Radiative Bc meson decays Bc → γud

A. K. Likhoded,1,2,* A. V. Luchinsky,1,3,† and S. V. Poslavsky1,3,‡1Institute for High Energy Physics, Protvino 142280, Russia

2Moscow Institute of Physics and Technology, Dolgoprundy 141700, Russia3SSC RF ITEP of NRC Kurchatov Institute, Institute for Theoretical and Experimental Physics,

Moscow 117218, Russia(Received 30 April 2014; published 18 August 2014)

Inclusive radiative decays Bc → γud are considered. It is shown that photon emission allows one tobypass the chiral suppression and leads the increase of the branching fraction by about 4 orders ofmagnitude. The estimates of exclusive decay branching fractions of Bc meson into different sets of πmesons are also given.

DOI: 10.1103/PhysRevD.90.034017 PACS numbers: 13.20.-v, 13.20.He

I. INTRODUCTION

Bc meson ðbcÞ takes a special place in the family ofheavy quarkonia. In contrast to ðccÞ and ðbbÞ systems thedecays of the ground Bc state are necessarily caused byweak interaction and can be divided onto three classes:

(i) b quark decays with spectator c;(ii) c quark decays with spectator b;(iii) annihilation decays Bc → eþνe, cs, ud, etc.Some decays of the first type, e.g., Bc → ψ ð0Þ þ X withX ¼ π, 3π, 5π, eν (see [1–5]), and only one decay of thesecond type (Bc → Bsπ) are experimentally measuredfor today.Annihilation decays are not yet observed. The total

branching fraction of these channels is about 10% [6]and the main contribution is given by Bc → τν and Bc →cs decays. Channels with light quarks in the final stateare almost negligible due to chirality suppression. Thissuppression, however, can be bypassed if one considersradiative decays, e.g., Bc → udþ γ. The probabilities ofthese decays are suppressed by fine structure constant α,but the chirality suppression factor ðmu;d=MBc

Þ2 is absent.Our article is devoted to studies of such decays.

II. LIGHT MESON PRODUCTION INRADIATIVE Bc DECAYS

Decays of Bc meson with only light particles in the finalstate necessarily require the weak annihilation of quark-antiquark pair cb. The effective Lagrangian of this inter-action has the form

Leff ¼ −GFffiffiffi2

p VbcVudðbγμð1þ γ5ÞcÞðuγνð1þ γ5ÞdÞ: ð1Þ

Higher order QCD corrections can raise this expression bya factor of a1ðmcÞ ≈ 1.14 [7].

Figure 1 shows Feynman diagrams corresponding to theinclusive radiative decay

BcðPÞ → γðkÞuðk1Þdðk2Þ

in the considered approximation (particles momenta areshown in the parentheses). It should be noted that at the sameorder the diagram with photon emission from the virtual Wboson should also be included, but the correspondingamplitude is suppressed by an additional Fermi constant,so in the following we will not take it into account.The corresponding amplitude can be written in the

form

M ¼ 4παGFa1ffiffiffi2

p VbcV�udϵ

μ½HμνðQ2ÞLν þHμLμνðQ2Þ�; ð2Þ

where ϵμ is the final photon polarization vector, Q2 ¼ðk1 þ k2Þ2 is the squared invariant mass of the quark-antiquark pair, and Hμν, Hν, Lμν, Lν are the amplitudes ofBc → γW, Bc → W, W → udγ, W → ud transitions,respectively. The last two amplitudes are equal to

Lν ¼ δijuðk1Þγνð1þ γ5Þdðk2Þ; ð3Þ

Lμν ¼ δij�eu

1

ðk1 þ kÞ2 uðk1Þγμðk1 þ kÞγνð1þ γ5Þdðk2Þ

þ ed1

ðk2 þ kÞ2 uðk1Þγνð1þ γ5Þðk2 þ kÞγμdðk2Þ�;

ð4Þ

where i, j are the quark’s color indices, eu ¼ 2=3,ed ¼ −1=3 are electric charges of u and d quarks,and we neglected the masses of final particles. Thevertices of Bc meson decays strongly depend on theirinternal structure and cannot be calculated in theframework of perturbation theory. It is convenient towrite them in the general form

*Anatolii.Likhoded@ihep.ru†Alexey.Luchinsky@ihep.ru‡stvlpos@mail.ru

PHYSICAL REVIEW D 90, 034017 (2014)

