# Resonance structures in kink-antikink scattering in a quantum vacuum

Mainak Mukhopadhyay<EMAIL_ADDRESS>Department of Physics; Department of
Astronomy & Astrophysics; Center for Multimessenger Astrophysics, Institute
for Gravitation and the Cosmos, The Pennsylvania State University, University
Park, PA 16802, USA.  Tanmay Vachaspati<EMAIL_ADDRESS>Physics Department,
Arizona State University, Tempe, AZ 85287, USA

###### Abstract

We investigate kink-antikink scattering in the $\lambda\phi^{4}$ model in the
presence of an additional scalar field, $\psi$, that is in its quantum vacuum
and interacts with $\phi$ via a $\xi\phi^{2}\psi^{2}$ term where $\xi$ is the
coupling. The final state of such a scattering is either a bound state with
eventual annihilation or a reflection of the kink-antikink pair. Without the
$\psi$ field, the outcome is known to depend fractally on the initial velocity
of the kink-antikink pair. In the quantum vacuum of the $\psi$ field, the
fractal dependence gets modified and disappears above a critical interaction
strength, $\xi\approx 0.1$.

##  I Introduction

The interactions and scattering of topological defects has been an important
area of research with applications to condensed matter systems [1, 2, 3, 4, 5,
6, 7, 8, 9, 10, 11, 12] and cosmology [13, 14, 15, 16]. Since topological
defects are solutions to classical equations of motion, most analyses have
considered classical scattering. This has led to important results such as the
universality of vortex reconnections [17] and fractal behavior in the
scattering of $\lambda\phi^{4}$ kinks in 1+1 dimensions [18, 19, 20, 21, 22,
23, 24, 25, 26]. Applications of kink-antikink scattering [27] include the
formation of abnormal nuclei [28, 29], domain walls in crystals [30, 31],
folding of protein chains [32, 33], and molecular dynamics [34, 35].

We are interested in examining how quantum effects modify kink scattering. The
inclusion of quantum effects presents some challenges as defects are described
as classical objects, while other excitations of interest are quantum in
nature. As a result, one has to couple classical and quantum degrees of
freedom and to include the quantum backreaction on the classical degrees of
freedom. Fortunately much work has been done in this area and now there is a
convenient framework called the “classical-quantum correspondence” (CQC) in
which certain quantum systems can be solved in terms of a classical system of
equations [36, 37, 38, 39]. Quantum backreaction on classical systems is in
general difficult to resolve satisfactorily and is also mired in
interpretations of quantum mechanics. However, the semiclassical approximation
offers a path forward and we utilize it in this work.

There are only two outcomes possible in the scattering of a kink and an
antikink in the $\lambda\phi^{4}$ model: (1) the formation of a bound state
(“bion”) and eventual annihilation, and (2) the kink and the antikink reflect
and escape to infinity. However, detailed analysis [20, 21, 25] shows that
which of these outcomes occurs depends sensitively on the initial scattering
velocity, although at a coarse level, low initial velocity kinks annihilate
and high initial velocity kinks reflect. For intermediate velocities, either
outcome can result depending on the precise value of the initial velocity:
there are windows of low initial velocities for which the kink-antikink
reflect off each other instead of annihilating and there is a “fractal”
structure in the space of initial velocities111This is an abuse of the word
“fractal” since that generally involves some self-similarity. The structure
here is similar to the stability bands in solutions of the Mathieu equation
[40].. The explanation of this non-trivial dependence involves the resonant
energy transfer mechanism amongst the internal modes (the zero mode and the
shape mode) of the kink (or antikink).

The minimal scheme to study quantum effects during kink-antikink scattering is
to include quantum fluctuations of the $\phi$ field of the $\lambda\phi^{4}$
model itself. In other words, we only have one scalar field $\phi$ that is
decomposed into a classical kink-antikink background ($\phi_{c}$) and quantum
excitations on top of this background ($\hat{\phi}$),

$\phi=\phi_{c}+{\hat{\phi}}.$ (1)

A straightforward analysis of this system, however, runs into trouble – we
find that $\phi_{c}\to 0$ with time. The difficulty can be traced to the use
of the semiclassical approximation in a double well potential since the
wavefunctional is not Gaussian and, in fact, may be bimodal. It is possible
that some of the problems we encountered may be due to the use of a finite
lattice. The full resolution of the difficulties is not clear to us. Hence we
study quantum effects of the vacuum of a second scalar field $\psi$ on kink-
antikink scattering of the classical field $\phi$. The main result of our
analysis is the change in the fractal structure of the scattering as a
function of the interaction strength between $\phi$ and $\psi$.

In Sec. II we setup the model and discuss the known results for the classical
scattering of kink-antikink in the $\lambda\phi^{4}$ model. We include the
quantum field in Sec. III and we describe the CQC. Finally in Sec. IV we solve
the entire system, including backreaction. Our results are summarized in Sec.
V. We discuss the main conclusions along with the importance and future
prospects related to this work in Sec. VI. We work in natural units, i.e.,
$\hbar=c=1$.

##  II Dynamics of the classical background

We consider a real classical scalar field $\phi$ in $1+1$ dimensions in the
$\lambda\phi^{4}$ theory. The Lagrangian density for the system is

$\mathcal{L}_{\phi}=\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\phi^{\prime
2}-\frac{\lambda}{4}\big{(}\phi^{2}-\eta^{2}\big{)}^{2}\,,$ (2)

where, $\lambda$ and $\eta$ are parameters of the theory. The
$\lambda\phi^{4}$ potential has two minima corresponding to $\phi=\pm\eta$.
The Lagrangian density in Eq. (2) is invariant under the transformation,
$\phi\rightarrow-\phi$ and hence has a _reflectional_ $Z_{2}$ symmetry222The
kinks (or antikinks) of this model are hence also known as $Z_{2}$ kinks (or
antikinks).. The dynamics of the field $\phi$ has been well-studied in the
literature [20, 21, 41] and is given by the equation of motion,

$\ddot{\phi}-\phi^{\prime\prime}+\lambda(\phi^{2}-\eta^{2})\phi=0$ (3)

By suitable rescalings, the parameters $\lambda$ and $\eta$ can be eliminated
from the equation of motion, effectively setting $\lambda=1$ and $\eta=1$.
From the Lagrangian density in Eq. (2) the conserved energy of the classical
background configuration ($E_{\phi}$) is given by,

