Shining light on magnetic monopoles through high-energy muon colliders

The search for magnetic monopoles has been a longstanding concern of the physics community for nearly a century. However, up to now, the existence of elementary magnetic monopoles remains an open question. The electroweak 't Hooft-Polyakov monopoles have been predicted with mass at the order of 10 TeV. This mass scale is unreachable at current colliders. Recently, the muon colliders have gained much attention in the community due to technological developments. The advantages of the muon beam encourage us to raise the high-energy option and consider the high-energy muon collider as a unique opportunity to search for magnetic monopoles. This letter discusses the production of magnetic monopoles via the annihilation process and proposes the search for magnetic monopoles at future high-energy muon colliders.


Introduction
Magnetic monopoles (MMs) with magnetic charge were motivated by electric-magnetic symmetry.Dirac first proposed that the existence of MMs could increase the symmetry of Maxwell's equations and explain the quantization of electric charge [1].It was realized that any UV completion theory of an interacting U (1) gauge field should contain MMs [2].Recent research shows that the virtual effect of MMs can provide axion potential if the axion couples to the electric-magnetic gauge field [3].MMs can also play as the role of dark matter candidate [4].The monopole can be treated as a hypothetical point particle.These point-like particles with unknown spin and mass carry a magnetic charge that is an integral multiple of the fundamental Dirac charge where α em ≈ 1/137 is the fine-structure constant, N is an integer, e is the electric charge, and g D = 68.5e is the fundamental Dirac charge.By extending the classical monopole with non-Abelian gauge theory by 't Hooft, Polyakov and others, composite monopoles naturally appear in the context of grand unified theories (GUT) [5,6].The mass of 't Hooft-Polyakov monopoles is determined by the scale of gauge symmetry breaking, which is of order 10 16 GeV or higher for GUT-scale monopole.As a result, they cannot be produced in a realistic experiment.Nevertheless, as some unification theories involve a number of symmetry-breaking scales, the electroweak symmetry-breaking can give rise to 't Hooft-Polyakov monopoles of mass at the order of M W /α em ∼ 10 TeV where M W is the W boson mass [7][8][9][10][11][12].Except for the electroweak monopole of 5.5 TeV proposed by Ref. [13], the 10 TeV monopole is not yet accessible to existing collider experiments.Thus, the heavy monopoles are mainly constrained by cosmological observations [14][15][16][17][18][19][20][21][22][23][24], the analysis of moon rocks [25] or terrestrial materials [26,27] exposed to cosmologically relic monopoles [28].
In recent years, composite MMs with masses as low as a few TeV have also been proposed in various cases [29][30][31].These theories offer the possibility of collider production of MMs, thus rekindling the enthusiasm of theorists and experimentalists in the search for MMs.Recent searches for TeV-scale monopoles have been carried out by ATLAS [32] and MoEDAL [33] at the LHC.These searches for MMs assume direct productions via the Drell-Yan/annihilation and Photon/Vector Boson Fusion (VBF) mechanism.Another method of producing MMs is through the dual Schwinger effect [34] in strong magnetic fields [35], which was performed by the MoEDAL experiment via Pb-Pb heavy-ion collisions at the LHC [36].In addition, recent studies proposed the production of MMs via collisions of cosmic rays bombarding the atmosphere and set leading robust bounds on the production cross-section of MMs in the 5 − 100 TeV mass range [37].It has been proposed that, the production and decay of monopolium bound state also provide sensitive probes of MMs at the LHC [38,39].
In this letter, we investigate the production mechanisms of MMs at the muon collider.This novel collider, operating at collision energies of 10-30 TeV [40][41][42][43][44], would be able to produce 10 TeV monopoles.Due to the large coupling constant between the photon and monopole, the interaction coupling is in the non-perturbative regime.There is no established theory that allows for direct calculation of MMs production cross-section so far.The electric-magnetic duality theory of Quantum Electrodynamics (QED) could be used as a basis to estimate monopole production cross-sections.The common approach is to consider possible benchmark scenarios via tree-level Feynman-like diagrams.The theoretical framework of this letter is based on electric-magnetic duality by considering effective field theories (EFTs) with only electromagnetic interactions.Perturbative unitarity is taken into account.The cross-section for production of monopole as well as the angular distribution are investigated.

