Critical behavior of gauge theories and Coulomb gases in three and four dimensions

TL;DR

The paper develops a controlled, non-perturbative framework to locate and characterize critical regions in SU($N$) gauge theories with $N_F$ fundamental fermions and a topological $\theta$ term, using calculable compactifications on $\mathbb{R}^3\times S^1$ and mixed anomaly constraints. It demonstrates that in 4D, for all $N_F\ge 1$, the gapless critical behavior occurs at isolated points in parameter space (with $N_F>1$ aligning with chiral-symmetry breaking expectations and $N_F=1$ connected to anomaly considerations), while establishing a mechanism for $ heta$-dependence transmutation between $\theta/N_F$ and $\theta/N$ in different regimes. A surprising byproduct is that 3D Coulomb gases can host gapless points when monopole amplitudes and Berry phases interplay under center-symmetry breaking, and that 3D parity-invariant QCD-like theories exhibit gapless intervals rather than points, with concrete implications for compact QED in three dimensions. The work unifies semiclassical calculus with anomaly matching to argue for adiabatic continuity from weak to strong coupling, providing a coherent picture across dimensions of how chiral dynamics, CP properties, and topological angles shape infrared behavior.

Abstract

Gauge theories with matter often have critical regions in their parameter space where gapless degrees of freedom emerge. Using controlled semiclassical calculations, we explore such critical regions in $SU(N)$ gauge theories with a topological $θ$ term and $N_F$ fundamental fermions in four dimensions, as well as related field theories in three dimensions. In four-dimensional theories, we find that for all $N_F \ge 1$ the critical behavior always occurs at a point in parameter space. For $N_F>1$ this is consistent with the standard QCD expectations, while for $N_F=1$ our results are consistent with recent observations concerning 't Hooft anomalies. We also show how the $N$-branched structure of observables transmutes into the $N_F$-branched structure seen in chiral Lagrangians as the mass parameter is dialed. As a side benefit, our analysis of these 4D theories implies the unexpected result that 3D Coulomb gases can have gapless critical points. We also consider QCD-like parity-invariant theories in three dimensions, and find that their critical behavior is quite different. In particular, we show that their gapless region is an interval in parameter space, rather than a point. Our results have non-trivial implications for the infrared behavior of three-dimensional compact QED.

Figures (4)

• Figure 1: A cartoon of the expected behavior of the massive fermion determinant $f(m)$ around a monopole-instanton as a function of the parameter $m = m_q/m_W$. We assume that the quark boundary conditions are such that the fermion zero modes would localize on the monopole-instanton in the limit $m \to 0$, so that the small-$m$ behavior is $f(m) \sim m$, and assume a renormalization prescription such that the determinant approaches unity in the pure YM limit $m\to \infty$. The solid parts of the curve summarize these limiting behaviors, while the dashed part is an interpolation, since the functional form of $f(m)$ has not yet been evaluated.
• Figure 2: Behavior of $V(\sigma')$ as a function of $m = m_q/m_W \in \mathbb{R}_{-}$, with $f(m)$ modeled as $f(m) \sim m$ for $|m| \lesssim |m^{*}|$. The theory becomes gapless at $m = m^{*}$.
• Figure 3: A plot of the squared masses of the four lightest pseudoscalar hadrons in adiabatically-compactified $N_F=1$ QCD with $N=5,7,10$ colors for small quark mass $m_q$, plotted as a function of $m = m_q/m_W$, with the sign of $m$ corresponding to whether $\theta = 0$ or $\theta = \pi$. The lightest mode, indicated by the solid red line, becomes gapless at smaller negative $m$ as $N$ increases. So this can be identified as the $\eta'$ meson. The dashed blue curves correspond to the modes whose masses are independent of $m$ at leading order in the semiclassical expansion, while the dot-dashed brown curves are pseudoscalar modes with masses that do depend on $m$. The $m$ dependence of these modes decreases as $N$ is increased.
• Figure 4: In four-dimensional QCD-like theories, we find that the critical region in mass-parameter space is a point, which is illustrated on the left for the $N_F=1$ theory. On the other hand, in three-dimensional theories, the critical region can be an interval as a function of a parity-invariant mass term, as is shown explicitly in the main text in a simple example.