1 Introduction
Tunneling through a barrier is a quintessentially quantum phenomenon. In quantum mechan-
ics (QM), tunneling has been studied analytically, numerically and experimentally, leading to
a consistent and comprehensive picture of when and how fast tunneling occurs. In quantum
field theory (QFT), much less is known. In QFT, one cannot solve the Schr¨odinger equation,
even numerically, due to the infinite dimensionality of the Hilbert space. The only approach
to calculating tunneling rates in QFT seems to be through the saddle point approximation of
the path integral [1–5]. This approach involves analytic continuation in an essential way. Be-
cause tunneling in QFT has important implications, such as for the stability of the Standard
Model vacuum [6–20] and because QFT tunneling rates are nearly impossible to measure
experimentally, it is critical to make sure the rather abstract formalism is actually capable
of calculating something physical.
A number of the subtleties in going from QM to QFT were resolved long ago, some more
recently, and some challenges still exist. For example, while tunneling rates are physical and
therefore should be gauge invariant, it has been challenging to check directly that this is the
case. Although exact non-perturbative proofs of gauge-invariance exist [21,22] and there have
been many investigations into gauge-dependence [18,23–36] it has not been shown that gauge-
invariance holds order-by-order in perturbation theory, as it does for S-matrix elements. For
some context, recall that for the simpler question of whether a state is absolutely stable in
the quantum theory, it was found that the corresponding bound was gauge-dependent with
then-current perturbative methods [17, 37, 38]. The problem was traced to an inconsistent
power counting and improper use of the renormalization group equations. A consistent
method was recently developed in [17, 38], with non-negligible implications for precision
top and Higgs-boson mass bounds in the Standard Model. Recently, progress was made in
understanding the gauge invariance of tunneling rates by Endo et al. [39, 40]; these authors
showed explicitly that the rate is gauge-invariant to one loop for general massive scalar scalar
field theory backgrounds and we build upon their results.
In fact, gauge invariance is the least of our worries. In order to produce a precision
calculation of the tunneling rate – or even the leading order rate with the correct units –
one must understand a whole slew of subtleties not relevant for the absolute stability bound.
First of all, there are suspicious elements in the common derivations [41–46] of the Callan-
Coleman decay rate formula [3, 4]. The leading-order confusion is that the rate is said to
be determined, even in QM, by taking the imaginary part of ha|e
−HT
|ai, a manifestly real
expression. The resolution of this paradox involves not analytic continuation of the potential,
as is often cited, but the specification of complex paths to be integrated over in the path
integral [47, 48]. A more physical derivation of a decay rate in QFT was presented recently
in [49, 50]. Some elements are reviewed in Section 2.
Even if we ignore gauge-dependence and trust the decay rate formulas, we encounter a
new roadblock in trying to evaluate tunneling rates in QFTs like the Standard Model, due to
classical scale invariance. The basic issue with scale invariance can be seen in the Gaussian
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