Advances in High Energy Physics

Volume 2018, Article ID 2396275, 9 pages

https://doi.org/10.1155/2018/2396275

## Magnetic Monopoles in Multivector Boson Theories

^{1}National Institute of Technology, Oshima College, Suooshima-cho, Yamaguchi 742-2193, Japan^{2}National Institute of Technology, Gifu College, Motosu-shi, Gifu 501-0495, Japan^{3}Graduate School of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi-shi, Yamaguchi 753-8512, Japan

Correspondence should be addressed to Kiyoshi Shiraishi; pj.ca.u-ihcugamay@hsiarihs

Received 1 February 2018; Revised 23 May 2018; Accepted 30 May 2018; Published 21 June 2018

Academic Editor: Luca Stanco

Copyright © 2018 Koichiro Kobayashi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The publication of this article was funded by SCOAP^{3}.

#### Abstract

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.

#### 1. Introduction

The existence of magnetic monopoles (for reviews, see [1–6]) has been discussed for many years, although monopoles have not yet been observed experimentally.

In 1931, Dirac [7] 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 [8] and Polyakov [9] 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 [10], 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 [10]: . 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.