1. 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 Ali Khorramian Institute for studies in theoretical Physics and Mathematics, (IPM) and Physics Department, Semnan University A Talk Given A Talk GivenBy MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

2. 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 g 1 p (x,Q 2 )  Results and conclusion Polarized parton distributions of the nucleon in improved valon model MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

3. UNPOLARIZED Parton Distributions and Structure Function 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 Q 2. A nucleon has three valons that carry all the momentum of the nucleon does not change with Q 2. 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 Q 2 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 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 y i are the momentum fractions of the U valons and D valon. The normalization factor g p is 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. This picture suggests that the structure function of a hadron involves a convolution of two distributions: Structure function of a v valon. It depends on Q 2 and the nature of the probe. Summation is over the three valons Describes the valon distribution in a proton. It independs on Q 2. Proton structure function  valon distributions in proton  quark distributions in a valon. In an unpolarized situation we may write: Proton structure function Proton structure function MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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

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

12. 12 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

13. 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 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 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

14. 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

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

16. we have the following sum rules 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  u v and  d v, 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. In valon model framework we have MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

17. 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 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 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

18. Descriptions of W function 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 Our motivation to predict this functional form is that For δW ’’ j (y) we choose the following form This term can controls the low-y behavior valon distribution The subscript j refers to U and D-valons This part adjusts valon distribution at large y values Polynomial factor 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. 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

19. Moments of polarized valon distributions in the proton Moments of polarized valon distributions in the proton Let us define the Mellin moments of any valon distribution δG j/p (y) as follows: Correspondingly in n-moment space we indicate the moments of polarized valon distributions Analysis of Moments in NLO Analysis of Moments in NLO MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

20. The Q 2 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 valon Moments of polarized parton distributions in valon MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

22. So in moment space for g 1 n (Q 2 ) 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 δu v, δd v, δ . and δg. NLO moments of PPDF’s and structure function MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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

24. Because for a given value of Q 2, 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 g 1 with p n,k (x) as weight functions, can be obtained in terms of odd and even moments where QCD fits to average of moments using Bernstein Polynomials QCD fits to average of moments using Bernstein Polynomials MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

25. Thus there are 16 parameters to be simultaneously fitted to the experimental g n,k (Q2) averages. The best fit is indicated by some sample carves in Fig.(1). The 41 Bernstein averages g n,k (Q 2 ) can be written in terms of odd and even moments To obtain these experimental averages from the exprimental data for xg 1, we fit xg 1 (x,Q 2 ) for each Q 2 separately, to the convenient phenomenological expression MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

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

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

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

34. Fig.9 Polarized proton structure function xg 1 p as a function of x which is compared with the experimental data for different Q 2 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

35. Fig. 10 Polarized structure function for some values of Q 2 as a function of x in NLO approximation. The solid curve is our model in NLO and dashed, dashed dot and long dashed curves are AAC, BB and GRSV model respectively. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

36. Fig. 10-continued Polarized structure function for some values of Q 2 as a function of x in NLO approximation. The solid curve is our model in NLO and dashed, dashed dot and long dashed curves are AAC, BB and GRSV model respectively. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian

37. Conclusion 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 1 p. MENU 04 - Beijing, Aug 29 -Sep. 4, 2004 Ali N. Khorramian