# Monopole production via photon fusion and Drell–Yan processes: MadGraph implementation and perturbativity via velocity-dependent coupling and magnetic moment as novel features

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## Abstract

In this work we consider point-like monopole production via photon-fusion and Drell–Yan processes in the framework of an effective *U*(1) gauge field theory obtained from conventional models describing the interaction of spin Open image in new window magnetically-charged fields with ordinary photons, upon *electric-magnetic dualisation*. We present arguments based on such dualities which support the conjecture of an effective monopole-velocity-dependent magnetic charge. For the cases of spin- Open image in new window and spin-1 monopoles, we also include a magnetic-moment term \(\kappa \), which is treated as a new phenomenological parameter and, together with the velocity-dependent coupling, allows for a perturbative treatment of the cross-section calculation. We discuss unitarity issues within these effective field theories, in particular we point out that in the spin-1 monopole case only the value \(\kappa =1\) may restore unitarity. However from an effective-field-theory point of view, this lack of unitarity should not be viewed as an impediment for the phenomenological studies and experimental searches of generic spin-1 monopoles, given that the potential appearance of new degrees of freedom in the ultraviolet completion of such models might restore it. The second part of the paper deals with an appropriate implementation of photon-fusion and Drell–Yan processes based on the above theoretical scenarios into MadGraph UFO models, aimed to serve as a useful tool in interpretations of monopole searches at colliders such as LHC, especially for photon fusion, given that it has not been considered by experimental collaborations so far. Moreover, the experimental implications of such perturbatively reliable monopole searches have been laid out.

## 1 Introduction

Eighty seven years since its concrete formulation by Dirac [1, 2] as a quantum mechanical source of magnetic poles, the magnetic monopole remains a hypothetical particle. Although there are concrete field-theoretical models beyond the Standard Model (SM) of particle physics which contain concrete monopole solutions [3, 4, 5, 6, 7, 8, 9, 10, 13], these are extended objects with complicated substructure, and their production at collider is either impossible, as their mass range is beyond the capabilities of the latter [3, 4, 13], or extremely suppressed, due to their underlying composite nature [14]. On the other hand, point-like monopoles, originally envisaged by Dirac, are sources of singular magnetic fields for which the underlying theory, if any, is completely unknown, even though in principle (due to their point-like nature) they could avoid suppression in production.

*electric-magnetic duality*. That is deriving the corresponding cross sections from perturbative field-theoretical models describing the interaction of fields of various spins, Open image in new window and 1 with photons, upon the replacement of the electric charge \(q_e\) by the magnetic charge

*g*, the latter obeying Dirac’s quantisation rule

*c*is the speed of light in vacuo, \(\hbar \) is the reduced Planck constant, \(\epsilon _0\) is the vacuum permittivity and \({\mathbb Z}\) is the set of integers (with \(n=0\) denoting the absence of magnetic charge). The quantity \(\xi \) depends on the system of units used, with \(\xi =0\) representing the CGS Gaussian system of units, and \(\xi =1\) the SI system of units. In natural SI units (\(\hbar =c = \epsilon _0=1\)), which we adopt here, the fine-structure constant at zero energy scales is given by \(\alpha = \frac{e^2}{4\pi } = \frac{1}{137}\) with \(e>0\) the positron charge, from which (1) yields

The electric-magnetic duality replacement, which obeys the quantisation rule (1), may be used as a basis for the evaluation of monopole-production cross sections from collisions of SM particles (quarks and leptons). Unfortunately, due to the large value of the magnetic charge (2), such a replacement renders the corresponding production process non-perturbative, consequently the strong-magnetic-coupling limit *dual* theory is not well defined. Nevertheless, one may attempt to set benchmark scenarios for the cross sections by using tree-level Feynman-like graphs from such dual theories. This is standard practice in all point-like monopole searches at colliders so far [15].

*M*, typical graphs participating in monopole-antimonopole pair production at LHC from proton-proton (

*pp*) collisions are given in Fig. 1. There are two kinds of such processes: the Drell–Yan (DY) (see Fig. 1b) and the photon-fusion (PF) induced production (see Fig. 1c, d). We also mention, to be complete, that in photon-photon production we have not only elastic but also semi-elastic and inelastic photon-fusion processes. For spin- Open image in new window monopoles relevant in this discussion, a comparison between the respective perturbative cross sections has been provided first in Ref. [16, 17] for \(p\bar{p}\) collisions at a centre-of-mass energy \(\sqrt{s}=1.96\ \hbox {TeV}\), and subsequently for

*pp*collisions of \(\sqrt{s}\) up to 14 TeV in Ref. [18]. The conclusion from such analyses was that for \(\sqrt{s}=1.96\ \hbox {TeV}\) the two cross sections are of comparable magnitude [18], whilst for \(\sqrt{s}=14\ \hbox {TeV}\) PF dominates DY by a factor \(> 50\), thus stressing the need to utilise the latter in monopole-search interpretations.

In this work we extend such combined (DY plus PF) studies to also incorporate spin-0 and spin-1 monopoles. In Ref. [16, 17], the authors have calculated (in appropriate dualised models) the total cross sections for monopole pair production by photon fusion for three different spin models, spins 0, Open image in new window and 1. Theoretically, the cross sections increase with increasing spin, for a fixed (common) value of the monopole mass. It should be stressed that the expressions for the cross sections are specific to particular definitions of the monopole interactions. Specifically, the spin- Open image in new window model fixes the particle in a minimally coupled theory, dual to standard QED, thus mirroring the observed behaviour of the electron with a gyromagnetic ratio \(g_{Re}=2\). The corresponding magnetic moment \(\kappa \) is assumed zero. The magnetically charged monopole with spin 1 considered in [16, 17], on the other hand, is characterised by a non zero magnetic moment term \(\kappa =1\), which is the value that characterises the charged \(W^{\pm }\) bosons in the SM. In fact the model mirrors the interactions of such bosons with a photon, but in the dual theory, where the electric charge is replaced by the magnetic charge. The value \(\kappa =1\) is the only one that respects unitarity [19]. For spin-0 monopoles, the dual theory of which resembles the Scalar Quantum Electrodynamics (SQED), no magnetic moment is allowed.

In this work, we generalise the discussion to include an arbitrary value for the magnetic dipole moment for monopoles with spin. The detailed reasoning for this is given in the next section. Hence, we shall treat \(\kappa \) as a *new phenomenological parameter*.^{1} We note that, setting the issue of unitarity aside, phenomenological models of charged *W*-bosons interacting with photons with \(\kappa \ne 1\) have been considered in the past [20], where it was demonstrated that the behaviour of the total cross section for the *W*-boson pair production for the \(\kappa =1\) case is quite distinct from the \(\kappa \ne 1\) cases. In the current work, we shall dualise such models to use them as effective theories for the \(\kappa \ne 1\) spin-1 monopole case, generalising the work of [16, 17]. As we shall see, one may allow for some formal large-\(\kappa \) limit where, despite the strong magnetic coupling, the associated monopole-pair production cross sections can be made finite. In fact one may give meaning – under some circumstances to be specified below (specifically, the production of slow monopoles) – to the perturbative tree-level Feynman-like graphs of the effective theory. The relevant formalism for arbitrary \(\kappa \) and various monopole spins will then be used as a guide for the construction of appropriate MadGraph [21] algorithms that can be used as tools for data analyses in monopole searches at colliders. We remark for completeness that, for fast relativistic monopoles (characterised by a relative velocity \(\beta \simeq 1\)) passing through materials, the (large) number of electron-positron pairs produced can be used as a signal for the presence of the monopole [22]. The perturbativity conditions discussed in the present article, which pertain to slowly moving monopoles (with \(\beta \ll 1\)), of relevance to MoEDAL-LHC searches [23, 24], do not apply to such cases, the study of which requires non-perturbative treatments.

