Magnetic monopoles in multi-vector boson theories

A classical solution for a magnetic monopole is found in a specific multi-vector boson theory. We consider the model whose $[SU(2)]^{N+1}$ gauge group is broken by sigma-model fields (\`a la dimensional deconstruction) and further spontaneously broken by an adjoint scalar (\`a la triplet Higgs mechanism). In this multi-vector boson theory, we find the solution for the monopole whose mass is $M_N\sim\frac{4\pi v}{g}\sqrt{N+1}$, where $g$ is the common gauge coupling constant and $v$ is the vacuum expectation value of the triplet Higgs field, by using a variational method with the simplest set of test functions.


I. INTRODUCTION
In many years, the existence of magnetic monopoles (for reviews, see [1][2][3][4][5][6]) has been discussed, though monopoles are not yet observed experimentally. Monopoles as solitonic objects in modern gauge field theories were first considered independently by 't Hooft [7] and Polyakov [8]. Their models are based on the Georgi-Glashow model [9], which uses spontaneous symmetry breaking of SU(2) gauge symmetry by a scalar field in the adjoint representation.
We consider a novel monopole in a multi-vector boson theory which is based on the dimensional deconstruction [10,11] and the Higgsless theories [12][13][14][15][16][17] 1 . The Higgsless theory is one of the theory including symmetry breaking of the electroweak symmetry. In the Higgsless theory, for example, the [SU (2)] N ⊗ U(1) gauge theory is considered. Such a theory yields N sets of massive vector fields besides one massless photon field.
In our model of the multi-vector boson theory, [SU (2)] N +1 gauge symmetry is assumed.
One of the SU(2) gauge group is broken by an adjoint scalar as in the Georgi-Glashow model. There remains one massless vector field due to the triplet Higgs mechanism. We can thus construct the 't Hooft-Polyakov-type monopole configuration in the model. We estimate the monopole mass M ∼ 4πv g √ N + 1, where v is the vacuum expectation value of the scalar field, and g is the coupling constant of the gauge field.
In Sec. II, we review the dimensional deconstruction and the Higgsless theory briefly.
Our model of the multi-vector boson theory is shown in Sec. III, which is a generalization of the gauge-field part the Higgsless theory. The mass spectrum in the multi-vector boson theories is investigated in Sec. IV. In Sec. V, we demonstrate construction of monopole configurations in the multi-vector boson theory. In order to treat many variables, we propose an approximation scheme by a variational method in this section. In Sec. VI, we discuss the magnetic charge of the monopole in the multi-vector boson theory. The final section (Sec. VII) is devoted to summary and discussion. A review of the theory of 't Hooft-Polyakov magnetic monopoles is given in Appendix, where the validity of our approximation method is also shown.

