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

TL;DR

This paper non-perturbatively investigates 4d non-commutative U(1) gauge theory, revealing a phase with broken translational symmetry and analyzing the IR behavior, which sheds light on the stability and continuum limit of such theories.

Contribution

It provides the first non-perturbative Monte Carlo analysis of 4d non-commutative U(1) gauge theory, demonstrating phase structure and continuum behaviors.

Findings

Identification of a phase with non-zero Wilson line vevs

Finite extent of non-commutative space in the continuum limit

Existence of Nambu-Goldstone mode in broken phase

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 $\theta$, 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 $\theta$ 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.