On absence of spontaneous parity breaking in QCD

We argue that the Vafa–Witten proof on absence of spontaneous parity breaking in QCD has the following two loopholes: first, when perturbed by an external field coupled to a parity-odd quark operator, the QCD vacuum energy could be a decreasing function of the field, contrary to the observation used in the proof. The difference comes from that the effective gluonic operator has in general a parity-even part which does not contribute to the path integral as a pure phase. Second, when perturbed by an external field coupled to a parity-odd gluonic operator, the vacuum energy is always a rising function of the field, independent of the existence of spontaneous parity breaking. This is because the parity-odd gluonic perturbations contribute to the path integral as a pure phase, and could not serve to select a physical vacuum should parity be spontaneously broken.