# Looking for monopolium excited states with electromagnetic detectors

Huner Fanchiotti  C.A. García Canal  IFLP(CONICET) and Department of Physics,
University of La Plata, C.C. 67 1900, La Plata, Argentina  Marco Traini
Dipartimento di Fisica, Università degli Studi di Trento, Via Sommarive 14,
I-38123 Povo (Trento), Italy  Vicente Vento  Departamento de Física Teórica
and IFIC, Universidad de Valencia - CSIC, E-46100 Burjassot (Valencia), Spain.

###### Abstract

We study the detection of particles with magnetic moment and no charge by
using coils and solenoids. The detector uses the Faraday-Lenz law to create a
current whose energy is measured. Our development can be applied to any
particle with magnetic moment. The neutron is ideal to show our detector
scheme for particles with long lifetimes and the result is encouraging.
However, our interest lies in the detection of monopolium states which decays
much faster than neutrons. The accelerator scenario with a relatively low
monopole mass $\sim 1000$ GeV requires the generalization of the scheme to
finite lifetimes. We use two descriptions, which we have called Dirac and
Schwinger, (also $\beta$-coupling), schemes, in our study. The theoretical
behavior of excited monopolia in coils teaches us many of their properties.
However, their small production cross sections, given the achievable LHC
luminosities, makes detection using coils difficult. Therefore, we generalize
the discussion to cosmological heavy monopolium. Our scheme becomes useful for
those and supplements detectors like MoEDAL and MAPP in distinguishing
monopolia from other long lived particles.

###### pacs:

04.20.Jb, 04.50+h, 04.50Gh,04.80-y

##  I Introduction

Monopoles and their detection has been a matter of much research since Dirac
discovered his now famous quantization condition Dirac:1931qs ; Dirac:1948um ,

$eg=\frac{N}{2}\;\;\;N=1,2,\ldots$ (1)

where $e$ is the electron charge, $g$ the monopole magnetic charge and we have
used natural units $\hbar=c=4\pi\varepsilon_{0}=1$.

In Dirac’s formulation, monopoles are assumed to exist as point-like particles
and quantum mechanical consistency conditions lead to establish the magnitude
of their magnetic charge. Monopole physics took a dramatic turn when ’t Hooft
tHooft:1974kcl  and Polyakov Polyakov:1974ek  independently discovered that
the SO(3) Georgi-Glashow model Georgi:1972cj  inevitably contains monopole
solutions. These topological monopoles are impossible to create in particle
collisions because of their huge GUT mass Preskill:1984gd  or in other models
with lower mass because of their complicated multi-particle structure
Drukier:1981fq .

Since the magnetic charge is conserved, monopoles will be produced
predominantly in monopole-antimonopole pairs. Inspired by the old idea of
Dirac and Zeldovich Dirac:1931qs ; Zeldovich:1978wj  that monopoles are not
seen free because they are confined by their strong magnetic forces we have
studied a monopole-antimonopole bound state that we have called monopolium
Vento:2007vy .

The discovery of monopole and dipole solutions in Kaluza Klein theories
Kaluza:1921tu ; Klein:1926tv  made these theories very exciting from a
theoretical point of view Gross:1983hb . In particular the dipole solution,
which is classically stable and therefore very long lived, forms a very
interesting state which we have also called monopolium in analogy with its
decaying cousin in gauge theories. The Kaluza Klein monopolium is extremely
massive with a mass of the order of the Planck mass and therefore impossible
to produce in laboratories. However, there might be clouds of them in the
cosmos which might enter our detectors Vento:2020vsq .

Much experimental research has been carried out into the search for monopoles
Cabrera:1982gz ; Milton:2006cp ; MoEDAL:2014ttp ; Acharya:2014nyr ;
Patrizii:2015uea . However, our interest here lies in the detection of
monopolium, who is charge-less but which has a magnetic moment in the presence
of magnetic fields or in excited deformed states Vento:2020vsq ; Vento:2019auh
. This is a phenomenologically oriented theoretical investigation and
therefore the study of the behavior of monopolium in the presence of coils
serves to determine interesting properties and energy regimes of excited
monopolia which may serve to guide more sophisticated experiments given the
progress of electromagnetic technologies at present.

In what follows we are going to discuss methods to detect magnetic moments and
how they might be used in the detection of monopolium. In Section II we
describe the effect of a magnetic moment traversing a coil. In Section III we
construct a detector for particles with magnetic moment and no charges. In
Section IV we will show some estimates analyzing the effect of a neutron beam
on the detector. In Section V we study the application to monopolium excited
states in accelerator and cosmological physics. Finally in Section VI we
collect the conclusions of our study.

##  II Effects due to a particle with magnetic moment passing through a
conducting coil

Our subject of study is a particle which has a permanent or an induced
magnetic dipole moment $\overrightarrow{\cal{M}}$. We call this particle
dipole for short. Assuming the size of the particle small compared with the
distance at which we are measuring the magnetic field its dipole moment will
produce a magnetic field Vento:2019auh

$\overrightarrow{B}_{d}(\vec{r})=\frac{3\overrightarrow{\cal{M}}\cdot\vec{r}\,\vec{r}-\overrightarrow{\cal{M}}\,r^{2}}{r^{5}},$
(2)

where $r=|\vec{r}|=\sqrt{(x-x_{0})^{2}+(y-y_{0})^{2}+(z-z_{0})^{2}}$,
$(x,y,z)$ is any point in space and $(x_{0},y_{0},z_{0})$ is the position of
the dipole considered point-like for this calculation.

Figure 1: We show our set up: a particle traveling with constant velocity
towards a conducting coil with its magnetic dipole moment perpendicular to the
plane of the coil. $S^{\prime}$ is the reference system associated to the
dipole and $S$ the one associated to the coil.

Let the dipole travel with a constant velocity towards a circular conducting
coil of radius $R$ with its magnetic moment perpendicular to the plane of the
coil as shown in Fig. 1. Let $S^{\prime}$ be the reference system associated
to the dipole and $S$ the reference system associated to the coil. We chose
the coil to be located in the $z=0$ plane, thus at time $t$ the dipole will be
located at $z=-vt$. The magnetic field felt by any point of space
$\vec{r^{\prime}}=(x^{\prime},y^{\prime},z^{\prime})$ when the dipole is
$\overrightarrow{\cal{M}}={\cal M}\,\hat{k^{\prime}}$ is

$\overrightarrow{B^{\prime}}_{d}(\vec{r^{\prime}})=\frac{3\mathcal{M}\hat{k^{\prime}}\cdot\vec{r^{\prime}}\,\vec{r^{\prime}}-{\mathcal{M}}\,\hat{k^{\prime}}\,r^{\prime
2}}{r^{\prime 5}},$ (3)

where $r^{\prime}=|\vec{r^{\prime}}|=\sqrt{x^{\prime 2}+y^{\prime 2}+z^{\prime
2}}$ and $\hat{k^{\prime}}$ is the unit vector in the $z$ direction. We need
the magnetic field at the coil. The Lorentz transformation from system
$S^{\prime}$ reference system to the $S$ reference system is given by,

$\displaystyle x^{\prime}=$ $\displaystyle x,$ $\displaystyle y^{\prime}=$
$\displaystyle y,$ $\displaystyle z^{\prime}=$ $\displaystyle\gamma(z-vt),$
$\displaystyle t^{\prime}=$ $\displaystyle\gamma(t-z),$

where $\gamma=1/\sqrt{1-v^{2}}$. Since we have no electric field present the
magnetic field changes Apyan:2007wq

$\displaystyle B^{\prime}_{x}$ $\displaystyle=\gamma B_{x},$ (4)
$\displaystyle B^{\prime}_{y}$ $\displaystyle=\gamma B_{y},$ (5)
$\displaystyle B^{\prime}_{z}$ $\displaystyle=B_{z}.$ (6)

With this geometry the flux through the coil is

$\Phi(t)=\int^{R}_{0}d\rho\int^{2\pi}_{0}d\varphi\,\overrightarrow{B}_{d}(x,y,-\gamma
vt)\cdot\hat{k},$ (7)

where $\rho$ and $\phi$ are the polar coordinates in the $z=0$ plane of the
coil and $R$ is the radius of the coil. This calculation can be performed
analytically and leads to

$\Phi(t)=-\frac{2{\cal M}\pi R^{2}}{((\gamma vt)^{2}+R^{2})^{3/2}}.$ (8)

Figure 2: Functional form of the voltage generated by a dipole with a magnetic
moment passing a conducting coil at uniform speed.

Applying the Faraday-Lenz law the induced Electro Motive Force (EMF) generated
by the magnetic field becomes

$V(\gamma\,v\,t,R)=\frac{6{\cal M}\pi R^{2}(\gamma v)^{2}t}{((\gamma
vt)^{2}+R^{2})^{5/2}}.$ (9)

This equation can be written using $z(t)=\gamma vt$, the area of the coil
$A=\pi R^{2}$ and the strength parameter $\chi=6{\mathcal{M}}\gamma v$ as

$V(z,A,\chi)=\frac{\chi Az}{(z^{2}+\frac{A}{\pi})^{\nicefrac{{5}}{{2}}}}.$
(10)

This function can be seen in Fig.2 for fixed $A$ and $\chi$.

We have assumed in the previous calculation that the dipole passes through the
axis of the coil. Ref. Apyan:2007wq  shows the result when the dipole does not
pass through the axis, which is non analytic. Off axis the shape of the pulse
is asymmetric and the time span shorter. For the purposes of the estimates
which will be calculated in this presentation Eq.(10) will be sufficient, with
the advantage of being analytic.

A simple mathematical calculation determines the position of the extrema at

$z=\pm\frac{1}{2}\sqrt{\frac{A}{\pi}}=\pm\frac{1}{2}R,$ (11)

where the voltage takes the value

$V(\pm\frac{1}{2}\sqrt{\frac{A}{\pi}},A,\chi)=\pm\frac{16\pi^{2}\chi}{25\sqrt{5}A}.$
(12)

The signal, a rise in the voltage, is governed by two parameters the strength
$\chi$ and the area of the coil, being proportional to the former and
inversely proportional to the latter, thus the ideal radius of the coil $R$
has to be as small as possible compatible with all the approximations.

