# The Path to Fault- and Intrusion-Resilient
Manycore Systems on a Chip

Ali Shoker Paulo Esteves-Verissimo and Marcus Völp and  RC3 Center, CEMSE
Division,
King Abdullah University of Science
and Technology (KAUST)
<EMAIL_ADDRESS><EMAIL_ADDRESS>University of Luxembourg
Interdisciplinary Center for Security,
Reliability and Trust (SnT) - CritiX group
<EMAIL_ADDRESS>
###### Abstract

The hardware computing landscape is changing. What used to be distributed
systems can now be found on a chip with highly configurable, diverse,
specialized and general purpose units. Such Systems-on-a-Chip (SoC) are used
to control today’s cyber-physical systems, being the building blocks of
critical infrastructures. They are deployed in harsh environments and are
connected to the cyberspace, which makes them exposed to both accidental
faults and targeted cyberattacks. This is in addition to the changing fault
landscape that continued technology scaling, emerging devices and novel
application scenarios will bring. In this paper, we discuss how the very
features—distributed, parallelized, reconfigurable, heterogeneous—that cause
many of the imminent and emerging security and resilience challenges, also
open avenues for their cure though SoC replication, diversity, rejuvenation,
adaptation, and hybridization. We show how to leverage these techniques at
different levels across the entire SoC hardware/software stack, calling for
more research on the topic.

###### Index Terms:

fault and intrusion tolerance, resilience, hardware, system on a chip, FPGA

##  I Opportunities for hardware resilience

Hardware chips continue to be the core building blocks of computing devices
due to their inherent immutability and speed, required in modern digital and
mission-critical systems like Cyber-Physical Systems, Healthcare, Fintech,
Automotive, and Space. This hardware can implement an entire monolithic system
or even be used as proof-of-trust anchors. Contrary to the common belief,
hardware is prone to unintentional (benign) and intentional/malicious
(intrusion or Byzantine [1]) faults. The former can be caused by the
fabrication (e.g., Silicon) material prone to dust, aging, and overheating, or
by design/implementation glitches [2, 3]. Malicious faults manifest in many
forms, prior- or post-fabrication, where stealthy logic, backdoors, trojans,
kill switches, and post-fab fabric editing are possible [4, 5, 6, 7]. In line
with this, the trends of building complex hardware out of smaller commercial-
off-the-shelf (COTS) components and introducing programmable/reconfigurable
hardware, e.g., FPGA [8, 9], are closing the gap with software systems:
hardware systems are no longer rigid, immutable, and fixed creatures. This
raises both new challenges and opportunities, which call to revisit the way
resilient and secure hardware systems are built.

The notable demand on hardware due to the automation and digitalization of
services in many sectors raised new challenges in the hardware fabrication
industry, where vendors need to maintain delivery on time and reduce
production costs. This resulted in a divide-and-conquer [10] production style:
a system is split into smaller and cheaper building blocks, i.e., components.
Components are developed in parallel to reduce the production cycle time. Each
block is likely developed by a dedicated specialized vendor, i.e., generating
COTS [11]. This means that the synthesising entity of these COTS can focus on
the technology it masters, rather than distributing its efforts on multiple
fronts. Despite this, these cheap components are becoming more prone to
failures and attacks [12], which can lead to drastic impacts on critical
sectors like Cyber-Physical Systems, health smart systems, mission-critical
space systems, etc. Our experience in software systems shows that building
resilient systems composed of small and cheap components can be more resilient
than a single complex monolithic system, that is usually very expensive.

There are ample opportunities for hardware resilience leveraging the above
advancements. To demonstrate this, we showcase in Fig. 1 different levels of
the chip development process, from low-level fine-grained gate logic blocks up
to multicore systems-on-chip (SoC). Literary works reveal some selected
resiliency techniques on most of these layers for constructing resilient clock
networks, replicated power domains, and lock-step coupling of cores [13, 14,
15, 16, 17, 18], which is a good starting point. We, however, advocate for
more systematic and comprehensive resiliency, probably leveraging hardware
hybrids to simplify the designs. This holistic view helps optimising SoC
designs by suggesting the right level of resiliency at each stage to reduce
the redundant complexity and cost.

In a nutshell, the lowest level, in Fig. 1, is building a single layer
microchip that constitutes a simple logical circuit of gates. Different gates
are known to have different resiliency levels [13, 17]. Recently, SiNW
transistors are used to bridge Source to Drain with multiple nanowires to
compensate manufacturing defects and aging [19]. While a typical design
process mainly considers the space, energy, and time metrics in the design,
making these circuits more resilient would mean trading these metrics for
resiliency, e.g., using backup gates, replicated parallel gates, or diverse
gates [17, 18]. On the other hand, single-layered circuits can today be
synthesized in a 3D fabric[20]. Layers typically have different complementary
functionalities. However, they can also have layers of identical functionality
from different vendors, which is useful to improve diversity in fault masking
scenarios (discussed later). It is also helpful to synthesize a monolithic
chip from multi-vendor layers to avoid vendor lock-in or potential aging
issues, backdoors, and kill switches [4, 2, 3]—so called Distribution attack
on the supply chain.

At a higher level, always depicted in Fig. 1, these 3D microchips can be
assembled to build a system-on-chip fabric [21]. Again, components of
identical functionalities can be used to build fault and intrusion masking SoC
fabric. This can be enriched with heterogeneous diverse microchips at a higher
level, thus building resilient Multicore Systems on Chip (MPSoC) [22, 9]. At
the higher layers, where a software stack complements the functionality of the
system to form a more programmable flexible hardware (discussed next), one can
take advantage of a remarkable body of research and practice to build
resilient soft-custom logic [23, 24, 25]. This can be done by exploiting
virtualization techniques to provide software-level containment and
replication. More complex systems can be built through networked systems of
systems on chip. First instances of networked SoC systems are already emerging
in the automotive, aeronautics, and CPS domain.

Across this spectrum, we foresee a need and opportunities to revisit how
resilient hardware is built:

  * •

building complex systems of systems and MPSoCs out of smaller COTS;

  * •

taking advantage of the programability and elasticity of modern hardware,
e.g., FPGA, GPGPU, to replicate, diversity and adapt; and

  * •

simplifying the design of secure robust systems using smaller hardware
hybrids—easy to design and verify, as resilient anchors.

Figure 1:  Resilience forms at the different (networked) hardware layers of
Multicore Systems on Chip.

##  II Programability, Elasticity, Plasticity

The genuine immutability properties of hardware components and elements, make
them ideal for security hardening and containment, i.e., by making the hard-
implemented logic tamper-resistant against both benign and intrusion faults.
Despite these facts, there is a continuous wave of relaxing these “rigid”
hardware designs through introducing programmable (including reconfigurable
and adaptable) fabric [8, 26]. The main reason is to improve hardware
flexibility and compatibility, i.e., making them application-agnostic, and to
facilitate the daunting design verification process prior to fabrication,
hence cutting off fabrication costs thereof. For this, programmable hardware
is considered a tradeoff between software logic—fully flexible, slow, and
mutable—and hard logic—fully rigid, fast, and immutable. We believe that there
are promising opportunities to boost the resilience of the programmable
platforms against faults and intrusions, although immutability is slightly
reduced. To explain these benefits, we consider two classes of programmable
hardware:

Soft Custom Logic Fabric (SCLF) : these are commonly known as software-defined
devices like PLC, ECU, and SDN devices [27, 28, 29, 24]. This hardware is
mostly domain-specialized, where computing is done using general-purpose
micro-controller or microprocessors, often managed by a full software stack:
hypervisors, RTOS/OS, drivers, libraries, and applications. Consequently,
these devices exhibit high programability features, analogous to IT computing,
although they have specialized roles and use domain specific peripherals,
e.g., sensors, actuators, and interfaces.

Hard Custom Logic Fabric (HCLF) : these are hardware chip fabrics, e.g., FPGA
[8] and GPGPU [26], composed of arrays of logical components, e.g., gates and
multiplexers, that are not “hard etched”, i.e., can be reprogrammed as needed.
The programming logic in this case is almost entirely implemented in hardware,
without the need for a software stack at runtime. Fabric is reprogrammed
through soft IP Cores [30, 31] (HDL code [32]) or through components
(softcores or blocks) synthesized on the chip as needed. This programability
feature is a very interesting tradeoff that retains the speed and security of
Application-Specific Integrated Circuit (ASIC) chips, while giving the
flexibility to support diverse applications and update implementations without
the need for costly and slow fabrication.

Although programability, in both classes, opens the door for tampering with
the system, and thus injecting surveillance circuits, intrusions and backdoors
[4, 3] after fabrication (though slightly compared with software systems),
there is a huge opportunity to leverage this programability to improve the
resilience of these systems through four main ingredients: replication,
diversity, rejuvenation, and adaptation.

