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

We consider the $O(N+1)/O(N)$ nonlinear 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 dynamically generates 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 exactly satisfies 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 behavior 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.

Figure 1

Feynman rules and diagrams at tree level, where dashed lines correspond to multiple pairs of pion lines.

Figure 2

Zero-temperature scattering amplitude.

Figure 3

Construction of the effective thermal tadpole vertex, where dashed lines correspond to multiple tadpole insertions.

Figure 4

Finite-temperature scattering amplitude.

Figure 5

Insertions of counterterm Lagrangian vertices for the renormalization of the scattering amplitude.

Figure 6

The $I=J=0$ channel phase shift for different fits, as explained in main text. The Peláez parametrization is given in [50], NA48 low-energy data in [51] and Grayer data in [49].

Figure 7

Surface levels for $|a_{00}^{II}(s;T){|}^2$ at $T=0$ with Peláez 1,2, Grayer and standard fit parameters, respectively. The elliptic regions show the positions of the pole in the upper half of the second Riemann sheet.

Figure 8

Mass of the $f_0(500)$ pole as functions of temperature when considering different fits in the large-$N$ framework. We also compare with the IAM approach in the chiral limit.

Figure 9

Width of the $f_0(500)$ pole as functions of temperature when considering different fits and the IAM in the chiral limit.

Figure 10

Normalized scalar mass squared (inverse scalar susceptibility) as a function of the temperature for different parameter sets and the IAM.

Figure 11

Diagrams up to $\mathcal{O}(s^3)$ for the renormalized amplitude at $T=0$. We plot the different topological configurations contributing, so that diagram (e) is multiplied by two, corresponding to the possible vertex insertions of $g_1$.

Figure 12

Diagrams up to $\mathcal{O}(s^3)$ for the renormalized amplitude at $T\ne 0$. We plot the different topological configurations contributing, so diagrams (e), (h), (j), (k), (l), (p) are multiplied by two, corresponding to the possible vertex insertions.