How sharp is the chiral crossover phenomenon for realistic meson masses?

The mass dependence of the chiral phase transition is studied in the linear SU(3)×SU(3) $\sigma$ model to leading order in a $\frac{1}{N_f}$-expansion, $N_f$ denoting the number of flavors. For realistic meson masses we find a smooth crossover between $T\sim 181.5 \mathrm{and} 192.6$ MeV. The crossover looks more rapid in the light quark condensate than in thermodynamic quantities such as the energy and entropy densities. The change in the light quark condensate in this temperature interval is ∼ 50% of the zero-temperature condensate value, while the entropy density increases by (5.5±0.8)×${10}^{-3\mathrm{}}$ Ge${\mathrm{V}}^3$. Since the numerical error is particularly large in this region, we cannot rule out a finite latent heat smaller than 0.2 GeV/${\mathrm{fm}}^3$. The chiral transition is washed out for an average pseudoscalar meson octet mass ≥ 203 MeV. This gives an upper bound on the first-order transition region in the meson mass parameter space. The corresponding ratio of critical to realistic light current quark masses $\frac{m_{u,d}^{\mathrm{crit}}}{m_{u,d}}$ is estimated as 0.26±0.08. This result is by an order of magnitude larger than the corresponding mean-field value. Therefore, the realistic quark or meson masses seem to lie less deeply in the crossover region than it is suggested by a mean-field calculation.