Recent results of the AAC analysis

Masanori HiraiTUS, Noda

(Asymmetry Analysis collaboration)Collaborators: S. Kumano and N. Saito

arXiv:0808.0413 [hep-ph]

2009 1 10,KEK

1/15

Contents

• Introduction

• AAC08 analysis– Global analysis of the polarized PDFs– Data sets

– Assumptions: positivity condition, antiquark SU(3)f symmetry

• Impact of RHIC 0 data – Discussion about a functional form of g(x)

• Impact of E07-011 data– Relative errors G/G ?

• Summary

2/15

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

– Unknown orbital angular moments Lq,g

• Quantitative value from sum rule of GPDFs• Qualitative understanding ?

– G is an important piece of the spin puzzle !• Undetermined G = 0.49 1.27 (AAC03)

– Experimental data from RHIC-Spin• 0 production [RUN05, PRD76, 051106 (2007)]

• AAC analysis– AAC00 [PRD62, 0304017 (2000)]: DIS– AAC04 [PRD69, 054021 (2004)]: DIS, uncertainty estimation– AAC06 [PRD74, 014015 (2006)]:DIS+0, uncertainty, G– AAC08 [arXiv:0808.0413 [hep-ph]]: DIS+0, uncertainty, E07-011(J-Lab)

3/15

4/15

• Cross section

• Hadron tensor (asymmetric part)

• Structure function g1

Polarized deeply inelastic scattering

2 2

2

,

, 0 12

P qQ q

M

Qx x

M

2 2

4' 2

d EL

d dE MW

Q E

1 2

( )A

q s q PW s x s

Mg g x

21

1

1

2( ) ( )

f

i i

n

i q gi

C x C xg x q x q x g xe

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

, coefficient func: t on

)

iq g

q x q x q x q x q x

C C

q x

1

x

dy x dyf g f g y

y y y

5/15

Spin asymmetry A1(x,Q2)

• Polarized DIS experiments– Proton : E130, E143, EMC, SMC, HERMES, E155,CLAS– Deuteron: E143, E155, SMC, HERMES, COMPASS, CLAS– Neutron : E142, E154, HERMES, J-Lab (Hall-A)

• Q2 dependence is obtained by the DGLAP equation

12

1

1

( ) ( )

( ) ( )

2 (1 )

d dA

d d

g

F

xg

R

F

2, T

L

R x Q

• L. W. Whitlow et al., Phys. Lett. B 250

(1990) 193

Total data 441 points

(Q2>1GeV2)

6/15

Spin asymmetry of production

• Cross section

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

• Consistent with unpolarized data– Fragmentation functions (FFs)

• Determined by e+e- data

• Ambiguity of Dg (z)

• Sing problem– gg process dominates at low pT

• ALL0 [g(x)]2, negative or positive ?

,

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

PRD76, 051106 (2007)

7/15

AAC analysis and uncertainty estimation

• Initial distribution of polarized PDF

• Constraint condition– Positivity condition:

• Imposing on positive g(x), but not impose on node type

– Antiquark SU(3)f symmetry:

• Fixed 1st moments: uv=0.926, dv= 0.341

• Fixed q=1.0: undetermined small-x behavior

• PDF uncertainty by Hessian method 2 2 1

,

( ) ( ) ( ) ij

i j i j

f x f xf x H

a a

( , ) ( , ),

, , ,

i i iii i

v v

f x Q x x x f x Q

i u d q g

2 20 0

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

( ) ( )f x f x

• Unpol PDF:GRV98

• Q02 = 1 GeV2

Value of 2

• Hessian method

–

• [K(N,s): 2 distribution ]

• 2 ~ N (N: number of parameters)– 1 error of normal distribution in

multi-parameter space– Not 1 error for PDF uncertainties– Correspondence between

uncertainties and variations of experimental data

• 2=1 : statistically correct– 1 error for 1 degree of freedom

• Asymmetry A1

• X-sections8/15

2

0( , ) 0.683K N s ds

2 2 1

,

( ) ( ) ( ) ij

i j i j

f x f xf x H

a a

Data-theory

AAC08 analysis

• Added new data– DIS: CLAS(p,d), COMPASS(d, renewal data)– 0production: RHIC Run 5 final data (10 points)

• Data sets– Set-A : DIS data only– Set-B : DIS + 0 data– Set-C : DIS + fake data of the E07-011 experiment

• Impact on the determination of g(x) – Set-A vs. Set-B: effect on the functional form, node or positive– Set-B vs. Set-C: an impact on determination of the gluon spin contribution G

