4 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of V A couplings The caption for the above figures are the same as figure 1

ndash 18 ndash

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Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

ndash 19 ndash

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Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

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Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

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Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

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Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

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Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

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Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

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Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

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Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 20: Model independent analysis of B →  (1430)μ+μ−decay

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Figure 5 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of V A couplings The caption for the above figures

are the same as figure 1

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Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

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Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

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Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

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Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

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Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

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Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

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Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

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Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

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Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

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(c)

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L

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L

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(f)

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L

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06

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

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Breg

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0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

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12acute 10-7

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L

0

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12acute 10-7

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15acute 10-7

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L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

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025

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FH

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005

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FH

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L

-01

-005

0

005

01

015

02

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FH

Breg

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Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

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Breg

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025

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075

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L

-05

-025

0

025

05

075

P LH

Breg

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0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

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L

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Breg

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L

-08

-06

-04

-02

0

02

04

P NH

Breg

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0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

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Breg

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(c) (d)

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L

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Breg

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L

0

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0 2 4 6 8 10 12 14q2HGeV2

L

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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06

08

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Breg

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0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

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01

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03P TH

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L

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1430L+Μ++Μ-L

(c)

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L

-08

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04

P NH

Breg

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1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 21: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

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1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 6 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of SP couplings inside their allowed region In figure

(a) only RS and RP are present figure (b) when only Rprime

S and Rprime

P are present figure (c) when both

RS Rprime

S and RP Rprime

P are present and in all figures the solid line represent the SM with light green

shaded region as uncertainty in SM

42 SP new-physics operators

In this section we shall consider the scenarios when (a) only RS and RprimeS couplings are

present (b) only RP and RprimeP couplings are present and (c) both types of couplings

are present

The differential branching ratio in figure 6 show that it is slightly disturbed by the

different values of SP parameters Figure 7 depicts that the lepton forward-backward

asymmetry is not very sensitive to the NP arising form SP operators There is a slight

shift in the zero position of the AFB(q2) towards left or right These mild effects can also

be inferred from the eq (312c) that SP effects on forward-backward asymmetry are ml

suppressed

Figure 8 displays the plots of the longitudinal lepton asymmetry in the presence of SP

operator couplings The S and P couplings separately (figures 8(ab)) have a mildly effect

on the longitudinal lepton polarization asymmetry while the effects increase three folds

when both the couplings are present figure 8(c) The dominant contribution comes from

the F9F10 term in eq (317c) which is enhanced by a factor of mBq2

ndash 20 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

06

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

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Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

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03P TH

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(c)

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1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 22: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

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Figure 7 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of SP couplings The caption for the above figures are the same as figure 6

(a) (b)

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Figure 8 The dependence of Longitudinal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of SP couplings The caption for the above figures are the same as figure 6

ndash 21 ndash

JHEP02(2012)045

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Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

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06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 32 ndash

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[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

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[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

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[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

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[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

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[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

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[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

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[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

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[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

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JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

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[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

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[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 23: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

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Figure 9 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of SP couplings The caption for the above figures are the same as figure 6

The normal lepton polarization asymmetry is very sensitive to the S and P couplings

In SM this asymmetry takes only negative values and there is no zero crossing but as

S and P couplings are introduced the curves are scattered in both positive and negative

regimes (Fig 9) When both SP couplings are present the effects are even more enhanced

The eq (318c) clearly shows that SP contribution is not ml suppressed like other V A

contributions to the normal polarization asymmetry So the normal lepton polarization

asymmetry is a very good tool to constraint the values of the SP couplings

In case of SP couplings the longitudinal and transverse helicity fractions are shown in

figure 10 The effects in this case are quite subtle especially in the large q2 region as seen

in figure 10 The deviation from SM amplifies when both S and P couplings are present

(see figures 10(cf))

43 T operators and interference of SP and T operators

In this section we shall consider the scenarios when (a) only CT and CTE couplings are

present (b) when S CT and CTE couplings are present and (c) when P CT and CTE

couplings are present

The differential branching ratio shown in figure 11 indicates that its value increases for

most values T coupling parameters These T type coupling also effects the zero crossing of

the AFB(q2) which can be seen in figure 12 By looking at the eq (312) one can see that

ndash 22 ndash

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Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

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Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

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Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

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Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

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Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

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Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

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0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

