# Magnetic Monopoles in Multivector Boson Theories

Authors: Koichiro Kobayashi; Nahomi Kan; Kiyoshi Shiraishi
Journal: Advances in High Energy Physics
 (2018)

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## Abstract

A classical solution for a magnetic monopole is found in a specific multivector boson theory. We consider the model whose[SU(2)]N+1 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 MN~(4πv/g)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.


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## Body

## 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 gm is related to the electric charge e by egm=nħc/2 with an integer n. 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 SU(2) 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 [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 multivector boson theory,[SU(2)]N+1 gauge symmetry is assumed. One of the SU(2) 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 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 Section2, 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 N+1 gauge fields A1μ,A2μ,…AN+1,μ. The field strength GIμν (I=1,2,…,N+1) is defined as(1)GIμν≡∂μAIν-∂νAIμ-igIAIμ,AIν,where gI is the I-th gauge coupling constant. The I-th field strength transforms as(2)GIμν⟶UIGIμνUI†1≤I≤N+1,according to the I-th gauge group transformation UI∈GI.In addition to the gauge fields, we introduceN scalar fields Σ1,Σ2,…ΣN, which would supply the Nambu-Goldstone fields as nonlinear-sigma model fields. The scalar field ΣI (I=1,2,…,N) transforms as in the bifundamental representation,(3)ΣI⟶UIΣIUI+1†1≤I≤N.(Here, we show the case of “linear moose”, and the different assignments of the transformation of ΣI yield the theory associated with various other types of moose diagrams [11–18].)Now, the Lagrangian density, which is invariant under the gauge transformation ofG1⊗G2⊗⋯⊗GN+1, is given by(4)L=-12∑I=1N+1trGIμνGIμν-∑I=1NtrDμΣI†DμΣI,where the covariant derivative of ΣI is(5)DμΣI≡∂μΣI-igIAIμΣI+igI+1ΣIAI+1,μ,and then its gauge transformation is(6)DμΣI⟶UIDμΣIUI+1†.In the usual dimensional deconstruction scheme, we consider thatG1=G2=⋯=GN+1=G and g1=g2=⋯=gN+1=g. We also assume that the absolute value of each nonlinear sigma model field ΣI has a common vacuum value, f. Then, the field ΣI is expressed as(7)ΣI=fexp⁡iπaTaf,where Ta is the generator in the adjoint representation of G and πa is the Nambu-Goldstone field, which is absorbed into the gauge fields. Taking the unitary gauge ΣI=f×(identitymatrix), we find that the kinetic terms of ΣI lead to the mass terms of the gauge fields as (provided that tr(TaTb)=(1/2)δab)(8)∑I=1NtrDμΣI†DμΣI=12g2f2∑I=1NAIμa-AI+1,μa2,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 G gauge theory with one-dimensional compactification on to S1/Z2 (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 G1=U(1) and G2=⋯=GN+1=SU(2). Then, the covariant derivative of Σ1 is(9)DμΣ1≡∂μΣ1-ig1A1μT3Σ1+igΣ1A2μ,where g1 is the U(1) gauge coupling constant, g is the common SU(2) gauge coupling constant, A1μ is the U(1) gauge field, and T3 is the third generator of SU(2). The nonzero vacuum expectation value of ΣI leads to symmetry breaking [SU(2)]N⊗U(1)→U(1) [13–18], and we get only one massless electromagnetic field and N 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[SU(2)]N⊗U(1) gauge group comes from the spontaneous symmetry breaking by an adjoint scalar [10]: [SU(2)]N+1→[SU(2)]N⊗U(1). The mechanism is now generally called the Higgs mechanism. The symmetry is broken into U(1) by the vacuum expectation value of the nonlinear sigma model field ΣI 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:(10)L=-12∑I=1N+1trGIμνGIμν-∑I=1NtrDμΣI†DμΣI-trDμϕ†Dμϕ-λ42trϕ†ϕ-v22,where GIμν(I=1,…,N+1) is the field strength of the SU(2)I gauge field AIμ(I=1,…,N+1) and ΣI(I=1,…,N) is the nonlinear sigma model fields in the bifundamental representation of SU(2)I⊗SU(2)I+1, 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 g.