Precision decay rate calculations in quantum field theory

TL;DR

The paper clarifies how vacuum decay rates are computed in quantum mechanics and quantum field theory, contrasting the traditional potential-deformation (Euclidean) approach with a direct, propagator-based method. It emphasizes precise definitions of decay rates, the pivotal role of bounce solutions, and the correct treatment of analytic continuations and multidimensional path integrals. It argues that derivative terms in the effective action are as important as potential terms at NLO, and cautions against naive use of the effective potential, especially in the Standard Model. The analysis highlights the SM vacuum's UV sensitivity, showing that high-scale physics can alter tunneling rates and that a complete, gauge-consistent calculation requires full effective actions and careful scale choices. Together, these insights sharpen predictions for our universe’s fate and guide robust, beyond-leading-order tunneling calculations.

Abstract

Tunneling in quantum field theory is worth understanding properly, not least because it controls the long term fate of our universe. There are however, a number of features of tunneling rate calculations which lack a desirable transparency, such as the necessity of analytic continuation, the appropriateness of using an effective instead of classical potential, and the sensitivity to short-distance physics. This paper attempts to review in pedagogical detail the physical origin of tunneling and its connection to the path integral. Both the traditional potential-deformation method and a recent more direct propagator-based method are discussed. Some new insights from using approximate semi-classical solutions are presented. In addition, we explore the sensitivity of the lifetime of our universe to short distance physics, such as quantum gravity, emphasizing a number of important subtleties.

Figures (19)

• Figure 1: On the left, an example of a physical potential with an metastable region $\text{FV}$, a destination region $\text{R}$, and a barrier. We label the local minimum inside the $\text{FV}$ region by $a$ and the turning point by $b$ (defined by $V(b)=V(a)$). On the right, the probability $P_\text{FV}(T)$ (see Eq. (\ref{['eqn:defineP']})) for this system (beginning in a Gaussian wavepacket centered at $a$) computed by numerically solving Schrödinger's equation. We see that the probability to find the particle in the false vacuum decays exponentially for intermediate times between the short timescale of sloshing inside the false vacuum and the long timescale on which the wavefunction begins to flow back into the false vacuum.
• Figure 2: The numerical evolution of a particle initially localized in the false vacuum. At each time step, the potential is shown (black), along with the probability $\left| \psi(x,t) \right|^2$ (red), and we also show the probability magnified by 50$\times$ (purple) so that we can see the small amount leaking through the barrier). By looking at the evolution of the wavefunction we see the sloshing behavior near the false vacuum, associated with the initial Gaussian state not being an exact resonance. In the first two rows the central value of the wavefunction can be seen moving back and forth within the false vacuum well. When it hits the right wall around times 3-4, the most wavefunction amplitude escapes through the barrier. In the third row we have jumped ahead to see the nonlinear behavior when there is enough wavefunction density in the outside region that it is no longer simply flowing out.
• Figure 3: Example of a potential that has a well region labeled $\text{FV}$, a barrier region B, and is constant in the region $\text{R}$ which extends to indefinitely to the right.
• Figure 4: Left: Generic potential with a false and true vacuum. Right: The inverted potential. The stationary path $\bar{x}(\tau)$ is the solution to the equations of motion of a ball rolling down the inverted potential with boundary conditions $x(0)=x_i$ and $x(\mathcal{T})=x_f$.
• Figure 5: Different solutions to the Euclidean equations of motion for they asymmetric double well.