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Constraint on g(x) from 0 production at RHIC

Masanori Hirai

Tokyo Tech

(Asymmetry Analysis collaboration)With S. Kumano and N. Saito

Phys.Rev.D74, 014015 (2006); hep-ph/0603217

2006, Oct 05 @Kyoto

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Introduction

• Origin of the nucleon spin 1/2– 1/2 = 1/2 + g + Lq,g

– Quark spin component from polarize DIS: 0.1-0.3

• Orbital angular momentum Lq,g ?– Jq= 1/2 +Lq: Generalized PDF from DVCS

• g is an important piece of the spin puzzle !– Undetermined g = 0.49 1.27 (AAC03)– Experimental data from RHIC-Spin

• Prompt photon, Jet, heavy quark production, et al.

– 0 production• PRL93, 202002 (2004)• RUN05

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Spin asymmetry of production

• Cross section

– gggg, ggqq,qgqg– qqqq, qq’qq’– qqqq, qqgg, ggqq

• Consistent with unpolarized data– Fragmentation functions (FFs)

• Determined by e+e- data• Ambiguity of Dg

(z)

,

ˆ( )

pp X abA Ba b

a bT T

d ddx dx f x f x D z

dp d dp d

1 2

1 1

1 2 1 2

PRL91, 241803 (2003)

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Initial distributions of polarized PDFs

• Initial distribution at Q02

– – Chose of the functional form is arbitrary– Source of model dependence

• Constraint condition:– Positivity condition:

• Constraint of f(x) at large x• Avoiding unphysical behavior for the asymmetry: A1(x) >1 (x1)

– Antiquark SUF(3) symmetry:• 0 asymmetry in low pT: gg process dominates

– Insensitive to flavor dependence of f(x)• Fixed 1st moments: uv=0.926, dv= 0.341• Fixed q=1.0: undetermined small-x behavior

( , ) ( , ), , , ,i i ii ii v vif x Q x x x f x Q i u d q g

2 20 0

( ) ( )f x f x

( ) ( ) ( ) ( ) ( )u x d x s x s x q x

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AAC global analysis

• Added new data – DIS A1[=g1*2x(1+R)/F2]

• COMPASS(d), HERMES(p,d), J-Lab(n) – 0 data

• PHENIX RUN05 preliminary data

• Total # of data: 421– DIS: 413 (Q2 1GeV2)– 0 : 8 (1<pT<10 GeV, R,F =pT)

• NLO analysis– DIS:MS scheme– 0: K factor:<Kpol>=1, <Kunopl>=1.6

• B. Jager, et al, PRD67,054005 (2000)

– 2(/d.o.f.) = 370.47 (0.90)• DIS:359.29• 0 : 11.18

• Q2 dependence of pol-PDFs– DGLAP eq– Q0

2 = 1GeV2

• Minimizing 2

– Scaling error of 0 data is not included

• Error estimation– Hessian method

– 2 (N=11) = 12.64

exp theo, ,

exp

exp

( , ) ( , ),

i i

i

sta sys

A x Q A x Q

22 2

1 122

2 2 2

2 2 1

,

( ) ( ) ( ) ij

i j i j

f x f xf x H

a a

2

0( , ) 0.683 (1 )K N s ds

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g from DIS and 0 data

• 1st moment g– 0.31 0.32 (DIS+0)– 0.47 1.08 (DIS only)

• Significant reduction of the g uncertainty

• Sign problem – gg process dominates

• g(x)

• Positive or negative g?• 2

0 : 11.18(g>0) vs. 11.05 (g<0)

(8 data points)

-0.6-0.4-0.2

00.20.40.60.8

11.21.4

0.001 0.01 0.1 1

x

xg(x)

DIS (DIS only)

DIS + 0 (g > 0)

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.08

0.1

1 2 3 4 5 6 7 8 9 10p

T (GeV)

RUN05

DIS + 0 (g < 0)

