Topologically stable, finite-energy electroweak-scale monopoles

The existence of a magnetic monopole, if it exists, remains elusive. Experimental searches have been carried out and are continuing in this quest. Of great uncertainty is the mass of the monopole which is model-dependent and ranging from some Grand Unified scale to the electroweak scale. In this paper, we propose a model where topologically stable, finite-energy monopoles {\em \`{a} la} 't Hooft-Polyakov could exist with a mass proportional to the electroweak scale. This comes about in a model of neutrino masses where right-handed neutrinos are {\em non-sterile} whose electroweak-scale Majorana masses are obtained by the coupling to a complex {\em triplet} Higgs field. Custodial symmetry which insures $M_W=M_Z \cos \theta_W$ requires the introduction of another triplet Higgs field but {\em real} this time. It is this {\em real Higgs triplet} that is at the core of our proposal.


INTRODUCTION
The fascinating idea of a magnetic monopole has been around since the time when Dirac was intrigued by why electric charges are quantized. Dirac's postulate of a point-like magnetic monopole with a semi-infinite "singular string" attached to it. The wave function for an electron going along a closed loop encircling the singular string is single valued if the Dirac quantization condition eg D = n/2 is satisfied with g D being the magnetic charge. Another way to say this is that, in order for the string to be undetected, the quantization condition has to be satisfied. Electric charges are quantized in units of 1/2g D in the presence of a Dirac point-like monopole.
't Hooft and Polyakov [1,2] discovered that, by embedding the "electromagnetic" U (1) em into the Georgi-Glashow [3] gauge group SO (3), there exists a stable, finite energy monopole-like solution where the singular string can be gauge-rotated away. It was subsequently found that, by embedding the SM into a Grand Unified (GUT) gauge group such as SU (5), one could construct topologically stable, finite-energy (TSFE) magnetic monopoles which are extremely heavy with a mass proportional to the GUT scale over the GUT fine structure constant [4]. One could also ask the question of whether or not the Standard Model (SM) could contain such a monopole with a mass which is now proportional to the electroweak scale and which could, in principle, be produced and detected. Unfortunately, it is well-known that the SM which contains only Higgs doublets does not have TSFE monopoles by topological arguments, as we shall briefly review below. An alternative way out of this conundrum is a proposal of Cho and Maison [5]which asserted that the SM monopole can exist by looking at a different topology. The Cho-Maison electroweak monopole however suffers from a divergence when one tries to compute its mass classically. Remedies to that problem, essentially by modifying the kinetic energy term of the U (1) Y gauge field, have been proposed [6]at the price of a non-negligible uncertainty in the monopole mass.
In this manuscript, we propose a model which contains topologically stable, finite-energy electroweak-scale monopolesà la 't Hooft-Polyakov. More precisely speaking, this is not a model proposed for the electroweak monopole but rather this solution is a consequence of a model proposed for something else: the possible existence of non-sterile right-handed neutrinos with electroweakscale masses in a seesaw mechanism for light neutrinos [7]. The seesaw mechanism within the framework of this model can be tested directly at colliders by searching for like-sign dileptons at displaced vertices, among many other collider signals. The main reason this model (EWν R model) can give rise to electroweak monopoles is as follows. As [7] has described, right-handed neutrinos acquire electroweak-scale Majorana masses M R ∝ Λ EW ∼ 246 GeV by coupling to a complex Higgs tripletχ. The Z-width requires that M R ≥ 46 GeV since ν R 's are nonsterile, which translate into χ = v M ∝ Λ EW . This however will grossly destroy the well-known and experimentally successful relationship M W = M Z cos θ W , a consequence of the so-called Custodial Symmetry, unless another Higgs real triplet ξ exists with ξ = χ = v M [8]. As we shall see below, it is this real triplet ξ which gives rise to the electroweak monopole solution. (For more details on the phenomenology of the EW-ν R model, please consult [7].) The logical chain of our arguments is as follows: M R ∝ Λ EW → Complex tripletχ, Custodial symmetry → Real triplet ξ → topologically stable, finiteenergy electroweak-scale monopoles. This chain shows the deep connection between neutrino masses and the possibility of electroweak monopoles.
