  The vacuum energy response of a quantum field theory is studied as a function
of complex external fields coupled to the symmetry-breaking observables of the
theory. It is found that the energy density is insensitive to spontaneous
symmetry breaking effects along a special direction in the complex plane. In
particular, in the case of vector-like gauge theories such as QCD, the change
in energy generated by coupling to pure-gluonic parity-odd operators is always
a growing function of the external fields along the real axis, independent of
the mode in which symmetry is realized. In light of this, although there is
little doubt that parity is not spontaneously broken in massive QCD, the
validity of the Vafa and Witten proof is, however, questionable.
