[Paper Review] Critical behavior of gauge theories and Coulomb gases in three and four dimensions

Aleksey Cherman, Mithat Ünsal|arXiv (Cornell University)|Nov 28, 2017

Black Holes and Theoretical Physics36 references19 citations

TL;DR

This paper uses semiclassical analysis via compactification on $\mathbb{R}^3 \times S^1$ to study critical behavior in 4D and 3D $SU(N)$ gauge theories with fermions and a $\theta$ term. It demonstrates that 3D Coulomb gases can host gapless critical points under complex fugacities, and shows that 3D QCD has a critical interval rather than a point, with implications for compact QED and chiral symmetry breaking.

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.

Motivation & Objective

• To understand the non-perturbative critical behavior of $SU(N)$ gauge theories with $N_F$ fundamental fermions and a $\theta$ term in 3D and 4D.
• To resolve the apparent paradox of a critical point in $N_F=1$ 4D QCD, where chiral symmetry breaking does not naively apply.
• To clarify the transmutation of $\theta$-angle dependence from $N_F$-branched in chiral Lagrangians to $N$-branched in pure Yang-Mills as quark masses are sent to infinity.
• To investigate the infrared structure of 3D compact QED and show that it can exhibit a critical interval due to spontaneous $U(1)$ symmetry breaking.

Proposed method

• Utilizes adiabatic continuity by compactifying 4D QCD on $\mathbb{R}^3 \times S^1$ with small $S^1$ circumference to access a weakly-coupled semiclassical regime.
• Applies mixed 't Hooft anomalies involving center symmetry and chiral symmetry to constrain the phase structure and critical behavior.
• Employs the Callias index theorem to compute fermionic zero modes on monopole backgrounds and track changes in monopole operator structure across phase boundaries.
• Analyzes the dual photon effective theory in 3D, identifying a gapless Nambu-Goldstone mode when $U(1)$ quark number symmetry is spontaneously broken.
• Uses the $\theta$-angle as a complex mass parameter to probe CP-odd physics and criticality in the $N_F=1$ case.
• Derives the monopole operator structure $\mathcal{M} = e^{-S_0} e^{i\sigma} \psi_1\psi_2$ for $|m_q| < gv$ and $\mathcal{M} = e^{-S_0} e^{i\sigma}$ for $|m_q| > gv$, showing a phase transition.

Experimental results

Research questions

• RQ1 Does 4D QCD with $N_F=1$ fundamental fermion exhibit a critical point at finite $\theta$, despite the absence of chiral symmetry breaking?
• RQ2 Can 3D Coulomb gases support a gapless critical point when monopole fugacities are complex due to a $\theta$-term or Berry phase?
• RQ3 How does the $\theta$-angle dependence in chiral Lagrangians transmute into the $N$-branched structure of pure Yang-Mills as quark masses are taken to infinity?
• RQ4 What is the nature of the infrared phase structure of 3D compact QCD with parity invariance—specifically, is the critical region a point or an interval in parameter space?
• RQ5 What are the implications of spontaneous $U(1)$ quark number symmetry breaking for the long-distance physics of 3D compact QED?

Key findings

• In 4D $SU(N)$ QCD with $N_F \geq 1$, a single gapless critical point exists at zero quark mass, consistent with $N_F > 1$ chiral symmetry breaking and with $N_F = 1$ anomaly constraints.
• For $N_F = 1$, the critical point at $m_q = m_q^*$ (corresponding to $\theta = \pi$) is shown to be real and calculable via compactification and semiclassical methods.
• The analysis reveals that 3D Coulomb gases with complex fugacities (induced by $\theta$-term or Berry phases) can support a gapless critical point, contradicting the historical belief of always being gapped.
• In 3D QCD with parity invariance, the critical region is an interval in parameter space, not a point, due to spontaneous $U(1)$ quark number symmetry breaking in a finite range of $m_q$.
• The monopole operator structure changes at $|m_q| = gv$, with a fermionic zero mode jump from 2 to 0, signaling a phase transition and enabling the formation of a Nambu-Goldstone boson.
• The dual photon acquires a shift symmetry under the diagonal $U(1)_{\rm diag}$, which is spontaneously broken by $\langle \sigma \rangle \neq 0$, leading to a gapless mode in the interval $|m_q| < gv$.