###  II-A Replication

Replication is often useful to build resilience against Benign or Byzantine
faults. Passive replication [33, 34] allows a failing system to failover into
a backup replica. This is a cheap solution that typically requires one passive
backup replica. However, recovery is slow, requires reliable detection and is
not seemless to the user, even if implemented entirely at transistor level.
For example, Razor [35] integrates detection capabilities, originally for
timing faults in sequential logic, but also for power instability [36] and
side channels [37], and re-injects stored state into the pipeline for re-
execution. Albeit functionally transparent, users may observe timing
differences and anomalies caused by them. Active replication masks faults
through building a deterministic replicated state machine [38], composed of
replicas of identical functionality, which execute an agreement protocol, e.g.
Paxos [39] or PBFT [1]. The number of required replicas is typically
$2f+1$/$3f+1$ in order to tolerate $f$ faults. Interestingly, several works
make use of hardware hybrids as root-of-trust to simplify these protocols to
build resilient broadcast and agreement abstractions for embedded real-time
systems [40, 41, 42] (requiring only $2f+1$ replicas to tolerate $f$ Byzantine
ones).

Replication in SCLF is analogous to software replication at the software
layer. While some literary works have studied this in some settings [24, 23,
25, 43], there are research opportunities in other real-time applications like
software-defined vehicles, UXVs, Smart Grid, etc. On the other hand,
replication in HCLF is today easier than ever. Using an FPGA, it is possible
to spawn replicas as soft cores or logical blocks, using off-the-shelf soft
IPs. This is a nice hardware feature that gives the flexibility to create
hard-replicas quickly and on-demand, using only one fabric, in a similar way
to creating virtual machines or containers at software level.

###  II-B Diversity

Resiliency through active replication is, however, only guaranteed as long as
the replicas fail independently [1, 38]. The second ingredient, diversity,
helps building replicas of the same functionality but with different
implementations. The aim is to avoid common-mode benign failures and
intrusions.

Since programability in both classes, SCLF and HCLF, open new avenues for
multi-vendor implementations and COTS, the likelihood of diversity is higher
than the case of monolithic hardware that require deep technology
capabilities. An interesting trend that would benefit this model greatly is
more standardization for architectures and APIs. For instance, the
introduction of the AutoSAR [44] standard has greatly enriched the automotive
market with multi-vendor implementations of the entire software and hardware
stack, which act as a blockboxes of identical functionalities. CUDA [26] and
OpenGL [45] provide standard APIs to implement accelerated parallel computing
logic on a GPGPU using COTS implementations. Open source hardware platforms
like RISC-V [46] also standardize the architectures provided by different
vendors, and enrich the market with diverse architectures.

Interestingly, FPGAs allow for hardware diversity through modifying the hard-
logic through using different implementations or specifications for the
softcore/block IP, possibly from different vendors, which is then used to
spawn computing cores. It would be interesting to study the case where IP
compilers can generate diverse versions of identical softcores to be used on
the fly. First approaches towards such a generation of morphable softcores has
been investigated in the context of organic computing [47].

###  II-C Rejuvenation

Rejuvenation is the third complementary ingredient to replication and
diversity. These latter techniques can only maintain resilience as long as the
assumed number of failing replicas $f$ is fixed. This assumption is
unfortunately hard due to benign faults and malicious behaviours. The first is
related to aging, which manifest in software [48] as memory leakage, failure
to release resources and locks, failure to garbage collect, data corruption,
etc. Surprisingly, aging occurs also in hardware, due to the deterioration of
hardware material under overuse and overheating, etc. The second reason is
recently getting more attention with the increasing attempts of Advanced
Persistent Attacks (APT)—where a big deal of time and effort is usually put to
identify vulnerabilities and exploit them. While this might be clear at the
software level, there are continuous concerns about hardware backdoors and
timed Trojans. Indeed, this is behind the recent agendas of acquiring chip
sovereignty or split manufacturing in many countries [49, 50].

SCLF reprogramability can greatly benefit from the huge body of research on
software rejuvenation, that is proven to mitigate failures. This would even be
more effective when rejuvenation is simultaneous with diversity, which allows
the rejuvenation to a different implementation with identical functionally, in
consequence, reducing the success rate of APTs. Using FPGAs, rejuvenation can
also happen at hardware level in HCLF [51]. An FPGA allows restarting or
spawning new soft cores and logical blocks at runtime—avoiding slow device
restarts. In fact, one can partially rejuvenate some soft cores while others
continue to run. FPGAs allow for even smarter techniques, e.g., to rejuvenate
to diverse softcore variants that are loaded in different FPGA spatial
locations, which can avoid potential backdoors in the FPGA grid fabric.

###  II-D Adaptation

Yet, another way to withstand a varying number of faults $f$ is to adapt the
resilient system accordingly. Among the adaptation forms are scaling out/in
the system when $f$ may change, e.g., upon experiencing more threats, or
switching to a backup protocol that is more adequate to the current conditions
[52, 53, 23, 54] (considering safety, liveness, performance, etc.). This would
require research on the aforementioned adaptation mechanisms and, importantly,
on severity detectors that can trigger adaptation actions once needed. As we
discussed above, both SCLF (e.g., virtualization) and HCLF (e.g., FPGAs)
provide tempo-spatial elasticity, which allows changing the number of replicas
and their locations on the fabric as needed. It will be interesting to study
these research questions from scratch or validating the feasibility of
existing ones (developed in the software realm).

###  II-E Resilient Reconfiguration

It should be evident that reconfiguration must be resilient to faults and
attacks, irrespective of the kind of adjustment performed (i.e., diverse
rejuvenation, relocation, or adaptation). This holds for both reconfiguration
of an FPGA grid fabric as well as multi-chip FPGAs—where the individual FPGA
chiplets are the unit of reconfiguration. We shall focus here exclusively on
internal, partial and dynamic reconfiguration, since the reliance on external
complex and non-configurable modules (e.g., CPUs) would induce a weak spot in
the system, which could contaminate its resilience or introduce downtimes.
Nevertheless, dependencies on external hybrids that are simple, and thus easy
to verify, are allowed if they simplify the design. Internal, partial and
dynamic mean respectively that reconfiguration (i) is driven from within the
FPGA, e.g., by an HCLF or softcore defining the configuration bitstream to be
loaded into a reconfigurable region (or frame) through interfaces like
internal configuration access ports, (ii) it is bound to the reconfigured area
and elements therein, and (iii) it happens while other parts of the FPGA
continue to execute.

Optimizing the mapping of blocks to the FPGA grid fabric and integrating the
configured block with the remaining blocks remain sufficiently complex tasks
to be executed by a software-level operating-system kernel. Disabling and
enabling configured circuits and frames constitute the critical operations,
which leaves writing the configuration memory and validating that a correct
bitstream is written as tasks that can be executed by the responsible kernel
or possibly even kernel replicas. Provided sufficient access controls are in
place at the internal configuration access ports, the actual configuration of
a frame can even be delegated to its current user. However, as shown in
Gouveia et al. [55], privilege change must remain a trusted operation executed
consensually and enforced by a trusted-trustworthy component. This leads to
the more general question of architectural hybridization, which we address
next.

##  III Architectural Hybridization

Differentiating how the individual hard- and software components of an MPSoC
architecture can fail, architectural hybridization aims at benefiting from
small easy-to-verify and therefore more trustworthy components, called
_hybrids_. The goal is to enable, simplify or improve the performance of the
overall system, by serving as trust anchors for these properties. These could
be components (registers, memory, trusted execution environments or networks)
such as USIG, A2M, TrInc, SGX and others, used in hybrid BFT-SMR protocols
[41, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65].

Realizing hybridization poses a challenge dual to the question whether SCLF or
HCLF leads to more reliable systems-on-a-chip. For software-only hybrids, we
used to equate simplicity (measured for example in lines-of-code required to
realize a certain functionality) with a low likelihood of failure and ease of
verification. However, at hardware level, this equation is not as obvious,
even if we consider lines-of-VHDL or another hardware description language.

We illustrate this using the USIG from the MinBFT protocol by Veronese et al.
[41] as example. USIG is essentially a sequential circuit, which is driven by
the counter register and a few additional registers, which provide as
constants the secret key for the HMAC and the ID of the replica. The lowest
complexity version of such a circuit will use normal registers. But then any
bitflip in the counter will have catastrophic effects on the consensus problem
at hand since it is reflected unchanged in the computed HMAC and USIG output.
ECC-registers on the other hand add extra bits and the logic required for
correction, which both increase the complexity of the circuit at the benefit
of tolerating a certain number of bitflips. We also see the converse effect
when the required complexity of producing a special purpose circuit for a
given functionality exceeds the complexity of a simple core that is able to
fetch, decode and execute software. Once the inherent complexity of such a
functionality exceeds this bound, software implementations become preferable
and hybridization amounts to providing such an isolated core.

