Finite Energy Electroweak Dyon

The recent MoEDAL experiment at LHC to detect the electroweak monopole makes the theoretical prediction of the monopole mass an urgent issue. We discuss different ways to estimate the mass of the electroweak monopole. We first present a scaling argument which indicates that the mass of the electroweak monopole to be around 4 TeV. To justify this we construct finite energy analytic dyon solutions which could be viewed as the regularized Cho-Maison dyon, modifying the coupling strengths of the electromagnetic interaction of $W$-boson in the standard model. Our result demonstrates that a genuine electroweak monopole whose mass scale is much smaller than the grand unification scale can exist, which can actually be detected at the present LHC.


I. INTRODUCTION
The recent "discovery" of the Higgs particle at LHC and Tevatron has reconfirmed that the electroweak theory of Weinberg describes the real world [1,2].If so, one might ask what would be the next hot subject after the Higgs particle in the standard model.Certainly there could be different opinions, but one thing must be clear.We must look for the electroweak monopole because the standard model provides the natural topology for the monopole [3,4].The existence of the monopole topology in the theory strongly implies that the electroweak monopole must exist.
In this sense it is really due and timely that the latest MoEDAL detector ("The Magnificient Seventh") at LHC is actively searching for such monopole [5].To detect the electroweak monopole experimentally, however, it is important to estimate the monopole mass in advance.The purpose of this paper is to provide an educated guess of the mass of the electroweak monopole.We show that the monopole mass could be around 4 to 7 TeV.
Ever since Dirac [6] has introduced the concept of the magnetic monopole, the monopoles have remained a fascinating subject.The Abelian monopole has been generalized to the non-Abelian monopoles by Wu and Yang [7,8] who showed that the pure SU (2) gauge theory allows a point-like monopole, and by 't Hooft and Polyakov [9,10] who have constructed a finite energy monopole solution in Georgi-Glashow model as a topological soliton.
In the interesting case of the electroweak theory of * Electronic address: ymcho7@konkuk.ac.krWeinberg and Salam, however, it has generally been asserted that there exists no topological monopole of physical interest [11].The basis for this "non-existence theorem" is, of course, that with the spontaneous symmetry breaking the quotient space SU (2) × U (1)/U (1) em allows no non-trivial second homotopy.This has led many people to believe that there is no monopole in Weinberg-Salam model which can be viewed as the generalization of the Dirac monopole.This claim, however, has been shown to be not true.Indeed some time ago Cho and Maison have proved that Weinberg-Salam model and Georgi-Glashow model have exactly the same topological structure, and demonstrated the existence of a new type of monopole and dyon solutions in the standard model [3].This was based on the observation that the Weinberg-Salam model, with the hypercharge U (1), could be viewed as a gauged CP 1 model in which the (normalized) Higgs doublet plays the role of the CP 1 field.So the Weinberg-Salam model does have exactly the same nontrivial second homotopy as the Georgi-Glashow model which allows topological monopoles.
Once this is understood, one could proceed to construct the desired monopole and dyon solutions in the Weinberg-Salam model.Originally the solutions of Cho and Maison were obtained by numerical integration.But a mathematically rigorous existence proof has since been established which endorses the numerical results, and the solutions are now referred to as Cho-Maison monopole and dyon [4].
It should be emphasized that the Cho-Maison monopole is completely different from the "electroweak monopole" derived from the Nambu's electroweak string.In his continued search for the string-like objects in physics, Nambu has demonstrated the existence of a rotating dumb bell made of the monopole anti-monopole pair connected by the neutral string of Z-boson flux (actually the SU (2) flux) in Weinberg-Salam model [12].Taking advantage of the Nambu's pioneering work, others claimed to have discovered another type of electroweak monopole, simply by making the string infinitely long and moving the anti-monopole to infinity [13].This "electroweak monopole", however, must carry a fractional magnetic charge and can not be isolated with finite energy.Moreover, this has no spherical symmetry which is manifest in the Cho-Maison monopole [3].
The existence of the electroweak monopole makes the experimental confirmation of the monopole an urgent issue [5].Till recently the experimental effort for the monopole detection has been on the Dirac's monopole [14].But the electroweak unification of the Maxwell's theory requires the modification of the Dirac monopole, and this modification changes the Dirac monopole to the Cho-Maison monopole.This means that the monopole which should exist in the real world is not likely to be the Dirac monopole but the electroweak monopole.
