Nonrenormalization Theorem for Gauge Coupling in 2+1D

TL;DR

In 2+1D gauge theories with Chern-Simons or topologically massive actions coupled to renormalizable matter, the gauge coupling does not run: the $β(g)$-function vanishes to all orders. The authors derive the conformal anomaly and show that a CS-density contribution to the trace is forbidden by gauge invariance/BRST symmetry, forcing $β(g)=0$; in the topologically massive case the Maxwell regulator yields $γ_A=0$ and thus $β(g)=0$ as well. This preserves the quantization of the CS level and eliminates RG flow for the gauge coupling, even in the presence of matter. The work highlights conformal anomaly analyses and operator-mixing constraints as robust tools for constraining RG behavior in lower-dimensional gauge theories.

Abstract

We prove that $\be$-function of the gauge coupling in $2+1D$ gauge theory coupled to any renormalizable system of spinor and scalar fields is zero. This result holds both when the gauge field action is the Chern-Simons action and when it is the topologically massive action.

Figures (1)

• Figure 1: Schwinger-Dyson equation for gluon self-energy. Boxes denote exact propagators and irreducible vertices. Curly lines correspond to gluons and solid lines correspond to ghosts and matter fields.