Gauge-Higgs Unification Models in Six Dimensions with 𝑆²/𝑍₂ Extra Space and GUT Gauge Symmetry

Abstract

We review gauge-Higgs unification models based on gauge theories defined on six-dimensional spacetime with 𝑆²/𝑍₂ topology in the extra spatial dimensions. Nontrivial boundary conditions are imposed on the extra 𝑆²/𝑍₂ space. This review considers two scenarios for constructing a four-dimensional theory from the six-dimensional model. One scheme utilizes the SO(12) gauge symmetry with a special symmetry condition imposed on the gauge field, whereas the other employs the E₆ gauge symmetry without requiring the additional symmetry condition. Both models lead to a standard model-like gauge theory with the SU(3)×SU(2)_𝐿×U(1)_𝑌(×U(1)²) symmetry and SM fermions in four dimensions. The Higgs sector of the model is also analyzed. The electroweak symmetry breaking can be realized, and the weak gauge boson and Higgs boson masses are obtained.

1. Introduction

The Higgs sector of the standard model (SM) plays an essential role in the spontaneous symmetry breaking (SSB) from the SU(3)_𝐶×SU(2)_𝐿×U(1)_𝑌 gauge group down to SU(3)_𝐶×U(1)_𝐸𝑀, thereby giving masses to the SM elementary particles. However, the SM does not address the most fundamental nature of the Higgs sector, such as the mass and self-coupling constant of the Higgs boson. Therefore, the Higgs sector is not only the last territory in the SM to be discovered, but will also provide key clues to new physics at higher energy scales.

Gauge-Higgs unification is one of many attractive approaches to physics beyond the SM in this regard [1–3] (for recent approaches, see [4–20]). In this approach, the Higgs particles originate from the extradimensional components of the gauge field defined on spacetime with the number of dimensions greater than four (for cases where). In other words, the Higgs sector is embraced into the gauge interactions in the higher-dimensional model, and many fundamental properties of Higgs boson are dictated by the gauge interactions.

In our recent studies, we have shown interesting properties of one type of gauge-Higgs unification models based on grand unified gauge theories defined on six-dimensional (6D) spacetime, with the extradimensional space having the topological structure of two-sphere orbifold 𝑆²/𝑍₂ [21, 22].

In the usual coset space dimensional reduction (CSDR) approach [1, 23–26], one imposes on the gauge fields the symmetry condition which identifies the gauge transformation as the isometry transformation of 𝑆² due to its coset space structure 𝑆²=SU(2)/U(1). The dimensional reduction is explicitly carried out by applying the solution of the symmetry condition. A background gauge field is introduced as part of the solution [1]. Such a background gauge field is also necessary for obtaining chiral fermions in four-dimensional (4D) spacetime, even without the symmetry condition. After the dimensional reduction, no Kaluza-Klein (KK) mode appears because of the imposed symmetry condition. The symmetry condition also restricts the gauge symmetry and the scalar contents originated from the extra gauge field components in the 4D spacetime. Moreover, a suitable potential for the scalar sector can be obtained to induce SSB at tree level.

In this paper, we consider two scenarios for constructing the 4D theory from a 6D model: one utilizing the symmetry condition for the gauge field with SO(12) symmetry [21], whereas the other without it for the gauge field with E₆ symmetry [22]. In the first scenario, however, we do not impose the condition on the fermions as used in other CSDR models. We then have massive KK modes for fermions but not the gauge and scalar fields in 4D. We can thus obtain a dark matter candidate under assumed KK parity. In the case without the symmetry condition, we find that the background gauge field is able to restrict the gauge symmetry and massless particle contents in 4D. Also, there are KK modes for each field, with the mass spectrum determined according to the model. Generically, massless modes do not appear in the KK mass spectrum because of the positive curvature of the 𝑆² space [27]. With the help of the background gauge field, however, we obtain massless KK modes for the gauge bosons and fermions.

In general, the gauge symmetry of a grand unified theory (GUT) tends to remain in 4D in these dimensional reduction approaches [24, 28–32]. Also, it is usually difficult to obtain an appropriate Higgs potential to break the GUT gauge symmetry to the SM-like one because of the gauge group structure. A GUT gauge symmetry can be broken to the SM-like gauge symmetry by imposing nontrivial boundary conditions (for cases with orbifold extra space, see, e.g., [4–8, 11, 12, 16–18, 33, 34]). Therefore, to solve the above-mentioned problems, we impose on the fields of the 6D model a set of nontrivial boundary conditions on the 𝑆²/𝑍₂ space. Therefore, the gauge symmetry, scalar contents, and massless fermions are determined by these boundary conditions and the background gauge field. We find that in both scenarios, with or without the symmetry condition for the gauge field, the electroweak symmetry breaking (EWSB) can be realized spontaneously. The Higgs boson mass is predicted by analyzing the Higgs potential in the respective models.

This paper is organized as follows. In Section 2, we review two schemes for constructing a 4D theory from gauge models defined on 6D spacetime whose extra space has the 𝑆²/𝑍₂ topology with a set of nontrivial boundary conditions. In Section 3, we show the models based on SO(12) and E₆ gauge symmetries, with the former being imposed with the symmetry condition on the gauge field and the latter without. We summarize our results in Section 4.

2. The 6D Gauge-Higgs Unification Model Construction Scheme with Extra 𝑆²/𝑍₂ Space

There are two schemes for constructing a 4D theory from a 6D gauge theory, where the extra space is a two-sphere orbifold 𝑆²/𝑍₂. Use of the symmetry condition is made on the first scheme but not the other. We apply nontrivial boundary condition in both schemes.

2.1. A Gauge Theory on 6D Spacetime with 𝑆²/𝑍₂ Extraspace

2.1.1. The 6D Spacetime with 𝑆₂/𝑍₂ Extraspace

We begin by considering a 6D spacetime 𝑀⁶ that is assumed to be a direct product of the 4D Minkowski spacetime 𝑀⁴ and two-sphere orbifold 𝑆²/𝑍₂, that is, 𝑀⁶=𝑀⁴×𝑆²/𝑍₂. The two-sphere 𝑆² is a unique two-dimensional coset space and can be written as 𝑆²=SU(2)_𝐼/U(1)_𝐼, where U(1)_𝐼 is a subgroup of SU(2)_𝐼. This coset space structure of 𝑆² requires that 𝑆² have the isometry group SU(2)_𝐼 and that the U(1)_𝐼 group be embedded in the group SO(2) which is in turn a subgroup of the full Lorentz group SO(1,5). We denote the coordinates of 𝑀⁶ by 𝑋^𝑀=(𝑥^𝜇,𝑦^𝜃=𝜃,𝑦^𝜙=𝜙), where 𝑥^𝜇 and {𝜃,𝜙} are 𝑀⁴ coordinates and 𝑆² spherical coordinates, respectively. The spacetime index 𝑀 runs over 𝜇∈{0,1,2,3} and 𝛼∈{𝜃,𝜙}. The orbifold 𝑆²/𝑍₂ is defined by the identification of (𝜃,𝜙) and (𝜋−𝜃,−𝜙) [35], leaving two fixed points: (𝜋/2,0) and (𝜋/2,𝜋). The metric 𝑔_𝑀𝑁 of 𝑀⁶ is written as𝑔_𝑀𝑁=𝜂_𝜇𝜈00−𝑔_𝛼𝛽, (2.1) where 𝜂_𝜇𝜈=diag(1,−1,−1,−1) and 𝑔_𝛼𝛽=𝑅²diag(1,sin²𝜃) are the metrics for 𝑀⁴ and 𝑆², respectively, and 𝑅 is the radius of 𝑆². We define the vielbeins 𝑒^𝑀_𝐴 that connect the metric of 𝑀⁶ and that of the tangent space of 𝑀⁶, denoted by ℎ_𝐴𝐵, through the relation 𝑔_𝑀𝑁=𝑒^𝐴_𝑀𝑒^𝐵_𝑁ℎ_𝐴𝐵. Here 𝐴=(𝜇,𝑎), where 𝑎∈{4,5}, is the index for the coordinates of tangent space of 𝑀⁶. The explicit forms of the vielbeins are𝑒⁴_𝜃=𝑅,𝑒⁵_𝜙=𝑅sin𝜃,𝑒⁴_𝜙=𝑒⁵_𝜃=0. (2.2) Also the nonzero components of the spin connection are𝑅_𝜙⁴⁵=−𝑅_𝜙⁵⁴=−cos𝜃. (2.3)

2.1.2. Lagrangian on 6D Spacetime with 𝑆²/𝑍₂ Extra Space

We now discuss the general structure of a gauge theory on 𝑀⁶. We first introduce a gauge field 𝐴_𝑀(𝑥,𝑦)=(𝐴_𝜇(𝑥,𝑦),𝐴_𝛼(𝑥,𝑦)), which belongs to the adjoint representation of a gauge group 𝐺, and fermions Ψ(𝑥,𝑦), which lies in a representation 𝐹 of 𝐺. The action of this theory is then given by 𝑆=𝑑𝑥⁴𝑅²sin𝜃𝑑𝜃𝑑𝜙Ψ𝑖Γ^𝜇𝐷_𝜇Ψ+Ψ𝑖Γ^𝑎𝑒^𝛼_𝑎𝐷_𝛼1Ψ−4𝑔²𝑔^𝑀𝑁𝑔^𝐾𝐿Tr𝐹_𝑀𝐾𝐹_𝑁𝐿, (2.4) where 𝐹_𝑀𝑁=𝜕_𝑀𝐴_𝑁(𝑋)−𝜕_𝑁𝐴_𝑀(𝑋)−[𝐴_𝑀(𝑋),𝐴_𝑁(𝑋)] is the field strength, 𝐷_𝑀 is the covariant derivative including the spin connection, and Γ_𝐴 represents the Dirac matrices satisfying the 6D Clifford algebra. Here 𝐷_𝑀 and Γ_𝐴 can be written explicitly as𝐷_𝜇=𝜕_𝜇−𝐴_𝜇,𝐷_𝜃=𝜕_𝜃−𝐴_𝜃,𝐷_𝜙=𝜕_𝜙Σ−𝑖³2cos𝜃−𝐴_𝜙,Γ_𝜇=𝛾_𝜇⊗𝐈₂,Γ₄=𝑖𝛾₅⊗𝜎₁,Γ₅=𝑖𝛾₅⊗𝜎₂, (2.5) where {𝛾_𝜇,𝛾₅} are the 4D Dirac matrices, 𝜎_𝑖(𝑖=1,2,3) are the Pauli matrices, 𝐈_𝑑 is the 𝑑×𝑑 identity matrix, and Σ₃=𝐈₄⊗𝜎₃. The covariant derivative 𝐷_𝜙 has the spin connection term 𝑖(Σ₃/2)cos𝜃 which is needed for space with a nonzero curvature- like 𝑆² and applied only to fermions. In 6D spacetime, one can define the chirality of fermions and the corresponding projection operators areΓ_±=1±Γ₇2, (2.6) where Γ₇≡𝛾₅⊗𝜎₃ is the chiral operator. The chiral fermions on 6D spacetime are thusΨ_±=Γ_±Ψ,Γ₇Ψ_±=±Ψ_±. (2.7) The 6D chiral fermions can be also written in terms of 4D chiral fermions 𝜓_𝐿(𝑅) as Ψ₊=𝜓_𝑅𝜓_𝐿,Ψ₋=𝜓_𝐿𝜓_𝑅. (2.8)(2.9) Here we note in passing that the mass dimensions of 𝐴_𝜇, 𝐴_𝛼, Ψ and 𝑔 in the 6D model are 1, 0, 5/2 and −1, respectively.