9/15

Fake data of the E07-011 experiment

• Update experiment of the J-Lab

• Expected errors for g1(x)– Converted to asymmetry errors

• Expected asymmetry error

10/15

212 2

1 1207 0111

07 011

,, * ,

,

d

d d

dE Set A

E

g x QA x Q A x Q

g x Q

Results

• Minimal 2

• 1st moments:

11/15

Set-A (436 point)Set-B (446

point)Set-C (441 point)

Positive Node Positive Node Positive Node

DIS 381.66 375.87 384.65 381.27 381.66 375.87

0 12.43 11.32

Total 2 381.66 375.87 397.08 392.60 381.66 375.87

2/d.o.f 0.90 0.88 0.91 0.90 0.89 0.87

Set-A Set-B Set-C

Positive Node Positive Node Positive Node

0.24 0.07

0.22 0.08

0.26 0.06

0.25 0.07

0.24 0.05

0.22 0.05

G 0.63 0.81

0.94 1.66

0.40 0.28

-0.12 1.78

0.63 0.45

0.94 1.09

1 2 2

0( ) , 1F f x dx Q GeV

Impact of RHIC 0 data

12/15

• Insensitive to uv(x) and dv (x)– Same center values and

uncertainties between Set-A and –B– gg process dominates

• g(x) becomes small• Gluon uncertainty is reduced by

RHIC data– RHIC data is sensitive to g(x)

• Node type (DIS+0)– Less 2 : 12.43(G>0) , 11.32 (G<0)– Changing sign at x=0.1– Large uncertainty in small-x region

• No data to constraint the behavior

Impact of E07-011 data

13/15

• Converged determination of the valence quark distributions– No changed these uncertainties

– Assuming SUf(3) symmetry

• Fixed 1st moments for uv and dv

• Significant reducing the uncertainty of g(x) – Error correlation with antiquark

distribution via Q2 evolution– Constraint via NLO gluon term

• Low Q2 data• Higher twist term ?

Impact of RHIC 0 and E07-011 data

• Positive type– DIS: large distribution– 0: small distribution

• Node type– DIS: small negative below x=0.02– 0 : large negative at small-x– Large uncertainty in small-x– Positive distribution at high-x

• Relative error G/G (x>0.1)

• Same impact as RHIC Run5 data14/15

Positive

Node

Set-B (DIS+0) 0.71 0.77

Set-C (E07-011)

0.73 0.54

Summary

• Polarized PDF from polarized DIS data– Well determined valence quark distributions– Undetermined anti-quark and gluon distributions

• Rather large uncertainties of these distributions• Reduction gluon uncertainty due to error correlation with antiquark

distribution• NLO gluon term < statistical errors

– Positive distribution of g(x) at medium and large-x region – E07-011 experiment (precise measurement )

• Constraint on g(x) via NLO gluon term• NLO gluon term > statistical errors• Same impact as RHIC run5 0 data

• g(x) from 0 production data– Node type, changing sign around x=0.1

• RHIC data: covering narrow x-region

– Undetermined small-x behavior• No data in the small-x region

• AAC08 numerical library: http://spin.riken.bnl.gov/aac/15/15

16/14

1st moments of g(x)

• Full range x=0~1

• 0.1< X <1

– CLAS data (x=0.175 ~0.55)

– 0 data (s=200 GeV, pT= 1.29 ~ 7.79 GeV)

17/14

Set-A Set-B Set-C

Positive Node Positive Node Positive Node

G 0.63 0.81

0.94 1.66

0.40 0.28

-0.12 1.78

0.63 0.45

0.94 1.09

Set-A Set-B Set-C

Positive Node Positive Node Positive Node

G 0.53 0.72

0.87 0.89

0.36 0.26

0.40 0.31

0.53 0.38

0.87 0.47

G/G 1.36 1.02 0.71 0.77 0.73 0.54

18/14

Comparison of FF for 0 production

• Model dependence of the FF global analysis ?– Kretzer’s FFs within LO uncertainty

– KKP’s FFs without the uncertainty at large-pT

• Cut of z for 2nd moments of FFs (z>0.05)• Heavy quark mass threshold

• Uncertainty from NLO FFs becomes 1/3

19/14

Constraint on large-x behavior of g(x)• Positive g(x)/g(x) at large-x

– HERMES A1d

• 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 g1D

– 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) aymmetry• 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

( ) ( )

( )gx

C x x g x

dz xC z x g

z z

1

-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)