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 $\wedge$ cusp singularities in the vacuum energy density and never $\vee$ cusps. This fact together with the Vafa-Witten diamagnetic inequality implies the vanishing of the density of Lee-Yang zeros at $\vartheta =0$ 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 ${\mathbb{C}\mathbb{P}}^N$ sigma model.