A non-perturbative study of 4d U(1) non-commutative gauge theory -- the fate of one-loop instability

TL;DR

This work addresses the non-perturbative fate of the tachyonic IR instability in 4d non-commutative U(1) gauge theory with two NC directions. It employs a lattice formulation and a twisted Eguchi–Kawai-based mapping to a 2d U(N) model to perform Monte Carlo simulations, uncovering a first-order phase transition to a broken phase where open Wilson lines condense and translational symmetry is spontaneously broken. In the broken phase, the dynamical NC space acquires a finite extent and a Nambu–Goldstone mode emerges, supporting a possible continuum limit at fixed θ; in the symmetric phase, the IR singularity prevents such a limit. The results illuminate the non-perturbative structure of NC gauge theories and hint at connections to string theory and observable consequences like θ-dependent photon dispersion.

Abstract

Recent perturbative studies show that in 4d non-commutative spaces, the trivial (classically stable) vacuum of gauge theories becomes unstable at the quantum level, unless one introduces sufficiently many fermionic degrees of freedom. This is due to a negative IR-singular term in the one-loop effective potential, which appears as a result of the UV/IR mixing. We study such a system non-perturbatively in the case of pure U(1) gauge theory in four dimensions, where two directions are non-commutative. Monte Carlo simulations are performed after mapping the regularized theory onto a U(N) lattice gauge theory in d=2. At intermediate coupling strength, we find a phase in which open Wilson lines acquire non-zero vacuum expectation values, which implies the spontaneous breakdown of translational invariance. In this phase, various physical quantities obey clear scaling behaviors in the continuum limit with a fixed non-commutativity parameter $θ$, which provides evidence for a possible continuum theory. The extent of the dynamically generated space in the non-commutative directions becomes finite in the above limit, and its dependence on $θ$ is evaluated explicitly. We also study the dispersion relation. In the weak coupling symmetric phase, it involves a negative IR-singular term, which is responsible for the observed phase transition. In the broken phase, it reveals the existence of the Nambu-Goldstone mode associated with the spontaneous symmetry breaking.