The objective of hardware-level hybridization is therefore to remain in this
middle-ground. Hardware hybrids, protected by ECC and other accidental- and
malicious-fault countermeasures, provide the desired functionality. This can
then be extended into the realm of software hybrids that are possibly executed
in a replicated manner and that vote to perform critical operations [55].

##  IV Conclusions and Call to Action

We emphasized that hardware architectures, and in particular multi- and
manycore systems-on-a-chip are not the robust, dependable and reliable
computing units we would like to have. We have subsequently started to
replicate entire systems, which has ultimately lead to the huge body of
knowledge on implementing resilient distributed systems. However, as we have
seen, the continuing miniaturization and integration of processing elements
into a single MPSoC, makes full system resilience increasingly costly, in
particular when a single system already provides all the processing power that
future critical applications need. We have shown how reconfiguration,
rejuvenation and adaptation already allow the hardware to repair itself, to
recover from faults and retain the resources classical resilience mechanisms
need, when applied entirely on chip. Hybridization rooted in exactly the
right-complexity circuits and applied to construct incrementally more complex
dependable systems will produce the next generation flexible, morphable and
highly trustable systems mission-critical systems will need. We therefore
appeal for more research to study the resilience of hardware-based systems,
systems of systems, and MPSoCs at different layers and cutting vertically
across layers, probably through validating the techniques developed in the
software Systems and Dependability areas.

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# Unraveling the bounce: a real time perspective on tunneling

Kfir Blum<EMAIL_ADDRESS>Weizmann Institute of Science, Rehovot,
Israel  Omri Rosner<EMAIL_ADDRESS>Weizmann Institute of Science,
Rehovot, Israel

###### Abstract

We study tunneling in one-dimensional quantum mechanics using the path
integral in real time, where solutions of the classical equation of motion
live in the complex plane. Analyzing solutions with small (complex) energy,
relevant for constructing the wave function after a long time, we unravel the
analytic structure of the action, and show explicitly how the imaginary time
bounce arises as a parameterization of the lowest order term in the energy
expansion. The real time calculation naturally extends to describe the wave
function in the free region of the potential, reproducing the usual WKB
approximation. The extension of our analysis to the semiclassical correction
due to fluctuations on the saddle is left for future work.

##  I Introduction

Vacuum decay through barrier penetration is typically considered in terms of
the survival probability associated with a state that is initially localized
in a false vacuum (FV) region of the potential Coleman:1977py ; Callan:1977pt
; Schulman:1981vu ; Kleinert:2004ev ; Andreassen:2016cvx :
$P_{FV}(t)=\int_{FV}dx|\psi(x,t)|^{2}$. Here $|\psi(t)\rangle$ is a
Schrödinger state and it is assumed that $P_{\rm FV}(0)\approx 1$. At late
time after transients fade out, but not so late that returning current matters
Andreassen:2016cvx , one finds exponential decay $P_{FV}(t)\propto e^{-\Gamma
t}$. Information about the tunneling process is contained in the wave function
$\psi(y,t)=\int dx\,\psi_{0}(x)K(t;y,x)$, with $\psi_{0}(x)=\langle
x|\psi(0)\rangle$ and the propagator $K(t;y,x)=\langle y|e^{-iHt}|x\rangle$.
In this paper we focus on the path integral representation of the propagator,

$\displaystyle K(t;y,x)$ $\displaystyle=$
$\displaystyle\underset{\scriptsize{\begin{array}[]{c}z(0)=x\\\
z(t)=y\end{array}}}{\int
Dz}e^{iS[z]},\;\;\;\;\;\;S[z]\,=\,\int_{0}^{t}dt^{\prime}\left(\frac{1}{2}\dot{z}^{2}-V(z)\right).$
(3)

We will restrict our attention to contributions to the propagator that become
relevant for the typical initial data $\psi_{0}(x)$ of the tunneling problem.

One learns to calculate $\Gamma$ by considering the FV-to-FV propagator
continued to imaginary time, $K(t;0,0)\to K(-i\tau;0,0)=K_{E}(\tau;0,0)$
(subscript E for Euclidean) Coleman:1977py ; Callan:1977pt ; Schulman:1981vu ;
Kleinert:2004ev . In a nutshell, in the limit of large $\tau$, the path
integral for $K_{E}(\tau;0,0)$ receives a contribution from a real-valued
saddle point function $r(\tau)$ that solves the equation of motion (EOM)
$\frac{d^{2}r}{d\tau^{2}}=V^{\prime}(r)$ with boundary conditions
$r(\pm\infty)\to 0$. The Euclidean action of this solution is twice
$S_{E}[r]=\int_{0}^{b}dr\sqrt{2V(r)}$, where $b$ is the classical turning
point of the potential (see Fig. 1). Including corrections due to fluctuations
around $r(\tau)$, one finds $\Gamma\propto e^{-2S_{E}}$.

Here, instead of going to imaginary time, we consider the path integral in
real time. Our goal is to investigate how the usual imaginary time result
arises from the real time path integral. The analysis allows us, at least in
principle, to calculate finite time corrections to the Euclidean result. It
also allows us to extend the calculation of the large-$t$ propagator to
arbitrary points outside the barrier, $y>b$, a calculation that is omitted in
the imaginary time formalism. Another way to state our exercise is as a real
time path integral derivation of the usual WKB tunneling wave function.

The semiclassical approximation expands around classical paths ${z_{c}}(t)$
that satisfy the real time EOM and boundary conditions:

$\displaystyle{\ddot{z}_{c}}+V^{\prime}$ $\displaystyle=$ $\displaystyle 0,$
(4) $\displaystyle{z_{c}}(0)$ $\displaystyle=$ $\displaystyle
x,\;\;\;{z_{c}}(t)=y.$ (5)

Solutions of Eq. (4) have a constant of motion, the energy $\epsilon$:

$\displaystyle\epsilon$ $\displaystyle=$
$\displaystyle\frac{1}{2}{\dot{z}_{c}}^{2}+V.$ (6)

The semiclassical approximation for the propagator is

$\displaystyle\underset{\scriptsize{\begin{array}[]{c}z(0)=x\\\
z(t)=y\end{array}}}{\int Dz}e^{iS[z]}$ $\displaystyle=$ $\displaystyle
e^{iS[{z_{c}}]}\mathcal{A}.$ (9)

The prefactor $\mathcal{A}$ contains the path integral over fluctuations on
the classical path. If more then one classical path exists, one needs to sum
$\sum_{j}e^{iS[{z_{c}}_{j}]}\mathcal{A}_{j}$.

We are interested in the action of small-$\epsilon$ solutions of the EOM, with
$|\epsilon|$ much smaller than the potential barrier. We call these tunneling
solutions. Tunneling solutions live in the complex $z$ plane, and have complex
$\epsilon$. Extending the path integral to complex paths requires a
restriction to guarantee that the dimensionality of the path space is
conserved. Picard-Lefschetz theory provides this machinery by arranging the
path integral as a sum over complex saddles, where the path integration in the
vicinity of each saddle is constrained by a flow equation Witten:2010zr ;
Witten:2010cx ; Basar:2013eka ; Cristoforetti:2013wha ; Tanizaki:2014xba ;
Cherman:2014sba ; Behtash:2015loa ; Alexandru:2016gsd ; Hertzberg:2019wgx ;
Ai:2019fri ; Mou:2019tck ; Mou:2019gyl ; Nishimura:2023dky 111Complex
solutions of the classical EOM were related by Bender:2008fr  to quantum
phenomena like energy quantization. The complexified path integral method
makes it natural to attribute these relations to the usual emergence of
classical mechanics from extremal action paths Turok:2013dfa ..

Ref. Tanizaki:2014xba  provided a useful summary of the formalism, and we
refer the reader there for details. The first stage of the calculation, which
yields the leading semiclassical factor $e^{iS[{z_{c}}]}$, is the usual task
of finding the saddle point paths and calculating their action. The only
change compared to the real path integral is that the saddle now extends to
complex configurations. More involved features of the theory arise in the
calculation of the integral over fluctuations. However, we leave this
interesting and important part of the calculation out of the current paper. We
plan to report the analysis of the fluctuation integral in subsequent work.
Leaving out the fluctuation analysis allows us to put the spotlight on the
leading semiclassical object, the real time parallel of Coleman’s bounce222To
be precise, we will focus on a solution that connects the FV to the free
region, so the Euclidean parallel is the instanton, or half of a bounce..

Many analyses of tunneling in the literature refer to real time path integrals
Turok:2013dfa ; Tanizaki:2014xba ; Cherman:2014sba ; Bramberger:2016yog ;
Hertzberg:2019wgx ; Ai:2019fri ; Mou:2019gyl ; Nishimura:2023dky 333See also
Levkov:2004ij  and Braden:2018tky  for related discussions.. Our analysis
differs from previous literature in that we do not explore the analytic
continuation of the amplitude w.r.t. the time variable. Instead, we stubbornly
stick to real time and unravel the complex bounce, exploring how the result of
the calculation maps to the imaginary time result by means of an explicit
expansion in powers of (complex) path energy; or (almost) equivalently,
inverse powers of real physical time. The analysis reveals how the basic WKB
Lorentzian-Euclidean factorization, highlighted in Bramberger:2016yog , arises
in the limit of large time.