$E_{\phi}=\int dx\left[\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\phi^{\prime
2}+\frac{\lambda}{4}\big{(}\phi^{2}-\eta^{2}\big{)}^{2}\right]\,.$ (4)

The equation of motion admit kink (and antikink) solutions
$\phi_{0K(\bar{K})}(t,x)$. One can easily Lorentz boost these solutions to
obtain,

$\phi_{0K(\bar{K})}(t,x)=\pm\eta\tanh{\Bigg{(}\eta\ \sqrt{\frac{\lambda}{2}}\
\gamma(x-vt)\Bigg{)}}\,,$ (5)

where, the sign in front corresponds to the kink (antikink) solution which
moves in the positive $x$ direction with a constant velocity $v$ and with
Lorentz factor $\gamma=1/\sqrt{1-v^{2}}$. The total energy of such a
configuration is given by,

$E_{0K(\bar{K})}=\frac{2\sqrt{2}}{3}\gamma\sqrt{\lambda}\,\eta^{3}\,.$ (6)

We are interested in scattering a kink and an antikink. Unlike in the
integrable sine-Gordon model, the $\lambda\phi^{4}$ model does not have a
closed form solution in which both a kink and an antikink are present.
However, assuming that the kink-antikink start out far enough from each other
the interaction energy between them is minimal and we can construct a field
configuration of the following form,

$\displaystyle\phi_{K\bar{K}}(t,x)$ $\displaystyle=$
$\displaystyle\eta\left[\tanh{\Bigg{(}\eta\ \sqrt{\frac{\lambda}{2}}\
\gamma(x+\Delta)\Bigg{)}}\right.$ (7) $\displaystyle-$
$\displaystyle\left.\tanh{\Bigg{(}\eta\ \sqrt{\frac{\lambda}{2}}\
\gamma(x-\Delta)\Bigg{)}}+1\right]\,,$

where, the kink and the antikink are initially displaced by a distance,
$\Delta=v_{\rm in}t_{0}$, where $t_{0}$ is negative (and is set to $-100$ in
our simulations)333The unit of time is $(\sqrt{\lambda}\eta)^{-1}$, and is set
to 1.. The kink collision occurs at $t=0$. The above configuration has a kink
and an antikink moving towards each other with an initial velocity $v_{\rm
in}$. When the kink and the antikink get close and start to interact, the
above configuration is no longer appropriate. One then needs to solve the
equation of motion in Eq. (3), with initial conditions specified by Eq. (7) to
get the dynamics of the field $\phi$. The kink-antikink pair collides, and
either forms a bound state (which is also called a bion) that eventually
decays into radiation or are reflected and travel away from each other. This
outcome depends on the initial velocity $v_{\rm in}$ and the non-integrability
of the $\lambda\phi^{4}$ model makes it difficult to intuit about this
dependence. Hence numerical simulations need to be used, the details of which
are discussed in Appendix. A.

###  II.1 Resonance structures in classical $\lambda\phi^{4}$ kink-antikink
scattering

Figure 1:  The ratio of the final velocity to the initial velocity ($v_{\rm
fin}/v_{\rm in}$) to the initial velocity ($v_{\rm in}$). Note that $v_{\rm
fin}=0$ denotes the formation of a bound state. The curves interpolate between
the actual data points. The parameters used are: $\lambda=1$, $\eta=1$,
$L=100$, $N=500$, and $t_{0}=-100$.

If a bion does not form, the kink and antikink reflect back with a definite
final velocity $v_{\rm fin}$. If a bion does form and the kink-antikink
annihilate, we define $v_{\rm fin}=0$. The ratio of the final velocity of the
kinks to the initial velocity ($v_{\rm fin}/v_{\rm in}$) depends on the
initial velocity ($v_{\rm in}$) and is shown in Fig. 1, a result that has been
obtained and discussed extensively in Refs. [20, 25]. We present it here using
our numerical calculations as a benchmark and most importantly to compare with
the evolution when quantum backreaction is included. Similar to prior
literature, we notice the fractal structures in the plot, which indicate
certain initial velocities for which the kink and the antikink reflect and
other windows of initial velocity in which the kinks annihilate. The
annihilation windows start out being broad but become thinner as the initial
velocity increases. Finally from $v_{\rm in}\sim 0.26$ the windows cease to
form and we have reflection for all $v_{\rm in}\gtrsim 0.26$. In Ref. [25] the
authors extended the initial velocities to $v_{\rm in}=0.9$ but did not see
any bound state formation post $v_{\rm in}\sim 0.26$.

This fractal structure has been extensively studied and well-understood in the
literature. The kinks (or antikinks) in the $\lambda\phi^{4}$ model have two
distinct excitation eigenmodes. One of them corresponds to translations of the
kink, known as the zero mode, and the other to the internal vibrations of the
kink (or antikink) known as the shape mode [42, 41]. In addition, there is a
continuum of radiative modes on the kink background.

When the kink and the antikink collide the kinetic energy is redistributed
between these modes and some of it is radiated. In cases, where the kinetic
energy is reduced to an extent where the kink-antikink do not have enough
energy left, leads to the formation of a bound state, whereas if the decrease
is not enough, they separate and reflect off each other. However, in some
cases there is a _resonant energy transfer_ where the shape mode returns some
of the kinetic energy to the zero mode leading the kink and the antikink to
separate and not form a bound state. This accounts for the windows we see in
Fig. 1. It may be worthwhile to note here given infinite precision, one would
find infinite number of subsequent narrower windows for $v_{\rm in}\lesssim
0.26$ [25].

##  III Dynamics of the quantum field without backreaction

We now focus on the dynamics of the quantum field. It is evident that since
the quantum field is coupled to the classical background, the non-adiabatic
dynamics of the background will lead to particle production of the quantum
field. We will use the well-studied “classical-quantum correspondence” (CQC)
[36, 37, 38, 43, 39, 44, 38] technique to address this problem.

