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Michel Garcon: An introduction to the Generalized Parton Distributions
ton in the initial and final states, a skewness parameter
which measures the difference between these two momen-
tum fractions, and the momentum transfer t to the target
nucleon. The above mentioned factorization theorems [7,
8] are applicable for tJ/Q2 < 1.
The physical content of the GPDs is quite rich: in sim-
ple terms, whereas ordinary parton distributions yield, for
example, the probability la(x) 2 that a quark carries a
fraction x of the momentum of a fast moving nucleon, the
GPDs measure the coherence b*(x - ) -pb(x+ ) between
two different (quark) momentum states of the nucleon and
in this way quark momentum correlations in the nucleon.
In quantum mechanics, one cannot reduce the knowledge
of a system formed of a superposition of states (in our
case, an infinite number of states described by the vari-
able x) to the sole probabilities that the system be in a
particular state. The interferences between these states,
or coherences, or off-diagonal elements, are necessary for
a full description of the system.
If one of the momentum fractions (x + or x -)
is negative, it is interpreted as representative of an anti-
quark. Meson-like qq configurations in the nucleon may
then be investigated. As mentioned in Sec. 3.2, the GPDs
are expected to exhibit a rich structure at IxI < which is
related in particular to the pion cloud around the nucleon.
Whereas x and characterize solely the longitudinal
momenta of the partons involved, the t-dependence of the
GPDs is related to their transverse momenta. By Fourier
transform, it is conceivable to access simultaneously the
longitudinal momentum fraction of quarks and their posi-
tion in the transverse plane [9]. This would open the way
to a femto-photography of the nucleon [10].
The richness of information contained in the GPDs
may be illustrated by several quite remarkable relations,
among which
- the forward limit:
lim H(x,Z,t) =q(x), or -q(-x) if x < 0
t-0, _
and
lim H(x, Z, t) =Aq(x), or Aq(-x).
As expected from the above discussion, the ordinary
parton distributions, both unpolarized q(x) and polar-
ized Aq(x), are but a limiting case of the GPDs. Note
that, though the GPDs are defined functions for 0
or t 0, these variables take only finite, non-zero val-
ues in any experiment. Note also that the functions
E and E have no connection with the ordinary par-
ton distributions. They are not constrained by deeply
inelastic scattering, which corresponds to this forward
limit.
- the form factor limit:
v, f H(x, , t) - dx F(t)
(4 relations of this type for each quark flavor). After
integration over x, the t-dependence of the GPDs isgiven by the nucleon elastic form factors (H -> Dirac,
E -> Pauli, f -> axial vector and E -> pseudoscalar
form factors).
- Ji's sum rule:
V, lim x - (H + E)(x, , t) - dx =J,
two 2 .Ja4
where Jq is the total angular momentum, i.e. the sum
of intrinsic spin and of orbital angular momentum, car-
ried by the quarks. This sum rule provides a way to
understand the origin of the nucleon spin. Indeed, the
contribution of the quark intrinsic spin ( AE) was
measured at CERN/SMC, SLACK and DESY/HER-
MES. The gluon contribution (AG) will be determined
at CERN/COMPASS, RHIC/STAR and SLACK. The
knowledge of GPDs allows one to isolate the contri-
bution of the quark orbital angular momentum to the
total spin of the nucleon: = Jq+ J= ( AE+ Lq)+
( AG + Lg).
The first two relations demonstrate the new unify-
ing frame given by the GPDs: observables as different as
elastic form factors and parton distributions are related
to each other through these new functions. However, the
knowledge of limits, or projections, does not imply a com-
plete knowledge of the GPDs themselves. Models are being
developed for a complete representation.
3 GPD calculations
We give here a short summary of selected calculations of
the GPDs. For a more complete account, see Refs. [3,5,6].
3.1 Constrained parametrizations
The "double distributions" [11] are well suited for the de-
scription of the GPDs. They incorporate the GPDs mathe-
matical properties (polynomiality of the Mellin moments)
and have a definite physical content. They satisfy positiv-
ity bounds, a number of inequalities that GPDs have been
shown to obey. Elastic form factors and parton distribu-
tions are included as limiting values.
An alternate, "dual" parametrization, based on a par-
tial wave expansion, has recently been proposed [12].
3.2 Model calculations
QCD-inspired models are very useful to understand how
non-perturbative mechanisms generate various structures
in GPDs.
In the simple bag model, GPDs have but a weak depen-
dence on , and show no structure for xl < . A number
of calculations in different constituent quark models are
being developed.
In contradistinction, in the chiral quark-soliton model,
where the quarks interact through a highly non-linear pion2
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Garcon, Michel. An introduction to the Generalized Parton Distributions, article, June 1, 2002; Newport News, Virginia. (https://digital.library.unt.edu/ark:/67531/metadc738098/m1/2/: accessed April 19, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.