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L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

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0341L+Μ++Μ-L

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L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

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L

0

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Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

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Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

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0341L+Μ++Μ-L

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L

-08

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-04

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0

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Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

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Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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1f LH

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

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0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 24: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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08

f LH

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

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02

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f LH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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0

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f LH

Breg

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0341L+Μ++Μ-L

(c)

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0

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f LH

Breg

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02

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f LH

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0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

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0

02

04

06

08

f TH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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f TH

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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f TH

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

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0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 10 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of SP couplings The caption for the above figures are

the same as figure 6

ndash 23 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

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12acute 10-7

dG qd2H

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0 2 4 6 8 10 12 14q2HGeV2

L

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2acute 10-8

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6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

dG qd2H

Breg

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0341L+Μ++Μ-L

Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

0

01

02

03

04

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

(d) (e)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

(f)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

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[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

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[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

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[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

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JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

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[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 25: Model independent analysis of B →  (1430)μ+μ−decay

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Figure 11 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of T-SP couplings inside their allowed region In figure

(a) only CT and CTE are present figure (b) when only CT CTE RS and Rprime

S are present figure

(c) when only CT CTE RP and Rprime

P are present and in all figures the solid line represent the SM

with light green shaded region as uncertainty in SM

(a) (b)

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Figure 12 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of T-SP couplings The caption for the above figures are the same as figure 11

ndash 24 ndash

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0 2 4 6 8 10 12 14q2HGeV2

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P LH

Breg

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Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

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Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

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Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

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Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

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Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

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Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

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Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

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Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

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Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

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[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 26: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

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Figure 13 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

the contribution from the T couplings is ml suppressed (cf eq (312e)) but after including

SP operators along with T operators it is no more ml suppressed It is therefore expected

that the value of AFB(q2) in this scenario will be large compared to T coupling only but

the sever constraints on the SP couplings do not allow AFB(q2) to become significantly

more than the SM in magnitude Also there is some parameter space of couplings where

AFB(q2) is negative every where ie there is no zero crossing (figure 12(bc))

In figure 13(a) the longitudinal lepton asymmetry is shown in the presence of T oper-

ator couplings From eq (317b) we can observe that the effect of the T couplings are m2l

suppressed and therefore we expect the small variation of longitudinal lepton polarization

asymmetry unless the value of T couplings is large and this can also be infer from figure 13

When SP couplings are also included the results hardly change (figure 13(bc)) since there

effect is also ml suppressed compared to SM results

The normal lepton polarization asymmetry (figure 14(a)) develops a zero crossing when

only T operator couplings are incorporated Adding the SP operators change the plots as

well The zero crossing of the normal lepton polarization asymmetry is shifted when the

SP couplings are introduces along with CT and CTE

The longitudinal and transverse helicity fractions in the presence of T operator cou-

plings are plotted in figure 15 Here the response to the different values of T couplings

ndash 25 ndash

JHEP02(2012)045

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Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

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Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

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Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

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Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

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Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

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02

04

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L

-08

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02

04

P NH

Breg

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0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

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L

0

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L

0

02

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(c) (d)

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L

0

02

04

06

08

f TH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

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0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

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1430L+Μ++Μ-L

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L

-04

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Breg

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1430L+Μ++Μ-L

(c)

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L

-08

-06

-04

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P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 27: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

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Figure 14 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus

on q2 for different values of T-SP couplings The caption for the above figures are the same as

figure 11

are visible both in the high and low q2 regimes The effects are more distinct when CT

coupling is present compared to when only CTE coupling is present When SP couplings

are added along with the T couplings the results are an emulsion of the effects coming from

T as well as high q2 effects of SP (figures 15(bcef))

44 All NP operators together

In presence of all new physics operator couplings of V A S P and T the results as expected

are pretty drastic The combination of all these couplings have most of the attributes of

each individual type of coupling The branching ration plots are shown in figure 16 increases

by a factor of two and depending upon the values of coupling this increase is four times

compared to SM branching ratio (figure 16(b))

The forward backward asymmetry gets scrambled results the zero crossing vanishes

for most values of NP couplings when new bound eq (43) is neglected (figure 17b) If this

new bound is incorporated then the zero position does not vanish but the shift in ZP is

significantly in both directions

The longitudinal lepton polarization varies a lot and takes on both positive and negative

values (figure 18) The normal lepton polarization crosses the zero and become positive for

some values of the NP couplings (figure 19) showing us the presence of SP couplings

ndash 26 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

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(c)

