1

95. Magnetic Monopoles

95. Magnetic Monopoles
Revised March 2024 by D. Milstead (Stockholm U.) and E.J. Weinberg (Columbia U.).

95.1 Theory of magnetic monopoles
The symmetry between electric and magnetic fields in the source-free Maxwell’s equations
naturally suggests that electric charges might have magnetic counterparts, known as magnetic
monopoles. Although the greatest interest has been in the supermassive monopoles that are a firmprediction of all grand unified theories, one cannot exclude the possibility of lighter monopoles.
In either case, the magnetic charge is constrained by a quantization condition first found by
Dirac [1]. Consider a monopole with magnetic charge QM and a Coulomb magnetic field
B=

QM r̂
.
4π r2

(95.1)

Any vector potential A whose curl is equal to B must be singular along some line running fromthe origin to spatial infinity. This Dirac string singularity could potentially be detected throughthe extra phase that the wavefunction of a particle with electric charge QE would acquire if it
moved along a loop encircling the string. For the string to be unobservable, this phase must be amultiple of 2π. Requiring that this be the case for any pair of electric and magnetic charges gives
the condition that all charges be integer multiples of minimum charges Qmin
and QminE
M obeyingmin
Qmin
E QM = 2π .

(95.2)(For monopoles which also carry an electric charge, called dyons [2], the quantization conditions
on their electric charges can be modified. However, the constraints on magnetic charges, as well as
those on all purely electric particles, will be unchanged.)
Another way to understand this result is to note that the conserved orbital angular momentumof a point electric charge moving in the field of a magnetic monopole has an additional component,with
L = mr × v − 4πQE QM r̂
(95.3)
Requiring the radial component of L to be quantized in half-integer units yields Eq. 95.2.
If there are unbroken gauge symmetries in addition to the U(1) of electromagnetism, the above
analysis must be modified [3, 4]. For example, a monopole could have both a U(1) magnetic charge
and a color magnetic charge. The latter could combine with the color charge of a quark to givean additional contribution to the phase factor associated with a loop around the Dirac string, sothat the U(1) charge could be the Dirac charge QD
M ≡ 2π/e, the result that would be obtained bysubstituting the electron charge into Eq. (95.2). On the other hand, for monopoles without colormagnetic charge, one would simply insert the quark electric charges into Eq. 95.2 and conclude
that QM must be a multiple of 6π/e.
The prediction of GUT monopoles arises from the work of ’t Hooft [5] and Polyakov [6], whoshowed that certain spontaneously broken gauge theories have nonsingular classical solutions thatlead to magnetic monopoles in the quantum theory. The simplest example occurs in a theory
where the vacuum expectation value of a triplet Higgs field φ breaks an SU(2) gauge symmetry
down to the U(1) of electromagnetism and gives a mass MV to two of the gauge bosons. In orderto have finite energy, φ must approach a vacuum value at infinity. However, there is a continuous
family of possible vacua, since the scalar field potential determines only the magnitude v of hφi,but not its orientation in the internal SU(2) space. In the monopole solution, the direction of φin internal space is correlated with the position in physical space; i.e., φa ∼ vr̂a . The stability of
S. Navas et al. (Particle Data Group), Phys. Rev. D 110, 030001 (2024)
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95. Magnetic Monopoles2

the solution follows from the fact that this twisting Higgs field cannot be smoothly deformed to a
spatially uniform vacuum configuration. Reducing the energetic cost of the spatial variation of φrequires a nonzero gauge potential, which turns out to yield the magnetic field corresponding to acharge QM = 4π/e. Numerical solution of the classical field equations shows that the mass of thismonopole is
4πMV
Mmon ∼.
(95.4)
e2
The essential ingredient here was the fact that the Higgs fields at spatial infinity could bearranged in a topologically nontrivial configuration. A discussion of the general conditions under which this is possible is beyond the scope of this review, so we restrict ourselves to the two
phenomenologically most important cases.
The first is the standard electroweak theory, with SU(2) × U(1) broken to U(1). There are no
topologically nontrivial configurations of the Higgs field, and hence no topologically stable monopole
solutions. Although electroweak scale monopoles are thus not required, there have been claims thatthey might not be ruled out. For example, there have been a variety of proposals in this direction
involving modifications of the Lagrangian.
The second case is when any simple Lie group is broken to a subgroup with a U(1) factor, a casethat includes all grand unified theories. Here the spectrum of states must include a topologicallystable monopole whose mass is determined by the mass scale of the symmetry breaking that allows
nontrivial topology. For example, an SU(5) model with
M

