Off-shellness in generalized parton distributions and form factors of the pion

TL;DR

This paper investigates how off-shell effects influence the generalized parton distributions and form factors of the pion, revealing substantial effects at the distribution level but minimal impact on observable electroproduction processes due to cancellations.

Contribution

It introduces a detailed analysis of off-shell effects in pion GPDs and form factors, including relations derived from Ward-Takahashi identities and their impact on electroproduction observables.

Findings

Off-shell effects significantly alter GPDs and form factors.

Cancellations reduce the impact of off-shellness on electroproduction cross-sections.

Analytic expressions for off-shell form factors are provided.

Abstract

We study the effects of off-shellness in the generalized parton distributions of the pion. On general grounds, these distributions exhibit a richer structure than in the on-shell case due to absence of the crossing symmetry. In particular, their moments involve additional terms odd in the skewness parameter, associated with new form factors. We bring up relations between the off-shell charge and gravitational form factors, as well as the pion form factor, and discuss their derivations based on the Ward-Takahashi identities. We illustrate the features at the (leading-$N_c$) one-quark-loop level with the help of the spectral quark model of the pion, constructed to embed the vector meson dominance. Simple analytic expressions for the form factors and the distributions follow. Thus obtained off-shell generalized parton distributions are evolved from the quark model scale to higher scales with the LO DGLAP equations. We evaluate the corresponding Compton amplitudes which enter the cross-section for the electroproduction of the pion off the proton (the Sullivan process). It is found in our model that the effects of off-shellness in the generalized parton distribution are substantial, however, they can be largely canceled by the corresponding off-shell corrections to the pion propagator. In particular, this is the case of the Compton form factors entering the deeply virtual Compton scattering amplitude. As a result, we expect small off-shellness effects in electroproduction reactions, such as the Sullivan process.