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(f)

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Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

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L

0

2acute 10-8

4acute 10-8

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dG qd2H

Breg

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Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

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K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

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02

04

06

08

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Breg

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025

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L

-05

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0

025

05

075

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Breg

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0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

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L

-08

-06

-04

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Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

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(c) (d)

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02

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L

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

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0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

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03P TH

Breg

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1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

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Breg

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1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

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04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 28: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

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(c)

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(f)

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0341L+Μ++Μ-L

Figure 15 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of T-SP couplings The caption for the above figures

are the same as figure 11

ndash 27 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

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dG qd2H

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Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

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Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

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05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

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ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 29: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

4acute 10-8

6acute 10-8

8acute 10-8

1acute 10-7

12acute 10-7

14acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

2acute 10-8

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12acute 10-7

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5acute 10-8

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15acute 10-7

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dG qd2H

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0

5acute 10-8

1acute 10-7

15acute 10-7

2acute 10-7

dG qd2H

Breg

K2 H

0341L+Μ++Μ-L

Figure 16 The dependence of decay rate for B0 rarr Klowast

2 (1430)micro+microminus on q2 where different curves

correspond to different choices of the values of all couplings are taken into account inside their

allowed region In all figures the solid line represent the SM with light green shaded region as

uncertainty in SM

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

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0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-005

0

005

01

015

02

025

AB

FH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

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-01

-005

0

005

01

015

02

AB

FH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

-01

-005

0

005

01

015

02

AB

FH

Breg

K2 H

0341L+Μ++Μ-L

Figure 17 The dependence of forward-backward asymmetry for B rarr Klowast

2 (1430)micro+microminus on q2 for

different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-05

-025

0

025

05

075

P LH

Breg

K2 H

0341L+Μ++Μ-L

Figure 18 The dependence of Longitudinal lepton polarization asymmetry for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

ndash 28 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

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08

1f LH

Breg

K2 H

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

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0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

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f TH

Breg

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0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

[1] BELLE collaboration J-T Wei et al Measurement of the Differential Branching Fraction

and Forward-Backword Asymmetry for B rarr K()ℓ+ℓminus Phys Rev Lett 103 (2009) 171801

[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

Phys Lett B 675 (2009) 59 [arXiv09033086] [INSPIRE]

[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

decay in the standard model and in supersymmetry JHEP 01 (2003) 074 [hep-ph0212158]

[INSPIRE]

[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

in the Decays B rarr K(lowast)ℓ+ℓminus Phys Rev Lett 102 (2009) 091803 [arXiv08074119]

[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

Asymmetries of B rarr Klowastmicro+microminus Decays in the Standard Model and Beyond

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[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

Klowast)ℓ+ℓminus in standard model and supersymmetric theories Phys Rev D 61 (2000) 074024

[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

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[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

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JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 30: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

0341L+Μ++Μ-L

Figure 19 The dependence of Normal lepton polarization asymmetry for B rarr Klowast

2 (1430)micro+microminus on

q2 for different values of All couplings The caption for the above figures are the same as figure 16

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

1f LH

Breg

K2 H

0341L+Μ++Μ-L

(c) (d)

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

0

02

04

06

08

f TH

Breg

K2 H

0341L+Μ++Μ-L

Figure 20 The dependence of Longitudinal and Transverse helicity fractions for B rarrKlowast

2 (1430)micro+microminus on q2 for different values of All couplings The caption for the above figures are the

same as figure 16

The longitudinal and transverse helicity fractions are shown in figure 20 for the case

when all couplings are present The deviation from SM values as expected are large since

now all couplings are present

Finally the transverse lepton polarization asymmetry PT acquires non-zero values

when T couplings are present (see eq (319)) In the SM this asymmetry is negligibly small

ndash 29 ndash

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

[1] BELLE collaboration J-T Wei et al Measurement of the Differential Branching Fraction

and Forward-Backword Asymmetry for B rarr K()ℓ+ℓminus Phys Rev Lett 103 (2009) 171801

[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

Phys Lett B 675 (2009) 59 [arXiv09033086] [INSPIRE]

