Parity and CT realization in QCD

TL;DR

This paper critically examines the assumptions behind Vafa and Witten's theorem on P and CT symmetry realization in QCD, highlighting that spontaneous symmetry breaking challenges the theorem's foundational premise.

Contribution

It demonstrates that the key assumption of a well-defined free energy density in the presence of symmetry breaking does not hold if the symmetry is spontaneously broken, questioning the theorem's applicability.

Findings

The free energy density assumption fails with spontaneous symmetry breaking.

Vafa and Witten's theorem lacks a formal proof under these conditions.

The existence of a free energy density is linked to symmetry realization in the vacuum.

Abstract

We show that an essential assumption in the Vafa and Witten's theorem on P and CT realization in vector-like theories, the existence of a free energy density in Euclidean space in the presence of any external hermitian symmetry breaking source, does not apply if the symmetry is spontaneously broken. The assumption that the free energy density is well defined requires the previous assumption that the symmetry is realized in the vacuum. Even if Vafa and Witten's conjecture is plausible, actually a theorem is still lacking.