The full Lagrangian density for the background $\phi$ coupled to a quantum
field $\psi$ in $1+1$ dimensions is given by,

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\phi}+\frac{1}{2}(\partial_{\mu}\psi)^{2}-\frac{1}{2}\mu^{2}\psi^{2}-\frac{\xi}{2}\phi^{2}\psi^{2}\,,$
(8)

where, $\mu$ is the mass of the quantum field, $\xi$ is the coupling strength,
and $\mathcal{L}_{\phi}$ is defined in Eq. (2). The truncated Lagrangian
density from Eq. (8) for the quantum field $\psi$ can be written as,

$\displaystyle\mathcal{L}_{\psi}=\frac{1}{2}\dot{\psi}^{2}$ $\displaystyle-$
$\displaystyle\frac{1}{2}\psi^{\prime
2}-\frac{1}{2}\mu^{2}\psi^{2}-\frac{\xi}{2}\phi^{2}\psi^{2}\,,$ (9)

where, the time-dependent background $\phi(t,x)$ is given by Eq. (3) if we
ignore the backreaction of the quantum field on the classical background.
Taking a closer look at the above equation it is evident that the Lagrangian
density represents a free scalar field with a space- and time-dependent mass-
squared $M^{2}(t,x)$,

$M^{2}(t,x)=\mu^{2}+\xi\phi^{2}(t,x)\,.$ (10)

The faster this term changes the more particle production occurs for the
quantum field. Just like for the classical background, we require numerical
simulations for having a quantitative understanding of the dynamics of the
quantum field. We now turn to doing that.

To begin with, the spatial dimension $x$ is discretized on a _circular_
lattice of length $L$. We consider $N$ evenly spaced points on the lattice,
leading to a lattice spacing of $a=L/N$. The discretized field values are
defined as, $\phi_{i}=\phi(t,-L/2+ia)$ and $\psi_{i}=\psi(t,-L/2+ia)$, where,
$i=1,2,\dots,N$. The discretized Lagrangian will now be,

$\displaystyle L_{\psi,\mathrm{disc.}}$ $\displaystyle=$ $\displaystyle
a\sum_{i=1}^{N}\left[\frac{1}{2}\right.\dot{\psi}_{i}^{2}+\frac{1}{2}\psi_{i}\Bigg{(}\frac{\psi_{i+1}-2\psi_{i}+\psi_{i-1}}{a^{2}}\Bigg{)}$
(11) $\displaystyle-$
$\displaystyle\frac{1}{2}\mu^{2}\psi_{i}^{2}-\left.\frac{\xi}{2}\phi_{i}^{2}\psi_{i}^{2}\right]\,.$

Note that we have performed a spatial integration by parts in writing the
gradient term above. Let us now switch to a more compact matrix notation,
where we can define a column vector for the discretized $\psi$ field as,
$\bm{\psi}=(\psi_{1},\psi_{2},\dots,\psi_{N})^{T}$. In this notation, the
discretized Lagrangian ($L_{\bm{\psi},\mathrm{disc.}}$) is,

$L_{\bm{\psi},\mathrm{disc.}}=\frac{a}{2}\dot{\bm{\psi}}^{T}\dot{\bm{\psi}}-\frac{a}{2}\bm{\psi}^{T}\bm{\Omega}^{2}\bm{\psi}\,,$
(12)

where $\bm{\Omega}^{2}$ is an $N\times N$ matrix defined as,

$[\bm{\Omega}^{2}]_{ij}=\begin{cases}+{2}/{a^{2}}+M^{2}(t,x),&i=j\\\
-{1}/{a^{2}},&i=j\pm 1\ (\text{mod}\ N)\\\ 0,&\text{otherwise}\,,\end{cases}$
(13)

where, $M^{2}(t,x)$ is defined in Eq. (10) and we now replace $\phi$ by the
discretized value of $\phi$ at a point $i$, $\phi_{i}$. The conjugate momentum
$\bm{\pi}$ can be calculated from Eq. (12) which allows us to define the
discretized Hamiltonian ($H_{\bm{\psi},\mathrm{disc.}}$) as,

$H_{\bm{\psi},\mathrm{disc.}}=\frac{a}{2}\bm{\pi}^{T}\bm{\pi}+\frac{a}{2}\bm{\psi}^{T}\bm{\Omega}^{2}\bm{\psi}\,.$
(14)

We now move on to quantizing the theory. This is achieved in the Heisenberg
picture by promoting the discretized field value $\psi_{i}$ to an operator
$\hat{\psi}_{i}$ at the lattice point $i$. Following [36, 37], we introduce
the complex time-dependent matrix $\bm{Z}$. The elements of $\bm{Z}$, $Z_{ij}$
satisfy the following relation,

$\hat{\psi}_{i}=Z_{ij}^{*}\hat{a}_{j}(t_{0})+Z_{ij}\hat{a}_{j}^{\dagger}(t_{0})\,,$
(15)

where, we assume that the background is static for any time $t\leq t_{0}$. The
complete dynamics of $\hat{\psi}_{i}$ is given by,

$\ddot{\bm{Z}}+\bm{\Omega}^{2}\bm{Z}=0\,,$ (16)

with the specific initial conditions,

$\bm{Z}(t_{0})=-\frac{i}{\sqrt{2a}}\bm{\Omega}(t_{0})^{-1/2},\
\dot{\bm{Z}}(t_{0})=\frac{1}{\sqrt{2a}}\bm{\Omega}(t_{0})^{1/2}.$ (17)

The above initial conditions come with a small caveat. The initial vacuum for
the quantum field is chosen around a static background that is given by the
initial conditions, but the background is not static and has some time
dependence due to the initial velocities of the kinks. This leads to some
unwanted but small excitations in the $\psi$ field, which are radiated out
even before the kinks have collided. This gives a small error $\lesssim 0.1\%$
of the total energy in the initial conditions for the range of initial
velocities $v_{\rm in}$ we have investigated.