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(11)Dμϕ≡∂μϕ-igA1μ,ϕ.In the last term in the Lagrangian density (10), 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 gaugeΣ1=⋯=ΣN=f×(theidentitymatrix). Then, the Lagrangian density is represented as follows:(12)L=-14∑I=1N+1GIμνaGIaμν-12g2f2∑I=1NAIμa-AI+1,μa2-12DμϕaDμϕa-λ4ϕaϕa-v22,where(13)GIμνa≡∂μAIνa-∂νAIμa+gεabcAIμbAIνcandDμϕa≡∂μϕa+gεabcA1μbϕc.Here, we use the component representations AIμ=AIμaTa, GIμν=GIμνaTa, ϕ=ϕaTa, and Dμϕ=DμϕaTa, and εabc is the totally antisymmetric symbol (a=1,2,3).Next, we consider the symmetry breakdown by the Higgs mechanism with respect to the adjoint scalar fieldϕ. We express the third component of the scalar field as ϕ3=v+φ. Then, the Lagrangian density is denoted by(14)L=-14∑I=1N+1GIμν1GI1μν-12g2f2∑I=1NAIμ1-AI+1,μ12-12g2v2A1μ1A11μ-14∑I=1N+1GIμν2GI2μν-12g2f2∑I=1NAIμ2-AI+1,μ22-12g2v2A1μ2A12μ-14∑I=1N+1GIμν3GI3μν-12g2f2∑I=1NAIμ3-AI+1,μ32-12∂μφ∂μφ-λv2φ2+interactionterms,where the labels a are explicitly represented. 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 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,[SU(2)]N+1→[SU(2)]N⊗U(1)→U(1) or [SU(2)]N+1→SU(2)→U(1), 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 a=1 is(15)Lmassterma=1=-12g2f2∑I=1NAIμ1-AI+1,μ12-12g2v2A1μ1A11μ=-12g2f2×A11μA21μA31μ⋯AN-11μAN1μAN+1μ1+v2f2-1-12-1-12-1⋱⋱⋱-12-1-12-1-11A1μ1A2μ1A3μ1⋮AN-1,μ1ANμ1AN+1,μ1.Therefore, for a=1, the mass-squared matrix (mass1)2 of the vector bosons is(16)mass12=g2f21+v2f2-1-12-1-12-1⋱⋱⋱-12-1-12-1-11≡g2f2M12.We consider the eigenvalue equation(17)M12A1=ME12A1,where A1 is the eigenvector(18)A1≡A1μ1A2μ1A3μ1⋮AN-1,μ1ANμ1AN+1,μ1,and (ME1)2 is the eigenvalue.We show theME1-v/f graphs in Figure 1. The highest eigenvalue behaves differently from the other eigenvalues. When v/f→∞, the highest eigenvalue becomes ME1~v/f, but the other eigenvalues asymptotically approach constant values that are less than two.Figure 1
The eigenvaluesME1 are shown as functions of v/f in the cases of N=10,5,1,and0.The mass term of gauge fields fora=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(19)mass32=g2f21-1-12-1-12-1⋱⋱⋱-12-1-12-1-11≡g2f2M32.The eigenvaluesME3 can be analytically obtained [13] as(20)ME3n=2sin⁡nπ2N+1n=0,…,N.Obviously, 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:(21)ϕa=δiaxigr2Hr,(22)AI0a=0,(23)AIia=εaijxjgr21-KIr,and the boundary conditions on the function of the radial coordinate r are(24)limr→0Hrr=0,limr→∞Hrr=gv,limr→0KIr=1,limr→∞KIr=0.The common form of AIia 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-L. Substituting the ansatz, we obtain the expression for total energy(25)EN=4πvg∫0∞dξ∑I=1N+1KI′2+12ξ21-KI22+f2v2∑I=1NKI-KI+12+12H′-Hξ2+1ξ2K12H2+λξ24g2H2ξ2-12where we set ξ≡gvr and the prime (′) denotes the derivative with respect to ξ.From this expression, we can obtain the following equations of motion by the variational principle:(26)ξ2K1′′=K1K12-1+H2K1+f2v2ξ2K1-K2,(27)ξ2KI′′=KIKI2-1+f2v2ξ22KI-KI-1-KI+12≤I≤N,(28)ξ2KN+1′′=KN+1KN+12-1+f2v2ξ2KN+1-KN,(29)ξ2H′′=2HK1+λg2HH2-ξ2.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 N 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:(30)KIξ=1+aIξexp⁡-aIξ1≤I≤N+1,(31)Hξξ=1-exp⁡-αξ,where both aI(1≤I≤N+1) and α are variational parameters. The functions KI(ξ) and H(ξ) 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.We substitute the expressions (30) and (31) into the energy EN and calculate the minimum value of the energy EN by varying the parameters aI and α.Each term is separately integrated as follows:(32)∫0∞dξKI′2+12ξ21-KI22=4164aI,(33)∫0∞dξKI-KI+12=54aI+54aI+1-4aI2+3aIaI+1+aI+12aI+aI+13,(34)∫0∞dξH′-Hξ2=14α,(35)∫0∞dξ1ξ2K12H2=α256a14+132a13α+111a12α2+39a1α3+5α44a1a1+α32a1+α3,(36)∫0∞dξξ2H2ξ2-12=635864α3.Therefore, the energy expressed by the variational parameters becomes(37)EN=4πvg4164∑I=1N+1aI+f2v2∑I=1N54aI+54aI+1-4aI2+3aIaI+1+aI+12aI+aI+13+18α+α256a14+132a13α+111a12α2+39a1α3+5α44a1a1+α32a1+α3+635λ3456g2α3.We evaluate the minimum value of this energy by numerical calculation withMathematica [26]. Thus, we get the approximate solution of KI(ξ) and H(ξ)/ξ, and the case of N=10 and λ=0 is shown in Figure 2.Figure 2