The structure of the article is the following: Sect. 2 is a review of the formal procedure to construct perturbative cross sections from the scattering amplitudes relevant to monopole production. Particular attention is given to a discussion of a rather unresolved current issue, regarding point-like monopole production through scattering of SM particles, namely the use of an effective magnetic charge coupling that depends on the relative velocity \(\beta \) between the monopole and the centre-of-mass of the producing particles (quarks in the case of interest here, see Fig. 1). We also motivate the introduction of the magnetic dipole moment for monopoles with spin. In Sect. 3, the differential and total cross sections for monopole-antimonopole pair production via PF and DY processes are derived for various monopole spins. Combined limits of large magnetic-moment parameter \(\kappa \) and small \(\beta \) can lead to finite perturbatively valid results, providing some support for the effective formalism. The pertinent Feynman rules are implemented in a dedicated MadGraph model, which is described in detail in Sect. 4. In Sect. 5, the monopole phenomenology at the LHC is discussed, utilising the MadGraph UFO models developed in this work, in the context of the above theoretical considerations. Conclusions and outlook are given in Sect. 6, while details on the calculations are provided in appendices A and B.

## 2 From amplitudes to kinematic distributions

We are interested in the electromagnetic interactions of a monopole of spin Open image in new window with ordinary photons. The corresponding theory is an effective *U*(1) gauge theory which is obtained after appropriate *dualisation* of the pertinent field theories describing the interactions of charged fields of spin-*S* with photons. However, there is a subtle issue here, which we now proceed to discuss, and which will be relevant to our subsequent studies. It concerns a potential dependence of the effective magnetic charge on the relative velocity of the monopole pair and the centre-of-mass of the producing particles, as noted in [25, 26].

*e*and mass

*m*, off a magnetic monopole, with magnetic charge

*g*and mass

*M*, reads [25, 27]:

*c*denotes the speed of light in vacuo and \(\theta \) is the scattering angle, given in [25]: \(\cos (\frac{\theta }{2} ) = \cos (\frac{\chi }{2}) \left| \sin \left( \frac{\pi /2}{\cos (\chi /2)}\right) \right| \). \(\mu = \frac{m\, M}{m + M} \) is the reduced mass of the two body problem at hand, with \(\mu \simeq m\) in the cases of monopoles where \(M \gg m\) which is of interest here. The angle \(\chi = 2 {{\mathrm{arccot}}}{(\mu v_0 b / |\kappa |)}\) defines the (Poincaré) cone on which the (classical) trajectory of the electron in the background of the monopole is confined, with

*b*the impact parameter. We note for completeness, that the cross section diverges in two occasions:

### 2.1 Velocity-dependent magnetic charge

*monopole-velocity dependent magnetic charge*has to be considered in the corresponding cross section formulae:

*both*a \(\beta \)-independent and a \(\beta \)-dependent magnetic charges, and then one may compare the corresponding bounds, as done in the recent searches by the MoEDAL Collaboration [32]. The monopole velocity \(\beta \) used in this work is given by the

*Lorentz invariant expression*in terms of the monopole mass

*M*and the Mandelstam variable

*s*(A8), where

*s*representing the square of the centre-of-mass energy of the fusing incoming particles (photons or (anti)quarks) (\(\sqrt{s}=2E_{\gamma /q}\)):

*S*field with photons, with the following substitutions:

An important remark is in order at this point. Since the ‘velocity’ \(\beta \) (7) is expressed in terms of Lorentz-invariant Mandelstam variables, the Lorentz invariance of the effective field theory action of the monopole is not affected by the introduction of the effective \(\beta \)-dependent magnetic charge (8). However, there is a well known paradox, due to Weinberg [33], who pointed out that the amplitude for a single photon exchange between and electric and a magnetic current (of relevance to DY monopole-antimonopole production processes) is *neither* Lorentz *nor gauge* invariant due to the rôle played by the Dirac string, which contradicts the fact that monopoles appear as consistent soliton solutions in Lorentz and gauge invariant field theories [3, 4]. It is only recently [34], that this paradox was arguably resolved, albeit within toy models of monopoles, coupled to photons, with perturbative electric and magnetic couplings. The resolution of Weinberg’s paradox in such models is provided by a resummation of soft photons, which was possible due to the pertubatively small magnetic charges involved. Such a resummation resulted in the exponentiation of the Lorentz (and gauge) non-invariant terms pointed out in [33] to a (Bohm-Aharonov type) phase factor in the respective amplitude. In this sense, the modulus of the amplitude (and hence the associated physical observables, such as cross sections) are Lorentz (and gauge) invariant, with the important result that, upon Dirac quantisation (1), the (resummed over soft photons) amplitude itself is Lorentz invariant. It is in such Lorentz (and gauge) invariant frameworks that, as we conjecture, the considerations of the effective, velocity-dependent magnetic charge (8), may apply, which by the way also implies perturbative magnetic couplings for sufficiently small production velocities of the monopoles in the laboratory frame.

### 2.2 The magnetic dipole moment as a novel free parameter for monopoles with spin

As already mentioned in the introductory Sect. 1, in previous effective-field theory treatments of spin-1 Dirac monopoles [16, 17], a magnetic dipole moment with the value \(\kappa = 1\) has been introduced mimicking the unitary SM case of \(W^{\pm }\) bosons (representing the monopoles), interacting with photons. Lacking an underlying microscopic model for point-like monopoles, the above restrictions in the value of the magnetic moment may not necessarily be applied to the monopole field. Indeed, for monopoles with spin, such a parameter may arise, e.g. by quantum corrections, in similar spirit to the electron case. The difference of course is that in the latter case it is the electric charge of the electron that would play a rôle, while in the monopole case it is the magnetic charge which, in view of its large value following the quantisation rule (2), cannot be treated perturbatively. Nonetheless, a non trivial (possibly large) magnetic moment might be induced in such a case, which might also be responsible for the restoration of unitarity of the effective theory. For example, one might hope that the apparent unitarity issues for generic \(\kappa \ne 1\) values in the case of spin-1 monopoles can be remedied by embedding the corresponding theory in microscopic ultraviolet complete models beyond the SM, in much the same way as unitarity is restored in the case of the \(W^\pm \) gauge bosons interacting with photons in the SM case.

*r*. One can readily confirm that \(\vec {\nabla } \cdot \vec {B}_\mathrm{D} = 0\).

*z*-axis, in which case the unit vector \(\widehat{n} =(0, 0, 1)\) also lies along that axis. The regularised form of the monopole’s magnetic field intensity yields the correct formula \(\vec {\nabla } \cdot \vec {B}^\mathrm{reg}_\mathrm{monopole} = 4\pi \, g \, \delta ^{(3)}(\mathbf {r})\), implying that the magnetic monopole is the source of a field.

If one considers a charged particle looping the Dirac string far away from the position of the monopole, one would then observe that it is the singular part of the magnetic field (10), \(\vec {B}_\mathrm{sing}\), which contributes to the phase of the electron wavefunction [27], \(q_e \oint _L \mathrm d\vec {x} \cdot \vec {A} = \int _{\Sigma (L)} \mathrm d\sigma \cdot \vec {B}_\mathrm{sing} = 4\pi q_e g \). The magnetic dipole moment does *not* contribute to the singular part of the magnetic field, and thus the charge quantisation (1) is not affected. We note that, as a result of the \(r^3\) suppression, the contributions of (9) would be subdominant, at large distances *r* from the monopole centre, as compared to those of (10).

It is worth remarking at this stage that one can also view [35, 36] the quantisation rule (1) itself as a consequence of representing the (non-physical) singular string solenoid, assumed in the original Dirac’s construction [1, 2], as a collection of (small) *fictitious* current loops (with an area perpendicular to the solenoid’s axis). Each one of these loops will induce a magnetic moment *IA*, with *I* the current and *A* the area of the loop (assumed vanishing in this case). Assuming a uniform magnetic moment per unit length \({\mathcal M}\) for the Dirac string, then, and taking into account that the solenoid may be viewed as the limiting case of a magnetic dipole of infinite length, one may apply the aforementioned formula (9) in this case to derive the singular magnetic field of the monopole itself, in which case the magnetic charge is obtained as \(g \propto {\mathcal M}\) [35, 36]. However, we stress, that, the contribution of the induced magnetic moment of the quantum effective theory of a monopole with spin, whose strength depends on the parameter \(\kappa \), which we shall discuss in this work, is independent of that due to the magnetic charge *g*, as explained above. Lacking though an underlying fundamental theory for the point-like monopole the determination of \(\kappa \) is at present not possible.

