# Structure of electroweak dumbbells

Teerthal Patel, Tanmay Vachaspati  ∗Physics Department, Arizona State
University, Tempe, Arizona 85287, USA.

###### Abstract

We analyze the magnetic field of electroweak dumbbells. While the magnetic
field of the untwisted dumbbell is given by the usual dipole formula and falls
of as $1/r^{3}$, dumbbells with twist have a novel twisted magnetic field that
only falls off as $\cos\theta/r^{2}$ (in spherical coordinates). We comment on
the relevance of twisted electroweak dumbbells for understanding the coherence
of the magnetic field generated at the electroweak phase transition.

##  I 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 (1977); Achucarro and Vachaspati (2000). The existence of such
non-perturbative field configurations in the electroweak model is of great
interest as they would 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 (2021).

Generally electroweak dumbbells are viewed as magnetic dipoles with the usual
dipolar $1/r^{3}$ fall off of the magnetic field strength with distance $r$
from the dipole, but the situation in the electroweak case is richer. There is
a one-parameter set of electroweak dumbbell configurations, all describing a
confined monopole-antimonopole pair but with additional structure called the
“twist”. Such twisted dumbbells are closely related to the electroweak
sphaleron as they also carry Chern-Simons number Vachaspati and Field (1994).

In the paper we investigate the structure of electroweak dumbbells using
“constrained relaxation”. We start with a field configuration that contains a
monopole and an antimonopole with a relative twist. The fields are then
relaxed subject to the constraint that the orientation of the Higgs field is
held fixed throughout the spatial volume, and this automatically fixes the
monopole-antimonopole positions and the twist.

For zero twist, we find the expected dipolar structure of the magnetic field
of the system. The results for non-zero twist are more unexpected. The
magnetic field lines do not connect the monopole to the antimonopole; instead
the field lines stretch to infinity, tending to pull the monopole and
antimonopole away from each other. At large distances from the dumbbell, the
magnetic field strength has a $\cos\theta/r^{2}$ behavior. The magnetic field
of a twisted dumbbell also has an azimuthal component, resembling the twisted
field pointed out for an electroweak sphaleron in Ref. Hindmarsh (1994);
Hindmarsh and James (1994).

We begin our analysis by describing the model and our initial configuration of
fields in Sec. II. The numerical relaxation scheme is described in Sec. III.
We define the electromagnetic field in Sec. IV and derive the magnetic field
configuration for the initial unrelaxed configuration. The results of our
constrained relaxation are given in Sec. V. We discuss consequences of our
findings in Sec. VI.

##  II Model

###  II.1 Electroweak model

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

$\displaystyle\mathcal{L}$
$\displaystyle=-\frac{1}{4}W^{a}_{\mu\nu}W^{a\,\mu\nu}-\frac{1}{4}Y_{\mu\nu}Y^{\mu\nu}$
(1) $\displaystyle\hskip
71.13188pt+|D_{\mu}\Phi|^{2}-\lambda(|\Phi|^{2}-\eta^{2})^{2}\,,$

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 Workman and Others (2022) are
$g=0.65$, $\sin^{2}\theta_{w}=0.22$, $g^{\prime}=g\tan\theta_{w}$,
$\lambda=0.129$ and $\eta=174{\text{G}eV}$.

We adopt the temporal gauge for convenience in numerical implementation, with
$W^{a}_{0}=B_{0}=0$. The 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)
$\displaystyle\partial_{\mu}Y^{\mu\nu}=g^{\prime}\,{\text{I}m}[\Phi^{\dagger}(D^{\nu}\Phi)]$
(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, the two
charged $W$ 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)

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

###  II.2 Initial field configuration

Here we setup a field configuration that describe the dumbbell. This
configuration will become our starting point for constrained numerical
relaxation, a process in which the total energy reduces while the monopole and
antimonopole are held fixed at their initial locations.

The angular distribution of the Higgs field for the monopole and antimonopole
can be taken to be Nambu (1977),

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

where the hat on $\Phi$ denotes that $|\hat{\Phi}|=1$, $(\theta_{m},\phi)$ are
spherical angular coordinates centered on the monopole, and
$(\theta_{\bar{m}},\phi)$ are corresponding angles centered on the anti-
monopole, as shown in Fig. 1. The combined monopole-antimonopole ansatz for
the Higgs field can be taken to be Vachaspati and Field (1994),

${{{{\hat{\Phi}}_{{m}{\bar{m}}}}}(\gamma)}=\left(\begin{array}[]{c}\sin(\theta_{m}/2)\sin(\theta_{\bar{m}}/2)e^{i\gamma}+\cos(\theta_{m}/2)\cos(\theta_{\bar{m}}/2)\\\
\sin(\theta_{m}/2)\cos(\theta_{\bar{m}}/2)e^{i\phi}-\cos(\theta_{m}/2)\sin(\theta_{\bar{m}}/2)e^{i(\phi-\gamma)}\end{array}\right)\,,$
(12)

where the monopole and antimonpole are located along the z-axis, as
illustrated in Fig. 1, and we have included a “twist” angle $\gamma$. In the
limit $\theta_{\bar{m}}\rightarrow 0$, (12), one recovers the monopole
configuration and, in the limit $\theta_{m}\rightarrow\pi$, one recovers the
antimonopole configuration in (12) but with a twist given by $\gamma$. The
effect of the relative twist $\gamma$ is more evident from the vector (10)
associated with the Higgs field. This is visualized in Cartesian space in
Figs. 2 and 3. Here, we see the projection
$\hat{n}-(\hat{n}\cdot\hat{j})\hat{j}$ of the vector in the $xz-$plane for
extreme values of twists $\gamma=0,\pi$.