For reasons to be given below, we shall call the topologically stable, finite-energy electroweak-scale monopoles by the name: γ-Z magnetic monopole.
We will show that the EW-ν R model contains very interesting non-perturbative solutions in the form of γ-Z magnetic monopoles upon a close examination of the global structure of the model. This provides another (non-perturbative) characteristic signal to be searched for at dedicated experiments such as MoEDAL and CMS,ATLAS, LHCb among others. As we will see below, the γ-Z magnetic monopole is a finite-energy soliton whose mass is concentrated in a core of radius ∼ 1/M Z . This mass is expected to be ∼ GeV, the γ-Z magnetic monopole mass is roughly between 900 GeV and 3 TeV. The long-range magnetic field appearing outside the core behaves like (1/g)(sin θ W )(1/r 2 ) = (1/e)(sin 2 θ W )(1/r 2 ). It is the appearance of sin 2 θ W that reveals the nature of the γ-Z magnetic monopole.

GLOBAL STRUCTURE OF THE EW-νR MODEL AND THE γ-Z MAGNETIC MONOPOLE
Since the 't Hooft-Polyakov monopole is crucial in our subsequent discussion, we will come back to it after a brief excursion into how topologically stable monopoles arise from consideration of results of homotopy groups of spheres.
To find a finite-energy field configuration at spatial infinity which corresponds to a monopole, one requires that the Higgs field approaches its minima, the so-called vacuum manifold, which forms a sphere in 3-dimensional internal space denoted by S 2 . That is one maps a 3dimensional spatial sphere to the sphere of vacuum manifold S 2 . In homotopy theory, this amounts to the second homotopy group π 2 (for 3-dimensional space) and π 2 (S 2 ) = Z, where Z = 0, 1, 2, ... First, Z or simply n = 0, the winding number, corresponds to the trivial vacuum with no monopole while n = 1 corresponds to the first non-trivial solution and so on. The monopole solution is topologically stable (i.e. the Higgs vacuum manifold forms a 2-sphere S 2 ) because of the fact that it takes an infinite amount of energy to go from the configuration n = 1 to n = 0 for example. An explicit example is the Georgi-Glashow model SO(3) ∼ SU (2) with a real Higgs triplet ξ = (ξ 0 , ξ 1 , ξ 2 ). In this model, the vacuum manifold is ξ 2 0 + ξ 2 1 + ξ 2 2 = v 2 M coresponding to S 2 and the model can accommodate a topologically-stable monopole.
Can SU (2) accommodate topologically-stable monopoles with different Higgs representations? We are particularly interested in a situation in which SU (2) contains, beside the real triplet ξ, a complex triplet χ and complex doublets φ i such as the case with the EW-ν R model [7]. A summary of some of the homotopy theory results is in order here.
π n (S n ) = Z; (1) where S 1 , .., S k denote spaces which, in our cases, represent different vacuum manifolds. (For pedagogical purposes, what is usually meant by a n-sphere S n is sim- The vacuum manifold of the SM with only a complex Higgs doublet (four independent degrees of freedom) is represented by This is a 3-sphere S 3 . From the above results, one has π 2 (S 3 ) = 0 and the SM has no topologically stable monopoles, a well-known result. This is true for any number of complex Higgs doublets. A complex Higgs triplet χ = (χ 0 , χ + , χ ++ ) has six real components and the vac- M is represented by a 5-sphere S 5 . One has π 2 (S 5 ) = 0. As we have stated above, a real Higgs triplet (ξ) vacuum manifold is represented by 2-sphere S 2 and π 2 (S 2 ) = Z.
