Phase transitions and theta vacuum energy

The analytic behaviour of $\theta$-vacuum energy is related to the existence of phase transitions in QCD and $\mathbb C\mathrm P^N$ sigma models. The absence of singularities different from Lee–Yang zeros only permits $\wedge$ cusp singularities in the vacuum energy density and never $\vee$ cusps. This fact, together with the Vafa–Witten diamagnetic inequality, provides a key missing link in the Vafa–Witten proof of parity symmetry conservation in vector-like gauge theories and $\mathbb C\mathrm P^N$ sigma models. However, this property does not exclude the existence of a first phase transition at $\theta=\pi$ or a second order phase transition at $\theta=0$, which might be very relevant for interpretation of the anomalous behaviour of the topological susceptibility in the $\mathbb C\mathrm P^1$ sigma model.