Vafa-Witten theorem and Lee-Yang singularities

TL;DR

This paper proves that the QCD partition function's analyticity and Lee-Yang zeros behavior prevent first-order phase transitions at theta=0, supporting parity symmetry conservation in vector-like gauge theories.

Contribution

It establishes the analyticity of the finite volume QCD partition function for complex theta and links Lee-Yang zeros to the Vafa-Witten theorem, extending understanding of phase transitions.

Findings

No Lee-Yang zeros at theta=0 imply no first-order phase transition.

Vacuum energy density exhibits cusp singularities only, not v cusps.

Supports parity symmetry conservation in vector-like gauge theories.

Abstract

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 ^ cusp singularities in the vacuum energy density and never v 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 vector-like gauge theories and follows from renormalizability, unitarity, positivity and existence of BPS bounds. Generalizations of this theorem to other physical systems are also discussed, with particular interest focused on the non-linear CPn sigma model.