Perturbative quantization of Yang-Mills theory with classical double as gauge algebra

TL;DR

This paper investigates the perturbative quantization of Yang-Mills theory with a nonsemisimple gauge algebra, demonstrating its one-loop UV divergence, renormalizability, and unitarity through cohomological analysis.

Contribution

It shows that Yang-Mills theory with a classical double gauge algebra is renormalizable at one loop and maintains unitarity despite the indefinite metric.

Findings

One-loop UV divergences are renormalized by coupling constant adjustment.

Higher-loop corrections are absent, indicating finiteness beyond one loop.

Unitarity is preserved due to cohomological triviality of problematic terms.

Abstract

Perturbative quantization of Yang-Mills theory with a gauge algebra given by the classical double of a semisimple Lie algebra is considered. The classical double of a real Lie algebra is a nonsemisimple real Lie algebra that admits a nonpositive definite invariant metric, the indefiniteness of the metric suggesting an apparent lack of unitarity. It is shown that the theory is UV divergent at one loop and that there are no radiative corrections at higher loops. One-loop UV divergences are removed through renormalization of the coupling constant, thus introducing a renormalization scale. The terms in the classical action that would spoil unitarity are proved to be cohomologically trivial with respect to the Slavnov-Taylor operator that controls gauge invariance for the quantum theory. Hence they do not contribute gauge invariant radiative corrections to the quantum effective action and the theory is unitary.