Phase transitions and theta vacuum energy

The analytic behaviour of θ-vacuum energy is related to the existence of phase transitions in QCD and ℂP N sigma models. The absence of singularities different from Lee-Yang zeros only permits ∧ cusp singularities in the vacuum energy density and never ∨ 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 ℂP N sigma models. However, this property does not exclude the existence of a first phase transition at θ = π or a second order phase transition at θ = 0, which might be very relevant for interpretation of the anomalous behaviour of the topological susceptibility in the ℂP1 sigma model.