Generalized Parton Distributions: Completed research 2002-2003

G.A. Miller and B.C. Tiburzi

We completed a detailed investigation of light-front current between bound states [23,24]. The investigation was undertaken to better understand the appearance of higher Fock components in the light-front Bethe-Salpeter formalism and to shed light on how these components could be modeled. We found that higher Fock components arise from the covariant formulation via light-cone energy poles of both the Bethe-Salpeter vertex and the electromagnetic vertex. Some authors miss this point (or assume vertex poles are absent) and calculate physical amplitudes as if higher Fock components can be integrated out. This assumption often is not valid for calculations involving Z-graphs where non-wave function type vertices appear and hence non-valence components of the bound state are required.

Using a $(1+1)$-dimensional model for simplicity, we showed [23] that non-wave-function vertices are supplanted by contributions from higher Fock states in light-front time-ordered perturbation theory (provided the interaction has light-cone time dependence). In essence, contributions from non-wave-function vertices are reducible and should only be used when the interaction is (or is approximately) instantaneous. This constitutes a replacement theorem which trivially extends to $(3+1)$ dimensions. The investigation was followed up with applications [24]. The immediate application for form factors (in a frame where the plus component of the momentum transfer is nonzero) is to compute generalized parton distributions [25]. The non-vanishing of these distributions at the crossover between kinematic regimes (where the struck quark rebounds with zero plus momentum) is tied to higher Fock components. Common lore about the vanishing of light-cone wave functions at end points is shown to be wrong. In general, only the valence wave function vanishes at zero plus momentum. Continuity of the generalized parton distributions holds due to relations between different Fock components with vanishing plus momentum.

Our research expanded to consider the double distribution formalism [26]. This was motivated by a recent paper [27] which used two-body light-front wave function models to calculate generalized parton distributions via double distributions. Initially we showed [28] that without modification of the quark distribution, the resulting generalized parton distributions violate positivity bounds and do not line up with the physical intuition provided by the Fock space representation. These inconsistencies are attributed to missing contributions from time-ordered diagrams corresponding to non-valence configurations which are absent when one uses non-covariant vertices. We discouraged the use of such models since there is no general way to conjure up a generalized parton distribution from a valence wave-function overlap.

Additionally we sought to test the ostensible construction of double distributions presented in [27] by using simple covariant models where the non-wave function troubles encountered above would not be present. Thus in [29], we constructed double distributions by appealing to the Lorentz invariance of the form factor. The resulting generalized parton distributions were found not to agree with those calculated directly on the light cone, moreover the positivity bounds need not be respected. This inconsistency was shown to arise from the ambiguity inherent in defining double distributions in the standard one-component formalism (even when the Polyakov-Weiss $D$-term [30] is absent. Expressing the moments of double distributions in the two component form suggested by [31], we were able to demonstrate that the correct model distributions can be calculated from non-diagonal matrix elements of twist-two operators. Thus we showed generalized parton distributions are far richer than the models commonly used to describe them.