Spin dependence of constituent quarks and structure function g1

1

Ali Khorramian

Institute for studies in theoretical Physics and Mathematics, (IPM)Tehran, IRAN

and

Physics Department, Semnan University

22

OutlineOutline

Valon model in unpolarized caseValon model in unpolarized case Proton structure functionProton structure function Convolution integral in polarized caseConvolution integral in polarized case The improvement of polarized valonsThe improvement of polarized valons NLO moments of PPDF’s and structure functionNLO moments of PPDF’s and structure function x-Space PPDF's and gx-Space PPDF's and g11

pp(x,Q(x,Q22))

Results and conclusionResults and conclusion

Spin dependence of constituent quarks and structure function gSpin dependence of constituent quarks and structure function g11

Spin04-Trieste-italia, October 10-16, 2004Spin04-Trieste-italia, October 10-16, 2004 Ali N. Khorramian Ali N. Khorramian

POLARIZEDPOLARIZED

Parton Distributions and Structure Function Parton Distributions and Structure Function


Unpolarized valon distributions in a proton Unpolarized valon distributions in a proton

In the valon model we assume that a proton consists of three valons (UUD) that separately

contain the three valence quarks (uud). The exclusive valon distribution function is

where yi are the momentum fractions of the U valons and D valon .The normalization factor gp is determined by this constrain

where B(m,n) is the beta function. The single-valon distributions are

05.1

76.1


The unpolarized valon distributions as a function of y.R. C. Hwa and C. B. Yang, Phys. Rev. C 66 (2002)


66

This picture suggests that the structure function of a hadronThis picture suggests that the structure function of a hadron

involves a convolution of two distributions:involves a convolution of two distributions:

Structure function of a v valon. It depends on Q2 and the nature of the probe.

Summation is over the three valons

Describes the valon distribution in a proton. It in dependent of Q2.

Proton structure function

valon distributions in proton valon distributions in proton quark distributions in a valon. quark distributions in a valon.

In an unpolarized situation we may write:In an unpolarized situation we may write:

Proton structure functionProton structure function


77

Using definition of unpolarized and polarized valon distributions according toUsing definition of unpolarized and polarized valon distributions according to

We haveWe have

As we can see the polarized quark distribution can be related to polarized valon distributionAs we can see the polarized quark distribution can be related to polarized valon distributionin a similar way like the unpolarized one.in a similar way like the unpolarized one.

Unpolarized quark distribution in proton

Polarized quark distribution in proton

Unpolarized and polarized valon distributionUnpolarized and polarized

quark distribution in a valon


Polarized valon distribution

Regarding to the existence of the difficulty we suggest the following solution.First we need to improve the definition of polarized valon distribution function as infollowing

The improvement of polarized valons

using the above ansatz we can write the first moment of polarized u, d and distribution functions in the improved forms as follows:

The above equations can help us to consider the constraint ofPolarized PDFs for the improved polarized valon model with an SU(3) Flavour symmetry assumption. These constrains have the same role as the unpolarized ones to control the amounts of the parameter values which will be appeared in polarized valon distributions.


Descriptions of Descriptions of W W functionfunctionNow, we proceed to reveal the actual Now, we proceed to reveal the actual yy-dependence in -dependence in W, W, functions. The chosen shapefunctions. The chosen shape

to parameterize the to parameterize the W W in in yy-space is as follows-space is as follows

For For δW δW ’’’’ j j ((yy) we choose the following form) we choose the following form

This term can controls the low-y behavior valon

distribution

The subscript The subscript j j refers to refers to U U

and and DD-valons-valons This part adjusts valon distribution at large y values

Polynomialfactor accounts

for the additional medium-y values

It can control the behavior of Singlet sector at very low-y values in such a way that we can extract the sea quarks contributions.

In these functions all of the parameters are unknown and we will get them from experimental data. In these functions all of the parameters are unknown and we will get them from experimental data. By using experimental data and using Bernstein polynomials we do a fitting, and can get the parameters By using experimental data and using Bernstein polynomials we do a fitting, and can get the parameters which are defined by unpolarized valon distributions which are defined by unpolarized valon distributions U U and and DD..


• Moments of polarized valon distributions in the protonMoments of polarized valon distributions in the proton

Let us define the Mellin moments of any valon distribution δGj/p(y) as follows:

Correspondingly in n-moment space we indicate the moments of polarized valon distributions

Analysis of Moments in NLOAnalysis of Moments in NLO


The Q2 evolutions of spliting function are given by their Mellin transformwhich admit an expansion as follows

The non-singlet (NS) part evolves according to

where

and the NLO running coupling is given by

The evolution in the flavor singlet and gluon sector are governed by 2x2 the anomalous dimension matrix with the explicit solution given by

• Moments of polarized parton distributions in valonMoments of polarized parton distributions in valon



So in moment space for g1n(Q2 ) we have some unknown parameters.

By having the moments of polarized valon distributions, the determination of the moments of parton distributions in a proton can be done strictly. The distributions that we shall calculate are δuv, δdv, δ. and δg.

• NLO moments of PPDF’s and structure function


E80, 130 (p) ; E142 (n)

E143 (p, d) ; E154 (n) ; E155 (p, d)

EMC, SMC (p, d)

HERMES (p, d, n)

Some experimental data for p, n, d


Fig. 10-continued Fig. 10-continued Polarized Polarized structure functionstructure function for some values of for some values of QQ22 as a function of as a function of x x in NLO approximation. The solid curve is our model in NLO approximation. The solid curve is our model in NLOin NLO and dashed and dashed, d, dashed dot ashed dot and long dashedand long dashed ccurveurvess are are AAC , AAC , BBBB and GRSand GRSVV model model respectively.respectively.


AA

The Q2 dependence of quark and gluon helicity and their orbital angular momentum.

ConclusionConclusion

Here we extended the idea of the valon model to the polarized case to Here we extended the idea of the valon model to the polarized case to describe the spin dependence of hadron structure function. describe the spin dependence of hadron structure function.

In this work the polarized valon distribution is derived from the In this work the polarized valon distribution is derived from the unpolarized valon distribution. In deriving polarized valon distribution unpolarized valon distribution. In deriving polarized valon distribution some unknown parameters are introduced which should be some unknown parameters are introduced which should be determined by fitting to experimental data. determined by fitting to experimental data.

After calculating polarized valon distributions and all parton After calculating polarized valon distributions and all parton distributions in a valon, polarized parton density in a proton are distributions in a valon, polarized parton density in a proton are calculable. The results are used to evaluate the spin components of calculable. The results are used to evaluate the spin components of the proton.the proton.

Our results for polarized structure functions are in good agreement Our results for polarized structure functions are in good agreement with all available experimental data on g with all available experimental data on g 11

pp..


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Spin dependence of constituent quarks and structure function g1