To be, or not to be finite? The Higgs potential in Gauge-Higgs Unification

TL;DR

This paper examines the finiteness of the Higgs effective potential in Gauge-Higgs Unification models, finding it finite at two loops but divergent at higher loops, with implications for model consistency.

Contribution

It demonstrates the two-loop finiteness of the Higgs potential in SU(N) GHU models and discusses divergence issues at three loops and beyond.

Findings

Higgs potential is finite at two-loop level.

Divergences appear at three-loop level and higher.

Counter terms are needed for divergences at three-loop level.

Abstract

In this paper, we investigate the finiteness of the Higgs effective potential in an ${\rm SU}(\mathcal N)$ Gauge-Higgs Unification (GHU) model defined on ${\bf M}^4\times S^1$. We obtain the Higgs effective potential at the two-loop level and find that it is finite. We also discuss that the Higgs effective potential is generically divergent for three- or higher-loop levels. As an example, we consider an ${\rm SU}(\mathcal N)$ gauge theory on ${\bf M}^5\times S^1$, where the one-loop corrections to the four-Fermi operators are divergent. We find that the Higgs effective potential depends on their counter terms at the three-loop level.