Figure 1: A general vector $\vec{x}$ in a Cartesian grid with the monopole and
antimonopole centered polar angles, $\theta_{m}$ and $\theta_{\bar{m}}$,
respectively.  Figure 2:  The projection of the vector $\hat{n}$ given by (10)
for the ansatz (12) in the $xz$ plane for $\gamma=0$. The blue and red dots
represent the monopole and antimonopole, respectively. Figure 3: The
projection of the vector $\hat{n}$ given by (10) for the ansatz (12) in the
$xz$ plane for $\gamma=\pi$. The blue and red dots represent the monopole and
antimonopole, respectively.

The gauge field configurations are obtained by setting the covariant
derivative of the Higgs field to vanish in the symmetry broken regions. This
procedure does not fix the gauge fields completely as it allows for an
arbitrary electromagnetic gauge field (since the generator of the
electromagnetic group annihilates $\Phi$). The electromagnetic gauge field is
defined as,

$A_{\mu}\equiv-\sin\theta_{w}{\hat{n}^{a}W^{a}_{\mu}}+\cos\theta_{w}Y_{\mu}\,.$
(13)

We completely fix the form of the gauge fields by requiring that $A_{\mu}=0$.
The gauge fields are then given by,

$\displaystyle gW^{a}_{\mu}$ $\displaystyle=$
$\displaystyle-\epsilon^{abc}n^{b}{\partial_{\mu}n^{c}}$ (14)
$\displaystyle-i\cos^{2}\theta_{w}n^{a}(\hat{\Phi}^{{\dagger}}\partial_{\mu}\hat{\Phi}-\partial_{\mu}\hat{\Phi}^{\dagger}\hat{\Phi})$
$\displaystyle g^{\prime}Y_{\mu}$ $\displaystyle=$
$\displaystyle-i\sin^{2}\theta_{w}(\hat{\Phi}^{\dagger}\partial_{\mu}\hat{\Phi}-\partial_{\mu}\hat{\Phi}^{\dagger}\hat{\Phi})$
(15)

To correctly account for the radial dependence of the Higgs field around the
monopole-antimonopole pair, we attach radial profiles. The general monopole
solution is given by

$\Phi=h(r)\hat{\Phi}\,,$ (16)

where $\hat{\Phi}$ are the normalized doublets in (11). Including profile
functions in the ansatz, the initial monpole-antimonopole scalar field
configuration is given by

${{\Phi_{m{\bar{m}}}}}=k({\vec{x}})h(r_{m})h(r_{{\bar{m}}}){{{\hat{\Phi}}_{{m}{\bar{m}}}}}\,,$
(17)

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

$r_{m}=|{\vec{x}}-{\vec{x}}_{m}|,\,\,\,\,r_{\bar{m}}=|{\vec{x}}-{\vec{x}}_{\bar{m}}|\,,$
(18)

where ${\vec{x}}_{m}=(0,0,d)$ and ${\vec{x}}_{\bar{m}}=(0,0,-d)$. While $h(r)$
represents the monopole profile function, $k({\vec{x}})$ in (17) is the
profile of the Z-string that connects the monopole and the antimonopole in the
dumbbell.

Similar to (17), we include radial profiles for the gauge fields as

$\displaystyle gW^{a}_{\mu}$ $\displaystyle=$ $\displaystyle
l({\vec{r}})j(r_{m})j(r_{\bar{m}})[-\epsilon^{abc}n^{b}{\partial_{\mu}n^{c}}$
(19) $\displaystyle\hskip
14.22636pt-i\cos^{2}\theta_{w}\,n^{a}({\hat{\Phi}}^{\dagger}\partial_{\mu}{\hat{\Phi}}-\partial_{\mu}{\hat{\Phi}}^{\dagger}{\hat{\Phi}})]$
$\displaystyle g^{\prime}Y_{\mu}$ $\displaystyle=$ $\displaystyle
l({\vec{r}})j(r_{m})j(r_{\bar{m}})[$ (20) $\displaystyle\hskip
14.22636pt-i\sin^{2}\theta_{w}\,({\hat{\Phi}}^{\dagger}\partial_{\mu}{\hat{\Phi}}-\partial_{\mu}{\hat{\Phi}}^{\dagger}{\hat{\Phi}})]$

Now that we have set up a field configuration that describes the initial
dumbbell in Eqs. (17), (19) and (20), we will perform a constrained numerical
relaxation, a process in which the total energy reduces while the monopole and
antimonopole are held fixed at their initial positions.

###  II.3 Profiles

The monopole profile functions are only known in closed form within the
context of the Bogomolnyi-Prasad-Sommerfield (BPS) limit ($\lambda\rightarrow
0$) Prasad and Sommerfield (1975) . Numerical solutions for the general case
have been outlined in Manton and Sutcliffe (2004). A functional form that
reduces to the BPS case is given by Vachaspati (2016),

$h(r)=\frac{1}{\tanh(\eta r)}-(1+mr)\frac{e^{-mr}}{\eta r}\,,$ (21)
$j(r)=1-\frac{\eta r}{\sinh(\eta r)}\,,$ (22)

where $m=2\sqrt{\lambda}\,\eta$ is the scalar mass and $r$ is the radial
coordinate centered around the monopole. In our application, we use these
radial profiles as initial guess functions for the monopole and the
antimonopole.

Figure 4: The component $k_{z}(z)$ of the string profile $k(\vec{x})$, given
by (24), as a function of $z$. The monopole and antimonopole are located at
$z=\pm 4\,\eta^{-1}$, respectively.  Figure 5: The contour plots for the
profile function $h(r_{m})h(r_{{\bar{m}}})$ in the $y=0$ plane. Here, the
monopole and antimonopole are located at $z=\pm 4$, respectively.  Figure 6:
The contour plots for the profile function $k(\vec{x})$ in the $y=0$ plane.
Here, the monopole and antimonopole are located at $z=\pm 4$, respectively.

The string profile functions have been studied in the context of the Abelian
Higgs model, known as Nielsen-Olesen strings Nielsen and Olesen (1973);
Vilenkin and Shellard (2000). The Nielsen-Olesen string profiles are not known
analytically, but have been studied numerically. Our string profile guess
functions $k({\vec{x}})$ and $l({\vec{x}})$ match the $\rho\rightarrow 0$ and
$\rho\rightarrow\infty$ behavior of Nielsen-Olesen string solutions, where
$\rho\equiv\sqrt{x^{2}+y^{2}}$. However, unlike the Nielsen-Olesen case, we
are dealing with a finite Z-string that joins the monopole-antimonopole pair.
Therefore, our guess functions depend on ${\vec{x}}$ and not just $\rho$,
allowing the string profile function to terminate smoothly at the monopole and
antimonopole. To this end, we define the string profile function by writing it
as

$k(\vec{x})=1-k_{\rho}(\rho)k_{z}(z)\,,$ (23)

where $k_{z}$ is the function that ensures that the Z-string terminates at the
monopole and antimonopole, while $k_{\rho}$ is the radial profile. These
functions are taken to be

$\displaystyle k_{z}(z)$
$\displaystyle=\frac{\tanh(\eta(-z+d))+\tanh(\eta(z+d))}{2\tanh(\eta d)}$ (24)
$\displaystyle k_{\rho}(\rho)$
$\displaystyle=\frac{\tanh(\eta(-\rho+\delta))+\tanh(\eta(\rho+\delta))}{2\tanh(\eta\delta)}\,.$
(25)

where $\delta$ is the lattice spacing. $k_{z}$ is illustrated in Fig. 4, and
the total string profile, $k(\vec{x})$, is plotted in Figure 6.

One issue is that the profile $k({\vec{x}})$ does not strictly vanish at the
location of the string. This is not an issue for us because in our numerical
work we place the string between lattice points and never have to evaluate the
profiles exactly at the center of the string.

##  III Numerical Relaxation

###  III.1 System of equations

The static equation of motion for $\Phi$ can be expanded as

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\partial_{0}^{2}\Phi=D_{i}D_{i}\Phi-2\lambda(|\Phi|^{2}-\eta^{2})^{2}\Phi$
(26)
$\displaystyle=\nabla^{2}\Phi-i\frac{g}{2}\sigma^{a}W^{a}_{i}\partial_{i}\Phi-i\frac{g^{\prime}}{2}Y_{i}\partial_{i}\Phi$
$\displaystyle-i\frac{g}{2}\sigma^{a}\Gamma^{a}_{i}\Phi-i\frac{g^{\prime}}{2}\Xi_{i}\Phi^{\alpha}-i\frac{g}{2}\sigma^{a}W^{a}_{i}(D_{i}\Phi)$
$\displaystyle-i\frac{g^{\prime}}{2}Y_{i}D_{i}\Phi-2\lambda(|\Phi|^{2}-\eta^{2})^{2}\Phi\,.$

Here, we introduced the notation $\Gamma^{a}_{i}=\partial_{i}W^{a}_{i}$ and
$\Xi_{i}=\partial_{i}Y_{i}$.Since we are computing the static configurations
for the monopole-antimonpole pair, we set all time derivatives to be 0.
Similarly, the static gauge field equations of motion lead to

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{2}W^{a}_{i}-\partial_{i}\Gamma^{a}_{k}-g\epsilon^{abc}(\partial_{k}W^{b}_{i})W^{c}_{k}-g\epsilon^{abc}W^{b}_{i}\Gamma^{c}_{k}$
(27) $\displaystyle\hskip
42.67912pt-g\epsilon^{abc}W^{b}_{k}W^{c}_{ik}+g{\text{Im}}[\Phi^{\dagger}\sigma^{a}(D_{i}\Phi)]\,$
$\displaystyle 0$ $\displaystyle=$
$\displaystyle\nabla^{2}Y_{i}-\partial_{i}\Xi_{k}+g^{\prime}{\text{Im}}[\Phi^{\dagger}\sigma^{a}(D_{i}\Phi)]\,.$
(28)

The relaxation scheme entails that we fix the monopole-antimonopole positions
when solving the system of equations (26)-(28). We implement this constraint
by utilizing the gauge freedom in the model. We work in the gauge where the
Higgs directions are chosen to be $\hat{\Phi}_{m\bar{m}}$ as given in (12).
Since the positions of the monopole and antimonopole are determined by the
orientation of the Higgs as discussed in Ref. Patel and Vachaspati (2022),
this fixes their positions for the entire relaxation process. Only the
magnitude of the Higgs field $|\Phi|$ and the gauge fields, $W^{a}_{i}$ and
$Y_{i}$ need to be relaxed to satisfy the equations of motion in (27) and
(28). Thus instead of solving for the two components of the Higgs doublet, we
solve for $|\Phi|$, where

$|\Phi|=\frac{1}{2}({\hat{\Phi}}^{\dagger}\Phi+\Phi^{\dagger}{\hat{\Phi}})\,.$

The Laplacian of $|\Phi|$ can be expressed as

$\displaystyle\nabla^{2}|\Phi|$ $\displaystyle=$
$\displaystyle\frac{1}{2}[(\nabla^{2}\hat{\Phi}^{\dagger})\Phi+\hat{\Phi}^{\dagger}(\nabla^{2}{\Phi})$
(29) $\displaystyle\hskip
14.22636pt+(\nabla^{2}{\Phi}^{\dagger})\hat{\Phi}+{\Phi}^{\dagger}(\nabla^{2}\hat{\Phi})$
$\displaystyle\hskip
28.45274pt+4|\Phi|\partial_{i}\hat{\Phi}^{\dagger}\partial_{i}\hat{\Phi}$

where we have used

$\partial_{i}\Phi=\partial_{i}|\Phi|\,\hat{\Phi}+|\Phi|\partial_{i}\hat{\Phi}.$
(30)

and $\partial_{i}(\hat{\Phi}^{\dagger}\hat{\Phi})=0$. An equation of motion
for $|\Phi|$ can thus be derived by using $\nabla^{2}{\Phi}$ and
$\nabla^{2}{\Phi}^{\dagger}$ from (26) and, using (12) to obtain $\hat{\Phi}$
and its derivatives since, as explained above, $\hat{\Phi}$ is held fixed
throughout the relaxation. Since, we are working in the temporal gauge, the
equations for $\Gamma^{a}_{i}$ and $\Xi_{i}$ are trivial.

The initial Higgs and gauge fields are given by the configurations in Sec.
II.2 and the guess profiles in Sec. II.3. Furthermore, we evaluated the exact
analytic first and second order derivatives of the Higgs and gauge fields for
the initial configuration.

###  III.2 Relaxation algorithm

We relax the initial field configuration in a cubic lattice. At any given
lattice point, the discretized system of equations is given by
$\mathbf{E}_{\alpha}[{f}]=0$, where $\mathbf{E}_{\alpha}[{f}]$ denote the
discretized static equations for $f\in\\{|\Phi|,W^{a}_{i},Y_{i}\\}$. The
equations are of the wave equation type

$\mathbf{E}_{\alpha}[{f}]=-\nabla^{2}f_{\alpha}+S_{\alpha}[f]=0$ (31)

where $S_{\alpha}$ denotes various other terms.

To illustrate the principle of numerical relaxation, consider the discretized
second order spatial derivative,

$\displaystyle\nabla^{2}f_{\alpha}$ $\displaystyle\rightarrow$
$\displaystyle-6\frac{f_{\alpha}}{\delta^{2}}+\frac{1}{\delta^{2}}[f_{\alpha}(i+1,j,k)+f_{\alpha}(i-1,j,k)$
(32) $\displaystyle\hskip 42.67912pt+f_{\alpha}(i,j+1,k)+f_{\alpha}(i,j-1,k)$
$\displaystyle\hskip 42.67912pt+f_{\alpha}(i,j,k+1)+f_{\alpha}(i,j,k-1)]$
$\displaystyle\equiv-6\frac{f_{\alpha}}{\delta^{2}}+\frac{\Delta_{\alpha}}{\delta^{2}}$

and so the equation of motion may be written as

$\displaystyle
f_{\alpha}(i,j,k)=\frac{\Delta_{\alpha}}{6}-\frac{\delta^{2}}{6}S_{\alpha}$
(33)

which can also be written as,

$f_{\alpha}(i,j,k)=\frac{\delta^{2}}{6}\mathbf{E}_{\alpha}[{f_{\beta}}]+f_{\alpha}(i,j,k)\,.$
(34)

The relaxation scheme is to take the left-hand side at the $n^{\rm th}$
iteration step and the right-hand side at the previous iteration step,

$f^{(n)}_{\alpha}(i,j,k)=\frac{\delta^{2}}{6}\mathbf{E}_{\alpha}[{f^{(n-1)}}]+f^{(n-1)}_{\alpha}(i,j,k)\,.$
(35)

Numerically, we iterate over the lattice points in a linear order to update
the field values. In our setup, the updated field values are made immediately
available for computing the field values at the next lattice site in the
computation. This is the Gauss-Seidel method and thus does not require an
additional array to hold field values from the previous iteration. We continue
iterating while the total energy of the configuration keeps decreasing. In the
example stated above, we used first order central finite differences which
results in the $1/6$ coefficient. In our numerical runs, we use sixth order
finite central differences and then the coefficient is $6/49$.

###  III.3 Numerical setup

For most of our numerical runs, we use a $740^{3}$ lattice with lattice
spacing $\delta=0.05\,\eta^{-1}$. We work in units of $\eta$. Thus a unit of
energy in our simulation corresponds to $\eta=174$ GeV. The monopole and
string radii are comparable to the inverse of the gauge boson masses,
$m_{W}^{-1}=\sqrt{2}\,\eta^{-1}/g\approx 44\,\delta$. Therefore there are
about 44 lattice points that resolve the radius of the monopole and a similar
number for the string.

The monopole and antimonopole are zeros of the Higgs field at their respective
centers. In addition, the coordinates along which the Z-string is centered are
also zeros of the Higgs field. These could lead to artificial numerical
singularities and so, we offset the center of the monopole and antimonopole
away from the $z$-axis in the $xy$ plane; that is to say that the monopole and
antimonopole are at the coordinates $x=y=\delta/2,z=\pm(d+\delta/2)$. This
also implies that the Z-string lies parallel to the $z$-axis, along
$x=y=\delta/2$. As the algorithm iterates over the lattice, the system
undergoes relaxation and slowly approaches the desired asymptotic solution
with the changes becoming infinitesimal, as the number of iterations performed
increases. Since this can prove to be impractical, we introduced an additional
constraint that the iterations are stopped once the consecutive fractional
difference of the total energy drops below a certain threshold. This threshold
was taken to be $10^{-6}$. Note that the number of total iterations required
to reach a satisfactory asymptotic scaling depends on both simulation
parameters, like the lattice size and spacing, as well as model parameters,
such as separation and twist. We run our algorithm for a range of separations
and twists. For the set of tested parameters, the number of iterations
performed ranges between $\sim 20000-70000$.

##  IV Magnetic field

We adopt the definition for the electromagnetic field strength tensor in the
symmetry broken phase ($|\Phi|=\eta$) ’t Hooft (1974); Vachaspati (1991),

$\displaystyle A_{\mu\nu}$ $\displaystyle\equiv$
$\displaystyle-\sin\theta_{w}\hat{n}^{a}W^{a}_{\mu\nu}+\cos\theta_{w}Y_{\mu\nu}$
(36)
$\displaystyle-i\frac{2\sin\theta_{w}}{g\eta^{2}}(D_{\mu}\Phi^{\dagger}D_{\nu}\Phi-
D_{\nu}\Phi^{\dagger}D_{\mu}\Phi)$ $\displaystyle=$
$\displaystyle\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}$
$\displaystyle-i\frac{2\sin\theta_{w}}{g\eta^{2}}(\partial_{\mu}\Phi^{\dagger}\partial_{\nu}\Phi-\partial_{\nu}\Phi^{\dagger}\partial_{\mu}\Phi)\,.$

This definition implies the presence of non-zero electromagnetic fields for
$A_{\mu}=0$ due to the Higgs gradient term. In the unitary gauge, the Higgs
gradient term vanishes and one recovers the standard expression for the
Maxwellian electromagnetic fields.

We now obtain analytic expressions for the magnetic field of the unrelaxed
electroweak dumbbell for arbitrary twist. The relaxation procedure will change
the detailed features of the magnetic field but still preserves the
qualitative features of the unrelaxed configuration as we will see in Sec. V.

Eqs. (19) and (20) ensure that $A_{\mu}=0$ and it is only necessary to
evaluate the last term in (36). The definition of $A_{\mu\nu}$ assumes that
electroweak symmetry is broken and the expression in (36) applies in regions
where $|\Phi|\approx\eta$. Hence we can replace $\Phi$ by $\Phi_{m\bar{m}}$ of
(12) and write,

$\partial_{i}\Phi\equiv
A\partial_{i}\theta_{m}+B\partial_{i}\theta_{\bar{m}}+C\partial_{i}\phi$ (37)

where $A$, $B$ and $C$ are derivatives of $\Phi$ with respect to $\theta_{m}$,
$\theta_{\bar{m}}$ and $\phi$ respectively. Then

$\displaystyle\partial_{[i}\Phi^{\dagger}\partial_{j]}\Phi$ $\displaystyle=$
$\displaystyle(A^{\dagger}B-B^{\dagger}A)\,\partial_{[i}\theta_{m}\partial_{j]}\theta_{\bar{m}}$
(38)
$\displaystyle\hskip-7.11317pt+(A^{\dagger}C-C^{\dagger}A)\,\partial_{[i}\theta_{m}\partial_{j]}\phi$
$\displaystyle\hskip-7.11317pt+(B^{\dagger}C-C^{\dagger}B)\,\partial_{[i}\theta_{\bar{m}}\partial_{j]}\phi$

and the square brackets in the indices denote antisymmetrization.

From Fig. 1 we see,

$\tan\theta_{m}=\frac{\rho}{z-d},\ \ \tan\theta_{\bar{m}}=\frac{\rho}{z+d}.$
(39)

where $\rho$ is the cylindrical radial coordinate. Therefore, after some
algebra,

$\partial_{[i}\theta_{m}\partial_{j]}\theta_{\bar{m}}=\frac{2d\rho}{r_{m}^{2}r_{\bar{m}}^{2}}\,\partial_{[i}\rho\partial_{j]}z$
(40)
$\partial_{[i}\theta_{m}\partial_{j]}\phi=\frac{1}{r_{m}^{2}}\left[(z-d)\,\partial_{[i}\rho\partial_{j]}\phi-\rho\,\partial_{[i}z\partial_{j]}\phi\right]$
(41)
$\partial_{[i}\theta_{\bar{m}}\partial_{j]}\phi=\frac{1}{r_{\bar{m}}^{2}}\left[(z+d)\,\partial_{[i}\rho\partial_{j]}\phi-\rho\,\partial_{[i}z\partial_{j]}\phi\right]$
(42)

