††institutetext:  ∗Physics Department, Arizona State University, Tempe,
Arizona 85287, USA.
†Theoretical Physics Department, CERN, 1211 Geneva 23, Switzerland.

# Annihilation of electroweak dumbbells

Teerthal Patel∗ Tanmay Vachaspati∗,†<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We study the annihilation of electroweak dumbbells and the dependence of their
dynamics on initial dumbbell length and twist. Untwisted dumbbells decay
rapidly while maximally twisted dumbbells collapse to form a compact
sphaleron-like object, before decaying into radiation. The decay products of a
dumbbell include electromagnetic magnetic fields with energy that is a few
percent of the initial energy. The magnetic field from the decay of twisted
dumbbells carries magnetic helicity with magnitude that depends on the twist,
and handedness that depends on the decay pathway.

CERN-TH-2023-200

##  1 Introduction

The “electroweak dumbbell” consists of a magnetic monopole and an antimonopole
of the standard electroweak model connected by a string made of $Z$ magnetic
field Nambu:1977ag ; Achucarro:1999it . The existence of such non-perturbative
field configurations in the electroweak model is of great interest as they can
provide the first evidence for (confined) magnetic monopoles. In a
cosmological context, dumbbells can source large-scale magnetic fields which
can seed galactic magnetic fields and play an important role in the
propagation of cosmic rays Vachaspati:2020blt .

Electroweak dumbbells are often regarded as magnetic dipoles, with the
magnetic field strength falling off as $1/r^{3}$ with the distance $r$ from
the dipole. However the situation is richer: there is a one-parameter set of
electroweak dumbbell configurations VachaspaticandField , all describing a
confined monopole-antimonopole pair, but with additional structure called the
“twist”. In our previous work Patel:2023sfm , we have shown that the magnetic
field strength of the maximally twisted dumbbell falls off asymptotically as
$\cos\theta/r^{2}$ (in spherical coordinates), a gross departure from the
usual dipolar magnetic field. The twisted dumbbell configuration was proposed
in VachaspaticandField  and is closely related to the electroweak sphaleron
Manton:1983nd ; Klinkhamer:1984di ; PhysRevD.46.3587 ; Akiba:1989xu .

The dynamics of electroweak sphalerons and dumbbells are of particular
interest in the efforts towards detecting them in experiments. In view of
ongoing experiments like the Monopole and Exotics Detector (MoEDAL) at the
Large Hadron Collider (LHC) Acharya_2022 , Ref. Ho_2020  recently studied the
production of the electroweak sphaleron in the presence of strong magnetic
fields that arise during heavy ion collisions. In the cosmological context,
simulations of the dynamical decay of electroweak sphalerons have been
conducted to study baryogenesis and magnetogenesis Copi_2008 ; Chu_2011 .
Thus, there have been several efforts to numerically resolve the configuration
and dynamics of electroweak sphalerons. The relevance of dynamics of
electroweak dumbbells was first alluded to by Nambu Nambu:1977ag  wherein he
showed that electroweak dumbbells could be stabilized by rotation, potentially
making them long lived enough to have significant implications in experimental
searches. This, along with a lack of detailed study of the dynamics of
electroweak dumbbells, has motivated our investigation into the dumbbell
configurations and their dynamics.

In this article, we investigate the dynamical evolution of an initially
stationary dumbbell configuration for a range of twists and initial dumbbell
lengths. The initial condition is found by numerically relaxing a “guess”
dumbbell configuration under the constraint that locations of the monopole and
antimonopole remain fixed, according to the method outlined in our previous
work Patel:2023sfm . The main quantities of interest that we analyze are the
dumbbell lifetimes and the magnetic field produced during the decay. The
untwisted dumbbells are found to be unstable, with the monopoles immediately
undergoing annihilation as expected. Twisted dumbbells, on the other hand,
lead to the creation of an intermediate sphaleron configuration after the
initial collapse, and subsequently decay into helical magnetic fields with
relatively stronger field strength.

The article begins with a description of the model and our initial
configuration in Section 2. The numerical simulation setup is described in
Section 3. Our results for the lifetimes and the relic magnetic fields are
given in Section 4, and we discuss our conclusions in Section 5.

##  2 Model

The Lagrangian for the bosonic sector of the electroweak theory is given by

$\mathcal{L}=-\frac{1}{4}W^{a}_{\mu\nu}W^{a\,\mu\nu}-\frac{1}{4}Y_{\mu\nu}Y^{\mu\nu}+|D_{\mu}\Phi|^{2}-\lambda(|\Phi|^{2}-\eta^{2})^{2}\,,$
(1)

where

$D_{\mu}\equiv\partial_{\mu}-\frac{i}{2}g\sigma^{a}W^{a}_{\mu}-\frac{i}{2}g^{\prime}Y_{\mu}\,.$
(2)

Here, $\Phi$ is the Higgs doublet, $W^{a}_{\mu}$ are the SU(2)-valued gauge
fields with $a=1,2,3$ and, $Y_{\mu}$ is the U(1) hypercharge gauge field. In
addition, $\sigma^{a}$ are the Pauli spin matrices with ${\rm
Tr}(\sigma^{a}\sigma^{b})=2\delta_{ab}$, and the experimentally measured
values of the parameters that we adopt from Particledatagroup2022  are
$g=0.65$, $\sin^{2}\theta_{w}=0.22$, $g^{\prime}=g\tan\theta_{w}=0.35$,
$\lambda=0.129$ and $\eta=174\,{\text{GeV}}$.

The classical Euler-Lagrange equations of motion for the model are given by

$\displaystyle D_{\mu}D^{\mu}\Phi+2\lambda(|\Phi|^{2}-\eta^{2})\Phi=0$ (3)
$\displaystyle\partial_{\mu}Y^{\mu\nu}=g^{\prime}\,{\rm
Im}[\Phi^{\dagger}(D^{\nu}\Phi)]$ (4)
$\displaystyle\partial_{\mu}W^{a\mu\nu}+g\epsilon^{abc}W^{b}_{\mu}W^{c\mu\nu}=g\,{\rm
Im}[\Phi^{\dagger}\sigma^{a}(D^{\nu}\Phi)]$ (5)

where the gauge field strengths are given by

$\displaystyle W^{a}_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}W^{a}_{\nu}-\partial_{\nu}W^{a}_{\mu}+g\epsilon^{abc}W^{b}_{\mu}W^{c}_{\nu}$
(6) $\displaystyle Y_{\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}Y_{\nu}-\partial_{\nu}Y_{\mu}\,.$ (7)

Electroweak symmetry breaking results in three massive gauge fields, namely
the two charged $W^{\pm}$ bosons and $Z_{\mu}$, and one massless gauge field,
$A_{\mu}$, that is the electromagnetic gauge field. We define

$\displaystyle Z_{\mu}$ $\displaystyle\equiv$
$\displaystyle\cos\theta_{w}n^{a}W^{a}_{\mu}+\sin\theta_{w}Y_{\mu},$ (8)
$\displaystyle A_{\mu}$ $\displaystyle\equiv$
$\displaystyle-\sin\theta_{w}n^{a}W^{a}_{\mu}+\cos\theta_{w}Y_{\mu}\,,$ (9)

where

$n^{a}\equiv\frac{\Phi^{\dagger}\sigma^{a}\Phi}{|\Phi|^{2}}\,$ (10)

are components of a unit three vector $\hat{n}$. The weak mixing angle,
$\theta_{w}$ is given by $\tan{\theta_{w}}=g^{\prime}/g$, the $Z$ coupling is
defined as $g_{Z}\equiv\sqrt{g^{2}+g^{\prime 2}}$, and the electric charge is
given by $e=g_{Z}\sin\theta_{w}\cos\theta_{w}$. The Higgs, $Z$ and $W$ boson
masses are given by $m_{H}\equiv 2\sqrt{\lambda}\eta=125\,{\text{GeV}}$,
$m_{Z}\equiv g_{Z}\eta/\sqrt{2}=91\,{\text{GeV}}$ and $m_{W}\equiv
g\eta/\sqrt{2}=80\,{\text{GeV}}$, respectively.

###  2.1 Initial dumbbell configuration

Figure 1: The location of the monopole (blue dot), antimonopole (red dot) and
the $Z$-string (purple line) in a cartesian grid.

We construct a suitable initial configuration for dumbbell dynamics by first
choosing a “guess” field configuration that contains a monopole and
antimonopole separated by a distance $2d$ and with relative twist $\gamma$
Achucarro:1999it . In the asymptotic region,

${{{{\hat{\Phi}}_{{m}{\bar{m}}}}}(\gamma)}=\left(\begin{array}[]{c}\sin(\theta_{m}/2)\sin(\theta_{\bar{m}}/2)e^{i\gamma}+\cos(\theta_{m}/2)\cos(\theta_{\bar{m}}/2)\\\
\sin(\theta_{m}/2)\cos(\theta_{\bar{m}}/2)e^{i\phi}-\cos(\theta_{m}/2)\sin(\theta_{\bar{m}}/2)e^{i(\phi-\gamma)}\end{array}\right)\,.$
(11)

The monopole and antimonpole are located along the z-axis at $z=\pm d$, with
$\theta_{m}$ and $\theta_{\bar{{m}}}$ being the spherical polar angles as
measured from the monopole and antimonpole, respectively; $\phi$ is the
azimuthal angle. The coordinate system is illustrated in Figure 1. In the
limit $\theta_{\bar{{m}}}\to 0$, (11) reduces to the monopole configuration,
and in the limit $\theta_{m}\to 0$ to an antimonopole. In between the monopole
and antimonopole, $\theta_{m}\to\pi$ and $\theta_{\bar{{m}}}\to 0$, the
configuration is that of a $Z-$string.

The asymptotic gauge field configurations are obtained by setting the
covariant derivative of the Higgs field to vanish,

$\displaystyle gW^{a}_{\mu}$ $\displaystyle=$
$\displaystyle-\epsilon^{abc}n^{b}{\partial_{\mu}n^{c}}+i\cos^{2}\theta_{w}n^{a}(\Phi^{\dagger}\partial_{\mu}\Phi-\partial_{\mu}\Phi^{\dagger}\Phi)$
(12) $\displaystyle g^{\prime}Y_{\mu}$ $\displaystyle=$
$\displaystyle-i\sin^{2}\theta_{w}(\Phi^{\dagger}\partial_{\mu}\Phi-\partial_{\mu}\Phi^{\dagger}\Phi)$
(13)

To correctly account for the spatial dependence of the Higgs field around the
monopole-antimonopole pair, we attach spatial profiles. Including this in the
ansatz, the initial monpole-antimonopole scalar field configuration is given
by

${{\Phi_{m{\bar{m}}}}}=k({\vec{r}}\,)h(r_{m})h(r_{{\bar{{m}}}}){{{\hat{\Phi}}_{{m}{\bar{m}}}}}\,,$
(14)

where $r_{m}$ and $r_{\bar{{m}}}$ are the radial coordinates centered on the
monopole and antimonpole, respectively, given by

$r_{m}=|{\mathbf{x}}-\mathbf{x}_{m}|,\,\,\,\,r_{\bar{m}}=|\mathbf{x}-\mathbf{x}_{\bar{m}}|\,.$
(15)

The function $k({\vec{r}}\,)$ is the Z-string profile. Similar to the Higgs,
we include spatial profiles for the gauge fields as

$\displaystyle gW^{a}_{\mu}$ $\displaystyle=$ $\displaystyle
l({\vec{r}}\,)j(r_{m})j(r_{\bar{{m}}})[-\epsilon^{abc}n^{b}{\partial_{\mu}n^{c}}+i\cos^{2}\theta_{w}n^{a}({\hat{\Phi}}^{\dagger}\partial_{\mu}{\hat{\Phi}}-\partial_{\mu}{\hat{\Phi}}^{\dagger}{\hat{\Phi}})]$
(16) $\displaystyle g^{\prime}Y_{\mu}$ $\displaystyle=$ $\displaystyle
l({\vec{r}}\,)j(r_{m})j(r_{\bar{{m}}})[-i\sin^{2}\theta_{w}({\hat{\Phi}}^{\dagger}\partial_{\mu}{\hat{\Phi}}-\partial_{\mu}{\hat{\Phi}}^{\dagger}{\hat{\Phi}})]$
(17)

We have previously used numerical relaxation to solve for the Higgs and gauge
field profile of a static dumbbell in with the constraint that the topology of
the monopole and antimonopole remain fixed during the relaxation process
Patel:2023sfm . We use the same procedure to find the initial field
configurations that we will then evolve to study dumbbell dynamics.

##  3 Numerical simulation

We have used a numerical relativity technique adapted from 2010nure.book…..B ,
and previously used in Vachaspati:2015ahr  to study monopole-antimonopole
scattering in the SO(3) model. Adopting the temporal gauge for convenience in
numerical implementation, $W^{a}_{0}=Y_{0}=0$, the evolution equations (3)-(5)
can be written as

$\displaystyle\partial_{0}^{2}\Phi$ $\displaystyle=$ $\displaystyle
D_{i}D_{i}\Phi-2\lambda(|\Phi|^{2}-\eta^{2})\Phi$ (18)
$\displaystyle\partial_{0}^{2}Y_{i}$ $\displaystyle=$
$\displaystyle-\partial_{j}Y_{ij}+g^{\prime}\,{\rm
Im}[\Phi^{\dagger}(D_{i}\Phi)]$ (19) $\displaystyle\partial_{0}^{2}W^{a}_{i}$
$\displaystyle=$
$\displaystyle-\partial_{j}W^{a}_{ij}-g\epsilon^{abc}W^{b}_{j}W^{c}_{ij}+g{\rm
Im}[\Phi^{\dagger}\sigma^{a}(D_{i}\Phi)]\,.$ (20)

Straightforward discretization of the evolution equations leads to numerical
instabilities. To control the instabilities, we introduce “Gauss constraint
variables” $\Xi=\partial_{i}Y_{i}$ and $\Gamma^{a}=\partial_{i}W^{a}_{i}$,
with their respective evolution equations

$\displaystyle\partial_{0}\Xi$ $\displaystyle=$
$\displaystyle\partial_{0}Y_{0i}-g_{p}^{2}\\{\partial_{0}Y_{0i}-g^{\prime}\,{\rm
Im}[\Phi^{\dagger}(\partial_{0}\Phi)]\\}$ (21)
$\displaystyle\partial_{0}\Gamma^{a}$ $\displaystyle=$
$\displaystyle\partial_{0}W^{a}_{0i}-g_{p}^{2}\\{\partial_{0}W^{a}_{0i}+g\epsilon^{abc}W^{b}_{i}\partial_{0}W^{c}_{i}-g{\rm
Im}[\Phi^{\dagger}\sigma^{a}(\partial_{0}\Phi)]\\}\,.$ (22)

Here, we have introduced the numerical stability parameter $g_{p}^{2}$. These
equations reduce to the Gauss constraints in the continuum, regardless of the
choice of $g_{p}^{2}$. However, once the system of equations is discretized
for numerical evolution, the term in the curly brackets do not always vanish,
and a non-zero value of $g_{p}^{2}$ ensures numerical stability as outlined in
2010nure.book…..B . The equations in (18)-(20) are now written with the Gauss
constraint variables as

$\displaystyle\partial_{0}^{2}\Phi$ $\displaystyle=$
$\displaystyle\nabla^{2}\Phi-i\frac{g}{2}\sigma^{a}W^{a}_{i}\partial_{i}\Phi-i\frac{g^{\prime}}{2}Y_{i}\partial_{i}\Phi-i\frac{g}{2}\sigma^{a}\Gamma^{a}\Phi-i\frac{g^{\prime}}{2}\Xi\Phi$
(23) $\displaystyle\hskip
14.22636pt-i\frac{g}{2}\sigma^{a}W^{a}_{i}(D_{i}\Phi)-i\frac{g^{\prime}}{2}Y_{i}(D_{i}\Phi)-2\lambda(|\Phi|^{2}-\eta^{2})^{2}\Phi$
$\displaystyle\partial_{0}^{2}Y_{i}$ $\displaystyle=$
$\displaystyle\nabla^{2}Y_{i}-\partial_{i}\Xi+g^{\prime}{\rm
Im}[\Phi^{\dagger}\sigma^{a}(D_{i}\Phi)]$ (24)
$\displaystyle\partial^{2}_{0}W^{a}_{i}$ $\displaystyle=$
$\displaystyle\nabla^{2}W^{a}_{i}+\partial_{i}\Gamma^{a}-g\epsilon^{abc}(\partial_{k}W^{b}_{i})W^{c}_{k}-g\epsilon^{abc}W^{b}_{i}\Gamma^{c}$
(25) $\displaystyle\hskip 14.22636pt-g\epsilon^{abc}W^{b}_{k}W^{c}_{ik}+g{\rm
Im}[\Phi^{\dagger}\sigma^{a}(D_{i}\Phi)]$

Our simulations are conducted on a regular cubic lattice with Dirichlet
boundary conditions and the fields are evolved in time using the explicit
Crank-Nicholson method with two iterations Teukolsky:1999rm . We adopted
phenomenological values for the electroweak model as given in Sec. 2. We
choose to work in units of $\eta$ and set its numerical value to $\eta=1$ in
our simulations. Then the Higgs mass of 125 GeV in these units is given by
$2\eta\sqrt{\lambda}=2\sqrt{\lambda}=0.72$. Similarly, the $W$ boson mass is
${m}_{W}=g\eta/\sqrt{2}=0.46$. For most of our runs, we use a lattice of size
$400^{3}$ with lattice spacing $\delta=0.25\eta^{-1}=0.25$, and time step
$dt=\delta/8$. The Compton radius of the $W$ boson is $m_{W}^{-1}=2.17$. This
is also approximately the radius of the monopole and string in the dumbbell,
implying that their profiles have a resolution of roughly 10 grid points in
our setup.

