Parity and CT realization in QCD

We show that an essential assumption in the Vafa-Witten theorem on P and CT realization in vectorlike theories, the existence of a free energy density in Euclidean space in the presence of any external Hermitian symmetry brea king source, does not apply if the symmetry is spontaneously broken. The as sumption that the free energy density is well defined requires the previous assumption that the symmetry is realized in the vacuum. Even if Vafa and W itten's conjecture is plausible, actually a theorem is still lacking.