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

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