AAC Global Analysis

Naohito Saito (KEK)

for

Shunzo Kumano

High Energy Accelerator Research Organization (KEK)

Graduate University for Advanced Studies (GUAS)

shunzo.kumano@kek.jp

http://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 AAC

AAC (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 → makes the determination of limx→ 0

Δq(x) difficult

→ quark spin content cannot be well determined

g lack of data in a wide range of Q2 at small x

→ makes the determination of Δg(x) difficult

→ gluon contribution to the proton spin cannot be determined

g lack of data in a wide range of Q2

→ difficult to remove the small Q2 (< 4 GeV2 ) data, which may contain higher-twist effects, from the anlysis data set

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 - q at small and large x issue

2004 version (AAC03) - uncertainty estimation (very large g uncertainty, impact of accurate g1

p (E155))

- error correlation between g and q 2006 version (AAC06) - include RHIC-Spin 0 (g uncertainty is significantly reduced) - g at large x ? (Q2 difference between HERMES and COMPASS) - g < 0 solution at x < 0.1 2008 ? - effects of future JLab data (g1

d) on g

σ λ+λN∝ ε λ

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

A1 ≡

σ1 2 −σ 3 2

σ1 2 +σ 3 2

=MνG1 −Q2G2

M 3W1

;g1

F1

=g1

2x 1+ R( )F2

νM 2

G1 = g1

ν 2

M 2G2 = g2

MW1 =F1

R ≡FL

2xF1

=F2 −2xF1

2xF1

General strategies for determining polarized PDFs

g1(x,Q2 ) =12

eq2

q∑ dy

yx

1

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

2 )2π

ΔCq(y) +⋅⋅⋅⎡

⎣⎢

⎤

⎦⎥

+12

eq2 dy

yx

1

∫ Δg(x / y,Q2 ) nf

αs(Q2 )

2πΔCg(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 )⎡⎣ ⎤⎦ δ(1−y) +αs(Q

2 )2π

Cq(2)(y) +⋅⋅⋅

⎡

⎣⎢

⎤

⎦⎥

+ x eq2 dy

y

x

1

∫ g(x / y,Q2 ) nf

αs(Q2 )

2πCg

(2)(y) +⋅⋅⋅⎡

⎣⎢

⎤

⎦⎥

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

Initial distributions

χ2 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 ) : expressed in terms of parameters

→ evolve them to experimental Q2 points

→ calculate g1

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

→ calculate F2

(3) R(x,Q2 ) : typical parametrization for explaining experimental data

(e.g. SLAC parametrization)

(4) calculate A1theo =g1

2x 1+ R( )F2

, then χ 2 =(A1i

exp −A1itheo)2

σ A1i

2i∑

(5) parameters are determined so as to minimize χ 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

A1

p

Q2(GeV2)

LO

NLO

E143

SMC

HERMESx = 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 g very difficult.

Figure from AAC00 analysis

“Spin content” DS

∫ ΔΣ=ΔΣ1

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

∝ xα

q (x → 0)

(from AAC00)

AAC03

(Some Highlights)

Error estimation Hessian method

2() is expanded around its minimum 0 ( =parameter)

where the Hessian matrix is defined by

€

χ 2(ξ0 + δξ ) = χ 2(ξ0) +∂χ 2(ξ0)

∂ξ i

δξ ii

∑ +12

∂ 2χ 2 (ξ0 )∂ξ i∂ξ j

δξ iδξ ji, j∑ + ⋅⋅⋅

In the 2 analysis, 1 standard error is

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

χ 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 Duv, Ddv

• Antiquark uncertainty is significantly reduced– g1

p 4Duv+ Ddv+12Dq

• Dg(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 = 1 GeV2

-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

xΔ (g x)

Q2 = 1 GeV2

xΔ (q x)

AAC03 uncertainties

AAC00 uncertainties

AAC00 AAC03 (with E155-p)

E155 data

(A1exp-A1

theo)/A1theo at the same Q2 point

Correlation between Dq(x) and Dg(x)