2.1.3. Nontrivial Boundary Conditions on the Two-Sphere Orbifold

On the two-sphere orbifold, one can consider parity operations 𝑃∶(𝜃,𝜙)→(𝜋−𝜃,−𝜙) and azimuthal translation 𝑇_𝜙∶(𝜃,𝜙)→(𝜃,𝜙+2𝜋). Notice that here the periodicity 𝜙→𝜙+2𝜋 is not associated with the orbifolding. We can impose the following two types of boundary conditions on both gauge and fermion fields under the two operations:𝐴_𝜇𝑥,𝜋−𝜃,−𝜙=𝑃₁𝐴_𝜇𝑥,𝜃,𝜙𝑃₁,𝐴_𝜇𝑥,𝜋−𝜃,2𝜋−𝜙=𝑃₂𝐴_𝜇𝑥,𝜃,𝜙𝑃₂,𝐴_𝛼𝑥,𝜋−𝜃,−𝜙=−𝑃₁𝐴_𝛼𝑥,𝜃,𝜙𝑃₁,𝐴_𝛼𝑥,𝜋−𝜃,2𝜋−𝜙=−𝑃₂𝐴_𝛼𝑥,𝜃,𝜙𝑃₂,Ψ𝑥,𝜋−𝜃,−𝜙=±𝛾₅𝑃₁Ψ𝑥,𝜃,𝜙,Ψ𝑥,𝜋−𝜃,2𝜋−𝜙=±𝛾₅𝑃₂Ψ𝑥,𝜃,𝜙, (2.10)(2.11)(2.12) or𝐴_𝜇𝑥,𝜋−𝜃,−𝜙=𝑃₁𝐴_𝜇𝑥,𝜃,𝜙𝑃₁,𝐴_𝜇𝑥,𝜃,𝜙+2𝜋=𝑃₂𝐴_𝜇𝑥,𝜃,𝜙𝑃₂,𝐴_𝛼𝑥,𝜋−𝜃,−𝜙=−𝑃₁𝐴_𝛼𝑥,𝜃,𝜙𝑃₁,𝐴_𝛼𝑥,𝜃,𝜙+2𝜋=𝑃₂𝐴_𝛼𝑥,𝜃,𝜙𝑃₂,Ψ𝑥,𝜋−𝜃,−𝜙=±𝛾₅𝑃₁Ψ𝑥,𝜃,𝜙,Ψ𝑥,𝜃,𝜙+2𝜋=±𝑃₂Ψ𝑥,𝜃,𝜙, (2.13)(2.14)(2.15) where the former conditions are associated with 𝑃 operation and combination of 𝑃 and 𝑇_𝜙 operations, while the latter conditions are associated with the 𝑃 or 𝑇_𝜙 operation individually. More explicitly, 𝑃₁,  𝑃₂, and𝑃₂ correspond to operations 𝑃, 𝑃𝑇_𝜙, and𝑇_𝜙, respectively. These boundary conditions are determined by requiring invariance of the 6D action under the transformation (𝜃,𝜙)→(𝜋−𝜃,−𝜙) and 𝜙→𝜙+2𝜋. Note that at the poles (sin𝜃=0), the coordinate 𝜙 is not well-defined and the translation 𝑇_𝜙 is irrelevant. Thus, only the components which are even under 𝜙→𝜙+2𝜋 can exist without contradiction.

The projection matrices 𝑃_1,2 act on the gauge group representation space and have eigenvalues ±1. They assign different parities for different representation components. For fermion boundary conditions, the sign in front of 𝛾₅ can be either + or − since the fermions always appear in bilinear forms in the action. The 4D action is then restricted by these parity assignments.

2.2. Dimensional Reduction Scheme with Symmetry Condition

Here we review the dimensional reduction scheme in which a symmetry condition is applied to the gauge field [21].

2.2.1. The Symmetry Condition

We impose on the gauge field 𝐴_𝑀(𝑋) the symmetry which connects SU(2)_𝐼 isometry transformation on 𝑆² and the gauge transformation of the field in order to carry out dimensional reduction. Moreover, the nontrivial boundary conditions of 𝑆²/𝑍₂ are also utilized to restrict the 4D theory. The symmetry demands that the SU(2)_𝐼 coordinate transformation should be compensated by a gauge transformation [1, 23]. It further leads to the following set of the symmetry condition on the gauge field:𝜉^𝛽_𝑖𝜕_𝛽𝐴_𝜇=𝜕_𝜇𝑊_𝑖+𝑊_𝑖,𝐴_𝜇,𝜉^𝛽_𝑖𝜕_𝛽𝐴_𝛼+𝜕_𝛼𝜉^𝛽_𝑖𝐴_𝛽=𝜕_𝛼𝑊_𝑖+𝑊_𝑖,𝐴_𝛼, (2.16) where 𝜉^𝛼_𝑖 are the killing vectors that generate the SU(2)_𝐼 symmetry, and 𝑊_𝑖 are some fields that generate an infinitesimal gauge transformation of 𝐺. Here the index 𝑖=1, 2, 3 corresponds to that of the SU(2) generators. The explicit forms of 𝜉^𝛼_𝑖s for 𝑆² are𝜉^𝜃₁=sin𝜙,𝜉^𝜙₁=cot𝜉𝜃cos𝜙,_𝜃2=−cos𝜙,𝜉^𝜙₂=cot𝜉𝜃sin𝜙,_𝜃3=0,𝜉^𝜙₃=−1. (2.17) The LHS’s and RHS’s of (2.16) are infinitesimal isometry transformations and the corresponding infinitesimal gauge transformations, respectively.

2.2.2. Dimensional Reduction and Lagrangian in 4D Spacetime

The dimensional reduction of the gauge sector is explicitly carried out by applying the solutions of the symmetry condition equations (2.16). These solutions are given by Manton [1]𝐴_𝜇=𝐴_𝜇(𝑥),𝐴_𝜃=−Φ₁(𝑥),𝐴_𝜙=Φ₂(𝑥)sin𝜃−Φ₃cos𝜃,𝑊₁=−Φ₃cos𝜙sin𝜃,𝑊₂=−Φ₃sin𝜙sin𝜃,𝑊₃=0, (2.18)(2.19) where Φ₁(𝑥) and Φ₂(𝑥) are scalar fields and the Φ₃ term for 𝐴_𝜙 corresponds to the background gauge field [36]. They satisfy the following constraints:Φ₃,𝐴_𝜇=0,−𝑖Φ₃,Φ_𝑖(𝑥)=𝑖𝜖_3𝑖𝑗Φ_𝑗(𝑥), (2.20)(2.21) where the LHS shows the gauge transformation associated with Φ₃ and the RHS shows the U(1)_𝐼 transformation embedded in Lorentz group SO(2). These constraints can be satisfied when U(1)_𝐼 is embedded in the gauge group 𝐺 and −𝑖Φ₃ should be chosen as the corresponding generator.

Substituting the solutions, (2.18), into 𝐴_𝑀(𝑋) in the action, (2.4), one can easily obtain the 4D action by integrating out coordinates 𝜃 and 𝜙 in the gauge sector.𝑆^(gauge)_4D=𝑑⁴𝑥−𝑅²4𝑔²Tr𝐹_𝜇𝜈𝐹^𝜇𝜈(𝑥−1)2𝑔²Tr𝐷^′_𝜇Φ₁(𝑥)𝐷^′𝜇Φ₁(𝑥)+𝐷^′_𝜇Φ₂(𝑥)𝐷^′𝜇Φ₂(𝑥−1)2𝑔²𝑅²TrΦ₃+Φ₁(𝑥),Φ₂(𝑥)Φ₃+Φ₁(𝑥),Φ₂(𝑥), (2.22) where 𝐷^′_𝜇Φ=𝜕_𝜇Φ−[𝐴_𝜇,Φ].

For fermions, we do not impose the symmetry condition. Then the gauge interaction term is not invariant under the coordinate transformation on 𝑆²/𝑍₂. The fermion sector of the 4D action is thus obtained by expanding fermions in terms of the normal modes of 𝑆²/𝑍₂ and then integrating out the 𝑆²/𝑍₂ coordinates in the 6D action. As a result, the fermions have massive KK modes which can provide a dark matter candidate. Generally, the KK modes do not contain massless modes because of the positive curvature of 𝑆² [27]. Nevertheless, we can show that the fermion components satisfying the condition−𝑖Φ₃ΣΨ=³2Ψ (2.23) do have massless modes. The squared masses of the KK modes are eigenvalues of the square of the extradimensional Dirac-operator −𝑖𝐷. In the 𝑆² case,−𝑖𝐷=−𝑖𝑒^𝛼𝑎Γ_𝑎𝐷_𝛼𝑖=−𝑅Σ₁𝜕_𝜃+cot𝜃2+Σ₂1𝜕sin𝜃^𝜙+Φ₃cot𝜃, (2.24) where Σ_𝑖=𝐈₄×𝜎_𝑖. Hence,−𝑖𝐷²1=−𝑅²1𝜕sin𝜃^𝜃sin𝜃𝜕_𝜃+1sin²𝜃𝜕²_𝜙+𝑖2−𝑖Φ₃−Σ₃cos𝜃sin²𝜃𝜕_𝜙−14−14sin²𝜃+Σ₃−𝑖Φ₃1sin²𝜃−−𝑖Φ₃²cot²𝜃. (2.25) By acting the above operator on a fermion Ψ(𝑋) that satisfies (2.23), we obtain the relation−𝑖𝐷²1Ψ=−𝑅²1𝜕sin𝜃^𝜃sin𝜃𝜕_𝜃+1sin²𝜃𝜕²_𝜙Ψ. (2.26) The eigenvalues of the operator on the RHS are less than or equal to zero. Hence, the fermion components satisfying (2.23) have massless modes, while other components have only massive KK modes. Note that the massless mode 𝜓₀ should be independent of 𝑆² coordinates 𝜃 and 𝜙, that is,𝜓₀=𝜓(𝑥). (2.27) The existence of massless fermions signifies the meaning and importance of the symmetry condition. Although the energy density of the gauge sector in the presence of the background field is higher than that with no background field, the massless fermions may help render a true ground state as a whole. In other words, the existence of the background field will give a positive contribution to the energy density of the gauge sector, indicating that the gauge sector with the background field alone is not at the ground state. Nevertheless, it gives rise to a negative contribution to the energy density of the fermion sector to induce massless fermions. We therefore expect that once both the gauge and fermion sectors are considered together, the existence of the background field renders the system at the ground state. We also note that one could impose symmetry condition on fermions [24, 37]. In that case, we obtain the massless condition equation (2.23) from the symmetry condition of fermion, and the solution of symmetry condition is independent of the 𝑆² coordinates: 𝜓=𝜓(𝑥) with no massive KK mode. Therefore, the same discussion as before can be applied for this case if one only focuses on the massless mode in our scheme.

2.2.3. Gauge Symmetry and Particle Contents in 4D Spacetime

The symmetry condition and the nontrivial boundary conditions substantially constrain the 4D gauge group and the representations of the particle contents.

