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Consistent Use of Effective Potentials

Anders Andreassen, William Frost, Matthew D. Schwartz

TL;DR

The paper addresses the gauge-dependence problem of the effective potential, showing that the value of the Coleman–Weinberg potential at its minimum is gauge-invariant when one uses a carefully constructed perturbative expansion that includes an infinite class of daisy diagrams. By performing a full two-loop calculation in general Rξ gauges and implementing daisy resummation, the authors demonstrate that expressing V_min at a special, gauge-invariant scale μ_X yields an unambiguous, gauge-invariant vacuum energy. They further develop an RG-improvement strategy that runs couplings to μ_X before evaluating the potential, and they extend the framework to higher-dimension operators, clarifying how to extract physical, gauge-invariant information from effective potentials in a consistent, order-by-order manner. The results have significant implications for precision studies of spontaneous symmetry breaking and for applying effective potentials in the Standard Model and beyond.

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.

Consistent Use of Effective Potentials

TL;DR

The paper addresses the gauge-dependence problem of the effective potential, showing that the value of the Coleman–Weinberg potential at its minimum is gauge-invariant when one uses a carefully constructed perturbative expansion that includes an infinite class of daisy diagrams. By performing a full two-loop calculation in general Rξ gauges and implementing daisy resummation, the authors demonstrate that expressing V_min at a special, gauge-invariant scale μ_X yields an unambiguous, gauge-invariant vacuum energy. They further develop an RG-improvement strategy that runs couplings to μ_X before evaluating the potential, and they extend the framework to higher-dimension operators, clarifying how to extract physical, gauge-invariant information from effective potentials in a consistent, order-by-order manner. The results have significant implications for precision studies of spontaneous symmetry breaking and for applying effective potentials in the Standard Model and beyond.

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.

Paper Structure

This paper contains 19 sections, 92 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Effective actions
  3. Gauge dependence
  4. Rescaling and calculation scale invariance
  5. Abelian Higgs Model
  6. Coleman-Weinberg Model
  7. The two-loop Coleman-Weinberg potential
  8. Counterterm contribution
  9. 2-loop contribution
  10. RGE cross-check
  11. Daisy resummation
  12. Gauge invariance of $V_{\text{min}}$
  13. Renormalization group improvement
  14. Calculation-scale invariance
  15. Discussion
  16. ...and 4 more sections

Figures (3)

  • Figure 1: Under the rescaling of the dependent variable, a function changes but its values at extrema do not Patel:2011th. This elementary mathematical fact explains why the effective potential can depend on the field normalization, but $V_{\text{min}}$ does not.
  • Figure 2: There are 12 prototype diagrams which contribute to the 2-loop Coleman-Weinberg potential in $R_\xi$ gauges. Only the 4 diagrams on the top row contribute at NLO (order $\hbar^2$) when $\lambda \sim \hbar$.
  • Figure 3: Example daisy graphs which contribute inverse powers of $\lambda$.