AAC analysis of polarized parton distributions with uncertainties

Shunzo Kumano, Saga University

kumanos@cc.saga-u.ac.jp, http://hs.phys.saga-u.ac.jp

12th International Workshops on Deep Inelastic Scattering (DIS04)

Strbske Pleso, Slovakia, April 14-18, 2004

Refs. (1) Y. Goto et. al., Phys. Rev. D62 (2000) 034017.

(2) M. Hirai, SK, N. Saito, Phys. Rev. D69 (2004) 054021.

April 15, 2004

Contents

Introduction

Polarized PDFs of AAC03 analysis

- global analysis

- uncertainty estimation

- analysis with g=0

Summary

• proton-spin issue

polarized e/-proton scattering measurement of g1 g

1

LO 1

2e i

2 (q ii q i)

proton, deuteron, 3He g1 data

with isospin symmetry valence and sea polarization uv ,dv ,q

quark spin content uv dv 6q

experimentally dx0

1

(x)0.1 0.3

rest of the spin ???

Papers on the polarized PDFs

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

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

T. Gehrmann and W. J. Stirling, Phys. Rev. D53 (1996) 6100.

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.

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 (2001) 479.

AAC (Y.Goto et al., M. Hirai et al.), Phys. Rev. D62 (2000) 034017; D69 (2004) 054021.

J. Bartelski and S. Tatur, Phys. Rev. D65 (2002) 034002.

J. Blümlein and H. Böttcher, Nucl. Phys. B636 (2002) 225.

Initial distributions fi(x,Q0

2) = A i x i (1 + i x i) fi(x,Q02)

i = uv, dv, q, g A i , i , i , i : parameters

χ2 fit to the data [p, n (3He), d]

A 1

g1

F1= g1

2 x (1 + R)F2

R =

FL

2 x F1=

F2 – 2 x F1

2 x F1

2 =

i

(A 1 idata – A 1 i

calc)2

(A 1 i

data)2

We analyzed with the following conditions.

• unpolarized PDF GRV98• initial Q2 Q0

2 = 1 GeV 2

• number of flavor Nf = 3

positivity f(x) f(x) (to be precise, )

antiquarkflavor: u = d = s

Experimental data on spin asymmetry A1(x,Q2)

• A1 data– Proton : E130, E143, EMC, SMC, HERMES, E155

– Deuteron: E143, E155, SMC

– Neutron : E142, E154, HERMES, JLab

Total data 399 points (Q2>1GeV2)

1

10

100

0.001 0.01 0.1 1

x

E130(p)

E143(p)

E155(p)

EMC(p)

SMC(p)

HERMES(p)

E143(d)

E155(d)

SMC(d)

E142(n)

E154

HERMES(n)

(GeV

2 )

Error estimation Hessian method

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

where the Hessian matrix is defined by

2(0 ) 2(0) 2(0) i

ii

12

2 2 (0 ) i j

i ji, j

In the 2 analysis, 1 standard error is

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

H ij 1

22 2(a0) i j

2 = 2(0 ) 2(0) i Hij ji, j

F( x)

2 2 F( x) i

ij– 1 F( x)

ji, j

P(s) N: 2(= s) distribution with N degrees of freedom

ds0

2

P(s) N = 11 = 0.6826 2=12.64 (N = 1 case, 2= 1)

Results • Total 2

2/d.o.f. = 0.893

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 1

x

AAC00

New

dg=0

Q2=1 GeV2

xuv(x)

xdv(x)

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.001 0.01 0.1 1

x

AAC00

New

dg=0 xg(x)

x6q(x)

Comparison of AAC03 with AAC00AAC00: Y.Goto et al., Phys. Rev. D62 (2000) 034017

AAC03: M.Hirai et al., Phys. Rev. D69 (2004) 054021

Spin asymmetry A1p

• Precise data (E155-proton) reduce asymmetry uncertainties• Agreement with polarized DIS data

– Quark and antiquark distributions are constrained– Gluon distribution need other constraint

-0.5

0

0.5

0.01 0.1 1

-0.5

0

0.5

0.001 0.01 0.1 1

-0.5

0

0.5

0.01 0.1 1

x

EMC (p)

SMC(p)

HERMES(p)

