Apparently noninvariant terms of $U(N)\times U(N)$ nonlinear sigma model in the one-loop approximation

TL;DR

This paper demonstrates that in the U(N) nonlinear sigma model, apparently noninvariant terms arising at one-loop level do not violate the model's symmetry, as shown through the Ward-Takahashi identity, extending previous SU(2) results.

Contribution

It generalizes the analysis of ANTs from SU(2) to U(N) nonlinear sigma models and confirms symmetry preservation at one-loop despite divergent terms.

Findings

ANTs are consistent with symmetry via Ward-Takahashi identity.

Divergent ANTs do not break the nonlinearly realized symmetry.

Symmetry is preserved at one-loop in U(N) nonlinear sigma models.

Abstract

We show how the Apparently Noninvariant Terms (ANTs), which emerge in perturbation theory of nonlinear sigma models, are consistent with the nonlinearly realized symmetry by employing the Ward-Takahashi identity (in the form of an inhomogeneous Zinn-Justin equation). In the literature the discussions on ANTs are confined to the SU(2) case. We generalize them to the U(N) case and demonstrate explicitly at the one-loop level that despite the presence of divergent ANTs in the effective action of the "pions", the symmetry is preserved.