^{2}to see the irrelevance of the magnetic dipole moment for the quantisation rule (1), which avoids the use of Dirac strings. To this end, one covers the three-space surrounding the monopole by two hemispheres, with appropriate gauge potentials defined in each of them, whose curl yields the corresponding magnetic field strengths. For the magnetic monopole gauge potential one has the expressions [27]:

*r*from the centre of the sphere where the monopole is located, is of the form

*z*-axis); this is not singular at the poles \(\theta =0,\pi \) (in fact it vanishes there). The total potential in each hemisphere is then given by the corresponding sum \(\vec {A}_i + \vec {A}_\mathrm{D}, i=S,N\). Hence, the magnetic moment does not contribute to the difference, and thus it does not affect the wavefunction phase, which is associated only with the monopole part (13).

## 3 Cross sections for spin-*S* monopole production

In this section we derive the pertinent Feynman rules and then proceed to give expressions for the associated differential and total production cross sections for monopole fields of various spins. The pertinent expressions are evaluated using the package FeynCalc [37, 38] in Mathematica. We commence the discussion with the well-studied cases of scalar (spin-0) and fermion (spin- Open image in new window ) monopole cases, but extend the fermion-monopole case to include an arbitrary magnetic moment term \(\kappa \ne 0\). Then we proceed to discuss the less studied case of a spin-1 monopole including an arbitrary magnetic moment term \(\kappa \). We consider both \(\beta \)-dependent and \(\beta \)-independent magnetic couplings. In an attempt to make some sense of the perturbative estimates, we discuss, where appropriate, various formal limits of weak \(\beta \) and large \(\kappa \) for which the pertinent cross sections remain finite. In each spin case we present both PF and DY cross sections, which will help us present a comparison at the end of the section.

### 3.1 Scalar monopole

*U*(1) gauge field representing the photon. Searches at the LHC [31, 32, 39, 40, 41] and other colliders have set upper cross-section limits is this scenario assuming Drell–Yan production. The Lagrangian describing the electromagnetic interactions of the monopole is given simply by a dualisation of the SQED Lagrangian

*U*(1) gauge field.

#### 3.1.1 Pair production of spin-0 monopoles via photon fusion

There are three possible graphs contributing to scalar monopole production by PF, a *t*-channel, *u*-channel and seagull graph shown in Fig. 2. Their respective matrix amplitudes are given by Eq. (B2) in “Appendix B”. *M* is the spin-0 boson mass, \(\varepsilon _{\lambda }(q_1)\) and \(\varepsilon _{\lambda ^{'}}(q_2)\) are the photon polarisations, \(p_{1}\) and \(p_{2}\) are the monopoles four-momenta such that \(p_{i_{\mu }}^{2}=M^{2}\), and \(q_{1}\) and \(q_{2}\) are the photons four-momenta such that \(q_{i_{\mu }}^{2}=0\), as defined in Fig. 2.

#### 3.1.2 Pair production of spin-0 monopoles via Drell–Yan

*s*-channel. The quark lines are supplemented by momentum 4-vectors \(q_{1\mu }\) and \(q_{2\mu }\), where \(q_{1,2}^2=m^2\) and the scalar monopole lines have momentum 4-vectors \(p_{1\mu }\) and \(p_{2\mu }\), where \(p_{1,2}^2=M^2\) on shell. The centre-of-mass energy of the colliding quarks is \(k_{\pi }k^{\pi }=s_{qq} \). The three-point vertex in this model is illustrated again in Fig. 30a, and the vertex for the \(q\overline{q}\mathcal {A}_{\mu }\) coupling in Fig. 30b.

### 3.2 Spin- Open image in new window point-like monopole with arbitrary magnetic moment term

*U*(1) gauge theory for a spinor field \(\psi \) representing the monopole interacting with the massless

*U*(1) gauge field \(A^{\mu }\), representing the photon. In the cases discussed in the existing literature, the effective Lagrangian describing the interactions of the spinor monopole with photons is taken from standard QED upon imposing electric-magnetic duality, in which there is no bare magnetic-moment term (at tree level). However for our analysis here, we shall insert in the Lagrangian a magnetic moment generating term,

^{3}In the event \(\kappa =0\), the Dirac Lagrangian is recovered. Thus, the effective Lagrangian for the spinor-monopole-photon interactions takes the formwhere \(F_{\mu \nu }\) is the electromagnetic field strength tensor, the total derivative is Open image in new window and \([\gamma ^{\mu },\gamma ^{\nu }]\) is a commutator of \(\gamma \) matrices. The magnetic coupling \(g(\beta )\) is given in (8), and is at most (depending on the case considered) linearly dependent on the monopole boost, \(\beta =|\vec {p}| / E_p\), where \(|\vec {p}|\) and \(E_p\) are the monopole momentum and energy, respectively. The effect of the magnetic-moment term is observable through its influence on the magnetic moment at tree level which is

*M*is the spinor-monopole mass, \(\hat{S}\) is the spin expectation value and the corresponding “gyromagnetic ratio” \(g_R = 2( 1 + 2\tilde{\kappa })\). The dimensionless constant \(\tilde{\kappa }\) is defined such that

^{4}

#### 3.2.1 Spinor pair production via photon fusion

*t*-channel and

*u*-channel processes as shown in Fig. 8, with the \(\kappa \)-dependent matrix amplitudes stated in Eq. (B16) of “Appendix B”. The photon momenta are \(q_1\), \(q_2\), the monopole momenta are \(p_1\), \(p_2\), whilst

*k*, \(\tilde{k}\) are the

*t*- and

*u*-channel exchange momenta, respectively.

*e*, i.e. restoring the Lagrangian to simple Dirac QED. This case is clearly distinctive as the only unitary and renormalisable case. It is observed that the vertical heights of the curves for the differential cross section change with \(\kappa \). Furthermore, the \(\kappa =1\) and \(\kappa =-1\) cases are totally equivalent, so a degeneration between positive and negative \(\kappa \) values is evident.

Hence, a unitarity requirement may isolate the \(\kappa =0\) model from the others as the only viable theory for the spin- Open image in new window monopole, unless the model is viewed as an effective field theory. In that case, the value of \(\kappa \) can be used as a window to extrapolate some characteristics of the extended model in which unitarity is restored. Also, as already mentioned, \(\kappa \) is not dimensionless, hence, a non-zero \(\kappa \) clearly makes this (effective) theory non-renormalisable.

#### 3.2.2 Pair production of spin- Open image in new window monopoles via Drell–Yan

*s*-channel is also possible for fermionic monopoles through the annihilation of quarks into a photon, which decays to the monopole-antimonopole pair. The relevant Feynman rules are displayed in Fig. 32 with the \(\kappa \)-dependent matrix amplitude given in Eq. (B20), both in “Appendix B”. The complete DY process is shown in Fig. 11. The exchange energy in the centre-of-mass frame is \(k_{\pi }k^{\pi }=s_{qq} \).

*for this process*is maintained for all values of \(\kappa \). This becomes apparent in the high energy limit \(s_{qq} \rightarrow \infty \). Taking the expression (30) to first order in \(\beta \) in this limit, we observe that it is finite for all \(\kappa \),

It is worth noting that monopole production in a high energy collider sees twice the production cross section for collisions with same-sign incoming beams, such as proton-proton collisions at the LHC, in contrast to opposite-sign hadron colliders, such as the Tevatron, which maintain the exact cross section as given in (32).

