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# Confinement Slingshot and Gravitational Waves

Maximilian Bachmaier ID<EMAIL_ADDRESS>Arnold
Sommerfeld Center, Ludwig-Maximilians-Universität, Theresienstraße 37, 80333
München, Germany  Max-Planck-Institut für Physik, Föhringer Ring 6, 80805
München, Germany  Gia Dvali  Arnold Sommerfeld Center, Ludwig-Maximilians-
Universität, Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut
für Physik, Föhringer Ring 6, 80805 München, Germany  Juan Sebastián Valbuena-
Bermúdez ID<EMAIL_ADDRESS>Arnold Sommerfeld Center, Ludwig-Maximilians-
Universität, Theresienstraße 37, 80333 München, Germany  Max-Planck-Institut
für Physik, Föhringer Ring 6, 80805 München, Germany  Michael Zantedeschi ID
<EMAIL_ADDRESS>Tsung-Dao Lee Institute and School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai, China  School of Physics
and Astronomy, Shanghai Jiao Tong University, Shanghai, China

(, )

###### Abstract

In this paper, we introduce and numerically simulate a quantum field theoretic
phenomenon called the gauge “slingshot” effect and study its production of
gravitational waves. The effect occurs when a source, such as a magnetic
monopole or a quark, crosses the boundary between the Coulomb and confining
phases. The corresponding gauge field of the source, either electric or
magnetic, gets confined into a flux tube stretching in the form of a string
(cosmic or a QCD type) that attaches the source to the domain wall separating
the two phases. The string tension accelerates the source towards the wall as
sort of a slingshot. The slingshot phenomenon is also exhibited by various
sources of other co-dimensionality, such as cosmic strings confined by domain
walls or vortices confined by $Z_{2}$ strings. Apart from the field-theoretic
value, the slingshot effect has important cosmological implications, as it
provides a distinct source for gravitational waves. The effect is expected to
be generic in various extensions of the standard model such as grand
unification.

Magnetic Monopoles, Domain Walls, Cosmic Strings, Confinement, Gravitational
Waves, Phase transition

##  I Introduction

Understanding the transition between the confining and deconfining regimes of
gauge theories remains one of the most fundamental challenges in physics. An
enlightening input in this direction can be provided by the study of systems
in which the different phases of a gauge theory can coexist in a controllable
manner. The early example is provided by the construction given in Dvali and
Shifman (1997) in which a domain wall (or a vacuum layer) supports a
deconfined (Coulomb) phase of a gauge theory that at the same time exhibits
the confining behavior in the bulk of space. Such coexistence of phases has
some important implications.

In particular, the layer of the deconfined phase localizes a massless $U(1)$
gauge field in the Coulomb regime. One effect of this localization is that the
charges (e.g., quarks), placed in a bulk of the confining vacuum, become
attached to the deconfining boundary (wall) by the QCD flux tubes. The string
stretches from the quark towards the boundary and opens up there. The flux
carried by the string spreads within the deconfined layer in the form of the
$U(1)$ Coulomb field. The system thereby realizes a field-theoretic analog to
a $D$-brane.

The dual setups, in which the analogous effect is exhibited by the magnetic
charges have been constructed in Dvali  _et al._ (1998); Dvali and Vilenkin
(2003); Dvali  _et al._ (2016). In these cases, it is the magnetic flux that
is confined in flux tubes (cosmic strings) in one of the vacuum domains. The
same flux, in the neighboring domain, gets deconfined and spreads in the form
of a Coulomb-magnetic field of a magnetic monopole.

In a system with coexisting phases, the interesting question is, what happens
when charges (either electric or magnetic) cross the boundary separating the
two phases?

In the present paper, we shall study this behavior. We first consider the case
of magnetic charges. For this purpose, we construct a prototype $SU(2)$ gauge
theory which admits two types of vacua. The vacuum in which $SU(2)$ is Higgsed
down to a $U(1)$ subgroup and the one in which the $U(1)$ is further Higgsed.
The first vacuum supports free ’t Hooft-Polyakov monopoles that are
magnetically charged under $U(1)$. These monopoles are in the Coulomb magnetic
phase.

In the second vacuum, the monopoles are confined, i.e., the monopoles and
antimonopoles are connected by magnetic flux tubes. These magnetic flux tubes
represent Nielsen-Olesen strings Nielsen and Olesen (1973) of the $U(1)$ gauge
theory (analogous to Abrikosov flux tubes Abrikosov (1957) in
superconductors).

A domain wall separates the two vacua. In this system, we study a scattering
process in which a monopole crosses from the magnetic-Coulomb to the magnetic-
confining phase. Due to the conservation of the magnetic charge, the magnetic
flux follows the monopole into the confining phase. However, the confinement
makes the flux trapped in a string.

The monopole thus becomes attached to the boundary wall by the string. The
string opens on the wall, releasing the entire flux into the Coulomb vacuum.
For an observer placed in the $U(1)$ Coulomb vacuum, the end-point of the flux
tube carries the entire magnetic charge of the ’t Hooft-Polyakov monopole. In
this way, the monopole that crosses from the Coulomb into a confining domain
leaves its “image” on the boundary. The image is represented by the throat of
the same flux tube via which the monopole is attached to the boundary from the
opposite side.

One important dynamical question is what happens when the energy in the
collision process is much larger as compared to the rest mass of the monopole?
A possible outcome one may consider is that the string breaks up by creating
monopole-antimonopole pairs. Then, instead of penetrating deeply into the
confining domain and stretching a long string, the energy of the collision is
released in the form of many monopole-antimonopole pairs connected by short
strings, which will soon annihilate into waves. In this case, one could say
that effectively the magnetic charge never enters the confining domain.

This (naive) intuition is supported by the study of the annihilation of
monopoles connected by a string Dvali  _et al._ (2023). As shown there, after
coming on top of each other, the pair does not oscillate even once. Instead,
it decays into the waves of Higgs and gauge fields. This effect was explained
by entropic arguments: the entropy of a monopole-antimonopole pair is much
lower than the entropy of waves. Once the monopole and antimonopole come on
top of each other, the system loses the memory of its pre-history of the
magnetic dipole. After this point, it simply evolves into the highest entropy
state, which is given by waves, as opposed to monopoles connected by a long
string. In the language of amplitudes, this can be understood as insufficient
entropy for compensating a strongly suppressed process of production of highly
coherent states Dvali (2021).

This outcome is characteristic of the phenomenon of defect “erasure”,
originally discussed in Dvali  _et al._ (1998) for the interaction of
monopoles and domain walls. In Dvali  _et al._ (2023) was argued that the same
effect must hold for heavy quark-antiquark pairs connected by the QCD strings.

As we shall show, in the present case, the situation is very different. The
reason is the existence of the net un-erased magnetic charge. That is, in
contrast with the cases of monopole-antimonopole Dvali  _et al._ (2023) and
monopole-wall Dvali  _et al._ (1998). The net magnetic charge is always point-
like. Due to this, the system is aware of its magnetic pre-history all the
time. Unlike the erasing antimonopole or an erasing domain wall, in the
present case, the magnetic charge neither cancels nor spreads.
Correspondingly, the string never breaks apart.

Thus the outcome is a formation of a monopole attached to a boundary by a long
string. The string stretches and absorbs the initial kinetic energy of the
monopole, gradually slowing it down. If the wall is static, after reaching a
certain maximal size, the string will start shrinking accelerating the
monopole towards the boundary. After reaching the boundary, the monopole will
be shot back into the Coulomb vacuum. We shall refer to this phenomenon as the
“slingshot” effect.

One of the important implications of the slingshot effect is the novel source
of production of gravitational waves. The monopole slingshot effect is
expected to be rather generic in the early cosmology of grand unified
theories. It is thereby important to understand the imprints of this effect in
the gravitational wave spectrum.

Due to this, we study the corresponding gravitational wave signal in detail in
our parameters space range. In particular, it is found that the energy
spectrum and the beaming angle of emission are analogous to the case of a
monopole-antimonopole connected by a string in the confined phase, complying
with the fact that most of the signal is due to the acceleration of the
monopole by the slingshot. In particular, the energy spectrum is found to
scale as the inverse frequency $\omega^{-1}$, which agrees with studies of a
confined monopole-antimonopole pair in the point-like approximation Martin and
Vilenkin (1997) and in the fully-fledged field theoretical case Dvali  _et
al._ (2023). Moreover, we also observe that the slingshot gravitational
radiation is emitted in a beaming angle $\theta$, measured from the
acceleration axis of the domain wall, scaling approximately as
$\omega^{-1/2}$.

The same type of gravitational wave signal is expected in the dual slingshot
case of the “electric” confinement. In this case, the role of a monopole is
played by a heavy quark that crosses over from the Coulomb to a confining
domain. Similarly to the monopole stretching a cosmic string, the quark
entering the confining domain stretches a QCD flux tube. For explicit analysis
we construct a model using the earlier setup discussed in Dvali and Vilenkin
(2003); Dvali  _et al._ (2008) for the study of the gauge field localization
mechanism of Dvali and Shifman (1997). In this setup, the two vacua represent
confining and deconfining phases of $SU(2)$ QCD. We assume that the quarks are
heavier than the corresponding QCD scale.

The QCD flux tubes connect these quarks in the confining domain. The flux
tubes can be exponentially long without the danger of breaking apart. In the
deconfining domain, $SU(2)$ is Higgsed down to $U(1)$, and the same quarks can
propagate freely and interact via the Coulomb $U(1)$ field. The two phases are
separated by a domain wall. The massless photon is “locked” in the $U(1)$
domain by the gauge field localization mechanism of Dvali and Shifman (1997).

We then consider a scattering process in which a heavy quark goes across the
wall from the $U(1)$ Coulomb to the $SU(2)$-confining phase. Transporting the
intuition from the monopole case of the dual theory, we shall argue that the
outcome is similar: the system exhibits a slingshot effect. Namely, the quark
stretches along the QCD string which connects it to the wall. Despite the
sufficient energy in the collision, the string does not break up into a
multiplicity of mesons and glueballs. The physical reason, as we shall argue,
is similar to the monopole case and has to do with the existence of the net
$U(1)$ charge measured by the Coulomb observer.

Just like the magnetic slingshot effect, its electric dual can be relevant for
cosmology in various extensions of the standard model, including grand
unification, since the coexistence of phases is rather generic. The
gravitational wave signal from the electric slingshot is rather similar to its
magnetic dual.

Finally, the slingshot effect generalizes to defects of other co-dimensions.
In particular, it can be exhibited by cosmic strings. When the string crosses
over into a phase in which it becomes a boundary of the domain wall, it
stretches the wall. The essence of the effect is captured by an effective
$2+1$-dimensional model. The model in which $Z_{2}$ vortices can be confined
was constructed earlier in Dvali (1991). We extend this model by allowing the
coexistence of two phases: the free phase with exact $Z_{2}$ symmetry, as well
as, the confining phase in which $Z_{2}$ is spontaneously broken. When the
vortex crosses over from the free into the confining phase, it results in a
slingshot effect.

The main findings of this paper are summarized in a companion letter Bachmaier
_et al._ .