Let us introduce a system with a large number of coils close to each other
forming a cylinder. There are two ways of creating this solenoid, either with
well separated coils or with closely winded coils. In the first case, as shown
in Fig. 3, the effect is simply to add the effect of each coil almost
independently.

Figure 3: The left figure shows the functional form of the voltage generated
by a particle with a magnetic moment when passing three independent coils at
uniform speed. The figure on the right shows the combined effect of the three
well separated coils forming a solenoid (solid curve) compared with the
independent effect of the three coils (dotted curve). The effect is basically
the same effect as that of the three coils independently.

If the coils are very close the effect is more sophisticated the advantage
being that we can put many coils in a shorter length to produce an enhancement
of the coil effect in shorter distance. This case requires a detailed
calculation that can be performed numerically:

$V_{N}(z,A,\chi,L)=\sum_{n=1}^{n=N}V(z-(n-1)\Delta z,A,\chi),$ (13)

where $V$ is the potential in Eq.(10), $N$ is the number of coils, $L$ is the
length of the coil cylinder and $\Delta z=\frac{L}{N}$. We show in Fig. 4 the
voltage divided by the number of coils generated by $50,500$ and $5000$ coils
located between $0\leq\frac{z}{R}\leq 5$. This result is only valid for a
large number of coils, when there is a cancellation between the positive and
negative potentials.

Figure 4: The left figure shows the functional form of the voltage generated
by a particle with a magnetic moment when passing a conducting coils’ system
of length $5R$ at uniform speed. The distance between the peaks is the length
of the coils’ system. The figure is composed by the calculation of the voltage
for $50,500$ and $5000$ loops divided by the number of loops showing that the
proportionality between number of loops and voltage persists for relatively
large lengths of the coils’ system. The figure on the right shows the same for
three different lengths of the coils’ system. The shorter the length the
greater the voltage.

Doing some numerical analysis we find the relation between the maxima of the
single coil potential and the $N$ coil potential for $N$ large

$\frac{V_{N}(max)}{N}\sim 0.1312\sqrt{A}\,V_{1}(max)\sim
0.3706\frac{\chi}{\sqrt{A}}.$ (14)

Moreover, we are able to find an approximate analytical solution valid for
large $N$

$\displaystyle\frac{V_{N\,approx}^{in}(z)}{N}$
$\displaystyle\sim\frac{-b}{(z^{2}+\frac{a}{\pi})^{\nicefrac{{5}}{{2}}}},$
$\displaystyle\frac{V_{N\,approx}^{out}(z)}{N}$
$\displaystyle\sim\frac{b}{((z-L)^{2}+\frac{a}{\pi})^{\nicefrac{{5}}{{2}}}},$
(15)

where $a=2A$ and $b=0.1198\,\chi\,A^{2}$. The $in$ potential describes the
approximation to the incoming minimum and the $out$ potential the
approximation to the outcoming maximum. In Fig.5 we show the result of the
exact calculation for $N=1000$ and compare it to the approximation for the
mentioned values of the parameters.

Figure 5: We show the functional form of the exact numerical calculation
Eq.(13)(solid) and the approximate one Eq.(15)(dashed) for $N=1000$, $L=5R$
$A=0.1R^{2}$ and $\chi=0.5$ V$\cdot R$.

It is interesting to note from the above expression that the peak of the
potential depends directly on the magnitude of the dipole moment and is
inversely proportional to the square root of the area of the coil. This means
that one does not need large coil sizes to construct a detector. Certainly
there are limitations to the sizes due to the approximations used, i.e for
circular coils $A=\pi R^{2}$ and $R$ has to be bigger than the size of the
system. Thus the ideal detector to see such dipole in the scenario presented
here seems to be the one shown in Fig. 6, a detector with a large number of
solenoids, each formed by a large number of coils, all linked in a circuit in
series to sum up the voltages. For a single particle the signal will be the
voltage structure shown in Fig. 4 multiplied by the number of coils, where the
separation between the voltage peaks is the length of the coil system. The
result of the calculation shows that the voltage scales nicely with the number
of coils and that the smaller the coil system the higher the potential for the
same number of coils. Once has thus to find a compromise between length and
number of coils. The width of the coil has not been taken into account in this
calculation, but its effect is irrelevant at this point as long as its size
$s$ is smaller than $R$.

Figure 6: The left figure shows the ensemble of coils of a possible detector
system where $W>>R$. The right figure shows the structure of a series of
closely located coils $R>>s$

##  III Determination of observables characterizing particles with magnetic
moment and no electromagnetic charges

In the first subsection we will calculate the current created in systems of
coils by the moving dipole as a means of characterizing particles with
magnetic moment and no charge, while in the second subsection we will comment
on the small effect on the particle motion of this induced current.

###  III.1 The induced current and the deposited energy in coil systems

A dipole traversing a coil creates a magnetic field due to the
${\overrightarrow{\cal M}}\cdot\hat{k}$ component of its magnetic dipole
moment whose flux in the coil generates, via the Faraday-Lenz law, an EMF
Eq.(9),

$V_{1}(z)=6{\mathcal{M}}_{z}\,\gamma\,v\,A\,z\frac{1}{(z^{2}+\frac{A}{\pi})^{\nicefrac{{5}}{{2}}}}.\\\
$ (16)

Only the $z$ component of the magnetic moment ${\overrightarrow{\cal
M}}\cdot\hat{k}={\cal M}_{z}={\cal M}\cos{\theta}$, where
$\cos{\theta}=\frac{{\overrightarrow{\cal M}}\cdot\hat{k}}{{\cal M}}$ is
active. Since we expect the dipoles initially in random directions in the
forward direction, the average ${\cal M}_{z}$ will be M multiplied by the
factor,

$\frac{\int_{0}^{2\pi}d\varphi\int_{-\frac{\pi}{2}}^{\frac{+\pi}{2}}d\theta\cos{\theta}}{\int_{0}^{2\pi}d\varphi\int_{-\frac{\pi}{2}}^{\frac{+\pi}{2}}d\theta}=\frac{2}{\pi}.$
(17)

and therefore Eq.(16) becomes

$V_{1}(z)=\frac{12}{\pi}{\mathcal{M}}\,\gamma\,v\,A\,z\frac{1}{(z^{2}+\frac{A}{\pi})^{\nicefrac{{5}}{{2}}}}.\\\
$ (18)

where ${\mathcal{M}}$ is the magnetic moment of the dipole.

For the $N$ coil solenoid, recalling Eq.(15), the EMF becomes

$\displaystyle\hskip 62.59596ptV_{N}^{in\;}(z)$
$\displaystyle\sim-0.4576\,{\mathcal{M}}\,\gamma\,v\,N\,A^{2}\frac{1}{(z^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}},$
$\displaystyle V_{N}^{out}(z)$
$\displaystyle\sim+0.4576\,{\mathcal{M}}\,\gamma\,v\,N\,A^{2}\frac{1}{((z-L)^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}},$
(19)

where $N$ is the number of coils in each coil system, $L$ its length, $A$ is
the area of the coil, $v$ the incoming velocity of the particle, $\gamma$ the
Lorentz factor and ${\mathcal{M}}$ is the magnetic moment.

The total EMF has to take into account the inductance, thus

$V_{T}=-\frac{d\Phi}{dt}-{\mathfrak{L}}\frac{d{\mathcal{I}}}{dt},$ (20)

where ${\mathfrak{L}}$ is the inductance and ${\mathcal{I}}$ the intensity of
the circuit. Ohm’s law then becomes the differential equation we have to solve
in order to obtain the intensity in the circuit

$\frac{d\Phi}{dt}+{\mathfrak{L}}\frac{d{\mathcal{I}}}{dt}+{\mathcal{I}}{\mathcal{R}}=0.$
(21)

where ${\mathcal{R}}=\frac{2\pi R}{S}\varrho N$, where $S$ is the surface of
the conductor and $\varrho$ the resistivity which changes from conventional
conductors, $\varrho=10^{-8}$ Ohm$\cdot$m, to superconductors
$\varrho=10^{-26}$ Ohm$\cdot$m.

If $\mathcal{R}$ corresponds to a conductor the inductance term is small and
only the induced EMF enters the equation, thus the intensity becomes
${\mathcal{I}}(z)=-\frac{1}{\mathcal{R}}\frac{d\Phi}{dt}=\frac{V(z)}{\mathcal{R}}$
and therefore from Eqs.(1) and (19) we get for one coil

${\mathcal{I}}_{1}(z)=\frac{12}{\pi}\frac{{\mathcal{M}}\,\gamma\,v\,A\,z}{{\mathcal{R}}}\frac{1}{(z^{2}+\frac{A}{\pi})^{\nicefrac{{5}}{{2}}}}=\frac{12}{\pi}\frac{{\mathcal{M}}\,\gamma\,v\,A\,z}{{\mathcal{R}}}\frac{1}{(z^{2}+R^{2})^{\nicefrac{{5}}{{2}}}},$
(22)

and for the solenoid

$\displaystyle\hskip 54.06006pt{\mathcal{I}}_{N}^{in\,}(z)$
$\displaystyle=-0.4576\,\frac{{\cal
M}\,\gamma\,v\,A^{2}\,N}{{\mathcal{R}}}\frac{1}{(z^{2}+2R^{2})^{\nicefrac{{5}}{{2}}}},$
$\displaystyle{\mathcal{I}}_{N}^{out}(z)$ $\displaystyle=+0.4576\,\frac{{\cal
M}\,\gamma\,v\,A^{2}\,N}{{\mathcal{R}}}\frac{1}{((z-L)^{2}+2R^{2})^{\nicefrac{{5}}{{2}}}}.$
(23)

If $\mathcal{R}$ corresponds to a superconductor then the term
${\mathcal{I}}{\mathcal{R}}$ is negligible and the intensity is given by
${\mathcal{I}}(z)=-\frac{\Phi}{{\mathfrak{L}}}=\frac{1}{\gamma\,v\,{\mathfrak{L}}}\int_{-\infty}^{z}V(z)dz$,
which becomes for the single coil

