1. Proton transversity and intrinsic motion of quarks Petr Závada Inst. of Physics, Prague

2. Introduction  Presented results follow from QPM, in which (valence) quarks are considered as quasifree fermions on mass shell, with effective mass x 0 =m/M. Momenta distributions describing intrinsic quark motion have spherical symmetry and constraint J=1/2 is applied. The model is constructed in consistently covariant way [for details see Phys.Rev. D65 054040(2002), D67 014019(2003), D70 054018(2004)]. In this talk properties of spin functions obtained in the model will be discussed:  Sum rules for g 1,g 2.  g 1,g 2 from valence quarks, comparison with experimental data, discussion on Γ 1.  Transversity – two ways of calculation, discussion of compliance with Soffer inequality.

3. Model Input:

4. Output:

5. Sum rules  Basis:

8. Valence quarks

9. Calculation - solid line, data - dashed line (left) and circles (right) E155

10. g 1 - analysis  Integrating g 1 gives:  …so, it seems: more motion=less spin? How to understand it? How to understand it? staticquarks masslessquarks

11. Lesson of QM  Forget structure functions for a while and calculate another task.  Remember, that angular momentum consists of j=l+s.  In relativistic case l,s are not conserved separately, only j is conserved. So, we can have pure states of j (j 2,j z ) only, which are represented by relativistic spherical waves:

14. Spin and intrinsic motion j=1/2 j=1/2 m=p 0 m<p 0 j=l+s 1≥ ‹ s › /j≥1/3 QM: For p 0 >m there must be some orbital momentum!

15. Transversity (P.Z.+A.Efremov, O.Teryaev, for details see Phys.Rev.D70 054018(2004))  First, remind our procedure for g 1, g 2 :

16.  Transversity may be related to auxiliary polarized process described by interference of axial vector and scalar currents. (see G.R. Goldstein, R.L. Jaffe and X.D. Ji, Phys. Rev. D 52, 5006 (1995); B.L. Ioffe and A. Khodjamirian, Phys. Rev. D 51, 3373 (1995)). We try to use simplest form of such vector, giving:

17.  Dashed line – from g 1  Full line – from q v 1 st way: interference effects are attributed to quark level only… comparison with previous expressions for, gives: expressions for g 1, g T gives:

18. Conflict with Soffer inequality?  But generally, obtained functions (in particular d- quarks) may not satisfy Soffer inequality. Why? One should consistently take into account interference nature of transversity…

19. Transversity based on the expression… satisfies Soffer bound, in fact it satisfies a new, more strict limit… We are able calculate only this new limit … We are able calculate only this new limit δq max …

20. 2 nd way: interference effects at parton-hadron transition stage are included…  Dashed line – Soffer bound  Full line – δq max  Both limits are equivalent either for static quarks or for pure states with polarization +.

21. Two ways are compared…  Dashed line – from g 1  Full line – from q v  Dotted – calculation by P.Schweitzer, D.Urbano, M.V.Polyakov, C.Weiss, P.V.Pobylitsa and K.Goeke, Phys.Rev. D 64, 034013 (2001).

22. PAX experiment: Q 2 =4GeV 2 Q 2 =5GeV 2 Efremov, Goeke, Schweitzer Eur.Phys.J. C35 (2004), 207: our calculation: 1 st way 2 nd way

23. Conclusion  Covariant version of QPM involving intrinsic motion was studied.  Model easily reproduces well known sum rules for g 1,g 2 : WW, ELT, BC.  Spin function g 1 depend on intrinsic motion rather significantly, this motion generates orbital momentum as a “obligatory” part of j.  Calculated g 1,g 2 are well compatible with experimental data.  Two ways for estimation of transversity were suggested: Interference on quark level only (V & S currents) Interference effects on quark-hadron transition stage are included  Also for transversity, intrinsic motion tends to reduce it.  Results on q(x) are compatible with calculation by P.Schweitzer et al.