# Chiral effective theories with a light scalar at one loop

## TL;DR

This paper develops a one-loop renormalization framework for chiral effective theories with a light scalar, providing tools to analyze theories where a scalar is nearly degenerate with Goldstone bosons, and discusses applications to chiral-scale perturbation theory.

## Contribution

It introduces a one-loop renormalization method for chiral theories with a light scalar, including a master formula applicable to chiral-scale perturbation theory.

## Key findings

- Derived the effective action at one loop using heat kernel expansion.
- Provided a renormalization scheme for theories with a light scalar in the spectrum.
- Discussed potential lattice simulation strategies to distinguish effective theories.

## Abstract

There are indications that some theories with spontaneous symmetry breaking also feature a light scalar in their spectrum, with a mass comparable to the one of the Goldstone modes. In this paper, we perform the one-loop renormalization of a theory of Goldstone modes invariant under a chiral $SU(n)\times SU(n)$ symmetry group coupled to a generic scalar singlet. We employ the background field method, together with the heat kernel expansion, to get an expression for the effective action at one loop and single out the anomalous dimensions, which can be read off from the second Seeley-DeWitt coefficient. As a relevant application, we use our master formula to renormalize chiral-scale perturbation theory, an alternative to $SU(3)$ chiral perturbation theory where the $f_0(500)$ meson is interpreted as a dilaton. Based on our results, we briefly discuss strategies to test and discern both effective field theories using lattice simulations.