[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

decay in the standard model and in supersymmetry JHEP 01 (2003) 074 [hep-ph0212158]

[INSPIRE]

[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

in the Decays B rarr K(lowast)ℓ+ℓminus Phys Rev Lett 102 (2009) 091803 [arXiv08074119]

[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

Asymmetries of B rarr Klowastmicro+microminus Decays in the Standard Model and Beyond

JHEP 01 (2009) 019 [arXiv08111214] [INSPIRE]

[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

Klowast)ℓ+ℓminus in standard model and supersymmetric theories Phys Rev D 61 (2000) 074024

[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

corrections Nucl Phys B 393 (1993) 23 [Erratum ibid B 439 (1995) 461] [INSPIRE]

[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 31: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

(a) (b)

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

03P TH

Breg

K2 H

1430L+Μ++Μ-L

0 2 4 6 8 10 12 14q2HGeV2

L

-04

-03

-02

-01

0

01

02

P TH

Breg

K2 H

1430L+Μ++Μ-L

(c)

0 2 4 6 8 10 12 14q2HGeV2

L

-08

-06

-04

-02

0

02

04

P NH

Breg

K2 H

1430L+Μ++Μ-L

Figure 21 The dependence of transverse lepton polarization asymmetry for B0 rarr Klowast

2 (1430)micro+microminus

on q2 where different curves correspond to different choices of the values of all couplings inside their

allowed region and the flat solid line at PT = 0 represent the SM with no uncertainty region In

figure (a) only tensor type couplings are present figures (bc) when all couplings are present

since all Wilson coefficients are predominantly real The transverse lepton polarization

asymmetry will be imaginary if all couplings are real(cf eq (319)) but for imaginary

values of CT CTE couplings it will become a real observable Figure 21 shows PT for

different values of imaginary tensor couplings while all other couplings are taken to be real

It can be seen that for the certain values of NP couplings the value of transverse lepton

polarization is significant It is therefore expected that its experimental observation in

future will help us to get constraints on CT CTE couplings In figure 21a only CT CTE

couplings are present and in figures 21(bc) all other couplings are present too hence the

values of PT become large

5 Conclusions

We carried out the study of the decay B rarr Klowast2 (1430)micro

+microminus in the presence of new physics

(NP) operators beyond SM We have performed a schematic model-independent analysis

allowing for new vector-axial vector (VA) scalar-pseudoscalar (SP) and tensor (T) operator

couplings We analyzed the effects on different physical observables for this channel in

the presence of these NP operator couplings To conclude this study we summarize the

important out comes as follows

ndash 30 ndash

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

[1] BELLE collaboration J-T Wei et al Measurement of the Differential Branching Fraction

and Forward-Backword Asymmetry for B rarr K()ℓ+ℓminus Phys Rev Lett 103 (2009) 171801

[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

Phys Lett B 675 (2009) 59 [arXiv09033086] [INSPIRE]

[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

decay in the standard model and in supersymmetry JHEP 01 (2003) 074 [hep-ph0212158]

[INSPIRE]

[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

in the Decays B rarr K(lowast)ℓ+ℓminus Phys Rev Lett 102 (2009) 091803 [arXiv08074119]

[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

Asymmetries of B rarr Klowastmicro+microminus Decays in the Standard Model and Beyond

JHEP 01 (2009) 019 [arXiv08111214] [INSPIRE]

[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

Klowast)ℓ+ℓminus in standard model and supersymmetric theories Phys Rev D 61 (2000) 074024

[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

corrections Nucl Phys B 393 (1993) 23 [Erratum ibid B 439 (1995) 461] [INSPIRE]

[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 32: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

bull The differential branching ratios generally increase for most of the parameter space

of NP couplings For V A couplings the BR increases considerably particularly when

both RVA and RprimeVA are involved the BR is enhanced by more than a factor of 4 and

the reason is that the new V A couplings are not ml suppressed compared to other

NP couplings On the other hand the BR for case of SP and T couplings fall mainly

in uncertainty region of SM The reason behind this lies in the fact that SP and T

operator couplings are severely constrained by the upper bound on B(Bs rarr micro+microminus)

as well as their contribution is ml suppressed

bull The forward-backward asymmetry of the final state leptons deviates sizable in the

presence of V A couplings When the new bound given in Eq 43 is ignored the zero

crossing even disappears and its values become all positives or all negative Similar

kind of trend was also witnessed for the decay B rarr Klowast(892)micro+microminus If we consider

the recent measurement of AFB for B rarr Klowast(892)micro+microminus decay at LHCb [15] we

can see that new VA couplings get constrained within a narrow band given in Eq

43 However the new SP couplings contribute delicately to the shifting of the zero

position of AFB for both decays B rarr Klowast(892)micro+microminus [52] and B rarr Klowast2 (1430)micro