##  II The Model

We shall now construct a simple prototype model that possesses a domain wall
separating the two vacua in which the magnetic field is in Coulomb and
confining phases respectively. Such examples were constructed earlier in Dvali
and Vilenkin (2003) as setups for realizing a dual (magnetic) version of the
gauge field localization mechanism of Dvali and Shifman (1997).
Correspondingly, in the construction of Dvali and Vilenkin (2003) there exist
Higgs and Coulomb phases of a $U(1)$ gauge theory are separated by a domain
wall. Within the $U(1)$ Higgs domain, the magnetic flux is trapped in the
tubes (cosmic strings). A string can terminate perpendicularly to the wall and
open up on the other side in the form of the sources of a magnetic Coulomb
field. In order to include the magnetic monopoles on both sides of the wall,
we embed the $U(1)$ as a subgroup of an $SU(2)$ gauge symmetry.

The model that we will analyze is an $SU(2)$ gauge theory with two scalar
fields. The first field $\phi$ transforms under the adjoint representation,
while the second field $\psi$ is in the fundamental representation. The
Lagrangian of the theory is

$\displaystyle\mathcal{L}=$
$\displaystyle\Tr\left((D_{\mu}\phi)^{\dagger}\,(D^{\mu}\phi)\right)+(D_{\mu}\psi)^{\dagger}(D^{\mu}\psi)$
$\displaystyle-\frac{1}{2}\Tr\left(G^{\mu\nu}G_{\mu\nu}\right)-U(\phi,\psi)\,,$
(1)

with the potential

$\displaystyle U(\phi,\psi)=$
$\displaystyle\lambda_{\phi}\left(\Tr(\phi^{\dagger}\phi)-\frac{v_{\phi}^{2}}{2}\right)^{2}$
$\displaystyle+\lambda_{\psi}\left(\psi^{\dagger}\psi-v_{\psi}^{2}\right)^{2}\
\psi^{\dagger}\psi+\beta\psi^{\dagger}\phi\psi\,.$ (2)

The field strength tensor and the covariant derivatives are given in the
conventional form

$\displaystyle G_{\mu\nu}$
$\displaystyle=\partial_{\mu}W_{\nu}-\partial_{\nu}W_{\mu}-ig[W_{\mu},W_{\nu}]\,,$
(3) $\displaystyle D_{\mu}\phi$ $\displaystyle=\partial_{\mu}\phi-
ig[W_{\mu},\phi]\,,$ (4) $\displaystyle D_{\mu}\psi$
$\displaystyle=\partial_{\mu}\psi-igW_{\mu}\psi\,.$ (5)

We can write the gauge field and the adjoint scalar field as
$W_{\mu}=W_{\mu}^{a}T^{a}$ and $\phi=\phi^{a}T^{a}$ respectively, where the
$SU(2)$ generators are normalized by $\Tr(T^{a}T^{b})=\frac{1}{2}\delta^{ab}$.

In this study, we consider a symmetry-breaking hierarchy characterized by
several distinct stages. Initially, the $SU(2)$ symmetry undergoes a Higgs
mechanism through the scalar field $\phi$, resulting in the reduction of
symmetry to $U(1)$. Subsequently, the $U(1)$ symmetry is Higgsed further down
by $\psi$. Schematically, the breaking pattern is

$SU(2)\rightarrow U(1)\rightarrow 1\,.$ (6)

During the first breaking process, two of the gauge bosons acquire a mass
$m_{v_{\phi}}=gv_{\phi}$, while one gauge boson, which we will refer to as the
photon, remains massless. The corresponding Higgs boson manifests a mass
$m_{h_{\phi}}=\sqrt{2\lambda_{\phi}}v_{\phi}$. Following the second symmetry
breaking, all gauge bosons, including the photon, acquire an additional
contribution to their mass denoted by $m_{v_{\psi}}=gv_{\psi}/\sqrt{2}$.
Additionally, the Higgs boson acquires a mass
$m_{h_{\psi}}=2\sqrt{\lambda_{\psi}}v_{\psi}^{2}$ in this subsequent stage. We
note here that although the potential (II) is non-renormalizable, it does not
concern our analysis since such a potential can be obtained from a
renormalizable theory by the introduction of an additional gauge singlet
field, as it was previously discussed in Dvali and Vilenkin (2003). Further
examples can be found in the same paper. The classical field equations of this
theory are

$\displaystyle(D_{\mu}G^{\mu\nu})^{a}=j^{a,\nu}_{\phi}+j^{a,\nu}_{\psi}\,,$
(7)
$\displaystyle(D_{\mu}D^{\mu}\phi)^{a}+\partialderivative{V(\phi,\psi)}{\phi^{a}}=0\,,$
(8) $\displaystyle
D_{\mu}D^{\mu}\psi+\partialderivative{V(\phi,\psi)}{\psi^{\dagger}}=0\,,$ (9)

where the currents are
$j^{a,\nu}_{\phi}=g\varepsilon^{abc}(D^{\nu}\phi)^{b}\phi^{c}$ and
$j^{a,\nu}_{\psi}=ig\psi^{\dagger}T^{a}(D^{\nu}\psi)+h.c.$.

For $\psi=0$, the $SU(2)$ symmetry is Higgsed down to $U(1)$. Consequently,
the theory encompasses a magnetic monopole solution characterized by the ’t
Hooft-Polyakov magnetic monopole ansatz ’t Hooft (1974a); Polyakov (1974)

$\displaystyle W^{a}_{i}$
$\displaystyle=\varepsilon_{aij}\frac{r^{j}}{r^{2}}\frac{1}{g}\left(1-K(r)\right),$
$\displaystyle W^{a}_{t}$ $\displaystyle=0\,,$ $\displaystyle\phi^{a}$
$\displaystyle=\frac{r^{a}}{r^{2}}\frac{1}{g}H(r)\,,$ (10)

where $K(r)$ and $H(r)$ are profile functions that depend on the parameters of
the theory.

The $SU(2)$ magnetic field can be defined by

$\displaystyle B^{a}_{k}=-\frac{1}{2}\varepsilon_{kij}G^{a}_{ij}\,.$ (11)

In order to obtain the $U(1)$ magnetic field, we can project out the component
that is parallel to $\phi$. This yields

$\displaystyle
B_{k}^{U(1)}=\frac{\phi^{a}}{\sqrt{\phi^{b}\phi^{b}}}B^{a}_{k}\,.$ (12)

With this definition, the magnetic field of the ’t Hooft-Polyakov magnetic
monopole in the limit of large $r$ is given by,

$\displaystyle B_{k}^{U(1)}\rightarrow\frac{1}{g}\frac{r^{k}}{r^{3}}\,.$ (13)

By substituting the ansatz (10) into the field equations (7) and (8), these
equations can be simplified to

$\displaystyle K^{\prime\prime}=$
$\displaystyle\frac{1}{r^{2}}(K^{3}-K+H^{2}K+J^{2}K)\,,$ (14) $\displaystyle
H^{\prime\prime}=$
$\displaystyle\frac{2}{r^{2}}HK^{2}+\frac{1}{2}m^{2}_{h_{\phi}}H\left(\frac{H^{2}}{m^{2}_{v_{\phi}}r^{2}}-1\right)\,.$
(15)

Note that we are still considering the case with $\psi=0$. The profile
functions can be determined analytically in the BPS limit
$m_{h_{\phi}}\rightarrow 0$ Bogomolny (1976); Prasad and Sommerfield (1975).
For other parameter choices, we employ numerical relaxation techniques. In
order to initiate the iteration procedure, we utilize the profile functions
obtained in the BPS limit as a starting point. The resulting profile functions
are visualized in Fig. 1.

Let us now shift our focus to the discussion regarding the $\psi$ field. For
the moment let us fix the $SU(2)$ direction to be $\psi=(\xi,0)^{T}$. For
$\beta=0$ the potential part corresponding to the $\xi$ field exhibits two
distinct vacua, the $U(1)$ Coulomb phase at $\xi^{\dagger}\xi=0$ and the
$U(1)$-Higgsed phase at $\xi^{\dagger}\xi=v_{\psi}^{2}$. The reason behind
this terminology will be further explained later. Since these two vacua are
disconnected, the model allows a domain wall solution interpolating between
them. By using the Bogomolny equation Bogomolny (1976) the solutions can be
found to be111See also Gani  _et al._ (2014); Dvali and Valbuena-Bermúdez
(2023).

$\displaystyle\xi_{(\pm v_{\psi},0)}(z)$ $\displaystyle=\frac{\pm
v_{\psi}}{\sqrt{1+e^{m_{h_{\psi}}z}}}\,,$ (16) $\displaystyle\xi_{(0,\pm
v_{\psi})}(z)$ $\displaystyle=\frac{\pm
v_{\psi}}{\sqrt{1+e^{-m_{h_{\psi}}z}}}\,.$ (17)

Therefore, the two phases, the $U(1)$ invariant and the $U(1)$-Higgsed phase
can coexist and are separated by these domain walls.

Figure 1: The profile function for a ’t Hooft-Polyakov magnetic monopole for
$m_{h_{\phi}}/m_{v_{\phi}}=1$.

When $\beta\neq 0$, the degeneracy of the two vacua is broken. The potential
difference between the $U(1)$ Coulomb vacuum and the $U(1)$-Higgsed vacuum
eliminates the possibility of a static domain wall. This potential difference
generates a pressure difference between the two sides of the domain wall,
causing it to accelerate toward the phase with higher potential energy. To
achieve higher relative collision velocities for our numerical analysis of the
interaction between a magnetic monopole and this type of domain wall, we
exploit this acceleration. Of course, the splitting of energies between
different vacua, and thereby the amount of the pressure difference acting on
the wall, can be controlled by the parameters of the Lagrangian. In
particular, the vacua can easily be kept to be exactly degenerate in energy,
resulting in the possibility of static domain walls.

As a final comment on the spectrum of the theory, we note that once the $U(1)$
is Higgsed down by $\psi$, the free monopole solutions no longer exist.
Instead, the monopoles get connected to antimonopoles by the cosmic strings.
These strings represent Nielsen-Olesen magnetic flux tubes Nielsen and Olesen
(1973) that carry the $U(1)$ magnetic field lines sourced by the monopoles.
Since $U(1)$ is embedded in $SU(2)$, the strings are not topologically-stable
and can break by quantum nucleation of monopole-antimonopole pairs Vilenkin
(1985). The process is exponentially suppressed by the ratio of the two
symmetry-breaking scales. As a result even for a mild hierarchy of scales, an
unperturbed segment of a long string is practically stable against such a
decay. In particular, this will be the case in our analysis.

##  III Initial Configuration

One generic phenomenon experienced by the monopoles in the confinement regime
is the annihilation of a monopole-antimonopole pair connected by a string.
During this process, the monopole and antimonopole are pulled together by the
string and subsequently annihilate. In the approximation of point-like
monopoles connected by a thin string, the system is allowed to perform several
oscillations. Since in this approximation the structures are not resolved, the
monopoles are permitted to pass through each other and stretch a long string
multiple times Martin and Vilenkin (1997). However, the fully resolved
analysis shows that this is not the case Dvali  _et al._ (2023). In the regime
of finite and comparable thicknesses of strings and monopoles, the system
decays after the first collision. In Dvali  _et al._ (2023) this is explained
by the loss of coherence Dvali  _et al._ (1998) during the collision and the
entropy suppression characteristic for the creation of low entropy solitons in
high energy collision processes Dvali (2021).

