aainstitutetext: Department of Physics, Arizona State University,
450 E. Tyler Mall, Tempe, AZ 85287-1504, U.S.A.bbinstitutetext: Institut de
Física d’Altes Energies (IFAE), The Barcelona Institute of Science and
Technology (BIST), Campus UAB, 08193 Bellaterra, Barcelona

#  Kink-antikink scattering in a quantum vacuum

Mainak Mukhopadhyay,111Corresponding author. b Evangelos I. Sfakianakis a
Tanmay Vachaspati b and George Zahariade<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study kink-antikink scattering in the sine-Gordon model in the presence of
interactions with an additional scalar field, $\psi$, that is in its quantum
vacuum. In contrast to the classical scattering, now there is quantum
radiation of $\psi$ quanta and the kink-antikink may form bound states that
resemble breathers of the sine-Gordon model. We quantify the rate of radiation
and map the parameters for which bound states are formed. Even these bound
states radiate and decay, and eventually there is a transition into long-lived
oscillons.

##  1 Introduction

Topological defects are expected to arise during phase transitions in the
early universe. These can include domain walls Saikawa:2020duz ;
Hiramatsu:2012sc ; Vilenkin:1982ks ; Sikivie:1982qv , cosmic strings
Vilenkin:1984ib ; Blanco-Pillado:2017rnf ; Dufaux:2010cf ; Damour:2000wa ;
Albrecht:1984xv ; Vachaspati:1984gt ; Hindmarsh:1994re ; Kibble:1976sj ,
monopoles tHooft:1974kcl ; Nambu:1974zg ; Gibbons:1986df ; Mavromatos:2020gwk
and textures Gueron:1997ck ; Notzold:1990jt ; Spergel:1992ff ; Durrer:1990qe ;
Turok:1989ai ; Brandenberger:2019txi , depending on the symmetry breaking
pattern. While topological defects have not yet been observed in a
cosmological context Planck:2013mgr , they are abundant in condensed matter
systems Zurek:1985qw ; PhysRevLett.119.013902 ; PhysRevLett.105.075701 ;
Chuang:1991zz ; Bowick:1992rz ; Hendry1994 ; Ruutu1996 ; Bauerle1996 ;
PhysRevLett.89.080603 ; PhysRevLett.84.4966 ; PhysRevLett.91.197001 ;
Beugnon2017 , with intriguing analogies between the two areas of research
Kibble:2001wj ; Vachaspati:1998vc ; Zurek:1996sj .

Topological defects are also an ideal setting for studying quantum effects in
spacetime dependent classical backgrounds. Significant effort has already gone
into the quantization of topological defects Rajaraman:1982is ;
Coleman:1985rnk ; Christ:1975wt ; Goldstone:1974gf ; Cahill:1974cs ;
Dashen:1974ci ; Dashen:1974cj ; Dashen:1974ck , in which case the topological
defects form space-dependent backgrounds for quantum fields. Less effort has
been invested in time-dependent problems that involve quantum fields in the
background of dynamical topological defects Dashen:1975hd . Going one step
further, the quantum fields will backreact on the dynamics of the topological
defects and lead to non-trivial interplay between the quantum and classical
degrees of freedom. This is the subject of the present paper.

The sine-Gordon equation in one spatial dimension holds a special place in the
soliton literature, since it is integrable and allows for the construction of
multi-soliton solutions by the use of the Bäcklund transformation. The
solitons (or kinks) of the sine-Gordon equation are solutions which
interpolate between two successive minima of the potential. The sine-Gordon
model has been analyzed in great detail Rajaraman:1982is ; Coleman:1985rnk ;
Vachaspati:2006zz  and kink-antikink scattering is known to be trivial – the
kink and antikink pass through each other albeit with a time delay – a result
that holds even when $\phi$ is treated in quantum field theory. scattering in
a model where the sine-Gordon scalar field, $\phi$, is coupled to a second
scalar field, $\psi$. The kink and antikink propagate in the quantum vacuum of
$\psi$, deforming the vacuum around them. (Alternately, the kink and antikink
get dressed by the $\psi$ field.) Upon kink-antikink scattering, $\psi$
particles (wavepackets) are radiated. Depending on the parameters, the quantum
radiation of $\psi$ particles can lead to the production of a kink-antikink
bound state that is known as a “breather” in the sine-Gordon literature
Rajaraman:1982is ; Coleman:1985rnk ; Vachaspati:2006zz . In this case, the
breather oscillates and continues to radiate $\psi$ particles. (This was more
specifically studied in Ref. Olle:2019skb , albeit in a slightly different
model.) Surprisingly we find that the breather eventually settles into another
oscillating configuration that radiates very slowly, resembling an “oscillon”.
Oscillons are long-lived localized configurations arising in a variety of
scalar field theories and have been shown to emerge naturally in scenarios
ranging from preheating to bubble collisions (see e.g. Amin:2011hj ;
Copeland:1995fq ; Gleiser:2009ys ; Farhi:2005rz ; Graham:2006vy ; Zhou:2013tsa
; Fodor:2008es ; Hiramatsu:2020obh ; VanDissel:2020umg ; Zhang:2020bec ;
Zhang:2020ntm ).

Our analysis is enabled by the “classical-quantum correspondence” (CQC) in
which the dynamics of free quantum fields in spacetime dependent classical
backgrounds has been shown to be equivalent to a completely classical problem
in higher dimensions Vachaspati:2018hcu ; Olle:2019skb ; Mukhopadhyay:2019hnb
. Quantum backreaction on the classical background is included in the analysis
in the semiclassical approximation in which quantum operators occurring in the
classical equations of motion are replaced by their dynamical expectation
values. Special attention is given to the renormalization issues that arise,
since a divergent part of the expectation values contributes to the mass
parameter of the background field.

In Sec. 2 we introduce the field theoretic model, as well as the equations
that will be solved numerically to determine the full dynamics of the kink-
antikink collision. We take care to distinguish the no backreaction case,
where the kink-antikink background is fixed, from the general case with
backreaction. In Secs. 3 and 4 we outline the results of our numerical
analysis: we determine the region of parameter space where a breather-like
bound state is formed after the collision and discuss the different phases of
its decay. We conclude in Sec. 5, where we discuss the implications of this
work, its limitations as well as several future prospects. We use natural
units where $\hbar=c=1$ throughout.

##  2 Setup

We work in 1+1 dimensions and consider a sine-Gordon field $\varphi(t,x)$
coupled to a massive scalar field $\psi(t,x)$ according to the Lagrangian
density

$\mathcal{L}=\frac{1}{2}\dot{\varphi}^{2}-\frac{1}{2}\varphi^{\prime
2}-\frac{m^{2}}{\kappa^{2}}(1-\cos(\kappa\varphi))+\frac{1}{2}\dot{\psi}^{2}-\frac{1}{2}\psi^{\prime
2}-\frac{1}{2}\mu^{2}\psi^{2}-\frac{\lambda}{2}(1-\cos(\kappa\varphi))\psi^{2}\,.$
(1)

Here $m$, $\mu$ are the masses of the two fields, $\kappa$ is a parameter
introduced for future convenience and $\lambda$ is a coupling constant. We
will assume that conditions are such that the $\varphi$ field can be treated
classically, while the $\psi$ field is treated fully quantum mechanically. In
general, the (potentially space and time-dependent) classical “background”
field $\varphi$ excites the quantum “radiation” field $\psi$, and the
excitations of $\psi$ in turn backreact on $\varphi$. In the following it will
be useful to make the field redefinition
$\varphi\rightarrow\phi=\kappa\varphi$ and to work with the rescaled field
$\phi(t,x)$. With this new field variable, the Lagrangian density becomes

$\mathcal{L}=\frac{1}{\kappa^{2}}\left[\frac{1}{2}\dot{\phi}^{2}-\frac{1}{2}\phi^{\prime
2}-m^{2}(1-\cos\phi)\right]+\frac{1}{2}\dot{\psi}^{2}-\frac{1}{2}\psi^{\prime
2}-\frac{1}{2}\mu^{2}\psi^{2}-\frac{\lambda}{2}(1-\cos\phi)\psi^{2}\,.$ (2)

We see that the coupling between the two fields $\phi$ and $\psi$ is such that
the discrete shift symmetry $\phi\to\phi+2\pi n$, $n\in\mathbb{Z}$, is
maintained.

###  2.1 Neglecting backreaction

####  2.1.1 Background dynamics

We start by discussing the limit $\kappa\rightarrow 0$ which corresponds to
the case where the dynamics of the $\phi$ field doesn’t feel the presence of
the radiation field $\psi$. If $\psi$ is entirely neglected, this is also the
limit when $\phi$ is classical because, in the path integral, $\kappa\to 0$ is
equivalent to $\hbar\to 0$. In this “no backreaction” case, the $\phi$ field
equation reduces to the sine-Gordon equation

$\ddot{\phi}-\phi^{\prime\prime}+m^{2}\sin\phi=0\,.$ (3)

It is worth mentioning that, in this case, one can define the (conserved)
energy of the sine-Gordon field unambiguously by

$E_{\phi}=\frac{1}{\kappa^{2}}\int
dx\left[\frac{1}{2}\dot{\phi}^{2}+\frac{1}{2}\phi^{\prime
2}+m^{2}(1-\cos\phi)\right]\,,$ (4)

and it is easy to see that the $\kappa\rightarrow 0$ limit is equivalent to
the energy of the sine-Gordon field $\phi$ being much larger than the
(renormalized) vacuum energy of the field $\psi$. We will come back to this
point in Sec. 4.