+microminus

SP operators couplings particularly S coupling shift the zero crossings both towards

left as well as towards right When T coupling is added to the scene the zero shifting is

enhanced and it even disappears for certain values of SPminusT couplings Qualitatively

the AFB in presence of SP and T couplings separately areml suppressed while SPminusTcouplings together are notml suppressed and hence show large NP effects on the decay

under consideration

bull The longitudinal lepton polarization asymmetry (PL) is also an important tool to

look for physics beyond SM Again the PL is very sensitive V A couplings its values

sway between the positive SM value to negative maximum value for defined range of

RVA and RprimeVA couplings The effects on PL are still mild when S and P couplings

are introduced individually but the effects are magnified when both SP couplings are

present The T couplings also disturb the values of PL and the effects get added up

when both SP and T couplings are in combination

bull The normal lepton polarization asymmetry is not as sensitive to V A operator cou-

plings like the other observables The V A effects on PN are more dominant in the low

q2 region while in high q2 regime they all asymptotically converge to zero The nor-

mal lepton polarization provides an ideal testing ground to test the existence of SP

new physics operators The distinct behavior of PN in the presence of SP couplings

is very important to test the existence of SP as well as SP minus T operators

bull The longitudinal and transverse helicity fraction of the final state meson Klowast2 (1430)

are also efficient tool to look for the NP effects In case of V A operators the there

is a noticeable change in helicity fractions of the final state meson For SP operator

couplings the effects are more prominent in the high q2 regime The tensor couplings

CT and CTE also allow the helicity fractions to deviate significantly from their SM

values

ndash 31 ndash

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

[1] BELLE collaboration J-T Wei et al Measurement of the Differential Branching Fraction

and Forward-Backword Asymmetry for B rarr K()ℓ+ℓminus Phys Rev Lett 103 (2009) 171801

[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

Phys Lett B 675 (2009) 59 [arXiv09033086] [INSPIRE]

[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

decay in the standard model and in supersymmetry JHEP 01 (2003) 074 [hep-ph0212158]

[INSPIRE]

[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

in the Decays B rarr K(lowast)ℓ+ℓminus Phys Rev Lett 102 (2009) 091803 [arXiv08074119]

[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

Asymmetries of B rarr Klowastmicro+microminus Decays in the Standard Model and Beyond

JHEP 01 (2009) 019 [arXiv08111214] [INSPIRE]

[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

Klowast)ℓ+ℓminus in standard model and supersymmetric theories Phys Rev D 61 (2000) 074024

[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

corrections Nucl Phys B 393 (1993) 23 [Erratum ibid B 439 (1995) 461] [INSPIRE]

[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 33: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

We hope that these results will be tested one some ongoing and future experiments

like LHCb and super B factories

Acknowledgments

The authors greatly acknowledge some helpful discussions with Prof Riazuddin and Prof

Fayyazuddin The authors would also like to thank Ali Paracha for some useful discus-

sions The author JA would like to thank the facilities provided by National Centre for

Physics and also support by Science and Engineering Research Canada (NSERC) during

the revision of this paper

References

[1] BELLE collaboration J-T Wei et al Measurement of the Differential Branching Fraction

and Forward-Backword Asymmetry for B rarr K()ℓ+ℓminus Phys Rev Lett 103 (2009) 171801

[arXiv09040770] [INSPIRE]

[2] BABAR collaboration B Aubert et al Measurements of branching fractions rate

asymmetries and angular distributions in the rare decays B rarr Kℓ+ℓminus and B rarr Klowastℓ+ℓminus

Phys Rev D 73 (2006) 092001 [hep-ex0604007] [INSPIRE]