M

X
W
SU(5) −→
SU(3) × SU(2) × U(1) −→SU(3) × U(1)(95.5)

has a monopole [7] with QM = 2π/e and mass
Mmon ∼

4πMX,
g2(95.6)

where g is the SU(5) gauge coupling. For a unification scale of 1016 GeV, these monopoles would
have a mass Mmon ∼ 1017 – 1018 GeV.
In theories with several stages of symmetry breaking, monopoles of different mass scales canarise. In an SO(10) theory with
M

M

1
2
SO(10) −→SU(4) × SU(2) × SU(2) −→SU(3) × SU(2) × U(1)(95.7)

there is monopole with QM = 2π/e and mass ∼ 4πM1 /g 2 and a much lighter monopole with
QM = 4π/e and mass ∼ 4πM2 /g 2 [8].
The central core of a GUT monopole contains the fields of the superheavy gauge bosons thatmediate baryon number violation, so one might expect that baryon number conservation could
be violated in baryon–monopole scattering. The surprising feature, pointed out by Callan [9]and Rubakov [10], is that these processes are not suppressed by powers of the gauge boson mass.
Instead, the cross-sections for catalysis processes such as p + monopole → e+ + π 0 + monopole are
essentially geometric; i.e., σ∆B β ∼ 10−27 cm2 , where β = v/c. Note, however, that this catalysis
is model-dependent and is not even a universal property of all GUT monopoles.

95.2 Production and Annihilation
GUT monopoles are far too massive to be produced in any foreseeable accelerator. However,they could have been produced in the early Universe as topological defects arising via the Kibble mechanism [11] in a symmetry-breaking phase transition. Estimates of the initial monopole
abundance, and of the degree to which it can be reduced by monopole-antimonopole annihilation,
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95. Magnetic Monopoles3

predict a present-day monopole abundance that exceeds by many orders of magnitude the astrophysical and experimental bounds described below [12]. Cosmological inflation and other proposedsolutions to this primordial monopole problem generically lead to present-day abundances exponentially smaller than could be plausibly detected, although potentially observable abundances canbe obtained in scenarios with carefully tuned parameters.
If monopoles light enough to be produced at colliders exist, one would expect that these could
be produced by analogs of the electromagnetic processes that produce pairs of electrically charged
particles. Because of the large size of the magnetic charge, this is a strong coupling problem forwhich perturbation theory cannot be trusted. Indeed, the problem of obtaining reliable quantitative
estimates of the production cross-sections remains an open one, on which there is no clear consensus.

95.3 Astrophysical and Cosmological Bounds
If there were no galactic magnetic field, one would expect monopoles in the galaxy to havetypical velocities of the order of 10−3 c, comparable to the virial velocity in the galaxy (relevant
if the monopoles cluster with the galaxy) and the peculiar velocity of the galaxy with respect to
the CMB rest frame (relevant if the monopoles are not bound to the galaxy). This situation ismodified by the existence of a galactic magnetic field B ∼ 3µG. A monopole with the Dirac charge
and mass M would be accelerated by this field to a velocity

vmag ∼

c,M . 1011 GeV ,1710−3 c 10 MGeV

1/2

,

M & 1011 GeV .

(95.8)

Accelerating these monopoles drains energy from the magnetic field. Parker [13] obtained an
upper bound on the flux of monopoles in the galaxy by requiring that the rate of this energy loss be
small compared to the time scale on which the galactic field can be regenerated. With reasonablechoices for the astrophysical parameters (see Ref. [14] for details), this Parker bound is10−15 cm−2 sr−1 sec−1 ,


F <
M
−15

cm−2 sr−1 sec−1 ,
10
17
10

GeV

M . 1017 GeV ,M & 1017 GeV .