To detect the electroweak monopole experimentally, it is important to estimate the most probable mass of the monopole theoretically.Unfortunately the Cho-Maison monopole carries an infinite energy at the classical level, so that the mass of the monopole is not determined.This is because it can be viewed as a hybrid between the Dirac monopole and the 'tHooft-Polyakov monopole, so that it has a U (1) point singularity at the center even though the SU (2) part is completely regular.
A priori there is nothing wrong with this, but this makes the experimental search for the monopole difficult.In this paper we show how to predict the mass of the electroweak monopole.Based on a scaling argument, we first predict the mass to be around 4 TeV.To backup this prediction we construct finite energy electroweak monopole and dyon solutions by regularizing the point singularity of the Cho-Maison dyon at the origin, and show that the energy of the regularized solution has the predicted value.Our result suggests that the electroweak monopole could have the mass around 4 to 7 TeV.This implies that there is a very good chance that the MoEDAL at the present LHC can detect the electroweak monopole.
The paper is organized as follows.In Section II we review the Cho-Maison dyon for later purpose.In Section III we provide a simple scaling argument which indicates that the mass of the electroweak monopole could be around 4 TeV.In Section IV we discuss the Abelian decomposition and gauge independent Abelianization of Weinberg-Salam model and Georgi-Glashow model to compare the Cho-Maison dyon with the Julia-Zee dyon.This teaches us how to regularize the Cho-Maison monopole.In Section V we show how to regularize the Cho-Maison dyon to obtain a finite energy electroweak dyon by modifying the coupling strengths of the magnetic moment interaction and quartic self-interaction of the W -boson.We suggest that this type of modifica-tion could come from the quantum correction or the unification of all interactions.In Section VI we show that we can make the Cho-Maison dyon regular by enlarging the gauge group SU (2) × U (1) to SU (2) × SU (2) Y .Finally in Section VII we discuss the physical implications of our results.

II. CHO-MAISON DYON IN WEINBERG-SALAM MODEL: A REVIEW
Before we construct a finite energy dyon solution in the electroweak theory we must understand how one can obtain the infinite energy Cho-Maison dyon solution first.Let us start with the Lagrangian which describes (the bosonic sector of) the Weinberg-Salam theory where φ is the Higgs doublet, F µν and G µν are the gauge field strengths of SU (2) and U (1) with the potentials A µ and B µ , and g and g ′ are the corresponding coupling constants.Notice that D µ describes the covariant derivative of the SU (2) subgroup only.With where ρ and ξ are the Higgs field and unit doublet, we have Notice that the hypercharge U (1) coupling of ξ makes the theory a gauge theory of CP 1 field [3].
From (1) one has the following equations of motion Now we choose the following ansatz in the spherical co-ordinates (t, r, θ, ϕ) Notice that ξ † τ ξ = −r.Moreover, A µ describes the Wu-Yang monopole when A(r) = f (r) = 0.So the ansatz is spherically symmetric.Of course, ξ and B µ have an apparent string singularity along the negative z-axis, but this singularity is a pure gauge artifact which can easily be removed making the hypercharge U (1) bundle nontrivial.So the above ansatz describes a most general spherically symmetric ansatz of an electroweak dyon.
Here we emphasize the importance of the non-trivial nature of U (1) gauge symmetry to make the ansatz spherically symmetric.Without the extra U (1) the Higgs doublet does not allow a spherically symmetric ansatz.This is because the spherical symmetry for the gauge field involves the embedding of the radial isotropy group SO(2) into the gauge group that requires the Higgs field to be invariant under the U (1) subgroup of SU (2).This is possible with a Higgs triplet, but not with a Higgs doublet [15].In fact, in the absence of the hypercharge U (1) degrees of freedom, the above ansatz describes the SU (2) sphaleron which is not spherically symmetric [16].
To see this, one might try to remove the string in ξ with the U (1) subgroup of SU (2).But this U (1) will necessarily change r and thus violate the spherical symmetry.This means that there is no SU (2) gauge transformation which can remove the string in ξ and at the same time keeps the spherical symmetry intact.The situation changes with the inclusion of the hypercharge U (1) in the standard model, which naturally makes ξ a CP 1 field [3].This allows the spherical symmetry for the Higgs doublet.