[3] BABAR collaboration B Aubert et al Angular Distributions in the Decays B rarr K ℓ+ℓminus

Phys Rev D 79 (2009) 031102 [arXiv08044412] [INSPIRE]

[4] E Lunghi and A Soni Hints for the scale of new CP-violating physics from B-CP

anomalies JHEP 08 (2009) 051 [arXiv09035059] [INSPIRE]

[5] Heavy Flavor Averaging Group collaboration E Barberio et al Averages of bminushadronand cminushadron Properties at the End of 2007 arXiv08081297 [INSPIRE]

[6] UTfit collaboration M Bona et al First Evidence of New Physics in blarrrarr s Transitions

PMC Phys A 3 (2009) 6 [arXiv08030659] [INSPIRE]

[7] S Baek C-W Chiang and D London The B rarr pi K Puzzle 2009 Update

Phys Lett B 675 (2009) 59 [arXiv09033086] [INSPIRE]

[8] T Feldmann and J Matias Forward backward and isospin asymmetry for B rarr K ℓ+ℓminus

decay in the standard model and in supersymmetry JHEP 01 (2003) 074 [hep-ph0212158]

[INSPIRE]

[9] BABAR collaboration B Aubert et al Direct CP Lepton Flavor and Isospin Asymmetries

in the Decays B rarr K(lowast)ℓ+ℓminus Phys Rev Lett 102 (2009) 091803 [arXiv08074119]

[INSPIRE]

[10] W Altmannshofer P Ball A Bharucha AJ Buras DM Straub et al Symmetries and

Asymmetries of B rarr Klowastmicro+microminus Decays in the Standard Model and Beyond

JHEP 01 (2009) 019 [arXiv08111214] [INSPIRE]

[11] G Burdman Short distance coefficients and the vanishing of the lepton asymmetry in B rarrV ℓ+ lepton- Phys Rev D 57 (1998) 4254 [hep-ph9710550] [INSPIRE]

[12] A Ali P Ball L Handoko and G Hiller A Comparative study of the decays B rarr (K

Klowast)ℓ+ℓminus in standard model and supersymmetric theories Phys Rev D 61 (2000) 074024

[hep-ph9910221] [INSPIRE]

ndash 32 ndash

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

corrections Nucl Phys B 393 (1993) 23 [Erratum ibid B 439 (1995) 461] [INSPIRE]

[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 34: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

[13] M Beneke T Feldmann and D Seidel Systematic approach to exclusive B rarr V ℓ+ℓminus V

gamma decays Nucl Phys B 612 (2001) 25 [hep-ph0106067] [INSPIRE]

[14] Belle Finds a Hint of New Physics in Extremely Rare B Decays reported in August 2009 (as

Press Release) httpwwwkekjpintra-epress2009BellePress14ehtml

[15] LHCb collaboration Angular analysis of B0 rarr Klowast0micro+microminus LHCb-CONF-2011-038 (Angular

analysis of B0 rarr Klowast0micro+microminus)

[16] AS Cornell N Gaur and SK Singh FB asymmetries in the B rarr K ℓ+ℓminus decay A Model

independent approach hep-ph0505136 [INSPIRE]

[17] A Ali T Mannel and T Morozumi Forward backward asymmetry of dilepton angular

distribution in the decay b rarr s ℓ+ℓminus Phys Lett B 273 (1991) 505 [INSPIRE]

[18] AJ Buras and M Munz Effective Hamiltonian for B rarr X(s) e+eminus beyond leading

logarithms in the NDR and HV schemes Phys Rev D 52 (1995) 186 [hep-ph9501281]

[INSPIRE]

[19] M Misiak The b rarr s e+eminus and b rarr s gamma decays with next-to-leading logarithmic QCD

corrections Nucl Phys B 393 (1993) 23 [Erratum ibid B 439 (1995) 461] [INSPIRE]

[20] F Kruger and E Lunghi Looking for novel CP-violating effects in B rarr Klowastℓ+ lepton-

Phys Rev D 63 (2001) 014013 [hep-ph0008210] [INSPIRE]

[21] A Ali E Lunghi C Greub and G Hiller Improved model independent analysis of

semileptonic and radiative rare B decays Phys Rev D 66 (2002) 034002 [hep-ph0112300]