### 3.3 Vector monopole with arbitrary magnetic moment term

Monopoles of spin-1 have been addressed for the first time in colliders recently by the MoEDAL experiment for the Drell–Yan production [32]. A monopole with a spin \(S=1\) is postulated as a *massive* vector meson \(W_{\mu }\) interacting only with a massless gauge field \(\mathcal {A}_{\mu }\) in the context of a gauge invariant Proca field theory. As mentioned previously, lacking a fundamental theory for point-like magnetic poles, we keep the treatment general by including a magnetic moment term in the effective Lagrangian, proportional to \(\kappa \), which is a free phenomenological parameter. Unlike the spin- Open image in new window monopole case, however, for the vector monopole the magnetic moment parameter \(\kappa \) is dimensionless. The case \(\kappa =0\) corresponds to a pure Proca Lagrangian, and \(\kappa =1\) to that of the SM \(W_{\mu }\) boson in a Yang–Mills theory with spontaneous symmetry breaking. In this respect, our approach resembles early phenomenological studies of charged \(W^{\pm }\)-boson production in the SM through PF, where the magnetic moment of the *W*-boson was kept free [20], different from the value \(\kappa =1\) dictated by unitarity. The aim of such analyses was to determine measurable (physical) quantities in purely electromagnetic SM processes, that were sensitive to the value of \(\kappa \), and more or less independent of the Higgs field and the neutral gauge boson \(Z^0\). These quantities were the angular distributions at sufficiently high energies, whose behaviour for the unitarity-imposed value \(\kappa =1\) was found to be quite distinct from the case \(\kappa \ne 1\). As we shall see in our case, for certain formal limits of large \(\kappa \) and slowly moving monopoles, one may also attempt to make sense of the perturbative DY or PF processes of monopole-antimonopole pair production, when velocity-dependent magnetic charges are employed.

*U*(1) covariant derivative, which provides the coupling of the (magnetically charged) vector field \(W_{\mu }\) to the gauge field \(\mathcal {A}_{\mu }\), playing the role of the ordinary photon. The parameter \(\xi \) is a gauge-fixing parameter. The magnetic coupling is considered in the general form of (8), so as to cover both the \(\beta \)-dependent and \(\beta \)-independent cases in a unified formalism. The tensor \(F^{\mu \nu }\) represents the Abelian electromagnetic field strength (Maxwell).

^{5}For \(\kappa \ne 1\), the theory is known to be non unitary, and is plagued by ultraviolet divergences in the self-energy loop graphs in this model, making the quadrupole moment infinite in a non-renormalisable way. To tackle such divergences, in the pre-SM era, Lee and Yang [19] proposed the effective Lagrangian (34), and demonstrated that such divergences are removed through the inclusion of the gauge fixing term with the gauge fixing parameter \(\xi \ne 0\), but at a cost of introducing a negative metric (and thus ghosts, reflecting the unitarity issue for \(\kappa \ne 1\)). In this “\(\xi \)-limiting formalism”, as it is called, the observables are evaluated from the \(\xi \)-dependent Lagrangian before taking the limit \(\xi \rightarrow 0\). The quadruple moment becomes finite at one-loop level [20]. Unitarity in this formalism is held only at energy scales \(E^2\le M^2 / \xi \), the rest mass energy of a single ghost state. For our purposes, this could be an acceptable assumption for an effective field theory considered valid up to a cut-off scale \(\Lambda ^2 = M^2 / \xi \). In general, lacking a concrete fundamental theory on magnetic poles, we shall ignore the unitarity issue when we consider the incorporation of an arbitrary magnetic moment \(\kappa \) in our construction.

*J*:

After this parenthesis, we come back to the spin-1 monopole-production process via PF within the context of the model (34). Restricting our attention to tree level, the gauge fixing parameter \(\xi \) is redundant and the \(\xi \)-independent interaction vertices are given in Fig. 33. These are used to evaluate the \(\kappa \)-dependent PF Born amplitudes, \(\mathcal {A}_{\mu }\mathcal {A}_{\nu }\rightarrow W_{\mu }W^{\dagger }_{\nu }\) and DY \(\psi \overline{\psi }\rightarrow W_{\mu }W^{\dagger }_{\nu }\). As already mentioned, we introduce a velocity-dependent magnetic coupling \(g=g(\beta )\) corresponding to a magnetic charge linearly dependent on the monopole boost \(\beta =|\vec {p}| / E_p\) where \(\vec {p}\) and \(E_p\) are the monopole momentum and energy, respectively.

#### 3.3.1 Pair production of spin-1 monopoles via photon fusion

Monopole-antimonopole pairs are generated at tree level by photons fusing in the *t*-channel, *u*-channel and at a 4-point vertex depicted by the Feynman-like graphs in Fig. 14. The matrix amplitude for each process is given in Eq. (B24) in Fig. 33 in “Appendix B”. *k* and \(\tilde{k}\) are the exchange momenta of the *t*- and *u*-channel processes, respectively. The monopoles have polarisation vectors \(\Upsilon (p_1)_{\kappa }, \Upsilon (p_2)_{\kappa '}\) and momentum 4-vectors \(p_{1\mu }\) and \(p_{2\mu }\), where (on mass-shell): \(p_{1,2}^2=M^2\). The photons polarisation vectors are \(\epsilon (q_1)_{\lambda },\epsilon (q_2)_{\lambda '}\) and their momentum 4-vectors are \(q_{1\mu }\) and \(q_{2\mu }\), \(q_{1,2}^2=0\). Details of the calculations of the analytic expressions for the kinematic distributions and cross section are given in “Appendix B”.

*pp*collisions including the (discernible here) \(\kappa =1\) case. For \(\kappa =1\) the expression (37) becomes:

*only finite*solution in the ultraviolet limit \(s_{\gamma \gamma } \rightarrow \infty \). Indeed, in the high energy limit, \(s_{\gamma \gamma } \rightarrow \infty \), one may approximate \(\beta ^4 \simeq 1\) and \(\beta \simeq 1-\frac{2M^2}{s_{\gamma \gamma }}\), implying that the angular distribution (38) falls off as \(s_{\gamma \gamma } ^{-1}\):

*only*the total cross section for \(\kappa =1\) is

*finite*, in similar spirit to the differential cross section behaviour:

#### 3.3.2 Pair production of spin-1 monopoles via Drell–Yan

*s*-channel, also known as Drell–Yan, drawn in Fig. 17, for which the relevant Feynman rules are given in Fig. 34. The quarks each have a mass

*m*considered small compared to the monopole mass,

*M*, and are characterised by momentum 4-vectors \(q_{1\mu }\) and \(q_{2\mu }\), where on mass shell one has \(q_{1,2}^2=m^2\). Similarly, the mesons have mass

*M*each and are characterised by momentum 4-vectors \(p_{1\mu }\) and \(p_{2\mu }\), where on mass-shell one has \(p_{1,2}^2=M^2\). The centre-of-mass energy of the quark-antiquark pair is \(k_{\nu }k^{\nu }=s_{qq} \).

*M*, the differential cross section reads:

### 3.4 Comparison of various spin models and production processes

Secondly, it is apparent that the cross section for monopole production increases with the spin of the monopole most of the mass range, as observed in Fig. 21, if the SM-like cases for the magnetic-moment parameters are chosen. This observation supports the findings of Ref. [18]. As shown in Figs. 10 and 13 for a fermionic monopole, the trend is maintained for \(\tilde{\kappa }>0\) for all masses. For a vector monopole, on the other hand, the cross-section ordering is not consistent across the monopole mass for varying \(\kappa \) values, as evident from Figs. 16 and 19. More discussion on the phenomenological implications of the magnetic-moment parameter will follow in Sect. 5.2, this time in the context of proton-proton collisions.

