Structure functions and intrinsic Structure functions and intrinsic quark orbital motionquark orbital motion

Structure functions and intrinsic Structure functions and intrinsic quark orbital motionquark orbital motion

Petr ZávadaInst. of Physics, Prague

IntroductionIntroduction

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].

ModelModel

e-e-

Structure functionsStructure functions

Input:Input:

3D distributionfunctions

Result:Result:

structure functions

(x=Bjorken xB !)

F1, F2 - manifestly covariant form:

g1, g2 - manifestly covariant form:

CommentsComments

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 obtainedfrom:

1) Relativistic covariance2) Spheric symmetry3) One photon exchange

In this talk In this talk m→0 is assumed.is assumed.

Comments

SStructure functions are represented by integrals from tructure functions are represented by integrals from probabilistic distributions:probabilistic distributions:

This form allows integral transforms:

1) g1 ↔ g2 or F1 ↔ F2 (rules mentioned above were example).2) With some additional assumptions also e.g. integral relation

g1 ↔ F2 can be obtained (illustration will be given).3) To invert the integrals and obtain G or G from F2 or g1 (main

aim of this talk).

g1, g2 from valence quarks

g1, g2 from valence quarks

Calculation - solid line, data - dashed lineCalculation - solid line, data - dashed line (left) and circles (right)(left) and circles (right)

E155E155

g1 fit of world data by E155 Coll., Phys.Lett B 493, 19 (2000).

TransversityTransversity In a similar way also the transversity was calculated; see 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 . Among others we obtainedobtained

- which follows- which follows only from covariant kinematics!only from covariant kinematics!

Obtained transversities were used for the calculation of double spin Obtained transversities were used for the calculation of double spin asymmetry in the lepton pair production in proposed PAX experiment; asymmetry in the lepton pair production in proposed PAX experiment; see see [A.Efremov, O.Teryaev and P.Z., arXiv: hep-ph/0512034)]. .

Double spin asymmetry in PAX experiment

1.1. 2.2.

Quark momenta distributions from structure functionsQuark momenta distributions from structure functions

1) Deconvolution of F2

Remarks:• G measures in d3p, 4p2MG 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, 112001 (1998).

Quark momenta distributions …

2) Deconvolution of g1

Remark:G=G+-G- represents subset of quarks giving net spin contribution - opposite polarizations are canceled out. Which F2

correspond to this subset?

Quark momenta distributions …

Calculation:

In this way, from F2 and g1 we obtain:

Quark momenta distributions …

Comments: Shape of ΔF2 similar to F2val Generic polarized and

unpolarized distributions G, 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 momentumIntrinsic 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:

j=1/2j=1/2

Spin and orbital motionSpin and orbital motion

<s>, <s>, ΓΓ11: two ways, one result: two ways, one result

-covariant approach is a common basis -covariant approach is a common basis

Comments

• are controlled by the factor , two extremes:

•massive and static quarks and

• 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.

•massless quarks and

• important role of the intrinsic quark orbital motion emerges as a direct consequence of the covariant approach

SummarySummary

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 rulesSum rules

Basis: