AAC Global AnalysisNaohito Saito (KEK)

for Shunzo Kumano

High Energy Accelerator Research Organization (KEK) Graduate University for Advanced Studies (GUAS)

shunzo.kumano@kek.jphttp://research.kek.jp/people/kumanos/

AAC: http://spin.riken.bnl.gov/aac/

Contents

Introduction to polarized PDFs Global analysis of longitudinally polarized PDFs

Polarized PDFs of AACAAC (Asymmetry Analysis Collaboration)

Papers on polarized PDFsT. Gehrmann and W. J. Stirling, Z. Phys. C65 (1995) 461; Phys. Rev. D53 (1996) 6100.

D. de Florian, L. N. Epele, H. Fanchiotti, C. A. Garcia Canal, and R. Sassot (+G. A. Navarro), Phys. Rev. D51 (1995) 37; D57 (1998) 5803; D62 (2000) 094025; D71 (2005) 094018.

M. Glück, E. Reya, M. Stratmann, and W. Vogelsang, Phys. Lett. B359 (1995) 201; Phys. Rev. D53 (1996) 4775; D63 (2001) 094005.

D. de Florian, R. Sassot, M. Stratmann, and W. Vogelsang, arXiv:0804.0422 [hep-ph].

G. Altarelli, R. D. Ball, S. Forte, and G. Ridolfi, Nucl. Phys. B496 (1997) 337; Acta Phys. Pol. B29 (1998) 1145.

C. Bourrely, F. Buccella, O. Pisanti, P. Santorelli, and J. Soffer, Prog. Theor. Phys. 99 (1998) 1017; Eur. Phys. J. C23 (2002) 487; C41 (2005) 327; Phys. Lett. B648 (2007) 39.

L. E. Gordon, M. Goshtasbpour, and G. P. Ramsey, Phys. Rev. D58 (1998) 094017.

SMC (B. Adeva et al.), Phys. Rev. D58 (1998) 112002.

E. Leader, A. V. Sidrov, and D. B. Stamenov, Phys. Rev. D58 (1998) 114028; Eur. Phys. J. C23 (2002) 479; PRD67 (2003) 074017; JHEP 0506 (2005) 033; PRD73 (2006) 034023; D75 (2007) 074027.

AAC (Y. Goto et al., M. Hirai et al.), PRD62 (2000) 034017; D69 (2004) 054021; D74 (2006) 014015. J. Bartelski and S. Tatur, Phys. Rev. D65 (2002) 034002.

J. Blümlein and H. Böttcher, Nucl. Phys. B636 (2002) 225. It is likely that I miss some papers!(Analyses of experimental groups)

Situation of data for polarized PDFs

Neutrino factory: ~10 years later

Small-x: EIC (eRHIC, ELIC) ?

Charm: EIC (eRHIC, ELIC) ?

RHIC

pp: RHIC

RHIC-Spin program should play an important role in determining polarized PDFs.

J-PARC GSI-FAIR

(MRST, hep/ph-9803445)

(from H1 and ZEUS, hep-ex/0502008)

F2 datafor the proton

[AAC, PRD74 (2006) 014015]

It may be possible to remove small Q2 datain an analysis of unpolarized PDFs, whereasit is difficult to remove them in g1 (or A1).

x =0.65

x=0.013

x=0.0005

Comments on the previous figure

g lack of small-x data → m akesthedeterm inationoflimx→ 0

Δq(x)difficult

→ quarkspincontentcannotbewelldeterm ined

g lack of data in a wide range of Q2 at small x → m akesthedeterm inationofΔg(x)difficult→ gluoncontributiontotheprotonspincannotbedeterm ined

g lack of data in a wide range of Q2

→ difficulttorem ovethesm allQ2(<4GeV 2 )data,whichm aycontainhigher-twisteffects,from theanlysisdataset

Three publications:(AAC00) Y. Goto et al., Phys. Rev. D62 (2000) 034017.(AAC03) M. Hirai, S. Kumano, N. Saito, Phys. Rev. D69 (2004) 054021;(AAC06) D74 (2006) 014015.

Original Members: Y. Goto, N. Hayashi, M. Hirai, H. Horikawa, S. Kumano, M. Miyama, T. Morii, N. Saito, T.-A. Shibata, E. Taniguchi, T. Yamanishi

(Theorist, Experimentalist)

http://spin.riken.bnl.gov/aac/Asymmetry Analysis Collaboration (AAC)

Active members now

2000 version (AAC00) - Q2 dependence of A1, positivity - Δq at small and large x Δ issue2004 version (AAC03) - uncertainty estimation (very large Δg uncertainty, impact of accurate g1

p (E155)) - error correlation between Δg and Δq2006 version (AAC06) - include RHIC-Spin p0 (Δg uncertainty is significantly reduced) - Δg at large x ? (Q2 difference between HERMES and COMPASS) - Δg < 0 solution at x < 0.12008 ? - effects of future JLab data (g1

d) on Δg

s l+lN∝ ε λ

μ∗ε λνWμν (λ N ) : photoabsorption cross section

A1 ≡

s1 2 −s 3 2

s1 2 +s 3 2

=MnG1 −Q

2G 2

M 3W1

;g1F1

=g12x 1+ R( )

F2

nM 2 G1 = g1

n 2

M 2 G2 = g2

MW1 =F1

R ≡FL2xF1

=F2 −2xF12xF1

General strategies for determining polarized PDFs

g1(x,Q2 ) =12

eq2

q∑ dy

yx

1

∫ Δq(x / y,Q2 )+ Δq(x / y,Q2 )⎡⎣ ⎤⎦ d(1−y)+as(Q

2 )2p

ΔCq(y)+⋅⋅⋅⎡⎣⎢

⎤⎦⎥

+12eq2 dy

yx

1

∫ Δg(x / y,Q2 ) nfas(Q

2 )2p

ΔCg(y)+⋅⋅⋅⎡⎣⎢

⎤⎦⎥ eq

2 =1nf

eq2

q∑

Leading Order (LO) Next to Leading Order (NLO)

ΔCq (ΔCg ) = quark (gluon) coefficient function

F2 (x,Q2 ) =x eq2

q∑ dy

yx

1

∫ q(x / y,Q2 )+ q(x / y,Q2 )⎡⎣ ⎤⎦ d(1−y)+as(Q

2 )2p

Cq(2)(y)+⋅⋅⋅

⎡⎣⎢

⎤⎦⎥

+ x eq2 dy

y

x

1

∫ g(x / y,Q2 ) nfas(Q

2 )2p

Cg(2)(y)+⋅⋅⋅

⎡⎣⎢

⎤⎦⎥

Unpolarized PDFsR(x,Q2 ) : taken from experimental measurements

Initial distributions

c2 fit to the data [p, n (3He), d]

We analyzed the data with the following conditions.• unpolarized PDF GRV98• initial Q2 Q0

2 = 1 GeV 2

• number of flavor Nf = 3

(1) Δq(x,Q02 ), Δq(x,Q0

2 ), Δg(x,Q02 ) : expressedinterm sofparam eters

→ evolvethem toexperim entalQ2points→ calculateg1

(2) q(x,Q2 ), q(x,Q2 ), g(x,Q2 ) : calculated by a library code of typical unpolarized PDFs → calculateF2

(3) R(x,Q2 ) : typical parametrization for explaining experimental data (e.g. SLAC parametrization)

(4) calculate A1theo =g1

2x 1+ R( )F2

, thenc 2 =(A1i

exp −A1itheo)2

sA1i2

i∑

(5) parameters are determined so as to minimize c 2

Analysis procedure

AAC00(Some Highlights)

Actual fit to A1 data(AAC00)

“neutron”

proton

deuteron

* Used data set as raw as Possible

Q2 dependence of A1 Q2 independence assumption was sometimes used for getting g1 from A1 data.

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.5 1 10 100

A 1p

Q2(GeV2)

LONLOE143SMCHERMES

x = 0.117

A1 is Q2 dependent especially at small Q2 (< 2 GeV2).

The lack of accurate Q2 dependent data at small x makes the determination of Δg very difficult.

Figure from AAC00 analysis

“Spin content” Δ

∫ Δ=Δ1

minmin

)()(x

dxxx

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.000001 0.00001 0.0001 0.001 0.01 0.1 1

xmin

SMC

LSS

aq=1.6

aq=1.0

aq=0.5

AAC (aq : free)

Δqq ∝ xαq (x→ 0)

(from AAC00)

AAC03 (Some Highlights)

Error estimation Hessian methodc2(x) is expanded around its minimum x 0 (x =parameter)

where the Hessian matrix is defined by

€

c 2(ξ0 + δξ ) = χ 2(ξ0) +∂χ 2(ξ0)

∂ξ iδξ i

i∑ +

12

∂ 2χ 2 (ξ0 )∂ξ i∂ξ j

δξ iδξ ji, j∑ + ⋅⋅⋅

In the c2 analysis, 1s standard error is

The error of a distribution F(x) is given by

c 2 =(A1i

exp − A1itheo )2

σ A1i

2i

∑

Polarized PDFs (AAC03)• PDF uncertainties are reduced by

including precise (E155-p) data

• Valence-quark distributions are well determined

– Small uncertainties of Δuv, Δdv

• Antiquark uncertainty is significantly reduced– g1

p 4Δuv+ Δdv+12Δq

• Δg(x) is not determined– Large uncertainty– Indirect contribution to g1

p – Correlation with antiquark

0

0.1

0.2

0.3

0.4

0.5

0.001 0.01 0.1 1

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.001 0.01 0.1 1

x

xΔuv(x)

xΔdv(x)