Then from (36) we get the (cylindrical) components of the magnetic field,

$\displaystyle B_{\rho}$ $\displaystyle=$
$\displaystyle-i\kappa\left[\frac{A^{\dagger}C-C^{\dagger}A}{r_{m}^{2}}+\frac{B^{\dagger}C-C^{\dagger}B}{r_{\bar{m}}^{2}}\right]$
(43) $\displaystyle B_{\phi}$ $\displaystyle=$ $\displaystyle
i\kappa\frac{2d\rho}{r_{m}^{2}r_{\bar{m}}^{2}}(A^{\dagger}B-B^{\dagger}A)$
(44) $\displaystyle B_{z}$ $\displaystyle=$
$\displaystyle-i\frac{\kappa}{\rho}\biggl{[}(z-d)\frac{A^{\dagger}C-C^{\dagger}A}{r_{m}^{2}}$
(45) $\displaystyle\hskip
56.9055pt+(z+d)\frac{B^{\dagger}C-C^{\dagger}B}{r_{\bar{m}}^{2}}\biggr{]}$

with $\kappa\equiv 2\sin\theta_{w}/g$.

The factors with the doublets $A$, $B$ and $C$ can be evaluated using
derivatives of (12) with respect to $\theta_{m}$, $\theta_{\bar{m}}$ and
$\phi$. The calculation simplifies if we write

$\Phi=\sin(\theta_{m}/2)e^{i\gamma}\Phi_{1}+\cos(\theta_{m}/2)\Phi_{2}$ (46)

where

$\Phi_{1}=\begin{pmatrix}\sin(\theta_{\bar{m}}/2)\\\
\cos(\theta_{\bar{m}}/2)e^{i(\phi-\gamma)}\end{pmatrix},$ (47)
$\Phi_{2}=\begin{pmatrix}\cos(\theta_{\bar{m}}/2)\\\
-\sin(\theta_{\bar{m}}/2)e^{i(\phi-\gamma)}\end{pmatrix}$ (48)

with the properties $|\Phi_{1}|=1=|\Phi_{2}|$, $\Phi_{1}^{\dagger}\Phi_{2}=0$,
$\Phi_{1}=-2\partial_{\theta_{\bar{m}}}\Phi_{2}$, and
$\Phi_{2}=2\partial_{\theta_{\bar{m}}}\Phi_{1}$. This gives

$A^{\dagger}B-B^{\dagger}A=\frac{i}{2}\sin\gamma\,\cos\theta_{m}.$

Similar calculations give

$A^{\dagger}C-C^{\dagger}A=\frac{i}{2}\left[\sin\theta_{m}\cos\theta_{\bar{m}}-\cos\gamma\cos\theta_{m}\sin\theta_{\bar{m}}\right],$
$B^{\dagger}C-C^{\dagger}B=\frac{i}{2}\left[\cos\theta_{m}\sin\theta_{\bar{m}}-\cos\gamma\sin\theta_{m}\cos\theta_{\bar{m}}\right],$

These expressions can now be inserted in (43), (44) and (45). The resulting
expressions are not transparent and we shall focus on a few interesting
features.

First of all consider the magnetic field twist given by $B_{\phi}$,

$B_{\phi}=-\kappa
d\sin\gamma\frac{\rho}{r_{m}^{2}r_{\bar{m}}^{2}}\cos\theta_{m}$ (49)

Note that the twisting is not symmetric in $\theta_{m}$ and
$\theta_{\bar{m}}$. For example, $B_{\phi}$ vanishes for $\theta_{m}=\pi/2$
but not when $\theta_{\bar{m}}=\pi/2$. This is a feature stemming from our
choice of the unrelaxed field. Upon relaxation the twist gets redistributed as
in apparent in Sec. V where we plot the magnetic field in the
$\theta_{m}=\pi/2$ plane. Also note that the twisting reverses direction under
$\gamma\to-\gamma$.

Far from the dumbbell $r_{m}\approx r_{\bar{m}}\approx r$,
$\theta_{m}\approx\theta_{\bar{m}}\approx\theta$, and $\rho=r\sin\theta$. Then

$B_{\phi}=-\kappa d\sin\gamma\frac{\sin\theta\cos\theta}{r^{3}}$ (50)

Hence the azimuthal field is non-vanishing only for $\gamma\neq 0$, falls off
as $1/r^{3}$, and the twisting is of opposite signs for $\cos\theta>0$ and
$\cos\theta<0$.

Next we consider the magnetic field in the $z=0$ plane. Then we have
$r_{m}=r_{\bar{m}}=\rho$ and $\theta_{m}+\theta_{\bar{m}}=\pi$. This gives us

$B_{\rho}(z=0)=\frac{\kappa}{2\rho^{2}}(1-\cos\gamma)\sin(\theta_{m}+\theta_{\bar{m}})=0$
(51) $B_{z}(z=0)=\frac{\kappa d}{2\rho^{3}}\sin(2\theta_{m})\,(1+\cos\gamma).$
(52)

Hence for maximal twist angle, $\gamma=\pi$, the magnetic field on the $z=0$
plane vanishes.

Finally we consider the asymptotic magnetic field. We have already calculated
the azimuthal component in (50). For the other components, note that once
again $r_{m}\approx r_{\bar{m}}\approx r$,
$\theta_{m}\approx\theta_{\bar{m}}\approx\theta$ and so

$B_{\rho}\bigl{|}_{r\gg
d}=\kappa(1-\cos\gamma)\frac{\sin\theta\cos\theta}{r^{2}}$ (53)

and

$B_{z}\bigl{|}_{r\gg d}=\kappa(1-\cos\gamma)\frac{\cos^{2}\theta}{r^{2}}$ (54)

Note that the magnetic field of the twisted dumbbells ($\gamma\neq 0$) falls
off as $1/r^{2}$, instead of the dipolar $1/r^{3}$.

Some more insight is gained by calculating the spherical radial component of
the magnetic field in the asymptotic region. The radial component is given by

$B_{r}=B_{\rho}\sin\theta+B_{z}\cos\theta.$ (55)

Using (53) and (54) we get

$B_{r}\bigl{|}_{r\gg d}=\kappa(1-\cos\gamma)\frac{\cos\theta}{r^{2}}.$ (56)

Therefore the radial field has the structure of a monopole’s magnetic field
that has been squeezed into the angular range $0\leq\theta\leq\pi/2$, and an
antimonopole’s magnetic field squeezed in the angular range
$\pi/2<\theta<\pi$. The magnetic field vanishes at $\theta=\pi/2$. The long
range magnetic field of a twisted dumbbell has a $1/r^{2}$ fall off, like that
of a monopole. Only in the untwisted ($\gamma=0$) case does this monopole
contribution vanish, and then the dipole $1/r^{3}$ term becomes the leading
contribution.

Many of the qualitative features of the initial magnetic field persist even
after relaxation as we now discuss.

##  V Results

As a check of our numerical relaxation scheme we have calculated energies of
the electroweak Z-string Vachaspati (1992) and the electroweak sphaleron. Our
relaxation procedure on a three dimensional lattice gives the energy per unit
length of the Z-string to be $1.023\pi\eta^{2}$. This is within $2\%$ of the
values previously calculated from numerical solutions of the radial
differential equations Nielsen and Olesen (1973); Vilenkin and Shellard
(2000). For the second check, we obtained the energy of the electroweak
sphaleron by using the configuration in (12) with twist $\gamma=\pi$ and zero
separation, i.e. $\theta_{m}=\theta_{\bar{m}}$. Then $\Phi_{m\bar{m}}$ has the
configuration of the $SU(2)$ ($\theta_{w}=0$) sphaleron. On relaxing the
configuration, we find the sphaleron energy to be $2.00\times 4\pi$ for
$\lambda=1/2$ and $\theta_{w}=0$, which is within $1\%$ of the result in
Manton and Sutcliffe (2004).

Figure 7: The energy of the configuration as a function of the number of
iterations for twist angle $n\pi/6$, $n=0,3,6$. The different colors and line
styles correspond to different values of $n$ and monopole-antimonopole
separation $\Delta z_{M\bar{M}}$, as indicated in the legends. The solid lines
are asymptotic fits to the converged energies.

In Fig. 7, we show the energies of electroweak dumbbell configurations for two
different monopole-antimonopole separations, and a few different twists as a
function of the number of iterations in our relaxation procedure. The solid
lines are asymptotic fits that show convergence of the energies of the
configurations. In Fig. 8 we show the energies of the relaxed electroweak
dumbbells for twist angle $\gamma\in[0,\pi]$, and for a range of separations.
At large separations, most of the energy is in the string and hence we see
linear growth. At small separations the string is less important and monopole-
antimonopole interactions become important. The flattening of the $\gamma=\pi$
curve at very small separations indicates the existence of an unstable
solution, which is precisely the sphaleron.

In Fig. 9, we show contours of the magnitude of the relaxed Higgs field
$|\Phi|$ in the $xz$-plane for twist $\pi$. There are no significant
differences in features for different values of the twist $\gamma$.

Figure 8: The energy of electroweak dumbbells as a function of monopole-
antimonopole separation for twist angle $\gamma=n\pi/6$ for various values of
$n$ as shown in the legend. The energies increase as we increase the twist.
The star at minimal separation and $\gamma=\pi$ denotes the sphaleron.  Figure
9: Contours of the Higgs field magnitude $|\Phi|$ in units of $\eta$ in the
$xz$-plane after relaxation for $\gamma=\pi$ and monopole-antimonopole
separation of $2d=120\delta=12\eta^{-1}$. The same contour for the $\gamma=0$
case has very similar features.

To find the magnetic structure of the electroweak dumbbell we use the
definition of the field strength in (36). This expression assumes
$|\Phi|=\eta$ and hence is strictly valid only far from the dumbbell. However,
we will apply it to the entire volume; points where $\Phi=0$ are avoided since
the dumbbell zeros are situated between lattice points. In Fig. 10 we show the
magnetic field strength (colors) and the magnetic field lines of an untwisted
dumbbell.

Figure 10:  Contours of the magnetic field magnitude $|B|$ (colors) in units
of $\eta^{2}$, and the projection of magnetic field lines in the $xz-$plane
after relaxation for $\gamma=0$, and monopole-antimonopole separation
$2d=120\delta=12\eta^{-1}$.

The magnetic structure of the twisted ($\gamma=\pi$) dumbbell is shown in Fig.
12 and, in marked contrast to the untwisted case, the magnetic field lines
flow away from the dumbbell. The structure agrees with the $\cos\theta/r^{2}$
expression given in (56). The magnetic field lines in this case tend to pull
the monopole and antimonopole apart, i.e. they provide a repulsive force
between the monopole and antimonopole.

Figure 11:  Contours of the magnetic field magnitude $|B|$ (colors) in units
of $\eta^{2}$, and the projection of magnetic field lines in the $xz-$plane
after relaxation for $\gamma=\pi$, and monopole-antimonopole separation
$2d=120\delta=12\eta^{-1}$.