Eq. (3) has well-known kink and anti-kink solutions,

$\phi_{\pm}(t,x)=\pm 4\arctan\left(e^{\gamma m(x-x_{0}-v(t-t_{0}))}\right)\,,$
(5)

which describe a soliton (or antisoliton depending on the sign) whose center
is at $x_{0}$ at time $t_{0}$ and which moves to the right with velocity $v$,
with $\gamma=1/{\sqrt{1-v^{2}}}$ being the Lorentz factor. It is worth noting
that the energy of this configuration is

$E_{\pm}(v)=\frac{8\gamma m}{\kappa^{2}}\,.$ (6)

The integrability properties of the sine-Gordon equation allow for the
analytical construction of more complicated solutions involving a kink and an
antikink. The first one is the so-called breather solution which can be
understood as a bound state of a kink and an antikink and which reads

$\phi_{\rm breather}(t,x)=4\arctan\left(\frac{\eta\sin(\omega
t)}{\cosh(\eta\omega x)}\right)\,.$ (7)

Here $\omega$ is the angular frequency of the breather and
$\eta={\sqrt{m^{2}-\omega^{2}}}/{\omega}$. The energy of the breather is

$E_{\rm
breather}=\frac{16\eta\omega}{\kappa^{2}}=\frac{16m}{\kappa^{2}}\sqrt{1-\frac{\omega^{2}}{m^{2}}}\,,$
(8)

and is seen to be less than the sum of the energy of a static kink and a
static antikink.

The other important solution, which will be the main focus of our attention in
the remainder of this paper, can be obtained from the breather solution by
making the formal substitution $\omega=im\gamma v$. It reads

$\phi_{K\bar{K}}(t,x)=4\arctan\left(\frac{\sinh(\gamma mvt)}{v\cosh(\gamma
mx)}\right)\,,$ (9)

and describes the elastic scattering of a kink and an antikink infinitely
separated at $t=-\infty$ and moving towards each other with velocity $v$. The
collision occurs at $t=x=0$ and the kink and antikink pass through each other
with a time delay. As expected, the energy of this field configuration is
simply the sum of the energies of a kink and an antikink moving with velocity
$v$,

$E_{K\bar{K}}=\frac{16\gamma m}{\kappa^{2}}=2E_{\pm}(v)\,.$ (10)

Having summarized some important features of the sine-Gordon equation, we are
ready to study the quantum field $\psi$ living in the background of the latter
solution.

####  2.1.2 Quantum radiation

Integrability of the sine-Gordon model implies that the scattering of a kink-
antikink pair is trivial i.e. never forms a bound state, and that a breather
never decays. These are classical properties of the sine-Gordon model but they
are maintained when taking the quantum fluctuations of the field into account.
However, in our case, the classical sine-Gordon field is coupled to an
external quantum scalar field and the energy exchange between the classical
field configuration and the quantum bath is expected to invalidate these
properties.

We thus turn our attention to the quantum radiation that occurs during kink-
antikink scattering, equivalently particle production in the time-dependent
background $\phi_{K\bar{K}}(t,x)$ given in Eq. (9), using the framework of the
classical- quantum correspondence (CQC) Vachaspati:2018llo ;
Vachaspati:2018hcu ; Olle:2019skb  that we briefly describe below. Since in
the limit $\kappa\rightarrow 0$ the dynamics of the background field $\phi$ is
insensitive to the presence of the field $\psi$, we will work with the
truncated Lagrangian density

${\mathcal{L}}_{\psi}=\frac{1}{2}\dot{\psi}^{2}-\frac{1}{2}\psi^{\prime
2}-\frac{1}{2}\left\\{\mu^{2}+\lambda\left[1-\cos\phi_{K\bar{K}}(t,x)\right]\right\\}\psi^{2}\,.$
(11)

It is easy to understand why this model leads to excitations of the quantum
field $\psi$. Indeed this Lagrangian density describes a free scalar field
with a space and time dependent mass-squared
$M^{2}(t,x)=\mu^{2}+\lambda\left[1-\cos\phi_{K\bar{K}}(t,x)\right]$. The non-
adiabatic variation of $M^{2}(t,x)$ will lead to particle production and is
expected to occur mostly at the time of kink-antikink collision at $t=0$.

To study the scattering more quantitatively using numerical methods, we start
by compactifying the spatial dimension on a circle of length $L$ which we then
discretize on a regular lattice consisting of $N$ evenly spaced points. The
lattice spacing is therefore $a={L}/{N}$. We can further define the
discretized field values $\psi_{i}=\psi(t,-L/2+ia)$ and
$\phi_{i}=\phi_{K\bar{K}}(t,-L/2+ia)$ for $i=1,\ldots,N$, and use the
following differencing scheme to estimate the second spatial derivative,

$\psi^{\prime\prime}(t,ia)\longrightarrow\frac{\psi_{i+1}-2\psi_{i}+\psi_{i-1}}{a^{2}}\,.$
(12)

With these conventions, and after a spatial integration by parts, the
Lagrangian of this discretized model can be written as

${L}_{\psi,\rm
disc.}=\frac{a}{2}\dot{\bm{\psi}}^{T}.\dot{\bm{\psi}}-\frac{a}{2}\bm{\psi}^{T}.\bm{\Omega}^{2}.\bm{\psi}\,,$
(13)

where we have arranged the discretized values of the field $\psi$ in a column
vector $\bm{\psi}=(\psi_{1},\psi_{2},\dots,\psi_{N})^{T}$ and introduced the
$N\times N$ matrix

$[\bm{\Omega}^{2}]_{ij}=\begin{cases}+{2}/{a^{2}}+\mu^{2}+\lambda\left[1-\cos{\phi}_{i}\right]\,,&i=j\\\
-{1}/{a^{2}}\,,&i=j\pm 1\ (\text{mod}\ N)\\\
0\,,&\text{otherwise}\,.\end{cases}$ (14)

In this form, the discretized Lagrangian is immediately seen to describe a
collection of $N$ harmonic oscillators (located at each lattice point) coupled
to each other via quadratic interactions. The total energy and energy density
at each lattice point $i$ for the $\psi$ field in this discretized model are
also well-defined and given by222Here, in order to get a more accurate
discretized estimate of the local gradient energy, the average between its
forward differencing and backward differencing approximations is used. This
allows us to integrate Eq. (16) into Eq. (15). Notice however that simply
choosing a forward or backward differencing scheme for the estimation of this
component of the energy density would also have been consistent.

$\displaystyle{H}_{\psi,\rm disc.}$ $\displaystyle=$
$\displaystyle\frac{a}{2}\dot{\bm{\psi}}^{T}.\dot{\bm{\psi}}+\frac{a}{2}\bm{\psi}^{T}.\bm{\Omega}^{2}.\bm{\psi}\,,$
(15) $\displaystyle{\mathcal{H}}_{\psi,\text{disc.},i}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\dot{\psi}_{i}^{2}+\frac{1}{4a^{2}}\left[(\psi_{i+1}-\psi_{i})^{2}+(\psi_{i}-\psi_{i-1})^{2}\right]+\frac{1}{2}\left\\{\mu^{2}+\lambda\left[1-\cos{\phi}_{i}\right]\right\\}\psi_{i}^{2}\,.$
(16)

The next step is to quantize the theory (in the Heisenberg picture) by
promoting the discretized field values $\psi_{i}$ to operators
$\hat{\psi}_{i}$ and introducing the time-dependent matrix $\bm{Z}$ whose
(complex) elements $Z_{ij}$ satisfy the relation

$\hat{\psi}_{i}=Z_{ij}^{*}\hat{a}_{j}(t_{0})+Z_{ij}\hat{a}^{\dagger}_{j}(t_{0})\,.$
(17)

Here the reference time $t_{0}$ is chosen so that the background for $t\leq
t_{0}$ can be approximated by a slowly-moving kink-antikink pair separated by
a large distance (much larger than their typical size). This is equivalent to
specifying a reference vacuum state $|0\rangle$, the state annihilated by all
the $\hat{a}_{i}(t_{0})$. (The ladder operators $\hat{a}_{i}(t_{0})$ and
$\hat{a}^{\dagger}_{i}(t_{0})$ refer to the quantum harmonic oscillators
located at each lattice point.) Now the dynamics of the field $\hat{\psi}$ is
simply given by the Heisenberg equations

$\ddot{\bm{Z}}+\bm{\Omega}^{2}.\bm{Z}=0\,,$ (18)

with initial conditions

$\bm{Z}(t_{0})=-\frac{i}{\sqrt{2a}}\bm{\Omega}(t_{0})^{-1/2}\quad\text{and}\quad\dot{\bm{Z}}(t_{0})=\frac{1}{\sqrt{2a}}\bm{\Omega}(t_{0})^{1/2}\,.$
(19)

Since $\Omega^{2}$ is a symmetric positive definite matrix, $\Omega^{\pm 1/2}$
is computed by first diagonalizing $\Omega^{2}$ and then applying the desired
power function to its positive eigenvalues. Notice that these initial
conditions only define the vacuum of the theory unambiguously when the
background $\bm{\Omega}^{2}(t)$ is approximately constant for $t\leq t_{0}$.
In other words if the time evolution of the background before the time $t_{0}$
is adiabatically slow, a different choice of initial time smaller than $t_{0}$
will not modify the quantum dynamics. More precisely the initial conditions
(19) define the $0$-th order adiabatic vacuum (which corresponds to the lowest
order approximation of the mode functions within the WKB approximation
Birrell:1982ix ). In particular, if the background is such that no $t_{0}$
obeying the required properties can be found, then we can expect spurious
excitations of the field $\psi$ to occur and one should strive to minimize
them. In the case at hand, the kink and antikink are moving towards each other
with velocity $v$ at $t=-\infty$ and we find ourselves exactly in the latter
situation. However, we expect that for small $v$ (in practice less than $0.3$)
the initial conditions (19) will still give reasonable results (and we will
come back to this later in Sec. 3).

