Free Energy of a Large-$N$ Pion Gas and Chiral Symmetry Restoration

We study thermal properties of a large-N massless pion gas using a low-energy QCD approach given by an $O(N+1)\mathrm{/}O(N)$ Nonlinear Sigma Model. We build diagrammatically the associated finite free energy to $O(T{M}^3)$ in the pion mass expansion through an effective vertex that considers all the contributions coming from closed diagrams. Subsequently, we calculate finite order parameters such as the quark condensate and its respective derivative, i.e., the scalar susceptibility, in the chiral limit, along with their associated critical exponents. These results are compared with our previous unitarized scattering analyses for the chiral transition universality class, thus showing a reasonable agreement both with lattice simulations and these resonant studies.