Formalism for magnetic monopoles pair production
The corresponding hypothesis is an effective U (1) gauge theory obtained after appropriate dualisation of the pertinent field theories describing the interactions of MMs with photons.By means of electric-magnetic duality, the charge and coupling of the MMs are often assumed to be velocity-dependent (in the following, the velocity is denoted as β).
After replacing e/v by g/c, the magnetic charge g v c ≡ g β would lead to the equivalence 2 of the electron-monopole scattering cross section with the corresponding Rutherford formula.This replacement was widely used when discussing MMs pair production through annihilation and VBF processes [29,31,33,[45][46][47][48][49].In the following, we use g(β) instead of g.When the β-dependent interaction is considered, g(β) = N g D β, or otherwise, g(β) = N g D , where N is the magnetic charge.The Lagrangians of the effective theories for the scalar, spinor and vector MMs are [29,50] where A µ is the photon field, ϕ, ψ and W are the scalar, spinor and vector monopole fields with masses M s,f,v , respectively, provides the coupling of the magnetically charged vector field W µ to the gauge field A µ , and κ is the interaction strength between the MMs and magnetic dipole moment.At the muon collider, MMs pairs are produced by both the µ + µ − annihilation and the VBF mechanism, as depicted in Fig. 1.For the µ + µ − annihilation into MMs pairs, the cross-section would sharply peak when the invariant mass of MMs pair is close to the collision energy.On the other hand, when the collision energy is much larger than the MMs pair mass, the VBF process dominates right after the threshold of MMs pair mass [29].Besides, in the latter case, the produced MMs typically have large velocities, possibly putting the unitarity into trouble.Therefore, the golden channel to detect MMs depends on the center-of-mass energy as well as the mass range of MMs.
Taking scalar MM for illustration, the µµ annihilation cross-section becomes σ ∼ (1 − 4M 2 s /s) 3/2 /s which is insensitive to the MM mass unless M s ≈ √ s/2.By contrast, the VBF process is very sensitive to the MM mass due to the threshold suppression ∼ 1/M 2 s .When the mass of MM approaches √ s/2, the annihilation process will again become dominant.By using Madgraph_@aMC toolkits [51][52][53], the results are shown in Fig. 2 for the scalar MM and β-independent coupling with N = 1 at a √ s = 30 TeV muon collider.The kinematic features are studied using MLAnalysis [54].Here 'VBF' represents µ + µ − → ℓ + ℓ − ϕ φ or pp → jjϕ φ including the contributions with same final states but not containing a VBF subprocess.
Since the purpose of this work is to illustrate the importance of the muon collider for a 't Hooft-Polyakov MM with mass of the order of 10 TeV, we focus only on MMs with masses above 10 TeV.In this region, the µµ annihilation overcomes the VBF processes as shown above.In the following, we will only consider the annihilation process.
At future pp colliders, a majority of the partons reside in the range of energies much lower than the collision energy, leading to a small luminosity of the partons with large enough energy to produce the MMs.The cross-sections at 27 TeV HE-LHC [55], 50 TeV FCC-hh [56] and 75 TeV SppC [57] with the parton distribution function as NNPDF2.3[58] are also shown in Fig. 2. When M s = 10 TeV, the most optimistic case is the VBF at the Sppc, which is about 1/5 of the annihilation at the muon collider.One can conclude that the future high-energy muon collider has a unique advantage in detecting MMs with large masses.

Partial wave unitarity
The coupling constant between the photon and monopole is too large to be valid for perturbation.Note that, with a none-zero κ f , the last term of L S= 1 2 is a dimension-5 operator.Meanwhile, it has been shown that, in the case of a vector MM, the unitarity is violated at large s with κ v ̸ = 0 for VBF processes.Therefore, Eq. ( 2) can only be viewed as an EFT.Whether the EFT is valid in the energy range of interest needs to be checked before meaningful results can be obtained from tree level calculations.
The breakdown of perturbative theory can be determined by the violation of partial wave unitarity [59],which has been widely used in the study of SMEFT.The 2 → 2 helicity amplitudes M i→f (θ, ϕ) can be expanded as where φ and θ are the azimuth and zenith angles of the MMs in the final state, λ i,f are helicity differences between the two particles in the initial and final states, d J λi,λ f (θ, φ) are Wigner-D functions, and T J,λi,λ f i→f are expansion coefficients In the case of inelastic scattering [60], where m 1,2 are masses of the two particles in the initial state or final state.Neglecting the masses of the particles in the initial state, one gets β i = 1.For the final state, is the velocity of the MM.The unitarity bound used in this letter is then Considering the case of annihilation process µ + µ − → M M ,we find the unitarity bound as β 3 e 2 g 2 (β)/(144π 2 ) ≤ 1 for the case of scalar MMs.It leads to β < 9/N 2 1/3 in the case of velocity independent interaction, and β < 9/N 2 1/5 for the velocity dependent interaction.Therefore, the unitarity bound is generally satisfied for N ≤ 3.In the case of spinor MMs, the unitarity bound becomes For simplicity, we define a dimensionless parameter κf ≡ κ f M f .For fixed s and N , Eq. ( 8) leads to the constraint on κf for a given note that, for β-dependent coupling, g(β) is also a function of M f .For vector MMs, we have the unitarity bound as follows A translation can be done similarly as Eq. ( 7), Using Eqs. ( 6) and ( 8), the upper bounds on β are presented in Figs. 3 and 4. It can be seen that the allowed β decreases rapidly along with increasing κ f,v , for both spinor and vector MMs.Since β is a monotonic function of the MM mass, an upper bound on β can be directly translated to a lower bound on M f,v for different √ s.Taking the β-independent coupling with N = 1 as an example, for the spinor MM case, the bound of β < 0.234 when κf = 3 results in 0. It turns out that a large κ leaves a small mass window to detect the MM.When the mass of MM is lower than this bound, the EFT no longer succeeds to describe the reactions perturbatively.In the following studies, the results are presented with unitarity taken into account.