1550-7998=2014=90(3)=034017(5) 034017-1 © 2014 American Physical Society

Hν ¼ fBcPν ð5Þ

HμνðQ2Þ ¼ Aggμν þ AppðQ2ÞPμPν þ AkkðQ2Þkμkνþ ApkðQ2ÞPμkν þ AkpðQ2ÞkμPν

þ iAeðQ2ÞeμναβPαkβ; ð6Þ

where fBcis the leptonic constant of Bc meson and

Ag;pp;kk;pk;kp;e are Bc → γW decay form factors. Let usfirst consider the consequences of gauge invariance. Inorder to satisfy this condition the vertices (3)–(6) shouldbe connected by the following relations:

kμLμν ¼ ðed − euÞLν; kμHμν ¼ ðec − ebÞHν: ð7Þ

The validity of the first equality can easily be checkedby direct calculations, and in order to satisfy the secondone the following identities should hold:

ðPkÞAppðQ2Þ ¼ ðeb − ecÞfBc;

AgðQ2Þ ¼ −ðPkÞApkðQ2Þ: ð8Þ

It should be stressed that these identities are analogs ofWard identities in QED and do not depend on the modelused for the description of Bc meson. It is interesting tocompare these results with radiative decay of neutralquarkonia ðccÞ, Bs, or Bd [8,9]. In the latter case, as it isclear from relation (7) after replacement of eb to ec, theconsidered diagrams are splitted into gauge invariantgroups and it is possible to determine the diagram thatgives the dominant contribution. The situation is

completely different if the radiative decay of a chargedmeson is considered, so all four diagrams should betaken into account [8,10].In order to obtain numerical results one should use some

explicit expressions for those presented in the vertex (6)form factors. The simplest model is the assumption thatheavy quarks stay at rest in the Bc meson rest frame, sotheir momenta are equal to

pb;c ¼mb;c

MP; ð9Þ

where mb;c are constituent masses of b and c quarks, andtotal mass of Bc meson is M ¼ mb þmc. In this approxi-mation the hadronization of the bc pair is described by theprojection operator

Π ¼ δij

2ffiffiffi3

p 1

4mbmc

ψð0ÞffiffiffiffiffiM

p ðpc þmcÞðPþMÞγ5ðpb −mbÞ;

ð10Þ

where i, j are quarks’ color indices, and Ψð0Þ is the Bcmeson wave function at the origin. The leptonic decayconstant fBc

in this approximation is equal to [11,12]

fBc¼ 2

ffiffiffiffiffi3

M

rΨð0Þ;

where Ψð0Þ is a radial part of the meson wave function atthe origin. Then, the form factors are determined by thefollowing expressions:

(a) (b)

(c) (d)

FIG. 1. Feynman diagrams for Bc → γud decay.

A. K. LIKHODED, A. V. LUCHINSKY, AND S. V. POSLAVSKY PHYSICAL REVIEW D 90, 034017 (2014)

034017-2

AδgðQ2Þ ¼ −

ffiffiffiffiffiffiffi3M

p

mbmca1jΨð0Þjðecmb − ebmcÞ;

AδppðQ2Þ ¼ 4

ffiffiffi3

pffiffiffiffiffiM

p a1ðeb − ecÞjΨð0Þj

M2 −Q2;

AδkkðQ2Þ ¼ 0;

AδpkðQ2Þ ¼ Aδ

kpðQ2Þ ¼ 2ffiffiffiffiffiffiffi3M

p

mbmca1

ðecmb − ebmcÞjΨð0ÞjM2 −Q2

;

AδeðQ2Þ ¼ 2

ffiffiffiffiffiffiffi3M

p

mbmca1

ðecmb þ ebmcÞjΨð0ÞjM2 −Q2

:

Differential and total widths of the considered decay in thisapproximation can be calculated analytically:

d2ΓdskdQ2

¼ 3α

16π2

�a1VbcVud

fBcGF

mc

�2

×ðebmc þ ecmbÞ2

M3

Q2

ðM2 −Q2Þ2

×

�Q4 þ 2s2Q2 þ s22

�1 −

m2c

m2b

�

− 2M2ðs2 þQ2Þ þM4

�; ð11Þ

dΓdQ2

¼ α

16π2

�a1VbcVud

fBcGF

mcmb

�2 ðebmc þmcebÞ2

M3

× ðm2b þm2

cÞQ2ðM2 −Q2Þ; ð12Þ

Γ ¼ α

96π2

�a1VbcVudfBc

GF

mbmc

�2

M3ðebmc þ ecmbÞ2; ð13Þ

where s2 ¼ ðP − k2Þ2. It is interesting to note that theseexpressions are free from collinear and infrared singular-ities and tend to zero in Q2 → 0 and Q2 → M2 limits.