##  II $K(t;y,0)$: propagator starting at the false minimum and ending in the
free region

We consider potentials that can be approximated by a parabola at the FV
minimum. A suitable example is

$\displaystyle V(z)$ $\displaystyle=$
$\displaystyle\frac{1}{2}z^{2}-\frac{1}{n}z^{n},$ (10)

assuming $n\geq 3$. The peak of this potential is at $x_{p}=1$, and we have
$V(x_{p})=\frac{1}{2}-\frac{1}{n}\sim\mathcal{O}(1)$. This is sufficiently
general for us to use as concrete example444Eq. (10) arises from the action
$S=\int
dt\left(\frac{1}{2}\dot{z}^{2}-\frac{m^{2}}{2}z^{2}+\frac{1}{n}l^{2-n}m^{2}z^{n}\right)$,
with mass and length parameters $m$ and $l$. Defining $t\to mt$ and $z\to lz$,
we have $S=(ml)^{2}\int
dt\left(\frac{1}{2}\dot{z}^{2}-\frac{1}{2}z^{2}+\frac{1}{n}z^{n}\right)$.. It
will become clear that the most important behaviour at small $\epsilon$ is
independent of the details of $V(z)$. A potential with $n=4$ is shown in Fig.
1.

Figure 1: Potential evaluated on the real axis, for $n=4$ (see Eq. (10)).

Coleman’s original derivation Coleman:1977py  focused on $K(t;0,0)$, a
restricted version of the FV-to-FV propagator. We will consider the slightly
different calculation of $K(t;y,0)$, that is, starting point $x\approx 0$ near
the FV minimum, but end point $y$ in the free region. These calculations
encode similar physics.

###  II.1 Choosing the correct saddle

Before entering any details of the calculation, the first point to note is
that for real potentials (in our case, potentials that are polynomials with
real coefficients), any solution $z_{c}(t)$ of the complexified EOM with real
boundary conditions $x,y$ always comes with a complex conjugate partner
solution $z_{c}^{*}(t)$. If the energy corresponding with $z_{c}$ is
$\epsilon$, the energy of $z^{*}_{c}$ is $\epsilon^{*}$. Similarly, the action
is also related by complex conjugation: $S[z^{*}_{c}]=S^{*}[z_{c}]$. Thus
$iS[z_{c}^{*}]=-(iS[z_{c}])^{*}$. Now, when the dust settles, our analysis
will (reassuringly) re-discover the usual WKB result that
$iS[z_{c}]=-S_{E}+iS_{\rm free}$, where $S_{E}$ and $S_{\rm free}$ are real
functions of $x$ and $y$, respectively. It follows that
$S_{E}[z_{c}^{*}]=-S_{E}[z_{c}]$: namely, one of the complex conjugate pair of
classical paths has a positive Euclidean action, and the other, negative.
Throughout this paper we focus on the solution with $S_{E}>0$, that we simply
denote by $z_{c}(t)$. Of the complex conjugate pair, it is only this solution,
with $S_{E}>0$, that takes part in the saddle point expansion of the path
integral. $z_{c}^{*}$ is discarded.

The classification of saddle points into relevant and irrelevant saddles is
discussed in Tanizaki:2014xba . In a sentence, the restriction of the domain
of functions that are included in the complexified path integral, is performed
by defining a downward flow: a prescription that guarantees that all paths in
the sum always posses a smaller value of ${\rm Re}\left(iS[z(t)]\right)$ than
that achieved for the set of real-valued paths. Since real valued paths $r(t)$
always have a real-valued action, ${\rm Re}\left(iS[r(t)]\right)=0$, it
follows that the functions participating in the complexified path integral
must have ${\rm Re}\left(iS[z(t)]\right)<0$. For the tunneling problem, this
maintains $z_{c}$ but removes $z_{c}^{*}$.

###  II.2 Basic features of the tunneling solution

We now discuss basic features of the tunneling solution $z_{c}(t)$.

For $n=4$, ${z_{c}}(t)$ can be found explicitly in terms of Jacobi functions
Turok:2013dfa , and is characterized by555The solution we plot here is written
in Mathematica by $z_{c}(t)=-\sqrt{\frac{2q}{1+q}}{\rm
JacobiSN\left[\frac{t}{\sqrt{1+q}},q\right]}$. We note that Ref. Turok:2013dfa
also studied the role of this solution for tunneling, with modified boundary
condition at $t=0$ following the choice to employ the saddle point
approximation on the wave function $\psi(y,t)=\int dx\,\psi_{0}(x)K(t;y,x)$,
rather than $K(t;y,x)$. $\epsilon=q/(1+q^{2})$ with complex $q$. An example
with $q=-0.06+0.06172i$ is shown in the left panel of Fig. 2. This solution
starts at $x=0$ and escapes in the positive real $z$ direction. Along the
green curves, the real part of ${\dot{z}_{c}}=\sqrt{2\epsilon-2V}$ vanishes;
along the orange curves, the imaginary part of ${\dot{z}_{c}}$ vanishes;
noting how the path changes direction in crossing these curves helps to
understand some features of the solution.

Figure 2: Left panel: tunneling solution ${z_{c}}(t)$ in the $({\rm
Re}\,z,{\rm Im}\,z$) plane (black curving line). The solution starts at $z=0$
and emerges from the barrier region in the negative real $z$ direction. Along
the green (orange) curves, the real (imaginary) part of
${\dot{z}_{c}}=\sqrt{2\epsilon-2V}$ vanishes. Middle panel: zoom on the
origin. Right panel: zoom on the exit point.

For small $\epsilon$, the path wraps multiple times around the points
$z_{\pm}$ that provide the small-$z$ solutions of the equation

$\displaystyle{\dot{z}_{c}}^{2}$ $\displaystyle=$ $\displaystyle
2\epsilon-2V(z)=0.$ (11)

For our potential, these zero points satisfy
$2\epsilon-z^{2}+\frac{2}{n}z^{n}=0$, and always contain a small-$z$ pair

$\displaystyle z_{\pm}$ $\displaystyle=$
$\displaystyle\pm\sqrt{2\epsilon}\left(1+\frac{\left(\pm\sqrt{2\epsilon}\right)^{n-2}}{n}+...\right).$
(12)

We focus on the small-$z$ region in the middle panel of Fig. 2. The points
$z_{\pm}$ can be identified on the plot as the contact points of the green and
orange curves.

Note that $z_{\pm}$, and in general, the set of points satisfying
${\dot{z}_{c}}^{2}=0$, are the branch points of the function
${\dot{z}_{c}}(z)=\sqrt{2\epsilon-2V(z)}$. A set of branch cuts can be
constructed that extends ${\dot{z}_{c}}(z)$ to an analytic function in the
complex $z$ plane. The branch cut associated to $z_{\pm}$ can be chosen as the
line connecting $z_{\pm}$. The path in Fig. 2 circles both $z_{\pm}$
simultaneously, thereby wrapping around the cut without crossing it. The path
does not wrap around the other branch points. This will become useful for us
later.

The density of the inner spiral increases as $\epsilon$ decreases (see
Turok:2013dfa ; Cherman:2014sba  for related discussions), and the outward
spiraling structure is driven by the nonlinear terms in $V(z)$. To see this,
note that neglecting the nonlinear terms (that is, approximating
$V(z)\approx\frac{1}{2}z^{2}$) we would have the harmonic oscillator solution
${z}_{\rm ho}(t)=x\cos t+s\sqrt{2\epsilon-x^{2}}\sin t$ with constant $x$ and
a sign choice $s=\pm 1$, describing a closed ellipse (or a line if $x=0$) with
period $2\pi$. We can set $x$ real and positive by choosing the phase of the
cycle. The orbit goes counter-clockwise if ${\rm
Im}\,s\sqrt{2\epsilon-x^{2}}>0$, and vice-verse. For a solution that starts at
the FV minimum ($x^{2}\ll|2\epsilon|$), the sense of rotation is fixed by
$s\,{\rm Im}\sqrt{\epsilon}$. To observe the slow outward drift of the spiral
at small $z$, write $z_{c}(t)=z_{\rm ho}(t)+f(t)$, and solve for $f$ with the
boundary condition $f(0)=0$. Letting $x\ll\sqrt{2|\epsilon|}$ to simplify the
analysis, the EOM reads ${\ddot{z}_{c}}+V^{\prime}=f+\ddot{f}-\left({z}_{\rm
ho}+f\right)^{n-1}\approx f+\ddot{f}-\left(\sqrt{2\epsilon}\sin
t+f\right)^{n-1}=0$, so at leading order in $\epsilon$ we have
$f\propto\left(\sqrt{2\epsilon}\right)^{n-1}$. It is straightforward to show
that for even $n$, $f$ contains a non-harmonic term
$f\supset\left(\sqrt{2\epsilon}\right)^{n-1}t\,\cos t$, in addition to sine
and cosine functions. The $\sim t\,\cos t$ term is responsible for the outward
motion. For odd $n$, the $t\,\cos t$ term arises only in the next order in the
expansion, and one finds $f\supset\left(\sqrt{2\epsilon}\right)^{2n-3}t\,\cos
t$. The result of this is that the full solution has the behavior
$z_{c}=z_{\rm ho}+a\,t\,\cos t+...$, where $(...)$ contains harmonic functions
with amplitude $\sim\left(\sqrt{2\epsilon}\right)^{n-1}$, and

