# Quantum tunneling from excited states:
Recovering imaginary-time instantons from a real-time analysis

Thomas Steingasser<EMAIL_ADDRESS>Department of Physics, Massachusetts
Institute of Technology, Cambridge, MA 02139, USA  Black Hole Initiative at
Harvard University, 20 Garden Street, Cambridge, MA 02138, USA  David I.
Kaiser<EMAIL_ADDRESS>Department of Physics, Massachusetts Institute of
Technology, Cambridge, MA 02139, USA

###### Abstract

We revisit the path integral description of quantum tunneling and show how it
can be generalized to excited states. For clarity, we focus on the simple toy
model of a point particle in a double-well potential, for which we perform all
steps explicitly. Instead of performing the familiar Wick rotation from
physical to imaginary time—which is inconsistent with the requisite boundary
conditions when treating tunneling from excited states—we regularize the path
integral by adding an infinitesimal complex contribution to the Hamiltonian,
while keeping time strictly real. We find that this gives rise to a complex
stationary-phase solution, in agreement with recent insights from Picard-
Lefshitz theory. We then show that there exists a class of analytic solutions
for the corresponding equations of motion, which can be made to match the
appropriate boundary conditions in the physically relevant limits of a
vanishing regulator and an infinite physical time. We provide a detailed
discussion of this non-trivial limit. We find that, for systems without an
explicit time-dependence, our approach reproduces the picture of an instanton-
like solution defined on a finite Euclidean-time interval. Lastly, we discuss
the generalization of our approach to broader classes of systems, for which it
serves as a reliable framework for high-precision calculations.

††preprint: MIT-CTP/5672

##  I Introduction

Tunneling is a universal feature of all quantum theories [1, 2, 3, 4]. Despite
being textbook knowledge for decades, it also remains a subject of active
research, driven by recent progress in the understanding of its path-integral
formulation. For the purpose of this article, two of these developments are of
particular interest. First, the direct approach to tunneling brought forward
recently in Refs. [5, 6, 7] and extended in Refs. [8, 9, 10] has clarified
several important properties of the tunneling-rate calculation. This approach
relies on the use of instantons, which are solutions of the equations of
motion in imaginary time. In this picture, the imaginary time arises through a
simple Wick rotation from physical time, allowing for a straightforward
identification of the infinitely long imaginary time interval with the limit
of an infinitely long physical time, relative to the typical time-scales of
the system. This limit has a strong physical motivation and it is necessary to
avoid so-called sloshing effects, as described in Refs. [5, 6]. Meanwhile, on
a practical level, the Wick rotation is necessary to realize the instanton’s
boundary conditions. Using a point particle for concreteness, the instanton
can be understood as its motion in the inverted potential, starting at rest
from the false vacuum at the initial (imaginary) time.

A straightforward generalization of this simple picture would suggest that the
decay rate out of an excited state should be determined by the Euclidean
action of a suitable instanton solution connecting the initial position of
interest with the emergence point on the other side of the potential barrier.
As the initial state is no longer a local extremum, this motion can be
expected to occur within some finite imaginary-time interval, whose duration
can be expected to be of the order of the potential’s natural time-scale. In
other words, for tunneling out of an excited state, one might expect that the
imaginary-time interval of interest would be determined by the form of the
potential and the initial state itself. This picture agrees with the result
obtained through the WKB approximation, in which the imaginary time appears as
a formal tool rather than a representation of the physical time [2, 1, 11,
12].

In the direct approach, however, the imaginary time is usually linked to the
real, physical time, from which it arises through a Wick rotation. This would
seem to suggest that the length of the Euclidean-time interval is determined
by the physical time that has passed since the particle started out from its
initial state. Taking seriously this interpretation, our previous observations
would suggest that the instanton only exists for one unique (physical) time,
of order of the system’s natural time-scale. This conclusion not only appears
counterintuitive on its own, but also disagrees with the necessity of a
physical time that is significantly longer than the system’s natural time-
scale, as found in Refs. [5, 6].

Clearly, the natural candidate for the origin of this issue is the Wick
rotation, which usually implicitly relies on the assumption that the system
approaches a minimum of the potential at asymptotic physical times. Thus, when
interested in the tunneling rate out of an excited state, one is forced to
work in real time. To make this manifest, as we demonstrate here, one may
instead regularize the path integral by introducing an infinitesimal imaginary
part of the Hamiltonian. For time-independent systems, this is equivalent to
performing an infinitesimal Wick rotation and eventually taking the
appropriate limit.111The interpretation of a complex energy arises naturally
when deriving the regularization from the properties of a physical initial
state (see, e.g., Refs. [13, 14]). The conceptual subtleties linked to the
common step of assigning the imaginary part to the time variable are
illustrated, e.g., in Ref. [15]. However, this equivalence is broken for
systems with an explicit time-dependence, for which the usual Wick rotation is
even more subtle.

Because our analysis relies entirely on real, physical time, it is ideally
suited for the investigation of such systems. Most important, we show that our
approach can be used to properly derive the length of the imaginary-time
interval on which the instanton is to be defined. This feature is crucial for
the investigation of more complicated systems, such as near evaporating black
holes, for which the precise imaginary-time description of the tunneling
process remains controversial [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27,
28, 29, 30]. By providing a simple tool with which to derive the decay rate as
well as the properties of the relevant Euclideanized spacetime, we aim to take
the first step towards providing clarity regarding this important matter.

While we perform our calculations exclusively in real time, the familiar
leading-order results imply that, when dealing with systems with no explicit
time-dependence, we should ultimately be able to express our results in terms
of an (imaginary-time) instanton. Making the connection between these two
perspectives requires an understanding of instanton-like solutions on time-
contours with an arbitrary Wick-rotation angle in the complex-time plane. For
tunneling out of a false vacuum, there has recently been significant progress
in this direction within the framework of Picard-Lefshetz theory [31, 32, 33,
34, 35, 36, 37, 38, 39, 40, 41]. In particular, it has been shown in great
detail in Ref. [39] that the decay rate of the false vacuum is independent of
the Wick-rotation angle.

Our computations are consistent with this result, while further clarifying
that the dynamics along the complex-time contour constitute a “shearing”
rather than a rotation: the convergence of the solution is controlled
primarily by the change of the complex-time variable along the imaginary-time
direction. In addition, we find that the limit of a vanishing angle between
the contour and the real-time axis is, in fact, not independent of the
physical time. In the case of an infinite imaginary time, this makes no
practical difference, since any contour in the complex-time plane with a
finite angle relative to the real-time axis will eventually lead to a
sufficiently large imaginary part to ensure convergence. For instantons
defined on a finite imaginary-time interval, on the other hand, this implies
that any contour along a finite angle will contain an infinite number of
instantons in the infinite-time limit, reproducing the problem already present
for the purely imaginary contour. We find that this problem can be cured by
restricting oneself to a particular subset of possible complex-time contours.
This step can be easily justified through our interpretation of the direct
approach, and provides a strong justification for the restriction of contours
to the so-called Lefshetz thimbles in the Picard-Lefshetz theory.

In Sec. II, we review the emergence of imaginary time in the leading-order
analysis of quantum tunneling, summarize the recently proposed direct
approach, and provide a first sketch of our results. Next, in Sec. III, we
review in more detail the derivation of the decay rate in the direct approach
and generalize it to an arbitrary initial state. We then discuss the
regularization scheme we use to evaluate the resulting path integrals in Sec.
IV, where we also provide a simple but helpful interpretation of the
corresponding saddle-point approximation in terms of interacting, non-
relativistic point particles.

In Sec. V, we analyze the equivalent of the instanton in our real-time
framework, which we dub a steadyon. We discuss in particular the ability of
these solutions to reproduce the boundary conditions of the path integral of
interest, and show how they can be used to deduce the tunneling rate in the
combined limit of a vanishing regulator and an infinite (real) physical time.
Crucially, we demonstrate how our approach can be used to derive the
imaginary-time picture for systems without an explicit time-dependence, to
leading order. An important feature of the direct approach, as we show, is
that it enables the computation of the appropriate normalization factor for
the path integral, in addition to identifying the terms associated with the
instanton. In Sec. VI, we show how our approach can be used to evaluate this
expression, and illustrate how each relevant term can be understood in terms
of imaginary-time dynamics. Finally, in Sec. VII, we review our work with a
special emphasis on its generalization to more complicated systems, such as
field theory or systems with an explicit time-dependence.

##  II Complex time in quantum tunneling

For simplicity, we consider a point particle of mass $m$ and energy $E$
trapped by some potential barrier in one spatial dimension. Assuming the
tunneling rate to be proportional to the transmission coefficient, it is
straightforward to use the WKB approximation to show that the tunneling rate
takes the general form [1, 2, 11, 12]

$\displaystyle\Gamma=Ae^{-B}\left[1+{\cal O}(\hbar)\right],$ (1)

where $B$ is the WKB factor

$\displaystyle B=2\int_{x_{i}}^{x_{s}}{\rm
d}x\left[2m(V(x)-E)\right]^{\frac{1}{2}}.$ (2)

Here $x_{i}$ is the classical turning point of the particle, while $x_{s}$ is
the point on the other side of the barrier for which the particle has the same
energy.

