Isentropic thermodynamics and scalar-mesons properties near the QCD critical end point

We investigate the QCD phase diagram and the location of the critical end point (CEP) in the SU(2) Polyakov$-$Nambu$-$Jona-Lasinio model with entanglement interaction giving special attention to the $\pi$ and $\sigma$-mesons properties, namely the decay widths $\sigma\rightarrow\pi\pi$, for several conditions around the CEP: we focus on the possible $\sigma\rightarrow\pi\pi$ decay along the isentropic trajectories close to the CEP since the hydrodynamical expansion of a heavy-ion collision fireball nearly follows trajectories of constant entropy. It is expected that the type of transition the dense medium goes through as it expands after the thermalization determines the behavior of this decay. It is shown that no pions are produced from the sigma decay in the chirally symmetric phase if the isentropic lines approach the first order line from chemical potentials above it. Near the CEP or above the $\sigma\rightarrow\pi\pi$ decay is possible with a high decay width.