To understand the physical content of the ansatz we perform the following gauge transformation on (5) and find that in this unitary gauge we have So introducing the electromagnetic and neutral Z-boson potentials A (em) µ and Z µ with the Weinberg angle θ w we can express the ansatz (5) in terms of the physical fields where W µ is the W -boson and e is the electric charge This clearly shows that the ansatz is for the electroweak dyon.
The spherically symmetric ansatz reduces the equations of motion to Obviously this has a trivial solution which describes the point monopole in Weinberg-Salam model This monopole has two remarkable features.First, this is the electroweak generalization of the Dirac's monopole, but not the Dirac's monopole.It has the electric charge 4π/e, not 2π/e [3].Second, this monople naturally admits a non-trivial dressing of weak bosons.Indeed, with the non-trivial dressing, the monopole becomes the Cho-Maison dyon.
To see this let us choose the following boundary condition Then we can show that the equation ( 10) admits a family of solutions labeled by the real parameter A 0 lying in the range [3,4] In this case all four functions f (r), ρ(r), A(r), and B(r) must be positive for r > 0, and A(r)/g 2 + B(r)/g ′2 and B(r) become increasing functions of r.So we have 0 ≤ b 0 ≤ A 0 .Furthermore, we have B(r) ≥ A(r) ≥ 0 for all range, and B(r) must approach to A(r) with an exponential damping.Notice that, with the experimental fact sin 2 θ w = 0.2312, ( 14) can be written as 0 ≤ A 0 < eρ 0 .
Near the origin the dyon solution has the following behavior, where Asymptotically it has the following behavior, where ω = (gρ 0 ) 2 /4 − A 2 0 , and ν = (g 2 + g ′2 )ρ 0 /2.The physical meaning of the asymptotic behavior must be clear.Obviously ρ, f , and A − B represent the Higgs boson, W -boson, and Z-boson whose masses are given by M H = √ 2µ = √ λρ 0 , M W = gρ 0 /2, and Notice that it is 1 − (A 0 /M W ) 2 M W , but not M W , which determines the exponential damping of the W -boson.This tells that the electric potential of the dyon slows down the exponential damping of the W -boson, which is reasonable.
The dyon has the following electromagnetic charges Also, the asymptotic condition (16) assures that the dyon does not carry any neutral charge, Furthermore, notice that the dyon equation ( 10) is invariant under the reflection This means that, for a given magnetic charge, there are always two dyon solutions which carry opposite electric charges ±q e .Clearly the signature of the electric charge of the dyon is determined by the signature of the boundary value A 0 .
With the ansatz (5) we have the following energy of the dyon The boundary condition (13) guarantees that E 1 is finite.
As for E 0 we can minimize it with the boundary condition f (0) = 1, but even with this E 0 becomes infinite.Of course the origin of this infinite energy is obvious, which is precisely due to the magnetic singularity of B µ at the origin.This means that one can not predict the mass of dyon.Physically it remains arbitrary.
Since the Cho-Maison solution is obtained numerically one might like to have a mathematically rigorous existence proof of the Cho-Maison dyon.The existence proof is non-trivial, because the equation of motion (10) is not the Euler-Lagrange equation of the positive definite energy (20), but that of the indefinite action Fortunately the existence proof has been established by Yang [4].

III. MASS OF THE ELECTROWEAK MONOPOLE: A SCALING ARGUMENT
To detect the electroweak monopole experimentally, we have to have a firm idea on the mass of the monopole.
Unfortunately, at the classical level we can not estimate the mass of the Cho-Maison monopole, because it has a point singularity at the center which makes the total energy infinite.This means that we can not predict the mass.It is undetermined.
To estimate of the monopole mass theoretically, we have to regularize the point singularity of the Cho-Maison dyon.One might try to do that introducing the gravitational interaction, in which case the mass is fixed by the asymptotic behavior of the gravitational potential.But the magnetic charge of the monopole is not likely to change the character of the singularity, so that asymptotically the leading order of the gravitational potential becomes of the Reissner-Nordstrom type [17].This implies the gravitational interaction may not help us to estimate the monopole mass.
A simple way to make the energy of the monopole finite is to introduce a UV-cutoff which can cure the divergence in E 0 .But introducing an explicit UV-cutoff is not easy to do.So, assuming a UV-cutoff, we use the Derrick's scaling argument which can tell us how to estimate the monopole mass.