In order to obtain a simple interpretation of this result, it was famously
proposed by the authors of Refs. [1, 2] to consider the motion of the same
point particle in imaginary time $t\to-i\tau$ along the classical path
$\bar{x}_{I}$ that connects the points $x_{i}$ and $x_{s}$, with no turning
points. The energy of the particle is then given by

$\displaystyle E=V(\bar{x}_{\rm I})-\frac{1}{2}m\dot{\bar{x}}_{\rm
I}^{2}(\tau),$ (3)

where dots denote derivatives with respect to $\tau$. It is straightforward to
show that the WKB factor $B$ is fully determined by this solution:

$\displaystyle B=$ $\displaystyle\,2\int_{x_{i}}^{x_{s}}{\rm
d}x\left[2m(V(x)-E)\right]^{\frac{1}{2}}$ $\displaystyle=$
$\displaystyle\,2m\int_{x_{i}}^{x_{s}}{\rm d}x\
\dot{\bar{x}}_{I}(\tau(x))=2m\int_{-\Delta\tau_{I}/2}^{0}{\rm d}\tau\
\dot{\bar{x}}_{I}^{2}$ $\displaystyle=$
$\displaystyle\int_{-\Delta\tau_{I}/2}^{\Delta\tau_{I}/2}{\rm d}\tau\
\left[\frac{m}{2}\dot{\bar{x}}_{I}^{2}+V(\bar{x}_{I}(\tau))-E\right]$
$\displaystyle=$ $\displaystyle\,S_{E}[\bar{x}_{I}]-E\cdot\Delta\tau_{I},$ (4)

where $\Delta\tau/2$ is defined as the (Euclidean) time necessary for the
particle to move from $x_{i}$ to $x_{s}$. In the second line, the
invertibility of $\bar{x}_{I}(\tau)$ has been used.

Eq. (4) implies that the exponent of the decay rate is, to leading order and
up to the term $E\cdot\Delta\tau_{I}$, given by the Euclidean action of the
solution $\bar{x}_{I}$. The latter can be recognized as the familiar
instanton, while the $E\cdot\Delta\tau_{I}$ term can be linked to the fact
that the initial and final states are energy eigenstates [42, 43, 44, 45]. For
our purpose, the most interesting insight gained from this discussion is that
the (finite) length of the Euclidean time interval $\Delta\tau_{I}$ is by
construction such that a suitable instanton connecting $x_{i}$ and $x_{s}$ can
exist.

Whereas this approach successfully captures the leading-order tunneling rate
in many situations, in particular out of the false vacuum, its range of
application is limited due to its reliance on the WKB approximation. For
example, on a conceptual level, there is no known procedure to extend this
approach to incorporate loop corrections or dynamical effects. Even at leading
order, the WKB approach breaks down in regions for which $V(x)\sim E$. This
includes, in particular, tunneling from an initial state that is relatively
close to the top of the potential barrier [11, 12]. This problem is of
particular significance for tunneling out of relatively flat minima.
Potentials with such features have recently gained interest in the context of
new approaches to beyond-the-Standard Model (BSM) model building [46].

A more rigorous formulation of quantum tunneling can be obtained in terms of
path integrals. Following Refs. [5, 6], the decay rate can be brought to the
form

$\displaystyle\Gamma(T)=$
$\displaystyle\frac{1}{Z}\int_{x_{1}(0)=x_{i}}^{x_{1}(T)=x_{s}}{\mathcal{D}}x_{1}\
e^{iS[x_{1}]}\delta\left(F_{x_{s}}[x_{1}]-T\right)$
$\displaystyle\quad\quad\times\left(\int_{x_{2}(0)=x_{i}}^{x_{2}(T)=x_{s}}{\mathcal{D}}x_{2}\
e^{iS[x_{2}]}\right)^{*}+c.c.,$ (5)

where $T$ denotes the physical time at which the tunneling rate is considered,
and the functional $F_{x_{s}}[x_{1}]$ identifies the time at which the path
$x_{1}$ first crosses $x_{s}$. (We provide a complete derivation of this
result in Sec. III.) In order to arrive at the familiar imaginary-time
picture, the authors of Refs. [5, 6] then propose to perform a Wick rotation,
$T\to i\tau$. An alternative derivation of the decay rate, leading to an
equivalent result, has been performed in Ref. [39]. The latter also relies on
a Wick rotation to reproduce the familiar imaginary-time results, while
further considering (incomplete) Wick rotations by some finite angle in the
complex-time plane, demonstrating that the final results are independent of
the rotation angle.

When considering tunneling out of the false vacuum, these results are
perfectly consistent with the one obtained from the WKB approximation. In
particular, they successfully reproduce the infinite imaginary-time interval
of the instanton in the limit of large physical times $T\to\infty$ motivated
in Refs. [5, 6]. However, it is easy to see that this procedure cannot be used
to describe tunneling out of an excited state, for which the Wick rotation
would seem to require an infinitely long imaginary-time interval, given the
infinite-$T$ limit. Comparing with Eq. (4), we see that taking the limit
$\Delta\tau_{I}\rightarrow\infty$ is inconsistent with the WKB approximation.

In addition, performing the usual Wick rotation and taking the limit
$\Delta\tau_{I}\rightarrow\infty$ would prevent us from using the saddle-point
approximation to evaluate the path integrals in the first place. The initial
condition linked to the excited state implies that the instanton begins and
ends its motion in points for which $V^{\prime}\neq 0$. This, in turn, implies
that the particle’s periodic motion occurs within some finite imaginary time,
of order of the system’s dynamical time-scale [42, 43, 44, 45]. In other
words, if one were to perform the usual Wick rotation to imaginary time, the
tunneling could only be described through the saddle-point approximation for
some unique (real) time $T$, and, in particular, not in the physically
relevant limit $T\to\infty$.222This issue cannot be avoided by taking into
account the possibility of multi-instantons, as the limit of an infinite
imaginary time would correspond to an infinite number of instantons, and
hence, an infinite Euclidean action.

Figure 1: The well-known kink solution describing tunneling out of the false
vacuum for a particle in a double-well potential (with vacua at $x_{\pm}$)
after a finite Wick rotation $\tau\to i(1-i\epsilon)t$. By considering
arbitrary rotation angles, the instanton solution $\bar{x}_{I}(\tau)$ (red
curve) becomes complex, and its convergence toward the desired initial and
final values is controlled by the combination $\epsilon\cdot t$. It was shown
in Ref. [39] that the decay rate obtained using this class of solutions agrees
with the imaginary-time result.

To understand the origin of this contradiction, we note that, on a qualitative
level, the cases of tunneling from the false vacuum and from an excited state
differ by the asymptotic behavior of the instanton: whereas in the first case
the instanton converges toward the false vacuum, the latter undergoes an
infinite, periodic motion, rendering the Wick rotation highly non-trivial. A
simple way around this issue would be to avoid the Wick rotation entirely and
evaluate the path integral in real time. This idea has been pursued in Ref.
[39], which analyzes tunneling out of a false vacuum by performing a Wick
rotation $t\to e^{-i\epsilon}t$ and taking the limit $\epsilon\to 0$. For the
special case of tunneling out of the false vacuum, and thus considering an
infinitely long time interval, these authors find that the decay rate is
independent of $\epsilon$.

This result hinges, however, on the infinite length of the time interval under
consideration. To understand this dependence, we may consider the tunneling of
a point particle in a potential with two (nearly) degenerate vacua $x_{\pm}$.
In imaginary time, the instanton describing this process to leading order is
the well-known kink, whose value at $\tau\to-\infty$ (or, equivalently,
$T\to-\infty$) describes the initial state of the system. Along a more general
complex-time contour, the solution itself becomes a complex function,

$\bar{x}_{I}(e^{i\epsilon}t)=\bar{x}_{\rm Re}(t)+i\bar{x}_{\rm Im}(t).$ (6)

In the limit $\epsilon\to 0$, the time evolution of $\bar{x}_{\rm Re}$ and
$\bar{x}_{\rm Im}$ can be decomposed into an oscillatory motion with the
system’s typical, classical frequency as well as a time evolution of the
amplitude, which is responsible for the convergence of these functions to
their desired initial values. See Fig. 1.

Most important for our purpose, we observe that the evolution of the amplitude
of the real part of the complex-valued instanton reproduces the behavior of
the imaginary-time instanton under the identification $\tau=\epsilon t$. In
simple terms, this suggests that the transition to a complex-time argument
involves a projection rather than a rotation, see Fig. 1. For tunneling out of
the false vacuum, this still ensures the convergence of the complex-time
solution, as the limit $T\to\infty$ amounts to $\tau\to\infty$ for any non-
vanishing $\epsilon$. As we will see, the generalization of this observation
will turn out to be crucial for the explanation of the finite imaginary-time
support of the instanton corresponding to tunneling out of an excited state.