[INSPIRE]

[22] A Ghinculov T Hurth G Isidori and Y Yao New NNLL results on the decay B rarr X(s)

ℓ+ℓminus Eur Phys J C 33 (2004) S288 [hep-ph0310187] [INSPIRE]

[23] C Bobeth T Ewerth F Kruger and J Urban Analysis of neutral Higgs boson

contributions to the decays B( s) rarr ℓ+ℓminus and B rarr Kℓ+ℓminus Phys Rev D 64 (2001) 074014

[hep-ph0104284] [INSPIRE]

[24] PH Chankowski and L Slawianowska Effects of the scalar FCNC in brarr sl+ sl- transitions

and supersymmetry Eur Phys J C 33 (2004) 123 [hep-ph0308032] [INSPIRE]

[25] G Hiller and F Kruger More model independent analysis of brarr s processes

Phys Rev D 69 (2004) 074020 [hep-ph0310219] [INSPIRE]

[26] AK Alok and S Sankar New physics upper bound on the branching ratio of Bs rarr ℓ+ℓminus

Phys Lett B 620 (2005) 61 [hep-ph0502120] [INSPIRE]

[27] AK Alok A Dighe and S Sankar Tension between scalarpseudoscalar new physics

contribution to B(s) rarr micro+microminus and B rarr K micro+microminus Mod Phys Lett A 25 (2010) 1099

[arXiv08033511] [INSPIRE]

[28] AK Alok A Dighe and S Sankar Probing extended Higgs sector through rare brarr smicro+microminus

transitions Phys Rev D 78 (2008) 034020 [arXiv08050354] [INSPIRE]

[29] A Hovhannisyan W-S Hou and N Mahajan B rarr K ℓ+ℓminus Forward-backward Asymmetry

and New Physics Phys Rev D 77 (2008) 014016 [hep-ph0701046] [INSPIRE]

[30] F Kruger and J Matias Probing new physics via the transverse amplitudes of B0 rarr K0 (rarrK- pi+) ℓ+ℓminus at large recoil Phys Rev D 71 (2005) 094009 [hep-ph0502060] [INSPIRE]

[31] E Lunghi and J Matias Huge right-handed current effects in B rarr K(K pi)ℓ+ℓminus in

supersymmetry JHEP 04 (2007) 058 [hep-ph0612166] [INSPIRE]

ndash 33 ndash

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 35: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

[32] U Egede T Hurth J Matias M Ramon and W Reece New observables in the decay mode

B(d)rarr anti- K0 ℓ+ℓminus JHEP 11 (2008) 032 [arXiv08072589] [INSPIRE]

[33] T Aliev V Bashiry and M Savci Double lepton polarization asymmetries in the B rarr K

ℓ+ℓminus decay beyond the standard model Eur Phys J C 35 (2004) 197 [hep-ph0311294]

[INSPIRE]

[34] T Aliev V Bashiry and M Savci Polarized lepton pair forward backward asymmetries in B

rarr K ℓ+ℓminus decay beyond the standard model JHEP 05 (2004) 037 [hep-ph0403282]

[INSPIRE]

[35] T Aliev M Cakmak A Ozpineci and M Savci New physics effects to the lepton

polarizations in the B rarr K lepton+ lepton- decay Phys Rev D 64 (2001) 055007

[hep-ph0103039] [INSPIRE]

[36] W Bensalem D London N Sinha and R Sinha Lepton polarization and forward backward

asymmetries in b rarr s tau+ tau- Phys Rev D 67 (2003) 034007 [hep-ph0209228]

[INSPIRE]

[37] AK Alok A Dighe D Ghosh D London J Matias et al New-physics contributions to

the forward-backward asymmetry in B rarr K micro+microminus JHEP 02 (2010) 053

[arXiv09121382] [INSPIRE]

[38] W Wang B to tensor meson form factors in the perturbative QCD approach

Phys Rev D 83 (2011) 014008 [arXiv10085326] [INSPIRE]

[39] K-C Yang B to Light Tensor Meson Form Factors Derived from Light-Cone Sum Rules

Phys Lett B 695 (2011) 444 [arXiv10102944] [INSPIRE]

