1. Structure functions and intrinsic quark orbital motion
Petr Závada
Inst. of Physics, Prague

2. Introduction In this talk:
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. [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) and arXiv: hep-ph/ ].
In this talk:
Relation between structure functions and 3D quark momenta distribution
Important role of quark orbital motion as a direct consequence of the covariant description
[full version in arXiv: hep-ph/ ].

3. Model

4. Structure functions Input: Result: 3D distribution functions structure
(x=Bjorken xB !)

5. F1, F2 - manifestly covariant form:

6. g1, g2 - manifestly covariant form:

7. Comments
In the limit of static quarks, for p→0, which is equivalent to the assumption p=xP, one gets usual relations between the structure and distribution functions like
Obtained structure functions for m→0 obey the known sum rules:
Sum rules were obtained
from:
1) Relativistic covariance
2) Spheric symmetry
3) One photon exchange
In this talk m→0 is assumed.

8. Comments
Structure functions are represented by integrals from probabilistic distributions:
This form allows integral transforms:
g1 ↔ g2 or F1 ↔ F2 (rules mentioned above were example).
With some additional assumptions also e.g. integral relation g1 ↔ F2 can be obtained (illustration will be given).
To invert the integrals and obtain G or DG from F2 or g1 (main aim of this talk).

9. g1, g2 from valence quarks

10. Calculation - solid line, data - dashed line
g1, g2 from valence quarks
E155
Calculation - solid line, data - dashed line
(left) and circles (right)
g1 fit of world data by E155 Coll., Phys.Lett B 493, 19 (2000).

11. Transversity
In a similar way also the transversity was calculated; see [A.Efremov, O.Teryaev and P.Z., Phys.Rev.D70, (2004)]. Among others we obtained
- which follows only from covariant kinematics!
Obtained transversities were used for the calculation of double spin asymmetry in the lepton pair production in proposed PAX experiment; see [A.Efremov, O.Teryaev and P.Z., arXiv: hep-ph/ )].

12. Double spin asymmetry in PAX experiment1. 2.

13. Quark momenta distributions from structure functions
1) Deconvolution of F2
Remarks:
G measures in d3p, 4pp2MG in the dp/M
pmax=M/2 – due to kinematics in the proton rest frame, ∑p=0
F2 fit of world data by SMC Coll., Phys.Rev. D 58, (1998).

14. Quark momenta distributions …
2) Deconvolution of g1
Remark:
DG=G+-G- represents subset of quarks giving net spin
contribution - opposite polarizations are canceled out. Which F2
correspond to this subset?

15. Quark momenta distributions …
Calculation:
In this way, from F2 and g1 we obtain:

16. Quark momenta distributions …
Comments:
Shape of ΔF2 similar to F2val
Generic polarized and unpolarized distributions DG, G and G+ are close together for higher momenta
Mean value:
Numerical calculation:
g1 fit of world data by E155 Coll., Phys.Lett B 493, 19 (2000).

17. Intrinsic motion and angular momentum
Forget structure functions for a moment…
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 the bispinor spherical waves:

18. j=1/2

19. Spin and orbital motion
<s>, Γ1: two ways, one result
-covariant approach is a common basis

20. Comments are controlled by the factor , two extremes:
for fixed j=1/2 both the quantities are almost equivalent:
more kinetic energy (in proton rest frame) generates more orbital motion and vice versa.
are controlled by the factor , two extremes:
massive and static quarks and
massless quarks and
important role of the intrinsic quark orbital motion emerges as a direct consequence of the covariant approach

21. Summary
Covariant version of QPM involving quark orbital motion was studied. New results:
Model allows to calculate 3D quark momenta distributions (in proton rest frame) from the structure functions.
Important role of quark orbital motion, which follows from covariant approach, was pointed out. Orbital momentum can represent as much as 2/3 j. The spin function g1 is reduced correspondingly.

22. Sum rules
Basis: