Magnetic Monopoles in Multivector Boson Theories
A classical solution for a magnetic monopole is found in a specific multivector boson theory. We consider the model whose gauge group is broken by sigma model fields (à la dimensional deconstruction) and further spontaneously broken by an adjoint scalar (à la triplet Higgs mechanism). In this multivector boson theory, we find the solution for the monopole whose mass is , where is the common gauge coupling constant and is the vacuum expectation value of the triplet Higgs field, by using a variational method with the simplest set of test functions.
In 1931, Dirac  reconsidered the duality in electromagnetism and showed that the quantum mechanics of an electrically charged particle can be consistently formulated in the presence of a point magnetic charge, provided that the magnetic charge is related to the electric charge by with an integer . In 1974, ‘t Hooft  and Polyakov  found that a nonsingular configuration arises from spontaneous symmetry breaking in a certain class of non-Abelian gauge theory. Their models are based on the Georgi-Glashow model , which uses spontaneous symmetry breaking of gauge symmetry by a scalar field in the adjoint representation. The ‘t Hooft–Polyakov monopoles are classical solutions, which are stable for topological reasons. Recently, the mathematical study of monopoles has focused on not only topology, but also integrable systems, supersymmetry, nonperturbative analyses, and so on.
In the present paper, we consider a novel monopole in a multivector boson theory, which is based on dimensional deconstruction [11, 12] and the Higgsless theories [13–18]. The Higgsless theory is one of the theories that include symmetry breaking of the electroweak symmetry. In the Higgsless theory, for example, the gauge theory is considered. Such a theory yields sets of massive vector fields besides one massless photon field.
In our model of the multivector boson theory, gauge symmetry is assumed. One of the gauge groups 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 , where is the vacuum expectation value of the scalar field, and is the coupling constant of the gauge field.
In Section 2, we briefly review dimensional deconstruction and the Higgsless theory. Our model of the multivector boson theory is shown in Section 3, which is a generalization of the gauge-field part of the Higgsless theory. The mass spectrum in the multivector boson theories is investigated in Section 4. In Section 5, we demonstrate the construction of monopole configurations in the multivector boson theory. In order to treat many variables, we propose an approximation scheme by a variational method in this section. In Section 6, we discuss the magnetic charge of the monopole in the multivector boson theory. The final section (Section 7) is devoted to summary and discussion.
2. Deconstruction and Higgsless Theory
We review the basic idea of dimensional deconstruction [11, 12] and the Higgsless theories [13–18] in this section. We consider gauge fields . The field strength () is defined aswhere is the -th gauge coupling constant. The -th field strength transforms asaccording to the -th gauge group transformation .
In addition to the gauge fields, we introduce scalar fields , which would supply the Nambu-Goldstone fields as nonlinear-sigma model fields. The scalar field () transforms as in the bifundamental representation,(Here, we show the case of “linear moose”, and the different assignments of the transformation of yield the theory associated with various other types of moose diagrams [11–18].)
Now, the Lagrangian density, which is invariant under the gauge transformation of , is given bywhere the covariant derivative of isand then its gauge transformation is
In the usual dimensional deconstruction scheme, we consider that and . We also assume that the absolute value of each nonlinear sigma model field has a common vacuum value, . Then, the field is expressed aswhere is the generator in the adjoint representation of and is the Nambu-Goldstone field, which is absorbed into the gauge fields. Taking the unitary gauge , we find that the kinetic terms of lead to the mass terms of the gauge fields as (provided that )and these produce the mass spectrum of vector bosons. It is known that a certain continuum limit of this model can be taken, which corresponds to the gauge theory with one-dimensional compactification on to (or an “interval”).
In the Higgsless theories, for example, the gauge group is adopted for explaining the electroweak sector in the particle theory. Namely, we set and . Then, the covariant derivative of iswhere is the gauge coupling constant, is the common gauge coupling constant, is the gauge field, and is the third generator of . The nonzero vacuum expectation value of leads to symmetry breaking [13–18], and we get only one massless electromagnetic field and sets of massive weak boson fields.
The original motivation for 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 [19, 20]. As a model of dark matter, the multivector boson theory describes a hidden sector of dark photons [21, 22] with mutual mixings. Therefore, we suppose that it is worth considering the theoretical models whose massive particle contents are rich and governed by certain symmetries.
3. Multivector Boson Theory from the Higgsless Theory Incorporating the Higgs Mechanism
Here, we consider the model whose gauge group comes from the spontaneous symmetry breaking by an adjoint scalar : . The mechanism is now generally called the Higgs mechanism. The symmetry is broken into by the vacuum expectation value of the nonlinear sigma model field 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 in the subsequent section, we show the mass spectrum of this model.
We consider the following Lagrangian density:where is the field strength of the gauge field and is the nonlinear sigma model fields in the bifundamental representation of , which connect the gauge fields at neighboring sites, as in the dimensionally deconstructed model reviewed in the previous section. For simplicity, all the coupling constants of the gauge fields are assumed to be the same .
Here, is a scalar field in the adjoint representation of , and the covariant derivative of the scalar field is given byIn the last term in the Lagrangian density (10), is a positive constant and the constant is the scalar field vacuum expectation value.
First, we consider the symmetry breaking by the sigma fields. We choose the unitary gauge . Then, the Lagrangian density is represented as follows:whereHere, we use the component representations , , , and , and is the totally antisymmetric symbol ().
Next, we consider the symmetry breakdown by the Higgs mechanism with respect to the adjoint scalar field . We express the third component of the scalar field as . Then, the Lagrangian density is denoted bywhere the labels are explicitly represented. We have only one massless symmetric gauge field in the third component. Therefore, we have obtained the symmetry breaking by using the Higgs mechanism. This type of symmetry breaking gives rise to the ‘t Hooft–Polyakov monopole configuration.
It should be noted that we do not discuss which sequences of symmetry breaking, that is, or , occurred in the universe, although the order may have an effect on the process of creation of monopoles in the early universe.
4. Mass Spectrum of Vector Bosons
In the Lagrangian density (14), the mass term of gauge fields for isTherefore, for , the mass-squared matrix of the vector bosons is
We consider the eigenvalue equationwhere is the eigenvectorand is the eigenvalue.
We show the - graphs in Figure 1. The highest eigenvalue behaves differently from the other eigenvalues. When , the highest eigenvalue becomes , but the other eigenvalues asymptotically approach constant values that are less than two.
The mass term of gauge fields for is the same as for , but the mass term is different for . The mass-squared matrix of gauge fields for is
The eigenvalues can be analytically obtained  asObviously, there is a zero mode, and we have only one massless vector field in the theory after symmetry breakdown.
5. Energy and Equations of Motion of the Monopole in the Multivector Boson Theory
In the multivector boson theory defined by the Lagrangian density (14), the ‘t Hooft–Polyakov-type monopole is expected.
Similar to the ‘t Hooft–Polyakov monopole, the static and spherically symmetric monopole solution in the multivector boson theory is considered to be specified by the following ansatz:and the boundary conditions on the function of the radial coordinate areThe common form of is due to the requirement of finite energy of the monopole, i.e., the contribution of the term (8) to the energy density vanishes at spatial infinity.
For the static case, the energy density is given by . Substituting the ansatz, we obtain the expression for total energywhere we set and the prime ) denotes the derivative with respect to .
From this expression, we can obtain the following equations of motion by the variational principle:
Analytical and semianalytical studies of the single ‘t Hooft–Polyakov monopole are found in [23–25]. Because it is hard to find a set of solutions for these coupled equations for large and because we are presently considering a simple toy model, we adopt a simple variational method to obtain approximate solutions in this paper. We have confirmed that this approach obtains a good solution for the ‘t Hooft–Polyakov monopole in the BPS limit.
For the approximation, we assume that the solutions take the following forms:where both and are variational parameters. The functions and with minimal number of parameters apparently satisfy the boundary conditions and are similar to those of the solutions in the ‘t Hooft–Polyakov monopole. This is the reason why we assume the simple form of solutions as shown above.
Each term is separately integrated as follows:Therefore, the energy expressed by the variational parameters becomes
The region of nonvanishing can be interpreted as the region where the -th massive vector bosons () condensate. For larger values of , the ranges of finite become narrower and degenerate, while the distance where becomes larger.
We obtain the energy of the monopole in the limiting case for the cases where and .From these results, we roughly estimate that the energy of the monopole () issince the difference that appears due to different is smaller than that due to different . We find that our approximate values of the static energies for are well fitted to , where is the Lambert -function, which is slightly smaller than for large . This is in contrast to the rather large dependence of the profiles of solutions for and on (Figure 2).
On the other hand, we know the exact value of the energy of the BPS limit [27, 28] for the ‘t Hooft–Polyakov monopole, which corresponds to for , asComparing these values, we find that the energy of the BPS monopole in the multivector boson theory is obtained by replacing in that of the usual BPS monopole.
Note that we only show the case of . However, we confirmed that the energy of the monopole changes at most factor two for a finite value of in general.
6. Magnetic Charge of the Monopole
In this section, we specify the magnetic charge of the monopole in the multivector boson theory obtained in the previous section. First of all, we should discuss the definition of electric charge. As in Section 3, if we choose , the massless gauge field satisfiesThe normalization factor is determined by the canonical form of the Lagrangian density of this zero-mode field. Therefore, if the charged matter field is virtually coupled only to , similar to that in the triplet Higgs field, the electric charge of the matter field becomesand the field strength satisfies .
Now, we consider the magnetic field far from the monopole. The projection of the vacuum expectation values of the field strength [1, 3, 6] iswhere . Then, the magnetic field is asymptoticallyComparing this magnetic field representation with the magnetic field created by a point magnetic charge the magnetic charge of our monopole isThis relation is the same as that for the ‘t Hooft–Polyakov monopole.
The static energy of the monopole in the multivector boson theory that was described in the previous section can be rewritten aswhich is the same as the mass of the ‘t Hooft–Polyakov monopole (or, the case of ).
7. Summary and Discussion
In this paper, we studied the static, spherically symmetric monopole solutions in the multivector boson theory with sets of vector bosons with the gauge coupling . The theory includes two mass scales and . We found that massive vector bosons and a single massless vector boson (of the electromagnetic field) appear according to the theory described in Section 4. We used a simple variational method to obtain approximate solutions in Section 5. The solution of 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 with respect to both and , the mass of the monopole is approximately , where is the electric charge defined in the theory. It is necessary to conduct a more accurate investigation for obtaining the precise dependence of mass of the monopole on .
The model used in this study is the simplest one; therefore, 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 [29–34].
Another possible connection to a phenomenological model can be considered in a model with symmetry breakdown by a Higgs doublet, as in the standard model. In 1997, Cho and Maison  found an electroweak monopole solution in the standard model. The Cho–Maison monopole and its generalization have been studied further [36, 37], and an experimental search for them is going on [38, 39]. We wish to investigate the multivector boson theory with a doublet Higgs field and compare the properties of its monopoles with those of the Cho–Maison monopoles.
We also wish to study a scenario in which the monopoles in the multivector boson theory represent the dark matter in the universe. Since the present model of multivector boson theory has two symmetry breaking scales and and there can be various mass spectra of massive vector bosons as seen in Section 4, we need to perform a detailed study on the process of symmetry breaking and (time-dependent) monopole production.
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
The authors thank Hideto Manjo for useful comments on numerical estimations.
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