In the present case, we wish to investigate another type of scattering process
involving the confined monopole. However, instead of being in the confinement
regime from the very beginning, initially, the monopole starts in the magnetic
Coulomb phase and only later enters the confinement domain with a relativistic
velocity.

Thus, we aim to determine the initial configuration for a specific scenario: a
magnetic monopole positioned within the $U(1)$ Coulomb phase, while elsewhere,
a domain wall separates the Coulomb phase and the Higgsed phase. We want to
analyze in a numerical simulation what happens when the monopole collides with
the domain wall.

In the phase where the $U(1)$ symmetry is Higgsed, the photon, that is
massless in the $U(1)$ Coulomb phase, receives a mass. Notice that the
magnetic charge is still fully conserved. However, in the $U(1)$ Higgs domain
the flux can only exist in the form of flux tubes. This is energetically
costly. Thereby, the lowest energy configuration with a single monopole placed
in the $U(1)$ Coulomb domain is the one in which the entire flux is spread
within the same domain. Upon reaching the wall, the magnetic flux lines are
repelled and spread parallel to the wall.

To include this effect in the initial configuration, we made use of the
monopole-antimonopole ansatz with a maximal twist Saurabh and Vachaspati
(2017). If we take only the monopole side of this ansatz, the magnetic field
lines resemble the right behavior. The general ansatz for $\hat{\phi}^{a}$,
where $\hat{\phi}^{a}=\phi^{a}/\sqrt{\phi^{b}\phi^{b}}$, for a monopole-
antimonopole configuration is Saurabh and Vachaspati (2017)

$\displaystyle\hat{\phi}_{1}=$
$\displaystyle\left(\sin\bar{\theta}\cos\theta-\sin\theta\cos\bar{\theta}\cos\alpha\right)\cos\left(\varphi-\alpha/2\right)$
$\displaystyle+\sin\theta\sin\alpha\sin\left(\varphi-\alpha/2\right),$
$\displaystyle\hat{\phi}_{2}=$
$\displaystyle\left(\sin\bar{\theta}\cos\theta-\sin\theta\cos\bar{\theta}\cos\alpha\right)\sin\left(\varphi-\alpha/2\right)$
$\displaystyle-\sin\theta\sin\alpha\sin\left(\varphi-\alpha/2\right),$
$\displaystyle\hat{\phi}_{3}=$
$\displaystyle-\cos\theta\cos\bar{\theta}-\sin\theta\sin\bar{\theta}\cos\alpha\,.$
(18)

The angle $\alpha$ represents the relative twist between the monopole and the
antimonopole. In our simulations, we took $\alpha=\pi$ to obtain a
configuration for which the magnetic field presents the right behavior.

We will take the monopoles to be located on the $z$-axis at $z_{\rm M}$ and
$z_{\rm\bar{M}}$. Thus, $\varphi$ is the azimuthal angle around the $z$-axis.
$\theta$ and $\bar{\theta}$ correspond to the angles between the $z$-axis and
the position vectors originating from the monopole and antimonopole,
respectively.

The ansatz that Saurabh and Vachaspati considered Saurabh and Vachaspati
(2017) is then given by

$\displaystyle\phi^{a}=$ $\displaystyle\frac{1}{g}\frac{H(r_{\rm M})}{r_{\rm
M}}\frac{H(r_{\rm\bar{M}})}{r_{\rm\bar{M}}}\hat{\phi}^{a},$ (19)
$\displaystyle W^{a}_{\mu}=$ $\displaystyle-\frac{1}{g}(1-K(r_{\rm
M}))(1-K(r_{\rm\bar{M}}))\varepsilon_{abc}\hat{\phi}^{b}\partial_{\mu}\hat{\phi}^{c}\,.$
(20)

In order to Lorentz boost this configuration we can replace $z-z_{\rm M}$ and
$z-z_{\rm\bar{M}}$ by $\gamma_{\rm M}(z-u_{\rm M}t-z_{\rm M})$ and
$\gamma_{\rm M}(z-u_{\rm\bar{M}}t-z_{\rm\bar{M}})$ respectively. For $t=0$, we
obtain the initial field values. The values for the time derivatives can be
determined numerically by using the field configuration at $t=0$ and
$t=\differential t$. We conducted the numerical simulations in the Lorenz
gauge.

Figure 2: A sketch of the magnetic field (top) and the scalar field vector
$(\phi^{3},-\phi^{2})^{T}$ (bottom) for the initial configuration. The color
in the background represents $\absolutevalue{\psi}$ ranging from
$\absolutevalue{\psi}=0$ (blue) to $\absolutevalue{\psi}=v_{\psi}$ (red).

In our configuration, we will incorporate a domain wall positioned at the
center between the monopole and the antimonopole. The domain wall is located
at $z=0$ and the monopole is on the $z<0$ side. To remove the antimonopole
from our setup, we modified our ansatz by

$\displaystyle\phi^{a}(x,y,z>0)$
$\displaystyle\rightarrow\phi^{a}(x,y,z=0)\,,$ (21) $\displaystyle
W^{a}_{\mu}(x,y,z>0)$ $\displaystyle\rightarrow
W^{a}_{\mu}(x,y,z=0)\frac{1}{1+e^{\gamma_{\rm D}m_{v_{\psi}}z}}\,,$ (22)

where $\gamma_{\text{D}}$ is the Lorentz factor of the domain wall. Note that
the suppression factor in equation (22) is an approximation that is in
accordance with the wall profile.

The ansatz for the $\psi$ field that includes the domain wall solution needs
to minimize the potential. In order to achieve this, we seek to extremize the
interaction term,

$\displaystyle\mathcal{L}_{int}=-\beta\psi^{\dagger}\phi\psi\,,$ (23)

by aligning $\psi^{\dagger}T^{a}\psi$ parallel to $\phi^{a}$. Therefore, the
ansatz for $\psi$ can be written as

$\displaystyle\psi^{1}=$
$\displaystyle-\frac{\xi_{(0,+v_{\psi})}}{\sqrt{2}}\frac{(\hat{\phi}^{1}-i\hat{\phi}^{2})}{\sqrt{1+\hat{\phi}^{3}}}\,,$
$\displaystyle\psi^{2}=$
$\displaystyle\frac{\xi_{(0,+v_{\psi})}}{\sqrt{2}}\sqrt{1+\hat{\phi}^{3}}\,.$
(24)

The Lorentz boosted configuration of the domain wall can be determined in a
similar manner to that of the magnetic monopole. Specifically, we replace the
variable $z$ with $\gamma_{\rm D}(z-u_{\rm D}t)$, where $\gamma_{\rm D}$
represents the Lorentz factor associated with the domain wall.

In Fig. 2, the scalar fields and the magnetic field are illustrated. Since the
ansatz we are employing is an approximation, we incorporated it into a
numerical relaxation procedure, as outlined in Saurabh and Vachaspati (2017),
to investigate the response of the field to the static field equations.
Notably, we observed that the deviations between the configuration before and
after the relaxation remained small, thus affirming its suitability for our
intended purpose.

##  IV Numerical Implementation

The numerical simulations were performed using the Python programming
language, leveraging the Numba package Lam  _et al._ (2015). Numba facilitates
the translation of Python code into efficient machine code, enabling faster
computations. Additionally, it offers a straightforward approach to
parallelizing the code, effectively utilizing the capabilities of multi-core
processors.

In order to improve the computation time, we took benefit of the axial
symmetry of the configuration, as we have previously done in the context of
magnetic monopole erasure Bachmaier  _et al._ (2023). The approach involved
utilizing only three lattice points in the $y$-direction, sufficient for
numerically calculating the second-order derivative appearing in the field
equations. At each time iteration step, we solved the field equations in the
$y=0$ plane and used the axial symmetry to determine the field values in the
two neighboring planes. This method, first employed in configurations of this
nature, was introduced in Pogosian and Vachaspati (2000).

From (18) we can find the axial symmetry of the $\phi^{a}$ field of a
monopole-antimonopole system for an arbitrary twist. This is given by

$\displaystyle\phi^{1}=$ $\displaystyle f_{1}x+f_{2}y\,,$
$\displaystyle\phi^{2}=$ $\displaystyle f_{1}y-f_{2}x\,,$
$\displaystyle\phi^{3}=$ $\displaystyle f_{3}\,,$ (25)

where the functions $f_{i}$ depend only on the time $t$, the radius around the
$z$-axis, and the $z$-coordinate. To find an ansatz for the gauge fields we
inserted (25) into $D_{\mu}\phi=0$. This gives us

$\displaystyle W^{1}_{x}=xyf_{4}+y^{2}f_{5}+f_{6}$ $\displaystyle
W^{1}_{y}=y^{2}f_{4}-f_{5}xy-f_{7}$ $\displaystyle
W^{2}_{x}=-x^{2}f_{4}-f_{5}xy+f_{7}$ $\displaystyle
W^{2}_{y}=-xyf_{4}+x^{2}f_{5}+f_{6}$ $\displaystyle W^{3}_{x}=xf_{8}+yf_{9}$
$\displaystyle W^{3}_{y}=yf_{8}-xf_{9}$ $\displaystyle
W^{1}_{z}=f_{10}x+f_{11}y$ $\displaystyle W^{1}_{t}=f_{12}x+f_{13}y$
$\displaystyle W^{2}_{z}=-f_{11}x+f_{10}y$ $\displaystyle
W^{2}_{t}=-f_{13}x+f_{12}y$ $\displaystyle W^{3}_{z}=0$ $\displaystyle
W^{3}_{t}=0\,.$ (26)

From equation (24) we can find the axial symmetric ansatz for the $\psi$ field

$\displaystyle\psi^{1}=f_{14}(x-iy)+f_{15}(y+ix)\,,$
$\displaystyle\psi^{2}=f_{16}\,.$ (27)

Notice that this axial symmetric ansatz presented here can be also used in the
analysis of head-on collisions between a monopole and an antimonopole like in
the situations described in Vachaspati (2016); Dvali  _et al._ (2023).

We employed the second iterative Crank-Nicolson method, as described in
Teukolsky (2000), to simulate the time evolution. We applied the axial
symmetry method described above every time we solved the field equation in the
$y=0$ plane. We used absorbing boundary conditions for $\phi^{a}$ and
$W_{\mu}$. For $\psi$ we chose Dirichlet boundaries in $z$-direction and
periodic boundaries in $x$-direction. Notice that for the twist $\alpha=\pi$,
the imaginary component of $\psi^{1}$ is anti-symmetric in $x$-direction.

