Phase diagram of chiral 2-flavor QCD based on an effective approach

Despite intense experimental and theoretical research, the QCD phase diagram at finite baryon density remains to a large extent unexplored. From the theoretical side, the obvious non-perturbative approach is lattice QCD simulations, which are, however, obstructed by a severe sign problem. Here we employ the O(4) non-linear $\sigma$-model as an effective theory for 2-flavor QCD in the chiral limit. The identical pattern of spontaneous symmetry breaking indicates that they belong to the same universality class. We assume high temperature dimensional reduction to the 3d O(4) model, with topological charge taking the role of the baryon number, along the lines of Skyrme's model. In this effective formulation, the baryonic chemical potential $\mu_{B}$ can be included in the lattice formulation without causing any sign problem in Monte Carlo simulations. This allows us to pin down the critical line, i.e. the critical temperature $T_{\rm c}(\mu_{B})$, which decreases monotonically for increasing $\mu_{B}$. In the range $0 < \mu_{B} \lesssim 309~{\rm MeV}$ and $132~{\rm MeV} \gtrsim T_{\rm c} \gtrsim 106~{\rm MeV}$, we do not find a Critical Endpoint (CEP), although there are hints for it to be in the vicinity of the maximal $\mu_{B}$-value that we could explore.