${\mathcal{I}}_{1}=\frac{4}{\pi}\frac{{\mathcal{M}}A}{\mathfrak{L}}\frac{1}{(z^{2}+\frac{A}{\pi})^{\nicefrac{{3}}{{2}}}}=\frac{4}{\pi}\frac{{\mathcal{M}}A}{\mathfrak{L}}\frac{1}{(z^{2}+R^{2})^{\nicefrac{{3}}{{2}}}},$
(24)

and for the $N$ coil solenoid

$\displaystyle\hskip 22.76228pt{\mathcal{I}}_{N}^{in}(z)$
$\displaystyle=-\frac{0.4576}{{\mathfrak{L}}}\,{\mathcal{M}}\,A^{2}\,N\int_{-\infty}^{z}\frac{1}{(z^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}}dz$
$\displaystyle=-\frac{0.3762}{{\mathfrak{L}}}\,{\mathcal{M}}\,N\,\frac{z(3R^{2}+z^{2})}{(z^{2}+2R^{2})^{\nicefrac{{3}}{{2}}}},$
$\displaystyle{\mathcal{I}}_{N}^{out}(z)$
$\displaystyle=\frac{0.4576}{{\mathfrak{L}}}\,{\mathcal{M}}\,A^{2}\,N\int_{-\infty}^{z}\frac{1}{((z-L)^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}}dz$
$\displaystyle=\frac{0.3762}{{\mathfrak{L}}}\,{\mathcal{M}}\,N\frac{(z-L)(3R^{2}+(z-L)^{2})}{((z-L)^{2}+2R^{2})^{\nicefrac{{3}}{{2}}}}.$
(25)

Let us study first the case of conducting coils, i.e. ${\mathcal{R}}\neq 0$
for stable particles111The case of decaying particles will be studied later
on.. The energy our detector can extract from the flux of dipoles is the
integral of the power in time

${\cal E}=\int_{-\infty}^{+\infty}{\frac{V^{2}(t)}{{\cal R}}dt}.$ (26)

Thus for the single coil the integral is immediate and we get

${\mathcal{E}}=\left(\frac{12}{\pi}\right)^{2}\frac{{\mathcal{M}}^{2}\gamma
v\,A^{2}}{{\mathcal{R}}}\int_{-\infty}^{+\infty}\frac{z^{2}}{(z^{2}+\frac{A}{\pi})^{5}}dz=\frac{45}{8\pi}\frac{{\mathcal{M}}^{2}\gamma
v}{R^{3}{\mathcal{R}}}.$ (27)

For the solenoid the integral expression is

${\mathcal{E}}\sim 0.2094\frac{{\cal M}^{2}\,\gamma\,v\,A^{4}\,N^{2}}{{\cal
R}}\int_{-\infty}^{+\infty}\left(\frac{1}{(z^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}}+\frac{1}{((z-L)^{2}+\frac{2A}{\pi})^{\nicefrac{{5}}{{2}}}}\right)^{2}dz,$
(28)

and its integration requires some discussion.

The integral of the square terms can be performed analytically while the cross
term not. In any case all of the three terms are very convergent and
numerically we can get the result quite fast. However, the square terms tend
to be much larger than the cross term, specially if $L$ is large compared to
the width of the potential bumps, since the cross term only takes into account
the overlap between the bumps. This argument has been checked numerically for
the scenarios of the present calculation and therefore we neglect the cross
term and get an analytic result which is

$\displaystyle{\mathcal{E}}$ $\displaystyle\sim$ $\displaystyle
0.2094\frac{{\mathcal{M}}^{2}\,\gamma\,v\,A^{4}\,N^{2}}{{\mathcal{R}}}\frac{35\pi}{64}\left(\frac{2A}{\pi}\right)^{-\nicefrac{{9}}{{2}}}$
(29) $\displaystyle\sim$ $\displaystyle
2.7451\frac{{\mathcal{M}}^{2}\,\gamma\,v\,N^{2}}{{\mathcal{R}}\sqrt{A}}.$

If we use the following units ${\cal M}$ (fm), $v$ in units of the velocity of
light in the vacuum, $S$ (mm2), $R$ (mm) and $\varrho$ (Ohm$\cdot$m) the
equation for the energy becomes for the single coil

${\mathcal{E}}=0.1682\,10^{-29}\,\frac{{\mathcal{M}}^{2}\gamma
v\,S}{R^{4}\varrho}\,\mbox{eV}$ (30)

and for the solenoid

$\hskip
14.22636pt{\mathcal{E}}=0.5130\,10^{-20}\,\frac{{\mathcal{M}}^{2}\,\gamma
v\,S\,N}{R^{2}\,\varrho}\,\mbox{ eV}.$ (31)

Let us now study the case of superconducting coils where the inductance term
is dominant. In this case

${\mathcal{E}}=\frac{{\mathfrak{L}}}{2}\int_{-\infty}^{+\infty}\frac{d{\mathcal{I}}_{N}^{2}(t)}{dt}dt=\frac{{\mathfrak{L}}}{2}\,(({\mathcal{I}}_{N}^{in})^{2}(+\infty)+({\mathcal{I}}_{N}^{out})^{2}(+\infty))=0.1415\,\frac{{\mathcal{M}}^{2}\,N^{2}}{{\mathfrak{L}}}$
(32)

For a thin long solenoid the ratio of inductance for superconducting and
normal solenoids is close to 1 HIRAKAWA1973287 , thus
${\mathfrak{L}}=\mu_{0}\frac{N^{2}A}{L}$ where ${\mathfrak{L}}$ is in H, if we
take $\mu_{0}=4\pi 10^{-7}$ H/m , $A$ the area of the coil m2, $L$ the length
of the coil in m. Recall that H = Ohm$\cdot$s, then

${\mathcal{E}}=0.2486\,10^{-16}\,\frac{{\mathcal{M}}^{2}L}{R^{2}}\,\mbox{eV}.$
(33)

where we are measuring ${\mathcal{M}}$ in fm, $R$ in mm and $L$ in m. This
choice of three units might seem strange at first, but it is related to the
natural scale of the physics and the apparatus: the magnetic moments of the
states of monopolium are microscopic (0-1000 fm), the most effective coils are
of mm size, and in order to have very large number of coils we need solenoids
lengths of meters.

The behavior of Eq.(31) and Eq.(33) are very different. For the former the
parameters at our disposal to increase the sensibility of the detector are the
number of coils $N$, the velocity of the dipole $v$, but specially the factor
$\gamma$, the radius of the coil $R$ and the resistivity of the material
$\varrho$. For the latter the length of the solenoid $L$ and the radius of the
coil $R$ but neither the velocity nor the number of coils play a role.

###  III.2 Effect of the induced current on the velocity of the particle

One might wonder if the induced current can produce a force which decelerates
considerably the initial velocity of the particle. For simplicity we will
perform a non relativistic calculation which is very transparent. Let us
assume a particle traveling with constant velocity towards a conducting coil
with its magnetic dipole moment perpendicular to the plane of the coil as
shown in Fig. 1. We would like to find out the motion of this particle due to
the effect of the Faraday-Lenz law which creates a magnetic field
$\vec{B}=B_{z}\hat{k}$ where

$B_{z}=\frac{2\pi R^{2}{\mathcal{I}}}{(z^{2}+R^{2})^{\nicefrac{{3}}{{2}}}},$
(34)

where ${\mathcal{I}}$ is the intensity running through the coil, $z$ is the
distance from the particle to the center of the coil and $R$ is the coil
radius. Newtons’s equation for the problem is given by

$M\frac{d^{2}z}{dt^{2}}=-{\mathcal{M}}\frac{dB_{z}}{dz}.$ (35)

In order to get the intensity we use Ohm’s law

$\frac{d\Phi}{dt}+{\mathcal{L}}\frac{d{\mathcal{I}}}{dt}+{\mathcal{I}}{\mathcal{R}}=0.$
(36)

where ${\mathcal{L}}$ is the inductance of the coil, ${\mathcal{R}}$ the
resistance and the magnetic flux $\Phi$ is given by

$\Phi(t)=-\frac{2{\cal M}\pi R^{2}}{(z^{2}+R^{2})^{3/2}}.$ (37)

We are dealing with a very good conductor, ${\mathcal{R}}=\frac{2\pi R}{\pi
r_{s}^{2}}\varrho$, where $r_{s}$ is the radius of the conductor and $\varrho$
the resistivity $\varrho=10^{-8}$ Ohm$\cdot$m and has a small inductance
${\mathcal{L}}=0.1$ nH. The set of equations that determine the motion of the
particle are

$\displaystyle M\frac{d^{2}z}{dt^{2}}$ $\displaystyle=$
$\displaystyle-6\pi{\mathcal{M}}R^{2}{\mathcal{I}}\frac{z}{(z^{2}+R^{2})^{\nicefrac{{3}}{{2}}}},$
(38)
$\displaystyle{\mathcal{L}}\frac{d{\mathcal{I}}}{dt}+{\mathcal{I}}{\mathcal{R}}$
$\displaystyle=$ $\displaystyle
6\pi{\mathcal{M}}R^{2}{\mathcal{I}}\frac{z}{(z^{2}+R^{2})^{\nicefrac{{3}}{{2}}}}\frac{dz}{dt}.$
(39)

This is a system of differential equations for $z(t)$ and ${\mathcal{I}}$ (t)
which has to be solved simultaneously. The initial conditions will be
$z_{i},v_{i}$ and ${\mathcal{I}}_{i}$. We shall take $z_{i}$ sufficiently far
from the coil so that the induced current is very small initially. The
parameters that influence the solution are
${\mathcal{L}},{\mathcal{R}},{\mathcal{M}}$ and $v_{i}$. The outcome of our
calculation is shown in Fig. 7.

Figure 7: A high excited monopolium (${\mathcal{M}}\sim 300$ fm) traveling
with constant velocity ($v=7\;10^{-5}$) from $z_{i}=-10$mm towards a
conducting coil located at $z=0$ with its magnetic dipole moment perpendicular
to the plane of the coil.The left figure shows the variation of the position
with time and the right figure the change in velocity with time.