First, we show the prescriptions to identify gauge symmetry and field components which satisfy the constraint equations (2.20), (2.21), and (2.23). The gauge group 𝐻 that satisfy the constraint equation (2.20) is identified as𝐻=𝐶_𝐺𝑈(1)_𝐼, (2.28) where 𝐶_𝐺(𝑈(1)_𝐼) denotes the centralizer of U(1)_𝐼 in 𝐺 [23]. Note that this implies 𝐺⊃𝐻=𝐻^′×U(1)_𝐼, where 𝐻^′ is some subgroup of 𝐺. In this way, the gauge group 𝐺 is reduced to its subgroup 𝐻=𝐻^′×U(1)_𝐼 by the symmetry condition.

Secondly, the scalar field components which satisfy the constraint equations (2.21) are specified by the following prescription. Suppose that the adjoint representations of SU(2)_𝐼 and 𝐺 are decomposed according to the embeddings SU(2)_𝐼⊃U(1)_𝐼 and 𝐺⊃𝐻^′×U(1)_𝐼 as3adjSU(2)=0adjU(1)_𝐼+(2)+(−2),adj𝐺=adj𝐻(0)+10adjU(1)_𝐼+_𝑔ℎ_𝑔𝑟_𝑔, (2.29)(2.30) where ℎ_𝑔ʼs denote representations of 𝐻^′, and 𝑟_𝑔ʼs denote the U(1)_𝐼 charges. Then the scalar components satisfying the constraints belong to ℎ_𝑔ʼs whose corresponding 𝑟_𝑔ʼs in (2.30) are ±2.

Thirdly, the fermion components which satisfy the constraint equations (2.23) are determined as follows [37]. Let the group U(1)_𝐼 be embedded in the Lorentz group SO(2) in such a way that the vector representation 2 of SO(2) is decomposed according to SO(2)⊃U(1)_𝐼 as2=(2)+(−2). (2.31) This embedding specifies a decomposition of the Weyl spinor representation 𝜎₆=4 of SO(1,5) according to SO(1,5)⊃SU(2)×SU(2)×U(1)_𝐼 as𝜎₆=(2,1)(1)+(1,2)(−1), (2.32) where the SU(2)×SU(2) representations (2,1) and (1,2) correspond to left-handed and right-handed spinors, respectively. We note that this decomposition corresponds to (2.8) [or (2.9)]. We then decompose 𝐹 according to 𝐺⊃𝐻^′×U(1)_𝐼 as𝐹=_𝑓ℎ_𝑓𝑟_𝑓. (2.33) Now the fermion components satisfying the constraints are identified as those ℎ_𝑓ʼs whose corresponding 𝑟_𝑓ʼs in (2.33) are +1 for left-handed fermions and −1 for right-handed fermions.

Finally, we show which gauge symmetry and field components remain in 4D spacetime by surveying the consistency between the boundary conditions (2.13)–(2.15), the solutions in (2.18), and the massless fermion modes equation (2.27). By applying (2.18) and (2.27) to (2.13)–(2.15), we obtain the parity conditions𝐴_𝜇(𝑥)=𝑃₁𝑃₂𝐴_𝜇(𝑥)𝑃₁𝑃₂,−Φ₁(𝑥)=−𝑃₁−Φ₁(𝑥)𝑃₁,−Φ₁(𝑥)=𝑃₂−Φ₁(𝑥)𝑃₂,Φ₂(𝑥)+Φ₃cos𝜃=−𝑃₁Φ₂(𝑥)𝑃₁+𝑃₁Φ₃𝑃₁Φcos𝜃,₂(𝑥)−Φ₃cos𝜃=𝑃₂Φ₂(𝑥)𝑃₂−𝑃₂Φ₃𝑃₂Ψ(𝑥)cos𝜃,=𝛾₅𝑃₁Ψ(𝑥),Ψ(𝑥)=𝑃₂Ψ(𝑥). (2.34) We find that the gauge fields, scalar fields, and massless fermions in 4D spacetime should be even for 𝑃₁𝐴_𝜇𝑃₁ and 𝑃₂𝐴_𝜇𝑃₂; −𝑃₁Φ_1,2𝑃₁ and 𝑃₂Φ_1,2𝑃₂; 𝛾₅𝑃₁Ψ and 𝑃₂Ψ, respectively. Φ₃ always remains in the spectrum because it is proportional to the U(1)_𝐼 generator and commutes with 𝑃(𝑃^′). Therefore, the particle spectrum contains those satisfying both the constraint equations (2.20), (2.21), and (2.23) and the parity conditions (2.34). The remaining 4D gauge symmetry can be readily identified by observing which components of the gauge field remain in the spectrum.

2.3. Dimensional Reduction Scheme without the Symmetry Condition

Here we review the dimensional reduction scheme which does not require the imposition of the symmetry condition on the gauge field [22].

2.3.1. Background Gauge Field and Gauge Group Reduction

Instead of utilizing the symmetry condition, we consider the background gauge field 𝐴^𝐵_𝜙≡𝐴^𝐵_𝜙sin𝜃 that corresponds to a Dirac monopole [36]𝐴^𝐵_𝜙=−𝑄cos𝜃∓1,𝜋sin𝜃−∶0≤𝜃<2𝜋,+∶2≤𝜃≤𝜋, (2.35) where 𝑄 is proportional to the generator of a U(1) subgroup of the original gauge group. The background gauge field 𝐴^𝐵_𝜙 corresponds to Φ₃cos𝜃⊂𝐴_𝜙 in (2.18).

Here we choose the background gauge field to belong to the U(1)_𝐼 group, which is a subgroup of original gauge group 𝐺:𝐺⊃𝐺_sub⊗U(1)_𝐼. (2.36) We find that there is no massless mode for gauge field components with a nonzero U(1)_𝐼 charge. In fact, these components acquire masses due to the background field from the term proportional to 𝐹_𝜇𝜙𝐹^𝜇_𝜙:−1Tr4𝐹_𝜇𝜈𝐹^𝜇𝜈+12𝑅²sin²𝜃𝐹_𝜇𝜙𝐹^𝜇_𝜙−1⟶Tr4𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇𝜕^𝜇𝐴^𝜈−𝜕^𝜈𝐴^𝜇−12𝑅²sin²𝜃𝐴_𝜇,𝐴^𝐵_𝜙𝐴^𝜇,𝐴^𝐵_𝜙. (2.37) For the components of 𝐴_𝜇 with nonzero U(1)_𝐼 charge, we have𝐴^𝑖_𝜇𝑄_𝑖+𝐴_𝑖𝜇𝑄^𝑖∈𝐴_𝜇, (2.38) where 𝑄_𝑖(𝑄^𝑖=𝑄^†_𝑖) are generators corresponding to distinct components in (3.30) that have nonzero U(1)_𝐼 charges, and 𝐴_𝑖𝜇(𝐴^𝑖_𝜇=𝐴^†_𝑖𝜇) are the corresponding components of 𝐴_𝜇. We find the term1sin²𝜃Tr𝐴_𝜇,𝐴^𝐵_𝜙𝐴^𝜇,𝐴^𝐵_𝜙=(cos𝜃∓1)²sin²𝜃Tr𝐴^𝑖_𝜇𝑄_𝑖+𝐴_𝑖𝜇𝑄^𝑖,𝑄𝐴^𝑖𝜇𝑄_𝑖+𝐴^𝜇_𝑖𝑄^𝑖𝑞,𝑄=−2²(cos𝜃∓1)²sin²𝜃𝐴^𝑖𝜇𝐴_𝑖𝜇, (2.39) where 𝑞 is the 𝑄 charge of the relevant component. Use of the facts that 𝐴^𝐵_𝜙 belongs to U(1)_𝐼 and that Tr[𝑄_𝑖𝑄^𝑖]=2 has been made in the above equation. A mass is thus associated with the lowest modes of those components of 𝐴_𝜇 with nonzero U(1)_𝐼 charges:−1𝑑ΩTr4𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇𝜕^𝜇𝐴^𝜈−𝜕^𝜈𝐴^𝜇−12𝑅²sin²𝜃𝐴_𝜇,𝐴_𝐵𝐴^𝜇,𝐴_𝐵lowest1⟶−2𝜕_𝜇𝐴_𝑖𝜈(𝑥)−𝜕_𝜈𝐴_𝑖𝜇(𝑥)𝜕^𝜇𝐴^𝑖𝜈(𝑥)−𝜕^𝜈𝐴^𝑖𝜇(𝑥)+𝑚²_𝐵𝐴^𝑖_𝜇(𝑥)𝐴^𝑖𝜇(𝑥), (2.40) where the subscript “lowest” means that only the lowest KK modes are kept. Here the lowest KK modes of 𝐴_𝜇 correspond to the term 𝐴_𝜇(𝑥)/√4𝜋 in the KK expansion. In short, any representation of 𝐴_𝜇 carrying a nonzero U(1)_𝐼 charge acquires a mass 𝑚_𝐵 from the background field contribution after one integrates over the extra spatial coordinates. More explicitly,𝑚²_𝐵=𝑞²4𝜋𝑅²()𝑑Ωcos𝜃∓1²sin²𝜃𝑞≃0.39²𝑅² (2.41) for the zero mode. Therefore the gauge group 𝐺 is reduced to 𝐺_sub⊗U(1)_𝐼 by the presence of the background gauge field. This condition is the same as the case with the symmetry condition.

2.3.2. Scalar Field Contents in 4D Spacetime

The scalar contents in 4D spacetime are obtained from the extradimensional components of the gauge field {𝐴_𝜃,𝐴_𝜙} after integrating out the extra spatial coordinates. The kinetic term and potential term of {𝐴_𝜃,𝐴_𝜙} are obtained from the gauge sector containing these components𝑆_scalar=𝑑𝑥⁴1𝑑Ω2𝑔²Tr𝐹_𝜇𝜃𝐹^𝜇_𝜃+12𝑔²sin²𝜃Tr𝐹_𝜇𝜙𝐹^𝜇_𝜙−12𝑔²𝑅²sin²𝜃Tr𝐹_𝜃𝜙𝐹_𝜃𝜙⟶𝑑𝑥⁴1𝑑Ω2𝑔²𝑇𝑟𝜕_𝜇𝐴_𝜃−𝑖𝐴_𝜇,𝐴_𝜃²+12𝑔²𝑇𝑟𝜕_𝜇𝐴_𝜙−𝑖𝐴_𝜇,𝐴_𝜙²−12𝑔²𝑅²1Tr𝜕sin𝜃^𝜃sin𝜃𝐴_𝜙+sin𝜃𝐴^𝐵_𝜙−1𝜕sin𝜃^𝜙𝐴_𝜃−𝑖𝐴_𝜃,𝐴_𝜙+𝐴^𝐵_𝜙², (2.42) where we have taken 𝐴_𝜙=𝐴_𝜙sin𝜃+𝐴^𝐵_𝜙sin𝜃. In the second step indicated by the arrow in (2.42), we have omitted terms which do not involve 𝐴_𝜃 and 𝐴_𝜙 from the right-hand side of the first equality. It is known that one generally cannot obtain massless modes for physical scalar components in 4D spacetime [14, 38]. One can see this by noting that the eigenfunction of the operator (1/sin𝜃)𝜕_𝜃sin𝜃 swith zero eigenvalue is not normalizable [14]. In other words, these fields have only KK modes. However, an interesting feature is that it is possible to obtain a negative squared mass when taking into account the interactions between the background gauge field 𝐴^𝐵_𝜙 and {𝐴_𝜃,𝐴_𝜙}. This happens when the component carries a nonzero U(1)_𝐼 charge, as the background gauge field belongs to U(1)_𝐼. In this case, the (𝑙=1, 𝑚=1) modes of these real scalar components are found to have a negative squared mass in 4D spacetime. They can be identified as the Higgs fields once they are shown to belong to the correct representation under the SM gauge group. Here the numbers (ℓ,𝑚) are the angular momentum quantum number on 𝑆²/𝑍₂, and each KK mode is characterized by these numbers. One can show that the (𝑙=1, 𝑚=0) mode has a positive squared mass and is not considered as the Higgs field. A discussion of the KK masses with general (ℓ,𝑚) will be given in Section 3.2.5.