The theory (II) contains six independent parameters which can be given in
terms of: $g$, the masses $m_{v_{\phi}}$, $m_{h_{\phi}}$, $m_{v_{\psi}}$,
$m_{h_{\psi}}$, and $\beta$. The first three parameters were set to $g=1$ and
$m_{v_{\phi}}/m_{h_{\phi}}=1$. We varied the latter three parameters in the
intervals $m_{v_{\psi}}\in[0.1\,m_{v_{\phi}},0.7\,m_{v_{\phi}}]$,
$m_{h_{\psi}}\in[0.1\,m_{v_{\phi}},1.0\,m_{v_{\phi}}]$, and
$\beta\in[0.001\,m_{v_{\phi}},0.1\,m_{v_{\phi}}]$. In the results section we
will focus especially on the case with $m_{v_{\psi}}=0.15\,m_{v_{\phi}}$,
$m_{h_{\psi}}=0.6\,m_{v_{\phi}}$ and $\beta=0.01\,m_{v_{\phi}}$.

Besides the aforementioned parameters, we have the flexibility to select the
initial velocities of the magnetic monopole ($u_{\rm M}$) and the domain wall
($u_{\rm D}$), as well as the distance between them. In the potential (II),
the interaction term between $\phi$ and $\psi$ causes the domain wall to
experience acceleration. Consequently, achieving a collision between the
monopole and the domain wall does not necessitate a Lorentz boost.
Nevertheless, we varied the initial velocities in the interval $u_{\rm
M},u_{\rm D}\in[0,0.98]$ (in units of $c=1$). Below we will specifically focus
on the scenario where the initial velocities are set to $u_{\rm M}=0.8$ and
$u_{\rm D}=0.8$ in opposite directions. The domain wall was located at $z=0$
and the magnetic monopole at $z=z_{\rm M}=-40\,m_{v_{\phi}}$.

For the numerical simulations, we used a lattice of the size
$[-60\,m_{v_{\phi}}^{-1},60\,m_{v_{\phi}}^{-1}]$ and
$[-180\,m_{v_{\phi}}^{-1},60\,m_{v_{\phi}}^{-1}]$ in $x$\- and $z$-direction
respectively. The lattice spacing was set to $0.25\,m_{v_{\phi}}^{-1}$ and the
time step we chose to be $0.1\,m_{v_{\phi}}^{-1}$. The time interval under
investigation was $[0,180\,m_{v_{\phi}}^{-1}]$.

##  V Results

During the time evolution of the initial setup outlined in the previous
section, we can observe that as the magnetic monopole approaches the domain
wall, a significant amount of magnetic energy density accumulates along the
wall. This phenomenon arises due to the presence of a mass for the photon on
the right-hand side of the domain wall. As a consequence, the penetration of
the photon, which carries the magnetic energy, into the Higgs vacuum is
exponentially suppressed.

Notice, however, that the magnetic field is repelled from the $U(1)$ Higgs
domain but not screened Dvali and Vilenkin (2003). This is analogous to the
Meissner effect in superconductors. Some of us already discussed the dual
case, in which the electric field is repelled while the magnetic field is
screened by a confining layer Bachmaier  _et al._ (2023). The penetration is
possible only in the form of a flux tube, which is costly in energy. This
repulsion leads to the concentration of energy density along the wall,
resulting in the observed phenomenon.

Upon collision with the wall, the monopole transitions to the right-hand side
and stretches a string, as this is the only way in which the monopole can
enter the $U(1)$ Higgs region. The end of the string opens up on the $U(1)$
Coulomb side of the wall, where the flux can spread out. Since the magnetic
charge is conserved, the integrated flux exactly matches the magnetic charge
of the monopole. Correspondingly, an observer located in the Coulomb vacuum
will effectively measure the same magnetic charge carried by the string
“throat” as the one taken by the original monopole.

This phenomenon can be seen in the magnetic energy density and the behavior of
the magnetic field as illustrated in Fig. 3. In addition to this figure, the
full-time evolution can be found in the video in the ancillary files or at the
following link:
https://youtu.be/IPJAPjo3nSc?si

Figure 3:  The illustration depicts the magnetic energy density and magnetic
field at time $t=115\,m_{v_{\phi}}^{-1}$ in the $y=0$ plane for the specific
case described in the numerical implementation section. The length values are
provided in units of $m_{v_{\phi}}^{-1}$, while the energy density values are
given in units of $m_{v_{\phi}}^{4}/g^{2}$. The black line represents the
contour corresponding to $\absolutevalue{\psi}=0.1\,m_{v_{\phi}}$, serving to
illustrate the presence of the domain wall. We observe that the magnetic
monopole has formed a string, connecting it to the domain wall. Both the
magnetic field lines and the magnetic energy density indicate the presence of
a localized magnetic flux within the string. In the Coulomb phase, the
magnetic field vectors point radially away from the point where the string
attaches to the domain wall, representing the magnetic field of a virtual
monopole located at that particular position.

In the time evolution, we can also see that the monopole decelerates during
the string stretching. At a certain point, the string approaches its maximum
length, and the entire configuration, with the monopole connected to the
domain wall via the string, moves collectively at the same velocity. Here it
is important to note that it is not exactly the same velocity but approaches
the same speed asymptotically. The reason for this is that the domain wall as
well as the monopole have a proper constant acceleration,

$\displaystyle a_{\rm DW}$ $\displaystyle\sim\frac{\delta}{\sigma_{\rm
DW}}\sim\beta\frac{m_{v_{\phi}}}{gm_{h_{\psi}}}\,,$ (28) $\displaystyle a_{\rm
M}$ $\displaystyle\sim\frac{\mu_{\rm string}}{M_{\rm
M}}\sim\frac{m_{v_{\psi}}^{2}}{m_{v_{\phi}}}\,,$ (29)

where $\delta$ is the potential energy difference between the $U(1)$-symmetric
and the $U(1)$-Higgsed vacua, $\sigma_{\rm DW}\sim m_{h_{\psi}}v_{\psi}^{2}$
is the domain wall tension, and $\mu_{\rm string}\sim m_{v_{\psi}}^{2}/g^{2}$
is the string tension. Of course, the accelerations in (28) may differ. In
some cases, it is even possible that the monopole is expelled from the Higgsed
phase and re-enters it as soon as the domain wall catches up with it again. It
is worth noting that the interaction term present in the potential equation
(II) plays a crucial role in this behavior. As previously mentioned, this term
introduces the vacuum energy difference between the Coulomb and Higgs phases.
Consequently, there is a constant acceleration of the domain wall. This
acceleration is essential for preventing the monopole from re-entering the
Coulomb phase since the tension of the string pulls it outward. Without the
domain wall’s acceleration, the monopole would be drawn back into the Coulomb
phase by the string slingshot effect.

Another notable observation regarding the magnetic energy density is the
emission of radiation during the interactions. When the monopole collides with
the wall, a significant amount of energy is invested in creating the string,
resulting in an extreme deceleration. This process generates electromagnetic
radiation in the form of a shock wave. We can also see that this radiation is
capable of penetrating into the Higgs phase, demonstrating its ability to
traverse regions with broken $U(1)$ symmetry.

In the parameter regime under consideration, the formation of a string was
observed to be nearly ubiquitous. However, when $m_{v_{\psi}}$ and
$m_{h_{\psi}}$ are sufficiently large, the energy gap at the domain wall
becomes too large for the string to form. As a result, the monopole remains
localized on the wall and moves together with it. Additionally, the thickness
of the string is dependent on the specific parameters of the theory. These
parameters and the initial velocities of the monopole and domain wall also
determine the maximum length of the string. When these objects possess higher
velocities, there is increased availability of energy, allowing the string to
extend to greater lengths. As a general estimate, assuming the point-like
limit for the monopole solution, the thin string, and the thin wall limit, the
maximal penetration is

$\ell_{\rm max}\sim\gamma_{c}\frac{M_{\rm M}}{\mu_{\rm string}}\,,$ (30)

where $\gamma_{c}$ is the relative gamma factor between the wall and the
monopole at the moment of the collision.

The natural question that arises is what is the fate of the extended string.
Energetically, it is theoretically possible to form monopole-antimonopole
pairs connected by strings after the collision. However, despite considering
various parameters in our classical simulation, we have not observed this
phenomenon. In our simulations, the magnetic monopole consistently stretches
the string and maintains its connection to the domain wall, as long as there
are no external influences present. Yet, if one perturbed the string, the only
way we found so far to disconnect the string from the domain wall is by
introducing an additional antimonopole.

In other simulations, we examined a specific configuration where a monopole
and an antimonopole, separated by a sufficiently large distance, enter the
$U(1)$-Higgsed phase successively along the $z$-axis. To ensure the correct
repulsion behavior of the magnetic field lines along the domain wall, we
combined two untwisted monopole-antimonopole pairs by introducing a twist
between them. This configuration was achieved using the following ansatz for
the scalar field

$\displaystyle\hat{\phi}=\begin{pmatrix}-\sin(\theta_{1}-\bar{\theta}_{1}+\theta_{2}-\bar{\theta}_{2})\sin\phi\\\
\sin(\theta_{1}-\bar{\theta}_{1}+\theta_{2}-\bar{\theta}_{2})\cos\phi\\\
-\cos(\theta_{1}-\bar{\theta}_{1}+\theta_{2}-\bar{\theta}_{2})\end{pmatrix},$
(31)

where $\theta_{1}$ and $\theta_{2}$ ($\bar{\theta}_{1}$ and
$\bar{\theta}_{2}$) correspond to the angles between the $z$-axis and the
position vectors stemming from the monopoles (antimonopoles).

The subsequent implementation followed a similar approach to the previously
described model. Our observations revealed that the initial monopole extended
a string, and later when the antimonopole entered the string, it caused the
detachment of the string from the domain wall as can be seen in Fig. 4.
Consequently, the monopole and antimonopole were drawn together with constant
acceleration until their annihilation occurred. The dynamics are analogous to
the one described in Dvali  _et al._ (2023). The energy stored in the strings
connecting them is transferred into kinetic energy of the monopole-
antimonopole, which turns them ultra-relativistic 222In Dvali  _et al._
(2021), this dynamics was applied to the production of primordial black holes.
In fact, for a long enough initial string, the system will find itself within
its own Schwarzschild radius well before the monopole-antimonopole
annihilation, therefore leading to the production of black holes..

Figure 4: The magnetic energy density and magnetic field for two magnetic
monopoles entering the confined phase. To the units, the same applies as in
Fig. 3. The first monopole enters and stretches a string. Afterward, the
antimonopole enters and detaches the string from the domain wall, leading to
the formation of a monopole-antimonopole pair connected by a string.

The full-time evolution for this configuration can be found in the ancillary
video, which is also available in the following link:
https://youtu.be/IPJAPjo3nSc?si

Monopoles connected by a string and their dynamics in this type of model have
been studied in great detail in Dvali  _et al._ (2023).

As already discussed, the string can break by the spontaneous creation of a
monopole-antimonopole pair on its world volume. As analysed by Vilenkin
Vilenkin (1985), for a classically-stable string this is a tunneling process
with extremely low probability. This analysis is equally applicable also to a
long string in our case after its formation.

The non-trivial question on which our analysis sheds light is, how probable is
the stretching of such a string in a monopole wall collision? We could imagine
that once the monopole collides with the wall, the entire energy gets
converted into radiation without ever stretching a string, as this was
observed to be the case in the collision of a confined monopole-antimonopole
pair Dvali  _et al._ (2023). There, in a head-on collision, the monopoles
would never pass each other and re-create a string. Instead, the system
decayed into waves after the first collision. In this respect, the two setups
give very different outcomes.