As can be seen in in Fig. 7, the particle approaches the coil at large
distances without changing its velocity, Close to the coil, for large
${\mathcal{M}}$, the coefficient of the derivative becomes sizeable,
generating an acceleration in the opposite direction of the motion, which
reduces the velocity. The velocity thus decreases steadily as the monopolium
traverses the coil but only in its proximity. However, the effect on the
velocity is very small as can be seen in the figure, where we have used
$z_{i}=-10$mm, $v_{i}=7.\,10^{-5}$, $t=10^{-6}$s, and a very high lying
Rydberg state with ${\mathcal{M}}\sim 300$ fm. It is interesting to note the
small oscillation in the velocity at the origin which is an effect of the
interplay between inductance and resistance. Thus the induced current does not
influence significantly the calculation and we do not take it into
consideration in what follows.

##  IV Application to conventional particle detection: neutrons

Let us apply our formalism to a well known particle, the neutron. We start
using a solenoid made of a conventional conductor, $\varrho\sim 10^{-}8$
Ohm$\cdot$m. The simplest way to get a flux of neutrons is from a reactor from
which one can obtain a beam of thermal neutrons with a velocity $v\sim
2000m/s$. In the case of the neutron ${\mathcal{M}}$ is given in natural units
by

${\mathcal{M}}\sim-1.91\frac{e\hbar}{2m_{p}}\sim-6.1\,10^{-2}\,\mbox{ fm}.$
(40)

Note that we have used in the presentation the classical approximation
$\frac{2}{\pi}$ which is very close to the quantum mechanical
$\frac{m}{s(s+1)}=\frac{1/2}{3/4}=2/3$ for the neutron.

Thus our experiment consists of a beam of neutrons hitting a solenoid. Thus
since $v\sim 2000$ m/s $\sim 0.67\,10^{-5}c$, $\gamma$ is very close to $1$.
We take a thousand coils ($N=1000$ ) of ten millimeter radius ($R=10\;mm$) and
a surface of the conductor of $S=\pi r_{s}^{2}$ and $r_{s}=1$ mm, then the
accumulated energy for one neutron is

${\mathcal{E}}\sim 0.4018\;10^{-18}\,\mbox{eV}.$ (41)

We send a beam of thermal neutrons from a reactor with a flux typically of
$10^{13}$ neutrons cm${}^{-2}\,$ s-1 to the detector. Let the coil system has
a surface covered by $10$ solenoids, as in Fig. 6 right, all connected in
series. In $1$ s the energy accumulated will be

${\mathcal{E}}(1s)\sim 0.4018\,10^{-4}\,\mbox{eV}.$ (42)

In a few days of running one accumulates ${\mathcal{E}}\sim$ eV which is
easily measurable. Since the velocity of the beam is known, ${\mathcal{M}}$
becomes the only unknown. Therefore the coil system could serve as a detector
to determine the magnetic moment of the neutron.

If we use a superconductor solenoid assuming the solenoids of $L=1$ m length,

${\mathcal{E}}(1s)\sim 0.9250\,10^{-8}\,\mbox{eV},$ (43)

the difference arises because there is no $N$ dependence in the
superconducting case, since the inductance grows also with $N^{2}$. A
superconducting material is not favored in this case.

For the proposed detector, even for a single coil, superconductors are not
favored over normal conductors, thus we will use the latter from now on.

##  V Application to unconventional particle detection: monopolium

We discuss two types of experimental scenarios: one related to a relatively
light monopolium ($M\sim 2000$ GeV) which can be created by particle
collisions Epele:2012jn ; Mavromatos:2020gwk  and one related to a heavy
monopolium ($M\sim M_{GUT}$ and $M\sim M_{Planck}$) that might be created in
cosmological scenarios Preskill:1984gd ; Vento:2020vsq .

###  V.1 Accelerator monopolium

Let us suppose that we have to measure a flux of light monopolia using their
magnetic moment. This case resembles the neutron case except by the fact that
we have considered the neutron as a stable state due to its long lifetime and
monopolia might decay faster, and therefore their lifetime has to be taken
into account. Let us implement this feature in our description. Monopolium is
produced in particle collisions with small cross sections Epele:2012jn ;
Mavromatos:2020gwk . To describe the momopole-antimonopole potential in
monopolium there are several models in the literature Goebel1970 ;
PhysRev.160.1257 ; Barrie:2016wxf ; Baines:2018ltl ; Barrie:2021rqa . For the
purpose of our present investigation the approximation to the potential of
Schiff and Goebel Goebel1970 ; PhysRev.160.1257

$V(r)=-g^{2}\frac{1-exp(-2r/r_{0})}{r},$ (44)

used in refs. Epele:2007ic ; Epele:2008un , will be sufficiently clarifying.
The approximation consists in substituting the true wave functions by Coulomb
wave functions of high $n$. For each $r_{0}$ a different value of $n$ will be
best suited. We use the equation

$\rho_{n}=48\alpha^{2}n^{2},$ (45)

to parametrize all expectation values in terms of $\rho_{n}$, where
$\rho_{n}=r_{M}/r_{classical}$, $r_{M}$, the radius of the monopolium state,
relation which results from the calculation of the expectation value of $r$ in
the $(n,0)$ Coulomb state; $\alpha$ is the electromagnetic fine structure
constant $\sim\frac{1}{137}$.

In terms of $\rho_{n_{min}}$ the mass of ground state monopolium becomes
Epele:2007ic ; Epele:2008un

$M=m\left(2-\frac{3}{4\rho_{n_{min}}}\right).$ (46)

Thus given a monopole mass $m$ and a monopolium ground state mass $M$ we
determine $\rho_{n_{min}}$ which characterizes $n_{min}$. The binding energy
of the ground state is thus $E_{n_{min}}=\frac{3m}{4\rho_{n{min}}}$. The
excited states correspond to $n>n_{min}$. Let us label the excited states only
by their principal quantum number $n$ since we are assuming in our simple
Coulomb potential model that all the $l$ corresponding to an $n$ are
degenerate, thus their mass will be

$M_{n}=m\left(2-\frac{3}{4\rho_{n}}\right),$ (47)

with $\rho_{n}=48\alpha^{2}n^{2}$ and their binding energy
$E_{n}=\frac{3m}{4\rho_{n}}$. The high lying deformed ($l\neq 0$) states will
have a larger magnetic moment and a larger size. Recall that in a Coulomb like
potential

$\frac{\langle
nl|r|nl\rangle}{r_{Bohr}}\sim(\frac{3}{2}n^{2}-\frac{1}{2}{l}({l}+1))\sim
n^{2},$ (48)

thus the high lying states will increase their size approximately as $n^{2}$.
We estimate their magnetic moment as,

${\mathcal{M}_{n}}\sim g\,\langle
nl|r|nl\rangle=g\,\rho_{n}\,r_{classical}=\frac{3}{32\,\alpha^{\nicefrac{{3}}{{2}}}\,E_{n}},$
(49)

where we have used the Dirac quantization condition (DQC) $g\,e=\frac{1}{2}$
and $r_{classical}=\alpha_{g}/m$. Looking at Eq.(49) it is clear that Rydberg
states will have the largest magnetic moments.

In the previous sections we have considered the effect on coils of permanent
magnetic moments, even in the case of the neutron, due to its long lifetime,
but since the monopolium states might have a shorter lifetime we have to
rephrase our formulation. The two photon production of the monopolium excited
states is determined by Epele:2008un ; Epele:2012jn

$\Gamma^{2ph}_{n}(E)=\frac{2\,\beta_{n}^{4}\,m^{3}}{\alpha^{2}M_{n}^{2}}\left(\frac{E_{n}}{m}\right)^{\nicefrac{{3}}{{2}}}.$
(50)

At leading order the dominant photo-production production rate is determined
by the wave function at the origin and is only non zero for S ($l=0$) states.
The $l\neq 0$ states will arise by de-excitation to lower $n$ levels.
Analogously, at leading order the dominant decay into two photons is
determined by the wave function at the origin, thus their decay rate is given
by Eq.(50). Note that the decay rate for the S states goes like $1/n^{3}$
Dirac:1930bga ; Ore:1949te .

Our interest here lies in the high lying deformed Rydberg states which are in
high $l$ states, with vanishing wave function at the origin, that is to say
that the monopole and antimonopole are far away from each other and therefore
they do not annihilate. They will de-excite by spontaneous emission of photons
to the lower levels where they will annihilate either as S states or to non
leading order as $l\neq 0$ state decays. In order to get an estimate for
spontaneous emission as a function of $n$ we use the naive Bohr model
Dicus:1983ry ,

$\Gamma^{se}_{n}(E)=4096\,\beta_{n}^{4}\,\alpha\,m\left(\frac{E_{n}}{m}\right)^{3}.$
(51)

Note that this dependence goes like $1/n^{6}$, therefore lifetimes of these
Rydberg states increase notably with $n$.

In Eqs. (50) and (51) $\beta_{n}=1$ for Dirac coupling $g$ and
$\beta_{n}=v_{n}$ for Schwinger’s $\beta$-coupling Schwinger:1976fr , $\alpha$
is the fine structure constant, which appears after using the DQC, $m$ for the
monopole mass, $v_{n}=\sqrt{1-\frac{M_{n}^{2}}{E^{2}}}$ is the velocity of the
monopolium state, which depends on the collision kinematics since $E$ is the
center of mass energy of the two photon producing the state.

####  V.1.1 The Dirac coupling scheme

The states with the largest magnetic moment , i.e. the most deformed states
will be Rydberg states with very small binding energy, $E_{n}<<2m$. The
distance travelled by monopolium with binding energy $E_{n}$ for high $n$
before disintegrating is

$d_{n}(E)=\frac{\gamma_{n}\,v_{n}}{\Gamma^{se}_{n}}\sim
64\frac{v_{n}}{\sqrt{1-v_{n}^{2}}}\,\frac{\alpha^{5}}{m}\,n^{6},$ (52)

since spontaneous emission dominates for high $n$ to annihilation. We plot
$d_{n}(E)$ in Fig.8 for $E\sim 2000$ GeV as a function of $E_{n}$ for Rydberg
states.

