Net quark number probability distribution near the chiral crossover transition

We investigate properties of the probability distribution of the net quark number near the chiral crossover transition in the quark-meson model. The calculations are performed within the functional renormalization group approach, as well as in the mean-field approximation. We find that there is a substantial influence of the underlying chiral phase transition on the properties of the probability distribution. In particular, for a physical pion mass, the distribution which includes the effect of mesonic fluctuations, differs considerably from both, the mean-field and Skellam distributions. The latter is considered as a reference for a noncritical behavior. A characteristic feature of the net quark number probability distribution is that, in the vicinity of the chiral crossover transition in the $O\left(4\right)$ universality class, it is narrower than the corresponding mean-field and Skellam function. We study the volume dependence of the probability distribution, as well as the resulting cumulants, and discuss their approximate scaling properties.

Figure 1

Probability distributions of the net quark number just below the pseudocritical temperature $T_{pc}$ (a) in the FRG approach and (b) in the mean-field (MF) approximation, compared with the corresponding Skellam distributions. The dots are the model results for different volumes while the lines show the Skellam distribution.

Figure 2

Probability distribution for different temperatures (a) in the FRG approach and (b) in the mean-field (MF) approximation. The dots are the model results while the lines show the corresponding Skellam distribution (see text).

Figure 3

The probability distribution obtained within the FRG approach (red, circles), the Skellam distribution (blue, dashed), and that of a free Fermi gas (green, solid). All distributions correspond to the same average quark and antiquark numbers, $b$ and $\overline{b}$, given by Eq. (14).

Figure 4

The ratios of the cumulants $c_n$ $(n=2,4,6)$ of the net charge obtained from the Skellam distribution with the mean multiplicities $b=\overline{b}=6.0$ using Eqs. (17, 18, 19, 20) for a given value of $N_{\max }$ to their corresponding exact values.

Figure 5

Cumulants $c_n$ $(n=2,4,6)$ obtained in the quark-meson model within the FRG approach using Eq. (16) (full line) and Eqs. (17, 18, 19, 20) (points) for several values of $N_{\max }$. Results for the corresponding Skellam distribution are also indicated by the lines with open circles.

Figure 6

Relative difference between cumulants $c_n$$(n=2,4,6)$ calculated in the quark-meson model within the FRG approach using Eq. (16) and Eqs. (17, 18, 19, 20) for two temperatures and for several values of the volume parameters $V$ (see text).

Figure 7

Volume scaling of the net quark number probability distribution at fixed $T/T_{pc}$ in the quark-meson model within the MF, the FRG approach, and for the corresponding Skellam distribution.