Structure functions and intrinsic
quark orbital motion
Petr Závada
Inst. of Physics, Prague
2nd Workshop on the QCD Structure of the Nucleon
June 12-16, 2006
Villa Mondragone Monte Porzio Catone, Rome, Italy
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
Model
e-
Structure functions
 Input:
3D distribution
functions
 Result:
structure
functions
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.
Comments
Structure functions are represented by integrals from
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 H from F2 or g1 (main
aim of this talk).
g1, g2 from valence quarks
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)].
 Transversities obtained above 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: hepph/0512034)].
Double spin asymmetry in PAX experiment
1.
2.
Quark momenta distributions from structure functions
1) Deconvolution of F2
Remarks:
• G measures in d3p, PG in the dp
• 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:
• H=D+-D- represents subset of quarks giving net spin contribution
– opposite polarizations are canceled out. Which F2 correspond to
this subset? One can calculate
Quark momenta distributions …
Comments:
 Shape of ΔF2 similar to F2val
 Generic polarized and unpolarized
distributions H 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 relativistic spherical waves:
j=1/2
Spin and orbital motion
<s>, Γ1: two ways, one result
-covariant approach is a common basis
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
•massless quarks
and
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:
Manifestly covariant form:
where