The Higgs field vanishes at the centers of the monopole, antimonopole and
$Z-$string. Since the initial dumbbell profile involves delicate cancellation
of zeros on the dumbbell, we offset the dumbbell location away from the
$z-$axis by half a lattice spacing. That is, the monopole and antimonopole are
at the coordinates $x=y=\delta/2,z=\pm(d+\delta/2)$, while the $Z-$string is
located at $x=y=\delta/2$ and parallel to the $z-$axis.

To ensure that the Dirichlet boundary conditions do not significantly affect
the dynamics of the annihilating dumbbell, we only consider initial dumbbell
lengths $2d$ that are sufficiently smaller than the lattice size. The maximum
separation considered in our simulations was slightly less than half the
lattice width and we ensure that the Higgs field is close to the vacuum
expectation value near the boundaries $|\Phi|\approx 1$. We run our
simulations for $\sim(5-10)\times T_{c}$, where $T_{c}$ is the dumbbell
lifetime, to study the relic energies in the different fields. We ensure that
there are no significant effects on the dumbbell dynamics due to reflections
from the Dirichlet boundary conditions in the time span considered here. We
have tested this by varying the lattice spacing $\delta$ and the total
simulation box size. Thus we ensure that our choice of numerical parameters
and boundary conditions do not affect the dumbbell dynamics.

##  4 Results

We ran the simulation for a range of initial dumbbell lengths and twists. We
show several snapshots of the energy density in the $xz$-plane for the
untwisted case ($\gamma=0$) in Figure 2, and for the twisted case
($\gamma=\pi$) in Figure 3. As can be inferred from these slices, the
untwisted dumbbell promptly undergoes annihilation. However, the twisted
dumbbell forms an intermediate object that appears as an over-density in the
center, before eventually decaying away. We will discuss the relevance of this
object (most likely the electroweak sphaleron, as discussed in Ref.
VachaspaticandField ) in the context of the dumbbell lifetimes and the relic
magnetic energies in the following sections.

Figure 2: The snapshots of energy density in $xz$-plane for $\gamma=0$ are
shown for times $0dt$, $1000dt$ and $5000dt$ in the left, middle and right
panels, respectively. The colors represent the energy density and the
corresponding values are given by the scale, in units of $\eta^{4}$. Here, the
simulation parameters are $dt=\delta/100$, with $\delta=0.25\eta^{-1}$, and
the initial dumbbell length is $2d=160\delta$. The horizontal ($z$) and
vertical ($x$) axis are in lattice units.  Figure 3: The energy density
snapshots, similar to Figure 2, for the twisted case ($\gamma=\pi$) and times
$0dt$, $5000dt$ and $10000dt$ in the left, middle and right panels,
respectively.

###  4.1 Estimating lifetimes

Figure 4: The evolution of ${\rm Min}[|\Phi|]$ in the simulation box for
twists $\gamma=0$ (left panel) and $\pi$ (right panel). The line colors
correspond to different values of dumbbell lengths expressed as a multiple of
the approximate monopole width $m_{W}^{-1}$. The hollow circles correspond to
the data points from the simulation and the solid curves are $4^{\rm th}$
order polynomial fits around the threshold ${\rm Min}[|\Phi|]=0.25$. We show
our threshold choice, ${\rm Min}[|\Phi|]=0.25$, as a dotted horizontal line.

The magnetic monopoles are zeros in the Higgs field and we leverage this to
estimate the lifetime of the dumbbell by tracking the zeros and finding when
they disappear in the simulation box. Since the dumbbell is offset by
$\delta/2$ in the positive $x$ and $y$ directions in our setup, the zeros of
the Higgs field are never located at lattice points. We instead borrow the
approach from Vachaspati:2015ahr  used in studying the creation of monopoles
via classical scattering. We track the evolution of the minimum value of the
Higgs field over the entire lattice, ${\rm Min}[|\Phi|]$. Once ${\rm
Min}[|\Phi|]$ exceeds a threshold, we tag the timestep in our simulation as
the dumbbell lifetime. The lifetime obtained by this criterion depends on the
chosen value of the threshold. As we will see, the result is sensitive to the
threshold in the untwisted case but is quite insensitive in the twisted case.
Additionally, we have tested the dependence of lifetimes on the spatial and
time resolution of the simulation and demonstrated that there were no
significant dependencies on the choice of our simulation parameters.

Figure 5: The lifetimes $T_{c}$ of the dumbbells as a function of their
initial separation are shown for twists $\gamma=0$ and $\pi$ in the left and
right panels, respectively.

In Figure 4, we show the evolution of ${\rm Min}[|\Phi|]$ for various values
of initial separation $2d$. Comparing the untwisted (left panel) and twisted
(right panel) cases, respectively, we see that the curves for the untwisted
case rise slowly, implying greater sensitivity of the dumbbell lifetime to the
chosen threshold. In the twisted case, the curves rise very sharply and the
dumbbell lifetime is not sensitive to our choice of threshold. After tagging
the timestep when the threshold condition on ${\rm Min}[|\Phi|]$ is satisfied
in our simulations, we interpolate the time evolution of ${\rm Min}[|\Phi|]$
in a small range around the tagged timestep. The interpolation was performed
via a fourth order polynomial curve fit, and we evaluate the interpolated
function at the threshold value (${\rm Min}[|\Phi|]=0.25$) to find the
lifetime. The interpolated ${\rm Min}[|\Phi|]$ are shown in Figure 4 as solid
lines.

