3D Field Theories with Chern--Simons Term for Large $N$ in the Weyl Gauge

TL;DR

This paper investigates the phase structure and properties of large N 3D U(N) gauge theories with Chern--Simons terms, revealing conditions for conformal and massive phases, and exploring spontaneous scale symmetry breaking and dualities.

Contribution

It provides a detailed analysis of the phase diagram, mass gap, and correlation functions in large N 3D Chern--Simons matter theories, including spontaneous scale symmetry breaking and fermion-boson duality.

Findings

Identified conditions for conformal and massive phases.

Calculated gauge-invariant correlation functions and mass gap.

Discovered a phase with spontaneously broken scale invariance and a dilaton.

Abstract

Three dimensional, $U(N)$ symmetric, field theory with fermion matter coupled to a topological Chern--Simons term, in the large $N$ limit is analyzed in details. We determine the conditions for the existence of a massless conformal invariant ground state as well as the conditions for a massive phase. We analyze the phase structure and calculate gauge invariant corelators comparing them in several cases to existing results. In addition to the non-critical explicitly broken scale invariance massive case we consider also a massive ground state where the scale symmetry is spontaneously broken. We show that such a phase appears only in the presence of a marginal deformation that is introduced by adding a certain scalar auxiliary field and discuss the fermion-boson dual mapping. The ground state contains in this case a massless $U(N)$ singlet bound state goldstone boson- the dilaton whose properties are determined. We employ here the temporal gauge which is at variance with respect to past calculations using the light-cone gauge and thus, a check (though limited) of gauge independence is at hand. The large $N$ properties are determined by using a field integral formalism and the steepest descent method. The saddle point equations, which take here the form of integral equations for non-local fields, determine the mass gap and the dressed fermion propagator. Vertex functions are calculated at leading order in $1/N$ as exact solutions of integral equations. From the vertex functions, we infer gauge invariant two-point correlation functions for scalar operators and a current. Indications about the consistency of the method are obtained by verifying that gauge-invariant quantities have a natural $O(3)$ covariant form. As a further verification, in several occasions, a few terms of the perturbative expansion are calculated and compared with the exact results in the appropriate order.