This insight, which represents one of the main results of this article, is
discussed in detail in Sec. V and summarized schematically in Fig. 9. In
essence, we argue that the Wick rotation angle $\epsilon$ is to be understood
as a regularization parameter, and hence it must be taken to zero. As we also
take the limit $T\to\infty$, it becomes possible to take these limits in such
a way that the combination $\epsilon\cdot T$ remains constant, ensuring the
existence of the complex-valued instanton solution. This solution is, however,
vastly different from the more familiar imaginary-time instanton. First,
taking the limit $\epsilon\to 0$ amounts to rotating the time contour
approximately onto the real-time axis. Second, due to the limit $T\to\infty$,
both the real and imaginary parts of the complex instanton solutions become
highly oscillatory functions, leading to a divergent real part of the (now
complex) action.

In Secs. V and VI, we demonstrate explicitly for our toy model how these
configurations lead to a sensible result for the decay rate, which agrees to
leading order with the naive expectation based on the imaginary-time picture.
We find, in particular, that the exponent of the decay rate is entirely
determined by the well-behaved imaginary part of the action, which in the
combined limit $\epsilon\to 0,\,T\to\infty$ converges to the Euclidean action
of the corresponding imaginary-time instanton.

##  III Tunneling rate from first principles

We consider a point particle in a potential with a false vacuum $x_{\rm FV}$,
surrounded by the false vacuum basin $\Omega_{\rm FV}$. In the most general
case, the initial state at $t=0$ of such a particle can be described through a
density matrix $\rho$, although we will restrict ourselves to the case of a
pure initial state, $\rho=|x_{i}\rangle\langle x_{i}|$. The probability to
recover the point particle within some region $\Omega$ (such as $\Omega_{\rm
FV}$) at some later time $t=T$ is simply given by [24]

$\displaystyle P_{\Omega}(T)=$
$\displaystyle\text{Tr}_{\Omega}\left[\rho\right](T)=\int_{\Omega}dx\langle
x,T|x_{i}\rangle\langle x_{i}|x,T\rangle.$ (7)

Here and throughout the remainder of our article, all states with no explicit
time label are to be understood as defined at $t=0$, e.g.
$|x_{i}\rangle\equiv|x_{i},0\rangle$.

Considering the case $\Omega=\Omega_{\rm FV}$, the probability can be expected
to decay as

$P_{\Omega_{\rm FV}}(T)=P_{\Omega_{\rm FV}}(0)\cdot e^{-\Gamma T},$ (8)

where $\Gamma$ is the decay rate we ultimately wish to compute, and we have
assumed that $\Gamma$ is approximately constant in the regime of interest, as
is the case, e.g., in the limit $T\to\infty$. It is straightforward to solve
Eq. (8) for $\Gamma$, yielding

$\Gamma(T)=-\frac{1}{P_{\Omega_{\rm
FV}}(T)}\frac{\text{d}}{\text{d}T}P_{\Omega_{\rm
FV}}(T)=\frac{1}{P_{\Omega_{\rm FV}}(T)}\frac{\text{d}}{\text{d}t}P_{R}(T),$
(9)

where $R$ is the region in position space into which the particle tunnels.
Henceforth we adopt the abbreviation $Z(T)\equiv P_{\Omega_{\rm FV}}(T)$. The
last of these factors can be represented through the point particle’s
propagator $D_{F}(x_{i},t_{i}|x_{f},t_{f})$ as

$\displaystyle P_{R}=$ $\displaystyle\int_{R}{\rm d}x_{f}\
D_{F}(x_{f},T|x_{i})D_{F}(x_{i}|x_{f},T).$ (10)

Next, following Eq. (7) we can make manifest that $x_{i}$ and $x_{f}$ lie in
different subsets of space by decomposing the propagator as

$\displaystyle D_{F}$
$\displaystyle(x_{i}|x_{f},t)=\underset{0}{\overset{t}{\int}}{\rm d}t_{s}\
\bar{D}_{F}(x_{i}|x_{s},t_{s})D_{F}(x_{s},t_{s}|x_{f},T),$ (11)

where $x_{s}$ is the closest point in $R$ for which the particle’s energy is
degenerate with its energy at point $x_{i}$.333A similar decomposition is
possible in the presence of more degrees of freedom, in which case one needs
to include an additional integral of $x_{s}$ over a suitable hypersurface. To
define the auxiliary quantity $\bar{D}_{F}$, we may first introduce the
functional $F_{x_{s}}[x]$, which (as noted above) maps any time-dependent path
$x$ onto the time when it first reaches $x_{s}\subset R$. In terms of this
object, $\bar{D}_{F}$ is defined as

$\bar{D}_{F}(x_{i}|x_{s},t_{s})\equiv\mathcal{N}\underset{x(0)=x_{i}}{\overset{x(t_{s})=x_{s}}{\int}}{\mathcal{D}}x\
e^{iS[x]}\delta\left(F_{x_{s}}[x]-t_{s}\right),$ (12)

with the usual path integral normalization factor $\mathcal{N}$. Moving
forward, we will refer to the condition $F_{x_{s}}[x]=t_{s}$ as the crossing
condition. The decomposition in Eq. (11) amounts to splitting the time
evolution of the particle’s motion from $x_{i}$ to $x_{f}$ into a piece
connecting $x_{i}$ with an energetically degenerate point in $R$ and the time
evolution thereafter. Using this representation, the probability $P_{R}$ can
be expressed as

$\displaystyle P_{R}=$
$\displaystyle\int_{0}^{T}\text{d}t_{s}\int_{0}^{T}\text{d}t_{s}^{\prime}\,\bar{D}_{F}(x_{i}|x_{s},t_{s})\bar{D}_{F}^{*}(x_{i}|x_{s},t_{s}^{\prime})$
$\displaystyle\times\int_{\smash[b]{\vphantom{\bigg{|}}}R}{\rm
d}x_{f}\,D_{F}(x_{s},t_{s}|x_{f},T)D_{F}(x_{f},T|x_{s},t_{s}^{\prime}).$ (13)

For the decay through a single tunneling event no back-tunneling occurs, which
allows us simplify Eq. (13) as

$\displaystyle\int_{R}{\rm d}x_{f}\,$ $\displaystyle
D_{F}(x_{s},t_{s}|x_{f},T)D_{F}(x_{f},T|x_{s},t_{s}^{\prime})$
$\displaystyle\simeq
D_{F}(x_{s},t_{s}|x_{s},t_{s}^{\prime})=D_{F}^{*}(x_{s},t_{s}^{\prime}|x_{s},t_{s}).$
(14)

Next, we can rewrite the time integrals as

$\int_{0}^{T}\text{d}s\int_{0}^{T}\text{d}s^{\prime}=\int_{0}^{T}\text{d}s\int_{0}^{s}\text{d}s^{\prime}+\int_{0}^{T}\text{d}s^{\prime}\int_{0}^{s^{\prime}}\text{d}s.$
(15)

Doing so allows us to use the second integral in each of these combinations to
recombine the propagator in Eq. (14) with one of the two factors of
$\bar{D}_{F}$ using Eq. (11). Eliminating the remaining time integral by
taking a derivative, we are ultimately left with

$\displaystyle\frac{{\rm d}P_{R}}{{\rm
d}t}=\bar{D}_{F}(x_{i}|x_{s},T)D_{F}^{*}(x_{s},T|x_{i})+c.c.$ (16)

Hence we find that the decay rate can be represented through the simple path-
integral expression

$\displaystyle\Gamma=$
$\displaystyle\lim_{T\to\infty}\frac{|\mathcal{N}|^{2}}{Z(T)}\underset{x_{1}(0)=x_{i}}{\overset{x_{1}(T)=x_{s}}{\int}}{\mathcal{D}}x_{1}\
e^{iS[x_{1}]}\delta\left(F_{x_{s}}[x_{1}]-T\right)$
$\displaystyle\quad\quad\quad\quad\quad\quad\times\left(\ \ \
\underset{x_{2}(0)=x_{i}}{\overset{x_{2}(T)=x_{s}}{\int}}{\mathcal{D}}x_{2}\
e^{iS[x_{2}]}\right)^{*}+c.c.$ (17)

Similarly, the normalization factor $Z$ can be represented as a product of two
path integrals:

$\displaystyle Z(T)=$ $\displaystyle P_{\Omega_{\rm FV}}(T)=\int_{\Omega_{\rm
FV}}{\rm d}x_{v}\langle x_{v},T|x_{i}\rangle\langle x_{i}|x_{v},T\rangle$
$\displaystyle=$ $\displaystyle\int_{\Omega_{\rm FV}}{\rm d}x_{v}\
D_{F}(x_{v},T|x_{i})D_{F}(x_{i}|x_{v},T)$ $\displaystyle=$
$\displaystyle|\mathcal{N}|^{2}\int_{\Omega_{\rm FV}}{\rm
d}x_{v}\underset{z_{1}(0)=x_{i}}{\overset{z_{1}(T)=x_{v}}{\int}}{\mathcal{D}}z_{1}\
e^{iS[z_{1}]}\left(\ \ \
\underset{z_{2}(0)=x_{i}}{\overset{z_{2}(T)=x_{v}}{\int}}{\mathcal{D}}z_{2}\
e^{iS[z_{2}]}\right)^{*}.$ (18)

In the cases $x_{i}=x_{\rm FV}$ and $z_{i}=x_{\rm FV}$, these expressions can
be simplified through Wick rotations, which translate the real-time actions to
Euclidean ones. Crucially, this suggests that one identify the physical time
$T$ with the duration of the imaginary-time interval $\tau$. Motivated by our
previous discussion, we will instead evaluate these path integrals through a
saddle-point approximation in real time.