2.3.3. Chiral Fermions in 4D Spacetime

We introduce fermions as the Weyl spinor fields of the 6D Lorentz group SO(1,5). They can be written in terms of the SO(1,3) Weyl spinors as (2.8) and (2.9). In general, fermions on the two spheres do not have massless KK modes because of the positive curvature of the two spheres. The massless modes can be obtained by incorporating the background gauge field (2.35) though, for it can cancel the contribution from the positive curvature. In this case, the condition for obtaining a massless fermion mode is1𝑄Ψ=±2Ψ, (2.43) where 𝑄 comes from the background gauge field and is proportional to the U(1)_𝐼 generator [35, 36, 38]. We observe that the upper [lower] component on the RHS of (2.8) [(2.9)] has a massless mode for the + [−] sign on the RHS of (2.43).

2.3.4. The Higgs Potential

The Lagrangian for the Higgs sector is derived from the gauge sector that contains extradimensional components of the gauge field {𝐴_𝜃,𝐴_𝜙}, as given in (2.42), by considering the lowest KK modes of them. The kinetic term and potential term are, respectively,𝐿_𝐾=12𝑔²𝑑ΩTr𝜕_𝜇𝐴_𝜃−𝑖𝐴_𝜇,𝐴_𝜃²+Tr𝜕_𝜇𝐴_𝜙−𝑖𝐴_𝜇,𝐴_𝜙²_lowest,1𝑉=2𝑔²𝑅²1𝑑ΩTr𝜕sin𝜃^𝜃sin𝜃𝐴_𝜙+sin𝜃𝐴^𝐵_𝜙−1𝜕sin𝜃^𝜙𝐴_𝜃−𝑖𝐴_𝜃,𝐴_𝜙+𝐴^𝐵_𝜙²_lowest. (2.44) In our model, scalar components other than the Higgs field have vanishing VEV because only the Higgs field has a negative mass-squared term, coming from the interaction with the background gauge field at tree level. Therefore, only the Higgs field contributes to the spontaneous symmetry breaking. Consider the (1,1) mode of the {(1,2)(3,−3,3)+h.c.} representation in (3.31) as argued in the previous section. The gauge fields are given by the following KK expansions:𝐴_𝜃1=−√2Φ₁(𝑥)𝜕_𝜃𝑌⁻₁₁𝜃,𝜙+Φ₂(𝑥)1𝜕sin𝜃^𝜙𝑌⁻₁₁𝜃,𝜙+⋯,𝐴_𝜙=1√2Φ₂(𝑥)𝜕_𝜃𝑌⁻₁₁𝜃,𝜙−Φ₁(𝑥)1𝜕sin𝜃^𝜙𝑌⁻₁₁𝜃,𝜙+⋯, (2.45)(2.46) where ⋯ represents higher KK mode terms [35]. The function 𝑌⁻₁₁=−1/√2[𝑌₁₁+𝑌₁₋₁] is odd under (𝜃,𝜙)→(𝜋/2−𝜃,−𝜙). We will discuss their higher KK modes and masses in the existence of the background gauge field in Section 3.2.5. With (2.45) and (2.46), the kinetic term becomes𝐿_𝐾(𝑥)=12𝑔²Tr𝐷_𝜇Φ₁(𝑥)𝐷^𝜇Φ₁(𝑥)+Tr𝐷_𝜇Φ₂(𝑥)𝐷^𝜇Φ₂(𝑥), (2.47) where 𝐷_𝜇Φ_1,2=𝜕_𝜇Φ_1,2−𝑖[𝐴_𝜇,Φ_1,2] is the covariant derivative acting on Φ_1,2. The potential term, on the other hand, is1𝑉=2𝑔²𝑅²𝑑ΩTr−√2𝑌⁻₁₁Φ₂(𝑥)+𝑖+𝑄2Φ₁(𝑥),Φ₂(𝑥)𝜕_𝜃𝑌⁻₁₁𝜕_𝜃𝑌⁻₁₁+1sin²𝜃𝜕_𝜙𝑌⁻₁₁𝜕_𝜙𝑌⁻₁₁+𝑖√2Φ₁(𝑥),𝐴^𝐵_𝜙𝜕_𝜃𝑌⁻₁₁+𝑖√2Φ₂(𝑥),𝐴^𝐵_𝜙1𝜕sin𝜃^𝜙𝑌⁻₁₁², (2.48) where 𝜕_𝜃(sin𝜃𝐴^𝐵_𝜙)=𝑄sin𝜃 from (2.35) is used. Expanding the square in the trace, we get1𝑉=2𝑔²𝑅²𝑑ΩTr2𝑌⁺₁₁²Φ²₂(𝑥)+𝑄²−14Φ₁(𝑥),Φ₂(𝑥)²𝜕_𝜃𝑌⁻₁₁𝜕_𝜃𝑌⁻₁₁+1sin²𝜃𝜕_𝜙𝑌⁻₁₁𝜕_𝜙𝑌⁻₁₁²−12Φ₁(𝑥),𝐴^𝐵_𝜙²𝜕_𝜃𝑌⁻₁₁²−12Φ₂(𝑥),𝐴^𝐵_𝜙²1𝜕sin𝜃^𝜙𝑌⁻₁₁²−2𝑖Φ₂(𝑥)Φ₁(𝑥),𝐴^𝐵_𝜙𝑌⁻₁₁𝜕_𝜃𝑌⁻₁₁−Φ₁(𝑥),𝐴^𝐵_𝜙Φ₂(𝑥),𝐴^𝐵_𝜙𝜕_𝜃𝑌⁻₁₁1𝜕sin𝜃^𝜙𝑌⁻₁₁+𝑖𝑄Φ₁(𝑥),Φ₂(𝑥)𝜕_𝜃𝑌⁻₁₁𝜕_𝜃𝑌⁻₁₁+1sin²𝜃𝜕_𝜙𝑌⁻₁₁𝜕_𝜙𝑌⁻₁₁, (2.49) where terms that vanish after the 𝑑Ω integration are directly omitted. In the end, the potential is simplified to1𝑉=2𝑔²𝑅²Tr2Φ²₂(𝑥)+4𝜋𝑄²−3Φ10𝜋¹(𝑥),Φ₂(𝑥)²+5𝑖2𝑄Φ₁(𝑥),Φ₂(𝑥)+𝜇₁𝑄,Φ₁(𝑥)²+𝜇₂𝑄,Φ₂(𝑥)², (2.50) where use of 𝐴^𝐵_𝜙=−𝑄(cos𝜃∓1)/sin𝜃 has been made and 𝜇₁=1−(3/2)ln2 and 𝜇₂=(3/4)(1−2ln2).

We now take the following linear combination of Φ₁ and Φ₂ to form a complex Higgs doublet,Φ(𝑥)=1√2Φ₁(𝑥)+𝑖Φ₂(𝑥),Φ(𝑥)^†=1√2Φ₁(𝑥)−𝑖Φ₂(𝑥). (2.51)(2.52) It is straightforward to see thatΦ₁(𝑥),Φ₂(𝑥)=𝑖Φ(𝑥),Φ^†(𝑥). (2.53) The kinetic term and the Higgs potential now become𝐿_𝐾=1𝑔²Tr𝐷_𝜇Φ^†(𝑥)𝐷^𝜇Φ(𝑥),1𝑉=2𝑔²𝑅²Tr2Φ²₂(𝑥)+4𝜋𝑄²+3Φ(𝑥)10𝜋,Φ^†(𝑥)²−52𝑄Φ(𝑥),Φ^†(𝑥)+𝜇₁𝑄,Φ₁(𝑥)²+𝜇₂𝑄,Φ₂(𝑥)². (2.54)(2.55) The last three terms in the potential are contributions to the squared mass term of the Higgs boson from the background gauge field and can lead to a negative value. This means that the existence of the background gauge field makes the minimum of Higgs potential lower.

3. The Models Based on Our Schemes

In this section, we show concrete models based on the scheme introduced in previous section. We review the model based on SO(12) gauge symmetry for the scheme with symmetry condition given in [21], and review the model based on E₆ gauge symmetry for the scheme without symmetry condition given in [22].

3.1. The SO(12) Model with Symmetry Condition

Here we show a model based on a gauge group 𝐺=SO(12) and a representation 𝐹=32 of SO(12) for fermions, under the scheme with symmetry condition [21]. The choice of 𝐺=SO(12) and 𝐹=32 is motivated by the study based on CSDR which leads to an SO(10)×U(1) gauge theory with one generation of fermion in 4D spacetime [28] (for SO(12) GUT see also [39]).

3.1.1. A Gauge Symmetry and Particle Contents

First, we show the particle contents in 4D spacetime without parities equations (2.13)–(2.15). We assume that U(1)_𝐼 is embedded into SO(12) such asSO(12)⊃SO(10)×U(1)_𝐼. (3.1) Thus we identify SO(10)×U(1)_𝐼 as the gauge group which satisfy the constraint equations (2.20), using (2.28). The SO(12) gauge group is reduced to SO(10)×U(1) by the symmetry condition. We identify the scalar components which satisfy (2.21) by decomposing adjoint representation of SO(12):SO(12)⊃SO(10)×U(1)_𝐼∶66=45(0)+1(0)+10(2)+10(−2). (3.2) According to the prescription below (2.28) in Section 2, the scalar components 10(2)+10(−2) remains in 4D spacetime. We also identify the fermion components which satisfy (2.23) by decomposing 32 representations of SO(12) asSO(12)⊃SO(10)×U(1)_𝐼∶32=16(1)+16(−1). (3.3) According to the prescription below (2.30) in Section 2, we have the fermion components as 16(1) for a left-handed fermion and 16(−1) for a right-handed fermion, respectively, in 4D spacetime.