### 3.5 Perturbatively consistent limiting case of large \(\kappa \) and small \(\beta \)

As discussed in Sect. 1, the non-perturbative nature of the large magnetic Dirac charge of the monopole invalidate any perturbative treatment based on Drell–Yan calculations of the pertinent cross sections and hence any result based on the latter is only indicative, due to the lack of any other concrete theoretical treatment. This situation may be resolved if thermal production in heavy-ion collisions – that does not rely on perturbation theory – is considered [44, 45, 46]. Another approach is discussed here involving a specific limit of the parameters \(\kappa \) and \(\beta \) of the effective models of vector and spinor monopoles, used above, in the case of a *velocity-dependent magnetic charge* (8). In this limit, the perturbative truncation of the monopole pair production processes, described by the Feynman-like graphs of Figs. 8, 11, 14 and 17, becomes meaningful provided the monopoles are slowly moving, that is \(\beta \ll 1\). In terms of the centre-of-mass energy \(\sqrt{s_{\gamma \gamma /qq}}\), such a condition on \(\beta \) implies, on account of Eq. (7), that the monopole mass is around \(2M \simeq \sqrt{s_{\gamma \gamma /qq}} + \mathcal O (\beta ^2)\). It should be noted at this point that, in collider production of monopole-antimonopole pairs considered in this work, \(s_{\gamma \gamma /qq}\) is not definite but follows a distribution, according to the parton (or photon) distribution function (PDF) for the DY or PF processes.

*large*(dimensionless) magnetic-moment-related parameters \(\kappa , \tilde{\kappa }\), relevant for the cases of vector and spinor monopoles, the situation changes drastically, as we shall now argue. To this end, we consider the limits

*absence of infrared*(\(\beta \rightarrow 0\))

*divergences*in the total cross sections, one may consistently arrange that the PF cross section (54) acquires a non-zero (finite) value as \(\beta \rightarrow 0\), whilst the DY cross section (55) vanishes in this limit:In such a limit, the quantity \(\kappa g \beta ^2 = |c_1|^{\frac{1}{4}} \, \beta ^{\frac{3}{4}} \, \xrightarrow {{\mathop {\kappa \rightarrow \infty }\limits ^{\beta \rightarrow 0}}} \, 0\), so (50) is trivially satisfied, and thus the perturbative nature of the magnetic moment coupling is guaranteed. Hence in this limiting case of velocity-dependent magnetic charge, large magnetic moment couplings and slowly moving vector monopoles, again the PF cross section is the dominant one relevant to searches in current colliders and can be relatively large (depending on the value of the phenomenological parameter \(c_1\)). This argument is successfully tested with simulated events in Sect. 5.2.

## 4 MadGraph implementation

The MadGraph generator [21] is used to simulate the generation of monopoles. In this section, we briefly present the development of the MadGraph Universal FeynRules Output (UFO) model [47] used to simulate different production mechanisms of monopole. This includes both the monopole velocity (\(\beta \)) dependent and independent photon-monopole-monopole coupling. Three different spin cases have been included: spins 0, Open image in new window and 1.

### 4.1 Monopole couplings

*g*is given in Eq. (2) in Gaussian units. However in MadGraph,

*Heaviside-Lorentz*units are used, where (2) becomes

*Heaviside-Lorentz*units simply becomes

*Heaviside-Lorentz*units, electric charge \(q_e\) is given by \(\sqrt{4\pi \alpha }\) where \(\alpha \) is the fine-structure constant. Hence, Eq. (58) turns out to be

*M*.

### 4.2 Implementation of the monopole Lagrangians in \(\textsc {MadGraph}\)

- 1.
Create a model

^{6}with all the user defined fields, parameters and interactions. Lately, the use of the UFO format [47] is strongly encouraged for such models. - 2.
In the MadGraph command prompt, import that model.

- 3.
Generate the process which will be simulated using the generate command and create an output folder.

- 4.
Fix the centre-of-mass energy, colliding particles, parton distribution functions in the run card.

- 5.
Fix the parameters (electric and magnetic charges, masses, etc) of the colliding and generated particles in the parameter card.

- 6.
Launch the output folder in order to compile the model and create the Les Houches Event (LHE) files [49].

- 7.
These LHE files will be used to produce simulated results.

### 4.3 Generating and validating the UFO models

Cross-section values obtained from theoretical calculations and from the MadGraph UFO model at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\) without PDF for monopoles of spin 0, Open image in new window , 1 and a \(\beta \)-dependent coupling through the photon-fusion production mechanism. The ratios simulation/theory prediction are also listed

Mass (GeV) | Spin 0 | Spin 1 | |||||||
---|---|---|---|---|---|---|---|---|---|

\(\gamma \gamma \rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio | \(\gamma \gamma \rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio | \(\gamma \gamma \rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio | ||||

UFO model | Theory | UFO/th. | UFO model | Theory | UFO/th. | UFO model | Theory | UFO/th. | |

1000 | \(1.4493\times 10^{4}\) | \(1.4336\times 10^{4}\) | 0.99 | \(1.364\times 10^5\) | \(1.358 \times 10^{5}\) | 1.004 | \(1.078\times 10^{7}\) | \(1.0781\times 10^{7}\) | 0.999 |

2000 | \(9.851\times 10^{3}\) | \(9.791\times 10^{3}\) | 1.006 | \(8.341\times 10^{4}\) | \(8.2551\times 10^{4}\) | 1.010 | \(2.277\times 10^{6}\) | \(2.2520\times 10^{6}\) | 1.011 |

3000 | \(5.685\times 10^{3}\) | \(5.640\times 10^{3}\) | 1.007 | \(4.803\times 10^{4}\) | \(4.7554\times 10^{4}\) | 1.010 | \(7.214\times 10^{5}\) | \(7.1290\times 10^5\) | 1.012 |

4000 | 2847 | 2810.5 | 1.013 | \(2.251\times 10^{4}\) | \(2.2156\times 10^{4}\) | 1.012 | \(2.275\times 10^{5}\) | \(2.2523\times 10^5\) | 1.010 |

5000 | 1094 | 1087 | 1.006 | 6362 | 6331 | 1.005 | \(5.256\times 10^{4}\) | \(5.1833\times 10^4\) | 1.014 |

6000 | 117.8 | 116.53 | 1.011 | 370 | 365.5 | 1.012 | \(3.034\times 10^{3}\) | \(3.014\times 10^3\) | 1.007 |

For scalar monopoles, the inclusion in the simulation of the four-particle vertex shown in Fig. 2c in addition to the *u*- and *t*-channel, shown in Figs. 2a, b respectively, led to the necessary use of UFO model written as a Python object and abandon the rather older method in Fortran code. The implementation of the four-vertex diagram proved to be non-trivial due to the \(g^2\) coupling. The Lagrangian, which takes the form given in (15), is rewritten in a Mathematica format so that FeynRules can understand the variables. We created the text file containing all the information related to the field, mass, spin and charge, etc, following the instructions given in Ref. [52].

To validate the MadGraph UFO model for monopoles, we compare the cross sections for the photon-fusion process from the theoretical calculation derived in Sect. 3 to those obtained from simulation. Since the theoretical calculations consider bare photon-to-photon scattering, we chose in MadGraph the no-PDF option, i.e. we assume direct \(\gamma \gamma \) collisions at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\). Also, the coupling used here is assumed to depend on \(\beta \).

The cross-section values for spin-0 monopoles are shown in the first columns of Table 1. The UFO-model-over-theory ratio values, also shown in the fourth column of the table, are very close to unity. This clearly shows the validity of spin-0 monopole UFO model.

In a similar fashion, spin- Open image in new window monopole Lagrangians (23) are also rewritten in a Mathematica format. The magnetic-moment parameter \(\tilde{\kappa }\) is also implemented in the model. No additional diagram was added to the *u* / *t*-channels already described in the UFO model. Again, the cross sections from theoretical calculations and MadGraph UFO models (for no PDF) for spin- Open image in new window monopoles were compared, are shown in Table 1. The comparison clearly shows the validity of the MadGraph UFO model for spin- Open image in new window monopoles.

