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Structure functions and intrinsic quark orbital motion

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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. [P.Z. Phys.Rev.D65, 054040(2002) and D67, 014019(2003)].
- Recently, this model was generalized to include the transversity distribution [A.Efremov, O.Teryaev and P.Z., Phys.Rev.D70, 054018(2004) and arXiv: hep-ph/0512034].

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/0609027].

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→0is assumed.

Comments

Structure functions are represented by integrals from probabilistic distributions:

- This form allows integral transforms:
- g1↔g2orF1↔ 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 F2or g1 (main aim of this talk).

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).

Transversity

- In a similar way also the transversity was calculated; see [A.Efremov, O.Teryaev and P.Z., Phys.Rev.D70, 054018(2004)]. Among others we obtained

- which followsonly 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/0512034)].

Quark momenta distributions from structure functions

1) Deconvolution of F2

- Remarks:
- G measures in d3p, 4pp2MGin 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, 112001 (1998).

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?

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).

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:

Comments

- 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

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.

Sum rules

Basis:

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