Q2=1GeV2

-2

-1

0

1

2

3

0.001 0.01 0.1 1

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.001 0.01 0.1 1

x

xΔg(x)

Q2=1GeV2

xΔq(x)

AAC03 uncertainties

AAC00 uncertainties

AAC00 AAC03 (with E155-p)

E155 data

(A1exp-A1

theo)/A1theo at the same Q2 point

Correlation between Δq(x) and Δg(x)

• analysis with Δg(x)=0 at Q2=1 GeV2

c2/d.o.f. = 0.915

• Δqv(x) uncertainties are not affected

• antiquark uncertainties are reduced

Strong correlation with Δg(x) Note: correlation with Δg(x) is almost terminated in the Δg=0 analysis

-2

-1

0

1

2

3

0.001 0.01 0.1 1

-0.05

-0.04

-0.03

-0.02

-0.01

0

0.01

0.02

0.001 0.01 0.1 1

x

xΔg(x)

Q2=1GeV2

xΔq(x)

0

0.1

0.2

0.3

0.4

0.5

0.001 0.01 0.1 1

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.001 0.01 0.1 1

x

xΔuv(x)

xΔdv(x)

Q2=1GeV2

The error band shrinks due to the correlation with Δg(x).

Δg=0 uncertainties

AAC03 uncertainties

Trial Fit!• Utilized NLO p0

calculation (thanks to Vogelsang and Stratmann) – to find “relevant-x”

for each pT bin

• Trial Fit to – A*x (GRSV ~ 1*x)– A*x*x

gba +

Δ+

Δ=

gg

ggALL

2

0pLLA

)GeV/( cpT

Lepton Scatt.model 2 ndf prob. 2 ndf prob. g

(0.73±0.39)x 10.8 7 0.15 6.2 5 0.29 0.30(-1.03±0.43)x 11.3 7 0.13 14.6 5 0.01 -0.42(5.74±3.10)x 2 10.8 7 0.15 5.3 5 0.38 0.35

(-9.64±2.99)x 2 10.5 7 0.16 18.3 5 0.00 -0.60

From PANIC05 talk

AAC06

Gluon polarization at large xAAC, PRD74 (2006) 014015: Analysis without higher-twist effects

QHERMES2 ~ 1 GeV2 QCOMPASS

2 ~ 6 GeV2

NLOΔCG=0

g1(x,Q2 ) =12

eq2

q∑ dz

zx

1

∫ Δq(x / z,Q2 )+ Δq(x / z,Q2 )⎡⎣ ⎤⎦

× d(1−z)+as(Q

2 )2p

ΔCq(z)+⋅⋅⋅⎡⎣⎢

⎤⎦⎥

+12eq2 dz

zx

1

∫ Δg(x / z,Q2 ) nfas(Q

2 )2p

ΔCg(z)+⋅⋅⋅⎡⎣⎢

⎤⎦⎥

This term is terminated.

x=0.001 x=0.05 x=0.3

Positive contribution to A1 comesfrom ΔCG ⊗Δgatx ~0.05.

Note: ΔCG ⊗Δg > 0ifΔg(0.05 / 0.2 =0.25)> 0Gluonpolarizationispositiveatlargex.

However, it may be a higher-twist effect.

LSS, PRD73 (2006) 034023.

LT fitLT+HT fit

Leading Twist (LT)Higher Twist (HT)

LT+HT fit, only LT term is shown

At this stage, we cannot conclude that the difference betweenthe HERMES and COMPASS data should be 100% HT orHT+ΔG(large x)>0, or 100% ΔG(large x)>0 effects.

A1(x,Q2 ) =g1(x,Q

2 )LT + h(x) /Q2

F2(x,Q2 )exp

2x 1+ R(x,Q2 )exp⎡⎣ ⎤⎦

dΔsdpT

= dhhm in

hm ax

∫ dxaxam in

1

∫a,b,c∑ Δfa(xa) dxbxb

m in

1

∫ Δfb(xb)∂(t, zc)∂(pT ,h)

dΔsdpT

Δcp (zc)

xamin =

x11−x2

, xbm in =

xax2xa −x1

, zc =x1xa

+x2xb, x1 =

xT2e+h , x2 =

xT2e−h , xT =

2pTs,h =−ln tan

qp

2⎛⎝⎜

⎞⎠⎟

⎡⎣⎢

⎤⎦⎥

Δs : Δfa(xa,Q

2)⊗ Δfb(xb,Q2)⊗ Δs(ab→ cX)⊗ Δc

p (z,Q2)a,b,c∑

Parton distribution functions

Parton interactions Fragmentation functions

Description of p0 production

( ) , ( ) , ,( , ) , ,

gg q g X qg q g X qq qXqq q g q X qq qX qq qX

→ → →′ ′ ′→ → →

Subprocesses

g + g→ q(g)+ Xprocessesaredom inantatsm allpT q + g→ q(g)+ XatlargepT

The p0 production process is suitable for findingthe gluon polarization Δg.

