Wigner and Husimi partonic distributions of the pion in a chiral quark model

TL;DR

This paper computes and analyzes the Wigner and Husimi distributions of quarks in the pion within a chiral quark model, revealing their features and non-positivity, with implications for understanding partonic structures.

Contribution

It provides analytic expressions for GTMDs and numerical evaluations of Wigner and Husimi distributions in a chiral quark model, highlighting their properties and the effects of coarse-graining.

Findings

Wigner distributions exhibit non-positivity.

Husimi distributions are nearly positive, with tiny negative regions.

Coarse-graining improves probabilistic interpretation.

Abstract

Generalized transverse momentum distributions (GTMDs), the Wigner, and the Husimi distributions of quarks in the pion are evaluated in a chiral quark model at the one-loop-level. Analytic expressions are obtained for GTMDs, allowing for a qualitative discussion of their features, whereas the Wigner and the Husimi distribution are obtained with numerical integration of simple formulas. We explain the features of the Wigner distributions, in particular their non-positivity. In our model, the Husimi distributions, which are interpreted as coarse-grained Wigner distributions, are not mathematically positive-definite, but the magnitude of their negative values is tiny and occurs at large transverse momenta and impact parameters. Hence, as expected, coarse-graining leads to better behaved functions from the point of view of the probabilistic interpretation.