Generalized quasi-, Ioffe-time-, and pseudodistributions of the pion in the Nambu–Jona-Lasinio model

Abstract

We analyze the generalized quasi, Ioffe-time, and pseudodistributions of the valence quarks in the pion at the quark model scale. We use the framework of the Nambu-Jona-Lasinio model and investigate the basic question of how fast the pion has to move to effectively reach the infinite momentum limit, where the approach can provide the information on the generalized parton distribution functions. We consider both the vector distributions and the transversity distributions, related to the spin densities. With the developed analytic expressions, we conclude that to effectively approach the infinite momentum limit in the Ioffe-time distributions for values of the Ioffe-time accessible in lattice QCD, one roughly needs the pion momenta of the order of $\ensuremath{\sim}3\text{ }\text{ }\mathrm{GeV}$. We explore polynomiality of the quasidistributions and study the generalized quasi form factors. The issue of separability of the transverse and longitudinal dynamics in the model is studied with the help of the generalized Ioffe-time distributions, with the conclusion that the breaking is not substantial, unless the momentum transfer $t$ is large. We also provide an estimate of the range of the Ioffe-time values needed to obtain the generalized parton distributions with a reasonable accuracy. Our model results, which are analytic or semianalytic, provide a valuable insight into the theoretical formalism and illustrate the intricate features of the investigated distributions.