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Parton distribution functions and quark orbital motion. P etr Z ávada Inst itute of Physics, Prague. The 6 th Circum-Pan-Pacific Symposium on High Energy Spin Physics July 30 - August 2, 2007 Vancouver BC. Introduction.

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parton distribution functions and quark orbital motion

Parton distribution functions and quark orbital motion

Petr Závada

Institute of Physics, Prague

The 6th Circum-Pan-Pacific Symposium on High Energy Spin Physics

July 30 - August 2, 2007

Vancouver BC

  • Presented results are based on the covariant QPM. Intrinsic motion, reflecting orbital momenta of quarks, is consistently taken into account. Due to covariance, transversal and longitudinal momenta appear on the same level. [P.Z. Phys.Rev.D65, 054040(2002) and D67, 014019(2003)].
  • In present LO version, no dynamics, but “exact” kinematics effective tool for separating effects due to dynamics (QCD) and kinematics. This viewpoint well supported by our previous results e.g:
    • sum rules WW, Efremov-Leader-Teryaev, Burkhard-Cottingham
    • the same set of assumptions implies substantial dependence Γ1 on kinematical effects
    • Calculation g1 and g2 from valence distributions – very good agreement with data
    • some new relations between structure functions, including transversity [A.Efremov, O.Teryaev and P.Z., Phys.Rev.D70, 054018(2004) and arXiv: hep-ph/0512034].
Previous papers:
  • What is the dependence of the structure functions on intrinsic motion of the quarks?

In this talk further questions:

  • How can one extract information about intrinsic motion from the structure functions?
  • What is the role of the orbital momentum of quarks, which is a particular case of intrinsic motion?

[full version in arXiv: hep-ph/0706.2988 and Eur.Phys.J. C – August2007].


Structure functions

  • Input:
  • 3D distribution
  • functions in the proton rest frame
  • 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 (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 followsfrom 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)].
momentum distributions from structure function f 2
Momentum distributions from structure function F2

Deconvolution of F2 :

  • Remarks:
  • G measures in d3p, Pin the dp/M
  • pmax=M/2 – due to kinematics in the proton rest frame, ∑p=0
  • Self-consistency test:
momentum distributions in the proton rest frame
Momentum distributions in the proton rest frame

Input q(x)




<pval>=0.11 (0.083) GeV/c for u (d) quarks

momentum distributions from structure function g 1
Momentum distributions from structure function g1

Deconvolution of g1 :

Since G=G++G- and ∆G=G++G-



… obtained from F2 ,g1 and represent distribution of quarks with polarization ±.

distribution functions f x
Distribution functions f±(x)

Let us note:



(equality takes place only in non-covariant IMF approach)

momentum distributions in the proton rest frame1
Momentum distributions in the proton rest frame

2) q(x) & Δq(x)



xΔfq(x) are similar to xqval(x)

spin contribution

comes dominatly from valence region

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
  • this scenario is clearly preferred for quarks with effective mass on scale of thousandths and momentum of tenths ofGeV.
  • important role of the intrinsic quark orbital motion emerges as a direct consequence of the covariant approach
proton spin
Proton spin

Second scenario:

implies, that a room for gluon contribution can be rather sensitive to the longitudinal polarization:

For ∆∑≈1/3, 0.3 and 0.2 gluon contribution represents 0, 10 and 40%. Value empirically known ∆∑≈0.2-0.35 does not exclude any of these possibilities.

CQSM-chiral quark soliton model:


Covariant version of QPM involving quark orbital motion was studied. New (LO) 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.
  • Important consequence for the composition of proton spin was suggested.