####  2.1.3 Observables

The evolution of $\psi$ is given by the system of equations given in matrix
form in Eq. (18). Initialized by Eqs. (19), they are particularly simple and
they can readily be solved numerically. Moreover, the $Z_{ij}$ variables allow
for the easy computation of various observable quantities of interest. For
example, the expression for the vacuum expectation value of the energy in the
radiation field can be written as

$E_{\psi}\equiv\langle 0|\hat{H}_{\psi,\rm
disc.}|0\rangle=\frac{a}{2}\text{Tr}\left[\dot{\bm{Z}}^{\dagger}.\dot{\bm{Z}}+\bm{Z}^{\dagger}.\bm{\Omega}^{2}.\bm{Z}\right]\,.$
(20)

This is equal to the total (classical) energy in the $Z_{ij}$ variables.
Similarly, the vacuum expectation value of the energy density at the $i$-th
lattice point is

$\displaystyle\rho_{\psi,i}$ $\displaystyle\equiv\langle
0|\hat{\mathcal{H}}_{\psi,\text{disc.},i}|0\rangle\,$
$\displaystyle=\sum_{j=1}^{N}\left(\frac{1}{2}|\dot{Z}_{ij}|^{2}+\frac{1}{4a^{2}}\left[|Z_{i+1,j}-Z_{ij}|^{2}+|Z_{ij}-Z_{i-1,j}|^{2}\right]+\frac{1}{2}\left\\{\mu^{2}+\lambda\left[1-\cos{\phi}_{i}\right]\right\\}|Z_{ij}|^{2}\right)\,.$
(21)

Moreover the spatial two-point function of the radiation field can be written
as

$C_{ij}\equiv\langle
0|\hat{\psi}_{i}\hat{\psi}_{j}|0\rangle=\sum_{k=1}^{N}Z^{*}_{ik}Z_{jk}\,.$
(22)

In order to render these expressions insensitive to the discretization scale
$a$, we first renormalize the spatial two-point function by subtracting its
$\lambda=0$ counterpart,

$C_{ij}^{(R)}=C_{ij}-\left.C_{ij}\right|_{\lambda=0}\,,$ (23)

and then use it to renormalize the $\lambda$-dependent part of the energy and
energy density. This procedure is closely related to point-splitting
Birrell:1982ix . The resulting renormalized quantities are thus

$\displaystyle\rho_{\psi,i}^{(R)}$ $\displaystyle=$
$\displaystyle\rho_{\psi,i}-\frac{1}{2}\lambda\left[1-\cos\phi_{i}\right]\left.C_{ii}\right|_{\lambda=0}-\left.\rho_{\psi,i}\right|_{\lambda=0}\,,$
(24) $\displaystyle E_{\psi}^{(R)}$ $\displaystyle=$ $\displaystyle
E_{\psi}-\sum_{i=1}^{N}\frac{a}{2}\lambda\left[1-\cos\phi_{i}\right]\left.C_{ii}\right|_{\lambda=0}-\left.E_{\psi}\right|_{\lambda=0}\,.$
(25)

In both these equations, the last term corresponds to the subtraction of the
(constant) zero-point energy.

We can therefore compute any quantity of interest for the study of $\psi$
particle production in the $\phi_{K\bar{K}}$ background by studying the
classical dynamics of the variables $Z_{ij}$ (with well-chosen initial
conditions). We thus trade $N$ real quantum variables (corresponding to the
discretized field values $\psi_{i}$) for $N\times N$ complex classical
variables (the $Z_{ij}$). This is the essence of the CQC and it is closely
related to the mode function and Bogoliubov coeffcient methods
Vachaspati:2018hcu .

####  2.1.4 Initial conditions and vacuum structure

The renormalized energy density observable $\rho_{\psi,i}^{(R)}$ allows us to
visualize the vacuum structure of the quantum field $\psi$. Fig. 1 shows the
renormalized energy density in the $\psi$ field at the initial time
superimposed over the background kink-antikink profile. (The parameter $a$
will always be chosen such that the lattice provides a good approximation of
the continuum limit and therefore we will liberally identify the discretized
quantities $\rho^{(R)}_{\psi,i}(t)$ and $\phi_{i}(t)$ to their continuous
counterparts $\rho^{(R)}_{\psi}(t,x)$ and $\phi(t,x)$ for $x=-L/2+ia$.) We see
that the presence of the background induces a dip in the energy density around
the kink and antikink, akin to two clouds of $\psi$ particles. The width of
these clouds is set by the kink and antikink width i.e. $1/m$. However, as
shown in Fig. 2, the depth of the underdensities depends on the coupling
$\lambda$ and the mass of the $\psi$ field $\mu$. In particular, the trough of
the $\psi$ energy density around the kink-antikink position increases for
larger $\lambda$ and smaller $\mu$. Moreover, for some parameters, the energy
density of $\psi$ shows some non-trivial features that reflect the relevance
of the additional length scales $\mu^{-1}$ and $\lambda^{-1/2}$.

Figure 1:  Example of the renormalized initial energy density in $\psi$ and
the initial kink-antikink profile. The kinks are dressed in $\psi$ particles
whose energy within the kink is lower than that outside the kink, i.e. they
form bound states. The kink-antikink profile is shown as a red dashed line
(vertical scale on the right) and the clouds of $\psi$ particles (represented
by the renormalized energy density of $\psi$) are shown in dark blue (vertical
scale on the left). The parameters are $L=100$, $N=500$, $m=1$, $v=0.1$,
$\mu=0.1$, $\lambda=0.3$, $\kappa\rightarrow 0$ and the initial time is chosen
as $t_{0}=-100$.

(a)

(b)

Figure 2: (a) Renormalized initial energy density in $\psi$
($\rho^{(R)}_{\psi}$) for various values of $\lambda$: 0.3 (solid green), 0.5
(dashed red), 0.7 (dotted dark blue) for $\mu=0.1$. (b) Renormalized initial
energy density in $\psi$ for various values of $\mu$: 0.1 (solid green), 0.3
(dotted brown), 0.5 (dashed purple), 0.7 (dot-dashed pink) for $\lambda=0.3$.
The other parameters are $L=100$, $N=500$, $m=1$, $v=0.1$ , $\kappa\rightarrow
0$ and the initial time is $t_{0}=-100$. Note that the solid green lines
correspond to the same set of parameters ($\lambda=0.3$, $\mu=0.1$) in both
the panels.

###  2.2 Including backreaction

The results above assumed that backreaction can be neglected. This is
certainly a good approximation when $\kappa$ is small but what happens when
backreaction can no longer be neglected? This constitutes the main part of
this work, to which we now turn.

We thus consider the case $\kappa\neq 0$ so that the field equation for the
$\phi$ field gets a contribution from the $\psi$ field:

$\ddot{\phi}-\phi^{\prime\prime}+\left(m^{2}+\frac{\lambda\kappa^{2}}{2}\psi^{2}\right)\sin\phi=0\,.$
(26)

Of course this equation is fully classical and one would need to decide how
the quantum excitations in the $\psi$ field couple to the classical background
$\phi$ by modifying the coupling term proportional to $\psi^{2}\sin\phi$. The
most straightforward way is to use the semiclassical approximation333 Another
procedure for incorporating backreaction is based on the stochastic method,
where the initial conditions for the field $\psi$ are sampled from a
distribution similar to Eq. (19). Several simulations, each with different
samples for the initial conditions, would be performed and then averaged. We
leave the interesting comparison between the two methods for future work.
which simply entails replacing $\psi(t,x)^{2}$ by $\langle
0|\hat{\psi}(t,x)^{2}|0\rangle$. Therefore the $\phi$ field equation becomes

$\ddot{\phi}-\phi^{\prime\prime}+\left(m^{2}+\frac{\lambda\kappa^{2}}{2}\langle
0|\hat{\psi}^{2}|0\rangle\right)\sin\phi=0\,.$ (27)

To make contact with the methods described in the previous section, we need
the discretized version of this equation,

$\ddot{\phi}_{i}-\frac{1}{a^{2}}(\phi_{i+1}-2\phi_{i}+\phi_{i-1})+\left(m^{2}+\frac{\lambda\kappa^{2}}{2}\langle
0|\hat{\psi}_{i}^{2}|0\rangle\right)\sin\phi_{i}=0\,,$ (28)

where $\phi_{i}=\phi(t,-L/2+ia)$ is not given by the non-dynamical kink-
antikink solution $\phi_{K\bar{K}}$ anymore, rather it will have to be solved
for subject to the initial conditions
$\phi_{i}(t_{0})=\phi_{K\bar{K}}(t_{0},-L/2+ia)$ and
$\dot{\phi}_{i}(t_{0})=\dot{\phi}_{K\bar{K}}(t_{0},-L/2+ia)$. Solving Eq. (28)
requires being able to compute the dynamics of the quantity $\langle
0|\hat{\psi}_{i}^{2}|0\rangle$. This can be done by using the results of the
previous section and in particular Eq. (22) to make the substitution

$\langle 0|\hat{\psi}_{i}^{2}|0\rangle=\sum_{j=1}^{N}|Z_{ij}|^{2}$ (29)

with $Z_{ij}$ given by Eqs. (18) and (19), where background field values
${\phi}_{i}$ in the definition of $\bm{\Omega}^{2}$ are replaced by the
corresponding dynamical values. Because $\sum_{j=1}^{N}|Z_{ij}|^{2}$ is a 1+1
dimensional two-point function in the coincident point limit, in other words
$C_{ii}$ given in Eq. (22), it is logarithmically sensitive to the
discretization scale $a$, and it would produce infinite backreaction in the
continuum limit Vachaspati:2018llo ; Vachaspati:2018hcu ; Olle:2019skb . This
can be remedied by noticing that the parameter $m$ appearing in the field
equations is actually the bare mass of the sine-Gordon field, and using it to
renormalize the divergence. We therefore define the physical mass $m_{\rm
phys}$ of the sine-Gordon field by

$m^{2}=m^{2}_{\rm
phys}-\frac{\lambda\kappa^{2}}{2}\sum_{j=1}^{N}\left.|Z_{ij}|^{2}\right|_{\lambda=0}=m^{2}_{\rm
phys}-\frac{\lambda\kappa^{2}}{4a}\left[\bm{\Omega}_{0}^{-1}\right]_{ii}=m^{2}_{\rm
phys}-\frac{\lambda\kappa^{2}}{4L}{\rm Tr}\,\bm{\Omega}_{0}^{-1}\,.$ (30)

Here, in the second equality, we used the fact that, when $\lambda=0$,
$\bm{\Omega}^{2}(t)\equiv\bm{\Omega}^{2}_{0}$ is a constant matrix, and
therefore Eq. (18) has the simple solution

$\bm{Z}(t)=-\frac{i}{\sqrt{2a}}e^{i\bm{\Omega}_{0}(t-t_{0})}\bm{\Omega}_{0}^{-1/2}\,.$
(31)

Moreover in the third equality of Eq. (30) we use the fact that all the
diagonal coefficients of the $\bm{\Omega}_{0}^{-1}$ matrix are equal to each
other (see Eq. (14)). Overall this procedure is equivalent to replacing the
two-point function $C_{ii}$ by its renormalized counterpart $C_{ii}^{(R)}$ in
Eq. (28), and $m$ by $m_{\rm phys}$ in every equation where it appears. Notice
that, in the non-backreacting case ($\kappa\rightarrow 0$), the bare mass $m$
and the physical mass $m_{\rm phys}$ are one and the same. Henceforth, we will
simply choose $\kappa=1$ when taking backreaction into account.

