Critical endpoint of QCD in a finite volume

We investigate the impact of finite volume and the corresponding restrictions on long-range correlations on the location of the critical endpoint in the QCD phase diagram. To this end, we employ a sophisticated combination of lattice Yang–Mills theory and a (truncated) version of Dyson–Schwinger equations in Landau gauge for $2+1$ quark flavors that has been studied extensively in the past. In the infinite-volume limit, this system predicts a critical endpoint at moderate temperature and large chemical potential. We study this system at small and intermediate volumes and determine the dependence of the location of the critical endpoint on the boundary conditions and the volume of a three-dimensional cube with edge length $L$. We demonstrate that noticeable volume effects of more than five percent occur only for $L\lesssim 5\mathrm{fm}$ and that volumes as large as $L^3\gtrsim (8\mathrm{fm}{)}^3$ are very close to the infinite-volume limit.

Figure 1

Two-dimensional ABC momentum grid. Red, solid circles represent spheres that are fully taken into account, while the blue, dashed one is excluded in our summations (see text for details).

Figure 2

Finite-volume effects without (left) and with (right) UV improvement. We display the dependence of the pseudocritical chiral transition temperature on the box size for ABC (blue circles), ${\mathrm{PBC}}^{*}$ (red squares), and PBC (green diamonds). The black, dotted line is the infinite-volume result. Data points are connected by lines to guide the eye.

Figure 3

Finite-volume effects without (left) and with (right) UV improvement. Here, we show the crossover lines and locations of the CEPs (symbols) in the QCD phase diagram for different box sizes. The phase diagrams are obtained with ABC (upper row), ${\mathrm{PBC}}^{*}$ (center row), and PBC (lower row).

Figure 4

UV-improved finite volume: $L$ dependence of the curvature of the crossover line for ABC (blue circles), ${\mathrm{PBC}}^{*}$ (red squares), and PBC (green diamonds). Data points are connected by lines to guide the eye.

Figure 5

Truncations without quark backcoupling onto the gluon in vacuum: $L$ dependence of the subtracted quark condensate normalized to its infinite-volume value with a Ball–Chiu-inspired (BC) and a bare vertex. Data points are connected by lines to guide the eye.