If a finite energy monopole does exist, it should be stable under the rescaling of its field configuration.So consider a static monopole configuration and let With the ansatz (5) we have (with A = B = 0) and Notice that K B makes the monopole energy infinite.Now, consider the spatial scale transformation With this the gauge fields and the scalar field undergo the following scaling transformations so that we have From this we have the following condition for the stable monopole configuration Although this relation holds only for the finite energy configuration, we can infer the value of K B from this since the other three terms are finite.
Now, for the Cho-Maison monopole we have (with M W ≃ 80. 4 GeV, M H ≃ 125 GeV, and sin 2 θ w = 0.2312) This, with (28), tells that From this we estimate the energy of the monopole to be Although this estimate is only an educated guess, this does imply that the electroweak monopole of mass around a few TeV could be possible.Moreover, this demonstrates that the scaling argument is very powerful in estimating the mass of the electroweak monopole.
One might wonder if there is any independent backup argument of the monopole mass which can support this estimate.In the following we will show that we can actually regularize the Cho-Maison monopole, and that the regularized monopole has the energy predicted by the above argument.To do that we have to understand the structure of the electroweak theory, in particular the Abelian decomposition of the electroweak theory.To understand this we have to know the Abelian decomposition of the SU (2) gauge theory first [18,19].So in the following we discuss the gauge independent Abelian decomposition of the SU (2) gauge theory.

IV. ABELIAN DECOMPOSITION OF THE ELECTROWEAK THEORY
Consider the Yang-Mills theory A best way to make the Abelian decomposition is to introduce a unit SU (2) triplet n which selects the Abelian direction at each space-time point, and impose the isometry on the gauge potential which determines the restricted potential Âµ [18,19] D µ n = 0, Notice that the restricted potential is precisely the connection which leaves n invariant under parallel transport.The restricted potential is called Cho connection or Cho-Duan-Ge connection [20][21][22].
With this we obtain the gauge independent Abelian decomposition of the SU (2) gauge potential adding the valence potential W µ which was excluded by the isometry [18,19] The Abelian decomposition has recently been referred to as Cho (also Cho-Duan-Ge or Cho-Faddeev-Niemi) decomposition [20][21][22].
Under the infinitesimal gauge transformation we have This tells that Âµ by itself describes an SU (2) connection which enjoys the full SU (2) gauge degrees of freedom.Furthermore the valence potential W µ forms a gauge covariant vector field under the gauge transformation.But what is really remarkable is that the decomposition is gauge independent.Once n is chosen, the decomposition follows automatically, regardless of the choice of gauge.
Notice that Âµ has a dual structure, Moreover, H µν always admits the potential because it satisfies the Bianchi identity.In fact, replacing n with a CP 1 field ξ by n = −ξ † τ ξ we have Of course Cµ is determined uniquely up to the U (1) gauge freedom which leaves n invariant.To understand the meaning of Cµ , notice that with n = r we have This is nothing but the Abelian monopole potential.The corresponding non-Abelian monopole potential is given by the Wu-Yang monopole [7,8] This justifies us to call A µ and Cµ the electric and magnetic potential.
The above analysis tells that Âµ retains all essential topological characteristics of the original non-Abelian potential.First, n defines π 2 (S 2 ) which describes the non-Abelian monopoles.Second, it characterizes the Hopf invariant π 3 (S 2 ) ≃ π 3 (S 3 ) which describes the topologically distinct vacua [26,27].Moreover, it provides the gauge independent separation of the monopole field from the generic non-Abelian gauge potential.
With the decomposition (34), one has so that the Yang-Mills Lagrangian is expressed as This shows that the Yang-Mills theory can be viewed as a restricted gauge theory made of the restricted potential, which has the valence gluons as its source [18,19].
An important advantage of the decomposition (34) is that it can actually Abelianize (or more precisely "dualize") the non-Abelian gauge theory gauge independently [18,19].To see this let(n 1 , n2 , n) be a right-handed orthonormal basis of SU (2) space and let With this we have so that with we can express the Lagrangian explicitly in terms of the dual potential A µ and the complex vector field W µ , where F µν = F µν + H µν and Dµ = ∂ µ + igA µ .This shows that we can indeed Abelianize the non-Abelian theory with our decomposition.