DIS + 0 (g > 0)

40% scaling error is not include

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Small-x behavior of g(x)

• 1st moment– g>0: 0.31 0.32 – g<0: 0.56 2.16

• Consistent results– 1st moment (0.1<xBj<1)

• g>0: 0.30 0.30 • g<0: 0.32 0.42

– DIS + 0 data covered

• Huge uncertainty of g<0– Ambiguity of small-x behavior – No constraint on the behavior

• g(x) is positive at large x

-1

-0.5

0

0.5

1

xg(x)

DIS + 0 (g < 0)

DIS + 0 (g > 0)

-1.2

-0.8

-0.4

0

0.4

0.8

1.2

0.001 0.01 0.1 1

x

HERMES

SMC

COMPASS Q2<1 GeV2

DIS + 0 (g < 0)

DIS + 0 (g > 0)

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Summary

• Gluon spin component g– DIS only: g = 0.47 1.08

• Small contribution through Q2 evolution (1<Q2<80 GeV2) and higher order correction

– DIS+0: g = 0.31 0.32• Direct contribution to 0 cross section in low and medium pT

• Significant reduction of its uncertainty• Assuming g > 0

• Sign problem:positive or negative g ?– g = 0.56 2.16 (DIS+0)

• gg process dominates in low pT: g(x)

• Ambiguity of the small-x behavior of g(x)• Need constraint on the behavior

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Large-x behavior of g(x)

• Positive g(x)/g(x) at large-x– HERMES A1

d

• 0.03 < xBj < 0.07, 1.2 < Q2 < 1.7 • COMPASS-d: 4.5 < Q2 < 8.6

– NLO gluon term• Positive contribution• Relative increasing for g1

D

– euv2 : 4/9(P) → 2.5/9(D)

• Positive g(x) at large-x

• Other DOF for HERMES-d ?– Higher Twist effects

• LSS: PRD73(2006)034023– Antiquark SUF(3) asymmetry

• D. de Florian, et al, PRD71(2005)094018

-0.02

0

0.02

0.04

0.06

0.08

0.1

0.01 0.1 0.3

x0.01 0.1 0.3

x

HERMES COMPASS

-0.2

-0.1

0

0.1

0.2

0.3

0.001 0.01 0.1 1

z

x=0.001

x=0.05

x=0.3

Cg(z)/10

x/zg(x/z) ( ) ( )

( )

g

gx

C x x g x

dz xC z x g

z z

1

.

( ) : . . ,

: . .

g

x

C z z

g x z x z

0 05

0 05 0 2

0 25 1

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Polarized PDFs by AAC analysis

• Improvements– dv(x): J-Lab(n) data– q(x): COMPASS and HERMES data– g(x): PHENIX RUN05 data

0.01

0.1

1

10

0.7 1 10 100 200

Q2[GeV2]

E130

E143

E155

EMC

SMC

HERMES

x = 0.75

x = 0.66

x = 0.55

x = 0.50

x = 0.35

x = 0.25

x = 0.175

x = 0.125

x = 0.08

x = 0.025

x = 0.007

x = 0.05

x = 0.45-0.1

0

0.1

0.2

0.3

0.001 0.01 0.1 1

AAC06

GRSV

BB

LSS

-0.2

0

0.2

0.4

0.6

0.8

0.001 0.01 0.1 1

x

xg(x)

Q2 = 1 GeV2

x(x)

0

0.1

0.2

0.3

0.4

0.5

0.001 0.01 0.1 1

-0.2

-0.1

0

0.001 0.01 0.1 1

x

xuv(x)

xdv(x)

Q2 = 1 GeV2

DIS (AAC03)

DIS + 0 (g > 0)

-1

0

1

2

0.001 0.01 0.1 1

xg(x)

DIS (AAC03)

DIS + 0 (g > 0)

-0.04

-0.03

-0.02

-0.01

0

0.01

0.001 0.01 0.1 1

x

xq(x)