On the relationship between gauge dependence and IR divergences in the \u210f-expansion of the effective potential

Andreas Ekstedt,Johan Löfgren

doi:10.1007/jhep01(2019)226

Abstract

Perturbative calculations of the effective potential evaluated at a broken minimum, Vmin, are plagued by difficulties. It is hard to get a finite and gauge invariant result for Vmin. In fact, the methods proposed to deal with gauge dependence and ir divergences are orthogonal in their approaches. Gauge dependence is dealt with through the ℏ-expansion, which establishes and maintains a strict loop-order separation of terms. On the other hand, ir divergences seem to require a resummation that mixes the different loop orders. In this paper we test these methods on Fermi gauge Abelian Higgs at two loops. We find that the resummation procedure is not capable of removing all divergences. Surprisingly, the ℏ-expansion seems to be able to deal with both the divergences and the gauge dependence. In order to isolate the physical part of Vmin, we are guided by the separation of scales that motivated the resummation procedure; the key result is that only hard momentum modes contribute to Vmin.

Highlights

One of the most important tools for studying spontaneous symmetry breaking within qft is the effective potential V, which can be considered as the quantum-corrected version of the classical potential V0
If the theory allows for spontaneous symmetry breaking through the scalar field φ, its vacuum expectation value, or vev, would be found by extremizing the effective potential ∂V |φ=φm = 0
We argue that in a general model with spontaneous symmetry breaking, the -expansion is capable of treating both the gauge invariance and the ir divergence issues

Summary

Introduction

One of the most important tools for studying spontaneous symmetry breaking within qft is the effective potential V , which can be considered as the quantum-corrected version of the classical potential V0. If the theory allows for spontaneous symmetry breaking through the scalar field φ, its vacuum expectation value, or vev, would be found by extremizing the effective potential ∂V |φ=φm = 0. In this way the effective potential allows us to find the quantum corrected minimum, φm, and the corresponding background energy density Vmin ≡ V (φm). The Abelian Higgs model is a useful toy-model because it exhibits the issues that we want to discuss: gauge dependence and ir divergences.

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Conclusion