In Secs. V and VI we show how to construct suitable stationary-phase solutions
for the equations of motions, with the boundary conditions imposed by the path
integral. However, even without having at hand the precise form of these
solutions, it is easy to see that they are fully determined by their boundary
conditions. Formally, this implies $\bar{x}_{1}=\bar{x}_{2}\equiv\bar{x}$,
where $\bar{x}_{1,2}$ is the stationary phase of the $x_{1,2}$ path integral.
As the same holds true for the path integrals in the denominator involving
paths $z_{1,2}$ with corresponding stationary phases $\bar{z}_{1,2}$
(restricted to paths that terminate within $\Omega_{\rm FV}$), we thus find
that the decay rate is, to leading, of the form

$\displaystyle\Gamma=$ $\displaystyle
A\cdot\frac{e^{iS[\bar{x}]-iS[\bar{x}]^{*}}}{e^{iS[\bar{z}]-iS[\bar{z}]^{*}}}=A\cdot
e^{-\left(2{\rm Im}(S[\bar{x}])-2{\rm Im}(S[\bar{z}])\right)}.$ (19)

Here $A$ denotes next-to-leading order corrections, which will not be
addressed in this work. In other words, to leading order the tunneling rate is
only sensitive to the imaginary part of the action evaluated on the
stationary-phase solutions, which may take complex values due to the
regularization of the path integral. In Secs. V and VI, we show that, in the
combined limit of a vanishing regulator and an infinite physical time, these
quantities reproduce the familiar leading-order results for a system without
an explicit time dependence,

$\displaystyle\lim_{T\to\infty}\Gamma(T)=A\cdot\frac{e^{-S_{E}[\bar{x}_{I}]}}{e^{-S_{E}[\bar{z}_{I}]}}.$
(20)

Here $\bar{x}_{I}$ denotes the well-known periodic instanton defined on some
Euclidean time interval $\Delta\tau_{I}$, while $\bar{z}_{I}$ plays a similar
role for the normalization factor.

##  IV Regularized real-time dynamics

In order to clarify the relation between real- and imaginary-time properties
of the instanton mediating the decay, we would like to evaluate the path
integrals in Eqs. (17) and (18) directly. To avoid the conceptual difficulties
linked to a Wick rotation, we will regularize the path integral while keeping
the time explicitly real. This can be achieved by introducing a small
imaginary part to the Hamiltonian, which amounts to the replacement

$H(t)\to(1-i\epsilon)H(t).$ (21)

For the case in which $H$ is independent of time, the replacement of Eq. (21)
is equivalent to performing a Wick rotation. On the other hand, we may perform
the substitution of Eq. (21) even for systems with explicit time dependence,
for which a simple Wick rotation is even more involved [15].

In terms of the point particle’s action, Eq. (21) amounts to the replacement

$\displaystyle S\to(1+i\epsilon)\int{\rm d}t\
\frac{m}{2}\dot{x}^{2}-(1-i\epsilon)^{2}\cdot V(x),$ (22)

where, in this section, an overdot denotes a derivative with respect to $t$.

For concreteness, we may consider a point particle in a double-well potential,
whose action we will take to be

$\displaystyle S=$ $\displaystyle(1+i\epsilon)\int_{0}^{t}{\rm d}t^{\prime}\
\frac{m}{2}\dot{x}^{2}-(1-i\epsilon)^{2}\cdot\frac{\lambda}{4}\left(x^{2}-x_{0}^{2}\right)^{2}$
$\displaystyle=$ $\displaystyle x_{0}^{3}\sqrt{\lambda m}(1+i\epsilon)\int{\rm
d}\tilde{t}^{\prime}\
\frac{1}{2}\dot{\tilde{x}}^{2}-\frac{1}{4}(1-i\epsilon)^{2}\left(\tilde{x}^{2}-1\right)^{2}$
$\displaystyle=$ $\displaystyle x_{0}^{3}\sqrt{\lambda m}\cdot\tilde{S}.$ (23)

In order to obtain the action in the second line, we have rescaled the time
and position variables as $x=\tilde{x}\cdot x_{0}$ and
$t=\sqrt{m/\lambda}\cdot\tilde{t}/x_{0}$. Moving forward, we will drop the
tilde on all variables for simplicity.

Using Eq. (23), it is straightforward to arrive at the equation of motion for
the (complex-valued) saddle point,

$\ddot{x}=-(1-2i\epsilon)x\left(x^{2}-1\right).$ (24)

This equation once again confirms the equivalence with the interpretation of
the imaginary part as a result of a complex-time contour, as the factor
$(1-2i\epsilon)$ on the right-hand side could equivalently be moved to the
left-hand side of the equation and absorbed into the time derivatives through
an infinitesimal Wick rotation $t^{\prime}\to(1-i\epsilon)\cdot t^{\prime}$.

The solutions of Eq. (24) that dominate the path integrals in Eqs. (17)–(18)
are now determined by their respective boundary conditions,

$\displaystyle\bar{x}_{1,2}(0)=x_{i},\ \ \ \bar{x}_{1,2}(T)=x_{s},$ (25)
$\displaystyle\bar{z}_{1,2}(0)=x_{i},\ \ \ \bar{z}_{1,2}(T)=x_{v},$ (26)

as well as the crossing condition for $\bar{x}_{1}$:
$F_{x_{s}}[\bar{x}_{1}]=T$.

Eq. (24) has well-known analytic solutions, which can be expressed in terms of
the Jacobian elliptic functions ${\rm sn}(u,k^{2})$ as

$x_{k}(t)=\sqrt{\frac{2k^{2}}{1+k^{2}}}\,{\rm
sn}\left(\frac{ie^{-i\epsilon}\cdot t-b}{\sqrt{1+k^{2}}},k^{2}\right).$ (27)

The parameter $k$, known as the elliptic modulus, controls both the solution’s
amplitude and oscillation frequency, while the parameter $b$ corresponds to a
translation.

For $\epsilon\to 0$, these solutions reproduce the behavior of a point
particle in the physical potential. In the limit $\epsilon\to\pi/2$,
meanwhile, they describe the motion of a particle in the inverted potential,
including, in particular, the periodic instantons obtained in the conventional
picture [42, 43, 44, 47, 45, 48].

A similarly simple interpretation can be obtained for the case of a small but
finite $\epsilon$, for which the path can be decomposed into a real and
imaginary part, $\bar{x}(t)=\bar{x}_{\rm Re}(t)+\bar{x}_{\rm Im}(t)$. Their
corresponding equations of motions are given by

$\displaystyle\ddot{\bar{x}}_{\rm Re}=$ $\displaystyle\bar{x}_{\rm
Re}\left(1-\bar{x}_{\rm Re}^{2}+3\bar{x}_{\rm
Im}^{2}\right)+2\epsilon\cdot\bar{x}_{\rm Im}\left(1+\bar{x}_{\rm
Im}^{2}-3\bar{x}_{\rm Re}^{2}\right),$ $\displaystyle\ddot{\bar{x}}_{\rm Im}=$
$\displaystyle\bar{x}_{\rm Im}\left(1+\bar{x}_{\rm Im}^{2}-3\bar{x}_{\rm
Re}^{2}\right)+2\epsilon\cdot\bar{x}_{\rm Re}\left(\bar{x}_{\rm
Re}^{2}-1-3\bar{x}_{\rm Im}^{2}\right).$

For small $t$ relative to the system’s typical time-scale, when $\bar{x}_{\rm
Im}(t)\simeq 0$, the system acts essentially classically, leading to an
oscillatory motion of $\bar{x}_{\rm Re}(t)$ in the false-vacuum basin. This
motion sources an excitation of the imaginary part $\bar{x}_{\rm Im}(t)$
through the interactions conveyed by a non-vanishing $\epsilon$, which in turn
backreacts on $\bar{x}_{\rm Re}(t)$. This process can unfold in two physically
relevant ways. In Sec. V, we construct the real-time equivalent of the well-
known instanton satisfying the boundary conditions of Eq. (25), which we refer
to as steadyons. Then, in Sec. VI, we discuss solutions satisfying the
boundary conditions in Eq. (26), which are imposed by the normalization factor
$Z$.