[40] R-H Li C-D Lu and W Wang Branching ratios forward-backward asymmetries and

angular distributions of B rarr Klowast

2 l+lminus in the standard model and new physics scenarios

Phys Rev D 83 (2011) 034034 [arXiv10122129] [INSPIRE]

[41] N Katirci and K Azizi B to strange tensor meson transition in a model with one universal

extra dimension JHEP 07 (2011) 043 [arXiv11053636] [INSPIRE]

[42] H Hatanaka and K-C Yang Radiative and Semileptonic B Decays Involving the Tensor

Meson K(2)(1430) in the Standard Model and Beyond Phys Rev D 79 (2009) 114008

[arXiv09031917] [INSPIRE]

[43] S Choudhury A Cornell and N Gaur Analysis of the B rarr K(2)(1430)ℓ+ℓminus decay

Phys Rev D 81 (2010) 094018 [arXiv09114783] [INSPIRE]

[44] S Choudhury A Cornell and J Roussos Analysis of the lepton polarisation asymmetries of

B rarr K2(1430) ℓ+ ℓminus decay Eur Phys J C 71 (2011) 1751 [arXiv11055901] [INSPIRE]

[45] M Junaid and M Aslam Complementarity of Semileptonic B to Klowast

2 (1430) and Klowast(892)

Decays in the Standard Model with Fourth Generation arXiv11033934 [INSPIRE]

[46] T Aliev and M Savci B rarr K2ℓ+ℓminus decay beyond the Standard Model

Phys Rev D 85 (2012) 015007 [arXiv11092738] 18 pages 5 figures LaTeX formatted

[INSPIRE]

[47] CDF collaboration T Aaltonen et al Search for B0s rarr micro+microminus and B0

d rarr micro+microminus decays with

2fbminus1 of pp collisions Phys Rev Lett 100 (2008) 101802 [arXiv07121708] [INSPIRE]

[48] Particle Data Group C Amsler et al Review of Particle Physics

Phys Lett B 667 (2008) 1

ndash 34 ndash

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions
Page 36: Model independent analysis of B →  (1430)μ+μ−decay

JHEP02(2012)045

[49] BABAR collaboration B Aubert et al Measurement of the B rarr Xsℓ+ℓminus branching

fraction with a sum over exclusive modes Phys Rev Lett 93 (2004) 081802

[hep-ex0404006] [INSPIRE]

[50] Belle collaboration M Iwasaki et al Improved measurement of the electroweak penguin

process B rarr X(s) ℓ+ℓminus Phys Rev D 72 (2005) 092005 [hep-ex0503044] [INSPIRE]

[51] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Conserving Observables JHEP 11 (2011) 121 [arXiv10082367] [INSPIRE]

[52] AK Alok A Datta A Dighe M Duraisamy D Ghosh et al New Physics in b rarr s

micro+microminus CP-Violating Observables JHEP 11 (2011) 122 [arXiv11035344] [INSPIRE]

[53] J-P Lee Radiative B decays to the axial K mesons at next-to-leading order

Phys Rev D 69 (2004) 114007 [hep-ph0403034] 17pages 4 figures Minor changes typos

corrected PRD accepted version [INSPIRE]

[54] M Beneke Hadronic B decays with QCD factorization Int J Mod Phys A 21 (2006) 642

[hep-ph0509297] [INSPIRE]

[55] TM Aliev and M Savci Λb rarr Λl+lminus decay in Universal Extra Dimensions

Eur Phys J C 50 (2007) 91 [hep-ph0606225] [INSPIRE]

[56] Particle Data Group collaboration K Nakamura et al Review of particle physics

J Phys G 37 (2010) 075021 [INSPIRE]

ndash 35 ndash

  • Introduction
  • B -gt K(2) (1430) mu+ mu- operators and matrix elements
    • Standard model
    • New physics operators
    • Parametrization of matrix elements and form factors
    • Numerical constraints on NP parameters
      • Decay rate forward-backward asymmetry and lepton polarization asymmetries for B-gt K(2) (1430) mu+ mu- decay
        • Differential decay rate
        • Forward-backward asymmetry
        • Lepton polarization asymmetries
        • Helicity fractions
          • Numerical Analysis
            • VA new-physics operators
            • SP new-physics operators
            • T operators and interference of SP and T operators
            • All NP operators together
              • Conclusions