Spontaneous parity violation in QCD at finite temperature: On the inapplicability of the Vafa-Witten theorem

The generalization of the Vafa-Witten theorem, ruling out parity violation to QCD at finite temperature is considered. It is shown that this generalization of the theorem rules out Lorentz-invariant parity violating operators from spontaneously acquiring vacuum expectation values. However, it does not rule out Lorentz-noninvariant parity-violating operators from acquiring expectation values. Other situations, where the theorem is inapplicable, are also discussed.