Conformality or confinement (II): One-flavor CFTs and mixed-representation QCD

TL;DR

The paper develops a controlled, semiclassical framework using deformation theory and twisted partition functions on $S^1\times\mathbb{R}^3$ to distinguish conformal versus confining IR behavior in QCD-like gauge theories. It uncovers a new class of one-flavor CFTs where confinement and chiral symmetry breaking at finite $L$ are driven by topological disorder operators, yet the theory flows to a CFT as $L\to\infty$, with a rigorous bound showing the mass gap vanishes in four dimensions. For mixed-representation theories, it refines the conformal window by deriving mass-gap scalings and bounds across various representations, including adjoint and fundamental, and extends Banks–Zaks-type ideas to these settings. A key innovation is the identification of abelian chiral symmetry breaking via topological operators and its interplay with dual photons, which yields concrete, testable predictions for how conformality emerges in non-supersymmetric gauge theories. The results provide a coherent, semi-analytic route to map conformal windows and offer insights that can guide lattice studies and further analytic approaches toward understanding conformality in gauge theories.

Abstract

We study QCD-like four dimensional theories in the theoretically controlled framework of deformation theory and/or twisted partition function on S*1 x R*3. By using duality, we show that a class of one-flavor theories exhibit new physical phenomena: discrete chiral symmetry breaking induced by the condensation of topological disorder operators, and confinement and the generation of mass gap due to new non-selfdual topological excitations. In the R*4 limit, we argue that the mass gap disappears, the chiral symmetry breaking vacua are of runaway type, and the theory flows to a CFT. We also study mixed-representation theories and find abelian chiral symmetry breaking by topological operators charged under abelian chiral symmetries. These are reminiscent to, but distinct, from Seiberg-Witten theory with matter, where 4d monopoles have non-abelian chiral charge. This examination also helps us refine our recent bounds on the conformal window. In an Addendum, we also discuss mixed vectorlike/chiral representation theories, obtain bounds on their conformal windows, and compare with the all-order beta function results of arXiv:0911.0931.

Figures (2)

• Figure 1: Possible behavior of the mass gap for gauge fluctuations in asymptotically free, center-symmetric theories as a function of the size $L$ of ${S}^1$. The semiclassical analysis is valid at $L N \Lambda\ll 1$, where $N$ is the number of colors, and $\Lambda$ is either the strong scale, for confining theories, or the scale where the running coupling saturates its IR fixed-point value.
• Figure 2: Conformal window estimates for QCD(F/AS/Adj/S) by using deformation theory and the mass gap criterion of this paper (solid lines, upper limits on the lower boundary) and the truncated Schwinger-Dyson approximation (dashed-lines). In order to not overcrowd the figure, we do not plot the estimates of other approaches; see Table 1.