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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., Comment: 6 pages

We show that the electromagnetic superconductivity of vacuum in strong magnetic field background is consistent with the Vafa-Witten theorem because the charged vector meson condensates lock relevant internal global symmetries of QCD with the electromagnetic gauge group., 2 pages

We revisit the proof provided by Vafa and Witten that there is no spontaneous parity breaking in theories with vector-like fermions. We argue that the various criticisms that have been leveled at the original proof do not invalidate it.

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., Minor typos corrected

We examine the Dyson-Schwinger equation for the quark propagator in quantum chromodynamics by employing the gauge technique with the object of investigating spontaneously broken flavor symmetry in the chiral limit of vanishing current quark masses. We show that dynamical breakdown of flavorSU V (N F ) symmetry does not occur, thus reiterating an assertion by Vafa and Witten. Our analysis is limited to three flavors but is generalizable to an arbitrary number of flavors,N F>3.