Two-Loop Analysis of Non-abelian Chern-Simons Theory

TL;DR

This paper investigates the perturbative renormalization of non-Abelian Chern-Simons theory, demonstrating the vanishing of the beta function to three loops and analyzing regularization methods' effects on gauge invariance and renormalization.

Contribution

It provides a detailed three-loop analysis of the renormalization properties of non-Abelian Chern-Simons theory using various regularization techniques, including a gauge-invariant variant.

Findings

Beta function for the Chern-Simons coefficient vanishes to three loops.

Dimensional regularization is not gauge invariant at two loops.

Regularization with a Yang-Mills term yields a finite integer renormalization at one loop.

Abstract

Perturbative renormalization of a non-Abelian Chern-Simons gauge theory is examined. It is demonstrated by explicit calculation that, in the pure Chern-Simons theory, the beta-function for the coefficient of the Chern-Simons term vanishes to three loop order. Both dimensional regularization and regularization by introducing a conventional Yang-Mills component in the action are used. It is shown that dimensional regularization is not gauge invariant at two loops. A variant of this procedure, similar to regularization by dimensional reduction used in supersymmetric field theories is shown to obey the Slavnov-Taylor identity to two loops and gives no renormalization of the Chern-Simons term. Regularization with Yang-Mills term yields a finite integer-valued renormalization of the coefficient of the Chern-Simons term at one loop, and we conjecture no renormalization at higher order. We also examine the renormalization of Chern-Simons theory coupled to matter. We show that in the non-abelian case the Chern-Simons gauge field as well as the matter fields require infinite renormalization at two loops and therefore obtain nontrivial anomalous dimensions. We show that the beta function for the gauge coupling constant is zero to two-loop order, consistent with the topological quantization condition for this constant.