(95.9)

Applying similar arguments to an earlier seed field that was the progenitor of the current galactic
field leads to a tighter bound [15],


F <

M
+ (3 × 10−6 ) 10−16 cm−2 sr−1 sec−1 .
17
10 GeV(95.10)Considering magnetic fields in galactic clusters gives a bound [16] which, although less secure, isabout three orders of magnitude lower than the Parker bound.
A flux bound can also be inferred from the total mass of monopoles in the Universe. If themonopole mass density is a fraction ΩM of the critical density, and the monopoles were uniformly
distributed throughout the Universe, there would be a monopole flux
−16

Funiform = 1.3 × 10

ΩM1017 GeVM!v
cm−2 sr−1 sec−1 .
10−3 c(95.11)

If we assume that ΩM ∼ 0.1, this gives a stronger constraint than the Parker bound for M ∼ 1015
GeV. However, monopoles with masses ∼ 1017 GeV are not ejected by the galactic field and canbe gravitationally bound to the galaxy. In this case their flux within the galaxy is increased by
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95. Magnetic Monopoles

4

about five orders of magnitude for a given value of ΩM , and the mass density bound only becomesstronger than the Parker bound for M ∼ 1018 GeV.
A much more stringent flux bound applies to GUT monopoles that catalyze baryon number
violation. The essential idea is that compact astrophysical objects would capture monopoles at a
rate proportional to the galactic flux. These monopoles would then catalyze proton decay, withthe energy released in the decay leading to an observable increase in the luminosity of the object.
A variety of bounds, based on neutron stars [17–21] , white dwarfs [22], and Jovian planets [23]have been obtained. These depend in the obvious manner on the catalysis cross section, but also
on the details of the astrophysical scenarios; e.g., on how much the accumulated density is reducedby monopole-antimonopole annihilation, and on whether monopoles accumulated in the progenitor
star survive its collapse to a white dwarf or neutron star. The bounds obtained in this manner lie
in the range


σ∆B β
F
∼ (10−18 − 10−29 )cm−2 sr−1 sec−1 .
(95.12)10−27 cm2
It is important to remember that not all GUT monopoles catalyze baryon number nonconservation. In particular, the intermediate mass monopoles that arise in some GUTs at later stages ofsymmetry-breaking are examples of theoretically motivated monopoles that are exempt from thebound of the above equation.

95.4 Searches for Magnetic Monopoles
To date there have been no confirmed observations of exotic particles possessing magnetic
charge. Precision measurements of the properties of known particles have led to tight limits on thevalues of magnetic charge they may possess. Using the induction method (see below), the electron’s−24 QD [24]. Furthermore, measurements of themagnetic charge has been found to be Qm
e < 10M
anomalous magnetic moment of the muon have been used to place a model dependent lower limitof 120 GeV on the monopole mass 1 [25]. Nevertheless, guided mainly by Dirac’s argument andthe predicted existence of monopoles from spontaneous symmetry breaking mechanisms, searches
have been routinely made for monopoles produced at accelerators, in cosmic rays, and boundin matter [26]. Although the resultant limits from such searches are usually made under theassumption of a particle possessing only magnetic charge, most of the searches are also sensitive todyons.

95.5 Search Techniques
Search strategies are determined by the expected interactions of monopoles as they pass throughmatter. These would give rise to a number of striking characteristic signatures. Since a complete
description of monopole search techniques falls outside of the scope of this minireview, only themost common methods are described below. More comprehensive descriptions of search techniques
can be found in Refs. [27, 28].
The induction method exploits the long-range electromagnetic interaction of the monopole with
the quantum state of a superconducting ring which would lead to a monopole which passes throughsuch a ring inducing a permanent current. The induction technique typically uses Superconducting
Quantum Interference Devices (SQUID) technology for detection and is employed for searches formonopoles in cosmic rays and matter. Another approach is to exploit the electromagnetic energy
loss of monopoles. Monopoles with Dirac charge would typically lose energy at a rate which isseveral thousand times larger than that expected from particles possessing the elementary electriccharge. Consequently, scintillators, gas chambers and nuclear track detectors (NTDs) have been
1

Where no ambiguity is likely to arise, a reference to a monopole implies a particle possessing Dirac charge.