Summing up, the backreacted dynamics (within the semiclassical approximation)
are given by the system of coupled differential equations

$\ddot{\phi}_{i}-\frac{1}{a^{2}}(\phi_{i+1}-2\phi_{i}+\phi_{i-1})+\left[m_{\rm
phys}^{2}+\frac{\lambda\kappa^{2}}{2}\sum_{j=1}^{N}\left(|Z_{ij}|^{2}-\left.|Z_{ij}|^{2}\right|_{\lambda=0}\right)\right]\sin\phi_{i}=0\,,$
(32)

and

$\ddot{\bm{Z}}+\bm{\Omega}^{2}\bm{Z}=0\,,$ (33)

with initial conditions

$\displaystyle\phi_{i}(t_{0})=\phi_{K\bar{K}}(t_{0},-L/2+ia)\,,\quad$
$\displaystyle\quad\dot{\phi}_{i}(t_{0})=\dot{\phi}_{K\bar{K}}(t_{0},-L/2+ia)\,,$
(34)
$\displaystyle\bm{Z}(t_{0})=-\frac{i}{\sqrt{2a}}\bm{\Omega}(t_{0})^{-1/2}\,,\quad$
$\displaystyle\quad\dot{\bm{Z}}(t_{0})=\frac{1}{\sqrt{2a}}\bm{\Omega}(t_{0})^{1/2}\,,$
(35)

where we recall that

$\phi_{K\bar{K}}(t,x)=4\arctan\left(\frac{\sinh(\gamma m_{\rm
phys}vt)}{v\cosh(\gamma m_{\rm phys}x)}\right)$ (36)

and where the matrix $\bm{\Omega}^{2}$ defined in Eq. (14) depends on the
fully dynamical background $\phi_{i}(t)$. Note that the mass of the $\phi$
field used in the initial conditions is the physical mass $m_{\rm phys}$, as
shown in the above equation.

Before discussing the results of our simulations, let us say a few words about
the observables that we are going to use in our study of particle production
and the associated backreaction during kink-antikink collisions. These need to
accurately account for the energy exchanged between the background and the
quantum radiation bath. Of course, in general, in a fully interacting theory
it is impossible to separate the energy of a particular subsystem. However, as
long as the coupling $\lambda$ is not too big (smaller than $m_{\rm phys}^{2}$
in practice) we can assume that Eqs. (25) and (24) still provide a useful
measure of the (renormalized) energy, respectively energy density, in the
quantum field $\psi$. Under these assumptions, the energy in the kink-antikink
background is given by

$E_{\phi}\equiv\frac{a}{\kappa^{2}}\sum_{i=1}^{N}\left[\frac{1}{2}\dot{\phi}_{i}^{2}+\frac{1}{4a^{2}}\left[(\phi_{i+1}-\phi_{i})^{2}+(\phi_{i}-\phi_{i-1})^{2}\right]+m_{\rm
phys}^{2}(1-\cos\phi_{i})\right]\,.$ (37)

Notice that the total (conserved) energy of the coupled system is given by

$E=E_{\phi}+E_{\psi}^{(R)}\,,$ (38)

where $E_{\phi}$ and $E_{\psi}^{(R)}$ are given by Eq. (37) and Eq. (25)
respectively.

##  3 Results

In this section we present our numerical results. We will always work in units
where $m_{\rm phys}=1$. which is equivalent to rescaling space, time, $\mu$
and $\lambda$ by $m_{\rm phys}$. Our lattice is a circle of physical size
$L=100$ and is sampled by $N=500$ equally spaced points ($a=0.2$). We use an
explicit Crank-Nicholson method with two iterations to solve the system of
coupled ordinary differential equations given in Eqs. (32) and (33) with the
initial conditions of Eqs. (34) and (35). The initial time is taken to be
$t_{0}=-100$. Because most interesting effects are expected to occur at or
around the time of collision, $t=0$, and to make sure that finite lattice size
effects don’t spoil any physical effects, we only evolve the equations for one
light-crossing time after the collision i.e. up to $t=L=100$. We have also
made sure that the results do not strongly depend on the UV and IR cutoffs
($a=L/N$ and $L$) and that our renormalization scheme is sufficient to
eliminate any UV sensitivity (see Fig. 16). Of course, decreasing $a$ would
improve the resolution of the simulations but, given the $N^{2}$ complexity of
the code, this is not feasible without significantly more computational cost.
Similarly a larger lattice would allow us to track the dynamics of the system
for a longer time but maintaining spatial resolution would again require a
corresponding increase in $N$. In the current work, energy non-conservation
due to numerical error over the whole duration of time evolution is of the
order of $0.1\%$.

The range over which we will vary the different parameters $\lambda$, $v$ and
$\mu$ will depend on numerical accuracy as well as on the intrinsic
limitations of our model. To avoid strong coupling effects we will be allowing
$\lambda$ to vary between 0 and 1. Since our initial conditions for the
$Z_{ij}$ variables are strictly only valid when the background is
adiabatically varying, we limit ourselves to velocities $v$ smaller than
0.3.444Because of this violation of adiabaticity at $t=t_{0}$, a small amount
of spurious particle production is observed for the $v=0.3$ case but the
corresponding dissipated energy is of the order of $0.1\%$ of $E_{\phi}$ which
can safely be neglected. Moreover, we require the kink-antikink pair to be
well-separated at the initial time $t_{0}$ which, because of the finite size
of our periodic lattice, limits the range of $v$ to the interval $(0.05,0.4)$.
Since the background field $\phi$ is assumed to behave classically, its mass
$m_{\rm phys}$ is expected to be larger than the mass of the radiation field
$\psi$. Thus $\mu$ is constrained to be smaller than 1.

We start by presenting the two qualitatively different possible outcomes of
the inelastic kink-antikink scattering studied in this paper: pure scattering,
or formation of a bound state, i.e. a breather-like structure. Then we will
discuss what role the different parameters of the model play in the occurrence
of these two outcomes.

###  3.1 Scattering or formation of a bound state

(a)

(b)

(c)

(d)

Figure 3: Time evolution of the renormalized energy density in $\psi$
($\rho^{(R)}_{\psi}$). (a) For, $\kappa\rightarrow 0$, i.e., no backreaction;
$v=0.1$, $\lambda=0.3$; (b) For, $\kappa=1$, i.e., with backreaction; $v=0.1$,
$\lambda=0.3$. (c) For, $\kappa\rightarrow 0$, i.e., no backreaction; $v=0.2$,
$\lambda=0.3$; (d) For, $\kappa=1$, i.e., with backreaction; $v=0.2$,
$\lambda=0.3$. The universal parameters are $L=100$, $N=500$, $\mu=0.1$,
$m_{\rm phys}=1$, and $t_{0}=-100$. The collision happens at $t=0$. The
animations corresponding to the different cases can be found at
https://sites.google.com/asu.edu/mainakm.

In Fig. 3 we show the renormalized energy density in the $\psi$ field as a
function of $x$ (horizontal axis) and $t$ (vertical axis) with and without
backreaction for two different sets of parameters. The color coding represents
the value of the energy density from its minimum value in dark blue to its
maximum value in bright red. Since the clouds of $\psi$ particles (thick dark
blue lines) track the kink and antikink perfectly, this representation is
particularly well suited for visualizing both the dynamics of the kink-
antikink background and any radiation bursts (thin orange lines) occurring
during their collision. It serves as an accurate spacetime diagram of the
collision. The first thing we notice, in Figs. 3a and 3c, is that when
$\kappa\rightarrow 0$ i.e. when backreaction is neglected, the kink and
antikink dynamics are unperturbed by the collision (as they should be); there
are however two bursts of radiation originating and $x=t=0$ i.e. at the
collision, and propagating away from the kink-antikink pair (one to the left
the other to the right) at or near the speed of light. When reaching the end
of the lattice they wrap around and come back towards the center by virtue of
the periodic boundary conditions. The origin of the bursts of radiation can be
explained intuitively by the high degree of non-adiabaticity of the background
at the moment of collision.

More interestingly, Figs. 3b and 3d, including backreaction effects, show two
radically distinct behaviors. In Fig. 3d, the parameters are such that the
collision does not lead to the formation of a bound state. Just like in the
non-backreacting case of Fig. 3c, the clouds of $\psi$ particles tracking the
kink-antikink pair describe an X-like pattern: they are seen to converge,
collide and subsequently diverge from one another. The collision is again
accompanied by the emission of two bursts of radiation. However, unlike in
Fig. 3c, the outgoing relative velocity of the kink-antikink pair in Fig. 3d
is smaller than the initial velocity as background kinetic energy has been
converted in quantum radiative energy. In Fig. 3b, the parameters are such
that the collision leads to the formation of a breather-like kink-antikink
bound state. The cactus (or caterpillar) pattern described by the clouds of
$\psi$ particles after the initial collision at $t=0$ represents the multiple
subsequent collisions that the kink and antikink undergo, each accompanied by
a burst of radiative energy. The first burst of energy depletes enough kinetic
energy that the kink and antikink are not able to break away from their mutual
attraction and must form a bound state. We will study the evolution of the
energy in $\phi$, $E_{\phi}$, in more detail in the next subsection.