This behavior is explained by mentioned above chiralitysuppression.In our paper we use the following values of model

parameters:

MBc¼ 6.2 GeV; mb ¼ 4.5 GeV; mc ¼ 1.7 GeV;

fBc¼ 400 MeV; Vbc ¼ 0.045; τBc

¼ 0.452 ps:

Transferred momentum distributions of Bc → γW formfactors and the Bc → γud branching fraction are shownin the left and right panels of Fig. 2. The integratedbranching fraction of the inclusive decay Bc → γud withthis choice of the parameters is equal to

Br½Bc → γud� ¼ 1.3 × 10−4:

Let us now consider the case of exclusive decays.According to the factorization theorem, for the exclusivedecay Bc → γR the distribution of the branching fractionover the squared invariant mass of light mesons system Q2

is connected with the width of the inclusive decay Bc →γud [13,14]:

dΓðBc → γRÞdQ2

¼Z

dQ2

2π

dΓðBc → γudÞdQ2

ρRðQ2Þ; ð14Þ

where the spectral function ρR describes the hadronizationof the ud pair into final state R.These spectral functions can be determined, for example,

from the analysis of the exclusive decays τ → ντR, as itwas done in [14]. The drawback of such approach is that thekinematical region of Q2 in our case is larger than in thecase of τ lepton decays. An alternative approach is toconsider the hadronization of the quark-antiquark pair bymesons in the Pythia event generator [15]. Using thismethod, one can determine the spectral functions over thewhole kinematic region. As it seen from Fig. 3 such an

10 20 30Q2, GeV2

0.2

0.1

0.1

0.2

0.3

A Q2 , GeV 1

10 20 30Q2, GeV2

1

2

3

4

Br Bc u d

Q2,

10 6

GeV2

(b)(a)

FIG. 2. (a) Bc → γW vertex form factors. Solid, dashed, and dotted lines correspond to App, Apk, and Ae form factors, respectively.The others are connected with them by the Ward identities (8). (b) Transferred momentum distribution of the Bc → γud decay branchingfraction.

RADIATIVE Bc MESON DECAYS … PHYSICAL REVIEW D 90, 034017 (2014)

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approach describes well the experimental data for meancharged particles multiplicity. It should be noted, however,that this approach does not allow one to control theconservation of some quantum numbers, e.g., isospin orcharge parity of the final state. For this reason presented

below results should be considered as rough estimated.The values of integrated branching fractions are presentedin Fig. 4.

III. CONCLUSION

In the presented paper the production of light mesonsin Bc decays with the emission of an additional photon isconsidered.It is well known that at leading order without photon

emission such decays are strongly suppressed by chiralityconservation. According to results presented in [6], thebranching fraction of Bc → ud decay is about 10−8 (suchsuppression is absent if charged Higgs boson contributions[18–20] are taken into account, so the considered decay canbe used to set an upper bound on its mass). In our paper weshow that consideration of higher order processes Bc →udγ allows one to bypass this rule, so the branchingfractions increase by about 4 orders of magnitude. It isclear also that the same effect can be observed in recentlydetected Bs → μþμ− decay [21–23].In our work we calculate analytical expressions for

differential and integrated branching fractions of inclusiveBc → udγ decay and present numerical results. In addition,using experimental data on τ-lepton decays and chargedmeson multiplicities we obtain the estimates for branchingfractions of exclusive decays.It should be noted that in our calculations the leading

order nonrelativistic quantum chromodynamic was used, sowe neglected the intrinsic motion of quarks in Bc meson.One can expect, however, that, similar to double charmo-nium production in electron-positron annihilation [24,25]and bottomonia decays [26–28], this effect could lead tosignificant increase of the branching fractions.

ACKNOWLEDGMENTS

The authors would like to thank V. V. Kiselev for fruitfuldiscussions. The work was financially supported by RFBR(#14-02-00096A) and FRRC grants.

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