The Price of an Electroweak Monopole

In a recent paper, Cho, Kim and Yoon (CKY) have proposed a version of the SU(2) $\times$ U(1) Standard Model with finite-energy monopole and dyon solutions. The CKY model postulates that the effective U(1) gauge coupling $\to \infty$ very rapidly as the Englert-Brout-Higgs vacuum expectation value $\to 0$, but in a way that is incompatible with LHC measurements of the Higgs boson $H \to \gamma \gamma$ decay rate. We construct generalizations of the CKY model that are compatible with the $H \to \gamma \gamma$ constraint, and calculate the corresponding values of the monopole and dyon masses. We find that the monopole mass could be $<5.5$ TeV, so that it could be pair-produced at the LHC and accessible to the MoEDAL experiment.


Introduction
Ever since Dirac first considered the possible existence of monopoles in QED [1], and Schwinger extended his considerations to dyons [2], theorists have explored the possible existence of finite-energy monopoles and dyons, and tried to estimate their masses. As pointed out by 't Hooft [3] and Polyakov [4], one very plausible scenario is that QED is embedded in a semi-simple unified group with coupling g U , in which case the core of the monopole/dyon is regularized and its mass is finite and O(V )/g U , where V is the vev of an Englert-Brout-Higgs field that breaks the unified group into pieces including a U (1) factor with a U(1) EM component.
However, physics at the electroweak scale is very well described by the Standard Model, which has an SU(2) × U(1) group structure that does not admit a finite-energy monopole or dyon solution unless its structure is modified [5,6], and there is no sign of an underlying semi-simple unified group that might be broken down to the Standard Model at any accessible energy scale. The question therefore arises whether there is any modification of the Standard Model that might contain a monopole or dyon solution with a mass O(v)/g, where g is a Standard Model gauge coupling and v the vev of the Standard Model Englert-Brout-Higgs field.
Cho, Kim and Yoon (CKY) [7] have recently proposed a scenario for modifying the Standard Model that includes a non-minimal coupling of its Englert-Brout-Higgs field to the square of its U(1) gauge coupling strength: (-1/4) L (|H|/v)B µν B µν . The coupling function is normalized so that (|H|/v) → 1 as |H| → v, in order to restore the conventional normalization of the U(1) gauge field in the standard electroweak vacuum.
Also, in order to have a finite-energy dyon solution, the coupling function should vanish as |H| → 0 like |H| n : n > 4 + 2 √ 3 7.46, so as to regularize the energy integral at the origin. Effectively, CKY create the possibility of a finite-energy dyon by postulating that the effective U(1) gauge coupling → ∞ sufficiently rapidly as |H| → 0.
CKY do not discuss an ultraviolet completion of the Standard Model that might lead to such behaviour, and nor do we. Our interest is limited to the question whether, in principle, the monopole mass could be regularized with a value low enough for it to be pair-produced at the LHC, and hence accessible to the MoEDAL experiment [8].
In their original model, CKY postulated a simple power law for the coupling function:  [11], namely that the quantity