Lastly, we note that, as for all path integrals, the stationary-phase method
we employ is only capable of capturing the contribution arising from the
neighborhood of one particular configuration, or at best a finite-dimensional
submanifold of path space in theories giving rise to collective coordinates.
This aspect becomes even more subtle in our picture. If one did not regularize
the path integral, there would not exist any stationary-phase solutions. For
this reason, the regularization of Eq. (21) can be understood as deforming the
theory in such a way as to “create” a viable stationary-phase solution at one
particular point in path space, which in turn allows for analytical access to
its neighborhood.444Something similar is already necessary to calculate the
decay rate of the electroweak vacuum in the Standard Model. As discussed in
Ref. [7], no instanton exists for the full potential of this theory due to the
Higgs field’s mass term, despite its small numerical value relative to the
relevant scales of the process. This problem can, however, be resolved through
a deformation of the theory by removing the problematic mass term entirely. It
can then be shown that this does, indeed, induce an error to the total decay
rate, which vanishes in the limit $m^{2}\to 0$.

In Sec. V.2, we show that our procedure can be used to construct stationary
phases along a one-parameter family of points in path space. In the imaginary-
time picture, these points correspond to different initial momenta of their
corresponding instantons.

##  V Steadyon contribution

The real-time equivalent of an instanton would correspond to a solution to the
equations of motion of the form in Eq. (27), together with the boundary
conditions in Eq. (25). Crucially, these conditions do not specify the initial
velocities of $\bar{x}_{\rm Re}(t)$ and $\bar{x}_{\rm Im}(t)$, much as the
initial velocity of an instanton solution is undetermined in the usual
imaginary-time picture. For tunneling out of the false vacuum, this degeneracy
is eliminated through the infinite length of the imaginary-time interval, as
laid out in great detail in Ref. [6]. It is now easy to see that, even if one
were to start out in the imaginary-time picture used in Ref. [6], the same
procedure cannot be applied to the case of interest in this article.

We therefore consider two different classes of solutions for the real-time
dynamics, which we dub steadyons. We adopt this term because such real-time
solutions describe a continuous transition from the initial state to the
desired final state, over an infinitely long physical time. We first consider
periodic steadyon solutions in Sec. V.1, corresponding to vanishing initial
velocity, before generalizing to the case of nonvanishing initial velocities
in Sec. V.2. As we will see, in the latter case, such solutions include the
dominant contribution to the decay rate.

###  V.1 Periodic steadyon solutions

In Sec. II, we found that the real- and imaginary-time versions of the
instanton describing tunneling out of the false vacuum in a system without
explicit time-dependence were related to one another through a simple analytic
continuation. For now focusing on the special case of a periodic solution,
this suggests the choice of parameters

$\displaystyle k^{2}=\frac{x_{i}^{2}}{2-x_{i}^{2}},$ (28) $\displaystyle
b=\sqrt{1+k^{2}}\int_{0}^{\frac{\pi}{2}}{\rm
d}\theta\frac{1}{\sqrt{1-k^{2}{\rm sin}(\theta^{2})}}.$ (29)

An example of such solutions is shown in Fig. 2.

Figure 2: The periodic steadyon solution for $x_{i}=-1/2$ and $b$ given by Eq.
(29).

The solutions in Fig. 2 can be understood easily in terms of our discussion in
the previous section. At $t=0$, the particle sets out at the appropriate
initial position with a vanishing velocity for both $\bar{x}_{\rm Re}$ and
$\bar{x}_{\rm Im}$. The oscillatory motion of $\bar{x}_{\rm Re}$ then causes
an excitation of $\bar{x}_{\rm Im}$, whose backreaction on $\bar{x}_{\rm Re}$
ultimately causes the latter to leave the false-vacuum basin, after which the
backreaction starts damping the oscillation of $\bar{x}_{\rm Re}$. Eventually,
$\bar{x}_{\rm Im}$ once again vanishes, as $\bar{x}_{\rm Re}$ transitions back
to its classical periodic motion, albeit now within the true-vacuum basin.
Physically, this suggests that the transition from one basin to the other is
controlled by the product $\epsilon\cdot t$. Understanding the regularization
as the introduction of a complex time variable, this combination can be
identified with the change in imaginary time along the complex-time contour,
in agreement with our observation regarding the tunneling out of the minimum
in Sec. II.

Indeed, when understood as analytic functions of a complex argument
$ie^{-i\epsilon}t$, the solutions of Eq. (27) are doubly-periodic in the
complex plane. This relates, in particular, to the period of the periodic
instanton $\Delta\tau_{I}$, which can immediately be identified with the
period of the solution in Eq. (27) along the imaginary-time axis. Meanwhile,
the period along the real-time axis can be recognized as the period of the
oscillatory motion within the false-vacuum basin in real time. Moving forward,
we will denote the period of this classical motion as $\Delta t_{R}$.

The transition of the particle from one basin to the other amounts to
evolution through one-half period of the periodic motion along the imaginary-
time direction. Together with the crossing condition, this implies that a
transition from $x_{i}$ to $x_{s}$ after the time $T$ would require

$\displaystyle\epsilon\cdot T=\frac{\Delta\tau_{I}}{2}\ \ {\rm and}$ (30)
$\displaystyle T=N\cdot\Delta t_{R},\ \ {\rm with}\ \ N\in\mathbb{N}.$ (31)

Here $\Delta\tau_{I}$ denotes the full period of the periodic (imaginary-time)
instanton connecting $x_{i}$ to $x_{s}$, while $\Delta t_{R}$ is the duration
of the particle’s classical oscillatory motion within the false-vacuum basin.
Note that, in general, both of these parameters depend on $x_{i}$.

These conditions are clearly not satisfied simultaneously for arbitrary times
$T$. In other words, for general $T$, the solutions described by Eq. (27) fail
to reproduce the correct boundary conditions of Eqs. (25)–(26), as illustrated
in Fig. 3. This behavior can be easily understood through the splitting of the
dynamics into an independent real and imaginary part. There is a priori no
reason for the oscillatory motion of the real part to arrive at the location
$x_{s}$ at the very moment when the imaginary part vanishes identically. Even
worse, as the period of the real part’s dynamics is independent of $\epsilon$,
Eq. (31) seems to suggest that the steadyon can only satisfy the appropriate
boundary condition during a countable subset of physical times.555This
behavior is a consequence of our choice of the initial state together with the
semi-classical picture, which assumes that the tunneling occurs out of the
classical motion’s turning point.

To address this issue, we first recall that the impossibility to perform the
stationary-phase approximation using the solutions of Eq. (27) for some
physical time $T$ does not have an immediate physical meaning—the decay rate
is still well-defined, we just cannot compute it in the way we were hoping
for. However, as we will see in the next subsection, knowledge of the decay
rate on a countable set of times is sufficient to deduce the physical decay
rate in the limit of interest. Starting from this premise, we can focus our
attention on those discrete times for which it is at least in principle
possible to satisfy the boundary conditions.

Figure 3: The periodic steadyon around $\Delta\tau_{I}/(2\epsilon)$ for
$x_{i}=-1/2$ and different values of $\epsilon$. For $t_{s}=N\cdot\Delta
t_{R}$ with $N\in\mathbb{N}$, $\bar{x}_{\rm Re}$ coincides with the value of
the corresponding instanton $\bar{x}_{I}$ evaluated at $\tau=\epsilon\cdot
t_{s}$, and $\bar{x}_{\rm Im}$ vanishes. These times, however, do not
necessarily agree with the physical time $T$, as would be required by the
boundary conditions in Eqs. (25)–(26). Thus, for any given $\epsilon$, we may
restrict ourselves to $T=t_{s}$. In that case, the boundary condition is not
satisfied exactly, but only up to some offset $\propto\epsilon$, which
vanishes in the limits of interest.

Following our previous discussion, the emergence of a non-vanishing imaginary
part $\bar{x}_{\rm Im}(t)$ can be understood as induced through its
interaction with the real part $\bar{x}_{\rm Re}(t)$, whose strength is
controlled by $\epsilon$. As an immediate consequence, taking the limit
$\epsilon\to 0$ implies that the amplitude of $\bar{x}_{\rm Im}(t)$ vanishes
in the vicinity of its root, as illustrated in Fig. 3. In other words, in the
limit $\epsilon\to 0$ the imaginary part $\bar{x}_{\rm Im}(t)$ vanishes within
a time interval roughly of length $\Delta t_{R}$. Thus, the boundary
conditions of Eqs. (25) and (26) can be approximately satisfied by identifying
$T$ with the time $t_{s}$ closest to $\Delta\tau_{I}/(2\epsilon)$ at which
$\bar{x}(t_{s})\simeq x_{s}$ with the highest accuracy. A sensible measure to
define the deviation of the boundary conditions is $\Delta
x_{s}^{2}=(\bar{x}_{\rm Re}(T)-x_{s})^{2}+\bar{x}_{\rm Im}(T)^{2}$.666This
choice, of course, is arbitrary, and in Sec. V.3 we demonstrate that our
result for the tunneling rate is independent of how we choose to quantify the
departure from exact boundary conditions, as long as $\Delta x_{s}^{2}\to 0$
in the appropriate limit. In cases in which there exist multiple values of
$t_{s}$ that yield a similar convergence rate, we choose the smallest such
value, as suggested by the crossing condition. And indeed, as shown in Fig. 4
for the concrete example corresponding to $x_{i}=-1/2$, taking the limit
$\epsilon\to 0$ and choosing $T=t_{s}$ leads to $\Delta x_{s}\to 0$.