##  IV Dynamics with backreaction

We now focus on the main aspect of this work - the dynamics of the classical
background coupled to the quantum field including backreaction. In this
scenario, the dynamics of the classical background $\phi$ also gets a
contribution from the $\psi$ field,

$\ddot{\phi}-\phi^{\prime\prime}+[\lambda(\phi^{2}-\eta^{2})+\xi\psi^{2}]\phi=0\,.$
(18)

To take the effects of the quantum field $\psi$ into account we can use the
semiclassical approximation in which $\psi^{2}$ is replaced by $\langle
0|\hat{\psi}^{2}|0\rangle\equiv\langle\hat{\psi}^{2}\rangle$, where the
expectation value is calculated in the vacuum state of the quantum field
$\hat{\psi}$. (States do not evolve in the Heisenberg picture.) The background
equation can now be written as,

$\ddot{\phi}-\phi^{\prime\prime}+[-m_{b}^{2}+\lambda\phi^{2}+\xi\langle\hat{\psi}^{2}\rangle]\phi=0\,.$
(19)

where $m_{b}^{2}\equiv\lambda\eta^{2}$. The parameters in (19) should be
thought of as bare parameters that will get renormalized due to the quantum
field $\hat{\psi}$. We only need to consider “mass renormalization” to the
order we are working in and we can write,

$\ddot{\phi}-\phi^{\prime\prime}+[-m^{2}+\lambda\phi^{2}+\xi(\langle\hat{\psi}^{2}\rangle-\langle\hat{\psi}^{2}\rangle_{\phi=\eta})]\phi=0\,.$
(20)

where the subscript $\phi=\eta$ implies that the expectation value is to be
taken in the ground state of $\hat{\psi}$ in the trivial background
$\phi=\eta$. Now $m^{2}$ is the physical mass parameter and is related to
$m_{b}^{2}$ by

$m^{2}=m_{b}^{2}-\xi\langle\hat{\psi}^{2}\rangle_{\phi=\eta}.$ (21)

In its discretized form Eq. (20) is,

$\displaystyle\ddot{\phi}_{i}-\frac{1}{a^{2}}(\phi_{i+1}-2\phi_{i}+\phi_{i-1})$
$\displaystyle\hskip-14.22636pt+\biggl{[}-m^{2}+\lambda\phi_{i}^{2}+\xi\sum_{j=1}^{N}\big{(}|Z_{ij}|^{2}-|Z_{ij}|^{2}_{\phi=\eta}\big{)}\biggr{]}\phi_{i}=0.$
(22)

where we have used [37],

$\langle\hat{\psi}_{i}^{2}\rangle=\sum_{j=1}^{N}|Z_{ij}|^{2}\,.$ (23)

Thus, Eq. (22) has to be solved for $\phi_{i}$ with initial conditions,

$\displaystyle\phi_{i}(t_{0})$ $\displaystyle=$
$\displaystyle\phi_{K\bar{K}}(t_{0},-L/2+ia)\,,$
$\displaystyle\dot{\phi}_{i}(t_{0})$ $\displaystyle=$
$\displaystyle\dot{\phi}_{K\bar{K}}(t_{0},-L/2+ia)\,,$ (24)

where, as before, $i=1,2,\dots,N$ and $\phi_{K\bar{K}}(t,x)$ is given by Eq.
(7).

The complete backreacted dynamics is given by Eqs. (22) and (16), with initial
conditions given by Eqs. (IV) and (17).

###  IV.1 Energy

The conserved total energy of the system ($E$) is given by,

$E=E_{\phi}+E_{\psi}^{(R)}\,,$ (25)

where, $E_{\phi}$ is the energy in the classical background and
$E_{\psi}^{(R)}$ is the renormalized energy in the quantum field. The total
energy in the classical background can be defined as,

$\displaystyle E_{\phi}$ $\displaystyle=$ $\displaystyle
a\sum_{i=1}^{N}\rho_{\phi,i}=a\sum_{i=1}^{N}\left[\frac{1}{2}\dot{\phi}_{i}^{2}+\frac{1}{4a^{2}}\big{(}(\phi_{i+1}-\phi_{i})^{2}\big{)}\right.$
(26) $\displaystyle\hskip
28.45274pt+(\phi_{i}-\phi_{i-1})^{2}+\left.\frac{\lambda}{4}\big{(}\phi_{i}^{2}-\eta^{2}\big{)}^{2}\right]\,.$

The energy in the quantum field is,

$E_{\psi}\equiv\langle
0|\hat{H}_{\psi,\text{disc.}}|0\rangle=\frac{a}{2}\text{Tr}\Bigg{(}\dot{\bm{Z}}^{\dagger}\dot{\bm{Z}}+\bm{Z}^{\dagger}\Omega^{2}\bm{Z}\Bigg{)}\,.$
(27)

The discretized energy density in $\hat{\psi}$ can be defined as,

$\displaystyle\rho_{\psi,i}=\sum_{j=1}^{N}\biggl{[}\frac{1}{2}|\dot{Z}_{ij}|^{2}+\frac{1}{4a^{2}}\biggl{\\{}|Z_{i+1,j}-Z_{ij}|^{2}$
$\displaystyle\hskip
28.45274pt+|Z_{ij}-Z_{i-1,j}|^{2}\biggr{\\}}+\frac{1}{2}\biggl{\\{}\mu^{2}+\xi\phi_{i}^{2}\biggr{\\}}|Z_{ij}|^{2}\biggr{]}.$
(28)

Similar to the renormalized mass defined in (21), the renormalized energy
density in the quantum field may be defined as,

$\rho_{\psi,i}^{(R)}=\rho_{\psi,i}-\rho_{\psi,i}|_{\phi=\eta}\,,$ (29)

where, we subtract the energy density of the quantum field in the trivial
vacuum ($\phi=\eta$) from the energy density. Hence, the renormalized total
energy in the quantum field is,

$E_{\psi}^{(R)}=E_{\psi}-E_{\psi}|_{\phi=\eta}.$ (30)

Figure 2:  The initial renormalized energy density in $\psi$ (solid dark blue
curve) and the initial profile of the kink-antikink background $\phi$ (dashed
red curve). We note that the kink and the antikink is dressed with the $\psi$
particles. The main parameters are: $v_{\rm in}=0.21$, $\xi=0.05$, $\mu=0.1$,
$\lambda=1$, $\eta=1$, $L=100$, $N=500$, and $t_{0}=-100$.

###  IV.2 Initial structure of the vacuum

The initial vacuum structure of the quantum field can be visualized from Fig.
2. In the figure, we show the initial renormalized energy density of $\psi$,
($\rho_{\psi,i}^{(R)}(t=t_{0},x)$). The figure also shows the background
$\phi$ at the initial time $\phi_{K\bar{K}}(t=t_{0},x)$ (see Eq. (IV)). In the
trivial vacuum, $\phi=\eta$, the renormalized energy density in $\psi$
vanishes, as in Fig. 2 at the boundaries of the lattice. However, at the
position of the kink and the antikink, there is a big dip in the energy
density of $\psi$ that depends on the parameters of the model The dips are the
ground state of $\psi$ in this particular (kink-antikink) background. As the
background changes, that is, as the kink-antikink move, the dips move along
with them. This will be evident in Sec. V.1 where we study dynamics of the
background.

