Vector-like symmetries and parity conservation in gauge theories with Yukawa couplings

Non-perturbative results in QCD-like theories can be derived employing positivity of the Euclidean path integral measure, as pioneered by Weingarten, Vafa, and Witten. We show that positivity of the measure can be generalized to parity-invariant theories with Yukawa couplings to fundamental scalars, provided the fermions are Dirac and carry a real representation of the gauge group. This result allows us to demonstrate the conservation of parity and vector-like flavor symmetries in such theories, as well as to derive exact inequalities among hadrons’ masses.