$\displaystyle a$ $\displaystyle\sim$
$\displaystyle\left\\{\begin{array}[]{cc}\left(\sqrt{2\epsilon}\right)^{n-1},&{\rm
n\;even}\\\ \left(\sqrt{2\epsilon}\right)^{2n-3},&{\rm
n\;odd}\end{array}\right\\}.$ (15)

We illustrate this behavior in Fig. 3, for $n=4$, where
$a=\frac{3}{2\sqrt{2}}\epsilon^{\frac{3}{2}}$.

Figure 3: Left panel: In blue, we show ${\rm Re}{z_{c}}$ vs. $t$, illustrating
the behavior at small $z$. In green we show ${\rm Re}{z_{c}}_{\rm ho}$, and in
cyan ${\rm Re}({z_{c}}_{\rm ho}+\dot{a}t\cos t)$, with constant $a$ taken from
Eq. (15). Dashed orange shows ${\rm Re}\sqrt{2|\epsilon|}$. Right panel: In
black we show ${\rm Re}({z_{c}}-{z_{c}}_{\rm ho})$ vs. $t$. In dashed red we
show $at\cos t$.

Eq. (15) shows that the modulus of out-spiraling solutions that start at
$x\lesssim\sqrt{2|\epsilon|}$ stays in the vicinity of $\sqrt{2|\epsilon|}$
for a long duration of time,
$t\sim\left|\sqrt{\epsilon}/a\right|\sim|\epsilon|^{1-\frac{n}{2}}$ for $n$
even, or $t\sim\left|\sqrt{\epsilon}/a\right|\sim|\epsilon|^{2-n}$ for $n$
odd. This estimate misses logarithmic corrections, as we will discuss further
below.

The analysis leading to Eq. (15) holds only as long as ${\rm
Im}\,\epsilon^{\frac{n}{2}-1}\neq 0$. Otherwise, one can show that no spiral
motion develops: the orbit is trapped in the FV region and does not escape. We
found it most convenient to explain this point using analysis tools that we
explain in the next section; we therefore defer the explanation to App. A.

The path in Fig. 2 exits the barrier along positive real $z$. We highlight the
exit point in the right panel. After escaping, the solution is sandwiched
between the real line and the curve ${\rm Im}\,{\dot{z}_{c}}=0$. At large
positive $z$ near the real axis, $|z_{r}|>1\gg|z_{i}|$ (with the notation
$z=z_{r}+iz_{i}$, and similarly for $\epsilon$), we have ${\rm
Im}\,{\dot{z}_{c}}\approx{\rm
Im}\sqrt{2\epsilon_{r}+\frac{2}{n}z_{r}^{2n}+2i(\epsilon_{i}+3z_{r}z_{i})}$,
so ${\rm Im}\,{\dot{z}_{c}}=0$ is described by
$z_{i}\approx-\epsilon_{i}/(3z_{r})$. Therefore, the solution approaches a
real endpoint $y$ as ${\rm Im}\,z\propto 1/y$.

###  II.3 Action calculation

We now show that $S[{z_{c}}]$ reduces to a simple generic expression when
$\epsilon$ is small. At leading order in $\epsilon$, there is no need to find
${z_{c}}(t)$ explicitly, in order to calculate the action. This conclusion
extends also to the fluctuation integral.

Rewriting the action as a contour integral666Ref. Behtash:2015loa  discussed a
related analysis, but there, the complexification is done for the path
integral of the Euclidean theory, with solutions calculating the spectrum of
states trapped in the potential, rather than the tunneling wave function.
along ${z_{c}}$, via $dt=dz/{\dot{z}_{c}}$, and using
$\frac{1}{2}{\dot{z}_{c}}^{2}-V=2\epsilon-2V-\epsilon$, we can write

$\displaystyle S[{z_{c}}]$ $\displaystyle=$
$\displaystyle\int_{\mathrm{{z_{c}}}}dz\sqrt{2\epsilon-2V(z)}-\epsilon t,$
(16)

with

$\displaystyle t$ $\displaystyle=$
$\displaystyle\int_{\mathrm{{z_{c}}}}\frac{dz}{{\dot{z}_{c}}}=\int_{\mathrm{{z_{c}}}}\frac{dz}{\sqrt{2\epsilon-2V(z)}}.$
(17)

We should make a couple of comments about the passage from time integral to
contour integral along $z_{c}$. First, note that the branch cuts of
$\dot{z}_{c}$ can be constructed such that the tunneling path never crosses a
cut. Of course, the limit $\epsilon\to 0$ needs to be handled with care,
because in this limit the path comes arbitrarily close to branch points.
Second, the tunneling path of least action is not periodic (in $t$), so it
defines a proper one-to-one map $t=t(z_{c})$. There are, in general, tunneling
paths that fold back upon themselves to include near exact cycles; for
example, a slight deformation of $\epsilon$ could reflect $z_{c}$ back on its
tracks when it hits the classical turning point $z\approx b$. But such paths
have an exponentially suppressed action in comparison to the “primary” $z_{c}$
we focus on, and consequently, we do not concern ourselves about them. (In the
imaginary time analysis, these are multi-bounce configurations.)

Consider the integral $\int_{\mathrm{{z_{c}}}}dz\sqrt{2\epsilon-2V(z)}$ in Eq.
(16). The starting point of the path is ${z_{c}}(0)=x$ and the end point is
${z_{c}}(t)=y$. We can add and subtract an integral from $x$ to $y$ along the
real $z$ axis; this splits ${z_{c}}$ into a closed contour starting and ending
at $z=x$ (call this contour $\mathcal{C}_{{z_{c}}}$), plus the integral on the
real line:

$\displaystyle\int_{\mathrm{{z_{c}}}}dz\sqrt{2\epsilon-2V(z)}$
$\displaystyle=$ $\displaystyle\mathcal{I}_{{z_{c}}}+\mathcal{S},$ (18)
$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\int_{x}^{y}dr\sqrt{2\epsilon-2V(r)},$ (19)
$\displaystyle\mathcal{I}_{{z_{c}}}$ $\displaystyle=$
$\displaystyle\oint_{\mathcal{C}_{{z_{c}}}}dz\sqrt{2\epsilon-2V(z)}.$ (20)

Let us first analyze $\mathcal{I}_{{z_{c}}}$. Recall that ${z_{c}}$ wraps
multiple times around the pair of zeros $z_{\pm}$ of ${\dot{z}_{c}}$ (see Eq.
(12)). $\mathcal{I}_{{z_{c}}}$ can therefore be evaluated as ${N}$ closed
cycles around $z_{\pm}$, where ${N}$ is a positive integer or half-integer
(${N}\gg 1$ if $t\gg 1$, so ${N}+0.5$ can be approximated as ${N}$). This is
illustrated in Fig. 4.

Figure 4: $\mathcal{C}_{{z_{c}}}$ with one cycle highlighted.

As long as we do not cross branch cuts, we can deform the contour of
integration. Thus, the $N$ cycles all give the same result. Each cycle can be
shrunk around $z_{\pm}$; denote the single small cycle
$\mathcal{C}^{1}_{\epsilon}$:

$\displaystyle\mathcal{I}_{{z_{c}}}$ $\displaystyle=$
$\displaystyle{N}\oint_{\mathcal{C}^{1}_{\epsilon}}dz\sqrt{2\epsilon-2V(z)}.$
(21)

Considering the result as an expansion in powers of $\epsilon$, we can expand
the potential to leading order, and then map to the unit circle via
$z=\sqrt{2\epsilon}\zeta$, noting that $|z_{\pm}|=\sqrt{2|\epsilon|}$ at
leading order in $\epsilon$. With this,

$\displaystyle\oint_{\mathcal{C}^{1}_{\epsilon}}dz\sqrt{2\epsilon-z^{2}}$
$\displaystyle\approx$ $\displaystyle
2\epsilon\oint_{\mathcal{C}^{1}_{1}}d\zeta\sqrt{1-\zeta^{2}}=2\pi\epsilon,$
(22)

where $\mathcal{C}^{1}_{1}$ is the unit circle. Here, we evaluated the
integral as two equal contributions, one for each side of the $(z_{-},z_{+})$
cut:
$\oint_{\mathcal{C}^{1}_{1}}d\zeta\sqrt{1-\zeta^{2}}=-2\int_{0}^{\pi}d\phi\sqrt{1-e^{2i\phi}}\left(-\sin\phi+i\cos\phi\right)=\pi$.
This exercise of extending the square-root on both sides of the cut is
equivalent to the physical requirement that the velocity of the trajectory be
continuous777The over-all sign on the RHS of Eq. (22) is somewhat tricky.
Obtaining it requires matching $dz$ to ${\dot{z}_{c}}$ along ${z_{c}}$, and
noting that the point $\zeta=e^{i0_{+}}$, from where we start the unit cycle
integration, is located just across the cut..