K I and H/ξ are shown for N=10, λ=0. In each graph, KI<KI+1(1≤I≤N=10) at any ξ. The three graphs correspond to the cases of f/v=0.5, f/v=1, and f/v=2, respectively.The region of nonvanishingKI can be interpreted as the region where the I-th massive vector bosons (a=1,2) condensate. For larger values of f/v, the ranges of finite KI become narrower and degenerate, while the distance where H/ξ~1 becomes larger.We obtain the energy of the monopole in the limiting caseλ/g2=0 for the cases where N=0,1,5,and10 and f/v=0.5,1,and2.(38)E0=4πvg×1.05⋯,BPSmonopole(39)E1=4πvg×1.41⋯,fv=0.5(40)E1=4πvg×1.47⋯,fv=1(41)E1=4πvg×1.48⋯,fv=2(42)E5=4πvg×2.07⋯,fv=0.5(43)E5=4πvg×2.37⋯,fv=1(44)E5=4πvg×2.51⋯,fv=2(45)E10=4πvg×2.47⋯,fv=0.5(46)E10=4πvg×2.99⋯,fv=1(47)E10=4πvg×3.32⋯,fv=2From these results, we roughly estimate that the energy of the monopole (λ=0) is(48)EN~4πvgN+1λ=0,since the difference that appears due to different f/v is smaller than that due to different N. We find that our approximate values of the static energies for λ=0 are well fitted to EN≈(4πv/g)×1.94×W(0.62N+0.96), where W(x) is the Lambert W-function, which is slightly smaller than (4πv/g)1+N for large N. This is in contrast to the rather large dependence of the profiles of solutions for KI and H/ξ on f/v (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 E0 for λ=0, as(49)Eλ=0=4πvg.Comparing these values, we find that the energy of the BPS monopole in the multivector boson theory is obtained by replacing g→g/N+1 in that of the usual BPS monopole.Note that we only show the case ofλ→0. However, we confirmed that the energy of the monopole changes at most factor two for a finite value of λ/g2 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 Section3, if we choose ϕ3=v, the massless gauge field satisfies(50)A1μ3=A1μ3=⋯=AN+1,μ3≡1N+1Aμ3.The 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 A1μ, similar to that in the triplet Higgs field, the electric charge of the matter field e becomes(51)e=gN+1,and the field strength satisfies G1μν3=G1μν3=⋯=GN+1,μν3≡(1/N+1)Gμν3.Now, we consider the magnetic field far from the monopole. The projection of the vacuum expectation values of the field strength [1, 3, 6] is(52)limr→∞Fij=limr→∞ϕ^aGija=limr→∞1N+1∑I=1N+1ϕ^aGIija=N+1g-εaijxar3,where ϕ^a=ϕa/v. Then, the magnetic field Bi is asymptotically(53)Bi=-N+1gxir3=-xier3.Comparing this magnetic field representation with the magnetic field created by a point magnetic charge gm(54)Bi=gm4πxir3,the magnetic charge gm of our monopole is(55)gm=-4πgN+1=-4πe.This 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 as(56)EN~4πvgN+1=4πve,which is the same as the mass of the ‘t Hooft–Polyakov monopole (or, the case of N=0).


## 7. Summary and Discussion

In this paper, we studied the static, spherically symmetric monopole solutions in the multivector boson theory withN+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 according to the theory described in Section 4. We used a simple variational method to obtain approximate solutions in Section 5. The solution of KI 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 N and f/v, the mass of the monopole is approximately EN~(4πv/g)N+1=4πv/e, where e 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 N.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 [35] 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 scalesf and v 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.



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