Anomalies and Dynamics in Strongly-coupled Gauge Theories. New Criteria for Different Phases and Lessons from Supersymmetric Theories

We review recent developments in our understanding of the dynamics of strongly-coupled chiral 𝑆𝑈(𝑁) gauge theories in four dimensions, problems which are potentially important in our quest to go beyond the standard 𝑆𝑈(3)𝑄𝐶𝐷 × (𝑆𝑈(2) ×𝑈(1))𝐺𝑊 𝑆 model of the fundamental interactions. The generalized symmetries and associated new ’t Hooft anomaly-matching constraints allow us to exclude, in a wide class of chiral gauge theories, confining vacuum with full flavor symmetries supported by a set of color-singlet massless composite fermions. The color-flavor-locked dynamical Higgs phase, dynamical Abelianization or more general symmetry breaking phase, appear as plausible IR dynamics, depending on the massless matter fermions present. We revisit and discuss critically several well-known confinement criteria in the literature, for both chiral and vectorlike gauge theories, and propose tentative, new criteria for discriminating different phases. Finally, we review an idea which might sound rather surprising at first, but is indeed realized in some softly-broken supersymmetric theories, that confinement in QCD is a small deformation (in the IR end of the renormalization-group flow) of a strongly-coupled, nonlocal, nonAbelian conformal fixed point.