###  3.2 Parameter dependence of the outcome of the collision

To understand the influence of the parameters of the model on the dynamics of
the collision (with backreaction taken into account), we turn our attention to
the energy in the kink-antikink pair. In Figs. 4, 5 and 6 we plot $E_{\phi}$
as a function of time for different values of the parameters $v$, $\lambda$
and $\mu$. We can make a few general comments before going into the details of
the plots. First we notice that before the collision $E_{\phi}$ is constant,
which is to be expected because the kink and antikink are effectively
decoupled until their relative distance becomes of the order of the kink width
$1/m_{\rm phys}$. At the collision, the energy decreases abruptly which
corresponds to the initial burst of radiation seen in the previous subsection.
The energy then quickly stabilizes to a new plateau. If the value of this new
energy plateau is larger than the energy of a static kink-antikink pair i.e.
$2E_{\pm}(0)=16m_{\rm phys}$, then a breather-like object does not form and
the kink-antikink pair remains unbound after the collision ($E_{\phi}$ remains
constant after the collision). On the contrary, if the first burst of energy
is large enough to make the value of the energy plateau fall below this
threshold energy, then a bound state is formed. This is followed by a cascade
of subsequent bursts of energy that lead to the decay of $E_{\phi}$ into lower
and lower plateaus of shorter and shorter duration. This cascade is readily
understood. After the first collision, the kinks separate out more than their
widths which leads to the formation of an energy plateau (because kinks
radiate only when they overlap). Eventually though, after a certain number of
subsequent collisions, they don’t manage to separate out and the plateau shape
is lost. The radiation thus transitions through different behaviors which we
will discuss in more detail in the next section.

Figure 4:  The energy in the background $\phi$ ($E_{\phi}$) as a function of
time for various values of $v$: 0.07 (Orange), 0.1 (Blue), 0.2 (Red) and 0.3
(Green). The other parameters are $L=100$, $N=500$, $\mu=0.1$, $m_{\rm
phys}=1$, $\lambda=0.3$ , $\kappa=1$ and $t_{0}=-100$. The gray dashed line
corresponds to $E_{\phi}=16$.

Figure 4 shows $E_{\phi}$ for $\lambda=0.3$, $\mu=0.1$, and varying $v$. The
higher the initial relative velocity $v$ the bigger the gap to the threshold
energy of bound state formation, $\Delta_{\rm gap}\equiv 16(\gamma-1)m_{\rm
phys}$, as seen from Eq. (10). We notice that the energy of the first
radiation burst doesn’t depend strongly on $v$ (see Appendix A) and therefore,
as long as it is larger than $\Delta_{\rm gap}$, a breather-like object is
formed. This happens in particular for $v=0.2$ and $v=0.3$.

Figure 5:  The energy in the background $\phi$ ($E_{\phi}$) as a function of
time for various values of $\lambda$: 0.3 (Orange), 0.5 (Green), 0.7 (Red) and
0.9 (Blue). The other parameters are $L=100$, $N=500$, $\mu=0.1$, $m_{\rm
phys}=1$, $v=0.2$ , $\kappa=1$ and $t_{0}=-100$. The gray dashed line
corresponds to $E_{\phi}=16$.

Figure 5 shows $E_{\phi}$ for $v=0.2$, $\mu=0.1$, and varying $\lambda$. Here,
the energy of the first radiation burst increases with $\lambda$ (it
approximately scales as $\lambda^{1.6}$ for the range of parameters considered
here) and exceeds $\Delta_{\rm gap}$ for $\lambda=0.5$ and $\lambda=0.7$, when
a breather-like object forms. In the $\lambda=0.7$ and $\lambda=0.9$ cases we
notice a peculiar change of behavior in $E_{\phi}$ around $t=70$ when the
cascading decay becomes an oscillatory decay with a smaller average slope. The
bound state then appears to settle to a long-lived, weakly radiating,
oscillon-like object. We will come back to this intriguing configuration in
the next section.

Figure 6:  The energy in the background $\phi$ ($E_{\phi}$) as a function time
for various values of $\mu$: 0.01(Purple), 0.1 (Orange), 0.3 (Blue), 0.5 (Red)
and 0.7 (Green). The other parameters are $L=100$, $N=500$, $\lambda=0.3$,
$m_{\rm phys}=1$, $v=0.1$ , $\kappa=1$ and $t_{0}=-100$. The gray dashed line
corresponds $E_{\phi}=16$.

Figure 6 represents $E_{\phi}$ for $v=0.1$, $\lambda=0.3$, and varying $\mu$.
Here we notice that the lighter the $\psi$ field, the stronger the initial
radiation burst after collision (it scales as $\exp(-1.8\mu)$). This is
readily understood, as a light field is more easily excited by the time-
dependent background. For $\mu\leq 0.5$, a breather-like object forms since
the kink-antikink pair releases an energy greater than $\Delta_{\rm gap}$ at
the collision.

(a)

(b)

(c)

(d)

Figure 7: Plots showing a parameter scan of the $\lambda-v$ plane for
different values of $\mu$: (a) $\mu=0.1$, (b) $\mu=0.3$, (c) $\mu=0.5$, and
(d) $\mu=0.7$. The dark blue points represent a breather-like object being
formed. The blank space denotes the region of uncertainty where a breather-
like object may or may not form. The red points denote where a breather-like
object does not form. The other parameters are $L=100$, $N=500$, $\kappa=1$,
and $m_{\rm phys}=1$. Because of the long computation time required to
generate these plots, we exceptionally choose $t_{0}=-50$.

We can even go further and determine the region of $(\lambda,v,\mu)$ parameter
space where a breather-like object forms, i.e. the region where the first
burst of energy at the collision is larger than $\Delta_{\rm gap}$. We expect
that the boundary between the region where a bound state forms and the one
where the kink-antikink pair remains unbound after collision, to approximately
look like a two-dimensional surface. However, since our determination of
$\Delta_{\rm gap}$ is imperfect, particularly because the first energy plateau
after collision exhibits small oscillations (as can be seen for instance in
Figs. 4, 5 and 6), the boundary will necessarily have a thickness. To estimate
this thickness we determine the characteristic amplitude of the small
oscillations around the first energy plateau in the worst case scenario (by
computing the standard deviation from the mean value $\sigma_{\rm max}$ in
this case) and we conservatively declare that a first energy burst equal to
$\Delta_{\rm gap}$ with a margin of error of $\pm 2\sigma_{\rm max}$ doesn’t
allow us to definitively determine whether a bound state forms or not. Here
$\sigma_{\rm max}=0.027$. In Fig. 7 we show the results of a parameter scan
with resolution of $0.25$ in $\lambda$, $0.01$ in $v$ and $0.2$ in $\mu$.

##  4 Decay of the bound state

We have already seen that, when the parameters of the problem are such that a
bound-state forms as a consequence of the kink-antikink collision, the
radiation goes through different phases. In this section we highlight two of
those phases: the fast decay phase where radiation is emitted via successive
energy bursts, and the oscillon phase which is weakly radiating and quasi-
stable.

###  4.1 The energy plateau phase: decay of the breather-like object

Figure 8:  Plot of the energy in $\phi$ ($E_{\phi}$) as a function of time
(solid dark blue) superimposed over a plot of $\phi(t,0)$ (dashed-red color).
The parameters correspond to the solid blue curve in Fig. 4. They are $L=100$,
$N=500$, $v=0.1$, $\mu=0.1$, $\lambda=0.3$, $m_{\rm phys}=1$, $\kappa=1$ and
$t_{0}=-100$.

(a)

(b)

Figure 9: (a) Plot of the energy of the $n$-th plateau of $E_{\phi}$ in Fig. 8
($E_{\phi}^{(n)}$) as a function of $n$ (purple); (b) Plot of the duration of
said plateau ($\tau_{n}$) as a function $n$ (orange). The parameters are
$L=100$, $N=500$, $v=0.1$, $\mu=0.1$, $\lambda=0.3$, $m_{\rm phys}=1$,
$\kappa=1$ and $t_{0}=-100$.  Figure 10:  Plot of the plateau energy for
$\phi$ ($E_{\phi}^{(n)}$) as a function of half period $\tau_{n}$ from
simulation (red), from theory (42) (solid blue) and the analytical fit (41)
(dashed orange). The parameters for the simulation points correspond to the
solid blue curve in Fig. 4. They are $L=100$, $N=500$, $v=0.1$, $\mu=0.1$,
$\lambda=0.3$, $m_{\rm phys}=1$, $\kappa=1$ and $t_{0}=-100$.

In cases where the kink and antikink scatter to form a bound state, we
observed that there is a cascade of bursts of radiation at decreasing time
intervals. In this subsection we focus on the blue curve in Fig. 4
corresponding to the choice of parameters $\lambda=0.3$, $\mu=0.1$ and
$v=0.1$. This case features particularly well-defined energy plateaus and
lends itself to further analysis. We plot $E_{\phi}^{(n)}$, the energy after
the $n^{\rm th}$ burst as a function of $n$ in Fig. 9a and find the fit

$E_{\phi}^{(n)}\simeq E_{\phi}^{(0)}-0.20\ n\,,$ (39)

where $E_{\phi}^{(0)}\simeq 16.04$ is the initial energy. Using Fig. 8 we can
also characterize the time intervals $\tau_{n}$ between the $n^{\rm th}$ and
$(n+1)^{\rm th}$ bursts (or between two successive zeros of $\phi(t,x=0)$), as
shown in Fig. 9b, to get the fit

$\tau_{n}\simeq\tau_{1}n^{-0.66}\,,$ (40)

where $\tau_{1}=26.75$ is the first burst interval. We can then eliminate $n$
to obtain the relation between $E_{\phi}^{(n)}$ and $\tau_{n}$,

$E_{\phi}^{(n)}\simeq
E_{\phi}^{(0)}-0.20\left(\frac{\tau_{1}}{\tau_{n}}\right)^{1.52}\,.$ (41)

Using the exact solution for the sine-Gordon breathers, the relation between
the energy $E_{\phi}$ and the half-period $\tau$ is

$E_{\phi}=16\sqrt{1-{\frac{\pi^{2}}{\tau^{2}}}}\,.$ (42)

Fig. 10 shows an excellent agreement of the numerical data with the analytic
sine-Gordon prediction for the relation between the plateau energy and the
half-period, showing that the $\phi$ field decay proceeds discretely through a
series of breather states. This decay of the energy continues for a finite
number of bursts and then, at a critical time, the system rapidly decays into
a new, more stable, oscillon-like phase (which we study in more detail in the
next subsection).

