Vafa-Witten theorem and Lee-Yang singularities

We prove the analyticity of the finite volume QCD partition function for complex values of the theta-vacuum parameter. The absence of singularities different from Lee-Yang zeros only permits boolean AND cusp singularities in the vacuum energy density and never boolean OR cusps. This fact together with the Vafa-Witten diamagnetic inequality implies the vanishing of the density of Lee-Yang zeros at Theta=0 and has an important consequence: the absence of a first order phase transition at theta=0. The result provides a key missing link in the Vafa-Witten proof of parity symmetry conservation in vectorlike gauge theories and follows from renormalizability, unitarity, positivity, and existence of Bogomol'nyi-Prasad-Sommerfield bounds. Generalizations of this theorem to other physical systems are also discussed, with particular interest focused on the nonlinear CPN sigma model.