The role of the thermal $f_0(500)$ in chiral symmetry restoration

Abstract: We show that the $\sigma/f_0(500)$ state with finite-temperature $T$ corrections to its spectral properties included, plays an essential role for the description of the scalar susceptibility $\chi_S$, signaling chiral symmetry restoration. First, we use the $O(4)$ Linear Sigma Model as a testbed to derive the connection between $\chi_S$ and the $\sigma$ propagator and to check the validity and reliability of the approach where $\chi_S$ is saturated by the $\sigma/f_0(500)$ inverse self-energy, which we calculate at finite $T$ to one loop. A more accurate phenomenological description is achieved by considering the saturation approach as given by the thermal $f_0(500)$ state generated in Unitarized Chiral Perturbation Theory. Such approach allows to describe fairly well recent lattice data within the uncertainty range given by the UChPT parameters. Finally, we compare the UChPT saturated description with one based on the Hadron Resonance Gas, for which the hadron mass dependences are extracted from recent theoretical analysis. Several fits to lattice data are performed, which confirm the validity of the thermal $f_0(500)$ saturated approach and hence the importance of that thermal state for chiral symmetry restoration.