###  4.2 Formation of a long-lived oscillon

(a)

(b)

(c)

(d)

Figure 11: Different observables for $\lambda=0.9$ to highlight the formation
of a long-lived oscillon: (a) Time evolution of the energy density in $\phi$;
(b) Time evolution of the energy density in $\psi$ ($\rho^{(R)}_{\psi}$); (c)
Time evolution of $\phi(t,x=0)$; (d) Log-log plot of the late-time evolution
of the energy in the $\phi$ ($E_{\phi}$). The parameters are $L=100$, $N=500$,
$v=0.2$, $\mu=0.1$ and $m_{\rm phys}=1$, $\kappa=1$ and $t_{0}=-100$. The
collision happens at $t=0$. The animation corresponding to this case can be
found at https://sites.google.com/asu.edu/mainakm.

In this subsection we focus on the particular case $v=0.2$, $\mu=0.1$ and
$\lambda=0.9$ (depicted in the blue curve of Fig. 5) which prominently
features a long-lived oscillon phase and allows us to study its formation and
decay. In Figs. 11a and 11b we show the time evolution of the energy densities
in $\phi$ and $\psi$ respectively. In both cases, we see a qualitative change
in the appearance of the breather-like structure occurring around $t=70$: the
spatial extension of the object ceases to decrease and its energy density
undergoes small oscillations in amplitude (visible on both plots). Fig. 13
shows that the oscillations in the energy of $\phi$ are accompanied with out-
of-phase oscillations in the localized energy of $\psi$ around the origin. As
can be seen in Fig. 11b the onset of this new regime is preceded by a large
burst of radiation that almost completely turns off at $t\sim 70$. Fig. 11c
shows the amplitude of the value of the $\phi$ field at the center of the
lattice as a function of time. This is in fact a good measure of the $\phi$
field profile amplitude. The same qualitative change of behavior is observed
on this plot: starting with the moment of collision, the $\phi$ field profile
undergoes oscillations of decaying amplitude until a new quasi-stable
oscillatory regime is reached around $t=70$. Although, the amplitude of
oscillation continues to decrease, it does so at a very slow rate which
justifies the long-lived oscillon nomenclature. Finally, Fig. 11d is simply a
zoomed in version of the blue curve in Fig. 5 allowing us to fit a power law
to the envelope of $E_{\phi}$ within the long-lived oscillon regime. We find a
$\sim t^{-0.6}$ power law decay. This is to be contrasted to the approximately
linear decay occurring during the immediately preceding phase, where, on
average, the energy decreases approximately linearly in time with a slope of
$-0.16$. Given this power-law decay, this object may not be a true “oscillon”,
since oscillons have been seen to remain highly stable for hundreds or
thousands of oscillation times. In order to distinguish it from the more
strongly radiating breather state that precedes it, we will adhere to the
“oscillon” nomenclature for the remainder of this work.

(a)

(b)

(c)

(d)

Figure 12:  (a) The numerically derived profiles of $\phi$ at the first 11
extrema of Fig. 11c (blue). The red-dashed curves correspond to the breather
profile of Eq. (7) with the frequency parameter $\omega$ chosen such that we
recover the maximum amplitude $\phi(x=0)$. We see excellent agreement for
almost all curves; (b) The shape of $\phi$ at the 11th, 12th and 15th extremum
(blue,orange and green respectively). The red, brown and black curves
correspond to the breather shape of Eq. (7); (c) The half-period $\tau$ (blue)
and maximum central field amplitude $\phi_{\rm max}$ (red) as a function of
the extremum number $N_{\rm ext}$, as in Fig. 11c; (d) The half-period as a
function of amplitude extracted from the simulation (blue dots) and computed
using the analytical breather solution (red curve).

Fig. 12 further showcases the behavior of the field $\phi$ in the two regimes.
We see that for early times, when there is significant energy loss every time
$\phi$ goes through zero, the maxima of the field $\phi$ match almost exactly
to the breather profile of the fully classical Sine-Gordon equation. However,
during the second, slowly radiating, part of the evolution, we can see a non-
negligible difference between the numerical profile for $\phi$ and that of a
sine-Gordon breather. We can thus conclude that, while the early time
evolution can be thought of a series of breathers, with the system “jumping”
from one to another every time $\phi$ crosses zero, the late time evolution
exhibits a deformed breather-like structure (which we denote as an oscillon),
which is much less radiating. The time between successive $\phi$ zero-
crossings matches rather well with the half-period of Sine-Gordon breathers,
as shown in Fig. 12d. We also see that both the frequency and the amplitude of
the oscillon-like structure remain almost constant.

Figure 13:  The energy in $\phi$ ($E_{\phi}$) in solid dark blue, and the
renormalized energy in $\psi$ within a window ranging from $x=-5$ to $x=5$
($E_{\psi,W}^{(R)}$) in solid brown, as a function of time for $\lambda=0.9$.
The sum of the two is denoted by the dashed black line. The parameters are
$L=100$, $N=500$, $v=0.2$, $\mu=0.1$, $m_{\rm phys}=1$ and $t_{0}=-100$.
Notice that the slight increase in $E_{\psi,W}^{(R)}$ for $t\gtrsim 90$ is due
to the emitted radiation reentering the window because of the periodic
boundary conditions.

One potential source of worry is whether the semi-classical approximation
remains valid long enough for the oscillon to form. In other words, can the
$\phi$ field still be treated as a classical field, even after a significant
amount of its energy has dissipated via quantum radiation? As mentioned in
Sec. 2, the approximation is valid as long as $E_{\phi}$ is larger than
$E_{\psi}^{(R)}$ minus the energy radiated away, or in other words, as long as
the classical energy in the $\phi$ field is larger than the quantum energy in
the clouds of $\psi$ particles “dressing up” the kink-antikink pair. As shown
in Fig. 13 we have checked that this is indeed the case for the whole duration
of the simulation and in particular during the onset of the long-lived
oscillon regime.

It would be interesting to understand the profound reasons behind the
formation of the long-lived oscillon. However, within the time constraints of
our simulation, which we can only evolve until $t\simeq 100$ (see Section 5),
only a small fraction of the parameter choices allow us to reach this
oscillon-like final state. Improved numerics, able to simulate this late-time
long-lived state, will allow us to compare it decay rate to analytical
estimates Hertzberg:2010yz , thereby further elucidating the truly quantum
mechanical nature of the slow decay. Overall, it is important to understand
the formation and eventual decay of this slowly radiating object and we will
return to this point in the future.

##  5 Discussion and conclusion

In this work we were able to simulate kink-antikink scattering in a model
where a classical sine-Gordon field $\phi$ (the background) is coupled to a
quantum field $\psi$ (the quantum radiation bath) via an interaction term
preserving the discrete shift symmetry. As expected the spacetime dependence
of the background kink-antikink configuration triggers particle production in
the $\psi$ field which can in turn backreact on the field $\phi$. As detailed
in Sec. 2 we used the Classical-Quantum Correspondence (CQC)
Vachaspati:2018llo ; Vachaspati:2018hcu  to study this system numerically.
When backreaction is taken into account within the semi-classical
approximation, the outcome of the scattering is either an outgoing kink-
antikink pair with reduced kinetic energy or a breather-like bound state,
which continues to radiate. In Sec. 3 we examine the dynamics of this
inelastic process and its dependence of the different parameters of the model.
Interestingly enough, in the case where a bound state forms, particle
production initially occurs in the form of a cascade of strong radiation
bursts, but after some time the system appears to settle in a long-lived,
weakly-radiating oscillon configuration. Understanding this final state would
provide valuable insight into the late-time evolution of realistic systems,
like bubble collisions in the early universe. Furthermore, it has been shown
that in some cases the presence of spectator fields can enhance the lifetimes
of oscillons Antusch:2015ziz . Recent work has also described the properties
of multi-component oscillons VanDissel:2020umg , which could be relevant for
understanding this configuration. A detailed analytical and numerical
investigation of these late-time oscillons falls outside the scope of our
current work and will be undertaken in the future.

A limitation of our numerical implementation arises from the fact that working
on a periodic lattice prevents us from evolving the dynamical system for a
time longer than one light-crossing time after the collision. Beyond that
point the emitted radiation comes back to interfere with the kink-antikink
pair and our results can no longer be fully trusted. To remedy this would
either require parallelizing the code so as to increase the size of the
lattice while maintaining spatial resolution, or implementing absorbing
boundary conditions. This is necessary for capturing phenomena that develop
over long time-scales (like oscillon evaporation) and is currently under
development.

One of the other limitations of our numerical setup lies in the choice of the
initial conditions given in Eq. (35) for the $Z_{ij}$ variables. As mentioned
in Section 2, these are technically only valid when the background is in a
quiescent state at time $t_{0}$, i.e. when its time variation can be
neglected. This is approximately true for non-relativistic collisions such as
the ones studied here. However, when $\gamma$ becomes large, we expect the
mismatch of initial conditions to violate adiabaticity strongly around
$t=t_{0}$ thus leading to spurious particle production. Fixing this issue
would require going beyond the choice of 0-th order adiabatic vacuum, for
example by pasting together the known vacuum modefunctions for a boosted kink
and for a boosted antikink morse1953methods ; Vachaspati:2006zz  propagating
in the opposite direction to obtain more accurate initial conditions for
$\bm{Z}$. Alternatively one could adiabatically turn on the relative velocity
in a well-separated kink-antikink pair but this again requires better control
over numerical error. An amusing fact is that collisions of classical solitons
have been analyzed in detail in the opposite regime, that of ultra-
relativistic velocities Amin:2013eqa . There, semi-analytic formulas where
derived, albeit neglecting any effect of quantum radiation. Extending the CQC
into this regime will allow us to capture the quantum radiation effects on
colliding relativistic domain walls in the early universe.

