Vafa-Witten theorem and Lee-Yang singularities

We show the absence of Lee-Yang singularities in the partition function of QCD close to the CP symmetric point theta=0 in the complex theta plane. Analyticity of the theta vacuum energy density at theta=0 provides a key missing link in the Vafa-Witten proof of absence of spontaneous breaking of parity symmetry in vector-like gauge theories. The result follows from renormalizability, unitarity, positivity and existence of BPS bounds. Generalisations of this theorem to other physical systems are also discussed with particular interest focused on the non-linear CP^N sigma model.