II. DECONSTRUCTION AND HIGGSLESS THEORY
We review a basic idea of dimensional deconstruction [10,11] and the Higgsless theories [12][13][14][15][16][17] in this section. We consider N + 1 gauge fields A 1µ , A 2µ , . . . A N +1,µ . The field strength G Iµν (I = 1, 2, . . . , N + 1) is defined as where g I is the I-th gauge coupling constant. The I-th field strength transforms as according to the I-th gauge group transformation U I ∈ G I .
In addition to the gauge fields, we introduce N scalar fields Σ 1 , Σ 2 , . . . Σ N , which would supply the Nambu-Goldstone fields as non-linear-sigma-model fields. The scalar field Σ I (I = 1, 2, . . . , N) transforms as in the bi-fundamental representation, Here we show the case of 'linear moose', and different assignments of the transformation of Σ I yield the theory associated with various other types of moose diagrams [10][11][12][13][14][15][16][17].) Now, the Lagrangian density which is invariant under the gauge transformation of G 1 ⊗ G 2 ⊗ · · · ⊗ G N +1 is given by where the covariant derivative of Σ I is and then its gauge transformation is In the usual dimensional deconstruction scheme, we consider that G 1 = G 2 = · · · = G N +1 = G and g 1 = g 2 = · · · = g N +1 = g. We also assume that the absolute value of each non-linear sigma model fields |Σ I | has a common vacuum value, f . Then the field Σ I is expressed as where T a is the generator in the adjoint representation of G and π a is the Nambu-Goldstone field, which is absorbed into the gauge fields. Taking the unitary gauge Σ I = f × (identity matrix), we find the kinetic terms of Σ I lead to the mass terms of the gauge fields as (provided that Tr ( and these produce the mass spectrum of vector bosons. It is known that a certain continuum limit of this model can be taken and corresponds to the G gauge theory with one-dimensional compactification on to S 1 /Z 2 (or an 'interval').
In the Higgsless theories, for example, the gauge group [SU (2)] N ⊗ U(1) is adopted for explaining the electroweak sector in the particle theory. Namely, we set G 1 = U(1) and where g 1 is the U(1) gauge coupling constant, A 1µ is the U(1) gauge field and T 3 is the third generator of SU (2). The non-zero vacuum expectation value of Σ I leads to symmetry breaking [SU (2)] N ⊗ U(1) → U(1) [12][13][14][15][16][17] and we get only one massless electromagnetic field and the N sets of massive weak boson fields.
The original motivation of the Higgsless theory has been abandoned after the discovery of the Higgs particles. Nevertheless, we would like to extend the standard model, since there might be a lack of unknown extra particles, which explain the dark matter problem [18,19].
Therefore, we suppose that it is worth considering the theoretical models whose massive particle contents are rich and governed by certain symmetries.

INCORPORATING THE HIGGS MECHANISM
We here consider the model whose [SU (2)] N ⊗ U(1) gauge group comes from the spontaneous symmetry breaking by an adjoint scalar [9]: [SU (2)] N +1 → [SU (2)] N ⊗ U(1). The mechanism is now generally called the Higgs mechanism. The symmetry is broken to U(1) by the vacuum expectation value of the non-linear sigma model field Σ I introduced in the previous section. As a consequence we have a monopole configuration; the construction of the monopole solution will be described in the next section. In this section, we define our model and show the mass spectrum of this model in the subsequent section.
We consider the following Lagrangian density: Here, φ is a scalar field in the adjoint representation of SU(2) 1 and the covariant derivative of the scalar field φ is given by In the last term in the Lagrangian density (3.1), λ is a positive constant and the constant v is the scalar field vacuum expectation value.
First we consider the symmetry breaking by the sigma fields. We choose the unitary gauge Σ 1 = · · · = Σ N = f × (the identity matrix). Then, the Lagrangian density is represented as follows: and we used the component representations, namely, and ε abc is the totally antisymmetric symbol (a = 1, 2, 3).
Next, we consider the symmetry break down by the Higgs mechanism with respect to the adjoint scalar field φ. Now, we express the third component of the scalar field as φ 3 = v + ϕ.
Then, the Lagrangian density is denoted by Further, if the labels a are explicitly represented, the Lagrangian density becomes We have only one massless U(1) symmetric gauge field in the third component. Therefore we have obtained the symmetry breaking SU(2) → U(1) by the Higgs mechanism. This type of symmetry breaking gives rise to the 't Hooft-Polyakov monopole configuration.
It is notable that we do not discuss which sequences of symmetry breaking [SU (2) the order may have an effect on the producing process of the monopoles in the early universe.