Next, we specify the parity assignment of 𝑃₁(𝑃₂) in order to identify the gauge symmetry and the particle contents that actually remain in 4D spacetime. We choose a parity assignment so as to break gauge symmetry as SO(12)⊃SO(10)×U(1)_𝐼⊃SU(5)×U(1)_𝑋×U(1)_𝐼⊃SU(3)×SU(2)_𝐿×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼 and to maintain Higgs-doublet in 4D spacetime. The parity assignment is written in 32 dimensional spinor basis of SO(12) such asSO(12)⊃SU(3)×SU(2)_𝐿×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼,32=(3,2)⁽⁺⁻⁾(1,−1,1)+(3,2)⁽⁺⁻⁾()+()−1,1,−13,1⁽⁻⁻⁾(4,1,−1)+3,1⁽⁻⁻⁾()+()−4,−1,13,1⁽⁻⁺⁾(−2,−3,−1)+3,1⁽⁻⁺⁾()+()2,3,11,2⁽⁺⁺⁾(3,−3,−1)+(1,2)⁽⁺⁺⁾()+()−3,3,11,1⁽⁻⁻⁾(6,−1,1)+(1,1)⁽⁻⁻⁾()+()−6,1,−11,1⁽⁻⁺⁾(0,−5,1)+(1,1)⁽⁻⁺⁾(0,5,−1), (3.4) where for example, (+,−) means that the parities (𝑃₁,𝑃₂) of the associated components are (even, odd). We find the gauge symmetry in 4D spacetime by surveying parity assignment for the gauge field. The parity assignments of the gauge field under 𝐴_𝜇→𝑃₁𝐴_𝜇𝑃₁(𝑃₂𝐴_𝜇𝑃₂) are66=(8,1)⁽⁺⁺⁾(0,0,0)+(1,3)⁽⁺⁺⁾(0,0,0)+(1,1)⁽⁺⁺⁾()+()0,0,01,1⁽⁺⁺⁾(0,0,0)+(1,1)⁽⁺⁺⁾()+)0,0,0(3,2⁽⁻⁺⁾(−5,0,0)+3,2⁽⁻⁺⁾()+()5,0,03,2⁽⁻⁻⁾(1,4,0)+3,2⁽⁻⁻⁾()+()−1,−4,03,1⁽⁺⁻⁾(4,−4,0)+3,1⁽⁺⁻⁾()+()−4,4,03,1⁽⁺⁻⁾(−2,2,2)+3,1⁽⁺⁻⁾(2,−2,−2)+(3,1)⁽⁺⁺⁾(−2,2,−2)+3,1⁽⁺⁺⁾(2,−2,2)+(1,2)⁽⁻⁻⁾(3,2,2)+(1,2)⁽⁻⁻⁾(−3,−2,−2)+(1,2)⁽⁻⁺⁾(3,2,−2)+(1,2)⁽⁻⁺⁾(−3,−2,2)+(1,1)⁽⁺⁻⁾(6,4,0)+(1,1)⁽⁺⁻⁾(−6,−4,0). (3.5) The components with an underline are originated from 10(2) and 10(−2) of SO(10)×U(1)_𝐼, which do not satisfy constraint equations (2.20), and hence these components do not remain in 4D spacetime. Thus we have the gauge fields with (+,+) parity components without an underline in 4D spacetime, and the gauge symmetry is SU(3)×SU(2)_𝐿×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼.

The scalar particle contents in 4D spacetime are determined by the parity assignments, under Φ_1,2→−𝑃₁Φ_1,2𝑃₁ and 𝑃₂Φ_1,2𝑃₂:66=(8,1)⁽⁻⁺⁾(0,0,0)+(1,3)⁽⁻⁺⁾(0,0,0)+(1,1)⁽⁻⁺⁾()+()0,0,01,1⁽⁻⁺⁾(0,0,0)+(1,1)⁽⁻⁺⁾()+)0,0,0(3,2⁽⁺⁺⁾(−5,0,0)+3,2⁽⁺⁺⁾()+()5,0,03,2⁽⁺⁻⁾(1,4,0)+3,2⁽⁺⁻⁾()+()−1,−4,03,1⁽⁻⁻⁾(4,−4,0)+3,1⁽⁻⁻⁾()+()−4,4,03,1⁽⁻⁻⁾(−2,2,2)+3,1⁽⁻⁻⁾(2,−2,−2)+(3,1)⁽⁻⁺⁾(−2,2,−2)+3,1⁽⁻⁺⁾(2,−2,2)+(1,2)⁽⁺⁻⁾(3,2,2)+(1,2)⁽⁺⁻⁾(−3,−2,−2)+(1,2)⁽⁺⁺⁾(3,2,−2)+(1,2)⁽⁺⁺⁾(−3,−2,2)+(1,1)⁽⁻⁻⁾(6,4,0)+(1,1)⁽⁻⁻⁾(−6,−4,0). (3.6) Note that the relative sign for the parity assignment of 𝑃₁ is different from (3.5), and that the only underlined parts satisfy the constraint equations (2.21). Thus the scalar components in 4D spacetime are (1,2)(3,2,−2)and(1,2)(−3,−2,2).

We specify the massless fermion contents in 4D spacetime, by surveying the parity assignments for each components of fermion fields. We introduce two types of left-handed Weyl fermions that belong to 32 representation of SO(12), which have parity assignments 𝜓⁽𝑃₂₎→𝛾₅𝑃₁𝜓⁽𝑃₂₎(𝑃₂𝜓⁽𝑃₂₎) and 𝜓⁽⁻𝑃₂₎→𝛾₅𝑃₁𝜓⁽⁻𝑃₂₎(−𝑃₂𝜓⁽⁻𝑃₂₎), respectively. They have the parity assignments as32⁽₂𝑃_)𝐿=(3,2)⁽⁻⁻⁾(1,−1,1)_𝐿+3,2⁽⁻⁻⁾(−1,1,−1)_𝐿+3,1⁽⁺⁻⁾(−4,−1,1)_𝐿+(3,1)⁽⁺⁻⁾(4,1,−1)_𝐿+3,1⁽⁺⁺⁾(2,3,1)_𝐿+(3,1)⁽⁺⁺⁾(−2,−3,−1)_𝐿+(1,2)⁽⁻⁺⁾(−3,3,1)_𝐿+(1,2)⁽⁻⁺⁾(3,−3,−1)_𝐿+(1,1)⁽⁺⁻⁾(6,−1,1)_𝐿+(1,1)⁽⁺⁻⁾(−6,1,−1)_𝐿+(1,1)⁽⁺⁺⁾(0,−5,1)_𝐿+(1,1)⁽⁺⁺⁾(0,5,−1)_𝐿,32⁽₂𝑃_)𝑅=(3,2)⁽⁺⁻⁾(1,−1,1)_𝑅+3,2⁽⁺⁻⁾(−1,1,−1)_𝑅+3,1⁽⁻⁻⁾(−4,−1,1)_𝑅+(3,1)⁽⁻⁻⁾(4,1,−1)_𝑅+3,1⁽⁻⁺⁾(2,3,1)_𝑅+(3,1)⁽⁻⁺⁾(−2,−3,−1)_𝑅+(1,2)⁽⁺⁺⁾(−3,3,1)_𝑅+(1,2)⁽⁺⁺⁾(3,−3,−1)_𝑅+(1,1)⁽⁻⁻⁾(6,−1,1)_𝑅+(1,1)⁽⁻⁻⁾(−6,1,−1)_𝑅+(1,1)⁽⁻⁺⁾(0,−5,1)_𝑅+(1,1)⁽⁻⁺⁾(0,5,−1)_𝑅,32⁽⁻₂𝑃_)𝐿=(3,2)⁽⁻⁺⁾(1,−1,1)_𝐿+3,2⁽⁻⁺⁾(−1,1,−1)_𝐿+3,1⁽⁺⁺⁾(−4,−1,1)_𝐿+(3,1)⁽⁺⁺⁾(4,1,−1)_𝐿+3,1⁽⁺⁻⁾(2,3,1)_𝐿+(3,1)⁽⁺⁻⁾(−2,−3,−1)_𝐿+(1,2)⁽⁻⁻⁾(−3,3,1)_𝐿+(1,2)⁽⁻⁻⁾(3,−3,−1)_𝐿+(1,1)⁽⁺⁺⁾(6,−1,1)_𝐿+(1,1)⁽⁺⁺⁾(−6,1,−1)_𝐿+(1,1)⁽⁺⁻⁾(0,−5,1)_𝐿+(1,1)⁽⁺⁻⁾(0,5,−1)_𝐿,32⁽⁻₂𝑃_)𝑅=(3,2)⁽⁺⁺⁾(1,−1,1)_𝑅+3,2⁽⁺⁺⁾(−1,1,−1)_𝑅+3,1⁽⁻⁺⁾(−4,−1,1)_𝑅+(3,1)⁽⁻⁺⁾(4,1,−1)_𝑅+3,1⁽⁻⁺⁾(2,3,1)_𝑅+(3,1)⁽⁻⁺⁾(−2,−3,−1)_𝑅+(1,2)⁽⁺⁻⁾(−3,3,1)_𝑅+(1,2)⁽⁺⁻⁾(3,−3,−1)_𝑅+(1,1)⁽⁻⁺⁾(6,−1,1)_𝑅+(1,1)⁽⁻⁺⁾(−6,1,−1)_𝑅+(1,1)⁽⁻⁻⁾(0,−5,1)_𝑅+(1,1)⁽⁻⁻⁾(0,5,−1)_𝑅, (3.7) where 𝐿(𝑅) means the left-handedness (right-handedness) of fermions in 4D spacetime, and the underlined parts correspond to the components which satisfy constraint equations (2.23). Note the relative sign for parity assignment of 𝑃₁ between left-handed fermion and right-handed fermion and that of 𝑃₂ between 32⁽𝑃₂₎ and 32⁽⁻𝑃₂₎. The difference between 32⁽𝑃₂₎ and 32⁽⁻𝑃₂₎ is allowed because of the bilinear form of the fermion sector. We thus find that the massless fermion components in 4D spacetime are one generation of SM-fermions with right-handed neutrino: {(3,2)(1,−1,1)_𝐿, (3,1)(4,1,−1)_𝑅, (3,1)(−2,−3,−1)_𝑅, (1,2)(−3,3,1)_𝐿, (1,1)(−6,1,−1)_𝑅, (1,1)(0,5,−1)_𝑅}.

3.1.2. The Higgs Sector of the Model

We analyze the Higgs-sector of our model. The Higgs-sector 𝐿_Higgs is the last two terms of(2.22)𝐿_Higgs1=−2𝑔²Tr𝐷^′_𝜇Φ₁(𝑥)𝐷^′𝜇Φ₁(𝑥)+𝐷^′_𝜇Φ₂(𝑥)𝐷^′𝜇Φ₂(𝑥−1)2𝑔²𝑅²TrΦ₃+Φ₁(𝑥),Φ₂(𝑥)Φ₃+Φ₁(𝑥),Φ₂(𝑥), (3.8) where the first term of RHS is the kinetic term of Higgs and the second term gives the Higgs potential. We rewrite the Higgs-sector in terms of genuine Higgs field in order to analyze it.

We first note that the Φ_𝑖s are written as Φ_𝑖=𝑖𝜙_𝑖=𝑖𝜙^𝑎_𝑖𝑄_𝑎, (3.9) where 𝑄_𝑎s are generators of gauge group SO(12), since Φ_𝑖s are originated from gauge fields 𝐴_𝛼=𝑖𝐴^𝑎_𝛼𝑄_𝑎; for the gauge group generator we assume the normalization Tr(𝑄_𝑎𝑄_𝑏)=−2𝛿_𝑎𝑏. Note that we assumed the −𝑖Φ₃ as the generator of U(1)_𝐼 embedded in SO(12),−𝑖Φ₃=𝑄_𝐼. (3.10) We change the notation of the scalar fields according to (2.29) such that,𝜙₊=12𝜙₁+𝑖𝜙₂,𝜙₋=12𝜙₁−𝑖𝜙₂, (3.11) in order to express solutions of the constraint equations (2.21) clearly. The constraint equations (2.21) then rewritten as𝑄_𝐼,𝜙₊=𝜙₊,𝑄_𝐼,𝜙₋=−𝜙₋. (3.12) The kinetic term 𝐿_𝐾𝐸 and potential 𝑉(𝜙) term are rewritten in terms of 𝜙₊ and 𝜙₋:𝐿_𝐾𝐸1=−𝑔²Tr𝐷^′_𝜇𝜙₊(𝑥)𝐷^′𝜇𝜙₋(𝑥),1𝑉=−2𝑔²𝑅²Tr𝑄²_𝐼−4𝑄_𝐼𝜙₊,𝜙₋+4𝜙₊,𝜙₋𝜙₊,𝜙₋, (3.13)(3.14) where covariant derivative 𝐷^′_𝜇 is 𝐷^′_𝜇𝜙_±=𝜕_𝜇𝜙_±−[𝐴_𝜇,𝜙_±].

