# Thermo-magnetic corrections to $\pi$-$\pi$ Scattering Lengths in the Linear Sigma Model

## TL;DR

This paper investigates how finite temperature and magnetic fields jointly influence pion-pion scattering lengths within the linear sigma model, revealing opposite effects of temperature and magnetic fields on these scattering parameters.

## Contribution

It introduces finite temperature effects into the analysis of pion-pion scattering lengths under magnetic fields within the linear sigma model, extending previous zero-temperature studies.

## Key findings

- Magnetic fields slightly increase scattering lengths.
- Finite temperature slightly decreases scattering lengths.
- Temperature and magnetic field effects oppose each other.

## Abstract

In this article, we extend our previous study of the $\pi$-$\pi$ scattering lengths under the presence of an external magnetic field, including finite temperature effects. The novelty of this work is precisely the introduction of temperature into the discussion, and its interplay with the magnetic field. As in the previous article, we base our analysis in the linear sigma-model, and our calculations are exact within this context. Although the effects are comparatively small, it is interesting to remark that magnetic field and temperature display opposite effects over the scattering lengths.