(from Torii’s talk at Pacific-Spin05)

Comparison of fragmentation functions in pion

• Gluon and light-quark fragmentation functions have large uncertainties, but they are within the uncertainty bands.® The functions of KKP, Kretzer, AKK, DSS, and HKNS are consistent with each other.

All the parametrizations agreein charm and bottom functions.

(KKP) Kniehl, Kramer, Pötter(AKK) Albino, Kniehl, Kramer(HKNS) Hirai, Kumano, Nagai, Sudoh(DSS) de Florian, Sassot, Stratmann

M. Hirai, SK, T.-H. Nagai, K. Sudoh, PRD75 (2007) 094009.A code is available athttp://research.kek.jp/people/kumanos/ffs.html

Analysis with RHIC pion data

(AAC06)

AAC03

AAC06(with PHENIX π0)

Possibility of negative gluon polarization Δg < 0

p0 production data do not distinguish between Δg < 0 and Δg > 0

Possibility of a node-type ∆g(x) for explaining pT=2.38 GeV (Type 3).

Roles of RHIC-pion data Polarized PDFs, especially Δg, are better determined by the additional RHIC-p0 data.

(AAC06)

Gluon polarization from lepton scattering

S. Koblitz@DIS07

(AAC06)

Situation of g1 measurementsand polarized PDF errors

Comparison with other parameterizations

1st moments

– GRSV01(Sta) [ Phys. Rev. D63 (2001) 094005 ]– LSS01 (MS) [ Eur.Phys.J. C23 (2002) 479 ]– BB02 (SET3) [ Nucl. Phys. B636 (2002) 225 ]– DNS05 (KKP) [ Phys. Rev. D71 (2005) 094018 ] (Q2=10 GeV2 )

Δg Δ

AAC06 0.31 0.32 0.27 0.07AAC03 0.50 1.27 0.21 0.14GRSV01 0.420 0.204LSS 0.680 0.210BB 1.026 0.138DNS 0.574 0.311

€

Δ =Δuv + Δdv + 6Δq( )

€

Q2 = 1 GeV2

AAC08 ?

Effects of expected JLab E07-011 data

“Expected data” from Xiaodong Jiang

AAC asymmetries

Analyses with / without E07-011 data in comparison with effects of RHIC π0 data

Analysis set Current DIS RHIC ÉŒ0 JLab E07-0111 ÅZ Å[ Å[2 ÅZ ÅZ Å[3 ÅZ Å[ ÅZ

Two initial functions for∆g(x)

“positive”

“node”

x

Effects of expected JLab E07-011 data

preliminary

Positivity is n

ot satisf

ied

at this st

age!

preliminary

∆g(x) with PHENIX run-5 or JLab E07-011 data preliminary

É¢g function First moment DIS DIS+RHIC ÉŒDIS+E07-011

positive É¢g 0.53 0.36 0.53positive d(É¢g) 0.72 0.26 0.38positive δ(É¢g)/É¢g 1.36 0.72 0.72

node É¢g 0.87 0.4 0.87node δ(É¢g) 0.89 0.31 0.47node δ(É¢g)/É¢g 1.02 0.72 0.54

Significant improvements

Note: π0 data are from run-5 although it may not be a good idea to compare future data with past ones.

JLab-E07-011 is comparableto RHIC run-5 π0 in determining ∆g(x).

preliminary

Positivity is n

ot satisf

ied.

Summary of the AAC global analysis for polarized PDFs – Δuv(x), Δdv(x) are determined well– Δ= 0.27 0.07 (Q2= 1 GeV2 )– Δg(x) could not be well constrained

AAC-polarized-pdfs code could be obtained fromhttp://spin.riken.bnl.gov/aac/

• Uncertainties of polarized PDFs • Effects of E155-proton data • Global analysis also with Δg=0• Error correlation between Δg and Δq • Better Δg determination by RHIC-pion• Δg determination at large x by scaling violation (HERMES, COMPASS)• Possibility of a node-type Δg (Δg>0 at x>0.1, Δg<0 at x<0.1)

AAC03

AAC06

AAC08 ? in progress

The End

The End