The EW-ν R model has the following Higgs content: 1) One real triplet ξ; 2) One complex tripletχ; 3) Four complex Higgs doublets, φ SM i and φ M i with i = 1, 2, which couple to the SM and mirror fermions respectively. (It also contains Higgs singlets φ S which are important for different reasons but are not relevant here.) (Notice that the proper vacuum alignment which guarantees the socalled custodial symmetry gives χ = ξ = v M [7].) The vacuum manifolds of that Higgs sector is From (3), one obtains the second homotopy group of the vacuum manifold of the EW-ν R model (4) as The EW-ν R model can accommodate a topologically stable monopole because of the existence of the real SU (2) triplet ξ! (It is amusing to note that custodial symmetry requires S 2 (related to ξ) has the same "radius" v M as that S 5 (related toχ).) In what follows, the treatment of the monopole in the EW-ν R model follows that of the 't Hooft-Polyakov monopole, except for the final expression of the magnetic field as we shall see below. Last but not least, let us again recall that the SM with only Higgs doublets has no finiteenergy monopoles for two reasons: 1) π 2 (S 3 ) = 0; 2) The unbroken subgroup of SU (2) × U (1) Y being U (1) em can only accommodate a Dirac monopole which is singular at the origin.
The next two steps that we would like to make is to first write down the 't Hooft-Polyakov ansätz and estimate the mass and size of the monopole. Next, we look at the perturbative spectrum by performing small fluctuations around the background of the 't Hooft-Polyakov solution. In particular, we would like to see how this perturbative spectrum interacts with the monopole. We first start with the case with only the real triplet ξ and include the other scalars (complex triplet χ, doublets φ i ) as parts of the perturbative spectrum. In this first step, This is where the t Hooft-Polyakov monopole enters our model. As we have argued above, this is also where the only non-perturbative solutions exist since other Higgs representations present in the model (a complex triplet and Higgs doublets) have no topologically stable monopole solutions.
1) The 't Hooft-Polyakov ansätz is given by Ξ a = r a gr 2 H(v M gr); W a n = ǫ aji where a = 1, 2, 3 and n = 1, 2, 3 are the group and space indices respectively. In (6), we use Ξ and W to denote the non-perturbative solutions and we shall use the lower cases to denote the perturbative spectrum: ξ and w. The boundary conditions at infinity for H and K are H → v M g r and K → 0 as r → ∞. At r = 0, K − 1 → 0 and H → 0. The differential equations for H and K are well-known in the literature and will not be repeated here. It suffices to state that only numerical solutions are known.They depend on the self-coupling of the scalars and were found to rapidly approach their asymptotic values for sufficiently large values of that coupling.
2) Since S 2 is part of the vacuum manifold coming from the real triplet ξ, "charge" quantization now involves the SU (2) coupling g instead of the electromagnetic coupling e. Here, one has the Dirac quantization condition where, in order to avoid confusion, we denote the magnetic charge byg instead of g which is used for the weak charge here. Notice that the quantization condition 7 is in terms of the monopole chargeg AND the weak charge g instead of the usual e of the Dirac quantization condition.
3) One of the most important results of the present manuscript is the estimate of the mass of the monopole and the size of its core.
• The calculation of that mass is well known. What is different here from the usual estimates is the value of the VEV of the real triplet which is less than the electroweak scale. One has where once again, in order to avoid confusion, g is the SU (2) gauge coupling and λ is the ξ self-coupling. It is well known that the function f (λ/g 2 ) varies between 1 for λ = 0 (Prasad-Sommerfield limit) and 1.78 for λ = ∞.
Notice that the monopole mass is proportional to the triplet VEV v M and so is the right-handed Majorana mass M R of ν R . The search for ν R is intrinsically linked to the search for the monopole. It goes without saying that the above estimate is for the sole purpose of showing that the monopole mass M ∼ O( TeV) and is, therefore, accessible experimentally. This mass is much smaller, by at least thirteen orders of magnitude, than a typical mass of a GUT monopole.
