Institute for studies in theoretical Physics and Mathematics,IPM Yazd University, IRAN

1. Polarized parton distributions in the approch of complete renormalization group improvment 16th International Syposium on Spin, ICTP, 10-16 Oct.04 . Institute for studies in theoretical Physics and Mathematics,IPM Yazd University, IRAN A Talk given by Abolfazl Mirjalili

2. Outline • RS and FS dependence in QCD predictions • Complete RG Improvement • Valon model • Unpolarized valon distribution in a proton • Polarized valon distribution • Parton distribution functions in moment space • Comparison of the Moments • Moments of PPDF’s and distribution function • Conclusion Spin04 Symposium-Italy

3. RS and FS dependence in QCD predictions For a single case of a dimensional observable R(Q) with perterbative series The RS can be labeled by the non-universal coefficients of the beta-function and ΛRS. 1 Spin04 Symposium-Italy

4. Self-consistency of perturbation theory that is the derivative of N-th order approximant R(n) With respect to ln(µ) is of higher order than the approximant itself, will yields where Which is which is a renormalization scale and scheme invariant.. 2 Spin04 Symposium-Italy

5. For other coefficients we will arrive first on following partial derivatives for r2 and for r3,r4,… we will have 3 Spin04 Symposium-Italy

6. on integration one finds and finally General structure is as follows Where Xn are Q-independent and RS-invariant and are unknown unless a complete NnLO calculation has been performed. 4 Spin04 Symposium-Italy

7. Given a NLO calculation of r1, the sum of all-orders of the RG-predictable term simplifies to a geometric progression : which is completely independent of renormalization µ scale Complete RG Improvement From pervious results for r1 , r2, … At NLO calculation r1 is known but X2, X3,…are unknown. Thus the complete subset of known terms in above at NLO is It is RS-invariant. Choose r1=0, c2=c3=…=cn=0 we obtain a0=a. 5 Spin04 Symposium-Italy

8. At NNLO calculation X2 is unknown. Further infinite subset of terms are known and can be resummed to all orders. Finally we will arrive at where a0=a(0,0,0, …) is the coupling in this scheme and satisfies The solution of this transcendental equation can be written in closed form in terms of the Lambert-W function where Lambert-W function is defined by 6 Spin04 Symposium-Italy

9. 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 type valons and one D valon. 7 Spin04 Symposium-Italy

10. Unpolarized valon distribution in a proton 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 inclusive valon distributions are obtained by integration 8 Spin04 Symposium-Italy

11. Polarized valon distribution Unpolarized U-valon distribution Polarized U-valon distribution 9 Spin04 Symposium-Italy

12. Parton distribution functions in moment space 10 Spin04 Symposium-Italy

13. 11 Spin04 Symposium-Italy

14. Comparison of the Moments Comparison of Non-singlet moments in standard perturbative QCD and CORGI approach. 12 Spin04 Symposium-Italy

15. Comparison of singlet moments in standard perturbative QCD and CORGI approach. 13 Spin04 Symposium-Italy

16. Moments of PPDF’s and distribution 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. 14 Spin04 Symposium-Italy

17. Comparison of polarized valance u and d quark distributions and polarized gluon distribution in standard perturbative QCD and in CORGI approach. 15 Spin04 Symposium-Italy

18. Conclusion • First we introduced the idea of complete RG- improvement and showed how we can reconstructed the QCD perurbatives series in this approach. • Valon model was defined and unpolarized and polarized valon distributions were obtained. • The nonsinglet and singlet moments of polarized valons were obtained and the results in standard perterbative series and the CORGI approch were comparied. • Using the convelution of moments in valon and the moments of valon in proton and finally the inverse melleion technique, the PDF’s were obtained and compared in Standard Perturbative series and in CORGI approch. • Since the perturbative terms in CORGI approach are scheme invariants and were obtained from a resummation of ultraviolet terms in specified order, we expect to get better consistency between polarized proton SF and available experimantal data • . This is our new task. 16 Spin04 Symposium-Italy