The reason for this difference is the following. First, as explained in Dvali
_et al._ (2023), in the case of monopole-antimonopole collision, after they
come on top of each other, the system completely “forgets” about the existence
of the magnetic charges. Basically, the monopole-antimonopole completely
erases one another. The collision also takes away some coherence, as this is
typical for the processes of defect erasure Dvali  _et al._ (1998); Dvali and
Valbuena-Bermúdez (2023). Correspondingly, in its further evolution, the
system has no “profit” in re-creating a highly coherent and low entropy state
of monopoles connected by a string. Such an outcome is exponentially
suppressed. It is much more probable to decay into a highly entropic state of
waves. This exponential suppression is generic for the transition amplitudes
into macroscopic final states of low entropy Dvali (2021); Dvali and Eisemann
(2022).

The situation in the present case is very different. The reason is that the
magnetic charge is conserved. Correspondingly, the system must maintain the
monopole, no matter what. The only question is the arrangement of its magnetic
flux. Since the monopole has sufficient kinetic energy for entering the
confining phase, the system has two choices: 1) accompany the monopole with a
long string; or 2) create at least one additional monopole-antimonopole pair
for breaking it apart. Since the pair-creation via quantum tunneling is
exponentially suppressed, the latter process would require a hard perturbation
which would force the adjoint field to vanish. This is not happening since the
monopole “sees” the wall through the change of the expectation value of the
fundamental field, which is a rather soft perturbation. Thus, the system
chooses the process of stretching the long string adiabatically. Due to this,
the outcome is a slingshot effect.

The string breaking could also occur via thermal fluctuations, which will be
the subject of future investigation.

##  VI Gravitational Waves

The slingshot mechanism provides a novel source of gravitational waves that
can be produced in the early universe. This adds to the list of previously
discussed sources of gravitational waves from various types of defects, such
as colliding bubbles Kosowsky  _et al._ (1992); Kamionkowski  _et al._ (1994),
monopole-antimonopole pairs confined by strings Martin and Vilenkin (1997);
Dvali  _et al._ (2023), cosmic string loops Vachaspati and Vilenkin (1985),
etc.

The interesting novelty of the slingshot source of gravity waves is that it is
expected to be rather generic in a grand unified phase transition, as such
transition often proceeds with the formation of domain walls separating the
phases of confined and free monopoles. For example, already the minimal grand
unified theory with Higgs fields in the adjoint $24_{\rm H}$ and fundamental
$5_{\rm H}$ representations, allows for the coexisting temporary phases such
as $SU(4)\times U(1)$ and $SU(4)$ separated by domain walls. The vacuum
expectation values in these two phases have the following forms: $\langle
24_{\rm H}\rangle\propto{\rm diag}(-4,1,1,1,1),~{}\langle 5_{\rm H}\rangle=0$
and $\langle 24_{\rm H}\rangle\propto\rm{diag}(-4,1,1,1,1),~{}\langle 5_{\rm
H}\rangle\propto(1,0,0,0,0)^{t}$ respectively. In these vacuum domains, the
magnetic monopoles are in the Coulomb and confining phases respectively.
Correspondingly, the interaction between monopoles and domain walls leads to
the slingshot effect.

Of course, the purpose of the present paper is not to study the full richness
of the grand unified phase portrait, which is also highly model-dependent. It
suffices to notice that the slingshot can even be a dominant source of
gravitational waves. In order to understand this, we can think about a single
spherically symmetric expanding bubble separating the two phases. Sweeping
away the monopoles by a slingshot mechanism produces gravity waves even in the
absence of bubble collisions with other bubbles. For this reason, we focus on
the generic aspects of the gravitational waves produced by the slingshot,
using a simple prototype example presented in previous sections.

Notice that in our simulation, we ignore the gravitational backreaction on the
dynamics of the source. Namely, we are assuming that the wall/string/monopole
dynamics is dominated by the string tension. This is a legitimate assumption,
provided that the tensions are below the Planck mass.

As was shown long ago Vilenkin (1983); Ipser and Sikivie (1984), the planar
infinite wall has repulsive gravity. It acts on a pointlike source of positive
mass $m$ with a repulsive linear potential, given by $V(r)\sim G\,\sigma_{\rm
DW}\,m\,r$, where $\sigma_{\rm DW}$ is the wall tension. In the present case,
this repulsion can compensate or even overtake the attractive potential due to
a string. It can also prevent the monopole from crossing the wall. Notice that
under the condition $\mu\gg G\,\sigma_{\rm DW}\,M_{\rm M}$ the slingshot
dynamics is negligibly affected by the gravitational field of the domain wall
which we assume throughout this work.

The radiated energy at frequency $\omega$ per unit frequency and per solid
angle, in direction $\mathbf{\hat{k}}$ ($\absolutevalue{\mathbf{k}}=\omega$)
can be calculated by Weinberg’s formula Weinberg (1972) (following the
conventions of Maggiore (2007))

$\displaystyle\frac{{\rm d}E}{{\rm d}\Omega\,{\rm
d}\omega}=\frac{G\,\omega^{2}}{2\pi^{2}}\Lambda_{ij,lm}(\mathbf{\hat{k}})T^{ij\ast}(\mathbf{k},\omega)T^{lm}(\mathbf{k},\omega)\,,$
(32)

where $\rm{d}\Omega$ is the differential solid angle and the Fourier transform
of the energy-momentum tensor is given by

$\displaystyle T_{\mu\nu}(\mathbf{k},\omega)=\int_{I_{t}}\differential
t\int_{V}\differential^{3}x\ e^{i\omega t-i\mathbf{k}\cdot\mathbf{x}}\
T_{\mu\nu}(\mathbf{x},t)\,,$ (33)

with $I_{t}$ and $V$ being the analyzed time interval and volume,
respectively. These were chosen around the time and length scales of the
dynamics of interest. The former corresponds to the duration of the source
$T\simeq 80m_{v_{\phi}}^{-1}$, while the latter is given by the volume spanned
by the system during its evolution. Given the relativistic motion involved,
$V\simeq T^{3}$ proved to be an optimal choice.

The operator $\Lambda_{ij,lm}$ projects a tensor into its transverse traceless
part and is defined as

$\displaystyle\Lambda_{ij,lm}(\mathbf{\hat{k}})\equiv
P_{il}(\mathbf{\hat{k}})P_{jm}(\mathbf{\hat{k}})-\frac{1}{2}P_{ij}(\mathbf{\hat{k}})P_{lm}(\mathbf{\hat{k}})\,,$
(34)

where $P_{ij}(\mathbf{\hat{k}})=\delta_{ij}-\hat{k}_{i}\hat{k}_{j}$ are
projectors into the orthogonal direction of $\mathbf{\hat{k}}$.

In the derivation of the equation (32), the divergenceless condition in
momentum space $k_{\mu}T^{\mu\nu}=0$ was assumed. The Fourier-transformed data
from the numerical simulations matches this condition well; thus, we can apply
formula (32).

Figure 5: The energy spectrum for the slingshot effect (red points). The blue-
dashed curve shows the scaling $\omega^{-1}$ for comparison.

Since we are working on a lattice with a finite resolution and the initial
configuration is an approximation (for example, the initial boost of the
monopole and the domain walls leads to fictitious sources), the presence of
noise in the gravitational energy spectrum, stemming from numerical
fluctuations, is anticipated. Moreover, both the domain wall and the monopole
are accelerated to relativistic velocities, which introduces an extra source
of background in the simulation due to the finiteness of the lattice spacing.
This imposes limitations on the available parameter space.

To ensure that such effects do not invalidate our analysis and that we capture
only the gravitational wave signal due to the slingshot, we execute a Lorentz
boost on the monopole in the opposing direction. With this strategy, we avoid
a collision with the domain wall in the considered time interval and we can
extract the magnitude of the background noise.

We observe that for $m_{v_{\psi}}=0.6m_{v_{\phi}}$ 333In the following
$m_{v_{\phi}}=1$ is used, and all dimensionful quantities are expressed in
units of it. (all the other parameters are kept unchanged), the background
noise in the energy spectrum is negligibly small - below $5\%$ of the energy
extracted in the presence of a slingshot. In this case, the length of the
string is comparable to the size of the magnetic monopole. Exploration of
alternative $m_{v_{\psi}}$ values reveals that numerical spurious effects
stops being negligible for $m_{v_{\psi}}\lesssim 0.4\,m_{v_{\phi}}$. Moreover,
for $m_{v_{\psi}}\gtrsim 0.7m_{v_{\phi}}$ the Lorenz gauge condition starts
being numerically violated by more than $10\%$.

The resulting energy spectrum, obtained upon integration over $\rm d\Omega$ is
shown in Fig. 5, where we fixed the Newton constant $G=1$ for
simplicity444Note that the instantaneously radiated power, according to the
notation of Weinberg (1972), can be obtained by multiplying
$\derivative{E}{\omega}$ by $\frac{2\pi}{T^{2}}$..

The energy spectrum is well characterized by the following scaling

$\derivative{E}{\omega}\propto\omega^{-1}\,.$ (35)

This is exemplified by the dashed blue line in Fig. 5. Unfortunately, the
finiteness of our numerical simulations does not permit a clear
characterization at higher frequencies. However, for sufficiently high
$\omega$ we expect the amplitude to be exponentially suppressed.

The direction of the emission is towards the bubble wall, as seen in Fig. 6.
Therein equation (32) is shown as a function of the axial angle $\theta$,
measured from the acceleration axis, and the frequency $\omega$. As it can be
seen, most of the radiation takes place in the direction of acceleration. In
particular, radiation is emitted in a beaming angle with frequency dependence
roughly approximated by

$\theta\propto\omega^{-1/2}\,,$ (36)

depicted by the dashed black line in the plot.

Figure 6: Angular dependence of the radiated gravitational energy as a
function of $\theta$ and $\omega$. The parameters chosen are the same as
outlined in the text. The angle $\theta$ gives the direction of radiation with
respect to the acceleration axis. $\theta=0$ corresponds to the direction of
acceleration (to the left).

In order to verify that the scalings (35) and (36) are due to the monopole
being accelerated by the flux tube attached to the domain wall, we performed a
separate analysis in which we isolated the slingshot dynamics from the initial
collision between the monopole and the domain wall. We found that indeed the
main contribution to the signal in Fig. 5 is due to the former.

The gravitational radiation due to the slingshot mechanism bears a close
resemblance to the one emitted by a confined pair of monopole-antimonopole. In
fact, also in that case the source of gravitational waves is due to monopoles
being accelerated by the flux tube. As shown by Ref. Martin and Vilenkin
(1997) in the limit of point-like monopoles and the zero thickness string,
(35) holds true also for that system. The result of Martin and Vilenkin was
confirmed by some of us for the case of fully-resolved confined $SU(2)$ ’t
Hooft-Polyakov monopoles Dvali  _et al._ (2023). The angular emission in the
point-like limit was instead found to scale according to (36) by Leblond  _et
al._ (2009).