• The monopole has a core of radius R c ∼ (gv M ) −1 ∼ 10 −16 cm, roughly a thousand times smaller than a proton "radius". Here one has gv M appearing in the denominator since that is the contribution to the W-boson mass from ξ. Inside the core are virtual W ± and Z. Far from the core, this monopole behaves like a Dirac monopole with some caveat as we shall see below.
In summary, due to the presence of the real triplet ξ of SU (2), a topologically stable monopole exists as a finite-energy soliton with finite size core (no singularity as opposed to a Dirac monopole) and with a mass of O( TeV).
5) The next step is to look at the interaction of the perturbative spectrum of the full SU (2) W × U (1) Y with the monopole i.e. small fluctuations of these fields in the presence of the monopole background. This procedure is reviewed in detail in an excellent review of Shnir [9] and we will adopt it here. We will denote the small fluctuations as w a µ and b µ for the gauge fields, and ξ, χ, φ SM i and φ M i (i = 1, 2) for the scalars. Let us start with the pure SU (2) case with the real triplet ξ. We have the following modifications for Eq. (6): ξ a = Ξ a + ξ a ;W a n = W a n + w a n ; W a 0 = w a 0 .
As it has been discussed in details in [9], the discussion of the dynamical equations governing the aforementioned small fluctuations is carried out in the simplest way in the Unitary gauge. Notice that, if we had only ξ which carries zero U (1) Y quantum number, its VEV would induce Let us first study this step before the U (1) W × U (1) Y symmetry is broken down to U (1) em by other Higgs fields. It is beyond the scope of this note to repeat well-known results for this case. Here, we just quote the essential points which can be found in [9]. We will not just look at the perturbative spectrum of SU (2) with a real scalar triplet but with the entire scalar spectrum, including a complex triplet χ and four doublets Φ SM i and Φ M i . Furthermore, the full gauge group is SU ( the familiar results for static field strengths are obtained and a well-known static radial magnetic field. Outside the core of radius R c ∼ (gv M ) −1 , one has a long-range magnetic field of strength 1/g. We now include the other Higgs fields: the complex tripletχ and the doublets, which spontaneously break SU (2) W × U (1) Y down to U (1) em . In the topological language, this means that we now include the spaces Mi and the vacuum manifold is described by S vac as shown in Eq. 4. As we have shown above, this vacuum manifold can still support the existence of a topologically stable monopole in light of Eq. (5). However, W 3 µ is no longer a mass eigenstate but is now written in terms of the Z-boson and photon fields as W 3 µ = cos θ W Z µ + sin θ W A µ . As a result, one has where F ij is the usual electromagnetic field strength tensor and Z ij is the Z field strength tensor. This is the reason why the name "γ-Z magnetic monopole " was chosen.
Since Z ij contains the static "electric" and "magnetic" Z fields, it has an exponential damping factor exp(−M Z r) where M Z denotes the Z-boson mass. One can now generalizes Eq. (13) as where e appearing in (15) denotes the usual electromagnetic coupling and where we have used the usual SM relationship e = g sin θ W . A few remarks are in order here. First, in the limit that the VEVs of all Higgs fields except for ξ vanish, M Z = 0, θ W = 0, B γZ i → B i and one recovers the 't Hooft from Eq. (15), one notices that, at large distances r ≫ R c , the magnetic field differs in strength from that of a point-like Dirac monopole by a factor sin 2 θ W . Third, the shortrange and long-range parts of B γZ i become comparable in strength at a distance r = 1 MZ ln(cot θ W ) ∼ 0.6/M Z . This is well inside the core of the monopole. The magnetic field for r ≫ R c is now simply A summary of the properties of the γ-Z magnetic monopole is in order here.
1) The existence in the EW-ν R model of a real Higgs triplet ξ gives rise to topologically-stable, finite-energy electroweak monopole; 2) The monopole mass, M = 4πvM g f (λ/g 2 ) ∼ 889 GeV − 2.993 TeV is intrinsically linked to the Majorana masses of the right-handed neutrinos.