##  V Results

Our numerical methods and checks are described in Appendix A and B.

###  V.1 Quantum kink-antikink scattering with backreaction

(a) (a)

(b) (b)

(c) (c)

(d) (d)

Figure 3:  Density plots showing the time evolution of the energy density
($\rho_{\phi,i}$) in the background (see Eq. (26)) - _Top Panel:_ Classical
evolution - (a) $v_{\rm in}=0.21$ where the final state is a bound state, (b)
$v_{\rm in}=0.3$ to show a case where the kink and the antikink get reflected
post-collision. _Bottom Panel:_ Evolution including coupling to the quantum
field and backreaction - (c) same as (a), (d) same as (b). The parameters are:
$\xi=0.05$, $\mu=0.1$, $\lambda=1$, $\eta=1$, $L=100$, $N=500$, and
$t_{0}=-100$ for all the cases.

In Fig. 3 we show the energy density in the background field $\phi$ – the
expression in rectangular brackets in Eq. (26) – as a function of time
(vertical axis). The top panels compare the classical evolution ($\xi=0$) for
$v_{\rm in}=0.21$, when a bion forms, and for $v_{\rm in}=0.30$, when the
kinks reflect. The bottom two panels show the corresponding evolution for
($\xi=0.05$). The bion is now tighter; the reflected kinks have smaller
velocity. The behavior is expected since the kinks in the $\xi\neq 0$ case
excite the quantum vacuum of $\hat{\psi}$, produce particles, and lose energy
during the evolution. The energy in $\hat{\psi}$ must come from the kinetic
energy of the kinks which results in a tighter bion for $v_{\rm in}=0.21$ and
for slower reflected kinks for $v_{\rm in}=0.30$.

(a) (a)

(b) (b)

Figure 4:  Time evolution of the value of the background $\phi$ at the center
of the lattice $\phi(t,x=0)$ contrasting the classical evolution (dashed) with
the case including quantum backreaction (solid) for - (a) $v_{\rm in}=0.21$
(brown and dark blue respectively) and (b) $v_{\rm in}=0.3$ (dark green and
purple respectively). The parameters used are the same as the ones stated in
Fig. 3.

In the case $v_{\rm in}=0.21$, where the final state is a bound state, we see
that the kink and the antikink travel towards each other, collide and get
stuck leading to a caterpillar-like structure in Fig. 3. Each segment of the
caterpillar shows that the kinks separate for a short duration but then come
back and collide again. This repeats for some time and finally the kink-
antikink do not have enough energy to separate and hence lose their individual
identity to form a bound state (or bion) that decays into radiation. It is
difficult to see the difference between Figs. 3a and 3c, since in both cases
the caterpillar structure exists and repeated collisions makes it difficult to
notice any differences. To clarify the differences, we plot the value of the
background $\phi$ at the center of the lattice, $\phi(t,x=0)$, in Fig. 4 for
$v_{\rm in}=0.21$ and for $v_{\rm in}=0.30$.

Figure 5:  Time evolution of the total energy in the background $\phi$
($E_{\phi}$) for the classical evolution (dashed) and the case including
quantum backreaction (solid) for - (a) $v_{\rm in}=0.21$ (brown and dark blue
respectively) and (b) $v_{\rm in}=0.3$ (dark green and purple respectively).
The parameters used are the same as the ones stated in Fig. 3.

(a) (a)

(b) (b)

Figure 6:  Time evolution of the renormalized energy density
$\rho_{\psi}^{(R)}$. (a) $v_{\rm in}=0.21$ where a bound state is formed and
(b) $v_{\rm in}=0.3$ where the kink and the antikink undergo reflection. The
parameters are the same as for Fig. 3.

In Fig. 5 we show the time dependence of the energy in the $\phi$ background
(see Eq. (26)). Before the collision, the energy in the background is constant
and is given by twice that of the single kink (or antikink) energy (see Eq.
6), which is basically the sum of the individual energies of the kink and the
antikink. For the classical evolution (dashed curves), $E_{\phi}$ is the total
energy and is conserved. In the presence of the quantum field, the outcome
depends on $v_{\rm in}$. When a bound state is formed (solid dark blue curve),
the repeated collisions lead to a cascading drop in $E_{\phi}$ and the lost
energy goes into radiating $\psi$ particles.

The loss of energy from the kinks to $\psi$ quanta can be visualized in Fig.
6. The plot shows the time evolution of the renormalized energy density in
$\psi$, denoted by $\rho_{\psi}^{(R)}$ (see Eqs. (28), (29)). As we have noted
in Fig. 2, there is a “cloud” of $\psi$ around the position of the kink. This
cloud perfectly follows the kink and the antikink at early times as can be
seen by comparing Figs. 3 and 6.

The collisions are visible as the segments of the caterpillar after $t=0$. In
the $v_{\rm in}=0.21$ case, as the kinks collide, separate and collide again,
we see bursts of radiation that are produced, which then propagate outwards
away from the center. By virtue of periodic boundary conditions, once the
radiation reaches the end of the lattice they wrap around and return back
towards the center. The bursts of radiation in the case of a bound state
formation is what is observed in Fig. 5 (solid dark blue curve), as a
cascading decrease in the total energy in the background.

When the kink-antikink pair is reflected, as in Fig. 6b, there is a burst of
$\psi$ radiation at the time of collision seen as the two yellow bands moving
out from the collision point in the figure After the collision, the $\psi$
clouds keep moving with the kink and the antikink but the kink internal modes
are excited and there are some weaker bursts of $\psi$ radiation. There are
the light yellow bands being radiated out as the kinks move away from each
other in Fig. 6b. The energy loss appears as the undulating features in the
$v_{\rm in}=0.30$ curve at late times in Fig. 5.

Figure 7:  The ratio of the final velocity to the initial velocity ($v_{\rm
fin}/v_{\rm in}$) post-collision to the initial velocity ($v_{\rm in}$). Note
that $v_{\rm fin}=0$ denotes the formation of a bound state. The dashed gray
line shows the classical evolution ($\xi=0$), whereas the solid lines denote
the evolution including quantum backreaction for different values of $\xi$:
$\xi=0.03$ (orange), $\xi=0.05$ (purple), $\xi=0.08$ (dark green), and
$\xi=0.1$ (red). The actual data points are also shown. The other parameters
used are: $\mu=0.1$, $\lambda=1$, $\eta=1$, $L=100$, $N=500$, and
$t_{0}=-100$.