The methods presented here in the context of the simple model of a sine-Gordon
potential in one spatial dimensions have a wide applicability to a variety of
other scenarios. For example, the analysis carries over directly to kink-
antikink collisions in the $\lambda\phi^{4}$ model Sugiyama:1979mi ;
Campbell:1983xu ; Anninos:1991un ; Dorey:2017dsn . Moreover its three-
dimensional extension would in fact describe domain wall collisions in the
early universe in the presence of quantum radiation. This presents an
intriguing opportunity to study possible signatures of the interplay between
classical and quantum degrees of freedom in a cosmological context.555Notice
also that the kink-antikink configuration discussed here provides an analogy
with the particle production phenomena thought to occur during gravitational
collapse, black hole production and evaporation. Furthermore, the formation of
oscillons after bubble collisions and their eventual decay will have to be
revisited, in order to encompass the backreaction of quantum radiation
Bond:2015zfa . We plan to address these open questions in future work.

##  Appendix A Structure of the radiation bursts

We provide a detailed look at the spatial form of the radiation burst that is
generated during the first kink-antikink collision. In order to get a “clean”
signal, we restrict the present analysis to the case where back-reaction of
the $\psi$ field on the $\phi$ background is neglected. This is an
increasingly good approximation for smaller values of $\lambda$.

Fig. 14 shows the resulting radiation bursts, where they are sufficiently far
away from the collision region. We choose to present two significantly
different values for each of the three parameters of the problem: the velocity
($v=0.1,0.3$), the $\psi$ field mass ($\mu=0.1,0.7$) and the coupling strength
($\lambda=0.1,0.9$).

An immediate realization concerns the effect of changing the initial kink-
antikink velocity $v$. We see that the radiation waveform is almost identical
for each pair of $(\mu,\lambda)$ when we change the velocity. Furthermore, the
energy contained in the radiation burst is almost equal for the two curves in
each panel of Fig. 14. Hence, our numerical results given in the main text are
corroborated by Fig. 14, showing that the quantum radiation emitted during the
kink-antikink collision essentially loses the memory of the initial velocity.

A further observation is related to the amplitude and shape of the
renormalized energy density $\rho_{\psi}^{(R)}$ in the radiation burst in each
of the four panels of Fig. 14, corresponding to the four different
combinations of $\lambda$ and $\mu$. Each radiation burst is seen to form a
wave packet with a spectral content that is highly dependent on the
parameters. Most importantly, we see that increasing $\lambda$ increases the
overall energy in $\psi$ radiation. This is to be expected, since $\lambda$ is
the coupling strength between the classical background field $\phi$ and the
quantum radiation field $\psi$. On the contrary, increasing $\mu$ leads to a
decreased energy in the radiation burst. This can also be qualitatively
understood. As shown in Fig. 2, the initial $\psi$ configuration deviates from
the trivial vacuum more for smaller values of $\mu$. Simply put, it is easier
to excite a lighter quantum field. Finally, we see that the front of the
wavepacket is at a slightly larger distance from the origin for $\mu=0.1$ than
for $\mu=0.7$. Since all panels correspond to snapshots taken at the same time
$\tau=40$, this translates into a larger velocity of the radiation burst in
the case of smaller $\psi$ mass ($\mu$), as one could have guessed from simple
kinematical arguments.

(a) $\mu=0.1$, $\lambda=0.1$, $E^{RB}_{0.1}=0.0161$, $E^{RB}_{0.3}=0.0136$

(b) $\mu=0.1,\lambda=0.9,E^{RB}_{0.1}=0.2799,E^{RB}_{0.3}=0.2698$

(c) $\mu=0.7,\lambda=0.1,E^{RB}_{0.1}=0.0033,E^{RB}_{0.3}=0.0030$

(d) $\mu=0.7,\lambda=0.9,E^{RB}_{0.1}=0.0908,E^{RB}_{0.3}=0.0848$

Figure 14: Snapshots (taken at time $\tau=40$) of the renormalized energy
density of the radiation bursts for different parameters in the case without
backreaction. The solid lines are for $v=0.1$ and the dashed lines are for
$v=0.3$. $E^{RB}_{v}$ is the energy in the radiation burst for $v=0.1$ and
$v=0.3$. The other parameters are $L=100$, $N=500$, $\kappa\rightarrow 0$,
$m_{\rm phys}=m=1$ and $t_{0}=-100$.

##  Appendix B Quality of numerics

In this appendix, we discuss the quality of the numerics in our simulations.
In Fig. 15, we illustrate the independence of the physical observables on the
choice of the temporal time step used in the simulation, $dt$. In Fig. 15a we
see that the total energy is conserved to an accuracy of $\sim 0.1\%$ over the
entire time of evolution (for a time step $dt=0.004$). As expected, energy is
conserved to an even better accuracy when dividing the time step by two.
However, given the trade-off in computational time we think $dt=0.004$ is
sufficient for the task at hand. Our choice is further justified by Fig. 15b
where we show the energy in $\phi$ ($E_{\phi}$) computed for two values of
$dt$, $0.004$ and $0.002$. Indeed the two plots superimpose each other.
Moreover, Fig. 16 clearly shows that, at least for the physical observables
that we are interested in (i.e. $E_{\phi}$), decreasing the lattice spacing
$a=L/N$ or increasing the size of the box $L$ do not sensibly change our
results. In other words, our choices of parameters are good enough and the
results shown in the main text are independent of both the UV and the IR
cutoffs.

(a)

(b)

Figure 15: (a) Time evolution of the total energy (38) for $dt=a/50=0.004$ and
$dt=a/100=0.002$. (b) Time evolution of the energy in $\phi$ ($E_{\phi}$) for
$dt=a/50=0.004$ (solid blue) and $dt=a/100=0.002$ (dashed red). The parameters
are $L=100$, $N=500$, $v=0.1$, $\mu=0.1$, $\lambda=0.3$ and $m_{\rm phys}=1$,
$\kappa=1$ and $t_{0}=-100$. The collision happens at $t=0$. Recall that
$a=L/N$. These plots illustrate the independence of our results on the choice
of time step.  Figure 16:  Time evolution of the energy in $\phi$ ($E_{\phi}$)
for different values of $L$ and $N$. The parameters are $v=0.1$, $\mu=0.1$,
$\lambda=0.3$ and $m_{\rm phys}=1$, $\kappa=1$ and $t_{0}=-50$. The collision
happens at $t=0$. Recall that $a=L/N$ and we have chosen $dt=a/50$. This
illustrates the independence of our results on the lattice spacing $a$ and on
the size of the box $L$.

###### Acknowledgements.

We would like to thank Oriol Pujolas for many useful discussions, and Fabio
van Dissel for independently verifying the robustness of our numerical
results. Computations for this work were performed on the Agave cluster at
Arizona State University. The research leading to these results has received
funding from the Spanish Ministry of Science and Innovation
(PID2020-115845GB-I00/AEI/10.13039/501100011033). IFAE is partially funded by
the CERCA program of the Generalitat de Catalunya. MM was supported by the
Fermi National Accelerator Laboratory (Fermilab) Award No. AWD00035045 and the
National Science Foundation grant numbers PHY-1613708 and PHY-2012195 during
this work. The work of EIS and GZ was supported by a fellowship from “La
Caixa” Foundation (ID 100010434) and from the European Union’s Horizon 2020
research and innovation programme under the Marie Skłodowska-Curie grant
agreement No 847648\. The fellowship code is LCF/BQ/PI20/11760021. TV is
supported by the U.S. Department of Energy, Office of High Energy Physics,
under Award DE-SC0019470 at ASU.

