On gauge theories for non-semisimple groups

TL;DR

This paper explores gauge theories based on non-semisimple Lie algebras, revealing their finiteness at one loop, potential unitarity through truncation, and connections to known gauge models like SU(2) and U(1).

Contribution

It introduces and analyzes 4D gauge theories for non-semisimple Lie algebras with invariant metrics, highlighting their finiteness and potential unitarity, and relates them to established gauge theories.

Findings

Quantum effective action is one-loop exact and finite.

Presence of a null direction leads to scale invariance.

The E^c_2 model relates to SU(2) gauge theory.

Abstract

We consider analogs of Yang-Mills theories for non-semisimple real Lie algebras which admit invariant non-degenerate metrics. These 4-dimensional theories have many similarities with corresponding WZW models in 2 dimensions and Chern-Simons theories in 3 dimensions. In particular, the quantum effective action contains only 1-loop term with the divergent part that can be eliminated by a field redefinition. The on-shell scattering amplitudes are thus finite (scale invariant). This is a consequence of the presence of a null direction in the field space metric: one of the field components is a Lagrange multiplier which `freezes out' quantum fluctuations of the `conjugate' field. The non-positivity of the metric implies that these theories are apparently non-unitary. However, the special structure of interaction terms (degenerate compared to non-compact YM theories) suggests that there may exist a unitary `truncation'. We discuss in detail the simplest theory based on 4-dimensional algebra E^c_2. The quantum part of its effective action is expressed in terms of 1-loop effective action of SU(2) gauge theory. The E^c_2 model can be also described as a special limit of SU(2) x U(1) YM theory with decoupled ghost-like U(1) field.