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

An alternative pattern of the chiral symmetry breaking, suggested recently, is investigated. It could be self-consistent provided that the chiral SU(Nf )×SU(Nf ) symmetry is broken spontaneously down to SU(Nf )× ZNf rather than to SU(Nf )V . The discrete axial ZNf 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. The lessons from supersymmetric gauge theories [1] teach us that gauge theories admit a broad spectrum of various dynamical regimes and, in particular, various patterns of the chiral symmetry breaking (CSB). Since QCD is not solved theoretically it is not unreasonable to search for viable alternatives to the standard CSB pattern ascending to the famous work of Gell-Mann, Oaks and Renner [2]. Recently an alternative regime has been suggested by Stern and collaborators who also performed an extensive analysis of possible phenomenological consequences [3]. The basic elements of this scenario are as follows. Consider QCD with Nf massless flavors. Then N f − 1 flavor-nonsinglet axial currents are coupled to N 2 f − 1 massless pions, 〈0|Aaμ|π 〉 = iqμ Fπ δ ab , (1) where Aaμ = ψ̄ λ 2 γμγ5ψ , λ are analogs of the Gell-Mann matrices acting in the flavor space (the color indices are suppressed), q is the pion momentum, and Fπ is the coupling constant, Fπ 6= 0. Correspondingly, the two-point current correlation function behaves in chiral limit as prescribed by the Goldstone theorem