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Ali N. Khorramian (a,b),

A. Mirjalili (a,c) and S. Atashbar Tehrani(d)

(a) (IPM) Institute for studies in theoretical Physics and Mathematics

(b) Physics Department, Semnan University, Semnan, Iran

(c) Physics Department, Yazd University, Yazd, Iran

(d) Physics Department, Persian Gulf University, Iran

khorramiana@theory.ipm.ac.ir

Polarized structure function in the valon model,

using QCD fits to Bernstein averages

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What is the valon model?What is the valon model?

The valon model describes the hadron structure relevant to multiparticle production.The valon is defined to be a dressed valence quark in QCD with the cloud of gluons and sea quarks which can be resolved by high Q^2 probes. At sufficiently low value of Q^2 the internal structure of a valon cannot be resolved and hence it behaves as a valence quark. For example in the proton we have 2 U-valons and one D valon.

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To facilitate the phenomenological analysis Hwa assumed a simple form for the exclusive valon distribution in unpolarized proton which is

Where yi is the momentum fraction of the i-th valon.The single valon distributions are obtained by integration

Unpolarized valon distribution in a proton

Valon model

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Unpolarized

U-valon distribution

PolarizedU-valon

distribution

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Polarized valon distribution

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Parton distribution functions in moment space

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In phenomenological investigations of structure functions, for a given value of Q2, only a limited number of experimental points, covering a partial range of values of x, are available. Therefore, one can not directly determine the moments.A method devised to deal with this situation is to take averages of the structure function weighted by suitable polynomials. We can compare theoretical predictions with experimental results for the Bernstein averages, which are defined by

Where are Bernstein polynomials,

which are normalized to unity.

)x(p k,n

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QCD fits to Bernstein averages

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Other restriction which we assume here, is that to ignore the effects of moments with high order n which are not very effective. To obtain these experimental averages from the E143 and SMC data for xg1, we fit polarized structure function for each bin in Q2 separately, to the convenient phenomenological expression

Some samples experimental Bernstein averages are plotted in Figure 1.

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2 4 6 8 10 12 14 16

Q2 (GeV2)

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

Fn,

k

F13, 7

F9,3 x 1.4

F7,3 x 1.2

2 4 6 8 10 12 14 16

Q2 (GeV2)

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

Fn,

k

F2,1

F6,2 x 1.4

F5,2 x 1.2

Table 1:Numerical values of fitting parameters for the best fit of Fig.1.

Fig. 2: Unpolarized and polarized valon distribution in a proton

Fig. 1: Fit to xg1 using Bernstein averages

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Polarized parton distributions and structure function

Now we want to compute the polarized structure function in a valon. Since we calculated before polarized valon distribution in a proton, by having the polarized structure function in a valon,it is possible to extract polarized parton structure in a proton. To obtain the z dependence of parton distributions in practical purposes from the n dependent exact analytical solutions in Mellin-moment space, one has to perform a numerical integral in order to invert the Mellin-transformation. The relationship between polarized quarks of proton and the polarized quarks in a valon is given by convolution:

and now we can get the following expressions for polarized parton distributions in a proton

By havinh the polarized parton distribution, we are going to use them to extract polarized structure function. To leading order (LO) in QCD, according to the quark model, polarized structure function can be written as a linear combination of

where eq are the electric charges of the (light) quark-flavors q=u,d,s. Furthermore, this equation can be decomposed into a flavor nonsinglet (NS) and singlet (S) component. In Fig 3, 4 and 5 we presented polarized parton distributions; xg 1,NS , xg1,S and xg1

p ; and polarized proton structure function.

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Fig. 3: Polarized parton distributions in proton at QFig. 3: Polarized parton distributions in proton at Q22=3 GeV=3 GeV22 as a function of x. The solid line is our model, dashed lineas a function of x. The solid line is our model, dashed line is AAC model (ISET=1), dashed-dotted line is BB modelis AAC model (ISET=1), dashed-dotted line is BB model (ISET=1) and long-dasheded line is GRSV model (ISET=3).(ISET=1) and long-dasheded line is GRSV model (ISET=3).

Fig. 4: The contribution of gFig. 4: The contribution of g1,S1,S and g and g1,NS 1,NS and and

combination of them as function of x and forcombination of them as function of x and for QQ22=5,50 GeV=5,50 GeV22..

Fig. 5: Polarized proton structure function as a function of Fig. 5: Polarized proton structure function as a function of x which is compared with the experimental data and for x which is compared with the experimental data and for different Qdifferent Q2 2 values.values.

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