Finally, the Lagrangian (34) for spin-1 monopoles is also written in Mathematica code. The possibility for choosing the value of the \(\kappa \) parameter in (34) exists, yet for validation purposes, the value of \(\xi \) is taken to be zero and the value of \(\kappa \) is taken to be one. The cross sections for spin-1 monopoles from the theoretical calculations and MadGraph UFO models (for no PDF), shown in Table 1, match which satisfactorily proved the validity of the MadGraph UFO model for the spin-1 monopole.

Cross-section values obtained from theoretical calculations and from the MadGraph UFO model at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\) without PDF for monopoles of spin 0, Open image in new window , 1 and a \(\beta \)-dependent coupling through the Drell–Yan production mechanism. The ratios simulation/theory prediction are also listed

Mass (GeV) | Spin 0 | Spin 1 | |||||||
---|---|---|---|---|---|---|---|---|---|

\(q\bar{q}\rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio UFO/th. | \(q\bar{q}\rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio UFO/th. | \(q\bar{q}\rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | Ratio UFO/th. | ||||

UFO model | Theory | UFO model | Theory | UFO model | Theory | ||||

1000 | 0.4223 | 0.4184 | 1.009 | 1.747 | 1.735 | 1.007 | 3362 | 3343.05 | 1.006 |

2000 | 0.3484 | 0.3465 | 1.005 | 1.614 | 1.603 | 1.007 | 230.6 | 228.872 | 1.007 |

3000 | 0.2463 | 0.2441 | 1.009 | 1.373 | 1.373 | 1.000 | 45.43 | 45.173 | 1.006 |

4000 | 0.1361 | 0.1352 | 1.007 | 1.039 | 1.0352 | 1.004 | 11.38 | 11.3162 | 1.006 |

5000 | 0.04724 | 0.0473 | 0.999 | 0.6029 | 0.601 | 1.003 | 2.299 | 2.282 | 1.007 |

6000 | 0.003745 | 0.00373 | 1.004 | 0.1454 | 0.1442 | 1.008 | 0.1206 | 0.1196 | 1.008 |

## 5 LHC phenomenology

### 5.1 Comparison between the photon fusion and the Drell–Yan production mechanisms

Apart from the total cross sections, it is important to study the angular distributions of the generated monopoles. This is of great interest to the interpretation of the monopole searches in collider experiments, given that the geometrical acceptance and efficiency of the detectors is not uniform as a function of the solid angle around the interaction point. The kinematic distributions for the direct \(\gamma \gamma \) and \(q\bar{q}\) scattering obtained with the UFO models were also compared against the calculated differential cross sections of Sect. 3 and showed good agreement with respect to the pseudorapidity, \(\eta \), and the transverse momentum, \(p_{\mathrm T}\), of the monopole. It is worth noting that the differential cross sections in Sect. 3 are plotted for a specific value of \(\beta \simeq 0.986\), while in this section we consider a range of monopole velocities connected to the ratio of the selected monopole mass over the proton-proton collision energy, thus some differences in the PF-vs-DY comparison are expected.

Here the kinematic distributions are compared between the photon-fusion (\(\gamma \gamma \)) and the Drell–Yan mechanisms. For this purpose, the \(\beta \)-dependent UFO monopole model was used in MadGraph. Monopole-antimonopole pair events have been generated for proton-proton collisions at \(\sqrt{s}=13\ \hbox {TeV}\), i.e. for the LHC Run-2 operating energy. The PDF was set to NNPDF23 [53] at LO for the Drell–Yan and LUXqed [54] for the photon-fusion mechanism. The latter choice is made due to the relatively small uncertainty in the photon distribution function in the proton provided by LUXqed [55]. The monopole magnetic charge is set to 1 \(g_{\text {D}}\), yet the kinematic spectra are insensitive to this parameter. The distributions are normalised to the same number of events, in order to facilitate the shape comparison.

^{7}spectra are shown in Fig. 23. We choose to show distributions of the kinetic energy because it is relevant for the monopole energy loss in the detector material, hence important for the detection efficiency. The kinetic-energy spectrum is slightly softer for PF than DY for scalar (left panel) and vector (right panel) monopoles, whereas it is significantly harder for fermions (central panel). This difference may be also due to the four-vertex diagram included in the bosonic monopole case. This observation is in agreement with the one made for \(\beta \) previously. We have also compared MadGraph predictions for kinetic-energy and \(p_{\mathrm T}\) distributions between with- and without-PDF cases, the latter also against analytical calculations (cf. Sect. 3) across different spins and production mechanisms. As expected, some features seen in the direct \(\gamma \gamma \) or \(q\bar{q}\) production are attenuated in the

*pp*production due to the sampling of different \(\beta \) values in the latter as opposed to the fixed value in the former.

*pp*(laboratory) frame and the event-by-event variation of the monopole velocity \(\beta \) that yields different event weight.

### 5.2 Perturbatively consistent limiting case of large \(\kappa \) and small \(\beta \) for photon fusion

In Sect. 3.5, the theoretical calculations show that in the perturbatively consistent limit of large \(\kappa \) and small \(\beta \), the cross sections are finite for both spin- Open image in new window (52) and spin-1 (56) cases. In this section, we focus on this aspect of the photon-fusion production mechanism, since it dominates at LHC energies. We first put to test this theoretical claim utilising the MadGraph implementation and later we discuss the kinematic distributions and comment on experimental aspects of a potential perturbatively-consistent search in colliders to follow in this context.

#### 5.2.1 Spin- Open image in new window case

*M*the mass of the monopole, is varied from zero (the SM scenario) to 10,000 for \(\gamma \gamma \) collisions at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\). The cross-section of the photon fusion process for \(\tilde{\kappa }=0\) is going to zero very fast as \(\beta \rightarrow 0\), as can be seen in the third column of Table 3. However for non-zero \(\tilde{\kappa }\), the cross-section values remain finite even if \(\beta \) goes to zero, as expected, as becomes evident from the last row of Table 3. The same conclusion is drawn from Fig. 26 (left), where the cross sections are plotted for

*pp*collisions at \(\sqrt{s}=13\ \hbox {TeV}\), i.e. with PDF. For masses \(M\gtrsim 6~\mathrm{TeV}\), the monopole production, although still rare, remains at detectable limits for the LHC experiments.

Photon-fusion production cross sections at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\) for spin- Open image in new window monopole, \(\beta \)-dependent coupling and various values of the \(\tilde{\kappa }\) parameter

Monopole mass (GeV) | \(\beta \) | \(\gamma \gamma \rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | |||
---|---|---|---|---|---|

\(\tilde{\kappa }=0\) | \(\tilde{\kappa }=10\) | \(\tilde{\kappa }=100\) | \(\tilde{\kappa }=10{,}000\) | ||

1000 | 0.9881 | \(1.37\times 10^{5} \pm 4.6\times 10^{2}\) | \(1.639\times 10^{24} \pm 3.3\times 10^{21}\) | \(1.639\times 10^{28} \pm 3.3\times 10^{25}\) | \(1.639\times 10^{36} \pm 3.3\times 10^{33}\) |

2000 | 0.9515 | \(8.303\times 10^{4} \pm 4.5\times 10^{2}\) | \(1.61\times 10^{24} \pm 3.1\times 10^{21}\) | \(1.61\times 10^{28} \pm 3.1\times 10^{25}\) | \(1.61\times 10^{36} \pm 3.1\times 10^{33}\) |

3000 | 0.8871 | \(4.78\times 10^{4} \pm 3.5\times 10^{2}\) | \(1.356\times 10^{24} \pm 2.5\times 10^{21}\) | \(1.356\times 10^{28} \pm 2.5\times 10^{25}\) | \(1.356\times 10^{36} \pm 2.5\times 10^{33}\) |