-0.5

0

0.5

0.01 0.1 1

-0.5

0

0.5

0.01 0.1 1

-0.5

0

0.5

0.01 0.1 1

x

E130 (p)

E143 (p)

E155 (p)

-0.2

0.2

0.6

1

1.4

1.8

0.001 0.01 0.1 1

x

AAC03

AAC00

Proton

Q2 = 5 GeV2

(Dat

a-th

eory

)

AAC03 uncertainties

AAC00 uncertainties

AAC00 AAC03 (with E155-p)

Polarized PDFs (AAC03)• PDF uncertainties reduced by

including precise (E155-p) data

• Valence-quark distributions are well determined

– Small uncertainty

of uv, dv

• Antiquark uncertainty is significantly reduced– g1

p 4uv+ dv+12q

• g(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

xuv(x)

xdv(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

xg(x)

Q2 = 1 GeV2

xq(x)

AAC03 uncertainties

AAC00 uncertainties

AAC00 AAC03 (with E155-p)

E155 data

Correlation between q(x) and g(x)

• analysis with g(x)=0 at Q2=1 GeV2

2/d.o.f. = 0.915

• qv(x) uncertainties are

not affected

• antiquark uncertainties

are reduced – strong correlation with g(x)– correlation with g(x) is almost terminated in the g=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

xg(x)

Q2 = 1 GeV2

xq(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

xuv(x)

xdv(x)

Q2 = 1 GeV2

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

g=0 uncertainties

AAC03 uncertainties

Comparison with other parameterizations

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

AAC03

GRSV

BB

LSS

xuv(x)

xdv(x)

Q2 = 1 GeV2

-1

0

1

2

0.001 0.01 0.1 1

AAC03

GRSV

BB

LSS

-0.04

-0.03

-0.02

-0.01

0

0.01

0.001 0.01 0.1 1

x

xg(x)

Q2 = 1 GeV2xq(x)

1st moments

– GRSV01 [ Phys. Rev. D63 (2001) 094005 ]– LSS01 [ Eur.Phys.J. C23 (2002) 479 ]– BB02 [ Nucl. Phys. B636 (2002) 225 ]

g

AAC03 0.499 1.268 0.213 0.138

GRSV01 0.420 0.204

LSS 0.680 0.210

BB 1.026 0.138

uv dv 6q

Q2 1 GeV2

W ( g qqq2 ) F1

ˆ p ˆ p pq F2 i

q p2pq F3 where ˆ p p

pqq2

q

i q spq g1i

q(pq s sq p )(pq)2 g2

ˆ p ˆ s ˆ s ˆ p

2pq sq ˆ p ˆ p

(pq)2

g3

sq ˆ p ˆ p (pq)2

g4 ( g qq

q2 ) sq pq g5

new structure functions g3, g4, g5

be careful about “various” definitions of g3, g4, g5 !

d( p 1CC p 1

CC )dx dy GF

2 Q2

(1Q2 / MW2 )2xy

{[ y(2 y)xg1CC (1 y)g4

CC y2xg5CC

]

2 x y M 2

Q2 [ x2y2 g1

CC 2 x2y g2CC 1 y x2y2 M 2

Q2

x g3

CC

x 1 3

2y x2y2 M2

Q2

g4

CC x2y2g5CC] }

0 at Q2>>M2

Polarized neutrino-proton scattering (CC)

Quark spin content

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-98

LSS-99

AAC-00-1

factory

1

minmin

)()(x

dxxx

e / scattering = 0 ~30%

It is not uniquely determined.

g1p + g1

p = (u + u) + (d + d)

+ (s + s) + (c + c)

in LO dx (g1p + g1

p) =

independent determination ofquark spin content !

scattering

on factorysee e.g. hep-ph/0310166

Summary • Global analysis for polarized PDFs

– uv(x), dv(x) are determined well– = 0.213 0.138 (Q2= 1

GeV2 )– g(x) could not be constrained

AAC03-polarized-pdfs code could be obtained fromhttp://spin.riken.bnl.gov/aac/

http://hs.phys.saga-u.ac.jp/aac.html (mirror site)

• Uncertainties of polarized

• Effects of E155-proton data

• Global analysis also with g=0

• Error correlation between g and q