Chiral symmetry breaking without bilinear condensates, unbroken axialZNsymmetry, and exact QCD inequalities

An alternative pattern of chiral symmetry breaking, suggested recently, is investigated. It could be self-consistent provided that the chiral $\mathrm{SU}{(N}_f)\times \mathrm{SU}{(N}_f)$ symmetry is broken spontaneously down to $\mathrm{SU}{(N}_f)\times Z_{N_f}$ rather than to $\mathrm{SU}{(N}_f{)}_V.$ The discrete axial $Z_{N_f}$ then would play a custodial role, preventing the quark bilinears from condensation. It is shown that this pattern of chiral symmetry breaking is ruled out in QCD by exact inequalities. It is not ruled out, however, in other gauge theories with scalar quarks and/or Yukawa couplings.