4000 | 0.7882 | \(2.237\times 10^{4} \pm 1.9\times 10^{2}\) | \(8.612\times 10^{23} \pm 2.1\times 10^{21}\) | \(8.613\times 10^{27} \pm 2.1\times 10^{25}\) | \(8.613\times 10^{35} \pm 2.1\times 10^{33}\) |

5000 | 0.639 | \(6396 \pm 61\) | \(3.154\times 10^{23} \pm 1.1\times 10^{21}\) | \(3.154\times 10^{27} \pm 1.1\times 10^{25}\) | \(3.154\times 10^{35} \pm 1.1\times 10^{33}\) |

5500 | 0.5329 | \(2256 \pm 22\) | \(1.247\times 10^{23} \pm 4.5\times 10^{20}\) | \(1.247\times 10^{27} \pm 4.5\times 10^{24}\) | \(1.247\times 10^{35} \pm 4.5\times 10^{32}\) |

5800 | 0.4514 | \(886.5 \pm 7.8\) | \(5.28\times 10^{22} \pm 2.5\times 10^{20}\) | \(5.28\times 10^{26} \pm 2.5\times 10^{24}\) | \(5.28\times 10^{34} \pm 2.5\times 10^{32}\) |

6000 | 0.3846 | \(367.2 \pm 3\) | \(2.294\times 10^{22} \pm 7.6\times 10^{19}\) | \(2.294\times 10^{26} \pm 7.6\times 10^{23}\) | \(2.294\times 10^{34} \pm 7.6\times 10^{31}\) |

6200 | 0.3003 | \(97.19 \pm 0.77\) | \(6.43\times 10^{21} \pm 3.3\times 10^{19}\) | \(6.43\times 10^{25} \pm 3.3\times 10^{23}\) | \(6.43\times 10^{33} \pm 3.3\times 10^{31}\) |

6400 | 0.1747 | \(5.846 \pm 0.025\) | \(4.065\times 10^{20} \pm 1.5\times 10^{18}\) | \(4.065\times 10^{24} \pm 1.5\times 10^{22}\) | \(4.065\times 10^{32} \pm 1.5\times 10^{30}\) |

6490 | 0.0554 | \(0.017\pm 2.27\times 10^{-5}\) | \(1.27\times 10^{18}\pm 8.74\times 10^{14}\) | \(1.27\times 10^{22}\pm 8.74\times 10^{18}\) | \(1.27\times 10^{30}\pm 8.74\times 10^{26}\) |

The central and right-hand-side plots of Fig. 26 depict a comparison of the \(p_{\mathrm T}\) and \(\eta \) distributions, respectively, between the SM value \(\tilde{\kappa }=0\) and much higher values up to \(\tilde{\kappa }=10^4\). The SM-like case is characterised by a distinguishably “softer” \(p_{\mathrm T}\) spectrum and a less central angular distribution than the large-\(\tilde{\kappa }\) case.^{8} The latter case, on the other hand, seems to converge to a common shape for the kinematic variables as \(\tilde{\kappa }\) increases to very large values. This is not the case for \(\tilde{\kappa }\) values distinct, yet near, the SM value, where the angular distributions are not considerably different, as shown in Fig. 9. The common kinematics among large \(\tilde{\kappa }\) values would greatly facilitate an experimental analysis targeting perturbatively reliable results. We note here that the DY process, which dominates the cross section for heavy monopoles *for the SM magnetic-moment value* (see Fig. 25), vanishes as \(\tilde{\kappa }\) acquires large values as shown in Sect. 3.5, rendering the study of photon-fusion process sufficient at the perturbative-coupling limit.

#### 5.2.2 Spin-1 case

*pp*collisions, instead of \(\gamma \gamma \) scattering, is considered. Indeed, Fig. 27 shows that for large values of \(\kappa \) and \(M \simeq \sqrt{s}/2\), which is equivalent to \(\beta \ll 1\), the cross section, although very small, remains finite.

Photon-fusion production cross sections at \(\sqrt{s_{\gamma \gamma }}=13~\mathrm{TeV}\) for spin-1 monopole, \(\beta \)-dependent coupling and various values of the \(\kappa \) parameter

Monopole mass (GeV) | \(\beta \) | \(\gamma \gamma \rightarrow M\bar{M},\;\;\sigma ~\mathrm{(pb)}\) | ||
---|---|---|---|---|

\(\kappa =1\) | \(\kappa =100\) | \(\kappa =10{,}000\) | ||

1000 | 0.9881 | \(1.086\times 10^{7} \pm 1.4\times 10^{5}\) | \(4.939\times 10^{15} \pm 1\times 10^{13}\) | \(5.033\times 10^{23} \pm 2.1\times 10^{21}\) |

2000 | 0.9515 | \(2.275\times 10^{6} \pm 1.6\times 10^{4}\) | \(2.844\times 10^{14} \pm 4.9\times 10^{11}\) | \(2.879\times 10^{22} \pm 9.8\times 10^{19}\) |

3000 | 0.8871 | \(7.198\times 10^{5} \pm 6.6\times 10^{3}\) | \(4.518\times 10^{13} \pm 1.5\times 10^{11}\) | \(4.536\times 10^{21} \pm 1.2\times 10^{19}\) |

4000 | 0.7882 | \(2.273\times 10^{5} \pm 2.2\times 10^{3}\) | \(9.079\times 10^{12} \pm 2.7\times 10^{10}\) | \(9.002\times 10^{20} \pm 3.2\times 10^{18}\) |

5000 | 0.639 | \(5.232\times 10^{4} \pm 4.9\times 10^{2}\) | \(1.513\times 10^{12} \pm 9.2\times 10^{9}\) | \(1.5\times 10^{20} \pm 9.3\times 10^{17}\) |

5500 | 0.5329 | \(1.785\times 10^{4} \pm 1.6\times 10^{2}\) | \(4.49\times 10^{11} \pm 1.7\times 10^{9}\) | \(4.466\times 10^{19} \pm 2.9\times 10^{17}\) |

5800 | 0.4514 | \(7118 \pm 62\) | \(1.658\times 10^{11} \pm 1.1\times 10^{9}\) | \(1.624\times 10^{19} \pm 8.4\times 10^{16}\) |

6000 | 0.3846 | \(3025 \pm 24\) | \(6.72\times 10^{10} \pm 2.5\times 10^{8}\) | \(6.627\times 10^{18} \pm 3.7\times 10^{16}\) |

6200 | 0.3003 | \(836.9 \pm 6.3\) | \(1.764\times 10^{10} \pm 1\times 10^{8}\) | \(1.733\times 10^{18} \pm 1\times 10^{16}\) |

6400 | 0.1747 | \(53.42 \pm 0.23\) | \(1.066\times 10^{9} \pm 3.9\times 10^{6}\) | \(1.05\times 10^{17} \pm 3.8\times 10^{14}\) |

6490 | 0.0554 | \(0.1694\pm 0.00065\) | \(3.293\times 10^{6} \pm 5.6\times 10^{3}\) | \(3.244\times 10^{14}\pm 5.6\times 10^{11}\) |

In Fig. 26, a comparison of the \(p_{\mathrm T}\) (centre) and \(\eta \) (right) distributions, between the SM value \(\kappa =1\) and much higher values up to \(\kappa =10^4\) is given for spin 1. As for spin Open image in new window , the large-\(\kappa \) curves converge to a single shape independent of the actual \(\kappa \) value, however the SM-case here *also* yield similar distributions as the for \(\kappa \gg 1\).^{9} Therefore the \(\eta \)-distribution features shown in Fig. 15 without PDF are completely smoothed out by folding with the photon distribution function in the proton. As discussed in Sect. 5.2.1 for fermionic monopoles, such an experimental analysis can be concentrated on the \(\kappa \)-dependence of the total cross section and the acceptance for very slow monopoles to provide perturbatively valid mass limits in case of non-observation of a monopole signal, since the kinematic distributions are \(\kappa \)-invariant in *pp* collisions. The MoEDAL experiment [23, 24], in particular, being sensitive to slow monopoles can make the best out of this new approach in the interpretation of monopole-search results.