Figure 4: Convergence of the boundary condition at $t_{s}$ in the limit
$\epsilon\to 0$ for the periodic steadyon. Figure 5: The relation between the
time $t_{s}$ and the regularization parameter $\epsilon$ for the periodic
steadyon.

This behavior can equivalently be understood by observing that $\bar{x}_{\rm
Re}(t)=\bar{x}_{I}(\epsilon\cdot t)$ whenever $\bar{x}_{\rm Im}(t)=0$. In the
limit $\epsilon\to 0$, the time $t_{s}$ coincides with the time closest to
$\Delta\tau_{I}/(2\epsilon)$ where this occurs. However, since the evolution
of $\bar{x}_{I}(t)$ is controlled by the combination $\epsilon\cdot t$, it is
easy to see that the residual difference between $\bar{x}_{\rm Re}(t_{s})$ and
$x_{s}$ satisfies

$\displaystyle\bar{x}_{\rm Re}(t_{s})=$
$\displaystyle\,\bar{x}_{I}(\epsilon\cdot t_{s})$ (32) $\displaystyle=$
$\displaystyle\,x_{s}-|\dot{\bar{x}}_{I}(\Delta\tau_{I}/2)|\cdot\epsilon\cdot|t_{s}-\Delta\tau_{I}/(2\epsilon)|$
$\displaystyle-\frac{1}{2}|\ddot{\bar{x}}_{I}(\Delta\tau_{I}/2)|\cdot\epsilon^{2}\cdot|t_{s}-\Delta\tau_{I}/(2\epsilon)|^{2}+\mathcal{O}(\epsilon^{3}).$

By construction $|t_{s}-\Delta\tau_{I}/(2\epsilon)|\leq\Delta t_{R}={\rm
constant}$, so we find that the boundary value of the steadyon at $t=t_{s}$
indeed converges toward the desired emergence point $x_{s}$. For the periodic
steadyons, this convergence is further accelerated, given that
$\dot{\bar{x}}_{I}(\Delta\tau_{I}/2)=0$.

We may now recall that one of the main motivations for our analysis was the
simple observation that the usual interpretation of imaginary time as Wick-
rotated physical time seemed to suggest that an instanton can only exist at
one finite physical time, of order of the system’s natural time-scale. At
first glance, this appears to be true even in our more careful treatment, as
there is still only one unique value for $t_{s}$ for any given $\epsilon$.
This apparent problem can be resolved thanks to two simple observations.
First, we note that the relation between $x_{i}$, $t_{s}$, and $\epsilon$
necessary to (approximately) satisfy the boundary condition can be understood
equivalently as determining $\epsilon$ in terms of the physical parameters
$x_{i}$ and $t_{s}$, with $\epsilon\sim\Delta\tau_{I}/(2t_{s})$. Second, we
recall that in our picture, $\epsilon$ is not an entirely free parameter, but
corresponds to a regularization of the Hamiltonian, and hence needs to be
taken to $0$ at the end of a given calculation. These two points suggest that
for any suitable time $t_{s}$ of interest, we may choose the “right”
regularization parameter $\epsilon$ to ensure the existence of an approximate
stationary phase. Inevitably, any such choice implies a finite error. However,
as we consider larger and larger physical times (in discrete steps), we also
lower the value of $\epsilon$ required, implying an increasingly better
approximation. In the limit $T=t_{s}\to\infty$, this automatically yields
$\epsilon\to 0$.

###  V.2 Non-periodic steadyon solutions

Figure 6: The steadyon describing the same tunneling process as in Fig. 2, but
with the initial velocity $\dot{\bar{x}}_{\rm Im}(0)\neq 0$ corresponding to
the largest contribution to the decay rate.

Having understood the special case of a periodic steadyon solution, we can now
generalize our previous discussion to cases with an initial velocity. An
important feature of the solutions of Eq. (27) is that their imaginary part
vanishes exactly at the turning points of the real part, as shown, e.g., in
Fig. 3. Therefore the boundary conditions of Eq. (25) require
$\dot{\bar{x}}_{\rm Re}(0)=\dot{\bar{x}}_{\rm Re}(T)$, leaving only
$\dot{\bar{x}}_{\rm Im}(0)$ as a free parameter.

Returning to our interpretation of the steadyon as two interacting degrees of
freedom, we can immediately understand the effect of this initial velocity.
For the periodic solution, we had found that the tunneling occurs as the
oscillations of $\bar{x}_{\rm Re}(t)$ and $\bar{x}_{\rm Im}(t)$ drive each
other until $\bar{x}_{\rm Re}(t)$ crosses from the false-vacuum basin
$\Omega_{\rm FV}$ into the other well $\Omega_{R}$. Upon giving $\bar{x}_{\rm
Im}(t)$ an initial velocity, this process will unfold faster for a given value
of $\epsilon$, also leading to a smaller $\Delta\tau_{I}$. See Fig. 6.

Again making use of the connection between steadyon solutions and their
corresponding instantons, this suggests that the steadyon with finite initial
velocity $\dot{\bar{x}}_{\rm Im}(0)$ can also be described by a solution of
the form in Eq. (27), albeit with a larger value of $k$. In Sec. V.3, we will
find that for any $x_{i}$ the largest contribution to the decay rate is given
by $k=1$, which corresponds to the false-vacuum instanton (or, equivalently,
steadyon). In other words, the solution leading to the most important
contribution to the overall decay rate is the same as for the tunneling out of
the false vacuum, defined on a shorter interval $\Delta\tau_{I}$ to match the
desired boundary conditions of Eq. (25). This is precisely the case
illustrated in Fig. 6.

The realization of these boundary conditions can be achieved in a similar way
as for the periodic case. For general physical times, no exact solution
exists. There exists, however, a countable number of physical times for which
$\bar{x}_{\rm Im}(t)$ vanishes and $\bar{x}_{\rm Re}(t)$ takes a value close
to the emergence point $x_{s}$, which agrees with the value of the instanton
at $\tau=\epsilon\cdot T$. This offset is controlled by $\epsilon$, and
therefore vanishes in the limit $\epsilon\to 0$, $T\to\infty$, consistent with
Eq. (32).

###  V.3 Real-time action and instanton limit

In the previous subsection, we have found that the steadyon only serves as a
stationary phase in the limit $\epsilon\to 0$, and even then only for a
countable number of points in time. First, we observe that the non-existence
of a stationary phase only implies that we cannot use the stationary-phase
approximation to evaluate the path integrals at hand, whereas the decay rate
itself nonetheless has a well-defined value, which could be calculated, e.g.,
on a lattice. Next, we are primarily interested in the rate in the combined
limit $\epsilon\to 0$, $T\to\infty$. For this purpose, being only able to
calculate the decay rate for a countable subset of times is sufficient as long
as our results allow us to identify an unambiguous asymptotic value.

Figure 7: The exponent of the decay rate arising from the periodic steadyon
contribution. We recover the Euclidean result in the appropriate limits.

Figure 8: The exponent of the decay rate arising from the steadyon
contribution with nonzero initial velocity corresponding to the largest
contribution to the decay rate. We again recover the Euclidean result in the
appropriate limits.

In order to illustrate this behavior, we calculate the imaginary part of the
steadyon action for multiple values of $\epsilon$ by simply inserting the
corresponding solutions into the regularized action of Eq. (22) with the
appropriate $t_{s}(\epsilon)$ for both the periodic steadyon as well as the
steadyon with nonzero initial velocity, which we will later find to dominate
the tunneling rate. And indeed, taking the limit in the suggested way leads to
a clear convergence. Crucially, we find that the limiting value is given
precisely by the Euclidean action of the corresponding instanton in the
imaginary-time picture. See Figs. 8 and 8.

To understand this behavior, we reconsider the regularized action of Eq. (22),
the resulting equation of motion in Eq. (24), and its analytic solutions in
Eq. (27). We note that, due to the time-independence of the Hamiltonian in
this case, we could formally assign the small imaginary part to the time
variable, which can be understood as evaluating all functions along the
complex-time path $\gamma_{\epsilon}(t)=(1-i\epsilon)t$ rather than along the
real-time axis. This suggests that we may understand the steadyon solutions,
as well as their Lagrange functions, as their analytic continuations evaluated
along this contour. Most important, these functions are regular and analytic
between $\gamma_{\epsilon}$ and the real- and imaginary-time axes. Thus, we
may deform the integration contour from $\gamma_{\epsilon}$ onto these axes,
as shown in Fig. 9.

$t_{s}$$t$$\tau$Im$\left(S\right)=0$$t_{s}\cdot\epsilon$$O(\epsilon)$$\Delta\tau_{I}/2$$\epsilon$
Figure 9: The exponent of the decay rate arises from the integral of the
steadyon’s complex Lagrange function over the diagonal time contour
$\gamma_{\epsilon}$. This contour can be deformed into pieces along the real-
and imaginary-time axis, respectively. The length of the imaginary-time piece
coincides with $\Delta\tau_{I}/2$ up to a factor of order of the
regularization parameter $\epsilon$.