Next, we change the notation of SO(12) generators 𝑄_𝑎 according to decomposition (3.5) such that𝑄_𝐺=𝑄_𝑖,𝑄_𝛼,𝑄_𝑌,𝑄,𝑄_𝐼,𝑄_{𝑎𝑥(−500)},𝑄^{𝑎𝑥(500)}𝑄_{𝑎𝑥(140)},𝑄^{𝑎𝑥(−1−40)},𝑄_𝑎(4−40),𝑄^𝑎(−440)𝑄_{𝑎(−22−2)},𝑄^𝑎(2−22),𝑄_𝑎(−222),𝑄^{𝑎(2−2−2)}𝑄_𝑥(322),𝑄^{𝑥(−3−2−2)},𝑄_𝑥(32−2),𝑄^{𝑥(−3−22)}𝑄₍₆₄₀₎,𝑄_(−6−40), (3.15) where the order of generators corresponds to (3.5), index 𝑖 = 1–8 corresponds to SU(3) adjoint rep, index 𝛼 = 1–3 corresponds to SU(2) adjoint rep, index 𝑎 = 1–3 corresponds to SU(3)-triplet, and index 𝑥=1, 2 corresponds to SU(2)-doublet. We write 𝜙_± in terms of the genuine Higgs field 𝜙_𝑥 which belongs to (1,2)(3,2,−2), such that 𝜙₊=𝜙_𝑥𝑄^{𝑥(−3−22)},𝜙₋=𝜙^𝑥𝑄_𝑥(32−2), (3.16) where 𝜙^𝑥=(𝜙_𝑥)^†. We also write gauge field 𝐴_𝜇(𝑥) in terms of 𝑄s in (3.38) as 𝐴_𝜇(𝑥)=𝑖𝐴^𝑖_𝜇𝑄_𝑖+𝐴^𝛼_𝜇𝑄_𝛼+𝐵_𝜇𝑄_𝑌+𝐶_𝜇𝑄+𝐸_𝜇𝑄_𝐼. (3.17) We need commutation relations of 𝑄^{𝑥(−3−22)}, 𝑄_𝑥(32−2), 𝑄_𝛼, 𝑄_𝑌, 𝑄, and𝑄_𝐼 in order to analyze the Higgs sector; we summarized them in Table 1.

Table 1 

Commutation relations of 𝑄^{𝑥(−3−22)}, 𝑄_𝑥(32−2), 𝑄_𝛼, 𝑄_𝑌, 𝑄, and 𝑄_𝐼.

Finally, we obtain the Higgs sector with genuine Higgs field by substituting (3.16)–(3.17) into (3.13) and (3.14) and rescaling the fields 𝜙→(𝑔/√2)𝜙 and 𝐴_𝜇→(𝑔/𝑅)𝐴_𝜇, and the couplings (√2/𝑅)𝑔=𝑔₂ and 6/(5𝑅²)𝑔=𝑔_𝑌,𝐿_Higgs=𝐷_𝜇𝜙_𝑥²−𝑉𝜙, (3.18) where the covariant derivative 𝐷_𝜇𝜙_𝑥 and potential 𝑉(𝜙) are𝐷_𝜇𝜙_𝑥=𝜕_𝜇𝜙_𝑥+𝑖𝑔₂12𝜎_𝛼^𝑦_𝑥𝐴_𝛼𝜇𝜙_𝑦+𝑖𝑔_𝑌12𝐵_𝜇𝜙_𝑥+𝑖15𝑔𝐶_𝜇𝜙_𝑥−𝑖𝑔𝐸_𝜇𝜙_𝑥,2𝑉=−𝑅²𝜙^𝑥𝜙_𝑥+3𝑔²2𝑅²𝜙^𝑥𝜙_𝑥², (3.19)(3.20) respectively.Notice that we omitted the constant term in the Higgs potential. We note that the SU(2)_𝐿×U(1)_𝑌 part of the Higgs sector has the same form as the SM Higgs sector.Therefore we obtain the electroweak symmetry breaking SU(2)_𝐿×U(1)_𝑌→U(1)_𝐸𝑀. The Higgs field 𝜙^𝑥 acquires vacuum expectation value (VEV) as𝜙=1√20𝑣,𝑣=431𝑔, (3.21) and 𝑊 boson mass 𝑚_𝑊 and Higgs mass 𝑚_𝐻 are given in terms of radius 𝑅𝑚_𝑊=𝑔₂𝑣2=231𝑅,𝑚_𝐻=√3𝑔𝑣𝑅=√41𝑅. (3.22) The ratio between 𝑚_𝑊 and 𝑚_𝐻 is predicted𝑚_𝐻𝑚_𝑊=√6. (3.23) We thus find 𝑚_𝐻~196 GeV in this model. The Weinberg angle is given bysin²𝜃_𝑊=𝑔²_𝑌𝑔²₂+𝑔²_𝑌=38, (3.24) which is same as SU(5) GUT case. The prediction for the Weinberg angle at tree level is not consistent with the electroweak measurements. One should also take into account quantum corrections including contributions from the KK modes. It is, however, beyond the scope of this paper.

In principle, one-loop power divergences in the Higgs potential would reappear since the operator linear in 𝐹_𝑎𝑏 is allowed, where {𝑎,𝑏} denote extraspatial components [40]. Such an operator would have the form𝐹^𝛼_𝜃𝜙(𝑥), (3.25) where 𝛼 corresponds to the index of the U(1) generator remaining in 4D. This operator is potentially dangerous since its coefficient can be divergent. We can readily avoid this by requiring parity invariance on 𝑆²/𝑍₂ as in the 𝑇²/𝑍₂ case [41].

First, consider the parity transformation 𝜃→𝜋−𝜃. The parity conditions for the fields are defined as𝐴_𝜇𝑥,𝜃,𝜙⟶𝐴_𝜇,𝐴𝑥,𝜋−𝜃,𝜙_𝜃𝑥,𝜃,𝜙⟶−𝐴_𝜃,𝐴𝑥,𝜋−𝜃,𝜙_𝜙𝑥,𝜃,𝜙⟶𝐴_𝜙(),Ψ𝑥,𝜋𝑥,𝜋−𝜃,𝜙𝑥,𝜃,𝜙⟶±Γ⁴Ψ𝑥,𝜋−𝜃,𝜙, (3.26) where Γ⁴=𝛾₅⊗𝜎₁. It is easy to see that the action in 6D, (2.4), is invariant under such a parity transformation.

Secondly, we check the consistency between the orbifold boundary conditions on 𝑆²/𝑍₂, (2.10)–(2.12), and the parity conditions, (3.26). By performing the parity transformation on both sides of the orbifold boundary conditions, (2.10)–(2.12), we obtain𝐴_𝜇𝑥,𝜃,−𝜙=𝑃₁𝐴_𝜇𝑥,𝜋−𝜃,𝜙𝑃₁,−𝐴_𝜃𝑥,𝜃,−𝜙=𝑃₁𝐴_𝜃𝑥,𝜋−𝜃,𝜙𝑃₁,𝐴_𝜙𝑥,𝜃,−𝜙=−𝑃₁𝐴_𝜙𝑥,𝜋−𝜃,𝜙𝑃₁,±Γ⁴Ψ𝑥,𝜃,−𝜙=±𝛾₅𝑃₁±Γ⁴Ψ,𝐴𝑥,𝜋−𝜃,𝜙_𝜇𝑥,𝜋−𝜃,𝜙+2𝜋=𝑃₂𝐴_𝜇𝑥,𝜋−𝜃,𝜙𝑃₂,−𝐴_𝜃𝑥,𝜋−𝜃,𝜙+2𝜋=𝑃₂𝐴_𝜃𝑥,𝜋−𝜃,𝜙𝑃₂,𝐴_𝜙𝑥,𝜋−𝜃,𝜙+2𝜋=−𝑃₂𝐴_𝜙𝑥,𝜋−𝜃,𝜙𝑃₂,±Γ⁴Ψ𝑥,𝜋−𝜃,𝜙+2𝜋=±𝛾₅𝑃₂±Γ⁴Ψ𝑥,𝜋−𝜃,𝜙. (3.27) Since (2.10)–(2.12) hold for any 𝜃 and 𝜙 and Γ⁴ commutes with 𝛾₅, we find that the orbifold boundary conditions still hold under the parity transformation with the identification of 𝜃=𝜋−𝜃′. In other words, the orbifold boundary conditions, (2.10)–(2.12), are parity invariant.

Finally, we find that under the parity, the operator 𝐹^𝛼_𝜃𝜙 transforms to −𝐹^𝛼_𝜃𝜙. Therefore, this operator is forbidden by parity invariance of the action. An explicit calculation of one-loop corrections to the Higgs potential to show that this operator vanishes, however, is beyond the scope of this paper.

3.2. The E₆ Model without Symmetry Condition

Here we show a model based on a gauge group 𝐺=E₆ with a representation 27 for a fermion, under the scheme without symmetry condition [22].

3.2.1. Gauge Group Reduction

We consider the following gauge group reduction E₆⊃SO(10)×U(1)_𝐼⊃SU(5)×U(1)_𝑋×U(1)_𝐼⊃SU(3)×SU(2)×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼. (3.28) The background gauge field in (2.35) is chosen to belong to the U(1)_𝐼 group. This choice is needed in order to obtain chiral SM fermions in 4D spacetime to be discussed later. There are two other symmetry reduction schemes. One can prove that the results in those two schemes are effectively the same as the one considered here once we require the correct U(1) combinations for the hypercharge and the background field.