An important point of the Abelian formalism is that it has the extra magnetic potential Cµ .In other words, the Abelian potential A µ is given by the sum of the electric and magnetic potentials A µ + Cµ .Clearly Cµ represents the topological degrees of the non-Abelian symmetry which does not show up in the naive Abelianization that one obtains by fixing the gauge [18,19].
Furthermore, this Abelianization is gauge independent, because here we have never fixed the gauge to obtain this Abelian formalism.So one might ask how the non-Abelian gauge symmetry is realized in this Abelian formalism.To discuss this let Certainly the Lagrangian (44) is invariant under the active (classical) gauge transformation (36) described by But it has another gauge invariance, the invariance under the following passive (quantum) gauge transformation Clearly this passive gauge transformation assures the desired non-Abelian gauge symmetry for the Abelian formalism.This tells that the Abelian theory not only retains the original gauge symmetry, but actually has an enlarged (both active and passive) gauge symmetries.
The reason for this extra (quantum) gauge symmetry is that the Abelian decomposition automatically put the theory in the background field formalism which doubles the gauge symmetry [28].This is because in this decomposition we can view the restricted and valence potentials as the classical and quantum potentials, so that we have freedom to assign the gauge symmetry either to the classical field or to the quantum field.This is why we have the extra gauge symmetry.
The Abelian decomposition has played a crucial role in QCD to demonstrate the Abelian dominance and the monopole condensation in color confinement [23][24][25].This is because it separates not only the Abelian potential but also the monopole potential gauge independently.Now, consider the Georgi-Glashow model where Φ is the Higgs triplet.With we have the Abelian decomposition of the Georgi-Glashow model, With this we can Abelianize it gauge independently, This clearly shows that the theory can be viewed as a (non-trivial) Abelian gauge theory which has a charged vector field as a source.
The Abelianized Lagrangian looks very much like the Georgi-Glashow Lagrangian written in the unitary gauge.But notice that we have derived (51) without any gauge fixing, so that (51) has nothing to do with the unitary gauge.As we have emphasized this is the gauge independent Abelianization which has the full (quantum) SU (2) gauge symmetry.
Obviously we can apply the same Abelian decomposition to the Weinberg-Salam theory Moreover, with we can Abelianize it gauge independently where . Again we emphasize that this is not the Weinberg-Salam Lagrangian in the unitary gauge.This is the gauge independent Abelianization which has the extra quantum (passive) non-Abelian gauge degrees of freedom.This provides us important piece of information.In the absence of the electromagnetic interaction (i.e., with A (em) µ = W µ = 0) the Weinberg-Salam model describes a spontaneously broken U (1) Z gauge theory, which is nothing but the Ginsburg-Landau theory of superconductivity.Furthermore, here M H and M Z corresponds to the coherence length (of the Higgs field) and the penetration length (of the magnetic field made of Zfield).So, when M H > M Z (or M H < M Z ), the theory describes a type II (or type I) superconductivity, which is well known to admit the Abrikosov-Nielsen-Olesen vortex solution [29].This confirms the existence of Nambu's string in Weinberg-Salam model.What Nambu showed was that he could make the string finite by attaching the fractionally charged monopole anti-monopole pair to this string [12].

V. COMPARISON WITH JULIA-ZEE DYON
The Cho-Maison dyon looks very much like the wellknown Julia-Zee dyon in the Georgi-Glashow model.Both can be viewed as the Wu-Yang monopole dressed by the weak boson(s).However, there is a crucial difference.The the Julia-Zee dyon is completely regular and has a finite energy, while the Cho-Maison dyon has a point singularity at the center which makes the energy infinite.
So, to regularize the Cho-Maison dyon it is important to understand the difference between the two dyons.To understand this notice that, in the absence of the Zboson, (54) reduces to This should be compared with (51), which shows that the two theories have exactly the same type of interaction in the absence of the Z-boson, if we identify F µν in (51) with F (em) µν in (56).The only difference is the coupling strengths of the W -boson quartic self-interaction and Higgs interaction of W -boson (responsible for the Higgs mechanism).This difference, of course, originates from the fact that the Weinberg-Salam model has two gauge coupling constants, while the Georgi-Glashow model has only one.This tells that, in spite of the fact that the Cho-Maison dyon has infinite energy, it is not much different from the Julia-Zee dyon.To amplify this point notice that the spherically symmetric ansatz of the Julia-Zee dyon can be written in the Abelian formalism as In the absence of the Z-boson this is identical to the ansatz (9).
