Recovering the chiral critical endpoint via delocalization of quark interactions

We show that for the lower branch of the quark condensate and values higher than approximately $-(250\text{MeV}{)}^3$ the chiral critical endpoint in the Nambu–Jona-Lasinio model does not occur in the phase diagram. By using lattice motivated nonlocal quark interactions, we demonstrate that the critical endpoint can be recovered. We study this behavior for a range of condensate values and find that the variation in the position of the critical endpoint is more pronounced as the condensate is increased.

Figure 1

We show the condensate as a function $\alpha$ for different reduced couplings $g$. Dots mark the minimal value separating the lower and the higher branch, see text.

Figure 2

We display contours of physical values of coupling $g$ as functions of $\alpha$, indicating the particular values of the condensate used. Note that every point on these curves represents a particular parametrization of the instantaneous NJL model. The variation of the critical couplings (dashed, black line) for chiral symmetry breaking, and for first order transition (full, light green line), $g_c$ and ${\overline{g}}_c$, respectively, is also shown.

Figure 3

The figure shows several chiral transition curves in the limit $m=0$ for $⟨\overline{q}q⟩=-(265.573\mathrm{MeV}{)}^3$. We use the parameter sets from Table 2 where we put $m=0$ by hand. The dashed (full) lines are the second (first) order phase transition. The case $\alpha =50$, where the CEP is located at $T=0$, is effectively the local NJL limit.

Figure 4

Each curve denotes the position of the CEP as a function of $\alpha$, for a particular value of the $⟨\overline{q}q⟩$. We use the same values of $⟨\overline{q}q⟩$ and the same line styles as defined in Fig. 2. The values of $⟨\overline{q}q⟩$ are decreased in magnitude as we proceed from the left-most to the right-most curve. The upper dots indicate the value of the CEP for minimal values of the condensate, see Table 1. The parameter $\alpha$ is varied continuously. The lower dots indicate the last integer value of $\alpha$ where the CEP occurs in the phase diagram.