Study of Hadrons Using the Gaussian Functional Method in the O(4) Linear $\sigma$ Model

TL;DR

This paper investigates hadron properties in the O(4) linear sigma model using the Gaussian functional method, accounting for meson fluctuations to improve the ground state description and analyze temperature-dependent masses.

Contribution

It introduces the Gaussian functional method to include meson fluctuations in the sigma model, providing more accurate meson mass calculations and insights into meson structure.

Findings

Recovered the Nambu-Goldstone theorem for pions.

Sigma meson exhibits a 4-quark structure.

Mass behaviors are similar to mean field results but with larger coupling constants.

Abstract

We study properties of hadrons in the O(4) linear $\sigma$ model, where we take into account fluctuations of mesons around their mean field values using the Gaussian functional (GF) method. In the GF method we calculate dressed $\sigma$ and $\pi$ masses, where we include the effect of fluctuations of mesons to find a better ground state wave function than the mean field approximation. Then we solve the Bethe-Salpeter equations and calculate physical $\sigma$ and $\pi$ masses. We recover the Nambu-Goldstone theorem for the physical pion mass to be zero in the chiral limit. The $\sigma$ meson is a strongly correlated meson-meson state, and has a 4 quark structure. We calculate $\sigma$ and $\pi$ masses as functions of temperature for the two cases of chiral limit and explicit chiral symmetry breaking. We get similar behaviors for the $\sigma$ and $\pi$ masses as the case of the mean field approximation, but the coupling constants are much larger than the values of the case of the mean field approximation.