• analysis with Dg(x)=0 at Q2=1 GeV2

2/d.o.f. = 0.915

• Dqv(x) uncertainties are not affected

• antiquark uncertainties are reduced

Strong correlation with Dg(x) Note: correlation with Dg(x) is almost terminated in the Dg=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

xΔ (g x)

Q2 = 1 GeV2

xΔ (q 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 = 1 GeV2

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

g=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

α

Δ

Δ

g

g

g

gALL

2

0πLLA

)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 x

AAC, PRD74 (2006) 014015: Analysis without higher-twist effects

QHERMES2 ~ 1 GeV2 QCOMPASS

2 ~ 6 GeV2

NLOCG=0

g1(x,Q2 ) =12

eq2

q∑ dz

zx

1

∫ Δq(x / z,Q2 ) + Δq(x / z,Q2 )⎡⎣ ⎤⎦

× δ(1−z) +αs(Q

2 )2π

ΔCq(z) +⋅⋅⋅⎡

⎣⎢

⎤

⎦⎥

+12

eq2 dz

zx

1

∫ Δg(x / z,Q2 ) nf

αs(Q2 )

2πΔCg(z) +⋅⋅⋅

⎡

⎣⎢

⎤

⎦⎥

This term is terminated.

x=0.001 x=0.05 x=0.3

Positive contribution to A1 comesfrom ΔCG ⊗Δg at x~0.05.

Note: ΔCG ⊗Δg> 0 if Δg(0.05 / 0.2 =0.25) > 0Gluon polarization is positive at large x.

However, it may be a higher-twist effect.

LSS, PRD73 (2006) 034023.

LT fit

LT+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+G(large x)>0, or 100% G(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⎡⎣ ⎤⎦

dΔσdpT

= dηηmin

ηmax

∫ dxaxamin

1

∫a,b,c∑ Δfa(xa) dxbxb

min

1

∫ Δfb(xb)∂(t̂,zc)∂(pT ,η)

dΔσ̂dpT

Dcπ (zc)

xamin =

x11−x2

, xbmin =

xax2

xa −x1, zc =

x1xa

+x2

xb

, x1 =xT

2e+η , x2 =

xT

2e−η , xT =

2pT

s, η =−ln tan

θπ

2⎛⎝⎜

⎞⎠⎟

⎡

⎣⎢

⎤

⎦⎥

Δσ : Δfa(xa,Q

2 )⊗ Δfb(xb,Q2 )⊗ Δσ̂ (ab→ cX)⊗ Dc

π (z,Q2 )a,b,c∑

Parton distribution functions

Parton interactions Fragmentation functions

Description of 0 production

( ) , ( ) , ,

( , ) , ,

gg q g X qg q g X qq qX

qq q g q X qq qX qq qX

→ → →′ ′ ′→ → →

Subprocesses

g + g→ q(g) + X processes are dominant at small pT q+ g→ q(g) + X at large pT

The 0 production process is suitable for findingthe gluon polarization g.

(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 g < 0

0 production data do not distinguish between g < 0 and g > 0

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

Roles of RHIC-pion data Polarized PDFs, especially g, are better determined by the additional RHIC-0 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 )

g DS

AAC06 0.31 0.32 0.27 0.07

AAC03 0.50 1.27 0.21 0.14GRSV01 0.420 0.204LSS 0.680 0.210BB 1.026 0.138DNS 0.574 0.311

€

ΔΣ =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 Ɍ0 JLab E07-011

1 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 not sa

tisfied

at this s

tage!

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 ɢg 0.53 0.36 0.53

positive δ(É¢g) 0.72 0.26 0.38

positive δ(É¢g)/É¢g 1.36 0.72 0.72

node ɢg 0.87 0.4 0.87

node δ(É¢g) 0.89 0.31 0.47

node 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 not sa

tisfied.

Summary of the AAC global analysis for polarized PDFs – Duv(x), Ddv(x) are determined well– DS = 0.27 0.07 (Q2= 1 GeV2 )– Dg(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 g=0• Error correlation between g and q • Better g determination by RHIC-pion• g determination at large x by scaling violation (HERMES, COMPASS)• Possibility of a node-type g (g>0 at x>0.1, g<0 at x<0.1)

AAC03

AAC06

AAC08 ? in progress

The End

The End