Finally we come to our main objective of examining how the fractal nature of
the classical scattering shown in Fig. 1 changes when the kinks scatter in the
quantum vacuum of a second field. The results are presented in Fig. 7. The
dashed gray curve shows the results of the classical scattering and is the
same as presented in Fig. 1. The other curves show the results of the
scattering as we turn up the interaction strength $\xi$. An increase in $\xi$
shifts the scattering peaks to higher $v_{\rm in}$ but also reduces the peak
$v_{\rm fin}/v_{\rm in}$. The critical $v_{\rm in}$ above which there is no
further fractal structure also increases with increasing $\xi$. For example,
for $\xi=0$ the critical $v_{\rm in}$ is $\sim 0.26$ while for $\xi=0.1$ it is
$\sim 0.37$. Some of these features can be qualitatively understood based on
our earlier discussion that the time-dependent background of the scattering
kinks excites $\psi$ radiation and causes the kinks to lose kinetic energy.
This process makes it easier for the kink-antikink to form a bound state and
annihilate and it takes higher initial velocities for the kinks to reflect.
Also, in cases that the kinks do reflect, the lost energy means that the final
velocities of the kinks will be lower.

We have checked that our results are insensitive to the choice of lattice
spacing, $a$, and to the size of the simulation box, $L$. A comparison of our
results for different choices of $a$ and $L$ is shown in Appendix. B.

##  VI Conclusions and Discussion

While kink-antikink scattering has been thoroughly investigated in the
literature [20, 21, 25], our focus in this work has been on quantum effects on
the scattering. Even if the kink-antikink propagate in a quantum vacuum, they
provide a time-dependent background that can produce particles and radiation,
which in turn depletes the kinetic energy of the kinks and modifies the
scattering outcomes. The main result of this work is summarized in Fig. 7
where we see changes in the fractal structure of the scattering as the
interaction strength ($\xi$) between the kinks and the quantum field
increases. For $\xi\gtrsim 0.1$, the fractal structure disappears and there is
simply annihilation for $v_{\rm in}\lesssim 0.37$ and reflection for $v_{\rm
in}\gtrsim 0.37$.

The main tool for our analysis was the CQC formalism that enabled us to
conveniently use classical equations of motion for the quantum fields. We
treated quantum backreaction on the classical background by employing the
semiclassical approximation. In future we hope to return to examining the
effect of quantum fluctuations of the background scalar field itself on the
scattering outcome, as mentioned in the introduction. Perhaps recent progress
in dealing with systems containing classical and quantum components will be
helpful [45, 46, 47].

###### Acknowledgements.

We thank Omer Faruk Albayrak, Fabio van Dissel and George Zahariade for useful
discussions. M. M. is supported by NSF Grant No. AST-2108466. M. M. also
acknowledges support from the Institute for Gravitation and the Cosmos (IGC)
Postdoctoral Fellowship. M. M. wishes to thank the Yukawa Institute for
Theoretical Physics (YITP), Kyoto University for hospitality where a major
part of this work was done. T. V. is supported by the U.S. Department of
Energy, Office of High Energy Physics, under Award No. DE-SC0019470.

##  Appendix A Numerical methods

We use the _Verlet_ method to solve the coupled differential equations for
this work. In particular, we use the position Verlet method as follows,

$\displaystyle x_{n+1/2}$ $\displaystyle=$ $\displaystyle x_{n}+\frac{1}{2}dt\
\dot{x}_{n}\,,$ $\displaystyle\dot{x}_{n+1}$ $\displaystyle=$
$\displaystyle\dot{x}_{n}+dt\ \ddot{x}(x_{n+1/2})\,,$ (31) $\displaystyle
x_{n+1}$ $\displaystyle=$ $\displaystyle x_{n+1/2}+\frac{1}{2}dt\
\dot{x}_{n+1}\,.$

Although fairly simple, this method is efficient, fast, and gives us accurate
results for the current problem. More advanced numerical techniques may be
used but given the $\mathcal{O}(N^{2})$ complexity of the problem, the
requirements of fine spatial and temporal resolution to capture the details of
the dynamics, and the necessity to evolve the system for long durations, the
current numerical scheme is ideal. We have also verified our benchmark results
using a more advanced numerical technique - the explicit Crank-Nicholson
method with two iterations, and found no significant difference in the
evolution and observable quantities. All the results presented are simulated
on a circular lattice of physical length $L=100$ which is sampled using
$N=500$ points. This implies a spatial resolution $a=L/N=0.2$. We use a
temporal spacing proportional to $a$, $dt=a/5$. A smaller $a$ would indeed
improve the resolution and help us to capture the dynamics even better, but
with the current lattice spacing we have a total energy non-conservation due
to numerical errors, over the _entire duration of time evolution_ of the order
of $0.2\%$. We have also checked that our results are fairly independent of
the UV ($a=L/N$) and IR ($L$) cutoffs. We start the evolution at the initial
time, $t_{0}=-100$. The collisions happen around $t\sim 0$. The periodic
boundary conditions make it necessary to ensure that the finite lattice size
does not significantly interfere with our results. Owing to our use of
periodic boundary conditions, the radiation propagating outwards from the
center eventually comes back. Hence, we only evolve the dynamics for one light
crossing time, $t=L=100$, such that our results are not affected by radiation
that re-enters the collision region.

For all the results form here on we assume, $\lambda=1$ and $\eta=1$. We
choose $\mu=0.1\ m$. This in the regime where $\mu<m$, which is required since
$m$ is the mass of the classical background which we expect to be larger than
the mass of the quantum field. The two main parameters that we vary are the
initial velocity, $v_{\rm in}$, and the interaction strength, $\xi$. The
initial incoming velocity of the kink-antikink configuration $v_{\rm in}$ is
varied between $0.15$ and $0.4$. We do not consider lower velocities since for
such low velocities the final state is fixed to be a bound state and no
resonance structures are present, and in this work we are interested in
studying quantum modifications to the resonance structures. We do not consider
$v_{\rm in}>0.4$ since kinks with high initial velocities get reflected. We
vary $\xi$ in the range $[0,0.1]$.

