Chiral Symmetry Breaking without Bilinear Condensates, Unbroken Axial Z_N Symmetry, and Exact QCD Inequalities

An alternative pattern of the chiral symmetry breaking, suggested recently by Stern, is investigated. It could be self-consistent provided that the chiral SU$(N_f)\times$SU$(N_f)$ symmetry is broken spontaneously down to SU$(N_f)\times Z_{N_f}$ rather than to 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 the 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.