IV. MASS SPECTRUM OF VECTOR BOSONS
In the Lagrangian density (3.6), the mass term of gauge fields for a = 1 is Therefore, for a = 1, the mass-squared matrix (mass 1 ) 2 of the vector bosons is We consider the eigenvalue equation where A 1 is the eigenvector Fig. 1. The highest eigenvalue behaves differently from other eigenvalues. When v/f → ∞, the highest eigenvalue becomes M 1 E ∼ v/f , but other eigenvalues are asymptotically approach constant values which are less than two.
The mass term of gauge fields for a = 2 is the same as for a = 1, but the mass term is different for a = 3. The mass-squared matrix of gauge fields for a = 3 is  Similarly to the 't Hooft-Polyakov monopole, the static and spherically symmetric monopole solution in the multi-vector boson theory is considered to be specified by the following ansatz The common form of A a Ii is due to the requirement of finite energy of the monopole, i.e., the contribution of the term (2.8) to the energy density vanishes at spatial infinity.
For the static case, the energy density is given by −L. Substituting the ansatz, we obtain the expression for total energy where we set ξ ≡ gvr and the prime ( ′ ) means the derivative with respect to ξ.
From this expression, we can obtain the following equations of motion by the variational principle: Because it is hard 2 to find a set of solutions of these coupled equations for large N, and because we presently treat a simple toy model, we adopt a simple variational method to obtain approximate solutions in this paper. 3 For the approximation, we assume that solutions takes the following forms: where both a I (1 ≤ I ≤ N + 1) and α are variational parameters. These K I (ξ) and H(ξ) functions with minimal number of parameters apparently satisfy the boundary conditions and are similar to these of the solutions in the 't Hooft-Polyakov monopole, as shown in Fig. 3 in Appendix A. This is the reason why we assume the simple form of solutions as above.
We substitute the expressions (5.10) and (5.11) into the energy E N and we calculate the minimum value of the energy E N by varying parameters a I and α.

The integration of each term separately is given by
Therefore the energy expressed by the variational parameters becomes We evaluate the minimum value of this energy by numerical calculation with Mathematica  We obtain the energy of the monopole in limiting case λ/g 2 = 0 in the cases of N = 0, 1, 5, and 10 and f /v = 0.5, 1, and 2.
From these results, we roughly estimate that the energy of the monopole (λ = 0) is since the difference which appears in different f /v is smaller than that due to the different N. 4 This fact is a contrast to the rather large dependence of the profiles of solutions for K I and H/ξ on f /v (Fig. 2).
On the other hand, we know the exact value of the energy of BPS limit [24,25] of 't Hooft-Polyakov monopole, which corresponds E 0 for λ = 0, as Comparing these values, we find that the energy of the BPS monopole in the multi-vector boson theory is obtained by replacing g → g/ √ N + 1 in that of the usual BPS monopole.
Note that we only show the case of λ → 0. But we confirmed that the finite λ/g 2 makes at most factor few of change in the energy of the monopole, as in the case of the 't Hooft-Polyakov monopole (Appendix A).

VI. MAGNETIC CHARGE OF THE MONOPOLE
In this section, we specify the magnetic charge of the monopole in the multi-vector boson theory obtained in the previous section. We should discuss on the definition of the electric charge, first of all. As in section III, if we choose the choice of φ 3 = v, the massless gauge field satisfies (6.1) 4 We find that our approximate values of the static energies for λ = 0 are well fitted to E N ≈ 4πv g × 1.94 × W (0.62N + 0.96), where W (x) is the Lambert W -function, which is slightly smaller than 4πv The normalization factor is determined by the canonical form of the Lagrangian density of this zero-mode field. Therefore, provided that the charged matter field is virtually coupled only to A 1µ , as similar to the triplet Higgs field, the electric charge of the matter field e becomes e = g √ N + 1 , (6.2) and the field strength similarly satisfies µν . Now, we turn to the magnetic field far from the monopole. The projection of the vacuum expectation values of the field strength [1,3,6] is whereφ a = φ a /v. Then, the magnetic field B i is asymptotically Comparing this magnetic field representation with the magnetic field created by a point magnetic charge g m the magnetic charge g m of our monopole is This relation is the same as that for the 't Hooft-Polyakov monopole.
The static energy of the monopole in the multi-vector boson theory in the previous section can be rewritten as and is the same as the mass of the 't Hooft-Polyakov monopole (or, the case of N = 0).