The emitted instantaneous power for a confined monopole-antimonopole pair is
given by $P\sim G\,\Lambda^{4}$ Martin and Vilenkin (1997). While in our
numerical simulation, we have little leverage on the string tension
$\mu=\Lambda^{2}$, we observe that the gravitational signal amplitude is
roughly compatible with this scaling as we changed the value of
$m_{v_{\psi}}$. Moreover, we observe that for low enough $m_{v_{\psi}}$ the
radiation from the impact could become comparable to the one from the
slingshot for a sufficiently short stretching of string.

In our analysis, we focus on monopoles as the representative objects on which
the confined flux can terminate. However, as we discuss below, the current
analysis is general and applies also to the case of confined heavy quarks
connected by gauge strings. For this latter case, the signal is produced in
the regime $M\gtrsim\Lambda$, with $\Lambda\sim\sqrt{\mu_{\rm string}}$ being
the confinement scale, and $M$ being the mass of the monopole or quark. In
particular, we showed that the energy spectrum decays as $\omega^{-1}$ and
that the beaming angle of emission displays a $\omega^{-1/2}$ behavior in the
considered parameter space. Therefore, a phase transition between an
unconfined and confined phase can provide a specific signal in the form of
gravitational waves coming from a slingshot effect.

##  VII Implications for QCD

Our analysis has direct implications for QCD-like gauge theories with
coexisting domains with confined and deconfined phases. Such a system was
originally considered in Dvali and Shifman (1997). This setup possesses a
domain wall on which the $SU(2)$ gauge theory is deconfined. Later in Dvali
_et al._ (2008) the domain wall was replaced by the vacuum layer the width of
which can be arbitrarily adjusted. This is the setup we shall consider now.
The Lagrangian has the following form,

$\displaystyle\mathcal{L}=$
$\displaystyle-\frac{1}{2}\Tr\left(G^{\mu\nu}G_{\mu\nu}\right)+\Tr\left((D_{\mu}\phi)^{\dagger}\,(D^{\mu}\phi)\right)-U(\phi)$
$\displaystyle+i\bar{Q}\gamma^{\mu}D_{\mu}Q-M_{Q}\bar{Q}Q\,,$ (37)

where $\phi$ is a Higgs field in the adjoint representation of the $SU(2)$
gauge symmetry with the potential

$\displaystyle
U(\phi)=\lambda\Tr\left(\phi^{2}\right)\left(\Tr\left(\phi^{2}\right)-\frac{v_{\phi}^{2}}{2}\right)^{2}.$
(38)

The $SU(2)$ gauge sector of the theory, as well as the corresponding
notations, are the same as in previous examples. The fermion content of the
theory consists of (for simplicity) a single flavor of a heavy quark $Q$, in
the fundamental representation of $SU(2)$. Under “heavy” we mean that the mass
of the quark $M_{Q}$ is above the confinement scale of the $SU(2)$ theory,
which we denote by $\Lambda$.

The potential $U(\phi)$ possesses the following two vacua. In the first vacuum
$\phi=0$, the perturbative spectrum of the theory consists of an adjoint
scalar of mass $m_{\phi}=\lambda v_{\phi}$, the fundamental quark of mass
$M_{Q}$, and a massless $SU(2)$ gauge field. The effective low energy theory
is therefore a massless Yang-Mills. As it is well-known, this theory becomes
confining and generates a mass gap at the corresponding QCD scale $\Lambda$.
Correspondingly, an electric flux of gluons confines into flux tubes that
represent QCD strings ’t Hooft (1974b, c) with tension $\mu_{\rm
string}\sim\Lambda^{2}$. The lowest mass excitations about this vacuum are
colorless glueballs, which can be thought of as closed QCD strings. The
spectrum also includes mesons, which represent quark-antiquark pairs connected
by flux tubes and open strings.

The effect of the adjoint scalar $\phi$ on the confinement can be consistently
ignored for $\Lambda\ll m_{\phi}$, which we assume for definiteness.

In the second vacuum, classically, we have $\phi=v_{\phi}$, and thus the
$SU(2)$ gauge group is Higgsed down to the $U(1)$ subgroup. The bosonic
spectrum of the theory consists of a real $U(1)$ neutral scalar of mass
$m_{\phi}=\sqrt{\lambda}v_{\phi}^{2}$, a charged (complex) massive gauge boson
of mass $m_{v_{\phi}}=gv_{\phi}$, and a massless Abelian $U(1)$ gauge field.
In addition, of course, there exists a massive fermion $Q$. For
$m_{v_{\phi}},m_{\phi}\gg\Lambda$, the quantum effects from the massive modes
can be safely ignored and the effective low-energy theory consists of a $U(1)$
gauge theory in the Coulomb phase.

The theory possesses a domain wall solution separating the two phases.
Classically the solution can be found exactly. In the above approximation, the
quantum effects of the shape of the solution are small, but they can lift the
degeneracy of the two vacua. The bias can create a pressure difference which
accelerates the wall. This does not change much in our discussion, and in
fact, the controlled acceleration of the wall can be welcome for the study of
the scattering as we have seen in the numerical simulations. The bias can be
controlled by proper adjustment of the parameters.

The long-distance physical effects of the heavy quark in the two domains are
different. In the $U(1)$ domain, the quark produces a $U(1)$ Coulomb electric
field. In the $SU(2)$ domain, the quark is a source of the flux tube. This
flux tube can either terminate on an antiquark or on the wall. In the latter
case, the QCD-electric flux flowing through the tube opens up in the form of
the $U(1)$ Coulomb flux on the other side of the wall.

Notice that a long QCD string can break by nucleation of quark-antiquark
pairs. However, this process is exponentially suppressed for $M_{Q}>\Lambda$,
and the string can be stable for all practical purposes. This suppression is
similar to the exponential suppression of the decay of a magnetic Nielsen-
Olesen string via nucleation of monopole-antimonopole pairs. In what follows
we shall assume the regime of such stability.

This structure makes it clear that in certain aspects the model (VII)
represents an electric “dual” of the previously discussed model (II). The role
of monopoles is played by the heavy quarks, whereas the role of the magnetic
field is taken up by the electric flux of QCD. In both cases, the wall
separates the confining and Coulomb phases for the given flux.

The problem of monopole scattering at the wall is mapped on the scattering
between the wall and a heavy quark. When a quark moves across the wall from
the $U(1)$ phase to the confining one the flux carried by it stretches in the
form of the string that opens up as a Coulomb flux at the entry point. By
conservation of the $U(1)$ flux, the charge measured by an observer in the
Coulomb vacuum must exactly match the $U(1)$ charge of the initial quark.

However, this conservation can be fulfilled in two ways. Upon entry into the
confining vacuum, the string may stretch without breaking up leaving the
initial quark as the source of the flux. Alternatively, the string may break
up by nucleating additional quark-antiquark pairs or closed strings. That is
the system can transfer most of its initial energy into mesons and glueballs.

The process is very similar to the scattering of a magnetic monopole at the
wall. In that case, we saw that the string never breaks up. If the analogy can
be trusted, we would conclude that the same must be true for the case of a
quark entering the confinement domain from the Coulomb one.

For $M_{Q}\gg\Lambda$ such a behavior is relatively easy to justify. The two
factors are defined: 1) The continuous memory of the initial state; and 2) the
softness of the process.

First, notice that by the gauge invariance, the $U(1)$ charge is fully
conserved. The generation of the mass gap in the confining domain does not
affect this conservation. Due to this, a charge was placed in the $U(1)$
Coulomb domain and never creates any image charges on the confining side and
the flux is repelled without any screening Dvali and Shifman (1997); Dvali
_et al._ (2008). This is the key to the localization of a massless photon by
the mechanism of Dvali and Shifman (1997).

Therefore, when the quark crosses the wall and enters the confining region,
the memory of the initial state is maintained in the form of the Coulomb
electric flux. The total flux is conserved and is exactly equal to the $U(1)$
charge carried by the quark. This flux can be monitored by measuring it on the
Coulomb side of the wall. Now, when the free heavy quark enters the confining
region, no hard collision takes place. The Coulomb-electric flux carried by
the quark gathers into a tube of the thickness $\Lambda^{-1}$, which is much
larger than the de Broglie and Compton wavelengths of the quark.
Correspondingly, the dynamics are soft and no processes with momentum transfer
exceeding the quark mass take place. Correspondingly, the probability of
quark-antiquark pair creation is exponentially suppressed. This results in a
slingshot effect during which a long thick string is stretched in the wake of
the quark. The resulting deceleration process is soft. The decay of a formed
long string via pair creation is exponentially suppressed due to the usual
reasons.

This reasoning must remain applicable also for the lower masses of quarks that
are closer to the QCD scale, $M_{Q}\gtrsim\Lambda$, as long as the exponential
suppression of the string decay is maintained. That is, provided the
parameters are such that the static long string is stable against the breakup
via pair-nucleation, the relativistic quark entering the confining domain is
expected to exhibit a slingshot effect.

Just like in the monopole case, this outcome is different from what is
expected from the collision of a quark-antiquark pair connected by a string.
In this case, due to the absence of a net $U(1)$ charge, upon annihilation of
quarks, the memory about the pre-existing charge dipole is gone. The system
then chooses to hadronize in a multiplicity of glueballs and mesons rather
than to stretch a long string.

Apart from its quantum field theoretic importance, the slingshot effect with
quarks can have equally interesting cosmological implications, since the
coexistence of confined and deconfined phases are generic in the cosmological
evolution of various extensions of the standard model, such as grand
unification. The quark slingshot effect can supplement the mechanism of the
primordial black hole formation proposed in Dvali  _et al._ (2021). Now,
instead of quarks connected by a string, the black hole can form by smashing a
highly energetic quark accelerated by a slingshot into a wall. In addition,
the quark slingshot effect can be the source of gravitational waves in a way
very similar to the monopole slingshot case discussed in the previous chapter.

##  VIII Slingshot of Confined Vortexes and strings

The slingshot effect is not limited to confined point-like sources, such as
monopoles or quarks with attached strings. Both are objects of co-dimension
$3$ confined by a connector of co-dimension $2$ (string). Objects of different
co-dimensionality can exhibit the slingshot effect. In general, sources of co-
dimension $d$ are confined by co-dimension $d-1$ agents. For example, in $3+1$
dimensions, strings that have co-dimension $2$ can be confined by domain walls
which are co-dimension $1$ objects. A well-known example of such confinement
is provided by strings bounding domain walls that stretch between them Kibble
_et al._ (1982); Vilenkin and Everett (1982). Similarly, in $2+1$ dimensions,
vortices can be confined by strings Dvali (1991).

In the current section, we shall study the slingshot effect for this case. For
this, we shall extend the model of confined vortices in $2+1$ dimensions
(strings in $3+1$ dimensions) introduced in Dvali (1991), by allowing an
additional vacuum in which vortices (strings) are not confined. The
$2+1$-dimensional model of this sort has double usefulness as on the one hand
it captures the dynamics of the string slingshot in $3+1$ dimensions and on
the other hand it represents a toy version of the monopole slingshot discussed
in the previous sections.

The key concept of the model involves replacing the adjoint and fundamental
scalar fields with complex scalar fields of different charges under an Abelian
symmetry. Instead of an $SU(2)$ gauge model, we consider a $U(1)$ gauge
theory. The symmetry-breaking mechanism involves two scalar fields. The first
scalar field of charge $q_{\phi}=g$, denoted as $\phi$, is breaking the $U(1)$
symmetry down to $Z_{2}$ symmetry. This discrete symmetry is broken further by
a second scalar field of charge $q_{\chi}=\frac{g}{2}$, referred to as $\chi$.
This field changes the sign under the $Z_{2}$ transformation.