With the ansatz we have the following equation This should be compared to the equation of motion (10) for the Cho-Maison dyon.They are not much different.
With the boundary condition one can integrate (59) and obtain the Julia-Zee dyon which has a finite energy.Notice that the boundary condition A(0) = 0 and f (0) = 1 is crucial to make the Julia-Zee dyon regular at the origin.This confirms that the Julia-Zee dyon is nothing but the Abelian monopole regularized by ρ and W µ , where the charged vector field adds an extra electric charge to the monopole.Again it must be clear from (59) that, for a given magnetic charge, there are always two dyons with opposite electric charges.
Moreover, for the monopole solution with A = 0, the equation reduces to the following Bogomol'nyi-Prasad-Sommerfield equation in the limit λ = 0 This has the analytic solution which describes the Prasad-Sommerfield monopole [10].
Of course, the Cho-Maison dyon has a non-trivial dressing of the Z-boson which is absent in the Julia-Zee dyon.But notice that the Z-boson plays no role in the Cho-Maison monopole.This tells that by modifying the coupling strengths of the Weinberg-Salam thory we could regularize the Cho-Maison monopole and obtain a finite energy electroweak monopole.

VI. ELECTROMAGNETIC REGULARIZATION OF CHO-MAISON DYON
Since the Cho-Maison dyon is the only dyon in the Weinberg-Salam model, it is impossible to regularize it within the model.However, the Weinberg-Salam theory is the "bare" theory which should change to the "effective" theory after the quantum correction.Besides, it (in particular the ultra-violet limit of the theory) has to be modified by the true unification of all interactions.And the "real" electroweak dyon must be the solution of such theory.This implies that the quantum correction or the unification of all interactions could regularize the Cho-Maison dyon.In this section we discuss how such modification could make the energy of the Cho-Maison dyon finite.
The importance of the quantum correction in classical solutions is best understood in QCD.The "bare" QCD Lagrangian has no explicit confinement, so that the classical solutions of this theory do not describe the real world and thus have no physical meaning.For example, the bare theory can never produce the classical solutions of (the bag model of) hadrons or the linear confining potential between q q.Only the effective theory can describe such solutions classically.Suppose we have the following modification of (52) from the quantum correction or the unification of all interactions where α, β, γ are the constants which are supposed to be fixed later.With this we have the following effective Lagrangian which we call the generalized Weinberg-Salam Lagrangian Clearly this Lagrangian can not be viewed to describe the true electroweak theory.Nevertheless, it is consistent with the spirit of the Weinberg-Salam model, in the sense that this does not introduce a new interaction.The corrections only modify the coupling strengths of the existing interactions.In this respect we could regard it an approximate effective theory of the electroweak interaction.
To understand the physical meaning of (64) notice that in the absence of the Z-boson the above Lagrangian reduces to (51) where the W -boson has an extra "anomalous" magnetic moment α when (1 + β) and (1 + γ) become e 2 /g 2 , if we identify the coupling constant g in the Georgi-Glashow model by the electromagnetic coupling constant e.Moreover, the ansatz (5) can be written as This shows that, for the monopole (i.e., when A = B = 0) the ansatz becomes formally identical to (57) if W µ is rescaled by a factor g/e.This tells that in the absence of the Z-boson the generalized Weinberg-Salam theory reduces to the Georgi-Glashow theory where the W -boson has an extra "anomalous" magnetic moment α when (1 + β) and (1 + γ) become e 2 /g 2 .With (64) the energy of the dyon is given by Notice that Ê1 remains finite with the modification.
To make Ê0 finite, we have to remove both O(1/r 2 ) and O(1/r) singularities at the origin.This requires us to have So we have the following condition for a finite energy solution where we have used 1 + g 2 /g ′2 = g 2 /e 2 .So among the three parameters α, β, and f (0), only one becomes arbitrary.Of course, (65) tells that at the origin A µ develops a singularity when f (0) = g/e.However, this is a harmless singularity which does not make the energy divergent.So we will keep f (0) arbitrary.With (68) the equation of motion is given by Now with the boundary condition we can integrate this numerically.The results are shown in Fig. 1 and Fig. 2. Of course, with a different f (0), we can still integrate (69) and have a finite energy solution.