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95. Magnetic Monopoles

Flux upper limit (cm-2s-1sr-1)

5

Mass (GeV)

Figure 95.1: Left: upper flux limits for GUT monopoles as a function of β. Right: upper flux
limits for monopoles as a function of monopole mass for β > 0.05.
used in cosmic ray and collider experiments. A further approach, which has been used at colliders,is to search for particles describing a non-helical path in a uniform magnetic field.
95.5.1 Searches for Monopoles Bound in Matter
Monopoles have been sought in a range of bulk materials which it is assumed would haveabsorbed incident cosmic ray monopoles over a long exposure time of order million years. Materials
which have been studied include moon rock, meteorites, manganese modules, and sea water [29,30].
A stringent upper limit on the monopoles per nucleon ratio of ∼10−29 has been obtained [30].
95.5.2 Searches in Cosmic Rays
Direct searches for monopoles in cosmic rays refer to those experiments in which the passage ofthe monopole is measured by an active detector. Searches made assuming a catalysis processes inwhich GUT monopoles could induce nucleon decay are discussed in the next section. To interpretthe results of the non-catalysis searches, the cross section for the catalysis process is typically either
set to zero [31] or assigned a modest value (1mb) [32].
Although early cosmic ray searches using the induction technique [33] and NTDs [34] observedmonopole candidates, none of these apparent observations have been confirmed. Recent experiments have typically employed large scale detectors. The MACRO experiment at the Gran Sasso
underground laboratory comprised three different types of detector: liquid scintillator, limited
stream tubes, and NTDs, which provided a total acceptance of ∼ 10000m2 for an isotropic flux.
As shown in Fig. 95.1, this experiment has so far provided the most extensive β-dependent flux
limits for GUT monopoles with Dirac charge [32]. Also shown are limits from an experiment at
the OHYA mine in Japan [31], which used a 2000m2 array of NTDs, and from the IceCube [35, 36]
and Antares [37] experiments which each employ the Cerenkov method.
In Fig. 95.1, upper flux limits are also shown as a function of monopole mass for monopole speed
β > 0.05. In addition to MACRO and OYHA flux limits, results from the SLIM [38] high-altitude
experiment are shown. The SLIM experiment provided a good sensitivity to intermediate massmonopoles (105 . M . 1012 GeV).
In addition to the results shown in Fig. 95.1, the NoVA experiment [39] has recently set an
upper limit on monopole flux of 2 × 10−14 cm−2 s−1 sr−1 for speeds 6 × 10−4 < β < 10−3 for masses
greater thsan 5 × 108 GeV. Stringent constraints on the flux of ultra-relativistic monopoles havebeen obtained at the Pierre Auger Observatory [40] which was sensitive to monopoles with γ values
31st May, 2024