We first consider the contribution from the real axis, along which the
steadyon reduces to the solution of the classical equation of motion. This
implies that its corresponding action is strictly real. Thus, this part of the
contour does not contribute to the leading-order exponent of the tunneling
rate, which, as we found in Eq. (19), depends only on ${\rm Im}(S[\bar{x}])$.

Along the imaginary-time axis, the steadyon reduces to the familiar instanton
solution, with an additional factor of $i$ coming from the integration along
the imaginary-time direction. Following our previous discussion, the
imaginary-time integral covers the interval
$[0,t_{s}(\epsilon)\cdot\epsilon]$. The double-periodicity of the Jacobian
elliptic functions then implies that the initial value of the instanton at
$\tau=0$ is given by $x_{i}$ in the limit $\epsilon\to 0$. Meanwhile, by
construction we have
$|t_{s}(\epsilon)\cdot\epsilon-\Delta\tau_{I}/2|=\mathcal{O}(\epsilon)$.
Altogether, this implies that the contribution of the imaginary-time contour
to the total action is strictly imaginary, and is equal to one-half of the
instanton’s Euclidean action, up to a factor of ${\cal O}(\epsilon)$. Hence,
in the limit $\epsilon\to 0$, we find that the imaginary part of the
steadyon’s complex action agrees with one-half of the instanton’s Euclidean
action, such that the exponent of the decay rate satisfies

$2\cdot{\rm Im}(S[\bar{x}])\to S_{E}[\bar{x}_{I}].$ (33)

Given our results for the periodic steadyon, it appears reasonable to expect a
similar behavior for the steadyon with initial velocity. As shown in Fig. 8,
this is indeed the case. To understand this result more generally, we observe
that much like the periodic steadyon, the solution with initial velocity can
be understood via its analytic continuation evaluated on the complex-time
contour $\gamma_{\epsilon}$. Thus, we may again decompose this complex contour
into its projection onto the real- and imaginary-time axes.

Figure 10: The analytic continuation onto the real-time axis of the steadyon
solution with nonvanishing initial velocity which yields the dominant
contribution to the overall decay rate.

Figure 11: The Lagrange function of the steadyon depicted in Fig. 11. Note
that its imaginary part is an anti-symmetric function around the center of the
real-time interval of length $\Delta t_{R}$.

However, unlike for the periodic steadyon, we find that the analytic
continuation onto the real-time axis is not a strictly real function linked to
classical behavior, as shown in Fig. 11. This, in turn, yields a complex
Lagrange density, see Fig. 11. First we note that the double-periodicity of
our more general solutions of Eq. (27) manifests, again, on the real-time
axis. Next we observe that within each period of length $\Delta t_{R}$, the
Lagrange function is asymmetric, such that its net contribution to the action
per interval of this length vanishes. Following our discussion around Eq.
(31), the boundary conditions imply that, in the limit $\epsilon\to\infty$,
$t_{s}\to N\cdot\Delta t_{R}$, with some positive integer $N$. Thus, although
the Lagrange function on the real-time axis is not manifestly real itself, its
overall contribution to ${\rm Im}(S[\bar{x}])$ vanishes in the appropriate
limit. As the contribution from the imaginary-time axis reproduces, by
construction, that of the instanton, we recover Eq. (33).

In the imaginary-time picture, it is straightforward to identify the
stationary-phase solution corresponding to the dominant contribution to the
decay rate. To do so, we solve the equation of motion in imaginary time for
varying values of $v_{0}\equiv\dot{\bar{x}}_{I}(0)$. This covers, in
particular, the bounded solutions giving rise to the solutions of Eq. (27), as
well as unbounded solutions not captured by this expression. Importantly, each
of these actions is defined on a different Euclidean-time interval
$\Delta\tau_{I}$, in agreement with our previous discussion.

In Fig. 12, we plot the Euclidean action as a function of $v_{0}$, again
describing the tunneling out of the initial position $x_{i}=-1/2$, the same
example we considered above.

Figure 12: The Euclidean action of the instanton describing tunneling out of
our previously considered example. The largest contribution arises from the
instanton with initial velocity $v_{0}\equiv\dot{\bar{x}}_{I}(0)\simeq 0.53$,
corresponding to the false-vacuum instanton with $k=1$.

##  VI Normalization factor

Having understood the numerator of the decay rate, it remains to evaluate the
denominator, i.e., the normalization factor $Z=P_{\Omega}$. Once again, we
wish to do so by means of the stationary-phase approximation, which requires
us to match the solutions of Eq. (27) with the boundary conditions of Eq.
(26). Before doing so for the real-time case, it is helpful for the
interpretation of our results to consider the imaginary-time picture, which we
will ultimately recover in the relevant limits.

###  VI.1 Pulls

In the limit $x_{i}\to x_{\rm FV}$, the saddle point of the normalization
factor is simply given by $\bar{z}_{I}(\tau)=x_{\rm FV}$. We furthermore
anticipate that the saddle-point solution will be defined on an imaginary-time
interval of the same length as the instanton, $\Delta\tau_{I}$.

This suggests that we consider solutions that begin at $x_{i}$ with an initial
velocity that is just large enough for the particle to reach some turning
point $x_{p}$ at $\tau=\Delta\tau_{I}/2$ and return to $x_{i}$ at
$\tau=\Delta\tau_{I}$, as in Fig. 13. We refer to such solutions as “pulls”:
they begin at $x_{i}$ and are “pulled back” to some point $x_{v}$ that is also
within the false-vacuum basin $\Omega_{\rm FV}$. We find that, for any pair
$\Delta\tau_{I}$ and $x_{i}$, these solutions minimize the Euclidean action,
that is, they give rise to the dominant contribution to the normalization
factor. This allows us to evaluate the integral over the steadyon’s end point
$x_{v}$ in Eq. (18) using a saddle-point approximation around the value of
$x_{v}$ corresponding to this behavior.

For the remainder of this section, we will focus on combinations of $x_{i}$,
$T$ and $\epsilon$ that give rise to the dominant stationary phase, i.e.,
$k=1$. In particular, these solutions require $\Delta\tau_{I}\to 0$ as $x_{i}$
approaches the top of the potential barrier.

Figure 13: The instanton linked to the dominant contribution to the decay
rate, $\bar{x}_{I}(\tau)$, together with the pull, $\bar{z}_{I}(\tau)$, for
our example with $x_{i}=-1/2$.

###  VI.2 Drags

Figure 14: The pull, $\bar{z}_{I}$, and the real and imaginary part of its
corresponding drag, $\bar{z}_{\rm Re}$ and $\bar{z}_{\rm Im}$ respectively,
for $x_{i}=-1/2$

In order to determine the real-time analogue of the pull—which we dub the
“drag”—we again begin with the general set of solutions given in Eq. (27).
First we recall that the instanton emerged from the time evolution of the
amplitude of the real part of the steadyon solution. This suggests that we
consider the complex solution of Eq. (27), which similarly reproduces the
pull, as shown in Fig. 14. Crucially, the parameters of this solution are the
same as those of the corresponding pull. In other words, $\epsilon$ is fixed
by the requirement that the steadyon solution exists, while $k$ and $b$ are
determined by $x_{i}$ and $T$.

The drag can also be understood along the lines of our discussion in Secs. V.1
and V.2. At $t=0$, the imaginary part of the particle’s motion begins with a
large velocity dampening the oscillatory motion of the real part within the
false-vacuum basin. At $t=t_{s}$, the amplitude of the latter reaches its
smallest value within this process, which is given by the turning point of the
pull.

Unlike the steadyon solution, the drag is not affected by the small but finite
mismatch between the analytic solution and the relevant boundary conditions,
as discussed (for the steadyon case) below Eqs. (30)–(31). First, unlike for
the steadyon, the value $x_{v}$ of the drag at $t=t_{s}$ is not fixed, but can
take any real value, since in the end we perform the remaining integral over
$x_{v}$. The boundary condition for the imaginary part, meanwhile, can be
satisfied in the limit $\epsilon\to 0$ by choosing the elliptic modulus $k$
corresponding to the pull, as shown in Fig. 14. The fact that the pull reaches
its turning point at $\tau\simeq t_{s}\cdot\epsilon$ translates to a roughly
constant oscillation amplitude for the real part of the drag, namely, an
oscillation of the particle around the false vacuum. In the limit $\epsilon\to
0$, this approximates a classically allowed motion, and hence $\bar{z}_{\rm
Im}(t_{s})\to 0$ in the limit $\epsilon\to 0$. Thus, the boundary conditions
of Eq. (26) are indeed satisfied by the drag with the choice of $k$
corresponding to the pull in the limit $\epsilon\to 0$.777Alternatively, this
allows one to understand the choice of $k$ as a consequence of the requirement
that the imaginary part of the steadyon vanishes at the emergence time
$t=t_{s}$.

###  VI.3 Real-time action and pull limit

In Sec. V.3, we demonstrated that the imaginary part of the action for both
periodic and non-periodic steadyon solutions converges to the Euclidean action
of the associated instanton in the appropriate limit, $\epsilon\to
0,\,T\to\infty$. As shown explicitly in Fig. 15, this is also the case for the
drag $\bar{z}(t)$ and its corresponding pull $\bar{z}_{I}(\tau)$. Just as for
the steadyon, this behavior can be understood by considering the analytic
continuation of $\bar{z}(t)$ onto the full complex-time plane.

