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A Consistent Calculation of Bubble-Nucleation Rates

A. Strumia, N. Tetradis

TL;DR

The paper addresses how to reliably compute bubble-nucleation rates in first-order phase transitions by introducing a coarse-grained, non-convex potential $U_k$ at a finite scale $k$. It develops both intuitive and rigorous renormalization-group formalisms via the effective average action $\Gamma_k$, deriving a scale-invariant flow for $U_k$ and showing that nucleation rates must be evaluated at $k>0$ so that the barrier remains meaningful. The main contributions are a practical framework to compute $S_k$ and a finite $A_k$ that cancels in the rate, plus a robust numerical procedure to evaluate the bounce and fluctuation determinants with regularization that avoids double counting; the results demonstrate near $k$-independence of the rate and delineate the regime where fluctuations overwhelm the semiclassical saddle, signaling limits of Langer’s picture. The approach connects high-temperature 4D field theories to their 3D effective descriptions, providing a consistent, non-perturbative method for tunnelling that is applicable to a wide class of scalar models and potentially to gauge theories with radiatively induced first-order transitions.

Abstract

We present a consistent picture of tunnelling in field theory. Our results apply both to high-temperature field theories in four dimensions and to zero-temperature three-dimensional ones. Our approach is based on the notion of a coarse-grained potential U_k that incorporates the effect of fluctuations with characteristic momenta above a given scale k. U_k is non-convex and becomes equal to the convex effective potential for k --> 0. We demonstrate that a consistent calculation of the nucleation rate must be performed at non-zero values of k, larger than the typical scale of the saddle-point configuration that dominates tunnelling. The nucleation rate is exponentially suppressed by the action S_k of this saddle point. The pre-exponential factor A_k, which includes the fluctuation determinant around the saddle-point configuration, is well-defined and finite. Both S_k and A_k are k-dependent, but this dependence cancels in the expression for the nucleation rate. This picture breaks down in the limit of very weakly first-order phase transitions, for which the pre-exponential factor compensates the exponential suppression.

A Consistent Calculation of Bubble-Nucleation Rates

TL;DR

The paper addresses how to reliably compute bubble-nucleation rates in first-order phase transitions by introducing a coarse-grained, non-convex potential at a finite scale . It develops both intuitive and rigorous renormalization-group formalisms via the effective average action , deriving a scale-invariant flow for and showing that nucleation rates must be evaluated at so that the barrier remains meaningful. The main contributions are a practical framework to compute and a finite that cancels in the rate, plus a robust numerical procedure to evaluate the bounce and fluctuation determinants with regularization that avoids double counting; the results demonstrate near -independence of the rate and delineate the regime where fluctuations overwhelm the semiclassical saddle, signaling limits of Langer’s picture. The approach connects high-temperature 4D field theories to their 3D effective descriptions, providing a consistent, non-perturbative method for tunnelling that is applicable to a wide class of scalar models and potentially to gauge theories with radiatively induced first-order transitions.

Abstract

We present a consistent picture of tunnelling in field theory. Our results apply both to high-temperature field theories in four dimensions and to zero-temperature three-dimensional ones. Our approach is based on the notion of a coarse-grained potential U_k that incorporates the effect of fluctuations with characteristic momenta above a given scale k. U_k is non-convex and becomes equal to the convex effective potential for k --> 0. We demonstrate that a consistent calculation of the nucleation rate must be performed at non-zero values of k, larger than the typical scale of the saddle-point configuration that dominates tunnelling. The nucleation rate is exponentially suppressed by the action S_k of this saddle point. The pre-exponential factor A_k, which includes the fluctuation determinant around the saddle-point configuration, is well-defined and finite. Both S_k and A_k are k-dependent, but this dependence cancels in the expression for the nucleation rate. This picture breaks down in the limit of very weakly first-order phase transitions, for which the pre-exponential factor compensates the exponential suppression.

Paper Structure

This paper contains 12 sections, 42 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Evolution equation for the potential
  3. Intuitive derivation
  4. Rigorous derivation
  5. Scale-invariant form of the evolution equation
  6. Explicit breaking of the $Z_2$ symmetry
  7. Calculation of the nucleation rate
  8. General formalism
  9. Details of the numerical computation
  10. Results
  11. Summary and conclusions
  12. Acknowledgements

Figures (2)

  • Figure 1: The steps in the computation of the nucleation rate for a model with $\mu^2_{k_0}/ k_0^2=-0.05$, $\lambda_{k_0}/k_0=0.1$, $\gamma_{k_0}/k_0^{3/2}=-0.0634$. The dimensionful quantities are given in units of $k_f=0.223~k_0$.
  • Figure 2: The behaviour of the nucleation rate $I$ for several values of the parameters of the model with $\lambda_{k_0} /k_0=0.1$. We show the values of $S_k$ (diamonds), $\ln (A_k/k_f^4)$ (stars) and $-\ln(I/k_f^4)$ (squares) as a function of $k/\sqrt{U_k"(\phi_t)}$.