We are going to need the leading correction to Eq. (22) at the next order in
$\epsilon$. Unlike the leading $2\pi\epsilon$ term, the correction depends on
the details of the nonlinear interactions in $V$. We can arrange the
correction as two contributions: one coming directly from the nonlinear term
$\sim z^{n}$ in $V$, and another coming from the $\epsilon$ corrections to
$z_{\pm}$ in Eq. (12). The latter effect is captured in the mapping of
$\mathcal{C}^{1}_{\epsilon}$ to $\mathcal{C}^{1}_{1}$, which should now read
$z=\sqrt{2\epsilon}\left(1+\frac{1}{n}\left(\sqrt{\pm
2\epsilon}\right)^{n-2}+...\right)\zeta$, where $(...)$ denotes terms at
higher orders in $\epsilon$. To simplify the derivation, let us assume that
$n$ is even – it is easy to generalize the result later on. For even $n$, we
have

$\displaystyle\oint_{\mathcal{C}^{1}_{\epsilon}}dz\sqrt{2\epsilon-z^{2}+\frac{2z^{n}}{n}}$
$\displaystyle=$ $\displaystyle
2\epsilon\left(1+\frac{1}{n}\left(\sqrt{2\epsilon}\right)^{n-2}\right)\oint_{\mathcal{C}^{1}_{1}}d\zeta\sqrt{1-\zeta^{2}+\frac{2}{n}\left(\sqrt{2\epsilon}\right)^{n-2}\left(\zeta^{n}-\zeta^{2}\right)}+...$
(23) $\displaystyle=$ $\displaystyle
2\pi\epsilon\left(1+\frac{2\Gamma\left(\frac{n}{2}+\frac{1}{2}\right)}{n\sqrt{\pi}\Gamma\left(\frac{n}{2}+1\right)}\left(\sqrt{2\epsilon}\right)^{n-2}+...\right),\;\;\;\;\;\;\;\;({\rm
for\;even}\;n)$

For odd $n$, one finds that the correction term in the brackets begins at
$\mathcal{O}\left(\epsilon^{n-2}\right)$, instead of the
$\mathcal{O}\left(\epsilon^{\frac{n}{2}-1}\right)$ of Eq. (23). We conclude
that

$\displaystyle\mathcal{I}_{{z_{c}}}$ $\displaystyle=$ $\displaystyle
2\pi{N}\epsilon\left(1+A_{\mathcal{I}}\epsilon^{\frac{n}{2}-1}+...\right),$
(24)

where the numerical coefficient $A_{\mathcal{I}}$ depends on the details of
the nonlinear terms in $V$.

Next, we analyze the $\epsilon$-dependence of $\mathcal{S}$ (Eq. (19)). It is
clear that $\mathcal{S}$ has a finite nonzero limit as $\epsilon\to 0$; this
limiting value will be seen to relate to Coleman’s Euclidean action. What we
are after, however, is a non-analytic piece
$\mathcal{S}\supset\epsilon\ln\epsilon$. Identifying this term will become
useful to clarifying the connection between $\epsilon$ and $t$.

In thinking about this integral, it is useful to define the distance scale
$r_{nl}$, below which nonlinear interactions in $V$ become unimportant in
${\dot{z}_{c}}$:

$\displaystyle r_{nl}$ $\displaystyle=$
$\displaystyle\left(n|\epsilon|\right)^{\frac{1}{n}}.$ (25)

We defined $r_{nl}$ such that at $r<r_{nl}$, the term $\frac{2}{n}r^{n}$ in
$2V(r)$ contributes less than $2|\epsilon|$. Note that for $n\geq 3$ we have
$r_{nl}\gg\sqrt{2|\epsilon|}$ at small $\epsilon$. Therefore in general we can
find some reference scale, call it $l$, that satisfies $\sqrt{2|\epsilon|}\ll
l\ll r_{nl}$. The $\mathcal{S}$ integral can then be written as two parts,
$r<l$ and $r>l$. In the former, nonlinear terms can be omitted:

$\displaystyle\mathcal{S}$ $\displaystyle\approx$
$\displaystyle\int_{x}^{l}dr\sqrt{2\epsilon-r^{2}}+\int_{l}^{y}dr\sqrt{2\epsilon-2V(r)}.$
(26)

Because the starting point $x$ of ${z_{c}}$ enters the calculation of
$\mathcal{S}$ explicitly, we tentatively allowed $x\neq 0$. We still think of
$x$ as small, specifically $x\ll r_{nl}$.

The $r<l$ integral gives

$\displaystyle\int_{x}^{l}dr\sqrt{2\epsilon-r^{2}}$ $\displaystyle=$
$\displaystyle
i\frac{l^{2}}{2}-\frac{x}{2}\sqrt{2\epsilon-x^{2}}+\frac{i\epsilon}{2}\left[\ln\left(\frac{-\frac{\epsilon^{2}}{2l^{2}}}{\epsilon-x^{2}-ix\sqrt{2\epsilon-x^{2}}}\right)-1\right].$
(27)

For $x\ll\sqrt{2|\epsilon|}$, we can neglect $x$ to obtain:

$\displaystyle\int_{x}^{l}dr\sqrt{2\epsilon-r^{2}}$ $\displaystyle=$
$\displaystyle\frac{i}{2}\left[\epsilon\ln\epsilon+\epsilon\left(i\pi-\ln(2l^{2})-1\right)+l^{2}+...\right],\,\,\,\,\,\,\left({\rm
for}\;\;x\ll\sqrt{2|\epsilon|}\right).$ (28)

This contains the $\epsilon\ln\epsilon$ term we were looking for. This term is
always present if $x$ is very small, specifically if $x\to 0$ as in Coleman’s
original analysis. For $x\gg\sqrt{2|\epsilon|}$, the explicit
$\epsilon\ln\epsilon$ term is not there. Instead we have

$\displaystyle\int_{x}^{l}dr\sqrt{2\epsilon-r^{2}}$ $\displaystyle=$
$\displaystyle
i\frac{l^{2}-x^{2}}{2}+i\epsilon\ln\frac{x}{l}+...,\,\,\,\,\,\,\left({\rm
for}\;\;x\gg\sqrt{2|\epsilon|}\right).$ (29)

We see that the logarithmic enhancement goes away for
$x\gg\sqrt{2|\epsilon|}$, and disappears for $x\sim r_{nl}$.

Considering the $r>l$ integral, it is not difficult to see that this does not
give an $\epsilon\ln\epsilon$ term: only a constant (in $\epsilon$) plus
$\mathcal{O}(\epsilon)$ terms. The details of these terms will not be needed
for us.

Altogether we conclude that the $\epsilon$ scaling of $\mathcal{S}$ is

$\displaystyle\mathcal{S}$ $\displaystyle=$
$\displaystyle\mathcal{S}_{0}+iA_{\mathcal{S}}\epsilon\ln\epsilon+B_{\mathcal{S}}\epsilon+...\,,$
(30)

where

$\displaystyle\mathcal{S}_{0}$ $\displaystyle=$
$\displaystyle\int_{x}^{y}dr\sqrt{-2V(r)}.$ (31)

For small $x\ll\sqrt{2|\epsilon|}$, the coefficient
$A_{\mathcal{S}}=\frac{1}{2}$ is independent of the details of $V$.