VII. SUMMARY AND DISCUSSION
In the present paper, we studied the static, spherically symmetric monopole solutions in the multi-vector boson theory with N + 1 sets of vector bosons with the gauge coupling g.
The theory includes two mass scales f and v. We found that 3N + 2 massive vector bosons and a single massless vector boson (of the electromagnetic field) appear in the theory in Sec. IV. We used a simple variational method to obtain approximate solutions in Sec. V.
The solution of K I shows that the regions of existence of massive vector fields have a multilayer structure, where massive bosons 'stratify'. Although the profile of condensation of the massive degrees of freedom is sensitive on both N and f /v, the mass of the monopole where e is the electric charge defined in the theory. It is necessary to demonstrate more accurate investigation for concluding the precise dependence of mass of the monopole on N.
The model in the present paper is the simplest one, so we would like to investigate more general models, which have different coupling constants for different gauge fields or have complicated mass matrices as in the clockwork theory [26][27][28][29][30][31].
We wish to study a scenario that the monopoles in the multi-vector boson theory represent the dark matter in the universe. Since the present model of multi-vector boson theory has two symmetry breaking scales f and v and there can be various mass spectrum of massive vector bosons as seen in Sec. IV, we need detailed study on the process of symmetry breaking and (time-dependent) monopole productions.

ACKNOWLEDGMENTS
We thank Hideto Manjo for useful comments on numerical estimations.

Appendix A: the 't Hooft-Polyakov monopole
If the electromagnetic U(1) symmetry is embedded in semi-simple group and described by an unbroken U(1) gauge theory after spontaneous symmetry breaking, the monopole will come out as a topologically non-trivial finite-energy solution. In this Appendix, we will describe such a solution discovered by G. 't Hooft [7] and A. M. Polyakov [8]. The simplest example is based on the SO(3) model due to Georgi and Glashow [9].
The Georgi-Glashow model is the SU(2) gauge theory with a triplet of the Higgs scalar φ = φ a T a , where a = 1, 2, 3 and T a is the generator of the adjoint representation of SU(2).
The Lagrangian density of the Georgi-Glashow model is where the field strength of the gauge field is G a µν ≡ ∂ µ A a ν − ∂ ν A a µ + gε abc A b µ A c ν and the covariant derivative of the scalar field is D µ φ a ≡ ∂ µ φ a + gε abc A b µ φ c . φ a is an isovector and A a µ is a gauge field. We define that the Greek indices take values µ, ν = 0, 1, 2, 3. The symbol ε abc denotes the totally antisymmetric symbol, g is the coupling constant of the gauge field and λ is a dimensionless positive constant.
Now we consider the energy in a static case (A a 0 = 0). Because the Lagrangian density does not include the time derivative terms in the static case, the energy density becomes where i = 1, 2, 3.
Substituting the static spherically symmetric ansatz into the Lagrangian density (A1), we obtain the energy of a spherical monopole as where ξ = gvr.
The variational principle yields equations of motion as second-order non-linear differential equations, As the simple case, we first consider the Bogomol'nyi-Prasad-Sommerfield (BPS) limit [24,25]. In the BPS limit, we take the limit λ → 0, i.e., the equations of motion become It is known that this set of equations is equivalent to the following set of first-order equations: A solution of these coupled equations is We consider two particular cases of λ → 0 and the λ → ∞. In each case, the static energy is known as [20] E(0) = 4πv g , (A9) Now, we evaluate the energy by a variational method. To this end, we consider two simple forms of test functions as K(ξ) = (1 + a I ξ) exp(−a I ξ) , where a and α are the variational parameters. The choice of the minimal number of parameters is taken here because the method can also be used in the multi-vector boson theory, where the number of parameters is inevitably large.
We could not obtain the energy in the case that the value of λ/g 2 is larger than 6 × 10 3 in our numerical calculation. From these results, our scheme is a good approximation.
We generalize the definition of the electromagnetic field F µν . We consider the projection of vacuum expectation values of the field strength [1,3,6], where we useφ a ≡ φ a / √ φ a φ a . Thus the magnetic field far from the monopole is Comparing magnetic fields by a point magnetic source, the magnetic charge g m is so the magnetic charge is inversely proportional to the gauge coupling constant. Therefore the smaller the value of g, the larger the value of g m is.