We then impose the parity assignments with respect to the fixed points, (2.10)–(2.15). The parity assignments for the fundamental representation of E₆ is chosen to be 𝟐𝟕=(1,2)(−3,−2,−2)^(+,+)+(1,2)(3,2,−2)^(−,−)+(1,2)(−3,3,1)^(+,−)+(1,1)(6,−1,1)^(+,+)+(1,1)(0,−5,1)^(−,−)+(1,1)(0,0,4)^(−,+)+(3,2)(1,−1,1)^(−,+)+(3,1)(−2,2,−2)^(+,−)+3,1(−4,−1,1)^(+,+)+3,1(2,3,1)^(+,+)+3,1(2,−2,−2)^(−,+), (3.29) where, for example, (+,−) means that the parities under 𝑃₁ and 𝑃₂ are (even, odd). By the requirement of consistency, we find that the components of 𝐴_𝜇 in the adjoint representation have the parities under 𝐴_𝜇→𝑃₁𝐴_𝜇𝑃₁(𝑃₂𝐴_𝜇𝑃₂) as follows: 𝟕𝟖|_𝐴𝜇=(8,1)(0,0,0)^(+,+)+(1,3)(0,0,0)^(+,+)+(1,1)(0,0,0)^(+,+)+(1,1)(0,0,0)^(+,+)+(1,1)(0,0,0)^(+,+)+(3,2)(−5,0,0)^(−,+)+3,2(5,0,0)^(−,+)+(3,2)(1,4,0)^(+,−)+3,2(−1,−4,0)^(+,−)+(3,1)(4,−4,0)^(−,−)+3,1(−4,4,0)^(−,−)+(1,1)(−6,−4,0)^(−,−)+(1,1)(6,4,0)^(−,−)+(3,2)(1,−1,−3)^(+,+)+3,2(−1,1,3)^(+,+)+(3,1)(4,1,3)^(−,+)+3,1(−4,−1,−3)^(−,+)+(3,1)(−2,−3,3)^(+,−)+3,1(2,3,−3)^(+,−)+(1,2)(−3,3,−3)^(−,−)+(1,2)(3,−3,3)^(−,−)+(1,1)(−6,1,3)^(−,+)+(1,1)(6,−1,−3)^(−,+)+(1,1)(0,−5,−3)^(+,−)+(1,1)(0,5,3)^(+,−), (3.30) where the underlined components correspond to the adjoint representations of SU(3)×SU(2)×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼, respectively. We note that the components with parity (+,+) can have massless zero modes in 4D spacetime. Such components include the adjoint representations of SU(3)×SU(2)×U(1)³,(3,2)(1,−1,−3) and its conjugate. The latter components seem problematic. Yet they do not appear in the low-energy spectrum due to nonzero U(1)_𝐼 charge. The zero modes of these components will get masses from the background field as in (2.41).

3.2.2. Scalar Field Contents in 4D Spacetime

With the parity assignments with respect to the fixed points, (2.11) and (2.14), we have for the 𝐴_𝜃 and 𝐴_𝜙 fields 𝟕𝟖|_{𝐴𝜃,𝜙}=(8,1)(0,0,0)^(−,−)+(1,3)(0,0,0)^(−,−)+(1,1)(0,0,0)^(−,−)+(1,1)(0,0,0)^(−,−)+(1,1)(0,0,0)^(−,−)+(3,2)(−5,0,0)^(+,−)+3,2(5,0,0)^(+,−)+(3,2)(1,4,0)^(−,+)+3,2(−1,−4,0)^(−,+)+(3,1)(4,−4,0)^(+,+)+3,1(−4,4,0)^(+,+)+(1,1)(−6,−4,0)^(+,+)+(1,1)(6,4,0)^(+,+)+(3,2)(1,−1,−3)^(−,−)+3,2(−1,1,3)^(−,−)+(3,1)(4,1,3)^(+,−)+3,1(−4,−1,−3)^(+,−)+(3,1)(−2,−3,3)^(−,+)+3,1(2,3,−3)^(−,+)+(1,2)(−3,3,−3)^(+,+)+(1,2)(3,−3,3)^(+,+)+(1,1)(−6,1,3)^(+,−)+(1,1)(6,−1,−3)^(+,−)+(1,1)(0,−5,−3)^(−,+)+(1,1)(0,5,3)^(−,+). (3.31) Components with (+,−) or (−,+) parity do not have KK modes since they are odd under𝜙→𝜙+2𝜋and the KK modes of gauge field are specified by integer angular momentum quantum numbers ℓ and 𝑚 on the two spheres. In the 𝑆¹/𝑍₂ case, the translation group on 𝑆¹ is U(1) and any quantum number is allowed. After orbifolding, we obtain the quantum numbers allowed by parity and they can be nonintegers. On the other hand, the translation group on 𝑆² is SU(2) and only integer quantum numbers are allowed because they correspond to quantized angular momenta. We then concentrate on the components which have either (+,+) or (−,−) parity and nonzero U(1)_𝐼 charges as the candidate for the Higgs field. These include {(1,2)(3,−3,3)+h.c.} and {(3,2)(1,−1,−3)+h.c.} with parities (+,+) and (−,−), respectively. The representations (1,2)(−3,3,−3) and (1,2)(3,−3,3) have the correct quantum numbers for the SM Higgs doublet. Therefore, we identify the (1,1) mode of these components as the SM Higgs fields in 4D spacetime.

3.2.3. Chiral Fermion Contents in 4D Spacetime

In our model, we choose the fermions as the Weyl fermions Ψ₋ belonging to the 27 representation of E₆. The 27 representation is decomposed as in (3.29) under the group reduction, (3.28). In this decomposition, we find that our choice of the background gauge field of U(1)_𝐼 is suitable for obtaining massless fermions since all such components have U(1)_𝐼 charge 1. In the fundemantal representation, the U(1)_𝐼 generator is𝑄_𝐼=16diag(−2,−2,−2,−2,1,1,1,1,4,1,1,1,1,1,1,−2,−2,−2,1,1,1,1,1,1,−2,−2,−2), (3.32) according to the decomposition equation (3.29). By identifying 𝑄=3𝑄_𝐼, we readily obtain the condition𝑄Ψ₋=12Ψ₋. (3.33) Therefore, the chiral fermions 𝜓_𝐿 in 4D spacetime have zero modes.

Next, we consider the parity assignments for the fermions with respect to the fixed points of 𝑆²/𝑍₂. The boundary conditions are given by (2.12) and (2.15). It turns out that four 27 fermion copies with different boundary conditions are needed in order to obtain an entire generation of massless SM fermions. They are denoted by Ψ^(1,2,3,4) with the following parity assignments:Ψ_±^(𝑖)𝑥,𝜋−𝜃,−𝜙=𝜉𝛾₅𝑃₁Ψ_±^(𝑖),Ψ𝑥,𝜃,𝜙_±(𝑖)𝑥,𝜋−𝜃,2𝜋−𝜙=𝜂𝛾₅𝑃₂Ψ_±^(𝑖)𝑥,𝜃,𝜙, (3.34) where 𝛾₅ is the chirality operator, and (𝜉,𝜂)=(+,+), (−,−), (+,−) and (−,+) for 𝑖=1,2,3,4, respectively. From these fermions we find that 𝜓_1,2,3,4 have the parity assignments𝟐𝟕_𝜓𝐿(1)=(1,2)(−3,−2,−2)^(−,−)+(1,2)(3,2,−2)^(+,+)+(1,2)(−3,3,1)^(−,+)+(1,1)(6,−1,1)^(−,−)+(1,1)(0,−5,1)^(+,+)+(1,1)(0,0,4)^(+,−)+(3,2)(1,−1,1)^(+,−)+(3,1)(−2,2,−2)^(−,+)+3,1(−4,−1,1)^(−,−)+3,1(2,3,1)^(−,−)+3,1(2,−2,−2)^(+,−),𝟐𝟕_𝜓𝐿(2)=(1,2)(−3,−2,−2)^(+,+)+(1,2)(3,2,−2)^(−,−)+(1,2)(−3,3,1)^(+,−)+(1,1)(6,−1,1)^(+,+)+(1,1)(0,−5,1)^(−,−)+(1,1)(0,0,4)^(−,+)+(3,2)(1,−1,1)^(−,+)+(3,1)(−2,2,−2)^(+,−)+3,1(−4,−1,1)^(+,+)+3,1(2,3,1)^(+,+)+3,1(2,−2,−2)^(−,+),𝟐𝟕_𝜓𝐿(3)=(1,2)(−3,−2,−2)^(−,+)+(1,2)(3,2,−2)^(+,−)+(1,2)(−3,3,1)^(−,−)+(1,1)(6,−1,1)^(−,+)+(1,1)(0,−5,1)^(+,−)+(1,1)(0,0,4)^(+,+)+(3,2)(1,−1,1)^(+,+)+(3,1)(−2,2,−2)^(−,−)+3,1(−4,−1,1)^(−,+)+3,1(2,3,1)^(−,+)+3,1(2,−2,−2)^(+,+),𝟐𝟕_𝜓𝐿(4)=(1,2)(−3,−2,−2)^(+,−)+(1,2)(3,2,−2)^(−,+)+(1,2)(−3,3,1)^(+,+)+(1,1)(6,−1,1)^(+,−)+(1,1)(0,−5,1)^(−,+)+(1,1)(0,0,4)^(−,−)+(3,2)(1,−1,1)^(−,−)+(3,1)(−2,2,−2)^(+,+)+3,1(−4,−1,1)^(+,−)+3,1(2,3,1)^(+,−)+3,1(2,−2,−2)^(−,−), (3.35) where the underlined components have even parities and U(1)_𝐼 charge 1. One can readily identify one generation of SM fermions, including a right-handed neutrino, as the zero modes of these components.

A long-standing problem in the gauge-Higgs unification framework is the Yukawa couplings of the Higgs boson to the matter fields. Here we discuss about the Yukawa couplings in our model. As mentioned before, the SM Higgs is the (ℓ=1, |𝑚|=1) KK mode of the extraspatial component of the gauge field, the Yukawa term at tree level has the following form: 𝐿_Yukawa⊃𝜓_𝐿⁰⁰Φ¹¹𝜓_𝑅^ℓ1+𝜓_𝐿^ℓ1Φ¹¹𝜓_𝑅⁰⁰+h.c., (3.36) where 𝜓^ℓ𝑚s are the fermionic KK modes with the (𝑙=0, 𝑚=0) modes appearing as the chiral fermions and Φ¹¹ denotes the SM Higgs field. We here identify the left-handed fermionic zero modes as SU(2) doublets and the right-handed fermionic zero modes as SU(2) singlets, as in the SM. Therefore, the (ℓ, |𝑚|=1) modes and the (ℓ=0, |𝑚|=0) modes mix after spontaneous symmetry breaking. One needs to diagonalize the mass terms to obtain physical eigenstates. The Yukawa couplings in our model are thus more complicated than other gauge-Higgs unification models in the sense that there is mixing between KK modes including the zero modes without a bulk mass term or fixed point localized term. However, similar mixing occurs in models on warped 5D spacetime or even in models with a flat metric if one takes into account the bulk mass term or fixed point localized term. In such cases, diagonalization is necessary.

The difficulty of obtaining a realistic fermion mass spectrum comes from the fact that the Yukawa couplings arise from gauge interactions. However, one can overcome the difficulty by introducing SM fermions localized at an orbifold fixed point and additional massive bulk fermions. The realistic Yukawa couplings would be obtained from nonlocal interactions of the fixed point localized fermions involving Wilson lines after integrating out the massive bulk fermions [41–43]. Another possible solution is to consider fermions in 6D spacetime belonging to a higher dimensional representation of the original E₆ gauge group, rendering more than one generation of SM fermions. In that case, mixing among generations will be obtained from gauge interactions and is given by Clebsch-Gordan coefficients. We expect that realistic Yukawa couplings could be obtained using these methods. A detailed analysis of this issue is beyond the scope of the paper and left for a future work.

