Large-$N$ pion scattering at finite temperature: the $f_0(500)$ and chiral restoration

TL;DR

This paper studies pion scattering at finite temperature within the large-N Non-Linear Sigma Model, demonstrating thermal unitarity, analyzing the $f_0(500)$ pole's temperature dependence, and linking it to chiral symmetry restoration.

Contribution

It provides a formal proof of thermal unitarity, updates zero-temperature scattering data analysis, and explores the $f_0(500)$ pole behavior at finite temperature in relation to chiral symmetry restoration.

Findings

Thermal unitarity is exactly satisfied in the model.

The $f_0(500)$ pole moves towards chiral restoration temperature.

The model's amplitude depends on few parameters after renormalization.

Abstract

We consider the $O(N+1)/O(N)$ Non-Linear Sigma Model for large $N$ as an effective theory for low-energy QCD at finite temperature $T$, in the chiral limit. At $T=0$ this formulation provides a good description of scattering data in the scalar channel and generates dynamically the $f_0(500)$ pole, the pole position lying within experimental determinations. Previous $T=0$ results with this model are updated using newer analysis of pion scattering data. We calculate the pion scattering amplitude at finite $T$ and show that it satisfies exactly thermal unitarity, which had been assumed but not formally proven in previous works. We discuss the main differences with the $T=0$ result and we show that one can define a proper renormalization scheme with $T=0$ counterterms such that the renormalized amplitude can be chosen to depend only on a few parameters. Next, we analyze the behaviour of the $f_0(500)$ pole at finite $T$, which is consistent with chiral symmetry restoration when the scalar susceptibility is saturated by the $f_0(500)$ state, in a second-order transition scenario and in accordance with lattice and theoretical analysis.