# Presentation: 688499

## QCD analysis of the nucleon spin structure function data in next to leading order in αSThe polarized gluon distribution in the nucleon

- Jechiel Lichtenstadt
- Tel Aviv University

## Outline

- World data of nucleon spin structure functions
- Q2 evolution with DGLAP equations
- Results
- Contribution of the polarized gluon distribution
- Direct measurements of the polarized gluon
-    distribution
- Prospective future measurements of the spin
-    structure functions
- Conclusions** **

**Outline**

## History

-   SLAC-YALE E80/E130
-   G. Baum et al. 1983
-         ***0.2****** < x < ******0.7***

## World data of theSpin Structure Functions

- Proton:EMC, SMC, SLAC-E143, E154,
-            HERMES, JLAB-CLAS
- Deuteron: SMC, SLAC-E143, E155, HERMES,
-                  JLAB-CLAS
- Neutron: SLAC-E142, HERMES,JLAB-MIT

## Spin structure function data

## QCD Evolution - DGLAP equations

- In the parton model, the spin structure function *g**1* is related to the polarized quark and gluon distributions in the nucleon by an integral equation involving quark and gluon coefficient functions Ciq and Cg
-    (Dokshitzer, Gribov, Lipatov, Altarelli  and Parisi)

## QCD Evolution - DGLAP equations (Cont.)

- Where:  *t *= ln(Q2/Λ2), and Λ is a scale parameter
-            Δ*G(x)* - the polarized gluon distribution
-                                                                     (singlet)
-                                                                                (non-singlet)

## QCD analysis- DGLAP equations (Cont.)

- Parameterization at initial scale (1 (GeV/c)2 )
-              ∆*f(x)= **η** N(**α**f**,**β**f**,a**f**,b**f**)** **x**α** **(1-x)**β**(1+ax+bx**1/2**)*
-       for ΔΣ(x), ΔG(x), ∆qpNS(x), ∆qnNS(x)
- Normalization
*   ∫**Nx**α**(1-x)**β**(1+ax+bx**1/2**)*d*x =1*
*   **η** *the first moment of the parton distribution
- Fit procedure:
-    Set parameters at initial scale
-    *Evolve parton distributions –using the  DGLAP equations *
*    **and calculate g**1**(x,Q**2**)  for each data*
-    Modify parameters and obtain best fit

## QCD analysis- DGLAP equations (Cont.)

- QCD analysis in the Adler-Bardeen scheme:
- First moment of *C**g **-*
- (G. Altarelli, R. Ball, DS. Forte and G. Ridolfi Nucl. Phys.  B 496 (1997) 337.)
- Compare also to the         scheme – *C**1**g **= 0.*
- The first moments of the singlet distributions differ by:
- (*a**0 *– the singlet axial current matrix element)

## QCD analysis in NLO- results

- Fit to data (shown at measured *Q**2**)*

##     QCD analysis in NLO - results

- 12 parameters
- 286 data
- 2=251 (C.L.=84%)
- at *Q**2**=1 (GeV/c)**2*

## ΔG/G from photon gluon fusion

- 1) _Open charm production_:
- 		**q = c**
**    **charm fragmentation:
- 	- D0, D* (60%)
- 	- D+ (20%)
- 	- DS+, ΛC+ (10% each)
- 2) _High-P__t__ hadron pair _    _production_:	**q=u,d,s**

*Δ**G/G* from photon gluon fusion

## Measurements of ΔG/G 

- Determination of Δ*G*/*G* from high PT hadron pairs

- -

**PGF**

- QCD Compton

- Leading order DIS

## Measurements of the gluon       polarization ∆G/G(x)

- Photoproduction of High PT hadron pairs
- SMC:  Phys. Rev C **D70** (2004) 012002
-    ∆G/G(xg=0.07; μ2=3 GeV2) =-0.20 ± 0.28 ± 0.10
- HERMES: Phys. Rev. Lett. **84** (2000) 2584
- ∆G/G(xg=0.17; μ2=2 GeV2) =0.41 ± 0.18 ± 0.03

## Determination of ΔG/G from open charm production

- D0            K-**+**BR = 4%

- D*+            Ds**+ ****~ **30%(*)
-                  K-**+**s**+**

**c**

**c**

- COMPASS at CERN

## Measurements of the gluon polarization ∆G/G(x) (Cont.)

- COMPASS:
-     High PT (low Q2)
-     ∆G/G(xg=0.085; μ2=3 GeV2) =0.016 ± 0.058 ± 0.055
-     High PT (high Q2)
-     ∆G/G(xg=0.13; μ2=3 GeV2) =0.06 ± 0.31 ± 0.06
-       Open Charm
-     ∆G/G(xg=0.15; μ2=13 GeV2) =-0.57 ± 0.41 ± 0.17
-     Use CTEQ parameterization of G(x) to obtain ∆G(x)

## Comparison of best-fit polarized gluon distribution with G measurements

- Best fit compatible with presently measured  *G*  values
- Including *G *data in fit has a very small effect.
- *G = 0.57 **** 0.11 (*stat.)