We plot the dumbbell lifetime against the initial dumbbell length for
untwisted and maximally twisted dumbbells in Figure 5. We find that the
lifetime grows with initial length but appears to saturate beyond a certain
length. To check the saturation we have performed one run for very long
dumbbells ($2dm_{W}=55.2$). These runs are computationally very expensive. The
saturation is clearer in the case of twisted dumbbells. We interpret these
plots recalling the instability analysis of Z-strings James:1992wb ;
Goodband:1995he : the untwisted dumbbell decays on a timescale $\sim
m_{W}^{-1}$ due to the instability, while the twisted dumbbell survives about
10 times longer, first collapsing to an electroweak sphaleron that eventually
decays due to its instability on a longer timescale. To verify that the
instability is dynamical and not a result of numerical artifacts, we followed
Achucarro:1999it ; Urrestilla:2001dd  and ran test simulations with large
values of $\sin^{2}\theta_{w}$ and small values of $m_{H}/m_{Z}$ for which
Z-strings are known to be stable James:1992wb ; Goodband:1995he . The results
of these simulations are given in Appendix A. In contrast to the simulations
with physical parameters, the Z-string instability is absent and the dynamics
of the monopoles is clearly visible in these “semilocal” simulations.

###  4.2 Relic Magnetic Fields

Figure 6:  The total magnetic energy as a function of time for twists
$\gamma=0$ and $\pi$ in the left and right panels, respectively. The time
along the horizontal axis is shifted by the lifetime $T_{c}$ and $t=T_{c}$ is
shown by a vertical dotted line. Different colored curves correspond to
different initial dumbbell lengths, as stated in the legends.

The definition for the electromagnetic field strength tensor in the symmetry
broken phase ($|\Phi|=\eta$) that reduces to the usual Maxwell definition in
unitary gauge is tHooft:1974kcl ; Vachaspati:1991nm ,

$\displaystyle A_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\sin\theta_{w}n^{a}W^{a}_{\mu\nu}+\cos\theta_{w}Y_{\mu\nu}-i\frac{2\sin\theta_{w}}{g\eta^{2}}(D_{\mu}\Phi^{\dagger}D_{\nu}\Phi-
D_{\nu}\Phi^{\dagger}D_{\mu}\Phi)$ (26) $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}-i\frac{2\sin\theta_{w}}{g\eta^{2}}(\partial_{\mu}\Phi^{\dagger}\partial_{\nu}\Phi-\partial_{\nu}\Phi^{\dagger}\partial_{\mu}\Phi)\,.$

This definition implies the presence of non-zero electromagnetic fields even
for $A_{\mu}=0$ due to the Higgs gradient term. In our study of the decay of
dumbbells, $\partial_{\mu}\Phi$ vanishes at late times and then the expression
in (26) agrees with the Maxwell definition.

The total magnetic field energy is given by

$E_{M}(t)=\frac{1}{2}\int d^{3}x\,B^{2}\,.$ (27)

We show the time evolution of magnetic energy $E_{M}(t)$ for twisted and
untwisted dumbbells of various initial lengths in Figure 6. After an initial
phase of annihilation, the total magnetic field energy reaches a steady value
that depends on the initial length of the dumbbell. An important observation
is that the relic magnetic energy depends on the twist, in addition to the
initial dumbbell lengths. In Figure 7 we plot the fractional magnetic field
energy at a late time after annihilation $T_{c}+\Delta t$, where we have
chosen $\Delta t=25\eta^{-1}$. Since we use Dirichlet boundary conditions, the
simulation time has a upper bound, after which reflections occurring at the
boundaries would affect quantities of interest. The time at which we evaluate
the relic magnetic field energy, $T_{c}+\Delta t$ is large enough such that
the relic magnetic energy has reached an asymptote but is still smaller than
the time at which a significant fraction of the energy is reflected. From
Figure 7, we infer the fractional magnetic energy, after annihilation, is
about twice as large for the twisted case when compared to the untwisted case,
for the same initial (large) dumbbell length.

Figure 7:  The magnetic energy at a late time, $E_{M}(T_{c}+25)$, as a
function of initial dumbbell length. The different colors correspond to
different values of twists $\gamma$ as stated in the legend.

In addition to the magnetic energy density, we are interested in the helicity
of the magnetic field, which has been shown to have significant implications
for cosmic baryogenesis and magnetogenesis Vachaspati_2001 ;
VachaspaticandField ; Vachaspati:2020blt . The total magnetic helicity is
defined as

$H_{M}(t)=\int d^{3}x\,{\bf A}\cdot{\bf B}\,,$ (28)

where ${\bf A}$ is given by the spatial components of the vector potential (9)
and ${\bf B}$ is derived from the EM tensor (26).

Figure 8:  In the left panel, the magnetic helicity $H_{M}(t)$ (blue curve),
and the physical helicity $H_{B}(t)$ (orange dashed curve) are shown for the
twisted case $\gamma=\pi$, with initial dumbbell length $2d=80\delta$. Here,
the annihilation time $T_{c}$ is marked with a black dotted vertical line. In
the right panel, we plot the physical helicity at late time,
$H_{B}(T_{c}+25\eta^{-1})$, as a function of the initial dumbbell length, with
different colors corresponding to the twist values shown in the legend.