## 6 Conclusions

The work described in this article consists of two parts. In the first part, we dealt with the computation of differential and total cross sections for pair production of monopoles of spin Open image in new window , through either photon-fusion or Drell–Yan processes. We have employed duality arguments to justify an effective monopole-velocity-dependent magnetic charge in monopole-matter scattering processes. Based on this, we conjecture that such \(\beta \)-dependent magnetic charges might also characterise monopole production.

A magnetic-moment term proportional to a new phenomenological parameter \(\kappa \) is added to the effective Lagrangians describing the interactions of these monopoles with photons for spins Open image in new window and 1. The lack of unitarity and/or renormalisability is restored when the monopole effective theory adopts a SM form, that is when the bare magnetic-moment parameter takes on the values \(\kappa =0\) for spin- Open image in new window monopoles, and \(\kappa =1\) for spin-1 monopoles. However we remark that the lack of unitarity and renormalisability is not necessarily an issue, from an effective-field-theory point of view. Indeed, given that the microscopic high-energy (ultraviolet) completion of the monopole models considered above is unknown, one might not exclude the possibility of restoration of unitarity in extended theoretical frameworks, where new degrees of freedom at a high-energy scale might play a role. In this sense, we consider the spin-1 monopole as a potentially viable phenomenological case worthy of further exploration.

The motivation behind the magnetic-moment introduction is to enrich the monopole phenomenology with the (undefined) correction terms to the monopole magnetic moment to be treated as free parameters potentially departing from the ones prescribed for the electron or \(W^{\pm }\) bosons in the SM. Lacking a fundamental microscopic theory of magnetic poles, such an addition appears reasonable. This creates a dependence of the scattering amplitudes of processes on this parameter, which is passed on to the total cross sections and, in some cases, to kinematic distributions. Therefore the parameter \(\kappa \) is proposed as a tool for monopole searches which can be tuned to explore different models.

Moreover, even more intriguing is the possibility to use the parameter \(\kappa \) in conjunction with the monopole velocity \(\beta \) to achieve a perturbative treatment of the monopole-photon coupling. Indeed, in general the large value of the magnetic charge prevents any perturbative treatment of the monopole interactions limiting us to a necessary truncation of the Feynman-like diagrams at the tree level. By limiting the discussion to very slow (\(\beta \ll 1\)) monopoles, the perturbativity is guaranteed, however, at the expense of a vanishing cross section. Nonetheless it turns out that the photon-fusion cross section remains finite *and* the coupling is perturbative at the formal limits \(\kappa \rightarrow \infty \) and \(\beta \rightarrow 0\). This ascertainment opens up the possibility to interpret the cross-section bounds set in collider experiments, such as MoEDAL, in a proper way, thus yielding sensible monopole-mass limits.

In the second part of this article, a complete implementation in MadGraph of the monopole production is performed both for the photon-fusion and the Drell–Yan processes, also including the magnetic-moment terms. The UFO models were successfully validated by comparing cross-section values obtained by the theoretical calculations and the MadGraph UFO models. Kinematic distributions, relevant for experimental analyses, were compared between the photon-fusion and the Drell–Yan production mechanism of spins 0, Open image in new window and 1 monopoles. This tool will allow to probe for the first time the – dominant at LHC energies – photon-fusion monopole production. Furthermore, the experimental aspects of a perturbatively valid monopole search for large values of the magnetic-moment parameters and slow-moving monopoles have also been outlined, based on these kinematic distributions.

## Footnotes

- 1.
In general one should also add electric dipole moment terms as well. In this work we shall ignore them, assuming them suppressed for brevity, although our analysis can be readily extended to include such terms.

- 2.
We thank V. Vento for a discussion on this point.

- 3.
For instance, it is known that such terms have a geometrical (gravitational) origin in 4-dimensional effective field theories obtained from (Kaluza-Klein) compactification of higher-dimensional theories, such as brane/string universes [42]. Moreover, as mentioned in the introduction, one could also include in the Lagrangian a CP-violating electric dipole moment (EDM) term for the spinor monopole, parametrised by a parameter \(\eta \), \({\mathcal L}_\mathrm{EDM} = \frac{1}{4}\, g(\beta )\eta F_{\mu \nu }\overline{\psi }\, \gamma ^5\, [\gamma ^{\mu },\gamma ^{\nu }]\psi \). In this work, we assume such EDM terms suppressed, compared to the magnetic-moment-\(\kappa \) terms, which can be arranged by assuming appropriate limits of parameters in the underlying microscopic theory, e.g. [42]. Nonetheless, our analysis can be extended appropriately to include both \(\kappa \) and \(\eta \) parameters.

- 4.
In case one adds an EDM \(\eta \)-term for the spinor monopole, the corresponding dimensionless parameter \(\tilde{\eta }\) can also be defined in analogy with (25), i.e. \(\eta =\frac{\tilde{\eta }}{M}\).

- 5.
Corrections to the magnetic moment of the monopole could also arise through anomalous spin interactions at a quantum level. For \(\beta \)-independent magnetic charges, these are uncontrollable, as perturbation theory fails. However, if one accepts velocity-dependent couplings (5), then for slowly-moving monopoles such loop corrections can be made subleading. Moreover, as for the case of spinor monopoles, one could also add an EDM term for the vector monopole [43], \({\mathcal L}_\mathrm{EDM} = i g(\beta )\eta \widetilde{F}^{\mu \nu }W^{\dagger }_{\mu }W_{\nu }\), where \(\widetilde{F}^{\mu \nu } = \frac{1}{2}\epsilon ^{\mu \nu \rho \sigma } F_{\rho \sigma }\), is the dual Maxwell tensor, with \(\epsilon ^{\mu \nu \rho \sigma }\) the totally antisymmetric Levi-Civita tensor in four space-time dimensions. Such terms are assumed suppressed in our analysis, although the latter can be straightforwardly extended to include them.

- 6.
The code of the model will be publicly available in the MadGraph web page [48].

- 7.
In the context of this work, the kinetic energy of a particle is defined as the scalar difference of its total energy and its mass.

- 8.
We should remark at this point, that in the absence of PDF, in the cases for the magnetic-moment parameter \(\tilde{\kappa }\ne 0\), we observe a fast-increasing distribution of events at high \(p_{\mathrm T}\)-values up to a cutoff of \(p_{\mathrm T} =\sqrt{s_{\gamma \gamma }}/2\), as expected from the non-unitarity of such cases.

- 9.
As in the spin- Open image in new window -monopole case, we also observe here that, in the absence of PDF, in the non-unitary cases \(\kappa \ne 1\), there is a fast-increasing distribution of events at high \(p_{\mathrm T}\)-values up to a cutoff of \(p_{\mathrm T} =\sqrt{s_{\gamma \gamma }}/2\).

## Notes

### Acknowledgements

We would like to thank J.R. Ellis, V. Vento, J. Bernabeu and D.V. Shoukavy for discussions on theoretical issues, Wendy Taylor, Manoj Kumar Mondal and Olivier Mattelaer for discussions on the MadGraph implementation, and fellow colleagues from the MoEDAL-LHC Collaboration for their interest. The work of S.B. is supported by an STFC (UK) doctoral studentship. The work of N.E.M. is partially supported by STFC (UK) under the research grant ST/P000258/1. N.E.M. also acknowledges a scientific associateship (“*Doctor Vinculado*”) at IFIC-CSIC-Valencia Univ. (Spain). V.A.M. and A.S. acknowledge support by the Generalitat Valenciana (GV) through the MoEDAL-supporting agreements, by the Spanish MINECO under the project FPA2015-65652-C4-1-R and by the Severo Ochoa Excellence Centre Project SEV-2014-0398. V.A.M. acknowledges support by the GV Excellence Project PROMETEO-II/2017/033 and by a 2017 Leonardo Grant for Researchers and Cultural Creators, BBVA Foundation.

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