1. Spin of the proton and orbital motion of quarks
Petr Závada
Inst. of Physics, Prague
Workshop on Nucleon Form Factors Frascati, October, 2005

2. Introduction
Presented results are based on the covariant QPM, in which quarks are considered as quasifree fermions on mass shell. Intrinsic quark motion, reflecting orbital momenta, is consistently taken into account. [for details see P.Z. Phys.Rev.D65, (2002) and D67, (2003)].
Recently, this model was generalized to include the transversity distribution [A.Efremov, O.Teryaev and P.Z., Phys.Rev.D70, (2004)].
In this talk the obtained spin functions will be discussed:
Sum rules for g1,g2.
Calculation of the spin functions, comparison with experimental data, discussion on Γ1.

3. Model
Input:

4. Spin structure functions
Output:

5. Sum rules
Basis:

6. …Sum rules

7. …Sum rules The rules were obtained from: Relativistic covariance
Spheric symmetry
One photon exchange

8. Valence quarks

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

10. g1 - analysis Integrating g1 gives:
massless
quarks
static
quarks
…so, it seems: more motion=less spin?
How to understand it?

11. Angular momentum Forget structure functions for a moment…
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 (j2,jz) only, which are represented by relativistic spherical waves:

12. j=1/2

13. Spin and orbital motion

14. Spin and orbital motion
j=1/2
j=1/2
j=l+s
1≥‹s›/j≥1/3
QM:
1) For p0>m there must be some orbital momentum!
2) For localized particles, there must be p0>m!
o r b i t a l m o m e n t u m
spin
spin
m<p0
m=p0

15. Recent experimental data:
Contribution of sea quarks is compatible with zero: HERMES Phys.Rev.Lett. 92, (2004); Phys.Rev. D71, (2005).
Contribution of gluons is compatible with zero, but with rather big errors: HERMES, SMC, COMPASS
Phys.Rev.Lett. 84, 2584 (2000); Phys.Rev. D70, (2004); hep-ex/

16. Support for dominant role of the orbital momentum:
Models, e.g.:
Casu, Sehgal, Phys.Rev.D58, 2644 (1996).
Wakamatsu, Watabe, Phys.Rev.D62, (2000).
Jefferson Lab Hall A Collaboration
Experimental analysis suggests significant presence of the orbital momentum: Phys.Rev.Lett.92, (2004); Phys.Rev.C70, (2004).

17. PAX experiment:
Polarized Drell-Yan process in this reaction is very effective for measuring the transversity, which can be extracted from the double spin asymmetry in μ+μ- pair. This was discussed in the previous papers
Efremov, Goeke, Schweitzer, Eur.Phys.J. C35 (2004), 207
Anselmino, Barone, Drago, Nikolaev, Phys. Lett. B 594 (2004) 97
We do the same, but as an input we use the transversity, which is calculated in the model.

18. Asymmetry corresponding to
Useful relations:
Asymmetry corresponding to
Double spin asymmetry defined as:
can be calculated from the transversity distributions:
where

19. Results
Our calculation based on the “1st way” - level of quark interference only (solid line)
Calculation based on the “2nd way”- interference effects at parton-hadron transition included (dashed line)
Calculation based on the quark-soliton model [Efremov…] (dotted line)
s=45GeV2, Q2=4-5GeV2

20. …Results
Our calculation based on the “1st way” - level of quark interference only (solid line)
Calculation based on “2nd way”- interference effects at parton-hadron transition included (dashed line)
Calculation based on the estimation by Anselmino… (dotted line)
s=45GeV2, Q2=4GeV2

21. Statistical errors
How many events one needs for discriminating among the curves? If
then
which means roughly
So for one needs at least 104 events in given subsample (bin, interval).

22. Summary
Covariant version of QPM involving intrinsic (orbital) motion was studied.
Model reproduces the well known sum rules for g1,g2 : WW, ELT, BC.
Spin function g1 depend on intrinsic orbital motion rather significantly, this motion generates orbital angular momentum, which can represent as much as 2/3 j.
Calculated g1,g2 from qval are well compatible with the experimental data.