Renormalization of Galilean Electrodynamics

TL;DR

This paper investigates the quantum behavior of a Galilean-invariant abelian gauge theory in 2+1 dimensions, revealing a non-renormalization property and a family of fixed points preserving non-relativistic conformal symmetry.

Contribution

It demonstrates the all-order vanishing of the gauge coupling beta function and systematically handles quantum corrections in Galilean electrodynamics.

Findings

Beta function of gauge coupling vanishes to all orders.

Infinite couplings generated at quantum level due to scalar field.

Existence of a continuous family of fixed points with conformal symmetry.

Abstract

We study the quantum properties of a Galilean-invariant abelian gauge theory coupled to a Schr\"odinger scalar in 2+1 dimensions. At the classical level, the theory with minimal coupling is obtained from a null-reduction of relativistic Maxwell theory coupled to a complex scalar field in 3+1 dimensions and is closely related to the Galilean electromagnetism of Le-Bellac and L\'evy-Leblond. Due to the presence of a dimensionless, gauge-invariant scalar field in the Galilean multiplet of the gauge-field, we find that at the quantum level an infinite number of couplings is generated. We explain how to handle the quantum corrections systematically using the background field method. Due to a non-renormalization theorem, the beta function of the gauge coupling is found to vanish to all orders in perturbation theory, leading to a continuous family of fixed points where the non-relativistic conformal symmetry is preserved.