Review of the Cho-Maison Monopole Solution
Before discussing the CKY construction [7] of a finite-energy monopole solution in the electroweak theory, we first review the structure of the (infinite-energy) Cho-Maison monopole solution. The Cho-Maison electroweak monopole [5] is a numerical solution of the Weinberg-Salam theory 2 . However, it suffers from a divergence in the energy due to a singularity at the centre of the configuration, r → 0, where r is the radial coordinate. As such, it cannot be considered as physical in the absence of a suitable ultraviolet completion. CKY [7] proposed a mechanism for rendering integrable the divergence at the monopole core, yielding a finite-energy solution that would be physical.
The starting-point of Cho and Maison and CKY is the Lagrangian describing the bosonic sector of the Weinberg-Salam theory, and H is the Englert-Brout-Higgs doublet. In the second line of (2.1) this is written as is essential for its interpretation as a CP 1 field with non-trivial second homotopy, making possible a topologically-stable monopole solution of the equations of motion [5]. Choosing the following Ansatz for the fields in spherical coordinates (t, r, θ, φ), one can find spherically-symmetric field configurations corresponding to electroweak monopoles 2 An analytical existence theorem for such monopole solutions can be established by appropriately adopting arguments by Yang [12]. and dyons 3 . With this Ansatz, the equations of motion take the form After an appropriate unitary gauge transformation U such that ξ → U ξ = 0 1 , one may obtain the physical gauge fields by rotating through the electroweak mixing angle where the electric charge e = g sin θ W = g cos θ W . The simplest non-trivial solution to the equations of motion with A(r) = B(r) = f (r) = 0 and ρ = ρ 0 ≡ √ 2µ/ √ λ describes a charge 4π/e point monopole with More general electroweak dyon solutions may be obtained for non-zero A, B and f . For example, with the boundary conditions where A 1 is a constant coefficient parametrising the 1/r asymptotic behaviour of A.
However, the Cho-Maison electroweak monpole and dyon [5] suffer from a non-integrable singularity in the energy density at the centre of the configuration when r → 0. This can be seen by calculating the total energy E of the dyon configuration, which has the form [7]: We see that, with the boundary conditions given by (2.5), E 1 is finite but the first term of E 0 is divergent at the origin.

Finite-Energy Monopoles and Dyons
The recent article by Cho, Kim and Yoon (CKY) [7] proposed, as one possibility, regularising the Cho-Maison monpole by modifying the Weinberg-Salam theory in such a way that the equations of motion have a finite-energy solution. The proposed modifications may be viewed as arising from unspecified dynamics that modify the form of the dielectric 'constant' in front of the U (1) Y hypercharge gauge kinetic term to become a non-trivial functional of the Englert-Brout-Higgs doublet, (H † H), a construction that preserves gauge invariance. Specifically, CKY considered the following form of effective Lagrangian that has a non-canonical kinetic term for the U (1) Y gauge field where (|H| 2 /v 2 ) is a positive dimensionless function of the Englert-Brout-Higgs doublet that approaches one asymptotically as |H| → v. Clearly modifies the permittivity of the U (1) Y gauge field, but the effective action still retains the SU (2)×U (1) Y gauge symmetry.
Moreover, since → 1 asymptotically, the effective action reproduces the Standard Model when the Englert-Brout-Higgs field adopts its canonical vacuum expectation value: |H| = v. However, the factor (|H| 2 /v 2 ) effectively changes the U (1) Y gauge coupling g to a "running" couplingḡ = g / √ that depends on |H|. This is because, with the rescaling of B µ → B µ /g , g changes to g / √ . By choosing so thatḡ → ∞ as |H| → 0, i.e., requiring to vanish at the origin, one can regularise the Cho-Maison monopole.
Such an ad hoc modification of the Standard Model is phenomenologically motivated as a way to render finite the energy integral, leading to a finite mass for the electroweak monopole. We leave open the question of how such a modification may occur in a 'topdown' approach, and pursue the question how light such a CKY monopole might be 4 . .
With such a regularisation, the equations of motion in the spherically-symmetric ansatz are modified toρ where we have defined ≡ d /dρ 2 . The original proposal [7] for regulating the infiniteenergy divergence was to consider a functional form where one must require n > 4 + 2 √ 3 in order for certain terms in the equations of motion to vanish fast enough as r → 0 that the energy remains finite. With the boundary conditions (2.5) the solution at the origin behaves as , and behaves asymptotically towards infinity as where ω = (gρ 0 ) 2 /4 − A 2 0 and ν = 1 2 g 2 + g 2 ρ 0 . These behaviours of the fields in the limits can be used together with the equations of motion (3.2) to obtain numerical solutions. We plot in Fig. 1 on the left the result for n = 8 when A = B = 0 (corresponding to a monopole with no electric charge) and on the right the general case with A 0 = M W /2 (corresponding to a dyon). Plugging the simplest A = B = 0 solution into the energy integral (2.7) with the appropriate form factor regularisation, we find a monopole mass of ∼ 5.7 TeV. The non-zero A, B solution yields a larger mass of ∼ 10.8 TeV for the dyon. An increase was to be expected, since non-vanishing forms of A and B will always contribute positively to the E 1 integral (2.7).
The topological stability of the lowest-lying monopole is guaranteed by the conservation of magnetic charge [5]. However, dyon solutions may be unstable if suitable decays into charged particles and a monopole are kinematically accessible, as is the case in this example.