Monopole production in muon collisions
In this section, we evaluate the production rate of MM pairs at future muon colliders.Taking ∆θ as the minimal zenith angle that can be covered, the cross-sections of MM  productions via the annihilation process are One can see that, for a small ∆θ, σ s,f,v are insensitive to ∆θ.Therefore, in numerical results, we take ∆θ = 0. We consider MMs with masses of the order of 10 TeV produced For N = 1 and β-dependent couplings, in the mass range of 10 TeV < M s,f,v < 14 TeV, we obtain 26.06 fb > σ s > 0.68 fb, 30.65 fb > σ min f > 0.13 fb and 150.33 fb > σ min v > 1.88 fb.The designed luminosity for muon colliders is expected to be 10 ∼ 90 ab −1 [42].Taking σ s or σ min f,v = 10 ab as a fiducial cross-section, we find the reachable masses are M s = 14.82TeV, M f = 14.52 TeV and M v = 14.88 TeV, respectively.
The angular distribution of the MM production is useful information for detecting the MMs.The results are independent of g(β) and are given by The scalar MMs are dominantly produced along the direction perpendicular to the beam line direction.For the spinor MM and vector MM, the production direction depends on the sign of A f,v .For the spinor MMs, A f < 0 when |κ f | < M f / √ s and they are produced along the perpendicular direction.Otherwise, one has A f > 0 and the spinor MMs are produced along the beam line direction.When κf = 1/2, A f reaches the minimum and dσ f / (σ f d cos θ) = 3 1 − cos 2 (θ) /4.When κf = 2M 2 f /s, A f reaches the maximum and dσ f / (σ f d cos θ) = 3 (3 + cos(2θ)) /16.For the vector MMs, when M v < s/6 (12.25 TeV at √ s = 30 TeV) and ), one has A v > 0 and the MMs are dominantly produced along the beam line direction.Otherwise, the MMs are produced along the perpendicular direction for A v < 0. When Various techniques have been used at the colliders to detect MMs based on different theoretical assumptions.The CDF experiment designed a special trigger based on the relativistic effects that MMs are accelerated along the uniform solenoidal magnetic field in a parabola slightly distorted [61].The MMs produced at the LHC exhibit characteristics of the long-lived highly ionizing particles which would quickly slow down and get trapped in the material surrounding the interaction points and produce a large number of highthreshold hits and a large number of δ rays emitted from the material [62,63].Based on the assumption that monopole are strongly bound to aluminum nuclei, MoEDAL [36] searched for MMs by looking for induced persistent currents after passage through a superconducting magnetometer [64].Whatever the assumptions are, the measurements at the LHC are determined by the ionization energy loss of the MMs which could be velocity dependent [62,63].

Summary
In this letter, we investigate the capability of the future muon colliders to detect the MMs with the mass of the order of 10 TeV in the framework of EFTs.We find that the annihilation process overcomes the VBF processes when probing a MM with mass close to √ s/2.Since the coupling constant is not in a perturbative region, the tree-level partial wave unitarity bounds are discussed, which result in upper bounds on the integral multiple of the fundamental Dirac charge N , and lower bounds on the mass of MMs, or upper bounds on magnetic dipole moments κ f,v .The cross-sections of the MM pair production via the annihilation process are calculated at the muon collider.Taking a fiducial cross-section of 10 ab, the reachable masses are M s = 14.82,M f = 14.52 and M v = 14.88 TeV, respectively.Compared with the LHC, the muon colliders provide a unique opportunity of the search for electroweak 't Hooft-Polyakov MMs.The angular distributions of the MM production are also investigated.Generally, the scalar MM, the spinor MM with a small |κ v | or the vector MM with a large M v can be produced along the perpendicular direction.
Finally, inspired by the measuring methods in experiments at LHC, we propose that the MMs detectors should be designed for the future muon collider because it provides us the first possible source of 't Hooft-Polyakov MMs at colliders.

Figure 1 :
Figure 1: Feynman-like diagrams for direct monopole pair production at leading order via the annihilation process (up); and VBF process (bottom a, b, c) at the muon collider.M ( M ) denotes the (anti-)MM.
VBF at Muon Collider Annihilation at Muon Collider VBF at HE-LHC Drell-Yan at HE-LHC VBF at FCC-hh Drell-Yan at FCC-hh VBF at Sppc Drell-Yan at Sppc

Figure 3 :
Figure 3: The upper bound on β for the case of a spinor MM.The cases of β-independent (β-dependent) g are shown in the left (right) panel.The solid, dotted and dashed lines are for N = 1, 2 and 3, respectively.

Figure 4 :
Figure 4: Same as Fig. 3 but for the case of a vector MM.

Figure 6 :
Figure 6: σ f,v as functions of M f,v and κf or κv.For illustration, only N = 1 case is shown.The top (bottom) panels correspond to σ f (σv) and κf (κv).The left (right) panels are for the β-independent (β-dependent) coupling.The regions that are forbidden by unitarity bounds are cropped out.