CP Symmetry, Lee-Yang zeros and Phase Transitions

We analyze the analytic properties of θ‐vacuum in QCD and its connection with spontaneous symmetry breaking of CP symmetry. A loss of analyticity in the θ‐vacuum energy density can only be due to the accumulation of Lee‐Yang zeros at some real values of θ. In the case of first order transitions these singularities are always associated to ∧ cusp singularities and never to ∨ cusps, which in the case θ = 0 are incompatible with the Vafa‐Witten diamagnetic inequality This fact provides a key missing link in the Vafa‐Witten proof of parity symmetry conservation in vector‐like gauge theories like QCD. The argument is very similar to that used in the derivation of Bank‐Casher formula for chiral symmetry breaking. However, the ∧ behavior does not exclude the existence of a first phase transition at θ = π, where a ∧ cusp singularity is not forbidden by any inequality; in this case the topological charge condensate is proportional to the density of Lee‐Yang zeros at θ = π. Moreover, Lee‐Yang zeros could give rise ...