Phenomenological Constraint from H → γγ Decay
However, the simple power-law functional form for the regulator that was chosen in [7] is phenomenologically excluded by data on Higgs decays to γγ [9].
In [10], dimension-six operators involving couplings of the Higgs field with the gauge sector of the Standard Model have been studied in an analysis of the data now available from the LHC. Among them, of interest to us here is the operator where we use the notation of Ref. [10] in which constraints are placed onc γ ≡ c γ M 2 W /Λ 2 . Based on a global fit to LHC data, mainly from the decay of the Higgs field H → γ γ, the best fit values ofc γ are in the range of 10 −3 and negative [10].
Expanding ρ near its vacuum expectation value ρ 0 ≡ √ 2µ/ √ λ: we may write the term (4.1) as an effective Lagrangian contribution of the form On the other hand, the -dependent modification (3.3) of the Lagrangian (3.1), when expanded around the vacuum expectation value, yields a term where we recall that finiteness of the monopole total energy/mass then requires for a simple power law that n ≥ 8 ∈ Z + .
Comparing (4.3) with (4.4), we see that to linear order inρ/ρ 0 , Since n ≥ 8 ⇒c γ −0.8 is strongly excluded by the 95% CL observed valuec γ 10 −3 [10], we conclude that the simple power-law modification of the U (1) Y permeability proposed in [7] cannot be valid all the way from the origin of the Englert-Brout-Higgs field ρ → 0 up to the region near the expectation value, ρ ρ 0 .
One needs therefore a modification of the Standard Model Lagrangian of the general form in (3.1), but with the U (1) Y permeability an interpolating functional having the following properties: where the superscript (n) indicates the n-th derivative with respect to ρ. In the following we impose the stronger conditionc γ = 0: relaxing this to |c γ | = O(10 −3 ) would not change our results significantly.

Figure 2:
Interpolating functions (ρ) that satisfy the required theoretical and phenomenological properties in solid brown, solid green, dotted blue, and dashed-dotted orange lines. The CKY regularisation [7] that is incompatible with LHC data [10] is shown in dashed red.

