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On the gauge dependence of the Standard Model vacuum instability scale

Luca Di Luzio, Luminita Mihaila

TL;DR

The paper investigates how gauge choice affects the Standard Model vacuum-instability scale Λ. Using the one-loop effective potential and Nielsen identities in linear gauges (notably the Fermi gauge) with RG improvement, the authors show that while the gauge-independent critical Higgs mass delineates stable from unstable phases, the instability scale itself is inherently gauge dependent, with Λ shifting by roughly two orders of magnitude under perturbative gauge-parameter variations. This underscores that Λ cannot be unambiguously identified with a physical threshold for new physics, even though the corresponding metastability lifetime can remain viable. The work emphasizes careful interpretation of high-scale implications and highlights the role of gauge choices in interpreting vacuum stability analyses, including extensions to background R_ξ gauges.

Abstract

After reviewing the calculation of the Standard Model one-loop effective potential in a class of linear gauges, we discuss the physical observables entering the vacuum stability analysis. While the electroweak-vacuum-stability bound on the Higgs boson mass can be formally proven to be gauge independent, the field value at which the effective potential turns negative (the so-called instability scale) is a gauge dependent quantity. By varying the gauge-fixing scheme and the gauge-fixing parameters in their perturbative domain, we find an irreducible theoretical uncertainty of at least two orders of magnitude on the scale at which the Standard Model vacuum becomes unstable.

On the gauge dependence of the Standard Model vacuum instability scale

TL;DR

The paper investigates how gauge choice affects the Standard Model vacuum-instability scale Λ. Using the one-loop effective potential and Nielsen identities in linear gauges (notably the Fermi gauge) with RG improvement, the authors show that while the gauge-independent critical Higgs mass delineates stable from unstable phases, the instability scale itself is inherently gauge dependent, with Λ shifting by roughly two orders of magnitude under perturbative gauge-parameter variations. This underscores that Λ cannot be unambiguously identified with a physical threshold for new physics, even though the corresponding metastability lifetime can remain viable. The work emphasizes careful interpretation of high-scale implications and highlights the role of gauge choices in interpreting vacuum stability analyses, including extensions to background R_ξ gauges.

Abstract

After reviewing the calculation of the Standard Model one-loop effective potential in a class of linear gauges, we discuss the physical observables entering the vacuum stability analysis. While the electroweak-vacuum-stability bound on the Higgs boson mass can be formally proven to be gauge independent, the field value at which the effective potential turns negative (the so-called instability scale) is a gauge dependent quantity. By varying the gauge-fixing scheme and the gauge-fixing parameters in their perturbative domain, we find an irreducible theoretical uncertainty of at least two orders of magnitude on the scale at which the Standard Model vacuum becomes unstable.

Paper Structure

This paper contains 14 sections, 91 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. The SM effective potential at one loop
  3. Fermi gauge
  4. Renormalization group improvement
  5. Physical observables in the vacuum stability analysis
  6. Gauge independence of the critical Higgs boson mass
  7. Gauge dependence of the extrema of the effective potential
  8. Gauge dependence of the SM vacuum instability scale
  9. Numerical analysis
  10. Discussion and conclusions
  11. Renormalization group equations
  12. On the UV behaviour of $\xi_B$ and $\xi_W$
  13. Background $R_\xi$ gauge
  14. Full result

Figures (4)

  • Figure 1: Schematic representation of the SM effective potential for different values of the Higgs boson mass. For $M_h < M_h^c$, the electroweak vacuum is unstable.
  • Figure 2: Two-loop running of the gauge-fixing parameters $\xi_W$ and $\xi_B$ in the Fermi gauge, for different values of $\xi \equiv \xi_W (M_t) = \xi_B (M_t)$: $\xi = 20$ (left panel) and $\xi = -5$ (right panel).
  • Figure 3: Two-loop running of $-\gamma$ (left panel) and $\Gamma$ (right panel) for different values of $\xi \equiv \xi_W (M_t) = \xi_B (M_t)$.
  • Figure 4: Instability scale as a function of $\xi \equiv \xi_W (M_t) = \xi_B (M_t)$ for the Fermi gauge. The dashed line corresponds to the case where the gauge-fixing parameters are not run. The full line encodes the resummation of the next-to-leading logs by means of two-loop RGEs.