## x dependence of the polarized gluon distribution

- Fit data with 11 parameters:
-    no mid-range node
- 286 data
***2**=261*
*   **(C.L.=72%)*
***G = 0.48****0.14*
- Not quite compatible with measured *G*
*(*C.L. -> 67%)
- Additional measurements of *G* are essential

## PHENIX at RHICAsymmetry in 0 production from p+p collisions 

- Calculations: NLO with pQCD
- Glück, Reya, Stratmann and Vogelsang
-   0.02 **< **Xg**<** 0.3
-    for each PT bin.

## Sensitivity to G x=(0.02-0.3)

- Compare data with
-   LL(*p**T**) *calculated with
-   different *G(x) *and

## ALL( 0) with G from best-fit to world g1 data

- Caculation by W. Vogelsang

## Systematic errors

-  **Preliminary** **(rough) estimates**
- Systematic errors in *g**1** ** *data ~ 0.10
- Factorization and renormalization
-    scales                                   ~  0.05
- Functional form                    ~  0.15
- Total estimate                       ~ 0.20

## Measurements with a future Electron Ion Collider

- Extend kinematic range:
- High luminosity
*E**e **= 10 GeV*
*E**p**= 250 GeV*
*∫**L =  *2fb-1

## Projected EIC data in QCD analysis

-  • Use best-fit distributions
-   to calculate asymmetries over EIC kinematic region
- Randomize data
- Repeat QCD analysis
-    with projected data

- *G = 0.64 **** 0.05 (*stat.)

## CONCLUSIONS (I)

- New measurements of the polarized gluon distribution are compatible with *g*1 data.
- Need more *g*1 data at low *x *(proton, deuteron).
- Need more precise *G* data to determine the *x* dependence of the gluon distribution
-    (Expected in near future)

## CONCLUSIONS (II)

- Determine the axial vector coupling constant – The Bjorken sum-rule.
- with
- Note: The non-singlet distribution evolves independently! (Measurements at same (*x,Q**2*)

## CONCLUSIONS (III) 

-     Remaining tasks:
- •   Evaluate systematic error.
-     Future tasks
- •  Introduce model-independent SF’s (Fourier-Bessel, Sum of Gaussians)  to get a realistic estimate on the uncertainty (***G *is not determined at low *x*)
-  • Incorporate RHIC measurement (***G**(x:0.02-0.3)*) in the analysis (iterative procedure)
- Fit strong coupling constant s

## The last slide

- Thanks to many people who were involved and contributed to this work:
- A. Deshpande
- W. Vogelsang
- R. Windmolders

## History – EMC 1988

- Measurement of the proton spin structure function and determination of its first moment: *Γ**= ∫*g1P(x)dx (Ellis and Jaffe Sum rule)

**0.01 ≤  *****X *****≤ 0.7**

## The t dependence of the distributions follows the DGLAP equations:
Gluon and singlet:

Note: The non-singlet evolves independently of the singlet and the gluon distributions:

The coefficient functions Ciq, Cg, and splitting functions Pi,j can be expanded in powers of αs. 

Use next to leading order (NLO) coefficient functions.

- Gluon and singlet:
- Note: The non-singlet evolves independently of the singlet and the gluon distributions:
- The coefficient functions Ciq, Cg, and splitting functions Pi,j can be expanded in powers of αs.
- Use next to leading order (NLO) coefficient functions.

- QCD Evolution - DGLAP equations (Cont.)

## QCD Evolution - DGLAP equations (Cont.)

- The t dependence of the distributions follows the DGLAP equations:
- Singlet and gluon distributions:

- The coefficient functions Ciq, Cg, and splitting functions Pi,j can be expanded in powers of αs.
- Use next to leading order (NLO) coefficient functions.

## Spin structure function data- deuteron

## Spin structure function data - proton

## Spin structure function data- deuteron (301)

## Spin structure function data - neutron

## ΔG/G from photon gluon fusion

- 1) _Open charm production_:
- 		**q = c**
**    **charm fragmentation:
- 	- D0, D* (60%)
- 	- D+ (20%)
- 	- DS+, ΛC+ (10% each)
- 2) _High-P__t__ hadron pair _    _production_:	**q=u,d,s**

*Δ**G/G* from photon gluon fusion