The QCD phase diagram is one of the most prominent outstanding puzzles within the Standard Model. Various experiments, which aim at its exploration beyond small baryon density, are operating or in preparation. From the theoretical side, this is an issue of non-perturbative QCD, and therefore of lattice simulations. However, a finite baryon density entails a technical problem (known as the “sign problem”), which has not been overcome so far. Here we present a study of an effective theory, the O(4) non-linear sigma model. It performs spontaneous symmetry breaking with the same Lie group structure as 2-flavor QCD in the chiral limit, which strongly suggests that they belong to the same universality class. Since we are interested in high temperature, we further assume dimensional reduction to the 3d O(4) model, which implies topological sectors. As pointed out by Skyrme, Wilczek and others, its topological charge takes the role of the baryon number. Hence the baryon chemical potential µB appears as an imaginary vacuum angle, which can be included in the lattice simulation without any sign problem. We present numerical results for the critical line in the chiral limit, and for the crossover in the presence of light quark masses. Their shapes are compatible with other predictions, but up to the value of about µB ≈ 300 MeV we do not find the notorious Critical Endpoint (CEP).

We present a phase diagram study of the O(4) model as an effective theory for 2-flavor QCD. In the chiral limit, both theories perform spontaneous symmetry breaking with isomorphic groups, which suggests that they belong to the same universality class. Since we are interested in high temperature, we further assume dimensional reduction to the 3d O(4) model, which implies topological sectors. According to Skyrme and others, the corresponding topological charge represents the baryon number. Hence the baryon chemical potential 𝜇 𝐵 appears as an imaginary vacuum angle, which can be included in the lattice simulations without any sign problem. We present simulation results for the critical line in the chiral limit, and for the crossover in the presence of light quark masses. The shapes of these lines are compatible with other conjectures, but up to about 𝜇 𝐵 ≈ 300 MeV we do not find a Critical Endpoint, although there are indications for it to be near-by.

We present a phase diagram study of the O(4) model as an effective theory for 2-flavor QCD. In the chiral limit, both theories perform spontaneous symmetry breaking with isomorphic groups, which suggests that they belong to the same universality class. Since we are interested in high temperature, we further assume dimensional reduction to the 3d O(4) model, which implies topological sectors. According to Skyrme and others, the corresponding topological charge represents the baryon number. Hence the baryon chemical potential µ B appears as an imaginary vacuum angle, which can be included in the lattice simulations without any sign problem. We present simulation results for the critical line in the chiral limit, and for the crossover in the presence of light quark masses. The shapes of these lines are compatible with other conjectures, but up to about µ B ≈ 300 MeV we do not find a Critical Endpoint, although there are indications for it to be near-by.