Figure 8: Distance in $\mu$m travelled by monopolium with binding Energy
$E_{n}$ produced at a center of mass energy $E=2000$ GeV before
disintegrating.

Despite the fact that the curve is divergent for $E_{n}\rightarrow 0$, i.e.
$n\rightarrow\infty$, the cross section of the very excited states is so small
that with reasonable accumulated luminosities they will not be produced. We
will calculate later on the highest reachable values of $n$.

Let us develop the formalism for states of finite lifetime, which might be
also applicable in other scenarios. For finite lifetime, the energy stored in
one coil during the lifetime of state $n$ is

${\mathcal{E}}_{n}\sim\left(\frac{12}{\pi}\right)^{2}\frac{{\mathcal{M}}_{n}^{2}\gamma_{n}\,v_{n}\,A^{2}}{{\mathcal{R}}}\int_{-z{n}}^{+z_{n}}\frac{z^{2}}{(z^{2}+\frac{A}{\pi})^{5}}dz\sim\frac{144\,{\mathcal{M}}_{n}^{2}\gamma_{n}\,v_{n}}{{\mathcal{R}}R^{3}}\int_{-a_{n}}^{+a_{n}}\frac{y^{2}}{(y^{2}+1)^{5}}dy.$
(53)

where $a_{n}=\frac{z_{n}}{R}$.

This integral can be solved analytically

${\mathcal{F}}(a)=\int_{-a}^{+a}\frac{y^{2}}{(y^{2}+1)^{5}}dy=\frac{1}{192}\left(\frac{15a^{7}+55a^{5}+73a^{3}-15a}{(1+a^{2})^{4}}+15\arctan{a}\right).$
(54)

In Fig.9 we show ${\mathcal{E}}_{n}$ as a function of the binding energy for
$\mathcal{R}=\frac{2\pi R{\varrho}}{\pi r_{s}^{2}}$ with the radius of the
coils $R=10$ mm, the radius of the section of the coil $r_{s}=1$ mm and the
resistivity ${\varrho}=10^{-8}$ Ohm$\cdot$m.

Figure 9: We show the energy deposited in one coil by the passing of a
monopolium in state $n$ produced at a center of mass energy $E=2m$ and a
monopole mass $m=1000$ GeV as a function of binding energy $E_{n}$ (KeV) for
Rydberg states.

We next assume an accumulated luminosity ${\mathcal{L}}$ and calculate the
effect of all the states produced during this running. The cross section for
monopolium production by photon fusion, which is the dominating process at
large energies, is given for each bound state $n$ by Epele:2007ic ;
Epele:2008un

$\sigma(2\gamma\rightarrow{\mathfrak{M}}_{n})(E)=\frac{4\pi}{E^{2}}\frac{M_{n}^{2}\,\Gamma^{2ph}_{n}(E)\Gamma_{M_{n}}}{(E^{2}-M_{n}^{2})^{2}+M_{n}^{2}\Gamma_{M_{n}}^{2}},$
(55)

where $M_{n}$ is the mass of the $n$th state, $\Gamma^{2ph}_{n}(E)$
characterizes the production rate and $\Gamma_{M_{n}}$ arises from the
softening of the delta function $\delta(E^{2}-M_{n}^{2})$. In the case of LHC
the photons originate from composite particles (protons) and this expression
has to be convoluted with their structure. Also an equivalent photon
approximation for the photon spectrum of the intermediate constituents has to
be used. This process has been described in detail for the calculation of the
ground state monopolium cross section by proton-proton collisions at LHC in
ref. Epele:2007ic ; Epele:2008un . We use the latter calculation to get an
estimate for the present calculations. Let $\sigma_{n_{min}}(E)$ be the cross
section calculated for the ground state then the cross section for the $n$
state will be approximately given by

$\sigma_{n}(E)\sim\frac{\sigma(2\gamma\rightarrow{\mathfrak{M}}_{n})(E)}{\sigma(2\gamma\rightarrow{\mathfrak{M}})(E)}\sigma_{n_{min}}(E).$
(56)

Ground state monopolium for a binding energy of $\frac{2m}{15}$ leads to
$n_{min}\sim 47$ and an estimate for $\sigma_{n_{min}}(E)\sim 50$ fb at the
$14$ TeV LHC in the region of interest, $E\sim 2m$, in the case of Dirac
coupling with a monopole mass $m\sim 1000$ GeV Epele:2007ic ; Epele:2008un .

Now for an accumulated luminosity ${\mathcal{L}}$ the number of states created
for a center of mass energy $E$ is $\sigma_{n}(E){\mathcal{L}}$. The maximum
excited state for an energy $E$ will be obtained by solving for $n$ the
following equation

$\sigma_{n}(E){\mathcal{L}}=1$ (57)

The outcome of the calculation, for accumulated luminosities of $1000$, $2000$
and $10000$ fb-1, leads to $n_{max}$ as a function of E as plotted in Fig. 10.

Figure 10: We show $n_{max}$ in the region of interest around $E=2m$ for
luminosities of $1000$ (solid), $2000$ (dashed) and $10000$ (dotted) fb-1, a
monopole mass $m=1000$ GeV as a function of center of mass energy $E$ (GeV).

If we now sum the contribution of all states to a single coil deposited energy
we get

${\mathcal{E}}_{total}(E)\sim\sum_{nmin}^{nmax}{\mathcal{E}}_{n}(E)n^{2}{\mathcal{L}}\sigma_{n}(E),$
(58)

where $n^{2}$ is associated to the degeneracy of the states.

Figure 11: We show the energy deposited in a single coil
${\mathcal{E}}_{total}(E)$ by the passing of all monopolia produced by
accumulated luminosities at LHC of $1000$ (solid), $2000$ (dashed) and $10000$
(dotted) fb-1, as a function of center of mass energy $E$ (for $m=1000$ GeV) .

The results for accumulated luminosities of $1000$, $2000$ and $10000$ fb-1
are shown in Fig. 11. ${\mathcal{E}}_{total}(E)$ saturates close to $E\sim
2m$, where the highest Rydberg states contribute and drops slowly above due to
the cross section drop away from the resonances. The signal for monopolia to
be expected is a sudden jump in the deposited energy, as shown in Fig.11,
followed by disintegration into photons a few micrometers away, that will be
captured by the LHC photon detectors. The current signal is small $\sim\mu eV$
but measurable. Moreover, the excited states produce two types of very
distinct photons, the ones originating from annihilation Epele:2012jn ;
Barrie:2016wxf ; Fanchiotti:2017nkk ; Barrie:2021rqa  and the photons produced
by cascading to lower states. The latter arising from Rydberg states will have
much lower energy than the annihilation photons. With the present parameters
the latter have energies energies of KeV while the former of GeV. Lastly,
another way to see these monopolium states would be by the deformation of the
proton beam Vento:2019auh  after having seen a jump in the coil detector.

To end this section it must be stressed that the luminosity requirement given
the small cross section of the high Rydberg states sets a huge limitation on
the coil energy and the distance before disintegration. We will come back to
this point in Section V.2.

####  V.1.2 The Schwinger coupling scheme

The velocity in Eq.(51) in Schwinger coupling makes the lifetime very short at
threshold and therefore we expect some interesting physics around the
threshold region. Let us calculate the value of $d_{n}$ at threshold

$\lim_{E\rightarrow
M_{n}}d_{n}(E)\sim\frac{m^{2}}{4096\,\alpha\,E_{n}^{3}}\left(\frac{M_{n}}{2(E-M_{n})}\right)^{\nicefrac{{3}}{{2}}}.$
(59)

The distance travelled by monopolia diverges at threshold and we expect
physics outside the beam region. Unluckily the production cross section tends
to zero in that region. Thus close to threshold we expect a small number of
slow moving excited monopolia traveling for long distances as shown in Fig.
12. Note that we are limiting the results of the figure to LHC luminosities.
For cosmologic monopolia the distances will be shown to be much greater.

Figure 12: We show the distance travelled by the highest Rydberg monopolia
states produced by LHC luminosities of $1000$ (solid), $2000$ (dashed) and
$10000$ (dotted) fb-1 as a function of the energy away from threshold.

Let us calculate now the size of the deposited energy. Using Eqs.(53) and (54)
we get

${\mathcal{E}}_{n}=1.650\,10^{-29}\frac{{\mathcal{M}}_{n}^{2}\gamma_{n}\,v_{n}r_{s}^{2}{\mathcal{F}}(a_{n})}{R^{4}\,\varrho}\;\;\mbox{eV},$
(60)

where $r_{s}$ and $R$ are in mm,
$\mathcal{M}_{n}\sim\frac{3}{32\alpha^{\nicefrac{{3}}{{2}}}E_{n}}$ in GeV-1
and $\varrho$ in Ohm$\cdot$m. At threshold
$a_{n}\sim\frac{d_{n}}{2\,R}\rightarrow\infty$, thus
${\mathcal{F}}(a_{n})\rightarrow\frac{5\pi}{128}$, and therefore

$\lim_{E\rightarrow M_{n}}{{\mathcal{E}}_{n}}\sim
1.450\,10^{-31}\frac{r_{s}^{2}}{R^{4}\alpha^{3}\varrho}{\mathcal{F}}(a_{n})\frac{1}{E_{n}^{2}}\left(\frac{E-M_{n}}{M_{n}}\right)^{\nicefrac{{1}}{{2}}}\sim
0.6471\,10^{-21}\frac{1}{E_{n}^{2}}\left(\frac{E-M_{n}}{E}\right)^{\nicefrac{{1}}{{2}}}\;\;\mbox{eV},$
(61)

where in the last expression we have used $R=10$ mm, $r_{s}=1$ mm,
$\varrho=10^{-8}$ Ohm$\cdot$m and $\alpha=\frac{1}{137}$. We note that the
deposited energy tends to zero at threshold. However if we look carefully, as
shown in Fig. 13, we see that there is a maximum of the deposited energy close
to threshold.

Figure 13: We show the energy deposited in one coil by a monopole state of
binding energy $E_{n}=100$ MeV as a function of $E-M_{n}$. The dashed line
shows the limiting value Eq.(61) while the dotted line shows the exact
calculation Eq.(60)

.