In Sec. III, we had found that the numerator of the decay rate is just the
time-derivative of its denominator. This suggests that one use the same
combination $(\epsilon,t_{s})$ for the drag and steadyon, with the latter
establishing a dependence $t_{s}(\epsilon)$. Thus, the action of the drag can
be understood as an integral over the same complex-time contour
$\gamma_{\epsilon}$ as the steadyon, which we can once again decompose into
its projections onto the real- and imaginary-time axes. Just as for the
steadyon solution with initial velocity, we find that the drag’s projection
onto the real-time axes does not describe a classically allowed motion. This
gives rise to a complex Lagrange function, which is again anti-symmetric
within each interval of length $\Delta t_{R}$.

However, since the drag $\bar{z}(t)$ corresponds to a different choice of the
elliptic modulus $k$ than for the steadyon solution $\bar{x}(t)$, the period
of the drag’s real-time projection does not necessarily match that of the
steadyon. Thus, in general, the value of the drag at $t=t_{s}$ is not
necessarily identical to the turning point $x_{p}$ of the pull. As the latter
serves as the initial value for dynamics along the imaginary-time axis, the
projection of the drag onto the imaginary-time axes does not necessarily
reproduce the pull, and the integral over the real-time axis contributes to
the imaginary part. The results shown in Fig. 15 suggest that the latter
compensates precisely for the deviation in the contribution from the
imaginary-time axis.

Figure 15: The imaginary part of the exponent of the normalization factor. We
again recover the desired Euclidean-time result in the limit $\epsilon\to 0$.

Figure 16: Left panel: The Euclidean action of the dominant instanton as a
function of the initial positon $x_{i}$. Middle panel: The Euclidean action of
the pull corresponding to the dominant instanton as a function of the initial
positon $x_{i}$. Right panel: The leading order exponent of the decay rate as
a function of the initial positon $x_{i}$.

Our previous reasoning can now also be used to quickly understand this
behavior. Whereas in general $x_{v}\neq x_{p}$, it is easy to see that for any
$x_{i}$ there exists a (countable) infinite number of choices for $\epsilon$
and $t_{s}$ for which $x_{v}\simeq x_{p}$ to arbitrary accuracy. For these,
the integral over the real-time axis contains a whole number of periods, each
of which gives rise to a strictly real action along the real-time axis. Having
$x_{v}\simeq x_{p}$ also implies that the solution along the imaginary-time
contour is, to very good accuracy, given by the pull, such that the
contribution from this branch is nothing but the Euclidean action of the pull
solution. As we can expect the path integral to converge to a well-defined
value, we can use this countable set of stationary-phase results to deduce
this value—and thus, by extension, also the value of the complex-valued action
for those values of $\epsilon$ for which the saddle point exists, but for
which the action takes a less straightforward form.

###  VI.4 Effect of the pull on the leading-order decay rate

We can distinguish two regimes for the pull, based on the value of $x_{p}$.
First, if $x_{i}$ is sufficiently close to $x_{\rm FV}$ and $\Delta\tau_{I}$
is large enough, the Euclidean action is minimized for values of $x_{p}$ close
to $x_{\rm FV}$. This implies that, for most of the time interval
$\Delta\tau_{I}$, the particle hovers near $x_{\rm FV}$, corresponding to a
near-vanishing Euclidean Lagrangian. In order to get to this point, the
particle needs a large enough initial velocity, giving rise to a contribution
to the Euclidean action from the kinetic term. If, on the other hand, $x_{i}$
is further away from $x_{\rm FV}$ and $\Delta\tau_{I}$ is small enough, the
reduction in Euclidean action achieved by approaching $x_{p}$ cannot
compensate for this additional contribution. In the most extreme case of a
near-vanishing $\Delta\tau_{I}$, this can lead to the pull essentially
approaching a constant function with $x_{p}\simeq x_{i}$, with a vanishing
Euclidean action.

We find that this last scenario is indeed realized in our toy-model, with
$\Delta\tau_{I}\to 0$ as $x_{i}$ approaches the top of the potential barrier
at $x_{i}=0$. See Fig. 16, in which we present the Euclidean actions of the
dominant instanton, its corresponding pull, and the overall leading-order
exponent of the tunneling rate. First, we observe that tunneling becomes
unsuppressed as we approach the top of the potential barrier
($S_{E}[\bar{x}]\to 0$), as expected. Next, we find that the normalization
factor gives rise to a significant contribution to the tunneling rate in the
intermediate regime.

##  VII Conclusion

We have derived a complete path-integral expression for the tunneling rate out
of an excited state from first principles. On a technical level, our most
important result is a strategy for the evaluation of path integrals in real
time. To avoid the complications linked to a Wick rotation by a finite angle,
we regularize the path integral by introducing a small imaginary part for the
Hamiltonian, controlled by a regularization parameter $\epsilon$. This
procedure is well-defined in both quantum mechanics and quantum field theory,
and has also been shown to produce meaningful results in curved backgrounds
and even for systems with an explicit time-dependence [15].

For systems without an explicit time-dependence, this step is equivalent to an
infinitesimal Wick rotation onto a complex-time contour $\gamma_{\epsilon}$.
This regularization allows us to identify a countable number of points in the
$(\epsilon,T)$-plane for which the path integral can be evaluated using the
stationary-phase method. We refer to the (approximate) stationary-phase
solutions as steadyons. Most important, these points approach the real-time
axis in the limit of a vanishing regulator. Assuming continuity of the decay
rate, the results obtained using the steadyons then converge to the physical
decay rate in the combined limit $\epsilon\to 0,\,T\to\infty$. We find that,
for systems without an explicit time-dependence, this reproduces the familiar
result obtained in the usual imaginary-time picture. This requires the
steadyon’s analytic continuation to be doubly-periodic and regular between the
real-time axis and $\gamma_{\epsilon}$. This can be expected for a wide class
of physical systems and is easily confirmed for the imaginary-time instanton,
allowing for a straightforward generalization of our result.

It is worth noting that these properties are, in general, non-trivial. For
field-theoretical systems, for example, dissipation effects can prevent the
existence of a periodic solution, e.g., if the initial state of interest is
some localized excitation, or if one is considering a system within a time-
dependent background. The complications linked to these properties, however,
would only affect the instanton limit—in other words, our general strategy for
calculating the decay rate using steadyons remains valid for such systems,
whereas the usual Wick-rotated instantons do not.

While motivated by the need to realize the boundary conditions describing
quantum tunneling, our method can also be used to evaluate more general path
integrals with non-trivial boundary conditions. This includes, in particular,
the normalization factor of the tunneling rate. In the real-time picture, the
latter is dominated by solutions which we called drags, which we have shown to
be linked to imaginary-time solutions called pulls in a similar way as the
steadyons are related to instantons. The “drag” contributions stem from
summing over paths that remain within the false-vacuum basin $\Omega_{\rm
FV}$.

Our work leaves open two important conceptual questions. First, it appears
evident that our technique can be readily generalized to more complicated
systems, in particular those with an explicit time-dependence. As our recovery
of the imaginary-time picture hinged on the absence of such a dependence,
considering such systems offers a promising path towards an even deeper
understanding of quantum tunneling phenomena across a broad range of
scenarios.

Second, our analysis so far captures only the leading-order contribution to
the decay rate. It is, however, well-known that many important aspects of the
tunneling rate can only be understood at the next-to-leading order. This
includes the prefactor of the exponent—that is, the term $A$ in Eq. (19)—as
well as the effect of possible zero modes. For the tunneling rate out of a
false vacuum, it is well-known that eliminating the latter introduces an
additional imaginary part to the decay rate, which, in some cases, can exactly
cancel the contributions of some individual stationary phases [6]. While this
task would be straightforward once one has transitioned to the imaginary-time
picture, it has recently been shown that this transition itself is already
quite non-trivial for the simpler case of tunneling out of a false vacuum
[39].888For a discussion of the functional determinants of periodic instantons
in this case, consider, e.g., Refs. [42, 43]. A similar discussion can be
found in Ref. [49]. Most important, these authors address the zero mode by
introducing a time-dependent collective coordinate, which also allows for the
preservation of the stricter boundary conditions emerging from our approach.
For a more general discussion of such degrees of freedom, consider Ref. [50].

###### Acknowledgements.

We thank Wen-Yuan Ai, Morgane König, and Matthew D. Schwartz for valuable
discussions. TS’s contributions to this work were made possible by the Walter
Benjamin Programme of the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) – 512630918. Portions of this work were conducted in
MIT’s Center for Theoretical Physics and partially supported by the U.S.
Department of Energy under Contract No. DE-SC0012567. This project was also
supported in part by the Black Hole Initiative at Harvard University, with
support from the Gordon and Betty Moore Foundation and the John Templeton
Foundation. The opinions expressed in this publication are those of the
author(s) and do not necessarily reflect the views of these Foundations.

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