Effect of fluctuations on the QCD critical point in a finite volume

We investigate the effect of a finite volume on the critical behavior of the theory of the strong interaction, Quantum Chromodynamics (QCD), by means of a quark-meson model for $N_{\mathrm{f}}=2$ quark flavors. In particular, we analyze the effect of a finite volume on the location of the critical point in the phase diagram existing in our model. In our analysis, we take into account the effect of long-range fluctuations with the aid of renormalization group techniques. We find that these quantum and thermal fluctuations, absent in mean-field studies, play an import role for the dynamics in a finite volume. We show that the critical point is shifted towards smaller temperatures and larger values of the quark chemical potential if the volume size is decreased. This behavior persists for antiperiodic as well as periodic boundary conditions for the quark fields as used in many lattice QCD simulations.

Figure 1

Chiral (longitudinal) susceptibilities ${\chi }_{\sigma }=\frac{1}{M_{\sigma }^2}$ as a function of the quark chemical potential $\mu$ for various temperatures and volume sizes ranging from $L\to \infty$ from the top left panel to $L=4\mathrm{fm}$ in the bottom right panel as obtained for periodic boundary conditions for the quark fields in spatial directions.

Figure 2

Contour plot of the magnitude of the chiral order parameter ${\sigma }_0=f_{\pi }$ in the plane spanned by temperature and quark chemical potential for $L$ towards infinity: The order parameter decreases from dark to light shading. The inlaid (red) dots show the behavior of the CEP [associated with a Z(2) symmetry] as a function of the volume size for periodic boundary conditions for the quark fields in spatial directions. We observe that the CEP is shifted towards larger values of the chemical potential but smaller values of the temperature for smaller volumes.