Amending the Vafa-Witten Theorem

TL;DR

This paper challenges the strong Vafa-Witten theorem by showing the zero condensate result is valid only for the symmetric vacuum, emphasizing the importance of the weak version related to Goldstone bosons in gauge theories.

Contribution

It clarifies the conditions under which the Vafa-Witten theorem holds, especially highlighting the role of the weak version in the presence of strong magnetic fields in QCD.

Findings

Zero condensate is valid only for the symmetric vacuum.

Charged -meson condensate can exist without violating the weak Vafa-Witten theorem.

The theorem's validity depends on the absence of Goldstone bosons in certain conditions.

Abstract

The strong version of the Vafa-Witten theorem is shown may not to hold because the zero condensate from a direct computation of the order parameter is found to be a result on the symmetric vacuum. The validity of the Vafa-Witten theorem relies then on its weak version, that the Goldstone boson is absent in vector-like gauge theories with vanishing \theta-angle. The existence of a charged \rho-meson condensate, which violates electromagnetic gauge symmetry, is consistent with this weak version of the Vafa-Witten theorem when applied to strong magnetic fields in QCD.