##  Appendix B Analysis of the quality of numerics

(a) (a)

(b) (b)

Figure 8:  (a) Time-evolution of the background energy ($E_{\phi}$) for
different values of $dt$. The two curves overlap and are not distinctly
visible. (b) Time-evolution of the renormalized total energy $E$ for different
values of $dt$. The plots illustrate the independence of our results to the
choice of time step. The other parameters considered are: $v_{\rm in}=0.21$,
$\xi=0.5$, $\mu=0.1$, $\lambda=1$, $\eta=1$, $t_{0}=-100$, $L=100$, and
$N=500$. The collision happens at $t=0$.

The choice of time-step ($dt$) should not affect our results in any
considerable way. This is shown in Fig. 8. The energy in the background does
not show any difference on using a smaller time step ($dt=a/10$, dashed red
curve) than what is used for this work ($dt=a/5$, solid blue curve). The total
energy non-conservation for the time step we use ($dt=a/5$, solid purple
curve) is of the order of $\sim 0.2\%$ over the _entire_ time of evolution.
This is more than sufficient for the current work. The accuracy of energy
conservation increases even more if we take a smaller $dt$ ($=a/10$, solid
orange curve), which is to be expected.

Figure 9:  Time-evolution of the background energy ($E_{\phi}$) for different
values of $L$ and $N$ to illustrate the UV- and IR-dependence of our results.
The other parameters considered are: $v_{\rm in}=0.21$, $\xi=0.5$, $\mu=0.1$,
$\lambda=1$, $\eta=1$, and $t_{0}=-100$. The collision happens at $t=0$. Note,
lattice spacing $a=L/N$ and $dt=a/5$.

One of the main physical observables we are interested in is the energy
($E_{\phi}$) in the background field $\phi$ and the total energy ($E$) (as
discussed in Sec. IV.1). As one might expect, these observables should not
depend on the discretization scales - the spacing of the lattice ($a$) and the
physical size of the lattice ($L$). In Fig. 9 we show such is the case, where
decreasing the lattice spacing or increasing the physical size of the box has
no effect on $E_{\phi}$.

