Generalized quasi, Ioffe-time, and pseudo quark distributions of the pion in the Nambu--Jona-Lasinio model

TL;DR

This paper investigates how the pion's generalized quasi, Ioffe-time, and pseudo distributions behave at the quark model scale within the Nambu-Jona-Lasinio model, focusing on the momentum needed to reach the infinite momentum limit and the properties of these distributions.

Contribution

It provides analytic expressions and insights into the behavior of various pion distributions, including the momentum threshold for effective infinite momentum limit and the polynomiality of quasi distributions.

Findings

Pion momenta of about 3 GeV are needed to approach the infinite momentum limit.

Transverse and longitudinal dynamics are not substantially broken at small momentum transfer.

Analytic results offer insights into the features of generalized distributions.

Abstract

We analyze the generalized quasi, Ioffe-time, and pseudo distributions 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, one roughly needs the pion momenta of the order of $\sim 3$GeV. We also explore polynomiality of the quasi distributions 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 semi-analytic, provide a valuable insight into the theoretical formalism and illustrate the intricate features of the investigated distributions.