Parity realization in Vector-like theories from Fermion Bilinears

TL;DR

This paper investigates whether discrete symmetries like Parity and CP are realized in the QCD vacuum from first principles, showing that under certain assumptions, all parity and CP odd fermion bilinear operators have zero vacuum expectation value in vector-like theories with multiple degenerate flavors.

Contribution

It demonstrates that, assuming conditions similar to Vafa and Witten's, parity and CP odd operators from fermion bilinears vanish in vector-like theories with multiple flavors, extending understanding of symmetry realization.

Findings

Parity and CP odd fermion bilinear operators have zero vacuum expectation value.

Vafa-Witten theorem supports the absence of spontaneous flavor symmetry breaking.

Results align with observed discrete symmetry realization in QCD.

Abstract

We reconsider in this paper the old aim of trying to understand if the observed realization of discrete symmetries as Parity or CP in the QCD vacuum can be satisfied from first principles. We show how under the appropriate assumptions implicitely done by Vafa and Witten in their old paper on parity realization in vector-like theories, all parity and CP odd operators constructed from fermion bilinears of the form $\bar\psi\tilde O\psi$ should take a vanishing vacuum expectation value in a vector-like theory with N degenerate flavours (N>1). In our analysis the Vafa-Witten theorem on the impossibility to break spontaneously the flavour symmetry in a vector-like theory plays a fundamental role.