The Lagrangian governing this model is expressed as follows Dvali (1991)

$\displaystyle\mathcal{L}=$
$\displaystyle(D_{\mu}\phi)^{\ast}(D^{\mu}\phi)+(D_{\mu}\raisebox{0.0pt}{$\chi$})^{\ast}(D^{\mu}\raisebox{0.0pt}{$\chi$})$
$\displaystyle-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}-U(\phi,\raisebox{0.0pt}{$\chi$})\,,$
(39)

with the potential

$\displaystyle U(\phi,\chi)=$
$\displaystyle\lambda_{\phi}(\absolutevalue{\phi}^{2}-v_{\phi}^{2})^{2}+\lambda_{\chi}(\absolutevalue{\raisebox{0.0pt}{$\chi$}}^{2}-v_{\chi}^{2})^{2}\absolutevalue{\raisebox{0.0pt}{$\chi$}}^{2}$
$\displaystyle+\beta\phi^{\ast}\raisebox{0.0pt}{$\chi$}^{2}+c.c.\,.$ (40)

The covariant derivatives and the field strength tensor are given by

$\displaystyle F_{\mu\nu}$
$\displaystyle=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\,,$ (41)
$\displaystyle D_{\mu}\phi$
$\displaystyle=\partial_{\mu}\phi+igA_{\mu}\phi\,,$ (42) $\displaystyle
D_{\mu}\raisebox{0.0pt}{$\chi$}$
$\displaystyle=\partial_{\mu}\raisebox{0.0pt}{$\chi$}+i\frac{g}{2}A_{\mu}\raisebox{0.0pt}{$\chi$}\,.$
(43)

Again, the novelty as compared to Dvali (1991) is that the potential for the
$\chi$ field is designed in such a way that in addition to the $Z_{2}$-Higgsed
phase, in which both fields have non-zero vacuum expectation values, there
coexists a $Z_{2}$-invariant phase in which the $\chi$ field vanishes. This is
possible as long as $\absolutevalue{\beta}$ is sufficiently small. Namely, if
$|2\beta v_{\phi}|<|\lambda_{\chi}v_{\chi}^{4}|$. Notice, that for simplicity
we have omitted the phase-independent interaction terms such as
$|\phi|^{2}|\raisebox{0.0pt}{$\chi$}|^{2}$. Such terms do not play any role in
the confinement of vortices. The crucial term in this respect is the phase-
dependent interaction term with the coefficient $\beta$. This term defines the
relative charges of the two fields.

Let us now discuss the properties of vortices in these two vacua. In the
$Z_{2}$-invariant vacuum, only the $\phi$ field has a non-zero vacuum
expectation value. Its absolute value is constrained to the field and is
$\langle|\phi|\rangle=v_{\phi}$, whereas the phase degree of freedom
$\theta_{\phi}$ becomes the longitudinal component of a massive vector field
through the usual Higgs effect.

Correspondingly, the spectrum of the theory contains a Nielsen-Olesen vortex
solution, given by the ansatz Nielsen and Olesen (1973)

$\displaystyle A_{i}(r,\theta)$
$\displaystyle=\frac{n}{g}\varepsilon_{ij}\frac{r^{j}}{r^{2}}K(r)\,,$ (44)
$\displaystyle\phi(r,\theta)$ $\displaystyle=v_{\phi}e^{in\theta}H(r)\,,$ (45)

where $n$ is the winding number, and $K(r)$ and $H(r)$ are the profile
functions that we found again by a numerical relaxation method by solving the
following differential equations

$\displaystyle K^{\prime\prime}$
$\displaystyle=\frac{K^{\prime}}{r}-m_{v_{\phi}}^{2}H^{2}(1-K)\,,$ (46)
$\displaystyle H^{\prime\prime}$
$\displaystyle=-\frac{H^{\prime}}{r}+\frac{(1-K)^{2}}{r^{2}}n^{2}H+\frac{m_{h_{\phi}}^{2}}{2}H(H^{2}-1)\,.$
(47)

In the $Z_{2}$-Higgsed phase also the $\chi$ field gets a non-zero vacuum
expectation value. For a small enough $\beta$-term, its absolute value is
approximately equal to $\langle|\raisebox{0.0pt}{$\chi$}|\rangle\simeq
v_{\chi}$. Due to this, the gauge field receives a further mass contribution
$m_{v_{\chi}}=v_{\chi}g/\sqrt{2}$. The Higgs masses are approximately given by
$m_{h_{\phi}}=2\sqrt{\lambda_{\phi}}v_{\phi}$ and
$m_{h_{\chi}}=2\sqrt{\lambda_{\chi}}v_{\chi}^{2}$.

The further breaking of the $Z_{2}$ symmetry by the vacuum expectation value
of $\chi$ puts the $\phi$ vortices in the confining phase Dvali (1991). The
dominant effect is due to the interaction $\beta$-term, which is phase-
dependent. Notice that without this term, no confinement would occur.

The reason is the following. For $\beta=0$, the theory would be invariant
under two independent global symmetries $U(1)_{\chi}\times U(1)_{\phi}$ with
only one subgroup being gauged. The gauged subgroup leaves the following
combination of the phases invariant,

$\Theta\equiv\theta_{\phi}-2\theta_{\chi}\,.$ (48)

This gauge-invariant phase shifts under the additional global $U(1)$ symmetry
which emerges for $\beta=0$. The breaking of this symmetry by the combination
of the two vacuum expectation values results in the emergence of a massless
Goldstone boson. In the regime $v_{\phi}\gg v_{\chi}$, this would-be-Goldstone
boson resides mostly in the phase $\theta_{\chi}$.

Correspondingly, for $\beta=0$, the vacuum expectation value of $\chi$ would
lead to the formation of a second type vortex. Around each vortex, the two
phases can in general have independent winding numbers.

Notice that some vortices would be “semi-global” Dvali and Senjanovic (1994).
In particular, a vortex around which both fields have unit winding numbers
would have a logarithmically divergent gradient energy since the gauge field
would be unable to compensate the winding of both phases simultaneously due to
the difference in their gauge charges.

We are interested in the regime of confined vortices which takes place for
$\beta\neq 0$. First notice that, since the $\beta$-term explicitly breaks the
global $U(1)$ symmetry, the would-be-Goldstone degree of freedom gets the mass

$m_{g}^{2}\simeq\absolutevalue{4\beta v_{\phi}}\,.$ (49)

Minimization of the $\beta$-term term forces the alignment in the phases of
the $\phi$ and $\chi$ fields. For $\beta<0$, the term is minimized for

$\theta_{\phi}=2\theta_{\chi}\,.$ (50)

However, such a relationship cannot be maintained everywhere around the $\phi$
vortex with winding number one around which the phase shift is
$\Delta\theta_{\phi}=2\pi$. In the light of (50), this would imply that the
corresponding change of the phase of $\chi$ around a closed path is
$\Delta\theta_{\phi}=\pi$, which violates a single-valuedness of the vacuum
expectation values.

To avoid the conflict, the field compromises: The presence of the $\beta$-term
makes sure that around the closed contour enclosing the $\phi$ vortex the
phase of $\chi$ experiences a jump (rapid change) from $\pi$ to $2\pi$ within
a region of thickness $\sim m_{g}^{-1}$. This region represents a string that
is attached to the $\phi$ vortex.

Far away from the vortex core, the corresponding configuration for the gauge
invariant combination of the two phases (48) can be found by solving the sine
Gordon equation,

$\Theta^{\prime\prime}-m_{g}^{2}\sin(\Theta)=0\,,$ (51)

where the derivative is taken with respect to a perpendicular coordinate $y$.
This equation has a well-known solution,

$\Theta(y)=4\tan^{-1}(e^{m_{g}y})\,,$ (52)

which interpolates from $\Theta=0$ to $\Theta=2\pi$.

In the vacuum with broken $Z_{2}$ symmetry, the string can terminate on
another vortex or an antivortex, and the two get confined. In the present
case, we have a separate domain with unbroken $Z_{2}$ symmetry. This gives a
possibility for the $Z_{2}$ string to terminate on a domain wall separating
the two phases.

The domains with free and confined $\phi$ vortices are separated by a domain
wall in which the $\chi$ field interpolates from $0$ to $v_{\chi}$. This
domain wall solution can be found by fixing the $U(1)$ direction, e.g.
$\xi=\real\raisebox{0.0pt}{$\chi$}$ and solving the Bogomolny equation
Bogomolny (1976)

$\displaystyle\xi(x)=\frac{v_{\chi}}{\sqrt{1+e^{m_{h_{\chi}}x}}}\,.$ (53)

This domain wall separates the $Z_{2}$-invariant phase from the
$Z_{2}$-Higgsed phase.

We now turn to the analysis of a slingshot effect experienced by a $\phi$
vortex that passes from the $Z_{2}$-invariant domain into the Higgsed $Z_{2}$
domain. Just like in the case of a monopole, the vortex stretches a string
that connects it to the boundary of the two phases.

Since the gauge field is massive in both regions, it has no long-range
effects. Its influence vanishes exponentially at distances larger than the
Compton wavelength of the photon. Consequently, a detailed study of its
behavior in close proximity to the domain wall is unnecessary, provided we
assume an initial configuration in which the vortex is far enough away from
the wall. This implies that the ansatz for the $\phi$ and $A_{\mu}$ fields in
the numerical simulation does not require adaptation to account for the
presence of the wall. The ansatz for the $\chi$ field can be written as

$\displaystyle\raisebox{0.0pt}{$\chi$}(x,y)=\xi(x)\,e^{in\theta/2}\,,$ (54)

where $\theta=\arctan\left(y/(x-x_{0})\right)$, $x_{0}$ being the position of
the vortex. The above choice for $\chi$ minimizes the $\beta$ coupling in the
broken $Z_{2}$ phase. Moreover, ansatz (54) ensures the single-valuedness of
$\chi$ since $\xi(x)$ is vanishing in the unbroken region.

In order to conduct the simulation, we employed the same numerical methods as
described earlier. However, since our current analysis is limited to two
dimensions, the axial symmetry method is not necessary. The lattice size,
lattice spacing, time step, and the investigated time interval remained the
same as in the magnetic monopole setup. Furthermore, the boundary conditions
stay similar. Absorbing boundaries were utilized for $\phi$ and $A_{\mu}$,
while the Dirichlet boundary condition was applied to $\chi$ in the
$x$-direction, accompanied by a periodic boundary condition in the
$y$-direction. Note that the imaginary part of $\chi$ is anti-symmetric in
$y$-direction.

We set $m_{v_{\phi}}$ and $m_{h_{\phi}}$ to one and $g=1/\sqrt{2}$.
Additionally, we took the following parameter values:
$m_{v_{\chi}}=0.3m_{v_{\phi}}$, $m_{h_{\chi}}=0.8m_{v_{\phi}}$, and
$\beta=-0.01m_{v_{\phi}}^{3/2}$.

