Volume dependence of baryon number cumulants and their ratios

We explore the influence of finite-volume effects on cumulants of baryon/quark number fluctuations in a nonperturbative chiral model. In order to account for soft modes, we use the functional renormalization group in a finite volume, using a smooth regulator function in momentum space. We compare the results for a smooth regulator with those for a sharp (or Litim) regulator, and show that in a finite volume, the latter produces spurious artifacts. In a finite volume there are only apparent critical points, about which we compute the ratio of the fourth- to the second-order cumulant of quark number fluctuations. When the volume is sufficiently small the system has two apparent critical points; as the system size decreases, the location of the apparent critical point can move to higher temperature and lower chemical potential.

Figure 1

The flow of the chiral condensate, $\sigma$, in the chiral limit as a function of the scale $k$ for the exponential and Litim regulators. The scale dependent condensate is obtained by minimizing the potential at each RG scale. The system size was chosen to be $L=3\mathrm{fm}$ and $L=\infty$, at zero temperature and chemical potential. The chiral condensate is normalized by the physical value of $f_{\pi }$.

Figure 2

The chiral condensate, $\sigma$, as a function of the system size, $L$, at zero temperature and chemical potential for the exponential and Litim regulators.

Figure 3

The chiral condensate as a function of the system size, $L=L_x=L_y$ for different values of the anisotropy parameter $A\equiv L_z/L_{x,y}$. The results are normalized by the corresponding values of the chiral condensate for the isotropic volume, $L_x=L_y=L_z=L$. Periodic boundary conditions are used.

Figure 4

The location of the ACPs as a function of the system size, $L$. Due to the numerical difficulties we were not able to resolve ACPs at temperatures below 5 MeV. The red squares continuously approach the true critical point in the limit of infinite volume, which is already well approximated by $L=5\mathrm{fm}$.

Figure 5

The potential at $\sigma =f_{\pi }$ and $\sigma =0$ as a function of the system size for different chemical potentials.

Figure 6

The ratio of the fourth- to the second-order susceptibilities as a function of temperature for different system sizes and the anisotropy parameter $A$; the results are computed at zero chemical potential.

Figure 7

The ratio of the fourth- to the second-order susceptibilities as a function of temperature for different system sizes and the anisotropy parameter $A$; the results are computed at $\mu /T=0.5$.

Figure 8

The ratio of the fourth- to the second-order susceptibilities as a function of temperature for different system sizes and the anisotropy parameter $A$; the results are computed at $\mu /T=1$.

Figure 9

The ratio of the fourth- to the second-order susceptibilities as a function of temperature for different system sizes and the anisotropy parameter $A$; the results are computed at $\mu /T=1.5$.

Figure 10

The dependence of the minimum of the potential on the FRG flow parameter $k$ for different system sizes in the chiral limit $h=0$. The calculations are performed at $T=10\mathrm{MeV}$. For higher $L$, the nontrivial minimum of the potential is preserved by the FRG evolution to lower values of $k$, as expected.