η ′ Mass and Chiral Symmetry Breaking at Large

We propose a method for implementing the large-Nc, large-Nf limit of QCD at the effective Lagrangian level. Depending on the value of the ratio Nf/Nc, different patterns of chiral symmetry breaking can arise, leading in particular to different behaviors of the η′-mass in the combined large-N limit. Typeset using REVTEX 1 1. Large-Nc considerations successfully explain many non-perturbative aspects of confining gauge theories [1,2]. However, there are at least two exceptions -both related to a strong OZI rule violationin which the 1/Nc expansion apparently fails: (i) In the scalar channel the spectrum is not dominated by a nonet of ideally mixed states and chiral symmetry breaking exhibits an important dependence on the number Nf of light quark flavors [3]; (ii) At large Nc, the η -field becomes massless due to its relation to the U(1) anomaly while Nature realizes it like a heavy state. In this note we reconsider these problems in the limit in which both Nf and Nc tend to infinity with fixed ratio [4]. Since, at least at lowest orders of perturbation theory, the (rescaled) QCD β-function (with gNc ≡ const.) only depends on the ratio Nf/Nc, we might expect that the hadronic spectrum resembles the physical one, in particular with chiral symmetry breakdown and ΛH ∼ 1 GeV. On the other hand, several hints (e.g. from the study of the conformal window, in QCD and its supersymmetric version), suggest a non trivial phase structure of the theory as a function of Nf and Nc. One can in principle distinguish three different phases, characterized by different symmetries of the vacuum, depending on the ratio Nf/Nc: (a) for low Nf/Nc only the SUV (Nf ) remains unbroken; (b) for higher Nf/Nc the vacuum is invariant under a larger group, SUV (Nf) × Zchiral(Nf ), where Zchiral(Nf ) is the center of the chiral symmetry group SUL(Nf )×SUR(Nf) [5,6]; (c) for high Nf/Nc no spontaneous symmetry breaking takes place (and hence no confinement) and the symmetry of the vacuum is the whole SUL(Nf)×SUR(Nf). Notice that case (b) corresponds to the maximal possible symmetry of the vacuum in a confining vector-like theory. The existence of this phase is an assumption related to the issue of the non-perturbative renormalization of the bare Weingarten’s inequalities comparing axial-axial and pseudoscalar-pseudoscalar two point functions [6,7]. We model the combined large-Nf , large-Nc limit by adding to the usual light flavors q = (u, d, s), a set of N auxiliary flavors Q = (Q1, . . . , QN) of common mass M ≫ mq, but still M ≪ ΛH . This mass M should be considered sufficiently small so that a power series expansion makes sense, but simultaneously much larger than any of the light quark masses mq, thus the auxiliary fields can be integrated out at sufficiently low-energy. We then formally deal with Nf = N +3 ≡ n → ∞ 2 flavors, but only the three lightest ones are physical. The rôle of these auxiliary flavors should be analogous to the one of the strange quark, when one considers the SU(2)×SU(2) chiral dynamics of u and d quarks. 2. Let Nf/Nc be subcritical, so that we are in the Zchiral(n)-asymmetric phase. The n 2−1 (pseudo) Goldstone bosons (GB) can be collected in a matrix Û(x) ∈ SU(n), (hereafter n×n matrices will be denoted by a hat) and their low-energy dynamics can be described by the effective Lagrangian Lsub = F 2 4 { 〈DμÛDÛ 〉+ 2B0〈Û χ̂+ χ̂Û〉 } . (1) where χ̂ is the scalar-pseudoscalar source, L χ̂ = −Ψ̄Lχ̂ΨR − Ψ̄Rχ̂ΨL, Ψ = 