Thermal Corrections to $\pi$-$\pi$ Scattering Lengths in the Linear Sigma Model

TL;DR

This paper investigates how the s-wave pion-pion scattering lengths in the linear sigma model change with temperature, revealing that one grows while the other diminishes as temperature increases.

Contribution

It provides a detailed calculation of thermal corrections to pion-pion scattering lengths using thermofield dynamics within the linear sigma model, focusing on temperature effects in the pion sector.

Findings

The isospin 0 scattering length $a_0^0$ increases with temperature.

The isospin 2 scattering length $a_0^2$ decreases with temperature.

Results are compared with other thermal behavior studies.

Abstract

In this article we explore the thermal evolution of the $\pi$-$\pi$ scattering lengths in the frame of the linear sigma model. We compute, using thermofield dynamics, all the one loop diagrams that contribute to $\pi$-$\pi$ scattering. We restrict ourselves to temperature correction only in the pion sector, due to the Boltzmann supression for the sigma meson. From this analysis we obtain the thermal behavior of $a_0^0$ and $a_0^2$, s-waves scattering lengths in the isospin channel I=0 and I=2 respectively. It turns out that $a_0^0$ grows whereas $a_0^2$ diminishes as function of temperature. We compare our results with other determinations of the thermal behavior of these magnitudes.