It is really remarkable that the finite energy solutions look almost identical to the Cho-Maison solutions, even though they no longer have the magnetic singularity at the origin.This is because they are obtained with a simple modification of the coupling strengths of the Wboson.
The solution must have the following behavior near the origin, where Notice that all four deltas are positive (as far as α > −1), so that the four functions are well behaved at the origin.But, as we have remarked, the potential A µ has a (harmless) singularity at the origin when α and b 0 are nonvanishing.
When f (0) = 1, the monopole energy is given by where we have used sin 2 θ w ≃ 0.2312 and M H /M W ≃ 1.56.This strongly supports our prediction of the monopole mass based on the scaling argument.
Clearly the energy of the dyon must be of the order of (4π/e 2 )M W , but depend on f (0) (or α and β).The energy dependence on f (0) is shown in Fig. 3.
As we have emphasized, in the absence of the Z-boson (64) reduces to the Georgi-Glashow theory with In this case (69) reduces to the following Bogomol'nyi-Prasad-Sommerfield equation in the limit λ = 0 [10] ρ ± This has the analytic monopole solution whose energy is given by the Bogomol'nyi bound From this we can confidently say that the mass of the electroweak monopole could be around 4 to 7 TeV.
This confirms that we can regularize the Cho-Maison dyon with a simple modification of the coupling strengths of the existing interactions in the standard model which could be caused by the quantum correction or by the unification of all interactions.This provides a most economic way to make the energy of the dyon finite, because here we use the existing interaction without introducing a new interaction.

VII. EMBEDDING U (1)Y TO SU (2)Y
There is another way to regularize the Cho-Maison dyon.As we have noticed the origin of the infinite energy of the Cho-Maison solutions was the magnetic singularity of U (1) em .On the other hand the ansatz (5) also suggests that this singularity really originates from the magnetic part of the hypercharge U (1) field B µ .So one could try to to obtain a finite energy monopole solution by regularizing this hypercharge U (1) singularity.This could be done by introducing a hypercharged vector field to the theory [3].
A simplest way to do this is, of course, to enlarge the hypercharge U (1) and embed it to another SU (2).This type of the generalization could naturally arise in the leftright symmetric grand unification models, in particular in the SO(10) grand unification, although the embedding of the hypercharge U (1) to a compact SU (2) may be too simple to be realistic.
To construct the desired solutions we introduce a hypercharged vector field X µ and a Higgs field σ, and generalize the Lagrangian (52) adding the following Lagrangian where Dµ = ∂ µ + ig ′ B µ .To understand the meaning of it let us introduce a hypercharge SU (2) gauge field B µ and a scalar triplet Φ, and consider the SU (2) Y Georgi-Glashow model Now we can have the Abelian decomposition of this Lagrangian with Φ = σn, and have (identifying B µ and X µ as the Abelian and valence parts) This clearly shows that Lagrangian (77) describes nothing but the embedding of the hypercharge U (1) to an SU (2) Georgi-Glashow model.Now for a static spherically symmetric ansatz we choose (5) and let With the spherically symmetric ansatz the equations of motion are reduced to Furthermore, the energy of the above configuration is given by where σ 0 = m 2 /κ, M X = g ′ σ 0 , C 1 and C 2 are constants of the order one.The boundary conditions for a regular field configuration can be chosen as Notice that this guarantees the analyticity of the solution everywhere, including the origin.
With the boundary condition (83) one may try to find the desired solution.From the physical point of view one could assume M X ≫ M W , where M X is an intermediate scale which lies somewhere between the grand unification scale and the electroweak scale.Now, let A = B = 0 for simplicity.Then (81) decouples to describe two independent systems so that the monopole solution has two cores, the one with the size O(1/M W ) and the other with the size O(1/M X ).With M X = 10M W we obtain the solution shown in Fig. 4 in the limit κ = 0 and M H /M W = 1.56.
In this limit we find C 1 = 1.53 and C 2 = 1 so that the energy of the solution is given by E = 4π e 2 cos 2 θ w + 0.153 sin 2 θ w M X ≃ 110.17 M X .