The initial distance between the vortex and the wall was chosen to be
$d=40m_{v_{\phi}}^{-1}$ and the velocities were $0.8$ and $-0.8$ for the
vortex and domain wall respectively.

From the simulation, we can observe that the vortex stretches a $Z_{2}$ string
when it enters the $Z_{2}$-Higgsed phase as can be seen in Fig. 7. The
formation happens very similarly to the magnetic monopole case. The
qualitative difference is that there is no magnetic flux inside the $Z_{2}$
string due to the short-range behavior of the vortex gauge field.

Figure 7: The scalar field vector $(\real\phi,\imaginary\phi)^{T}$ (top) and
$(\real\chi,\imaginary\chi)^{T}$ (bottom) at time $t=100m_{v_{\phi}}^{-1}$.
The length values are given in units of $m_{v_{\phi}}^{-1}$. The red line
represents the contour $\absolutevalue{\chi}=v_{\chi}/2$ and the black circle
the contour $\absolutevalue{\phi}=v_{\phi}/2$. We can observe that the vortex
that is entering the $Z_{2}$-Higgsed phase is connected to the domain wall by
a $Z_{2}$ string.

The minimization of the interaction term results in
$\raisebox{0.0pt}{$\chi$}^{2}$ being proportional to $\phi$. Given the
vortex’s winding number of one, this proportionality implies a winding of
$1/2$ in the $\chi$ field at the end of the string. Consequently, the field
vectors exhibit a rotation by $\pi$ around the string’s end. Within the
string, the phase is changing according to equation (52). This rotational
behavior explains why the $Z_{2}$ string does not detach, as a rotation by
$2\pi$ is necessary for the detaching to occur. Therefore, the formation of
the $Z_{2}$ string is purely explained by topology.

Unlike in the case of monopoles or quarks in $3+1$ dimensions, the slingshot
effect of vortices (strings) happens without the confinement of the gauge
flux. Instead, what confines within the string connecting two vortices is the
flux of gradient of the Goldstone field which in the $\beta=0$ limit becomes
uniformly distributed around the vortex resulting in $2+1$-dimensional Coulomb
interaction between them. For vortices separated by a distance $r$ the
interaction potential is $\propto\ln(r)$. If $\beta\neq 0$, for distances
$r\gg m_{g}^{-1}$, the potential is converted into a linear confining
potential $\propto r$.

Again, we can add an antivortex that enters the string later, leading to the
breaking of the string and subsequent annihilation of the vortex-pair.
However, the $2+1$-dimensional model possesses a distinctive feature not
present in the magnetic monopole model. In this case, it is possible for a
second vortex to enter the system instead of an antivortex. As a result, the
string breaks, causing the two vortices to be drawn together until they form a
bound state. This bound state exhibits a winding number of two in the $\phi$
field and a winding number of one in the $\chi$ field.

During the collision, we observe that the two vortices scatter at an angle of
$\pi/2$. This scattering behavior has been previously explained and analyzed
using the moduli space approximation in Myers  _et al._ (1992); Shellard and
Ruback (1988). Due to the binding effect of the $\chi$ field on the two
vortices, this right-angle scattering occurs repeatedly. In Fig. 8 two moments
of this bound state are illustrated.

Figure 8: The scalar field vector $(\real\phi,\imaginary\phi)^{T}$ at times
$t=120m_{v_{\phi}}^{-1}$ and $t=180m_{v_{\phi}}^{-1}$. The frames show moments
after two vortices of the same winding entered one after one the
$Z_{2}$-Higgsed phase. We observe that the two vortices are connected by a
string and form an oscillating bound state.

The results of the simulations can be also found in the video attached as an
ancillary file or at the following link:
https://youtu.be/IPJAPjo3nSc?si

The behavior observed in the $2+1$-dimensional model can be seamlessly
extended to the three-dimensional case. The $\phi$-vortices are lifted in
strings that extend in an additional dimension. Furthermore, the $Z_{2}$
string that in $2+1$ confines vortices, in $3+1$ is lifted into a domain wall
that confines strings.

In a manner analogous to the connected vortex-vortex pair and vortex-
antivortex pair, we can now have a string-string pair and string-antistring
pair connected by a domain wall. Within the string-string scenario, the
entities align to create a bound state, adopting a cable-like configuration
characterized by identical right-angle oscillations as witnessed in the case
of vortices. In the string-antistring case, however, they will annihilate.

Just like in the monopole/quarks case, the string/vortex slingshot effect can
have cosmological implications as it is expected to take place in various
extensions of the standard model.

##  IX Conclusion and Outlook

In this paper, we introduced and numerically studied the slingshot effect and
its implications such as gravitational waves.

In the first example, we studied the scattering process in which a magnetic
monopole crosses a domain wall separating the vacua of magnetic-Coulomb and
magnetic-confining phases. The setup is achieved by variants of an
$SU(2)$-symmetric model with coexisting phases of the type discussed earlier
Dvali and Vilenkin (2003). It possesses two vacuum states. In one of them, the
$SU(2)$ is Higgsed down to $U(1)$ and the spectrum contains ’t Hooft-Polyakov
monopoles. In the neighboring vacuum, the $U(1)$ symmetry is further Higgsed
and the photon has a non-zero mass. In this vacuum, the monopoles can only
exist in a confined form. The magnetic flux of the monopole is trapped in a
tube, the Nielsen-Olesen string Nielsen and Olesen (1973). The two vacua are
separated by a domain wall.

We study a process in which the monopole with a high kinetic energy crosses
over from the $U(1)$ Coulomb phase into the Higgs phase. We observe that upon
entering the $U(1)$ Higgs domain, the monopole becomes connected to the wall
by a long string. The string carries the magnetic flux of the monopole which
opens up on the other side of the wall in the form of the Coulomb-magnetic
field.

Despite the fact that the conservation laws permit the disposal of the kinetic
energy of the monopole in the form of waves, without stretching a long string,
this does not happen. Instead, the system creates a string that follows the
monopole. The string tension tends to pull the monopole back towards the wall,
exhibiting a sort of slingshot effect.

This outcome is different from the previously studied case Dvali  _et al._
(2023) of scattering of a monopole-antimonopole pair connected by a string. In
the point-like approximation, which does not resolve the structure of
monopoles and strings, one cannot exclude that monopoles pass through each
other, oscillating and re-stretching the string multiple times Martin and
Vilenkin (1997). However, the simulation of the fully resolved system Dvali
_et al._ (2023) showed that in a head-on collision, the monopoles never pass
through each other. Instead, they decay into waves.

In Dvali  _et al._ (2023) this behavior was explained by the following
factors. First when the monopole and antimonopole overlap, they effectively
erase the magnetic charges of each other and the system forgets about the
magnetic dipole. Also, as in the generic cases of the erasure of defects Dvali
_et al._ (1998), the coherence is lost. From this point on, the system evolves
into the highest entropy configuration which is given by the waves, as opposed
to monopoles connected by a long string. The latter configuration carries much
lower entropy. The outcome can be interpreted as a particular case of a
generic phenomenon the essence of which is an exponential suppression of the
creation of the low-entropy macroscopic objects in collision processes Dvali
(2021); Dvali and Eisemann (2022).

We explained that in the present case, the situation is very different due to
the conservation of the net magnetic charge and the softness of the monopole-
wall collision. At no point, the monopole encounters a phase in which the
expectation value of the adjoint Higgs vanishes. Therefore, neither the
coherence nor the memory of the preexisting state is lost. The monopole, due
to its high kinetic energy, enters the confining phase softly and its magnetic
flux stretches in the form of a string.

We argued that similarly to the earlier discussed analogy between confined
quarks and monopoles Dvali  _et al._ (2023), the current behavior must also be
shared by a dual QCD-like theory.

In order to make the mapping more precise, as an electric dual version of the
present model, we have used the construction analogous to Dvali and Vilenkin
(2003); Dvali  _et al._ (2008). This gauge theory represents the $SU(2)$ QCD
which possesses two vacua. In one vacuum, the theory confines at a scale
$\Lambda$, and quarks are connected by the QCD flux tubes. In the other
vacuum, $SU(2)$ is Higgsed down to $U(1)$ and the theory is in the $U(1)$
Coulomb phase mediated by a massless photon. The theory possesses a domain
wall separating the two phases. Due to the mass gap $\Lambda$ in the confining
vacuum, the massless photon is repelled from there and is localized within the
Coulomb vacuum via the dual-Meissner mechanism of Dvali and Shifman (1997).

In analogy with the monopole case, we consider a scattering process in which
an energetic heavy quark crosses over from the $U(1)$ Coulomb domain into the
$SU(2)$-confining one. We argued that the same behavior is expected as in the
case of a monopole in the dual theory. That is, upon entering the confining
phase, the quark will softly stretch a long QCD string. The string transports
the electric flux of the quark to the wall and spreads it out in the other
domain in the form of the $U(1)$ Coulomb field. This should be the likely
outcome as opposed to hadronizing into a high multiplicity of mesons and
glueballs.

Our reasoning is the same as in the monopole case. The conservation of the
$U(1)$ charge forces the system to maintain the quark. The creation of
additional quark-antiquark pairs that would break the string, requires
collisions with high momentum transfer. These are absent since the quark-wall
collision is soft. Correspondingly, the system chooses the QCD slingshot
effect as the likely outcome.

Our results have a number of implications. First, as discussed, it allows us
to capture certain important parallels between the behaviors of confined
monopoles and quarks. In particular, in processes involving traversing the
domain walls between confined and deconfined phases, both are expected to
exhibit the slingshot effects.

This effect can also have a number of important cosmological consequences
since the above phases are expected to coexist at several stages of the
universe’s evolution. One observable imprint can occur in the form of
gravitational waves. Within this paper, we scrutinized the energy spectrum and
emission direction of radiation from the slingshot scenario. Our observations
reveal that the spectrum exhibits an $\omega^{-1}$ trend within our region of
parameter space, akin to the behavior arising from the evolution of a cosmic
string connecting a magnetic monopole and antimonopole Martin and Vilenkin
(1997); Dvali  _et al._ (2023). Moreover, the emission takes place in a
beaming angle in the direction of acceleration and scales, as a function of
frequency as $\theta\propto\omega^{-1/2}$ within our range of parameters.

In the last part of this work, we investigated the slingshot effect for the
case of a vortex in $2+1$ dimensions, which can be extended to a theory with a
cosmic string in $3+1$ dimensions that are confined by domain walls.

Considering that cosmic strings are objects that can occur during a phase
transition in the early universe, this scenario may also leave relevant marks
in the gravitational wave background. Further explorations into this direction
are left for future studies.

## Acknowledgements

This work was supported in part by the Humboldt Foundation under Humboldt
Professorship Award, by the European Research Council Gravities Horizon Grant
AO number: 850 173-6, by the Deutsche Forschungsgemeinschaft (DFG, German
Research Foundation) under Germany’s Excellence Strategy - EXC-2111 -
390814868, and Germany’s Excellence Strategy under Excellence Cluster Origins.
Disclaimer: Funded by the European Union. Views and opinions expressed are
however those of the authors only and do not necessarily reflect those of the
European Union or European Research Council. Neither the European Union nor
the granting authority can be held responsible for them.

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