Before we return to calculating the action, we pause to derive a useful
relation. From Eqs. (17), (24), and (30), we
find888$\mathcal{I}_{z_{c}}+\mathcal{S}$ is not an analytic function of
$\epsilon$. Indeed, $\mathcal{S}\supset\epsilon\ln\epsilon$, and
$\mathcal{I}_{z_{c}}$ contains a simple pole. Expressing $t$ as an
$\epsilon$-derivative of this function may seem awkward, but this shortcut
expression works because we are interested precisely in extracting the leading
divergence of the derivative as $\epsilon$ becomes small (but never truly
zero).:

$\displaystyle t$ $\displaystyle=$
$\displaystyle\frac{d}{d\epsilon}\left(\mathcal{I}_{{z_{c}}}+\mathcal{S}\right)$
(32) $\displaystyle=$ $\displaystyle
2\pi{N}\left(1+\frac{n}{2}A_{\mathcal{I}}\epsilon^{\frac{n}{2}-1}\right)+iA_{\mathcal{S}}\ln\epsilon+iA_{\mathcal{S}}+B_{\mathcal{S}}+...\,.$
(33)

Imposing ${\rm Im}\,t=0$, we have

$\displaystyle{N}$ $\displaystyle=$ $\displaystyle\frac{A_{\mathcal{S}}}{n\pi
A_{\mathcal{I}}}\frac{-\ln|\epsilon|}{{\rm
Im}\epsilon^{\frac{n}{2}-1}}+...\,.$ (34)

We have already seen, from the analysis of the inner spiraling structure of
${z_{c}}$ (Sec. II.2), that ${N}\sim\epsilon^{1-\frac{n}{2}}$ if the starting
point $x$ is close to the FV minimum. Eq. (34) sharpens this result including
log corrections.

Finally, we turn to the action. Using Eqs. (24), (30), and (34), we find

$\displaystyle S[{z_{c}}]$ $\displaystyle=$
$\displaystyle\left(1-\epsilon\frac{d}{d\epsilon}\right)\left(\mathcal{I}_{{z_{c}}}+\mathcal{S}\right)$
(35) $\displaystyle=$
$\displaystyle\mathcal{S}_{0}-iA_{\mathcal{S}}\epsilon-2\pi\left(\frac{n}{2}-1\right)A_{\mathcal{I}}{N}\epsilon^{\frac{n}{2}}$
$\displaystyle=$
$\displaystyle\mathcal{S}_{0}+\mathcal{O}\left(\epsilon\ln\epsilon\right).$

In the last line, we have assumed that the real part of $\epsilon$ is not
parametrically large compared with the imaginary part999We did not find a
simple argument to justify this assumption. Indeed, we suspect that saddle
point solutions with small $|\epsilon|$ but large ${\rm Re}\,\epsilon/{\rm
Im}\,\epsilon$ exist, related to the decay of excited states of the FV region.
Nevertheless, we also expect that the decay of states near to the ground level
of the FV region does not exhibit this kind of hierarchy, and it would be such
solutions that dominate the large $t$ wave function..

Inspecting $\mathcal{S}_{0}$ (Eq. (31)), we can summarize that the
small-$\epsilon$ saddle point contribution to the propagator is constructed
from a part that produces the usual exponential suppression, coming from the
integral between the starting point $x$ and the classical turning point $b$;
and a pure phase part corresponding to the action of a free particle rolling
with zero energy from $b$ to the endpoint $y$:

$\displaystyle iS[{z_{c}}]$ $\displaystyle\approx$ $\displaystyle-
S_{E}+iS_{\rm free},$ (36) $\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\int_{x}^{b}dr\sqrt{2V(r)},$ (37) $\displaystyle S_{\rm free}$
$\displaystyle=$ $\displaystyle\int_{b}^{y}dr\sqrt{-2V(r)}.$ (38)

Both $S_{E}$ and $S_{\rm free}$ are real and positive. $S_{E}$ is, of course,
a generalization of Coleman’s Euclidean action Coleman:1977py . Altogether,
Eq. (36) coincides with the usual WKB expression for the wave function. This
is the bottom line we were getting at, and concludes our exercise of
unraveling the bounce.

##  III Discussion

We now discuss a few aspects of our derivation.

  1. 1.

Decay law. Eq. (36) reproduces the exponential decay law. A quick (and
standard) way to see this, is to extend the FV probability $P_{\rm FV}$ (see
Sec. I) to the interval $(-\infty,y)$, namely, count the total probability to
find the particle to the left of $y$. Call this

$\displaystyle P_{<y}$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{y}dx|\psi(x,t)|^{2}.$ (39)

In terms of the probability current $j(x,t)={\rm
Im}\,\psi^{*}\partial_{x}\psi$, we have

$\displaystyle\dot{P}_{<y}$ $\displaystyle=$ $\displaystyle-j(y,t),$ (40)

so as long as $P_{<y}(t)$ is of order unity, we can extract the decay rate
from $\Gamma=j$. Using our result for the semiclassical propagator, the wave
function is given by

$\displaystyle\psi(y,t)$ $\displaystyle=$ $\displaystyle\left[\int
dx\psi_{0}(x)\mathcal{A}e^{-S_{E}}\right]e^{iS_{\rm free}}.$ (41)

To directly compare our results to Coleman’s formalism, we can let
$\psi_{0}(x)\approx\delta(x)$; in that case $\int
dx\psi_{0}(x)\mathcal{A}e^{-S_{E}}\approx\mathcal{A}e^{-S_{E}}$ evaluated
at101010More generally, we would have $j(y,t)={\rm Im}\int dx^{\prime}\int
dx\,\psi^{*}_{0}(x^{\prime})\psi_{0}(x)e^{-S_{E}(x)-S_{E}(x^{\prime})}\mathcal{A}_{x^{\prime}y}^{*}\mathcal{A}_{xy}\left(\frac{\partial_{y}\mathcal{A}_{xy}}{\mathcal{A}_{xy}}+ip(y)\right)$,
where we manifest explicitly the $x$ dependence of $S_{E}$ from Eq. (37), and
the $x$ and $y$ dependence of $\mathcal{A}$. $x=0$. Neglecting the $y$
dependence of the fluctuation term $\mathcal{A}$ compared with the
exponential, we find

$\displaystyle j(y,t)$ $\displaystyle\approx$ $\displaystyle
2\left|\mathcal{A}\right|^{2}p(y)e^{-2S_{E}},\;\;\;\;\;p(y)=\partial_{y}S_{\rm
free}=\sqrt{-2V(y)}.$ (42)

Note that $p(y)$ is the classical momentum of a zero energy particle rolling
down the potential from $b$ to $y$. We see $\Gamma=j\approx
2\left|\mathcal{A}\right|^{2}p\,e^{-2S_{E}}$.

We have not calculated the fluctuation prefactor $\mathcal{A}$. From the
standard (Schrödinger-based) WKB analysis, we can expect
$|\mathcal{A}(y)|^{2}\propto 1/\sqrt{-2V(y)}=1/\sqrt{p(y)}$. This scaling is
associated with current conservation: the flux crossing any point $y$ must be
independent of $y$ at large $t$.

Neglecting $\partial_{y}\mathcal{A}\sim\frac{\partial_{y}p}{p}\mathcal{A}$ as
compared to $\mathcal{A}\partial_{y}S_{\rm free}\sim p\mathcal{A}$ is also a
standard step in the semiclassical approximation. This does not mean that this
step is generally valid. Its validity boils down to the requirement
$p^{2}\gg\partial_{y}p$: for our potential $p(y)\sim y^{\frac{n}{2}}$ outside
of the barrier, so the approximation is valid at sufficiently large $y$. Near
the exit point, $y\sim b$, we have $p\to 0$ while $\partial_{y}p\neq 0$, and
the WKB approximation is potentially poor. This reinforces the intuition that
the semiclassical approximation falls apart at the edge of the barrier, but
improves as the particles roll further downhill becoming more and more
“classical” Guth:1985ya .

  2. 2.

Exploring $x\neq 0$ in Eq. (37). The essential parts of our analysis hold also
if we allow $x\neq 0$ in Eq. (37), at least for small
$|x|\sim\sqrt{2|\epsilon|}$. Importantly, the lower integration limit of Eq.
(37) combines with initial data $\psi_{0}(x)\sim e^{-\frac{x^{2}}{2}}$
(namely, with the wave function of the would-be ground state of the FV) to
give
$\psi_{0}(x)e^{-\int_{x}^{b}dr\sqrt{2V(r)}}\approx\psi_{0}(0)e^{-\int_{0}^{b}dr\sqrt{2V(r)}}$,
independent of $x$. This is of course not an accident: the would-be FV ground
state wave function can be estimated with a small twist on the real time
analysis we reported (or equally well, in the standard imaginary time
technique) to scale as $\psi_{0}(x)\sim e^{-\int_{0}^{x}dr\sqrt{2V(r)}}$,
which precisely complements the tunneling term
$e^{-\int_{x}^{b}dr\sqrt{2V(r)}}$ in the propagator. Neglecting the
$x$-dependence of $\mathcal{A}$ (which by symmetry reasons, must have
vanishing first derivative at $x=0$, exactly for even $n$, and at lowest order
in the interactions for any $n$), this gives

$\displaystyle\int dx\psi_{0}(x)\mathcal{A}e^{-\int_{x}^{b}dr\sqrt{2V(r)}}$
$\displaystyle\sim$
$\displaystyle\left(\psi_{0}\mathcal{A}e^{-S_{E}}\right)_{x=0}.$ (43)

  3. 3.

