Conformality or confinement: (IR)relevance of topological excitations

TL;DR

This work advances a nonperturbative diagnostic for confinement versus conformality in non-supersymmetric Yang–Mills theories by analyzing the gauge-fluctuation mass gap on ${\bf R}^3\times{\bf S}^1$ using twisted partition functions and deformation theory. By stabilizing center symmetry and identifying the relevant topological excitations (monopoles, magnetic bions, triplets, quintets), the authors derive semiclassical mass-gap expressions that depend on the radius $L$, matter content $N_f$, and representation. They show that, for small $N_f$, the mass gap grows with $L$ (favoring confinement), while for large $N_f$ it decreases (indicating IR-CFT), enabling estimates of the lower boundary of the conformal window across vectorlike and chiral theories; in some cases, these bounds align with ladder or NSVZ-inspired predictions, and for certain chiral quivers they provide unique estimates. The approach yields a unifying framework applicable to complex representations and chiral theories, complements lattice results, and, in select cases, yields solvable semiclassical limits at any ${\bf S}^1$ scale, thereby offering new insight into how confinement and conformality emerge from topological dynamics. This has potential implications for model-building in near-conformal dynamics and for understanding the nonperturbative structure of strongly coupled gauge theories.

Abstract

We study aspects of the conformality to confinement transition for non-supersymmetric Yang-Mills theories with fermions in arbitrary chiral or vectorlike representations. We use the presence or absence of mass gap for gauge fluctuations as an identifier of the infrared behavior. Present-day understanding does not allow the mass gap for gauge fluctuations to be computed on R*4. However, recent progress allows its non-perturbative computation on R*3xS*1 by using either the twisted partition function or deformation theory, for a range of S*1 sizes depending on the theory. For small number of fermions, Nf, we show that the mass gap increases with increasing radius, due to the non-dilution of monopoles and bions, the topological excitations relevant for confinement on R*3xS*1. For sufficiently large Nf, we show that the mass gap decreases with increasing radius. In a class of theories, we claim that the decompactification limit can be taken while remaining within the region of validity of semi-classical techniques, giving the first examples of semiclassically solvable Yang-Mills theories at any size S*1. For general non-supersymmetric vectorlike or chiral theories, we conjecture that the change in the behavior of the mass gap on R*3xS*1 as a function of the radius occurs near the lower boundary of the conformal window and give non-perturbative estimates of its value. For vectorlike theories, we compare our estimates of the conformal window with existing lattice results, truncations of the Schwinger-Dyson equations, NSVZ beta function-inspired estimates, and degree of freedom counting criteria. For multi-generation chiral gauge theories, to the best of our knowledge, our estimates of the conformal window are the only known ones.

Figures (3)

• Figure 1: Possible behavior of the mass gap for gauge fluctuations in asymptotically free, center-symmetric theories as a function of the radius of $\bf{S}^1$: a.)$N_f$ small: mass gap increases in the semiclassical domain of abelian confinement and saturates to its ${\bf R}^4$ value in the non-abelian confinement domain. b.)$N_f$ sufficiently large, perhaps just below $N_f^{AF}$: mass gap is a decreasing function of radius. There are theories in this class for which semiclassical analysis applies at any size ${\bf S}^1$. c.) Mass gap may start decreasing in the semiclassical domain, but may possibly saturate to a finite value on ${\bf R}^4$. This may happen, for example, if $\chi$SB takes place on the way. d.) Mass gap starts to increase in the semiclassical domain, but before reaching $\Lambda L \sim 1$, the coupling reaches a fixed point value without triggering $\chi$SB and the mass gap decreases to zero at larger scales. We will argue that a.) and b.) occur at small and large $N_f$, respectively, in all classes of theories we consider. We do not know whether c.) and d.) occur in any of the theories we consider and our semiclassical methods are of no help in this regard. We note, however, that c.) and d.) are mutually exclusive---if either kind of behavior exists in a given class of theories for some $N_f$, the other kind is not expected to occur, see the text.
• Figure 2: (a) Monopole operators ${\cal M}_{1}$, ${\cal M}_{2}$, ${\cal M}_{3}$, with fermionic zero modes dictated by the index theorem, appearing at order $e^{-S_0}$ in the semiclassical expansion. Monopoles cannot induce confinement due to fermionic zero modes. (b) Magnetic bion ${\cal{B}}_1 \sim {\cal M}_1 { \overline{\cal M}_2}$, which appears at order $e^{-2S_0}$. (c) The two magnetic triplet operators ${\cal T}_1$ and ${\cal T}_2$, appearing at order $e^{-3S_0}$. A combination of bions and triplets leads to a mass gap for the dual photons and confinement in QCD(S).
• Figure 3: Phase diagram of the center stabilized QCD*(F) theories as a function of $N_f$ and $L$. Shown are only $N_f \ge 2$ theories with a continuous chiral symmetry, which may exhibit a $\chi$SB transition as a function of $L$. a.) QCD* theories with $N_f< N_f^*$ exhibit confinement with and without chiral symmetry breaking as a function of radius. QCD* theories with $N_f> N_f^*$ and fixed point at weak coupling exhibit confinement without $\chi$SB at any finite radius (however, at large $L$, the onset scale of confinement is so large that any foreseeable simulations will identify this phase as abelian Coulomb phase). At $L=\infty$, all QCD* theories with $N_f> N_f^*$ flow to a CFT. b.) depicts the main idea of the paper that the mass gap induced by topological excitations becomes IR relevant or irrelevant for the two class of theories. The dashed center line is to guide to eye, there is no transition there and everything is smooth. $1/(N\Lambda)$ is the natural scale (somehow counter-intuitively) of chiral symmetry breaking.