We find in one-loop calculations and spectator models that twist-3 GPDs exhibit discontinuities. In the forward limit these discontinuities grow into Dirac delta functions which are essential to satisfy the sum rules involving twist-3 PDFs. We calculate twist-3 quasi-PDFs as a function of longitudinal momentum and identify the Dirac delta function terms with momentum components in the nucleon state that do not scale as the nucleon is boosted to the infinite momentum frame.PACS numbers:
I. INTRODUCTIONA complementary picture of the nucleon structure is obtained by simultaneous information on both transverse spatial and longitudinal momentum distributions of partons. The relevant physical observables are generalized parton distributions (GPDs) [1][2][3][4]. Theoretically they are calculated from nonforward matrix elements of local operators, and experimentally they are accessible through exclusive deep inelastic scattering experiments, such as deeply virtual Compton scattering (DVCS) [3,5]. GPDs give information about the spin, momentum and spatial distribution of the quarks, anti-quarks and gluons within a fast moving nucleon [6-8] and therefore provide a remarkable insight on its inner structure.One property to classify the GPDs, is their twist [9]. Twist determines the order in Q 2 (squared four-momentum transfer) at which a matrix element contributes to the physical amplitude of a given hard process. With increasing twist the number of partons which participate in that matrix element also tend to increase. At leading twist, twist-2 GPDs describe two-particle correlations in the nucleon while the next leading twist, twist-3 GPDs also involve three-particle correlations such as quark-gluon-quark (qgq). It is advantageous to define the twist in the infinite momentum frame (IMF) where the nucleon has a large momentum in the longitudinal direction (direction of the nucleon propagation), i.e. P + M and nearly zero momentum in the transverse direction, i.e. P T ≈ 0 [10]. In the IMF the twist of a distribution can be identified with its behavior under a longitudinal momentum boost. While twist-2 distributions are invariant under the boosts along the longitudinal direction, twist-3 distributions scale as 1/P + .In the Bjorken limit, the matrix elements are dominated by twist-2 operators [49]. Even though they are mostly relevant for subleading corrections, there are several motivations to study twist-3 GPDs.
While twist-2 GPDs allow for a determination of the distribution of partons on the transverse plane, twist-3 GPDs contain quark-gluon correlations that provide information about the average transverse color Lorentz force acting on quarks. As an example, we use the nonforward generalization of g T (x), to illustrate how twist-3 GPDs can provide transverse position information about that force.
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