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

Parity realization in Vector-like theories from Fermion Bilinears