Scaling violation and the magnetic equation of state in chiral models

The scaling behavior of the order parameter at the chiral phase transition, the so-called magnetic equation of state, of strongly interacting matter is studied within effective models. We explore universal and nonuniversal structures near the critical point. These include the scaling functions, the leading corrections to scaling, and the corresponding size of the scaling window as well as their dependence on an external symmetry breaking field. We consider two models in the mean-field approximation, the quark-meson and the Polyakov loop extended quark-meson (PQM) models, and compare their critical properties with a purely bosonic theory, the $O(N)$ linear sigma model in the $N\to \infty$ limit. In these models the order parameter scaling function is found analytically using the high temperature expansion of the thermodynamic potential. The effects of a gluonic background on the nonuniversal scaling parameters are studied within the PQM model.

Figure 1

The only contribution to ${\mathrm{\Gamma }}_2$ in the $1/N$ expansion of the $O(N)$ sigma model for $N\to \infty$. The dashed lines are pion propagators, whereas the filled dot depicts the four-point vertex, with coupling strength $\lambda /N$.

Figure 2

The scaling function of the order parameter in mean-field models, the $O(N)$ linear sigma model in the $N\to \infty$ limit, and in the $O(4)$ universality class.

Figure 3

The magnetic equation of state for the QM (left) and PQM (right) models. The black line corresponds to the universal scaling curve, which coincides in these two models. In both models the sigma mass was fixed to $m_{\sigma }=400\mathrm{MeV}$, whereas the constituent quark mass in the vacuum was set to $m_q=300\mathrm{MeV}$.

Figure 4

The magnetic equation of state for the $O(N)$ linear sigma model in the $N\to \infty$ limit. The black line corresponds to the universal scaling curve of the $O(N=\infty )$ universality class. The end points of the curves at negative $z$ values correspond to $T=0$. In this calculation, $m_{\sigma }=400\mathrm{MeV}$.

Figure 5

Left: The universal scaling part of the chiral susceptibility in the QM model calculated within the mean-field dynamics and in the $O(N)$ LS model obtained in the $N\to \infty$ limit within the high-temperature expansion. Right: The chiral susceptibilities in the LS model in the $N\to \infty$ limit and in the QM model under the mean-field approximation calculated at physical and at ten times lower pion mass.

Figure 6

The chiral order parameters in the QM and LS models calculated with different inputs for the vacuum sigma mass. The data points are lattice results from Ref. [60]. The pseudocritical temperature $T_{pc}$ for these data is taken as the inflection point of the order parameter.