Implementing the H → γγ Constraint
An acceptable form of the interpolating functional may be found by making an Ansatz with two or more parameters, for which the simplest possibility is Solving for the coefficients C 1 and C 2 using the constraints (4.5) we find C 1 = 5 and C 2 = −4. This 1 regularisation is plotted in dotted blue in Fig. 2, with the original CKY denoted by a dashed red line for comparison. The solution for ρ and f with this 1 regularisation is obtained numerically and plotted in the left panel of Fig. 3, and gives a monopole mass of M ≡ E 0 + E 1 6.6 TeV when integrated in (2.7). We focus here on the A = B = 0 monopole case, as this minimises the total energy, with the lowest possible dyon mass being significantly larger.
The powers n = 8, 10 chosen in (5.1) are the lowest powers of ρ that are consistent with convergence of the energy integral and analyticity in |H| 2 . The monopole mass is larger than for the CKY Ansatz, because the energy integrand must be larger at intermediate values of ρ/ρ 0 in order that be able to approach unity with a very small derivative as ρ → ρ 0 . On the other hand, the fact that n = 8 is barely integrable suggests that a smaller value of the monopole mass might be found for larger values of n. 2) on the right that satisfy all theoretical and phenomenological constraints. The solid blue (green) line represents the solution for ρ (f ), where ρ is normalised by ρ 0 . We also plot the one-coefficient solution that is excluded by Higgs data using dashed lines.
Accordingly, we have examined a second Ansatz, 2 , that is a combination of n = 10 and 12: where the values of the coefficients have again been chosen so that 2 → 1 with 2 → 0 when ρ → ρ 0 . The solution in this case is shown as a solid green line in Fig. 2. Solving the equations of motion numerically once more, we plot the result in the right panel of Fig. 3. Plugging this solution into the energy integral we find that the energy in this case is lowered to M 6.2 TeV, as anticipated because the regulating function gives faster convergence as we approach the origin. One can also consider more complicated functional forms for , that need not be polynomial in ρ. Even if one considers just polynomials with more coefficients, one has too many parameters to be determined by the number of conditions to be satisfied.
Finding the minimum of the energy integral over a multi-dimensional space is impractical. Another possibility is to apply the principle of maximum entropy (PME) method [11] to determine the coefficients. For example, one may consider the following Ansatz with three coefficients: where two combinations of coefficients can be determined by the two constraint equations obtained (4.5), and one may solve for the remaining combination of coefficients by We also plot the one-coefficient solution that is excluded by Higgs data using dashed lines.
requiring that the entropy function be maximised. In this case we find the three-coefficient function that satisfies all these properties to be which is plotted as an orange dash-dotted line in Fig. 2. The numerical solution to the equations of motion for this three-coefficient 2 regularisation is plotted for A = B = 0 on the left in Fig. 4. Plugging this solution into the energy integral yields a slightly higher monopole mass than the two-coefficient case, namely M 6.8 TeV. Thus, we do not find a lowering of the monopole mass with this simplest generalisation to more coefficients of higher powers.
A lower monopole mass can be obtained by considering higher powers of n. For example, a two-coefficient polynomial regularisation with n = 14, 16 yields a monopole mass of M 5.7 TeV. One may also consider non-polynomial functional forms for . As an example, we consider the following: which is plotted as a solid brown line in Fig. 2. This regularisation converges faster, due to the higher powers involved and the logarithm that modifies the behaviour of  the function away from the vacuum expectation value of ρ while vanishing when ρ = ρ 0 . This solution is plotted in the right panel of Fig. 4, and gives M 5.4 TeV. As expected from the improved convergence, this is the lowest monopole mass of all the regularisation functions that we have considered here. The effect of the logarithm relative to the improvement due solely to the higher powers n = 14, 16 is seen in the reduction of the monopole mass from M 5.7 TeV.
For the reader's convenience, our results for the specific regularisations that we have studied are summarised in Table 1. They lead to monopole masses ranging from M ∼ 6.8 TeV down to ∼ 5.4 TeV, with a larger mass ∼ 10.8 TeV for the dyon case we consider.
We may expect that a lower-mass monopole could be found in a more exhaustive survey of parameter space, particularly if attention was restricted to powers n ≥ 10.

Conclusion
Earlier papers by Cho and Maison [5] and by Cho, Kim and Yoon [7] have indicated how finite-mass electroweak monopole and dyon solutions may be found in suitable modifications of the Standard Model. In particular, it was shown in [7] that an appropriate non-trivial permittivity in the U(1) sector of the Standard Model could regularise the monopole and dyon energy integrals. However, the simplest example of such a scenario proposed in [7] is incompatible with data on H → γγ decay from the LHC [9, 10]. Nevertheless, we have shown in this paper how to generalise their construction in a way that is compatible with these data and yields finite-mass monopole solutions.
The lowest monopole mass found in illustrative examples is 5.4 TeV, and one may expect that a smaller mass could be found in a more complete study of generalisations of our construction. However, a more exhaustive study should perhaps be contingent upon stronger theoretical indications what type of modification of the U(1) permittivity might arise in which completion of the Standard Model.
From our point of view, the most important conclusion of this exploratory study has been that there is a possibility, compatible with the present constraints on the Standard Model [9, 10], that there may exist an electroweak magnetic monopole with a mass < 5.5 TeV, which could therefore be pair-produced at the LHC. We recall that the MoEDAL experiment [8] dedicated to searches for monopoles and other heavily-ionising particles has been installed at the LHC, and has started taking data at 13 TeV in the centre of mass. Our analysis reinforces the motivation to pursue the monopole search with MoEDAL.