Effective potential at three loops

TL;DR

This paper computes the effective potential at three-loop order for general theories, with applications to the Wess-Zumino model and the Standard Model, including checks for consistency and resummation of divergences.

Contribution

It provides the first complete three-loop effective potential calculation for general renormalizable theories in the \\MSbar scheme, with specific applications to supersymmetry and the Standard Model.

Findings

Renormalization group scale invariance confirmed

Vacuum energy vanishing checked in Wess-Zumino model

Higgs vacuum expectation value minimized at three-loop order

Abstract

I present the effective potential at three-loop order for a general renormalizable theory, using the \MSbar renormalization scheme and Landau gauge fixing. As applications and illustrative points of reference, the results are specialized to the supersymmetric Wess-Zumino model and to the Standard Model. In each case, renormalization group scale invariance provides a consistency check. In the Wess-Zumino model, the required vanishing of the minimum vacuum energy yields an additional check. For the Standard Model, I carry out the resummation of Goldstone boson contributions, which provides yet more opportunities for non-trivial checks, and obtain the minimization condition for the Higgs vacuum expectation value at full three-loop order. An infrared divergence due to doubled photon propagators appears in the three-loop Standard Model effective potential, but it does not affect the minimization condition or physical observables and can be eliminated by resummation.