The evolution of total magnetic helicity for an initially twisted dumbbell is
shown by the blue curve in the left panel of Figure 8. It is clear that the
helicity does not approach a specific value within the span of our simulation.
A possible explanation for the behavior is that the definition of $H_{M}$ is
gauge independent only if the magnetic field is orthogonal to the areal
vectors everywhere on the boundaries of the simulation domain. (Alternately,
if the magnetic field vanishes on the boundaries.) This is certainly not true
in our simulations. Hence we cannot assign physical meaning to our evaluation
of $H_{M}$ at late times when the magnetic field is not small at the
boundaries. Nonetheless, we can infer that the relic magnetic field has
significant net helicity when compared to the untwisted case, which had a
helicity $H\sim 10^{-13}$, consistent with numerical roundoff.

An alternative measure of the parity violation in the magnetic field is
provided by the “physical helicity” defined as,

$H_{B}=\int d^{3}x\,{\bf B}\cdot(\nabla\times{\bf B})\,.$ (29)

In contrast to the magnetic helicity, the physical helicity does not have
issues with gauge invariance. It is also better behaved in a finite domain as
the magnetic field ${\bf B}$ falls off faster than the gauge field ${\bf A}$.
The evolution of $H_{B}$ is shown in Figure 8 for $\gamma=\pi$, and it
asymptotes to a constant value at late times. We also plot $H_{B}$ at late
times after annihilation for various initial dumbbell lengths and twists in
the right panel of Figure 8. From this, we can infer that the $H_{B}$ for the
untwisted case $\gamma=0$ is consistent with zero. In the twisted case with
$\gamma=\pi$, it takes on finite positive values, $\sim 3\eta^{2}$, as shown
in Figure 8. One feature, that is immediately clear, is that the signs of the
physical helicities, $H_{B}$, for $\gamma=\pi$ and $0<\gamma<\pi$ are opposite
to each other. The opposite signs are related to the untwisting dynamics of
the dumbbell configuration as it annihilates. Dumbbells with twist
$\gamma=\pi$ are maximally twisted and carry maximal energy as a function of
twist Patel:2023sfm . Dumbbells with twist slightly less than $\pi$ will tend
to untwist by reducing the twist angle from $\pi-\to 0$, while those with
twist somewhat greater than $\pi$ will untwist by increasing the twist angle
from $\pi+\to 2\pi$. These decay modes lead to magnetic field helicity of
opposite signs. We tested this argument by running the simulation for twists
slightly larger and smaller than $\pi$, and confirmed that the signs of the
physical helicity are indeed opposite to each other. The plots for the tests
we conducted, for two different initial dumbbell lengths, are shown in Figure
9, where we see that the evolution of $H_{B}$ for twist $\pi+$ mirrors the one
for twist $\pi-$.

Figure 9:  The evolution of physical helicity $H_{B}$ for twists
$\gamma=5.5\pi/6$ (red curve) and $6.5\pi/6$ (blue curve) are shown for
initial dumbbell lengths $2d=80\delta$ and $160\delta$ in the left and right
panels, respectively. The annihilation time $T_{c}$ is marked with a black
dotted vertical line. The horizontal red and blue curves correspond to the
physical helicities at time $t=20m_{W}^{-1}$. Here, the time step size in the
simulation is $dt=\delta/8$, with $\delta=0.25\eta^{-1}$.

##  5 Discussion and Conclusion

We have studied the collapse, annihilation, and decay products of electroweak
dumbbells as a function of their length and twist.

The untwisted case has the expected dynamics, where the dumbbell collapses in
a time that is very short, $\sim m_{W}^{-1}$, and comparable to the
instability time scale of the Z-string James:1992wb ; Goodband:1995he . The
energy of the dumbbell that is converted into magnetic field energy depends on
the length of the dumbbell. For short untwisted dumbbells, the magnetic field
energy can be $\sim 0.1\%$; for longer dumbbells it is $\sim 1\%$. The
magnetic fields produced in the untwisted case are not helical.

Twisted dumbbells collapse on a time-scale that saturates to $\sim
10m_{W}^{-1}$ for long dumbbells. They collapse to form a long-lived object
that presumably is an electroweak sphaleron, see Figure 3. Eventually, the
sphaleron decays as well. The decay products include magnetic field energy
that is large compared to the untwisted case. The fractional energy converted
to magnetic field energy is roughly independent of the dumbbell length and is
$\sim 4\%$. The produced magnetic field is helical though our calculation of
the magnetic helicity is not reliable, especially at late times, due to the
finiteness of the simulation box. As an alternative we have calculated the
integrated physical helicity and this asymptotes to a constant ($\sim
3\eta^{2}$), at late times for the maximally twisted dumbbell.

In future work, we propose to study the dynamics of rotating dumbbells as this
pertains to their production and detection in a laboratory setting as first
discussed by Nambu Nambu:1977ag . We expect the study to be technically
challenging as it would require significant improvements in implementing
boundary conditions, especially if rotating dumbbells survive for a long time.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy, Office of High
Energy Physics, under Award No. DE-SC0019470. The authors acknowledge Research
Computing at Arizona State University for providing access to high performance
computing and storage resources on the Sol Supercomputer that have contributed
to the research results reported within this paper.

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##  Appendix A Semilocal Dumbbell simulations

The Z-string has been shown to be stable in the semilocal limit
Achucarro:1999it , which corresponds to $\sin^{2}\theta_{w}=1$ and
$\beta\equiv m_{H}^{2}/m_{Z}^{2}<1$. As a test of our simulation code, we ran
an instance of the dumbbell simulation for the parameters
$\sin^{2}\theta_{w}=0.995$ and $\beta=0.01$, which lie in the stable regime
stated in Achucarro:1999it  and studied for a cosmological distribution of
dumbbells in Urrestilla:2001dd . We show several snapshots of the energy
density in the $xz$-plane for the untwisted case ($\gamma=0$) in Figure 10,
and for the twisted case ($\gamma=\pi$) in Figure 11. Unlike the electroweak
case, shown in Figures 2 and 3, it can be seen that the Z-string is stable,
and the monopole-antimonopole pair move towards each, eventually undergoing
annihilation. For the twisted case, we once again observe the formation of an
intermediate stable object which eventually decays, and can be seen in Figure
11. The evolution of ${\rm Min}[|\Phi|]$ are shown in Figure 12, and the
lifetimes are $T_{c}m_{W}=4.9$ and $12.0$, for the untwisted ($\gamma=0$) and
twisted ($\pi$) cases, respectively. These lifetimes are much longer than
those of the dumbbell in the electroweak case for the same lengths;
$T_{c}m_{W}=0.4$ and $7.3$, for twists $\gamma=0$ and $\pi$, respectively.

