Linear $\Sigma$ Model in the Gaussian Functional Approximation

TL;DR

This paper applies a Gaussian functional approximation to the linear sigma model, confirming key chiral symmetry identities and relations, and numerically analyzing the solutions and particle content.

Contribution

It introduces a self-consistent variational approach to the linear sigma model, verifying chiral Ward-Takahashi identities and solving the equations numerically.

Findings

Confirmed chiral Ward-Takahashi identities

Validated the Nambu-Goldstone theorem and axial current conservation

Numerically analyzed the particle spectrum and solutions

Abstract

We apply a self-consistent relativistic mean-field variational ``Gaussian functional'' (or Hartree) approximation to the linear $\sigma$ model with spontaneously and explicitly broken chiral O(4) symmetry. We set up the self-consistency, or ``gap'' and the Bethe-Salpeter equations. We check and confirm the chiral Ward-Takahashi identities, among them the Nambu-Goldstone theorem and the (partial) axial current conservation [CAC], both in and away from the chiral limit. With explicit chiral symmetry breaking we confirm the Dashen relation for the pion mass and partial CAC. We solve numerically the gap and Bethe-Salpeter equations, discuss the solutions' properties and the particle content of the theory.