Eq.(60) is divergent if we let $n$ run to infinity. However, as we saw in the
Dirac case the higher lying states are not produced once we fixed the
accumulated luminosity leading to a finite $n_{max}$ in the expression for the
total energy,

${\mathcal{E}}_{total}(E)\sim\sum_{nmin}^{nmax}{\mathcal{E}}_{n}(E)n^{2}{\mathcal{L}}\sigma_{n}(E).$
(62)

We follow the same procedure as before to obtain $n_{max}$. By using the cross
section for $E$ slightly above $M_{n}$ and recalling that in this case the
ground state monopolium cross section for Schwinger coupling is much smaller
($\sim 1$fb). We show in Fig. 14 three cases of $n_{max}$ as a function of the
difference in energy between threshold and measured. The cases correspond to
${\mathcal{L}}=1000,2000$ and $10000$ fb-1. We see that $n_{max}$ stabilizes
beyond 10 $GeV$ away from threshold around $\sim 460,580,1000$. Since
$n_{max}$ describes the Rydberg states it is clear that the higher the
luminosity the higher Rydberg states will be excited and therefore the larger
will be the magnetic moment and the distance travelled by the monopolia before
decaying. Compared with the Dirac case, since the coupling now is much weaker
close to threshold, the values of $n_{max}$ are smaller and therefore the
deposited energy is smaller despite the fact that the distance travelled is
larger.

Figure 14: We show the variation of $n_{max}$ way from threshold for three
accumulated luminosities of $1000$ (solid) , $2000$ (dashed) and $10000$
(dotted) fb-1.

Let us next show the total deposited energy of Eq.(62) in Fig. 15 for the same
three luminosities.

Figure 15: We show the energy deposited in a single coil by the monopole
excited states for accumulated luminosities of $1000$ (solid), $2000$ (dashed)
and $10000$ (dotted) fb-1.

The values are very small due to the relative low values of $n_{max}$. In this
case the distances travelled by monopolia for the three luminosities are shown
in Fig.12 and we notice that the distances where the values of $n_{max}$ are
maximum are small, thus the use of a solenoid of a large number of coils, as
discussed in Section II, is out of question. The deposited energy is therefore
too small to be detectable by actual means in this case. However, one might
still look for cascading and decay photons. The former in this case will have
energies of MeV.

If we sweep across $2m$ we see the characteristic feature of Schwinger
coupling: peaks at the threshold values and the fall off away from the
thresholds as shown in Fig. 16. This characteristic feature, for higher values
of the deposited energy, will occur in those scenarios with very large values
of $n_{max}$.

Figure 16: We show the energy deposited in a single coil by the monopole
excited states for an accumulated luminosity of $1000$ (solid) , $2000$
(dashed) and $10000$ (dotted) fb-1 as we sweep across the monopole-
antimonopole thresholds and above $2m$.

####  V.1.3  Summary of accelerator experiments

Let us summarize our findings in both subsections of collider experiments. The
experimental scheme discussed thus far is based on two crucial properties of
monopolia, namely their large magnetic moments in deformed states and their
decay into photons. An ideal experimental scenario would be to have a large
flux of Rydberg states of monopolium with very high $n$, this would produce a
large current in the coils and large decay distances. Unluckily this does not
happen in collider experiments due to the small cross sections of the high
lying Rydberg states and the limited luminosity. However, this study has
deepen our understanding of the properties of monopolia. We have done so by
discussing two schemes under the name of Dirac and Schwinger coupling
scenarios. The Dirac scenario is characterized by a strong coupling at
threshold and therefore the distance travelled by monopolia before decaying is
small, however it leads to relatively high $n$ excited states. The outcome of
our calculation shows that in this scenario we are on the verge of detection
at colliders if one is able to measure deposited energies at the coils of $\mu
eV$. Monopolia in this case decay inside the beam region with decay distances
of $\mu m$. Thus the signal for monopolia in colliders is a jump in the
deposited energy followed by a decay into photons.The photons are of two
types: cascading photons, which can be of relatively low energy, and
annihilation photons which are of very high energy. The Schwinger coupling
scenario on the other hand might lead to larger decay distances, mm, close to
threshold but the deposited energies are small and difficult to detect (feV).
This features have to do with the fact that the excited Rydberg states are not
very high $n$ states due to their low cross sections near threshold. Away from
threshold, i.e. were $v_{n}\sim 1$ both Schwinger and Dirac schemes are
identical. In the Schwinger scenario the coil scheme is not suitable for
collider experiments. One has then to resort only to the study of photon
decays as done sometime ago Epele:2012jn ; Barrie:2016wxf , but in the case of
excited states one has to look not only for photons of very high energies,
GeV, arising from disintegration, but also at cascading photons with lower
energies, KeV to MeV, for the highest Rydberg states. If by other means one
could create very high $n$ Rydberg states they would have have long decay
paths and a combination of solenoids with the MAPP detector Mitsou:2020hmt
would be an ideal experimental tool for their detection and characterization.
We see next that this might happen in astronomical observations. To conclude,
Figs. 11 and 16 summarize the different behavior of monopolia in the two
coupling schemes: the Dirac scheme shows a sudden jump about the zero binding
energy point ($E=2m$), while the Schwinger scheme shows peaks for each
threshold ($E=M_{n}$) accumulating as we approach the zero binding energy
point.

###  V.2 Cosmic monopolium

Let us pursue now the observation of cosmological monopolium states. Our setup
is particularly suited for heavy monopolia originating from GUT or Kaluza-
Klein theories, which cannot be produced in accelerators. However, we will
extend their detection analysis also to light monopolia because the very high
$n$ Rydberg states of light monopolium before dissociation in the Earth’s
magnetic field have large lifetimes as we have already seen. The discussion
related to the effect of the induced current on the particle velocity can be
easily generalized to cosmic monopolium. The huge mass of the monopolium
states leads to small accelerations and therefore the effect on the traveling
particle velocity is here also negligible.

####  V.2.1 Light monopolium

In the case of light monopolia ($m\sim 1000$ GeV) we have to generalize the
calculation of their velocity in a galaxy and find a reasonable flux bound.
Monopolia will interact with the cosmological magnetic field through their
magnetic moment, thus the force acting on them will be given by

$\vec{F}=-\vec{{\mathcal{M}}}\cdot\vec{\nabla}\vec{B},$ (63)

where ${\mathcal{M}}\sim gd$ is the magnetic moment, $d$ represents the
distance between the poles and $\vec{B}$ the cosmological magnetic field.

$\displaystyle B(R)=$
$\displaystyle\;\,B_{0}\;\tanh{\left(\frac{R-R_{C}}{R_{1}}\right)}\exp{\left(-\left|\frac{R-R_{C}}{R_{0}}\right|\right)}\hskip
42.67912pt$ $\displaystyle{\mbox{f}or}\;R>0,$ $\displaystyle=$ $\displaystyle-
B_{0}\;\tanh{\left(\frac{R+R_{C}}{R_{1}}\right)}\exp{\left(-\left|\frac{R+R_{C}}{R_{0}}\right|\right)}\hskip
42.67912pt$ $\displaystyle{\mbox{f}or}\;R<0.$

In the left hand side of Fig. 17 the model for the magnetic field of a galaxy
described in Eq. LABEL:galaxyEq is shown Vento:2020vsq  and in the right hand
side its gradient along the model galaxy.

Figure 17: We show left our one dimensional model for the magnetic field of a
galaxy and right its gradient. $M_{C}$ represents a monopolium cloud traveling
towards the center of the galaxy.

The mathematical structure of the field makes the analysis of the velocity of
monopolium complicated. In order to get an estimate we substitute the force
field by two simplified forces of adequate strength as shown in Fig. 18
compared to the actual force field. With this constant force field the
velocity becomes, after crossing the two accelerating structures, and using
relativistic kinematics

$v_{M}\sim\sqrt{1-\frac{1}{(\frac{gdB_{0}L}{2MR_{1}}+1)^{2}}}\sim\sqrt{1-\frac{1}{(0.32\,10^{-26}\,d+1)^{2}}}\,.$
(65)

The last expression has been calculated for a high $n$ Rydberg state, $M\sim
2000$ GeV, and for conventional values of the galaxy parameters $B_{0}=10\mu$G
$\sim 1.95\;10^{-25}$ GeV2, $R_{1}=0.5$ Kpc, where the width of the force
field $L\sim 2.2$ Kpc, with $d$ measured in fm. Despite the huge sizes of
Rydberg states ($d\sim 10^{9}$ fm $\sim 10^{-3}$ mm) the small prefactor in
the formula cannot be overcome and the velocity originating from the
gravitational magnetic fields will be much smaller than the galactic
gravitational velocity $v_{M}\sim 10^{-3}$. Moreover, latter is mass
independent Preskill:1984gd .

Figure 18: The simplified force field used in the calculation of the velocity
is in grey over imposed on the real force field. $L$ is the width of the
simplified force field.

The Rydberg states with long lifetimes and large magnetic moments are lightly
bound monopole-antimonopole pairs with binding energies of the order of eVs
and the small Parker bound can be obviated and detection is possible
Dicus:1983ry . The Parker bound is based on the monopoles taking energy from
the intergalactic magnetic field. However, monopolia below the dissociation
energy do not take energy from the magnetic field since they are almost
neutral, i.e., magnetic moment interactions are much weaker than magnetic
charge interactions. However, they might be dissociated by the much stronger
Earth’s magnetic field leading to free monopole-antimonopole pairs which might
annihilate or ionize metal rods depending on its production kinematics.
Monopolia below the dissociation limit survive and those close to the
dissociation limit have long lifetimes, of the order of minutes for Dirac
coupling, and much greater lifetimes for Schwinger coupling close to
threshold. Thus in this case their paths before annihilation will be very long
and we can use long solenoids with many coils increasing detection efficiency
enormously.