Figure 10:  The snapshots of energy density, for dumbbell evolution in the
semilocal limit, in $xz$-plane and twist $\gamma=0$ are shown here. The left,
middle and right panels correspond to times $0dt$, $200dt$ and $500dt$,
respectively. The colors represent the energy density and the corresponding
values are given by the scale, in units of $\eta^{4}$. Here, the simulation
parameters are $dt=\delta/8$, with $\delta=0.25\eta^{-1}$, and the initial
dumbbell length is $2d=80\delta$. The horizontal ($z$) and vertical ($x$) axis
are in lattice units.  Figure 11:  The energy density snapshots, similar to
Figure 2, for the twisted case ($\gamma=\pi$) and times $0dt$, $450dt$ and
$1000dt$ in the left, middle and right panels, respectively.  Figure 12: The
evolution of ${\rm Min}[|\Phi|]$ in the simulation box for twists $\gamma=0$
(red curve) and $\pi$ (blue curve). The hollow circles correspond to the data
points from the simulation and the solid lines are $4^{\rm th}$ order
polynomial fits around the threshold ${\rm Min}[|\Phi|]=0.25$. We show our
threshold choice, ${\rm Min}[|\Phi|]=0.25$, as a dotted horizontal line.


Let $O_\tau$ be an observed random variable taking values in $\{1, ..., |O|\}$ indexing all possible observations. The prior distribution over $O_\tau$ is a categorical distribution conditioned on $S_\tau$ and represented by the random matrix $\bm{A}$. There is no posterior distribution over $O_\tau$ because $O_\tau$ is observed, i.e., realised.

Let $O_I$ be a random variable taking values in $\{1, ..., |O|\}$ indexing all possible observations. The prior distribution over $O_I$ is a categorical distribution conditioned on $S_I$ and represented by the random matrix $\bm{\bar{A}}$. Note that $O_I$ refers to an observation in the future and is therefore a hidden variable. The posterior distribution over $O_I$ is a categorical distribution represented by $\bm{\hat{E}}_I$.

Let $S_I$ be an random variable taking values in $\{1, ..., |S|\}$ indexing all possible states. The prior distribution over $S_I$ is a categorical distribution conditioned on $S_\IdMLast{I}$ and represented by the matrix $\bm{\bar{B}}_I \delequal \bm{\bar{B}}(\bigcdot, \bigcdot, I_{last})$. The posterior distribution over $S_I$ is a categorical distribution represented by $\bm{\hat{D}}_I$.

Let $\bm{A}$ be a $|O| \times |S|$ random matrix defining the probability of an observation $O_\tau$ given the hidden state $S_\tau$. The prior distribution over $\bm{A}$ is a product of Dirichlet distributions whose parameters are stored in a $|S| \times |O|$ matrix $\bm{a}$. The posterior distributions over $\bm{A}$ is also a product of Dirichlet distribution, but the parameters are stored in a $|S| \times |O|$ matrix $\bm{\hat{a}}$.

Let $\bm{B}$ be a $|S| \times |S| \times |U|$ random 3-tensor defining the probability of transiting from $S_\tau$ to $S_{\tau+1}$ when taking action $U_\tau$. The prior distribution over $\bm{B}$ is a product of Dirichlet distributions whose parameters are stored in a $|S| \times |U| \times |S|$ 3-tensor $\bm{b}$. The posterior distribution over $\bm{B}$ is also a product of Dirichlet distributions, but the parameters are stored in a $|S| \times |U| \times |S|$ 3-tensor $\bm{\hat{b}}$.

Let $\bm{D}$ be a random vector of size $|S|$ defining the probability of the initial state $S_0$. The prior distribution over $\bm{D}$ is a Dirichlet distribution whose parameters are stored in a vector $\bm{d}$ of size $|S|$. The posterior distribution over $\bm{D}$ is also a Dirichlet distribution, but the parameters are stored in a vector $\bm{\hat{d}}$ of size $|S|$.

Let $\bm{\Theta}_\tau \,\, \forall \tau \in \{0, ..., t-1\}$ be a random vector of size $|U|$ defining the probability of the action $U_\tau$. The prior distribution over $\bm{\Theta}_\tau$ is a Dirichlet distribution whose parameters are stored in a vector $\bm{\theta}_\tau$ of size $|U|$. The posterior distribution over $\bm{\Theta}_\tau$ is also a Dirichlet distribution, but the parameters are stored in a vector $\bm{\hat{\theta}}_\tau$ of size $|U|$.

Let $\gamma$ be a random variable taking values in $\mathbb{R}_{>0}$. The prior distribution over $\gamma$ is a gamma distribution with shape parameter $\bm{\alpha} = 1$ and rate parameter $\bm{\beta} \in \mathbb{R}_{>0}$. The posterior distribution over $\gamma$ is a gamma distribution with shape parameter $\bm{\hat{\alpha}} = 1$ and rate parameter $\bm{\hat{\beta}} \in \mathbb{R}_{>0}$.

Let $\pi$ be a random variable taking values in $\{1, ..., |\pi|\}$ indexing all possible policies. The prior distribution over $\pi$ is a softmax function of the vector $\bm{G}$ multiplied by minus the precision $\gamma$. Note that $\bm{G}$ is a vector of size $|\pi|$ whose $i$-th element is the expected free energy of the $i$-th policy. The posterior distribution over $\pi$ is a categorical distribution whose parameters are stored in a vector $\bm{\hat{\pi}}$ of size $|\pi|$.

