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

TL;DR

This work analyzes the perturbative renormalization of non-Abelian Chern-Simons theory in three dimensions across three regularization schemes. It shows that pure CS theory is finite up to two loops and that the gauge coupling beta function vanishes to at least two to three loops, with dimensional reduction preserving Slavnov-Taylor identities and potentially yielding no renormalization at all orders. When matter is included, both gauge and matter fields acquire infinite renormalization and anomalous dimensions at two loops, yet the gauge coupling remains non-running (β_g = 0) at this order, aligning with conformal symmetry for massless cases. The results underscore the critical role of regularization in maintaining gauge/diffeomorphism invariance and reference the topological quantization constraint as a guiding principle for renormalization.

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.