Clearly the solution describes the Cho-Maison monopole whose singularity is regularized by a Prasad-Sommerfield monopole of the size O(1/M X ).
It must be emphasized that, even though the energy of the monopole is fixed by the intermediate scale, the size of the monopole is fixed by the electroweak scale.Furthermore from the outside the monopole looks exactly the same as the Cho-Maison monopole.Only the inner core is regularized by the hypercharged vector field.

VIII. CONCLUSIONS
It has generally been believed that the finite energy monopole could exist only at the grand unification scale [30].But our result tells we can have the electroweak monopole whose mass is much smaller.In this paper we have discussed two ways to estimate the mass of the electroweak monopole.We first use the scaling argument to predict the mass, and provide approximate solutions of the Cho-Maison dyon which have finite energy to back up this prediction.
Our result implies that the genuine electroweak monopole of mass around 4 to 7 TeV could exist, which is within the range of the present LHC.This strongly implies that there is an excellent chance that MoEDAL could actually detect such monopole.
The importance of the electroweak monopole is that it is the electroweak generalization of the Dirac monopole, and that it is the only realistic monopole which can be produced and detected.A remarkable aspect of this monopole is that mathematically it can be viewed as a hybrid between the Dirac monopole and the 'tHooft-Polyakov monopole.However, unlike the Dirac monopole, the magnetic charge of the electroweak monopole must satisfy the Schwinger quantization condition q m = 4πn/e.This is because the electroweak generalization requires us to embed the electromagnetic U (1) to the U (1) subgroup of SU (2), which has the period of 4π.So the magnetic charge of the electroweak monopole has the unit 4π/e.
It must be emphasized that we are not claiming the finite energy solutions to be the solutions of the Weinberg-Salam model.Our point here is that a simple modification of the model which could come from quantum correction or from the unification of all interactions can make their energy finite.Moreover, from the physical point of view there is no doubt that the finite energy solutions should be interpreted as the regularized Cho-Maison dyons whose mass (and size) is fixed by the electroweak scale.
Before we close it is worth to compare our dyon with other classical objects which exist in the standard model.As we have pointed out, long time ago Nambu has shown the existence of the electroweak string in the standard model carrying the Z-flux which has the fractionally charged monopole anti-monopole pair at the ends [12].Based on this one could try to construct a fractionally charged "electroweak monopole" carrying the charge qm = (4π/g) × sin θ w stretching the string to the infinite [13], or a vortex ring ("vorticon") closing the string [31].Moreover, the theory has the sphaleron which could be viewed as a twisted monopole anti-monopole pair of Nambu's string in the limit the string shrinks to a point, or similar objects [16,32].
To construct the fractionally charged monopoles, however, we have to pump in an infinite energy.In other words, the fractionally charged monopoles, just like the quarks in QCD, can only be paired to be confined.Moreover, along the string the Higgs field vanishes, so that asymptotically the Higgs field does not approach to its vacuum value.So they can not be identified as the electroweak monopoles.
Other objects like the sphalerons, although interesting, are known to be unstable [16,32].In comparison our monopole and dyon have no such defects, and can exist in nature and be detected.The only theoretical issue here is whether the hypercharge U (1) is non-trivial or not.If the U (1) is non-trivial, the Cho-Maison dyon must exist.In this sense the experimental detection of the Cho-Maison monopole could be the final test of the standard model.
Certainly the existence of the finite energy electroweak monopole should have important physical implications [33].In particular, it could have important implications in cosmology because it can be produced after inflation.The physical implications of the electroweak monopole will be discussed in a separate paper [34].
and M Z determine the exponential damping of the Higgs boson, W -boson, and Z-boson to their vacuum expectation values asymptotically.

FIG. 1 :
FIG. 1: The finite energy electroweak monopole solution.The solid line represents the finite energy monopole and dotted line represents the Cho-Maison monopole, where we have chosen sin 2 θw = 0.2312, MH /MW = 1.56.

FIG. 2 :
FIG. 2: The finite energy electroweak dyon solution.The solid line represents the finite energy dyon and dotted line represents the Cho-Maison dyon, where Z = A − B and we have chosen f (0) = 1 and A(∞) = MW /2.

FIG. 3 :
FIG.3:The energy dependence of the electroweak monopole on α, β, or f (0).The red and green curves represents the α and β dependence, and the blue curve represents the f (0) dependence.