structure functions and intrinsic quark orbital motion
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Structure functions and intrinsic quark orbital motion. P etr Z ávada Inst. of Physics, Prague. Introduction.

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structure functions and intrinsic quark orbital motion

Structure functions and intrinsic quark orbital motion

Petr Závada

Inst. of Physics, Prague

  • 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].


Structure functions

  • Input:
  • 3D distribution
  • functions
  • Result:
  • structure
  • functions
  • (x=Bjorken xB !)
  • 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


1) Relativistic covariance

2) Spheric symmetry

3) One photon exchange

  • In this talk m→0is assumed.

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).
g 1 g 2 from valence quarks1
g1, g2 from valence quarks


Calculation - solid line, data - dashed line

(left) and circles (right)

  • g1 fit of world data by E155 Coll., Phys.Lett B 493, 19 (2000).
  • 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
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
Quark momenta distributions …

2) Deconvolution of g1


DG=G+-G- represents subset of quarks giving net spin

contribution - opposite polarizations are canceled out. Which F2

correspond to this subset?

quark momenta distributions1
Quark momenta distributions …


In this way, from F2 and g1 we obtain:

quark momenta distributions2
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
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:
spin and orbital motion

<s>, Γ1: two ways, one result

-covariant approach is a common basis

Spin and orbital motion
  • 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

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
Sum rules