Let us calculate the maximum $n$ before dissociation by equating the Energy of
the Rydberg state, $E_{n}$ with the dissociation energy Dicus:1983ry

$E_{n}\sim gB_{E}r_{n}.$ (66)

Using the $E_{n}=\frac{3m}{4\rho_{n}}=\frac{3g^{2}}{4r_{n}}$ and the Earth’s
magnetic field $B_{E}\sim 1$ G $\sim 1.95\,10^{-20}$ GeV2, we get for the
maximum allowed size $r_{n}\lesssim\sqrt{\frac{3}{8eB_{E}}}\sim 3\,10^{-3}$
mm. Using now $r_{n}\sim\rho_{n}r_{classical}$ we get the $n$ for dissociation
$n\sim\sqrt{\frac{r_{n}m}{12\alpha}}$, where m is the monopole mass. For our
light monopole $m\sim 10^{3}$ GeV, $n\lesssim 10^{7}$ and a binding energy
$E_{n}\gtrsim 1eV$

Let us assume that we have one of those monopolia of $d\sim 10^{-3}$mm
reaching a solenoid of $10$ m long with $N\sim 10^{5}$ no tight coils made of
a good conductor $\varrho\sim 10^{-8}$ Ohm$\cdot$m, with $R\sim 10$ mm and
$r_{s}\sim 1$ mm. At a speed $v_{M}\sim 10^{-3}$ the solenoid will be
transversed since the decay distances are very large, as shown in Fig. 19 .
The solenoid will get a deposited energy for the highest Rydberg states of
$10^{-6}-10^{-1}$ eV per monopolia as shown in Fig. 19. This experimental
setup is suited to detect single events in the line with the original monopole
detectors Cabrera:1982gz ; Milton:2006cp ; MoEDAL:2014ttp ; Acharya:2014nyr ;
Patrizii:2015uea . A more sophisticated MoEDAL type detector, implemented with
solenoids and MAPP MoEDAL:2014ttp ; Mitsou:2020hmt  would be an ideal
detector. The Earth’s magnetic field might break up some high lying monopolia
Rydberg states and pairs of monopole-antimonopoles would be produced. These
could be detected by MoEDAL’s traditional design. If the pair annihilates into
photons MAPP could detect these photons. However, if the Rydberg states are
sufficiently bound and not broken by the Earth’s magnetic field, they produce
a sudden jump of the energy deposited in the solenoids, and MAPP will detect
the low energy photons from the cascading Rydberg states. Note that given
their long lifetimes annihilation, high energy photons, occurs beyond MAPP.
The cascading photons between the upper Rydberg states will be of low energy
($1-100$ eV) and easy to detect. To determine the mass, since the velocity of
monopolium is known, one has to resort to kinematics locating an adequate
target in the MAPP detector, and having monopolia interact with charged
particles which are easy to detect. Once their mass is known a spectroscopic
analysis will determine the dynamics and the mass of the monopole.

Figure 19: Left: The decay distance travelled by monopolia of high $n$: Dirac
coupling (solid) and Schwinger coupling (dashed), as a function of binding
energy. Right:The deposited energy on a solenoid of the characteristics
explained in the text as a function of the binding energy. For fixed binding
energy the deposited energy is the same for both Dirac and Schwinger
couplings.

####  V.2.2 Heavy monopolium

This scenario is thought for extremely heavy monopoles and therefore very
heavy monopolia. For GUT monopoles the Parker bound can be overcome when
monopole-antimonopole bind. In this case, by equating the binding energy to
the dissociation energy as above but using a mass for the monopole of
$10^{16}$ GeV, we get a limiting value for monopolia of $n\lesssim 5\,10^{13}$
which corresponds to binding energies $E_{n}\gtrsim 1$ eV and sizes of
$d\lesssim 10^{-3}$ mm as before. The velocity of the monopolia, in this case,
is also the intergalactic gravitational velocity $v_{M}\sim 10^{-3}$ which is
independent of mass. The crucial ingredients of our previous calculation,
binding energy and size, remain the same and the analysis follows previous
calculation for the very high Rydberg states leading to results similar to
those shown in Fig.19 for the deposited energy. The decay distance is much
larger since the ratio $(E_{n}/m)^{3}$ entering the spontaneous emission
lifetime, Eq. (51), is much smaller. Despite the difference the decay
distances are so large that for both light and GUT monopolia their decay will
occur far outside the apparatus. The low energy cascading photons are of
similar energy. Thus we see no new physics arising for the GUT monopoles
except their mass which should be measured by a kinematical detector within
our setup.

The idea behind Kaluza-Klein (KK) theories is that the world has more than
three spatial dimensions and some of them are curled up to form a circle so
small as to be unobservable Kaluza:1921tu ; Klein:1926tv . KK theories have
been the subject of revived interest in recent years since many standard model
extensions in extra dimensions yield KK theories. KK theories contain a very
rich topological structure, which includes very heavy monopoles whose mass is
around the Planck mass Gross:1983hb ; Sorkin:1983ns . Most important they also
contain other soliton solutions in different topological sectors. In
particular the dipole, which has the quantum numbers of a monopole-
antimonopole bound state. As we have seen above, in conventional gauge
theories monopolium has vacuum quantum numbers and annihilates. However, in KK
theories monopolium does not belong to the topological sector of the vacuum
and therefore it is classically stable Gross:1983hb . In realistic scenarios
the dipole does not have vacuum quantum numbers Newman:1963yy ; Hawking:1976jb
; Gibbons:1979xm ; Vento:2020vsq  and its structure is described in terms of
some parameter $d$ (distance between center of the poles), which approaches
the monopole-antimonopole structure as $d$ becomes large. The dipole can thus
be interpreted as a bound state with a mass smaller than twice the monopole
mass. The parameter $d$ describes the magnetic moment of the dipole. The
ground state (smallest mass) dipole, i.e. when $d\rightarrow 0$, has
vanishingly small dipole moment and therefore it is electromagnetically
neutral and thus has gravitational interaction only. When $d>0$ we can ascribe
these states to excited states of monopolium and in the large $d$ limit we are
discussing, the equivalent of Rydberg states. since those states are very much
like an elongated monopole-antimonopole structure separated by a distance $d$
the magnetic moment is

${\cal M}\sim g\,d.$ (67)

Once this similarity is exploited, again the parameters that play a role in
the calculation are the monopole mass $m\sim
M_{Plack}\sim\sqrt{\frac{\hbar\,c}{G}}\sim 10^{19}GeV$ where $G$ is the
gravitational constant, the binding energy, $E_{d}=2M-M_{d}$ and the mass
independent gravitational velocity $v_{d}\sim 10^{-3}$. In this case again
binding energies and sizes will be determined by avoiding dissociation. Let us
use for our calculation the mass formula for the dipole of the Gross-Perry
model Gross:1983hb ; Vento:2020vsq

$M_{d}=2\left(m-\frac{M_{d}^{2}}{\sqrt{4M_{2}^{2}+\frac{d^{2}}{G^{2}}}}\right),$
(68)

where $G$ is Newton’s constant $1.32\,10^{-39}$ fm/GeV,
$m\sim\frac{M_{Planck}}{4\sqrt{\alpha}}$ the monopole mass, with
$M_{Planck}\sim 1.22\,10^{19}$ GeV, $M_{d}$ the monopolia mass and $d$ is
given in fm and the masses in GeV. For large $d$ we obtain the binding energy
approximation for the Rydberg states

$E\sim\frac{8Gm^{2}}{d}.$ (69)

Equating this energy to the dissociation energy
$E=gB_{E}d\sim\frac{gB_{E}d}{0.197GeV\cdot fm}$, where $B_{E}$ is measured in
GeV2 and $d$ in fm, we get the maximum monopolium size in the Earth at
$d=\sqrt{\frac{1.58GeV\cdot fmGm^{2}}{gB_{E}}}$ which turns out to be $d\sim
10^{-2}$ mm with a binding energy is $E\sim 1$ eV. Size and binding energy are
very similar to the ones above. The physics is going to be the same, and
therefore the experimental setup is also suitable for this monopole dynmics.
Only a kinematic detector will distinguish between the three monopoles
studied. The remaining detectors determine the presence of monopoles or
monopolia but are not able to distinguish which type of monopoles are being
detected. The fact that they KK monopolia are stable does not make any
difference for detection purposes because the decay distance in gauge theories
is very large, thus from the point of view of our setups all monopolia look
stable. We foresee in this case two mechanisms of production given their huge
mass, either a primordial mechanism which took place before inflation
Vento:2020vsq  or a cataclysmic collision of monopolium clouds after
inflation. The former will produce mostly ground state or low excited
monopolia with small values of $d$, very difficult to detect with the present
setup. The latter will produce highly excited states with large $d$, but very
few in number. Similarly to our previous discussion we need detectors which
look at extraordinary events and the presented setup, i.e. MoEDAL monopole
detector together with by solenoids and MAPP supplemented by some kinematic
detector, is well suited MoEDAL:2014ttp ; Mitsou:2020hmt  .

##  VI Concluding remarks

Monopolium, a bound state of monopole-antimonopole, has no magnetic charge and
in its ground state no magnetic moment being very difficult to detect
directly. In conventional gauge theories it has the quantum numbers of the
vacuum and annihilates into photons, and these disintegrations have been
intensively studied Epele:2007ic ; Epele:2008un ; Epele:2012jn ;
Barrie:2016wxf ; Fanchiotti:2017nkk ; Barrie:2021rqa . In Kaluza-Klein
theories monopolium is classically stable and therefore very long lived
Gross:1983hb ; Vento:2020vsq  but its detection problems still persist since
it only interacts gravitationally in vacuum.

What happens with excited monopolium states? This paper deals with the study
of the excited states of monopolium. Excited states have permanent magnetic
moments and their interaction with magnetic and electric fields is known. We
have used a theoretical analysis of magnetic dipole detection to learn about
excited monopolium states. Analyzing the detailed characteristics of excited
monopoliun states we have discovered many properties, in particular of the
high $n$ Rydberg states.

The first part of the analysis deals with a theoretical description of the
interaction of magnetic moment with coils and solenoids both for conductors
and superconductors. Given that the difference between the two is not large
and that our aim is to learn about the behavior of Rydberg states more than to
perform a detailed experimental analysis of detection, we have continued only
with conductors. We have applied our analysis to the neutron, which for the
purposes of the analysis has been treated as a stable particle, describing a
possible experiment to measure the magnetic moment of the neutron.

