Two-loop analysis of non-Abelian Chern-Simons theory

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 β 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 term 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 the 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 β function for the gauge coupling constant is zero to two-loop order, consistent with the topological quantization condition for this constant.