§ APPENDIX G: MULTI-ARMED BANDIT PROBLEM

In the multi-armed bandit problem, the agent is prompted with $K$ actions (one for each bandit's arm). Pulling the $i$-th arm returns a reward sampled from the reward distribution $P_i(X)$ associated to this arm. Let $\mu_i$ be the mean of the $i$-th reward distribution and $T_i(n)$ be the number of times the $i$-th bandit has been selected after $n$ plays. To solve the bandit problem, one needs to come up with an allocation strategy that selects the action that minimises the agent's regret defined as:

\begin{align}
R_n = \mu^*n  - \sum_{i=1}^K \mu_i \mathbb{E} [T_i(n)],
\end{align}

where $\mu^*$ is the average reward of the best action. Note that an upper bound of $\mathbb{E} [T_i(n)]$ is derived by first upper bounding $T_i(n)$, and then using: the Chernoff-Hoeffing bound, the Bernstein inequality and some properties of $p$-series, c.f., proof of Theorem 1 in Auer et al, 2002 for details. So, if we first assume that,

\begin{align}
UCB1_i = \underbrace{{\color{white}\sqrt{\frac{1}{1}}} \bar{X}_i{\color{white}\sqrt{\frac{1}{1}}}}_{\text{exploitation}} + \underbrace{\sqrt{\frac{2 \ln n}{n_i}}}_{\text{exploration}},
\end{align}

where $n_i$ is the number of times the $i$-th action has been selected, and $\bar{X}_i$ is the average reward received after taking the $i$-th action. Then, the main result of Auer et al, 2002 was to show that if an allocation strategy was using the UCB1 criterion to select the next action, the expected regret of this allocation strategy will grow at most logarithmically in the number of plays $n$, i.e., $\mathcal{O}(\ln n)$. In addition, since it is known that the expected regret of the (best) allocation strategy grows at least logarithmically in $n$ [Lai and Robbins, 1985], we say that the UCB1 criterion resolves the exploration / exploitation trade-off, i.e., the UCB1 criterion ensures that the expected regret grows as slowly as possible.

§ APPENDIX H: THE EXPONENTIAL COMPLEXITY CLASS

In this appendix, we precisely pinpoint the exponential complexity class that is addressed in this paper, but first, we introduce a multi-index notation. Multi-indices will help us to refer to hidden states in the future. Naturally enough the indexes inside the multi-indices will correspond to the actions the agent will have to perform to reach the hidden state, e.g., $S_{(123)}$ corresponds to a hidden state at time $t + 3$ obtained by performing action 1 at time $t$, 2 at time $t + 1$ and 3 at time $t + 2$. Using this notation, Figure <ref> depicts all the possible policies up to two time steps in the future and the associated hidden states. Importantly, Figure <ref> shows that the number of policies grows exponentially with the number of time steps for which the agent tries to plan. Therefore, the definition of the prior over the policies, i.e., $P(\pi|\gamma) = \sigma(-\gamma \bm{G})$, exhibits an exponential space and time complexity class because the agent needs to store and compute the $|U|^T$ parameters of $P(\pi|\gamma)$, where $T$ is the time-horizon.

Illustration of all possible policies up to two time steps in the future when $|U| = 2$. The state at the current time step is denoted by $S_t$. Additionally, each branch of the tree corresponds to a possible policy, and each node $S_I$ is indexed by a multi-index (e.g. $I=(12)$) representing the sequence of actions that led to this state. This should make it clear that for one time step in the future, there are $|U|$ possible policies, after two time steps there are $|U|$ times more policies, and so on until the time-horizon $T$ where there are a total of $|U|^T$ possible policies, i.e., the number of possible policies grows exponentially with the number of time steps for which the agent tries to plan.

To show that this exponential explosion is not only a theoretical problem and also appears in practice, we modified the $DEMO\_MDP\_maze.m$ of the SPM[Statistical parametric mapping (SPM) is a software package created by the Wellcome Department of Imaging Neuroscience at University College London, which was initially developed to carry out statistical analyses of functional neuroimaging data. Today, SPM also contains MatLab simulations implementing active inference agents (among other things), c.f. <https://www.fil.ion.ucl.ac.uk/spm/>. ] package in two ways. First, we allowed the agent to plan $N$ time steps in the future as follows:
c V = Allowable policies (N moves into the future), nu = number of actions
nV p= p[];
kfor ni1 p= l+m+mi1p:nnup, ni2 p= l+m+mi1p:nnup, p.., niN p= l+m+mi1p:nnu
nVp(:,kend o+ l+m+mi1p) p= p[ni1p;ni2p; p.. p;niNp];

Second, we used the “tic” and “toc” functions provided by Matlab to track the execution time required to execute the function “$spm\_maze\_search$” where the parameters of the model are assumed to be known already by the agent:

ntic c Start a timer used to evaluate the execution time of spmmazesearch
nMDP p= nspmmazesearchp(nmdpp,l+m+mi8p,nSTARTp,nENDp,l+m+mi128p,l+m+mi1p);
ntoc c Display the time elapsed since the last call to tic

Figure <ref> presents the results of our simulations for $N$ from 2 to 6. Under a logarithmic scale on the time axis, the experimental results show that the graph is almost a perfect line, which provides empirical evidence for an exponential time explosion. Note that the simulation for $N = 7$ crashed after trying to allocate an array of 9.5GB (space explosion). In section <ref>, we presented an approach proposed to deal with the exponential complexity class that arises during planing, yet is fundamentally similiar to active inference. This effectively means that our paper shows how standard active inference can be made more efficient and scale to longer time horizons.

This figure shows the time required to execute the function “$spm\_maze\_search$” when the agent is allowed to plan $N$ time steps in the future (for $N$ from 2 to 6). A logarithmic scale is used on the time axis.