## References

  * Zurek [1985] W. Zurek, Cosmological Experiments in Superfluid Helium?, Nature 317, 505 (1985).
  * Pal  _et al._ [2017] V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, Observing dissipative topological defects with coupled lasers, Phys. Rev. Lett. 119, 013902 (2017).
  * del Campo  _et al._ [2010] A. del Campo, G. De Chiara, G. Morigi, M. B. Plenio, and A. Retzker, Structural defects in ion chains by quenching the external potential: The inhomogeneous kibble-zurek mechanism, Phys. Rev. Lett. 105, 075701 (2010).
  * Chuang  _et al._ [1991] I. Chuang, B. Yurke, R. Durrer, and N. Turok, Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals, Science 251, 1336 (1991).
  * Bowick  _et al._ [1994] M. J. Bowick, L. Chandar, E. A. Schiff, and A. M. Srivastava, The Cosmological Kibble mechanism in the laboratory: String formation in liquid crystals, Science 263, 943 (1994), arXiv:hep-ph/9208233 .
  * Hendry  _et al._ [1994] P. C. Hendry, N. S. Lawson, R. A. M. Lee, P. V. E. McClintock, and C. D. H. Williams, Generation of defects in superfluid 4he as an analogue of the formation of cosmic strings, Nature 368, 315 (1994), https://www.nature.com/articles/368315a0.pdf .
  * Ruutu  _et al._ [1996] V. M. H. Ruutu, V. B. Eltsov, A. J. Gill, T. W. B. Kibble, M. Krusius, Y. G. Makhlin, B. Plaçais, G. E. Volovik, and W. Xu, Vortex formation in neutron-irradiated superfluid 3he as an analogue of cosmological defect formation, Nature 382, 334 (1996), https://www.nature.com/articles/382334a0.pdf .
  * Bäuerle  _et al._ [1996] C. Bäuerle, Y. M. Bunkov, S. N. Fisher, H. Godfrin, and G. R. Pickett, Laboratory simulation of cosmic string formation in the early universe using superfluid 3he, Nature 382, 332 (1996), https://www.nature.com/articles/382332a0.pdf .
  * Monaco  _et al._ [2002] R. Monaco, J. Mygind, and R. J. Rivers, Zurek-kibble domain structures: The dynamics of spontaneous vortex formation in annular josephson tunnel junctions, Phys. Rev. Lett. 89, 080603 (2002).
  * Carmi  _et al._ [2000] R. Carmi, E. Polturak, and G. Koren, Observation of spontaneous flux generation in a multi-josephson-junction loop, Phys. Rev. Lett. 84, 4966 (2000).
  * Maniv  _et al._ [2003] A. Maniv, E. Polturak, and G. Koren, Observation of magnetic flux generated spontaneously during a rapid quench of superconducting films, Phys. Rev. Lett. 91, 197001 (2003).
  * Beugnon and Navon [2017] J. Beugnon and N. Navon, Exploring the kibble–zurek mechanism with homogeneous bose gases, Journal of Physics B: Atomic, Molecular and Optical Physics 50, 022002 (2017).
  * Zurek [1996] W. H. Zurek, Cosmological experiments in condensed matter systems, Phys. Rept. 276, 177 (1996), arXiv:cond-mat/9607135 .
  * Vachaspati [1998] T. Vachaspati, Topological defects in the cosmos and lab, Contemp. Phys. 39, 225 (1998), arXiv:hep-ph/9802311 .
  * Kibble [2002] T. W. B. Kibble, Testing cosmological defect formation in the laboratory, Physica C 369, 87 (2002), arXiv:cond-mat/0111082 .
  * Ade  _et al._ [2014] P. A. R. Ade _et al._ (Planck), Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, Astron. Astrophys. 571, A25 (2014), arXiv:1303.5085 [astro-ph.CO] .
  * Vilenkin and Shellard [2000] A. Vilenkin and E. P. S. Shellard,  _Cosmic Strings and Other Topological Defects_ (Cambridge University Press, 2000).
  * Kudryavtsev [1975] A. Kudryavtsev,  _Solitonlike solutions for a Higgs scalar field_ , Tech. Rep. (Institute of Theoretical and Experimental Physics, 1975).
  * Sugiyama [1979] T. Sugiyama, Kink-antikink collisions in the two-dimensional $\varphi$4 model, Progress of Theoretical Physics 61, 1550 (1979).
  * Campbell  _et al._ [1983] D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Resonance Structure in Kink - Antikink Interactions in $\phi^{4}$ Theory, Physica D 9, 1 (1983).
  * Anninos  _et al._ [1991] P. Anninos, S. Oliveira, and R. A. Matzner, Fractal structure in the scalar lambda (phi**2-1)**2 theory, Phys. Rev. D 44, 1147 (1991).
  * Fei  _et al._ [1992] Z. Fei, Y. S. Kivshar, and L. Vázquez, Resonant kink-impurity interactions in the ${\mathrm{\varphi}}^{4}$ model, Phys. Rev. A 46, 5214 (1992).
  * Goodman and Haberman [2005] R. H. Goodman and R. Haberman, Kink-antikink collisions in the $phi^{4}$ equation: The n-bounce resonance and the separatrix map, SIAM Journal on Applied Dynamical Systems 4, 1195 (2005), https://doi.org/10.1137/050632981 .
  * Marjaneh  _et al._ [2017] A. M. Marjaneh, D. Saadatmand, K. Zhou, S. V. Dmitriev, and M. E. Zomorrodian, High energy density in the collision of $N$ kinks in the $\phi^{4}$ model, Commun. Nonlinear Sci. Numer. Simul. 49, 30 (2017), arXiv:1605.09767 [nlin.PS] .
  * Alonso Izquierdo  _et al._ [2021] A. Alonso Izquierdo, J. Queiroga-Nunes, and L. M. Nieto, Scattering between wobbling kinks, Phys. Rev. D 103, 045003 (2021), arXiv:2007.15517 [hep-th] .
  * Adam  _et al._ [2020] C. Adam, K. Oles, T. Romanczukiewicz, and A. Wereszczynski, Kink-antikink collisions in a weakly interacting $\phi^{4}$ model, Phys. Rev. E 102, 062214 (2020), arXiv:1912.09371 [hep-th] .
  * Lizunova and van Wezel [2021] M. Lizunova and J. van Wezel, An introduction to kinks in $\varphi^{4}$-theory, SciPost Phys. Lect. Notes 23, 1 (2021), arXiv:2009.00355 [nlin.PS] .
  * Lee and Wick [1974] T. D. Lee and G. C. Wick, Vacuum Stability and Vacuum Excitation in a Spin 0 Field Theory, Phys. Rev. D 9, 2291 (1974).
  * Boguta [1983] J. Boguta, ABNORMAL NUCLEI, Phys. Lett. B 128, 19 (1983).
  * Wada and Schrieffer [1978] Y. Wada and J. R. Schrieffer, Brownian motion of a domain wall and the diffusion constants, Phys. Rev. B 18, 3897 (1978).
  * Yamaletdinov  _et al._ [2017] R. D. Yamaletdinov, V. A. Slipko, and Y. V. Pershin, Kinks and antikinks of buckled graphene: A testing ground for the $\phi^{4}$ field model, Phys. Rev. B 96, 094306 (2017), arXiv:1705.10684 [cond-mat.mes-hall] .
  * Hu  _et al._ [2011a] S. Hu, M. Lundgren, and A. J. Niemi, Discrete frenet frame, inflection point solitons, and curve visualization with applications to folded proteins, Phys. Rev. E 83, 061908 (2011a).
  * Hu  _et al._ [2011b] S. Hu, A. Krokhotin, A. J. Niemi, and X. Peng, Towards quantitative classification of folded proteins in terms of elementary functions, Phys. Rev. E 83, 041907 (2011b).
  * Bishop [1980] A. Bishop, Defect states in polyacetylene and polydiacetylene, Solid State Communications 33, 955 (1980).
  * Rice and Mele [1980] M. Rice and E. Mele, Phenomenological theory of soliton formation in lightly-doped polyacetylene, Solid State Communications 35, 487 (1980).
  * Vachaspati and Zahariade [2018] T. Vachaspati and G. Zahariade, Classical-quantum correspondence and backreaction, Phys. Rev. D 98, 065002 (2018), arXiv:1806.05196 [hep-th] .
  * Vachaspati and Zahariade [2019a] T. Vachaspati and G. Zahariade, Classical-Quantum Correspondence for Fields, JCAP 09, 015, arXiv:1807.10282 [hep-th] .
  * Vachaspati and Zahariade [2019b] T. Vachaspati and G. Zahariade, Classical-Quantum Correspondence and Hawking Radiation, JCAP 04, 013, arXiv:1803.08919 [hep-th] .
  * Mukhopadhyay and Vachaspati [2019] M. Mukhopadhyay and T. Vachaspati, Rolling classical scalar field in a linear potential coupled to a quantum field, Phys. Rev. D 100, 096018 (2019), arXiv:1907.03762 [hep-th] .
  * Morse and Feshbach [1953] P. Morse and H. Feshbach,  _Methods of Theoretical Physics_, International series in pure and applied physics (McGraw-Hill, 1953).
  * Vachaspati [2010] T. Vachaspati,  _Kinks and domain walls: An introduction to classical and quantum solitons_ (Cambridge University Press, 2010).
  * Rajaraman [1982] R. Rajaraman,  _Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory_, North-Holland personal library (North-Holland Publishing Company, 1982).
  * Ollé  _et al._ [2019] J. Ollé, O. Pujolas, T. Vachaspati, and G. Zahariade, Quantum Evaporation of Classical Breathers, Phys. Rev. D 100, 045011 (2019), arXiv:1904.12962 [hep-th] .
  * Mukhopadhyay  _et al._ [2022] M. Mukhopadhyay, E. I. Sfakianakis, T. Vachaspati, and G. Zahariade, Kink-antikink scattering in a quantum vacuum, JHEP 04, 118, arXiv:2110.08277 [hep-th] .
  * Bojowald and Ding [2021] M. Bojowald and D. Ding, Canonical description of cosmological backreaction, JCAP 03, 083, arXiv:2011.03018 [gr-qc] .
  * Oppenheim and Weller-Davies [2023a] J. Oppenheim and Z. Weller-Davies, Path integrals for classical-quantum dynamics, (2023a), arXiv:2301.04677 [quant-ph] .
  * Oppenheim and Weller-Davies [2023b] J. Oppenheim and Z. Weller-Davies, Covariant path integrals for quantum fields back-reacting on classical space-time, (2023b), arXiv:2302.07283 [gr-qc] .