We continue with the application to the detection of excited monopolium in
accelerators which requires a reformulation of the previous analysis to adapt
to states with finite lifetimes which affect the behavior of the coils. We
have done the calculation for the two most used schemes, the Dirac scheme and
the Schwinger or $\beta$ scheme. We have seen that, due to the low production
cross section of excited monopolium, the detection via the Faraday-Lenz
analysis is very difficult. There might be a slight chance of detection in
Dirac coupling, if one is able to measure relative low coil energies, but no
chance in Schwinger coupling. The Dirac coupling has the inconvenience of very
short decay distances which implies that the decay will occur inside the beam.
In both cases the excited Rydberg states will de-excite via photons of
energies in the KeV to MeV range, which compared to the decaying photons which
are at the level of hundreds of GeV, are clear signals of excited monopolia.
Thus besides the deposited energy, the detection of these cascading photons
together with the energetic annihilating ones, are clear signatures of excited
monopolia.

Very heavy monopoles cannot be produced in accelerators. However, nature might
provide us with them in the Cosmos. Since bound states do not absorb energy
from the galactic magnetic fields, monopolia can therefore overcome the Parker
bound. When these excited monopolia reach the Earth, the Earth’s magnetic
field might break them giving rise to monopole-antimonopole pairs, which might
not annihilate since the excited states are very large, $10^{-3}-10^{-2}$ mm.
Typical magnetic monopole detectors can be successful in seeing them
Cabrera:1982gz ; Milton:2006cp ; MoEDAL:2014ttp ; Acharya:2014nyr ;
Patrizii:2015uea . However, for larger binding energies the monopolium Rydberg
states remain and they are ideal candidates for the setup presented here.
These states, as we have shown, have large decay distances and therefore
appear as stable for the detectors. Thus, we can use large solenoids which
increase the deposited energy. These states emit extremely soft cascading
photons $\sim eV$. All these properties point that a Cosmological detector of
MoEDAL type, supplemented by MAPP and Faraday-Lenz solenoids could be ideal to
detect monopoles and/or excited monopolia.

## Acknowledgement

Vicente Vento would like to thank Nick Mavromatos, Vassia Mitsou and Jim
Pinfold for useful conversations. This work was supported in part by the
MICINN and UE Feder under contract FPA2016-77177-C2-1-P.

## References

  * (1) P. A. M. Dirac, “Quantised singularities in the electromagnetic field,,” Proc. R. Soc. Lond. A, vol. 133, pp. 60–72, 1931.
  * (2) P. A. M. Dirac, “The Theory of magnetic poles,” Phys. Rev., vol. 74, pp. 817–830, 1948.
  * (3) G. ’t Hooft, “Magnetic Monopoles in Unified Gauge Theories,” Nucl. Phys. B, vol. 79, pp. 276–284, 1974.
  * (4) A. M. Polyakov, “Particle Spectrum in the Quantum Field Theory,” JETP Lett., vol. 20, pp. 194–195, 1974.
  * (5) H. Georgi and S. L. Glashow, “Unified weak and electromagnetic interactions without neutral currents,” Phys. Rev. Lett., vol. 28, p. 1494, 1972.
  * (6) J. Preskill, “MAGNETIC MONOPOLES,” Ann. Rev. Nucl. Part. Sci., vol. 34, pp. 461–530, 1984.
  * (7) A. K. Drukier and S. Nussinov, “Monopole Pair Creation in Energetic Collisions: Is It Possible?,” Phys. Rev. Lett., vol. 49, p. 102, 1982\.
  * (8) Y. B. Zeldovich and M. Y. Khlopov, “On the Concentration of Relic Magnetic Monopoles in the Universe,” Phys. Lett. B, vol. 79, pp. 239–241, 1978\.
  * (9) V. Vento, “Hidden Dirac Monopoles,” Int. J. Mod. Phys. A, vol. 23, pp. 4023–4037, 2008.
  * (10) T. Kaluza, “Zum Unitätsproblem der Physik,” Int. J. Mod. Phys. D, vol. 27, no. 14, p. 1870001, 2018.
  * (11) O. Klein, “Quantum Theory and Five-Dimensional Theory of Relativity. (In German and English),” Z. Phys., vol. 37, pp. 895–906, 1926.
  * (12) D. J. Gross and M. J. Perry, “Magnetic Monopoles in Kaluza-Klein Theories,” Nucl. Phys. B, vol. 226, pp. 29–48, 1983.
  * (13) V. Vento, “Primordial monopolium as dark matter,” Eur. Phys. J. C, vol. 81, no. 3, p. 229, 2021.
  * (14) B. Cabrera, “First Results from a Superconductive Detector for Moving Magnetic Monopoles,” Phys. Rev. Lett., vol. 48, pp. 1378–1380, 1982.
  * (15) K. A. Milton, “Theoretical and experimental status of magnetic monopoles,” Rept. Prog. Phys., vol. 69, pp. 1637–1712, 2006.
  * (16) B. Acharya et al., “The Physics Programme Of The MoEDAL Experiment At The LHC,” Int. J. Mod. Phys. A, vol. 29, p. 1430050, 2014.
  * (17) B. Acharya et al., “The Physics Programme Of The MoEDAL Experiment At The LHC,” Int. J. Mod. Phys. A, vol. 29, p. 1430050, 2014.
  * (18) L. Patrizii and M. Spurio, “Status of Searches for Magnetic Monopoles,” Ann. Rev. Nucl. Part. Sci., vol. 65, pp. 279–302, 2015.
  * (19) V. Vento and M. Traini, “Scattering of charged particles off monopole–anti-monopole pairs,” Eur. Phys. J. C, vol. 80, no. 1, p. 62, 2020.
  * (20) A. Apyan, A. B. Apyan, and M. Schmitt, “Detecting neutrino magnetic moments with conducting loops,” Phys. Rev. D, vol. 77, p. 037901, 2008.
  * (21) H. Hirakawa and A. Futakawa, “Inductance of superconducting solenoids,” Cryogenics, vol. 13, no. 5, pp. 287–289, 1973.
  * (22) L. N. Epele, H. Fanchiotti, C. A. G. Canal, V. A. Mitsou, and V. Vento, “Looking for magnetic monopoles at LHC with diphoton events,” Eur. Phys. J. Plus, vol. 127, p. 60, 2012.
  * (23) N. E. Mavromatos and V. A. Mitsou, “Magnetic monopoles revisited: Models and searches at colliders and in the Cosmos,” Int. J. Mod. Phys. A, vol. 35, no. 23, p. 2030012, 2020.
  * (24) C. Goebel, Quanta, In: Essays in Theoretical Physics, ed. by P.G.O. Freund, C.J. Goebel, Y. Nambu.  Chicago: University of Chicago Press, 1970.
  * (25) L. I. Schiff, “Quarks and magnetic poles,” Phys. Rev., vol. 160, pp. 1257–1262, Aug 1967.
  * (26) N. D. Barrie, A. Sugamoto, and K. Yamashita, “Construction of a model of monopolium and its search via multiphoton channels at LHC,” PTEP, vol. 2016, no. 11, p. 113B02, 2016.
  * (27) S. Baines, N. E. Mavromatos, V. A. Mitsou, J. L. Pinfold, and A. Santra, “Monopole production via photon fusion and Drell–Yan processes: MadGraph implementation and perturbativity via velocity-dependent coupling and magnetic moment as novel features,” Eur. Phys. J. C, vol. 78, no. 11, p. 966, 2018.  [Erratum: Eur.Phys.J.C 79, 166 (2019)].
  * (28) N. D. Barrie, A. Sugamoto, M. Talia, and K. Yamashita, “Searching for Monopoles via Monopolium Multiphoton Decays,” 4 2021.
  * (29) L. N. Epele, H. Fanchiotti, C. A. Garcia Canal, and V. Vento, “Monopolium: The Key to monopoles,” Eur. Phys. J. C, vol. 56, pp. 87–95, 2008.
  * (30) L. N. Epele, H. Fanchiotti, C. A. G. Canal, and V. Vento, “Monopolium production from photon fusion at the Large Hadron Collider,” Eur. Phys. J. C, vol. 62, pp. 587–592, 2009.
  * (31) P. A. M. Dirac, “On the Annihilation of Electrons and Protons,” Proc. Cambridge Phil. Soc., vol. 26, pp. 361–375, 1930.
  * (32) A. Ore and J. L. Powell, “Three photon annihilation of an electron - positron pair,” Phys. Rev., vol. 75, pp. 1696–1699, 1949.
  * (33) D. A. Dicus and V. L. Teplitz, “Circumvention of Parker’s bound on galactic magnetic monopoles,” Nature, vol. 303, pp. 408–409, 1983.
  * (34) J. S. Schwinger, K. A. Milton, W.-y. Tsai, L. L. DeRaad, Jr., and D. C. Clark, “Nonrelativistic Dyon-Dyon Scattering,” Annals Phys., vol. 101, p. 451, 1976.
  * (35) H. Fanchiotti, C. A. García Canal, and V. Vento, “Multiphoton annihilation of monopolium,” Int. J. Mod. Phys. A, vol. 32, no. 35, p. 1750202, 2017.
  * (36) V. A. Mitsou, “MoEDAL physics results and future plans,” PoS, vol. CORFU2019, p. 009, 2020.
  * (37) R. d. Sorkin, “Kaluza-Klein Monopole,” Phys. Rev. Lett., vol. 51, pp. 87–90, 1983.
  * (38) E. Newman, L. Tamburino, and T. Unti, “Empty space generalization of the Schwarzschild metric,” J. Math. Phys., vol. 4, p. 915, 1963.
  * (39) S. W. Hawking, “Gravitational Instantons,” Phys. Lett. A, vol. 60, p. 81, 1977.
  * (40) G. W. Gibbons and S. W. Hawking, “Classification of Gravitational Instanton Symmetries,” Commun. Math. Phys., vol. 66, pp. 291–310, 1979.