The field lines in Fig. 12 show the structure of the magnetic field projected
on to the $xz-$plane: the field lines shown are given by the direction of
$\vec{B}-(\vec{B}\cdot\hat{y})\hat{y}$ and suppresses the component out of the
page (in the $y-$direction). In Fig. 12 we show the projected field lines in
the $z=d$ plane. Here we clearly see the twist in the magnetic field lines
first discussed in the context of the sphaleron in Ref. Hindmarsh (1994);
Hindmarsh and James (1994).

Figure 12:  Contour plots of the magnetic field magnitude $|B|$ (colors) in
units of $\eta^{2}$ in the $xy-$plane containing the monopole $(z=d)$, and the
projection of magnetic field lines (black curves):
$\vec{B}-(\vec{B}\cdot\hat{k})\hat{k}$, for $\gamma=\pi$, $2d=12\eta^{-1}$.

##  VI Conclusions and Discussion

We have developed a numerical technique to study the magnetic structure of
electroweak dumbbells in which the positions of the monopole and antimonopole
are held fixed. We have studied the constrained solution as a function of the
monopole-antimonopole separation and the twist angle. As expected, the energy
grows linearly with separation at large separations, while monopole-
antimonopole interactions become important at small separations. For maximum
twist, the dumbbell energy approaches the electroweak sphaleron energy as the
separation goes to zero.

The magnetic field of the electroweak dumbbell at zero twist resembles that of
an ordinary bar magnet. Then the magnetic field strength has the usual dipolar
$1/r^{3}$ fall off at large distances. However the magnetic field in the case
of non-zero twist has an unexpected distribution – the magnetic field lines
emanating from the monopole, instead of terminating at the antimonopole, are
directed towards spatial infinity and pull the monopole away from the
antimonopole. The magnetic field strength at large distances has a
$\cos\theta/r^{2}$ fall off. In addition, the magnetic field lines are twisted
in the azimuthal direction.

For a general electroweak dumbbell formed during electroweak symmetry
breaking, the twist angle will be non-zero and the magnetic field lines
emanating from a monopole will terminate on an antimonopole of some other
dumbbell. After the dumbbells have annihilated, the remaining field lines will
perform a random walk in three dimensions and will not close on themselves.
This is likely to have consequences for the correlation length of magnetic
fields leftover from the electroweak epoch Vachaspati (2021). The situation
may be similar to that of cosmic strings in which most of the energy of the
cosmic string network is in infinite strings and not in closed loops. We plan
to examine this scenario in more detail in future work.

Another outcome of our work is in the context of Nambu’s calculation of the
lifetime of rotating electroweak dumbbells Nambu (1977). Our relaxation
methods have provided the structure of the dumbbells which we can feed into an
evolution code and study their lifetime as a function of energy and angular
momentum.

###### 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 Agave and Sol Supercomputer that have
contributed to the research results reported within this paper.

## References

  * Nambu (1977) Y. Nambu, Nucl. Phys. B 130, 505 (1977).
  * Achucarro and Vachaspati (2000) A. Achucarro and T. Vachaspati, Phys. Rept. 327, 347 (2000), eprint hep-ph/9904229.
  * Vachaspati (2021) T. Vachaspati, Rept. Prog. Phys. 84, 074901 (2021), eprint 2010.10525.
  * Vachaspati and Field (1994) T. Vachaspati and G. B. Field, Phys. Rev. Lett. 73, 373 (1994), URL https://link.aps.org/doi/10.1103/PhysRevLett.73.373.
  * Hindmarsh (1994) M. Hindmarsh, in _1st International Conference on Strong and Electroweak Matter_ (1994), eprint hep-ph/9408241.
  * Hindmarsh and James (1994) M. Hindmarsh and M. James, Phys. Rev. D 49, 6109 (1994), eprint hep-ph/9307205.
  * Workman and Others (2022) R. L. Workman and Others (Particle Data Group), PTEP 2022, 083C01 (2022).
  * Prasad and Sommerfield (1975) M. K. Prasad and C. M. Sommerfield, Phys. Rev. Lett. 35, 760 (1975), URL https://link.aps.org/doi/10.1103/PhysRevLett.35.760.
  * Manton and Sutcliffe (2004) N. S. Manton and P. Sutcliffe, _Topological solitons_ , Cambridge Monographs on Mathematical Physics (Cambridge University Press, 2004), ISBN 978-0-521-04096-9, 978-0-521-83836-8, 978-0-511-20783-9.
  * Vachaspati (2016) T. Vachaspati, Phys. Rev. D 93, 045008 (2016), eprint 1511.05095.
  * Nielsen and Olesen (1973) H. B. Nielsen and P. Olesen, Nucl. Phys. B 61, 45 (1973).
  * Vilenkin and Shellard (2000) A. Vilenkin and E. P. S. Shellard, _Cosmic Strings and Other Topological Defects_ (Cambridge University Press, 2000), ISBN 978-0-521-65476-0.
  * Patel and Vachaspati (2022) T. Patel and T. Vachaspati, JHEP 01, 059 (2022), eprint 2108.05357.
  * ’t Hooft (1974) G. ’t Hooft, Nucl. Phys. B 79, 276 (1974).
  * Vachaspati (1991) T. Vachaspati, Phys. Lett. B 265, 258 (1991).
  * Vachaspati (1992) T. Vachaspati, Phys. Rev. Lett. 68, 1977 (1992), URL https://link.aps.org/doi/10.1103/PhysRevLett.68.1977.