3) The monopole is a finite-energy soliton with a core of radius R c ∼ (gv M ) −1 ∼ 10 −16 cm, with virtual W ± and Z inside the core.
4) This γ-Z magnetic monopole has a long-range magnetic field B i ≈ sin 2 θW er 2ri at distances larger than the core radius and looking like a Dirac monopole with a strength reduced by sin 2 θ W .
Last but not least, if the last assumption of Eq. (6), namely W a 0 = 0, is replaced by W a 0 = W a 0 (r) = 0, one can obtain a dyon solution which carries both magnetic and electric charges. It is beyond the scope of this paper to discuss such a possibility. It will be treated elsewhere.

PRODUCTION AND DETECTION OF THE γ-Z MAGNETIC MONOPOLE
We will make a few remarks in this section and postpone a more detailed treatment of the search for the γ-Z magnetic monopole to a longer version. This could be considered to be a very brief overview of the search for monopoles.
The first thing to notice is the coupling strength of the monopoleg. From the quantization condition (7) and from e = g sin θ W , one obtainsg = (e/2α em ) sin θ W ≈ 32.85 e. This is roughly half of the usual estimate g D = e/2α em ≈ 68.6 e because of the factor sin θ W . Because of the fact that the coupling is large, there is a large uncertainty in the calculation of the production cross section of MM , where M stands for a monopole. It goes without saying that a non-perturbative treatment is needed. It is useful nevertheless to get some rough ideas about what one might expect from colliders such as the LHC.
There are great uncertainties in making an estimate for the production cross section. In general, this estimate is highly non-perturbative. In addition, it has been argued that the production process depends on the initial states which could be, for instance, p-p collisions at the LHC or heavy-ion collisions, also at the LHC. In the first case of p-p collisions, arguments have been given for why the cross section for the production of a pair of composite 't Hooft-Polyakov-like monopoles is exponentiallysuppressed as σ ∼ exp(−4/α = −548) and thus ruling out the production of such monopoles even if they are "light" enough to be pair-produced [10]. In a nutshell, the argument implies that the energy carried by a few degrees of freedom of the initial p-p states has to be distributed among a large number O(1/α) of coherent states that the composite monopoles carry (unlike the point-like Dirac monopoles) and hence an exponential suppression factor in the cross section. One may, however, suspect that such an argument might be insufficient considering the nonperturbative nature of the production process and is not a definite proof. It goes without saying that a more comprehensive analysis of this topic is needed. An alternative proposal [11] was to use heavy-ion collision because the production process is very different from that of a p-p collision, coming mainly from a thermal Schwinger thermal pair production process. It has been argued that this process is valid for both cases of point-like Dirac monopoles or soliton 't Hooft-Polyakov-like monopoles. Taking into account the aforementioned caveat, let us go ahead and NAIVELY estimate what most likely is an upper bound on the expected number of monopoles of a given mass at the LHC.
Leading-order calculations for the production cross sections through Drell-Yan and γγ fusion at the 13 GeV LHC have been carried out [12]. Notice that the DY cross sections estimated in [12] will be down by sin 2 θ W here while the photon-fusion cross sections will be down by a factor sin 4 θ W . Taking that remark into consideration, a rough estimate for a 2-TeV monopole production cross section is ∼ 10f b, assuming a spin-0 monopole and assuming it is point-like. It is obvious that this estimate is for the sole purpose of illustration. With the projected luminosity of the high luminosity LHC (HL-LHC) to be around 250f b −1 /year, one might expect 2500 monopole events. These numbers are, of course, to be taken with a BIG GRAIN OF SALT. As mentioned above, heavy-ion collisions are another venue that one can explore.
There have been extensive discussions on the various methods of detection of electroweak-scale monopoles. These include highly ionizing tracks in detectors such as ATLAS or LHCb. Monopoles could be trapped by Al nuclei in the MoEDAL detector [12] or the Beryllium CMS beam pipe. It goes without saying that this kind of search is sufficiently important to motivate additional novel techniques.