a talk given by

11

MENU04MENU04

Beijing, Aug 29 -Sep. 4, 2004Beijing, Aug 29 -Sep. 4, 2004 Polarized parton distributions of the nucleon Polarized parton distributions of the nucleon

in improved valon modelin improved valon model

Ali KhorramianAli Khorramian

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

and and

Physics Department, Semnan University Physics Department, Semnan University

A Talk GivenA Talk GivenByBy

MENU 04 - MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. KhorramianBeijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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

Polarized parton distributions of the nucleon in improved valon modelPolarized parton distributions of the nucleon in improved valon model


33

UNPOLARIZEDUNPOLARIZED Parton Distributions and Structure Function Parton Distributions and Structure Function


Topics points in unpolarized case

• The model under discussion is the valon model.

• Valons play a role in scattering problems as the constituent quarks do in bound-state problems.

• In the model it is assumed that the valons stand at a level in between hadrons and partons and that the structure of a hadron in terms of the valons is independent of Q2.

• A nucleon has three valons that carry all the momentum of the nucleon does not change with Q2.

• Each valon may be viewed as a parton cluster associated with one and only one valence quark, so the flavor quantum numbers of a valon are those of a valence quark.

• At sufficiently low value of Q2 the internal structure of a valon cannot be resolved.


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). Let the exclusive valon distribution function be

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)


77

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 independs on 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


CTEQ4M


POLARIZEDPOLARIZED

Parton Distributions and Structure Function Parton Distributions and Structure Function


1212

Why we use the valon model? Why we use the valon model?

Now valon model which is very helpful toNow valon model which is very helpful toobtain unpolarized parton distributions andobtain unpolarized parton distributions andhadron structure, can help us as well to get hadron structure, can help us as well to get polarized parton distribution and polarizedpolarized parton distribution and polarizedstructure function.structure function.


Convolution integral in polarized caseConvolution integral in polarized case

To describe quark distribution To describe quark distribution qq((xx) ) in valon model, one can imagine in valon model, one can imagine q↑ q↑ or or q↓ q↓ can be related can be related to valon distribution to valon distribution G↑ G↑ and and G↓.G↓. In the case we have two quantities, unpolarized and polarized distribution, there is a choiceIn the case we have two quantities, unpolarized and polarized distribution, there is a choice of which linear combination exhibits more physical content. Therefore in our calculation weof which linear combination exhibits more physical content. Therefore in our calculation weassumed a linear combination of assumed a linear combination of G↑ G↑ and and G↓ G↓ to determine the unpolarized to determine the unpolarized ((GG) ) and polarizedand polarized((δGδG) valon distributions respectively.) valon distributions respectively.

To indicate this reality that To indicate this reality that q↑ q↑ and and q↓q↓ to be related to both to be related to both G↑ G↑ and and G↓G↓ we can consider linear we can consider linear combinations as followscombinations as follows

here here q q ↑↑↑↑ and and q q ↑↑↓↓ denotes respectively the probability of finding denotes respectively the probability of finding qq--upup and and qq--downdown in in GG--upup

valon and etc. valon and etc.

If we add and subtract above equations we can determine unpolarized and polarized quark If we add and subtract above equations we can determine unpolarized and polarized quark

distributions as following:distributions as following:


Since it is acceptable to assume that Since it is acceptable to assume that q ↑↑ q ↑↑ = = q ↓↓q ↓↓ and and q ↑↓ q ↑↓ = = q ↓↑q ↓↑ then to reach to unpolarized and then to reach to unpolarized and polarized quark distributions in proton we need to chose polarized quark distributions in proton we need to chose α α = = α α = = β β = = β β = 1.= 1.Consequently we will haveConsequently we will have


Now by using definition of unpolarized and polarized valon distributions according toNow by using 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


we have the following sum rules

In the above equation 2 denotes to the existence of 2 U -valons in proton. As we can seethe first moment of the polarized valence quark distributions is equal to the first momentof polarized valon distributions. Since the sea quark contribution arises from diference

between and sum of uv and dv, we can see there is no contribution for the firstmoments of sea quarks.

• Considering the role of these quantities in the spin contribution of proton, we try to calculatethe polarized sea quarks distribution in frame work of improved valon model.

In valon model framework we have


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 down the first moment of polarized u, d and distribution functions in the improved forms as follows:

These constrains have the same role as the unpolarized ones to control theamounts 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

Our motivation to predict this Our motivation to predict this functional form is thatfunctional form is that

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 are governed by the anomalous dimension

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


Because for a given value of Q2, only a limited number of experimental points, covering a partial range of x values are available, one can not use the moments directly. A method device to deal to this situation is that to take averages of structure functions with Bernstein polynomials.We define these polynomials as

Thus we can compare theoretical predictions with experimental results for the Bernstein averages, which are defined by

Using the binomial expansion, it follows that the averages of g1 with pn,k (x) as weight functions, can be obtained in terms of odd and even moments

where

QCD fits to average of moments using Bernstein PolynomialsQCD fits to average of moments using Bernstein Polynomials


Thus there are 16 parameters to be simultaneously fitted to the experimental gn,k (Q2) averages.

The best fit is indicated by some sample carves in Fig.(1).

The 41 Bernstein averages gn,k (Q2) can be written in terms of odd and even moments

To obtain these experimental averages from the exprimental data for xg1 ,we fit xg1(x,Q2) for each Q2 separately, to the convenient phenomenological expression


29 MENU 04 - MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. KhorramianBeijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

By using convolution integral as following

we can reach to the PPDF's in the proton in x-space.

To obtain the z-dependence of structure functions and parton distributions, usually required for practical purposes, from the above n-dependent exact analytical solutions in Mellin-moment space, one has to perform a numerical integral in order to invert the Mellin-transformation in according to

X-space PPDF's and polarized proton structure functionX-space PPDF's and polarized proton structure function


Fig.7 Polarized parton distributions at Q 2 = 3 GeV 2 as a function of x in NLO approximation. The solid curve is our model and dashed, dashed dot and long dashed curves are AAC,BB and GRSV model respectively.

Y. Goto`and et al., Phys. Rev. D 62 (2000) 34017. J. Blumlein, H. Bottcher, Nucl. Phys. B 636(2002) 225. M. Gl¨uck, E. Reya, M. Stratmann and W. Vogelsang, Phys. Rev. D 63(2001) 094005.


Fig.9 Polarized proton structure function xgFig.9 Polarized proton structure function xg11p p as a function of x which as a function of x which

is compared with the experimental data for different Qis compared with the experimental data for different Q22 values. The values. The solid line is our model in NLO and dashed line is LO approximation.solid line is our model in NLO and dashed line is LO approximation.


Fig. 10 Fig. 10 Polarized Polarized structure functionstructure function for some values of for some values of Q Q 22 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 NLO in NLO and dashed and dashed, , ddashed dot ashed dot and long dashedand long dashed ccurveurvess are are AAC , AAC , BBBB and GRSand GRSVV model model respectively.respectively.


Fig. 10-continued Fig. 10-continued Polarized Polarized structure functionstructure function for some values of for some values of Q Q 22 as a as a function of function of x x in NLO approximation. The solid curve is our model in NLO approximation. The solid curve is our model in NLO in NLO and and dasheddashed, d, dashed dot ashed dot and long dashedand long dashed ccurveurvess are are AAC , AAC , BBBB and GRSand GRSVV model model respectively.respectively.


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..


a talk given by

Documents

v valon

ali khorramian institute

khorramian menu

proton quark distributions

khorramian outlinevalon

proton structure functionmenu

nucleonin improved valon

improved valon modelmenu