Abstract
We discuss consequences of the ’t Hooft anomaly matching condition for Quantum Chromodynamics (QCD) with massless fundamental quarks. We derive the new discrete ’t Hooft anomaly of massless QCD for generic numbers of color Nc and flavor Nf , and an exotic chiral-symmetry broken phase without quark-bilinear condensate is ruled out from possible QCD vacua. We show that the U(1)B baryon number symmetry is anomalously broken when the {left({mathrm{mathbb{Z}}}_{2{N}_{mathrm{f}}}right)}_{mathrm{A}} discrete axial symmetry and the flavor symmetry are gauged. In the ordinary chiral symmetry breaking, the Skyrmion current turns out to reproduce this ’t Hooft anomaly of massless QCD. In the exotic chiral symmetry breaking, however, the anomalous breaking of U(1)B does not take the correct form, and it is inconsistent with anomaly matching. This no-go theorem is based only on symmetries and anomalies, and thus has a wider range of applicability to the QCD phase diagram than the previous one obtained by QCD inequalities. Lastly, as another application, we check that duality of mathcal{N}=1 supersymmetric QCD with Nf ≥Nc + 1 satisfies the new anomaly matching.
Highlights
Background gauge fields andUV regularization of quark fieldsWe introduce the G-gauge field in order to detect the ’t Hooft anomaly
The original ’t Hooft anomaly matching is only sensitive to the infinitesimal part of G around identity, and we show that non-trivial topology of G provides a new anomaly matching condition
In order to properly gauge the symmetry, we have to introduce both one-form and two-form gauge fields in order to take into account the quotient structure of the symmetry group
Summary
We consider the four-dimensional gauge theory with the gauge group SU(Nc) coupled to Nf massless Dirac fermions in the fundamental representation (Nf -flavor massless QCD). A is the SU(Nc) gauge field (a† = −a), D(a) = d + a is the covariant derivative, Fc(a) = D(a) ∧ D(a) = da + a ∧ a is the SU(Nc) gauge field strength, ψ is the quark field realized as Nc ×Nf Grassmannian variables, ψ is the conjugate field of ψ, and trc represents the trace over color indices in the defining representation When it is evident, we write D = D(a) and Fc = Fc(a). 2.1 Symmetry group of massless Nf -flavor QCD The (internal) global symmetry of this theory is given by This is the correct global symmetry of massless QCD, in the sense that G has the faithful representation on the physical Hilbert space.. The vector rotation by e2πi/Nc must be regarded as the identity, and the faithful flavor symmetry has to be divided by ZNc. We obtain the physical symmetry group as
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