The bounce. Coleman’s bounce Coleman:1977py  can be used to parameterize Eq.
(37) by defining “imaginary time” $\tau(r)$ via
$\tau(r)=\int_{r}^{b}\frac{dr^{\prime}}{\sqrt{2V(r^{\prime})}}$ for $r$ in the
range $(0,b)$. The inverted function $r(\tau)$, monotonically decreasing from
$r=b$ at $\tau=0$ to $r\to 0$ at $\tau\to\infty$, satisfies Coleman’s bounce
equations $\frac{1}{2}\dot{r}^{2}-V(r)=0$ and $\ddot{r}-\partial_{r}V=0$. Eq.
(37) (with $x\to 0$) becomes

$\displaystyle S_{E}$ $\displaystyle=$
$\displaystyle\int_{0}^{b}dr\sqrt{2V(r)}=\int_{0}^{\infty}d\tau\left(\frac{1}{2}\dot{r}^{2}+V(r)\right).$
(44)

The calculation we did is not Coleman’s FV-to-FV calculation, but FV-to-free
region. However, our analysis carries to Coleman’s if we let $y\to x\to 0$ and
note that the lowest $\epsilon$ real time solution resembles the path in the
left panel of Fig. 2, apart from that instead of monotonously out-spiraling
towards the exit point $b$, it strikes $b$ earlier on, and inspirals back to
the origin. This solution has action $2S_{E}$, up to finite-$\epsilon$
corrections as we calculated.

  4. 4.

WKB-like factorization, $\epsilon$ corrections. The factorization of the
propagator into an “imaginary time” piece and a ”real time” piece was
discussed in Bramberger:2016yog . Of course, this factorization was also
expected from the usual WKB calculation. Our derivation, apart from providing
a somewhat different pedagogical perspective, may add to this analysis the
ability to incorporate finite-time corrections via higher-order $\epsilon$
terms. For the simple unbounded polynomial potential we considered, Eq. (35)
shows that

$\displaystyle S[z_{c}]$ $\displaystyle=$
$\displaystyle\mathcal{S}_{0}+\left(\frac{1}{2}-\frac{1}{n}\right)\frac{\epsilon^{\frac{n}{2}}\ln|\epsilon|}{{\rm
Im}\epsilon^{\frac{n}{2}-1}}+{\rm higher\;powers\;of\;}\epsilon.$ (45)

  5. 5.

Multi-instanton configurations. It is natural to guess that multi-instanton
solutions extend the “fundamental” solution we analyzed. These solutions would
resemble the path in the left panel of Fig. 2, but recoil back and forth
between $b$ and the origin before finally exiting. In the small $\epsilon$
limit, $m$ back-and-forth detours before final exit would contribute a factor
of $e^{-2mS_{E}}$ to the propagator, the usual multi-instanton suppression.

If this picture is correct, then the $\epsilon$ expansion may help to test the
validity of the $e^{-2mS_{E}}$ approximation. An $m$-instanton configuration
must still make it to the final destination $y$ by time $t$. Since each single
cycle around the FV region lasts $\Delta t\approx 2\pi$, the total number of
cycles $N$ must be the same as for the $m=0$ “fundamental” solution. This
means that an $m$-instanton path out-spirals from $x\approx 0$ to $x\approx b$
in $N/(2m+1)$ cycles. This leads to a modified version of Eq. (34):
$N\approx(2m+1)\frac{A_{\mathcal{S}}}{n\pi
A_{\mathcal{I}}}\frac{-\ln|\epsilon_{m}|}{{\rm
Im}\epsilon_{m}^{\frac{n}{2}-1}}$. Inverting this relation shows that the
$\epsilon_{m}$ of $m$-configurations is larger than the $\epsilon$ of the
fundamental solution. For example, considering the quartic potential $n=4$, we
expect $\epsilon_{m}\sim(2m+1)\epsilon$, up to log corrections. Referring back
to Eq. (45), we expect that finite-$\epsilon$ corrections start to clatter the
imaginary time limit for sufficiently high-order (high $m$) multi-instanton
configurations.

  6. 6.

Finite-time expansion. $\epsilon$-corrections map to finite-time corrections
via $t\approx 2\pi N$ and Eq. (34); e.g., for $n=4$,
$\mathcal{O}\left(\frac{\ln\epsilon}{\epsilon}\right)=\mathcal{O}\left(t\right)$
(see Turok:2013dfa  for closely related discussion of this point, including a
consistent derivation of the time–energy relation). Up to the possibility of
parametric hierarchy between ${\rm Re}\,\epsilon$ and ${\rm Im}\,\epsilon$ (a
hierarchy that – we should note – we were not completely able to exclude),
this could allow one to organize the analysis of multi-instanton corrections
in terms of a $1/t$ expansion (see Ref. Pimentel:2019otp  for related
discussion).

##  IV Summary

We presented an analysis of the saddle point tunneling solution of the
complexified classical equations of motion (EOM), that dominates the wave
function at large times when calculated using the real time path integral. Our
goal was to examine how this saddle point unravels to give Coleman’s imaginary
time result; or, similarly, the Schrödinger-based stationary WKB result; while
keeping tabs on finite-time corrections. We did this exercise by organizing
the calculation in powers of the energy $\epsilon$ characterizing the path.
Our analysis differs from previous literature in that we track the real time
analytically-continued complex path, rather than performing the analytic
continuation w.r.t. to the time variable itself.

Apart from some pedagogical value (we think!), our derivation may also be
useful for the analysis of finite-time corrections to the tunneling wave
function. For example, although we did not explore this in detail, our
approach may help to study the breakdown of naive multi-instanton resummation.

Extending our analysis to the fluctuation determinant is left for future work.
At the time of writing, similar arguments to those presented above seem to
successfully identify the usual imaginary time fluctuation integral at lowest
order in the $\epsilon$ expansion, extending it in a natural way out to the
free region of the potential. However, we are still bogged down by some
questions related to the analyticity properties of complexified fluctuations.

###### Acknowledgements.

We thank Ofer Aharony, Shimon Levit, Ohad Mamroud, Mehrdad Mirbabayi, Yossi
Nir, Gui Pimentel, Adam Schwimmer, Amit Sever, and Giovanni Villadoro for
useful discussions. This work was supported by the Israel Science Foundation
grant 1784/20, and by MINERVA grant 714123.

##  Appendix A Constraint on the phase of $\epsilon$ required for out-
spiraling structure of $z_{c}$

Let us calculate the time $\Delta t_{1}$ it takes $z_{c}$ to propagate from
one crossing of the real $z$ axis, say at $z=r$, to the next crossing,
$z=r+\Delta r$ (see Fig. 4). As in the main text, we can write $\Delta t_{1}$
as the sum of a closed loop integral (going through the entire marked cycle in
Fig. 4), plus a short integral of length $\Delta r$ along the real $z$ axis.
With an analysis similar to the main text, we readily obtain, for even $n$:

$\displaystyle\Delta t_{1}$ $\displaystyle=$ $\displaystyle\int_{r}^{r+\Delta
r}\frac{dr}{\sqrt{2\epsilon-2V(r)}}+\oint_{\mathcal{C}^{1}_{\epsilon}}\frac{dz}{\sqrt{2\epsilon-2V(z)}}$
$\displaystyle=$ $\displaystyle 2\pi+n\pi A_{\mathcal{I}}{\rm
Re}\,\epsilon^{\frac{n}{2}-1}-\frac{1}{2}{\rm
Arg}\left(\frac{\epsilon-(r+\Delta r)^{2}-i(r+\Delta
r)\sqrt{2\epsilon-(r+\Delta
r)^{2}}}{\epsilon-r^{2}-ir\sqrt{2\epsilon-r^{2}}}\right)$ $\displaystyle+$
$\displaystyle i\left[n\pi A_{\mathcal{I}}{\rm
Im}\,\epsilon^{\frac{n}{2}-1}+\frac{1}{2}\ln\left|\frac{\epsilon-(r+\Delta
r)^{2}-i(r+\Delta r)\sqrt{2\epsilon-(r+\Delta
r)^{2}}}{\epsilon-r^{2}-ir\sqrt{2\epsilon-r^{2}}}\right|\right].$

Now, what we are calculating here is real time across the motion, so ${\rm
Im}\,\Delta t_{1}=0$ must hold. This says that if ${\rm
Im}\,\epsilon^{\frac{n}{2}-1}=0$, then the $\ln|...|$ term in the last line of
Eq. (A) must vanish, namely, we must have $\Delta r=0$. Thus, for ${\rm
Im}\,\epsilon^{\frac{n}{2}-1}=0$ the path must close-in on itself whenever it
completes a cycle, and there cannot be any outward motion.

The analysis of the odd $n$ case is very similar, and leads to the same
constraint: ${\rm Im}\,\epsilon^{\frac{n}{2}-1}\neq 0$ is needed for outward
motion.

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