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A Talk Given By

MENU04 Beijing, Aug 29 -Sep. 4, 2004 Polarized parton distributions of the nucleon in improved valon model. Ali Khorramian Institute for studies in theoretical Physics and Mathematics, (IPM) and Physics Department, Semnan University. A Talk Given By.

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A Talk Given By

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  1. MENU04Beijing, Aug 29 -Sep. 4, 2004 Polarized parton distributions of the nucleonin improved valon model Ali KhorramianInstitute for studies in theoretical Physics and Mathematics, (IPM) and Physics Department, Semnan University A Talk Given By MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  2. Polarized parton distributions of the nucleon in improved valon model Outline • Valon model in unpolarized case • Proton structure function • Convolution integral in polarized case • The improvement of polarized valons • NLO moments of PPDF’s and structure function • x-Space PPDF's and g1p(x,Q2) • Results and conclusion MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  3. UNPOLARIZED Parton Distributions and Structure Function MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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

  5. 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 yiare the momentum fractions of the U valons and D valon . The normalization factor gpis determined by this constrain where B(m,n) is the beta function. The single-valon distributions are MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  6. The unpolarized valon distributions as a function of y. R. C. Hwa and C. B. Yang, Phys. Rev. C 66 (2002) MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  7. Proton structure function • valon distributions in proton • quark distributions in a valon. In an unpolarized situation we may write: This picture suggests that the structure function of a hadron involves a convolution of two distributions: Proton structure function Structure function of a v valon. It depends on Q2 and the nature of the probe. Describes the valon distribution in a proton. It independs on Q2. Summation is over the three valons MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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  9. CTEQ4M MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  10. POLARIZED Parton Distributions and Structure Function MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  11. Why we use the valon model? Now valon model which is very helpful to obtain unpolarized parton distributions and hadron structure, can help us as well to get polarized parton distribution and polarized structure function. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  12. Convolution integral in polarized case To describe quark distribution q(x) in valon model, one can imagine q↑ or q↓ can be related to valon distribution G↑ and G↓. In 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 we assumed a linear combination of G↑ and G↓ to determine the unpolarized (G) and polarized (δG) valon distributions respectively. To indicate this reality that q↑ and q↓to be related to both G↑ and G↓we can consider linear combinations as follows here q ↑↑and q ↑↓denotes respectively the probability of finding q-up and q-down in G-up valon and etc. If we add and subtract above equations we can determine unpolarized and polarized quark distributions as following: MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  13. Since it is acceptable to assume that q ↑↑ = q ↓↓and q ↑↓ = q ↓↑then to reach to unpolarized and polarized quark distributions in proton we need to chose α = α = β = β = 1. Consequently we will have MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  14. Now by using definition of unpolarized and polarized valon distributions according to We have Unpolarized quark distribution in proton Unpolarized and polarized valon distribution Polarized quark distribution in proton Unpolarized and polarized quark distribution in a valon As we can see the polarized quark distribution can be related to polarized valon distribution in a similar way like the unpolarized one. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  15. we have the following sum rules In valon model framework we have • In the above equation 2 denotes to the existence of 2 U -valons in proton. As we can see • the first moment of the polarized valence quark distributions is equal to the first moment • of polarized valon distributions. Since the sea quark contribution arises from diference • between  and sum of uvand dv, we can see there is no contribution for the first • moments of sea quarks. • Considering the role of these quantities in the spin contribution of proton, we try to calculate • the polarized sea quarks distribution in frame work of improved valon model. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  16. The improvement of polarized valons 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 in following 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 the amounts of the parameter values which will be appeared in polarized valon distributions. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  17. Descriptions of W function Our motivation to predict this functional form is that Now, we proceed to reveal the actual y-dependence in W, functions. The chosen shape to parameterize the W in y-space is as follows The subscript j refers to U and D-valons This part adjusts valon distributionat large y values Polynomial factor accounts for the additional medium-y values This term can controls the low-y behavior valon distribution 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. For δW ’’ j(y) we choose the following form 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 which are defined by unpolarized valon distributions U and D. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  18. Analysis of Moments in NLO • Moments 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 MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  19. Moments of polarized parton distributions in valon 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 MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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  21. NLO moments of PPDF’s and structure function 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. So in moment space for g1n(Q2 ) we have some unknown parameters. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  22. Some experimental data for p, n, d E80, 130 (p) ; E142 (n) E143 (p, d) ; E154 (n) ; E155 (p, d) EMC, SMC (p, d) HERMES (p, d, n) MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  23. QCD fits to average of moments using Bernstein Polynomials 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 MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  24. 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 The 41 Bernstein averages gn,k (Q2) can be written in terms of odd and even moments 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). MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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  29. X-space PPDF's and polarized proton structure function 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 MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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  31. Fig.7 Polarized parton distributions at Q2 = 3GeV2 as a function ofx in NLOapproximation.The solid curve is our model and dashed, dashed dotand long dashedcurvesareAAC,BBandGRSV 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. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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  33. Fig.9 Polarized proton structure function xg1p as a function of x which is compared with the experimental data for different Q2 values. The solid line is our model in NLO and dashed line is LO approximation. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  34. Fig. 10 Polarized structure function for some values of Q2 as a function ofx in NLOapproximation.The solid curve is our model in NLO and dashed, dashed dotand long dashedcurvesareAAC , BBandGRSV model respectively. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  35. Fig. 10-continued Polarized structure function for some values of Q2 as a function ofx in NLOapproximation.The solid curve is our model in NLO and dashed, dashed dotand long dashedcurvesareAAC , BBandGRSV model respectively. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

  36. Conclusion • Here we extended the idea of the valon model to the polarized case to describe the spin dependence of hadron structure function. • In this work the polarized valon distribution is derived from the unpolarized valon distribution. In deriving polarized valon distribution some unknown parameters are introduced which should be determined by fitting to experimental data. • After calculating polarized valon distributions and all parton distributions in a valon, polarized parton density in a proton are calculable. The results are used to evaluate the spin components of the proton. • Our results for polarized structure functions are in good agreement with all available experimental data on g 1p. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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