## References

  * (1) K. Saikawa, Gravitational waves from cosmic domain walls: a mini-review, J. Phys. Conf. Ser. 1586 (2020), no. 1 012039.
  * (2) T. Hiramatsu, M. Kawasaki, K. Saikawa, and T. Sekiguchi, Axion cosmology with long-lived domain walls, JCAP 01 (2013) 001, [arXiv:1207.3166].
  * (3) A. Vilenkin and A. E. Everett, Cosmic Strings and Domain Walls in Models with Goldstone and PseudoGoldstone Bosons, Phys. Rev. Lett. 48 (1982) 1867–1870.
  * (4) P. Sikivie, Of Axions, Domain Walls and the Early Universe, Phys. Rev. Lett. 48 (1982) 1156–1159.
  * (5) A. Vilenkin, Cosmic Strings and Domain Walls, Phys. Rept. 121 (1985) 263–315.
  * (6) J. J. Blanco-Pillado, K. D. Olum, and X. Siemens, New limits on cosmic strings from gravitational wave observation, Phys. Lett. B 778 (2018) 392–396, [arXiv:1709.02434].
  * (7) J.-F. Dufaux, D. G. Figueroa, and J. Garcia-Bellido, Gravitational Waves from Abelian Gauge Fields and Cosmic Strings at Preheating, Phys. Rev. D 82 (2010) 083518, [arXiv:1006.0217].
  * (8) T. Damour and A. Vilenkin, Gravitational wave bursts from cosmic strings, Phys. Rev. Lett. 85 (2000) 3761–3764, [gr-qc/0004075].
  * (9) T. Vachaspati and A. Vilenkin, Formation and Evolution of Cosmic Strings, Phys. Rev. D 30 (1984) 2036.
  * (10) T. Vachaspati and A. Vilenkin, Gravitational Radiation from Cosmic Strings, Phys. Rev. D 31 (1985) 3052.
  * (11) M. B. Hindmarsh and T. W. B. Kibble, Cosmic strings, Rept. Prog. Phys. 58 (1995) 477–562, [hep-ph/9411342].
  * (12) T. W. B. Kibble, Topology of Cosmic Domains and Strings, J. Phys. A 9 (1976) 1387–1398.
  * (13) G. ’t Hooft, Magnetic Monopoles in Unified Gauge Theories, Nucl. Phys. B 79 (1974) 276–284.
  * (14) Y. Nambu, Strings, Monopoles and Gauge Fields, Phys. Rev. D 10 (1974) 4262.
  * (15) G. W. Gibbons and N. S. Manton, Classical and Quantum Dynamics of BPS Monopoles, Nucl. Phys. B 274 (1986) 183–224.
  * (16) N. E. Mavromatos and V. A. Mitsou, Magnetic monopoles revisited: Models and searches at colliders and in the Cosmos, Int. J. Mod. Phys. A 35 (2020), no. 23 2030012, [arXiv:2005.05100].
  * (17) E. Gueron and P. S. Letelier, Textures and Newtonian gravity, Phys. Rev. D 56 (1997) 5272–5275, [gr-qc/9708052].
  * (18) D. Notzold, Gravitational Effects of Global Textures, Phys. Rev. D 43 (1991) R961–R964.
  * (19) D. N. Spergel and N. G. Turok, Textures and cosmic structure, Sci. Am. 266 (1992) 36–43.
  * (20) R. Durrer, M. Heusler, P. Jetzer, and N. Straumann, General relativistic collapse of textures, Phys. Lett. B 259 (1991) 48–52.
  * (21) N. Turok, Global Texture as the Origin of Cosmic Structure, Phys. Rev. Lett. 63 (1989) 2625.
  * (22) R. Brandenberger and H. Jiao, Cosmic Textures and Global Monopoles as Seeds for Super-Massive Black Holes, JCAP 02 (2020) 002, [arXiv:1908.04585].
  * (23) Planck Collaboration, P. A. R. Ade et al., Planck 2013 results. XXV. Searches for cosmic strings and other topological defects, Astron. Astrophys. 571 (2014) A25, [arXiv:1303.5085].
  * (24) W. Zurek, Cosmological Experiments in Superfluid Helium?, Nature 317 (1985) 505–508.
  * (25) V. Pal, C. Tradonsky, R. Chriki, A. A. Friesem, and N. Davidson, Observing dissipative topological defects with coupled lasers, Phys. Rev. Lett. 119 (Jul, 2017) 013902.
  * (26) 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 (Aug, 2010) 075701.
  * (27) I. Chuang, B. Yurke, R. Durrer, and N. Turok, Cosmology in the Laboratory: Defect Dynamics in Liquid Crystals, Science 251 (1991) 1336–1342.
  * (28) 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 (1994) 943–945, [hep-ph/9208233].
  * (29) 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 (1994), no. 6469 315–317, [https://www.nature.com/articles/368315a0.pdf].
  * (30) 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 (1996), no. 6589 334–336, [https://www.nature.com/articles/382334a0.pdf].
  * (31) 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 (1996), no. 6589 332–334, [https://www.nature.com/articles/382332a0.pdf].
  * (32) 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 (Aug, 2002) 080603.
  * (33) R. Carmi, E. Polturak, and G. Koren, Observation of spontaneous flux generation in a multi-josephson-junction loop, Phys. Rev. Lett. 84 (May, 2000) 4966–4969.
  * (34) A. Maniv, E. Polturak, and G. Koren, Observation of magnetic flux generated spontaneously during a rapid quench of superconducting films, Phys. Rev. Lett. 91 (Nov, 2003) 197001.
  * (35) J. Beugnon and N. Navon, Exploring the kibble–zurek mechanism with homogeneous bose gases, Journal of Physics B: Atomic, Molecular and Optical Physics 50 (jan, 2017) 022002.
  * (36) T. W. B. Kibble, Testing cosmological defect formation in the laboratory, Physica C 369 (2002) 87, [cond-mat/0111082].
  * (37) T. Vachaspati, Topological defects in the cosmos and lab, Contemp. Phys. 39 (1998) 225–237, [hep-ph/9802311].
  * (38) W. H. Zurek, Cosmological experiments in condensed matter systems, Phys. Rept. 276 (1996) 177–221, [cond-mat/9607135].
  * (39) R. Rajaraman, Solitons and Instantons: An Introduction to Solitons and Instantons in Quantum Field Theory.  North-Holland personal library. North-Holland Publishing Company, 1982\.
  * (40) S. Coleman, Aspects of Symmetry: Selected Erice Lectures.  Cambridge University Press, Cambridge, U.K., 1985.
  * (41) N. H. Christ and T. D. Lee, Quantum Expansion of Soliton Solutions, Phys. Rev. D 12 (1975) 1606.
  * (42) J. Goldstone and R. Jackiw, Quantization of Nonlinear Waves, Phys. Rev. D 11 (1975) 1486–1498.
  * (43) K. E. Cahill, Extended Particles and Solitons, Phys. Lett. B 53 (1974) 174–176.
  * (44) R. F. Dashen, B. Hasslacher, and A. Neveu, Nonperturbative Methods and Extended Hadron Models in Field Theory 1. Semiclassical Functional Methods, Phys. Rev. D 10 (1974) 4114.
  * (45) R. F. Dashen, B. Hasslacher, and A. Neveu, Nonperturbative Methods and Extended Hadron Models in Field Theory 2. Two-Dimensional Models and Extended Hadrons, Phys. Rev. D 10 (1974) 4130–4138.
  * (46) R. F. Dashen, B. Hasslacher, and A. Neveu, Nonperturbative Methods and Extended Hadron Models in Field Theory. 3. Four-Dimensional Nonabelian Models, Phys. Rev. D 10 (1974) 4138.
  * (47) R. F. Dashen, B. Hasslacher, and A. Neveu, The Particle Spectrum in Model Field Theories from Semiclassical Functional Integral Techniques, Phys. Rev. D 11 (1975) 3424.
  * (48) T. Vachaspati, Kinks and domain walls: An introduction to classical and quantum solitons.  Cambridge University Press, 4, 2010.
  * (49) J. Ollé, O. Pujolas, T. Vachaspati, and G. Zahariade, Quantum Evaporation of Classical Breathers, Phys. Rev. D 100 (2019), no. 4 045011, [arXiv:1904.12962].
  * (50) M. A. Amin, R. Easther, H. Finkel, R. Flauger, and M. P. Hertzberg, Oscillons After Inflation, Phys. Rev. Lett. 108 (2012) 241302, [arXiv:1106.3335].
  * (51) E. J. Copeland, M. Gleiser, and H. R. Muller, Oscillons: Resonant configurations during bubble collapse, Phys. Rev. D 52 (1995) 1920–1933, [hep-ph/9503217].
  * (52) M. Gleiser and D. Sicilia, A General Theory of Oscillon Dynamics, Phys. Rev. D 80 (2009) 125037, [arXiv:0910.5922].
  * (53) E. Farhi, N. Graham, V. Khemani, R. Markov, and R. Rosales, An Oscillon in the SU(2) gauged Higgs model, Phys. Rev. D 72 (2005) 101701, [hep-th/0505273].
  * (54) N. Graham, An Electroweak oscillon, Phys. Rev. Lett. 98 (2007) 101801, [hep-th/0610267]. [Erratum: Phys.Rev.Lett. 98, 189904 (2007)].
  * (55) S.-Y. Zhou, E. J. Copeland, R. Easther, H. Finkel, Z.-G. Mou, and P. M. Saffin, Gravitational Waves from Oscillon Preheating, JHEP 10 (2013) 026, [arXiv:1304.6094].
  * (56) G. Fodor, P. Forgacs, Z. Horvath, and A. Lukacs, Small amplitude quasi-breathers and oscillons, Phys. Rev. D 78 (2008) 025003, [arXiv:0802.3525].
  * (57) T. Hiramatsu, E. I. Sfakianakis, and M. Yamaguchi, Gravitational wave spectra from oscillon formation after inflation, JHEP 03 (2021) 021, [arXiv:2011.12201].
  * (58) F. Van Dissel and E. I. Sfakianakis, Symmetric multi-field oscillons, arXiv:2010.07789.
  * (59) H.-Y. Zhang, M. A. Amin, E. J. Copeland, P. M. Saffin, and K. D. Lozanov, Classical Decay Rates of Oscillons, JCAP 07 (2020) 055, [arXiv:2004.01202].
  * (60) H.-Y. Zhang, Gravitational effects on oscillon lifetimes, JCAP 03 (2021) 102, [arXiv:2011.11720].
  * (61) T. Vachaspati and G. Zahariade, Classical-Quantum Correspondence for Fields, JCAP 09 (2019) 015, [arXiv:1807.10282].
  * (62) M. Mukhopadhyay and T. Vachaspati, Rolling classical scalar field in a linear potential coupled to a quantum field, Phys. Rev. D 100 (2019), no. 9 096018, [arXiv:1907.03762].
  * (63) T. Vachaspati and G. Zahariade, Classical-quantum correspondence and backreaction, Phys. Rev. D 98 (2018), no. 6 065002, [arXiv:1806.05196].
  * (64) N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.  Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge, UK, 2, 1984.
  * (65) M. P. Hertzberg, Quantum Radiation of Oscillons, Phys. Rev. D 82 (2010) 045022, [arXiv:1003.3459].
  * (66) S. Antusch and S. Orani, Impact of other scalar fields on oscillons after hilltop inflation, JCAP 03 (2016) 026, [arXiv:1511.02336].
  * (67) P. Morse and H. Feshbach, Methods of Theoretical Physics.  International series in pure and applied physics. McGraw-Hill, 1953.
  * (68) M. A. Amin, E. A. Lim, and I.-S. Yang, A scattering theory of ultrarelativistic solitons, Phys. Rev. D 88 (2013), no. 10 105024, [arXiv:1308.0606].
  * (69) T. Sugiyama, KINK - ANTIKINK COLLISIONS IN THE TWO-DIMENSIONAL PHI**4 MODEL, Prog. Theor. Phys. 61 (1979) 1550–1563.
  * (70) D. K. Campbell, J. F. Schonfeld, and C. A. Wingate, Resonance Structure in Kink - Antikink Interactions in $\phi^{4}$ Theory, Physica D 9 (1983) 1.
  * (71) P. Anninos, S. Oliveira, and R. A. Matzner, Fractal structure in the scalar lambda (phi**2-1)**2 theory, Phys. Rev. D 44 (1991) 1147–1160.
  * (72) P. Dorey and T. Romańczukiewicz, Resonant kink-antikink scattering through quasinormal modes, Phys. Lett. B 779 (2018) 117–123, [arXiv:1712.10235].
  * (73) J. R. Bond, J. Braden, and L. Mersini-Houghton, Cosmic bubble and domain wall instabilities III: The role of oscillons in three-dimensional bubble collisions, JCAP 09 (2015) 004, [arXiv:1505.02162].