695. Magnetic Monopoles

ranging from 109 to 1012 , leading to flux limits in the range 10−15 − 2.5 × 10−21 cm−2 s−1 sr−1 .
The RICE [41] and ANITA-II experiments [42] at the South Pole have also sought ultra-relativistic
monopoles with γ values of 107 . γ . 1012 and 109 . γ . 1013 , respectively, and which produced
flux limits as low as 2.5 × 10−21 cm−2 s−1 sr−1 .
95.5.3 Searches via the Catalysis of Nucleon-Decay
Searches have been performed for evidence of the catalysed decay of a nucleon by a monopole,
as predicted by the Callan-Rubakov mechanism. The searches are thus sensitive to the assumed
value of the catalysis decay cross section. Searches have been made with the Soudan [43] andMacro [44] experiments, using tracking detectors. Searches at IMB [45], the underwater Lake Baikal
experiment [46, 47] and the IceCube experiment [48] which exploit the Cerenkov effect have also
been made. The resulting β-dependent flux limits from these experiments typically vary between∼ 10−18 and ∼ 10−14 cm−2 sr−1 s−1 . A search for low energy neutrinos (assumed to be produced frominduced proton decay in the sun) was made at Super-Kamiokande [49]. A model- and β-dependentof limit of 6.3 × 10−24 ( 10β−3 )2 cm−2 sr−1 s−1 was obtained.
95.5.4 Searches at Colliders
Searches have been performed at hadron-hadron, electron-positron and lepton-hadron experiments. Collider searches can be broadly classed as being direct or indirect. In a direct search,evidence of the passage of a monopole through material, such as a charged particle track, is sought.
In indirect searches, virtual monopole processes are assumed to influence the production rates ofcertain final states.
95.5.4.1 Direct Searches at Colliders
Collider experiments typically express their results in terms of upper limits on a production
cross section and/or monopole mass. To calculate these limits, ansatzes are used to model the
kinematics of monopole-antimonopole pair production processes since perturbative field theory
cannot be used to calculate the rate and kinematic properties of produced monopoles. Limits
therefore suffer from a degree of model-dependence, implying that a comparison between the resultsof different experiments can be problematic, in particular when this concerns excluded mass regions.
A conservative approach with as little model-dependence as possible is thus to present representativevalues of the upper cross-section limits as a function of the centre-of-mass energy of the collisions,as shown in Fig. 95.2 for recent results from high energy colliders.
Searches for monopoles produced at the highest available energies in hadron-hadron collisions
were made in pp collisions at the LHC by the ATLAS [50–53] and MoEDAL [54–57] experiments.
ATLAS looked for highly ionising particles leaving a characteristic energy deposition profile in the
tracker and calorimeter. MOEDAL used a NTD to tag highly ionising particles and the induction
method to search for stopped monopoles. The charge-dependent mass limits extend up to around
4 TeV. The ATLAS experiment has considered hypotheses of monopoles with charges between
D
D
D
0.5QD
M and 2QM while MoEDAL has quoted limits for monopoles in the charge range QM to 5QM .
Models considered include monopole-pair production via photon fusion along with, as is commonlyused in hadron-hadron collisions, Drell-Yan processes [58]. MoEDAL has also looked for monopole
produced via the Schwinger mechanism in the intense magnetic field created in PbPb collisions [57].
Tevatron searches have also been carried out by the CDF [59] and E882 [60] experiments. The
CDF experiment used a dedicated time-of-flight system whereas the E882 experiment employed
the induction technique to search for stopped monopoles in discarded detector material which
had been part of the CDF and D0 detectors using periods of luminosity. Earlier searches at theTevatron, such as [61], used NTDs and were based on comparatively modest amounts of integrated
luminosity. Lower energy hadron-hadron experiments have employed a variety of search techniques
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95. Magnetic Monopoles

7

1.0E+01

Cross section limit (pb)

1.0E+00
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E+01

1.0E+02

1.0E+03

1.0E+04

Centre of mass energy (GeV)

Figure 95.2: Upper limits on the production cross sections of monopoles from various colliderbased experiments. Results from lepton-hadron (triangle), electron-positron (filled circle), protonantiproton (star) and proton-proton (filled square) experiments are shown.

including plastic track detectors [62] and searches for trapped monopoles [63].
The only LEP-2 search was made by OPAL [64] which quoted cross section limits for the
production of monopoles possessing masses up to around 103 GeV. At LEP-1, searches were madewith NTDs deployed around an interaction region. This allowed a range of charges to be sought
for masses up to ∼ 45 GeV. The L6-MODAL experiment [65] gave limits for monopoles withD
charges in the range 0.9QD
M and 3.6QM , whilst an earlier search by the MODAL experiment wassensitive to monopoles with charges as low as 0.1QD
M [66]. The deployment of NTDs around thebeam interaction point was also used at earlier e+ e− colliders such as KEK [67] and PETRA [68].
Searches at e+ e− facilities have also been made for particles following non-helical trajectories [69,70].
There has so far been one search for monopole production in lepton-hadron scattering. Using theinduction method, monopoles were sought which could have stopped in the aluminium beampipe
which had been used by the H1 experiment at HERA [71]. Cross section limits were set forDmonopoles with charges in the range QD
M − 6QM for masses up to around 140 GeV.
95.5.4.2 Indirect Searches at Colliders
It has been proposed that virtual monopoles can mediate processes which give rise to multiphoton final-states [72, 73]. Photon-based searches were made by the D0 [74] and L3 [75] experiments. The D0 work led to spin-dependent lower mass limits of between 610 and 1580 GeV, while
L3 reported a lower mass limit of 510 GeV. However, it should be stressed that uncertainties on
the theoretical calculations which were used to derive these limits are difficult to estimate.
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