Amending the Vafa-Witten theorem

The strong version of the Vafa-Witten theorem is shown to likely not hold true because a direct computation of the order parameter has the zero condensate as a result for the symmetric vacuum. Thus, the validity of the Vafa-Witten theorem relies on its weak version, in which the Goldstone boson is absent in the vector-like gauge theories that have a vanishing θ angle. The existence of a charged ρ meson condensate, which violates electromagnetic gauge symmetry, is consistent with this weak version of the Vafa-Witten theorem when it is applied to strong magnetic fields in QCD.