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Quantum Fluctuations and New Instantons II: Quartic Unbounded Potential

Viatcheslav Mukhanov, Eliezer Rabinovici, Alexander Sorin

TL;DR

This work investigates false vacuum decay when the scalar potential has a local minimum and a quartic unbounded tail, showing that the Coleman $O(4)$ instanton does not exist under the standard boundary conditions. By incorporating quantum fluctuations, the authors regularize singular classical instantons and construct a new class of regularized $O(4)$-invariant instantons, deriving their explicit form and the associated decay rate. The analysis yields closed-form and asymptotic expressions for the instanton action across different parameter regimes, revealing that the dominant decay contribution comes from a specific $E_-$ window and that the rate can be exponentially suppressed even without a Coleman instanton. These results extend the understanding of vacuum decay to quartic unbounded potentials, with implications for Standard Model–like landscapes and potential generalizations to higher-order unbounded potentials.

Abstract

We study the fate of a false vacuum in the case of a potential that contains a portion which is quartic and unbounded. We first prove that an $O(4)$ invariant instanton with the Coleman boundary conditions does not exist in this case. This, however, does not imply that the false vacuum does not decay. We show how the quantum fluctuations may regularize the singular classical solutions. This gives rise to a new class of $O(4)$ invariant regularized instantons which describe the vacuum instability in the absence of the Coleman instanton. We derive the corresponding solutions and calculate the decay rate they induce.

Quantum Fluctuations and New Instantons II: Quartic Unbounded Potential

TL;DR

This work investigates false vacuum decay when the scalar potential has a local minimum and a quartic unbounded tail, showing that the Coleman instanton does not exist under the standard boundary conditions. By incorporating quantum fluctuations, the authors regularize singular classical instantons and construct a new class of regularized -invariant instantons, deriving their explicit form and the associated decay rate. The analysis yields closed-form and asymptotic expressions for the instanton action across different parameter regimes, revealing that the dominant decay contribution comes from a specific window and that the rate can be exponentially suppressed even without a Coleman instanton. These results extend the understanding of vacuum decay to quartic unbounded potentials, with implications for Standard Model–like landscapes and potential generalizations to higher-order unbounded potentials.

Abstract

We study the fate of a false vacuum in the case of a potential that contains a portion which is quartic and unbounded. We first prove that an invariant instanton with the Coleman boundary conditions does not exist in this case. This, however, does not imply that the false vacuum does not decay. We show how the quantum fluctuations may regularize the singular classical solutions. This gives rise to a new class of invariant regularized instantons which describe the vacuum instability in the absence of the Coleman instanton. We derive the corresponding solutions and calculate the decay rate they induce.

Paper Structure

This paper contains 7 sections, 67 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Model
  3. Absence of the Coleman Instanton
  4. Quantum Fluctuations
  5. New Instantons
  6. Asymptotics
  7. Conclusions

Figures (2)

  • Figure 1: A potential that contains a false vacuum and a quartic unbounded portion is displayed for the case $\lambda _{+}\ll \lambda _{-}$. The values indicated on the lower part of vertical axis are the values of the potential at the core of the instanton. The arrows point to the different values of this potential associated with different values of the parameter $E_-$. The decay probabilities $\Gamma$ correspond to the results obtained in equation (\ref{['48c']}).
  • Figure 2: A potential in the case $\lambda _{-}\ll \lambda _{+}$. The values indicated on the lower part of vertical axis are the values of the potential at the core of the instanton. The corresponding values of the decay probability $\Gamma$ are given in equation (\ref{['El_fig_2']}).