Consistent Use of Effective Potentials

TL;DR

This paper demonstrates how summing an infinite class of loops in massless scalar QED yields a gauge-invariant potential minimum, clarifying the gauge dependence and proposing a self-consistent perturbative method to extract physical quantities.

Contribution

It shows that an infinite class of loops must be summed to achieve gauge invariance of the potential minimum and introduces a method for consistent perturbative calculations of physical quantities.

Findings

Summing infinite loops yields a gauge-invariant potential minimum.

The potential depends on scale and normalization, but the vacuum energy does not.

A self-consistent perturbative method with renormalization group improvement is proposed.

Abstract

It is well known that effective potentials can be gauge-dependent while their values at extrema should be gauge-invariant. Unfortunately, establishing this invariance in perturbation theory is not straightforward, since contributions from arbitrarily high- order loops can be of the same size. We show in massless scalar QED that an infinite class of loops can be summed (and must be summed) to give a gauge invariant value for the potential at its minimum. In addition, we show that the exact potential depends on both the scale at which it is calculated and the normalization of the fields, but the vacuum energy does not. Using these insights, we propose a method to extract some physical quantities from effective potentials which is self-consistent order-by-order in perturbation theory, including improvement with the renormalization group.