3.2.4. Higgs Potential of the Model

Here we analyze the Higgs potential for the E₆ model. To further simplify the Higgs potential, we need to find out the algebra of the gauge group generators. Note that the E₆ generators are chosen according to the decomposition of the adjoint representation given in   (3.30)𝑄_𝑖,𝑄_𝛼,𝑄_𝑌,𝑄_𝑋,𝑄_𝐼,𝑄_{𝑎𝑥(−5,0,0)},𝑄^{𝑎𝑥(5,0,0)},𝑄_{𝑎𝑥(1,4,0)},𝑄^{𝑎𝑥(−1,−4,0)},𝑄_{𝑎(4,−4,0)},𝑄^{𝑎(−4,4,0)},𝑄_{(−6,−4,0)},𝑄_(6,4,0),𝑄_{𝑎𝑥(1,−1,−3)},𝑄^{𝑎𝑥(−1,1,3)},𝑄_𝑎(4,1,3),𝑄^{𝑎(−4,−1,−3)},𝑄_{𝑎(−2,−3,3)},𝑄^{𝑎(2,3,−3)},𝑄_{𝑥(3,−3,3)},𝑄^{𝑥(−3,3,−3)},𝑄_(−6,1,3),𝑄_{(6,−1,−3)},𝑄_{(0,−5,−3)},𝑄_(0,5,3), (3.37) where the generators are listed in the corresponding order of the terms in (3.30) and the indices𝑖=1,…,8∶SU(3)adjrepindex⟹𝑄_𝑖∶SU(3)generators,𝛼=1,2,3∶SU(2)adjrepindex⟹𝑄_𝛼∶SU(2)generators,𝑄_{𝑋,𝑌,𝐼}∶U(1)_{𝑋,𝑌,𝐼}generators,𝑥=1,2∶SU(2)doubletindex,𝑎=1,2,3∶SU(3)colorindex. (3.38)(3.39)(3.40)(3.41)(3.42) Here we take the normalization for generators, Tr[𝑄𝑄^†]=2 which is taken from [24]. The Higgs fields are in the representations of (1,2)(3,−3,3) and (1,2)(−3,3,−3). We writeΦ(𝑥)=𝜙^𝑥𝑄_{𝑥(3,−3,3)}Φ^†(𝑥)=𝜙_𝑥𝑄^{𝑥(−3,3,−3)}. (3.43) Likewise, the gauge field 𝐴_𝜇(𝑥) in terms of the 𝑄’s in (3.38) is 𝐴_𝜇(𝑥)=𝐴^𝑖_𝜇𝑄_𝑖+𝐴^𝛼_𝜇𝑄_𝛼+𝐵_𝜇𝑄_𝑌+𝐶_𝜇𝑄_𝑋+𝐸_𝜇𝑄_𝐼. (3.44) The commutation relations between the generators 𝑄_𝛼, 𝑄_{𝑋,𝑌,𝐼}, 𝑄_{𝑥(3,−3,3)}, and 𝑄^{𝑥(−3,3,−3)} are summarized in Table 2.

Table 2 

Commutation relations of 𝑄_𝛼, 𝑄_{𝑋,𝑌,𝐼}, 𝑄_{𝑥(3,−3,3)} and 𝑄^{𝑥(−3,3,−3)}, where 𝜎_𝑖 are the Pauli matrices.

Finally, we obtain the Lagrangian associated with the Higgs field by applying (3.43) and (3.44) to (2.54) and (2.55) and carrying out the trace. Furthermore, to obtain the canonical form of kinetic terms, the Higgs field, the gauge field, and the gauge coupling need to be rescaled in the following way:𝑔𝜙⟶√2𝐴𝜙,_𝜇⟶𝑔𝑅𝐴_𝜇,𝑔√6𝜋𝑅²=𝑔₂, (3.45) where 𝑔₂ denotes the SU(2) gauge coupling. The Higgs sector is then given byℒ_Higgs=𝐷_𝜇𝜙²−𝑉𝜙, (3.46) where𝐷_𝜇𝜙=𝜕_𝜇+𝑖𝑔₂𝜎_𝛼2𝐴_𝛼𝜇1+𝑖𝑔√40𝜋𝑅²𝐵_𝜇1−𝑖𝑔2320𝜋𝑅²𝐶_𝜇1+𝑖𝑔2√4𝜋𝑅²𝐸_𝜇𝜙,𝜒𝑉=−8𝑅²𝜙^†𝜙+3𝑔²40𝜋𝑅²𝜙^†𝜙², (3.47)(3.48) where 𝜒=7+9𝜇₁+9𝜇₂. The numerical values 𝜇_1,2 are given by 𝜇₁=1−(3/2)ln2 and 𝜇₂=(3/4)(1−2ln2) as in Section 2.3.4. We have omitted the constant term in the Higgs potential. Comparing the potential derived above with the standard form 𝜇²𝜙^†𝜙+𝜆(𝜙^†𝜙)² in the SM, we see that the model has a tree-level 𝜇² term that is negative and proportional to 𝑅⁻². The negative contribution to the squared mass term comes from the interaction between background gauge field and 𝜙 as seen in Section 2.3.4. Moreover, the quartic coupling 𝜆=3𝑔²/(40𝜋𝑅²) is related to the 6D gauge coupling 𝑔 and grants perturbative calculations because it is about 0.16, using the value of 𝑅 to be extracted in the next section. Therefore, the order parameter in this model is controlled by a single parameter 𝑅, the compactification scale.

In fact, the (1,1) mode of the {(3,2)(1,−1,−3)+h.c.} representation also has a negative squared mass term because it has the same 𝑄_𝐼 charge as the {(1,2)(3,−3,3)+h.c.} representation. Therefore, it would induce not only electroweak symmetry breaking but also color symmetry breaking. This undesirable feature can be cured by adding brane terms𝛼𝑅²sin²𝜃𝐹^𝑎_𝜃𝜙𝐹^𝑎𝜃𝜙𝛿𝜋𝜃−2𝛿𝜙+𝛿𝜙−𝜋, (3.49) where 𝑎 denotes the group index of the {(3,2)(1,−1,−3)+h.c.} representation. These brane terms preserve the 𝑍^′₂ symmetry which corresponds to the symmetry under the transformation (𝜙→𝜙+𝜋). With an appropriate choice of the dimensionless constant 𝛼, the squared mass of the (1,1) can be lifted to become positive and sufficiently large. We need to forbid a similar brane term for the SU(2) doublet component, and it can be achieved by imposing some additional discrete symmetry. However, here we simply assume that such a brane term for the SU(2) doublet component does not exist.

Due to a negative mass term, the Higgs potential in (3.48) can induce the spontaneous symmetry breakdown: SU(2)×U(1)_𝑌→U(1)_𝐸𝑀 in the SM. The Higgs field acquires a vacuum expectation value (VEV):𝜙=1√20𝑣with𝑣=5𝜋𝜒31𝑔≃4.6𝑔. (3.50) One immediately finds that the 𝑊 boson mass:𝑚_𝑊=𝑔₂21𝑣=65𝜒21𝑅≃0.53𝑅, (3.51) from which the compactification scale 𝑅⁻¹≃152 GeV is inferred. Moreover, the Higgs boson mass at the tree level is𝑚_𝐻=320𝜋𝑔𝑣𝑅=325𝑚_𝑊=√𝜒21𝑅, (3.52) which is about 152 GeV, numerically very close to the compactification scale. Since the hypercharge of the Higgs field is 1/2, the U(1)_𝑌 gauge coupling is derived from (3.47) as𝑔_𝑌=𝑔√10𝜋𝑅². (3.53) The Weinberg angle is thus given bysin²𝜃_𝑊=𝑔²_𝑌𝑔²₂+𝑔²_𝑌=38, (3.54) and the 𝑍 boson mass𝑚_𝑍=𝑚_𝑊cos𝜃_𝑊=𝑚_𝑊85, (3.55) both at the tree level. These relations are the same as the SU(5) GUT at the unification scale. This is not surprising because this part only depends on the group structure. Again, this Weinberg angle is not consistent with experimental measurements, and we need to take into account quantum corrections.

We can repeat the discussion in Section 3.1.2 about the one-loop power divergence in the Higgs potential associated with the linear operator 𝐹_𝑎𝑏. The operator 𝐹^𝛼_𝜃𝜙 transform to −𝐹^𝛼_𝜃𝜙 under the parity transformation 𝜃→𝜋−𝜃. Hence, this operator is forbidden by parity invariance of the action. In this case, we check the consistency between the orbifold boundary conditions on 𝑆²/𝑍₂, (2.13)–(2.15), and the parity conditions, (3.26). By performing the parity transformation on both sides of the orbifold boundary conditions, (2.13)–(2.15), we obtain 𝐴_𝜇𝑥,𝜃,−𝜙=𝑃₁𝐴_𝜇𝑥,𝜋−𝜃,𝜙𝑃₁,−𝐴_𝜃𝑥,𝜃,−𝜙=𝑃₁𝐴_𝜃𝑥,𝜋−𝜃,𝜙𝑃₁,𝐴_𝜙𝑥,𝜃,−𝜙=−𝑃₁𝐴_𝜙𝑥,𝜋−𝜃,𝜙𝑃₁,±Γ⁴Ψ𝑥,𝜃,−𝜙=±𝛾₅𝑃₁±Γ⁴Ψ,𝐴𝑥,𝜋−𝜃,𝜙_𝜇𝑥,𝜃,2𝜋−𝜙=𝑃₂𝐴_𝜇𝑥,𝜋−𝜃,𝜙𝑃₂,−𝐴_𝜃𝑥,𝜃,2𝜋−𝜙=𝑃₂𝐴_𝜃𝑥,𝜋−𝜃,𝜙𝑃₂,𝐴_𝜙𝑥,𝜃,2𝜋−𝜙=−𝑃₂𝐴_𝜙𝑥,𝜋−𝜃,𝜙𝑃₂,±Γ⁴Ψ𝑥,𝜃,2𝜋−𝜙=±𝛾₅𝑃₂±Γ⁴Ψ𝑥,𝜋−𝜃,𝜙. (3.56) Since (2.13)–(2.15) hold for any 𝜃 and 𝜙 and Γ⁴ commutes with 𝛾₅, we find that the orbifold boundary conditions still hold under the parity transformation with the identification of 𝜃=𝜋−𝜃′. In other words, the orbifold boundary conditions, (2.13)–(2.15), are parity invariant.

3.2.5. KK Mode Spectrum of Each Field

Since we did not impose symmetry condition, we have KK modes for each field in this model. Here we show KK mass spectrum under the existence of background field for our E₆ model. The masses are basically controlled by the compactification radius 𝑅 of the two spheres. They receive two kinds of contributions: one arising from the angular momentum in the 𝑆² space and the other coming from the interactions with the background field.

The KK masses for fermions have been given in [35, 36, 38]. We give them in terms of our notation here:𝑀^KK_ℓ𝑚𝜓_𝐿=1𝑅ℓ(ℓ+1)−4𝑞²−14, (3.57) where 𝑞 is proportional to the U(1)_𝐼 charge of a fermion and determined by the action of 𝑄=3𝑄_𝐼 on fermions as 𝑄Ψ=𝑞Ψ=3𝑞_𝐼Ψ. Note that the mass does not depend on the quantum number 𝑚. The lightest KK mass, corresponding to ℓ=1 and 𝑞_𝐼=1/6, is about 214 GeV at the tree level. The range of ℓ is2𝑞±12≤ℓ+∶for𝜓_𝑅(𝐿)inΨ₊₍₋₎,−∶for𝜓_𝐿(𝑅)inΨ₋₍₊₎. (3.58) We thus can have zero mode for 𝑄Ψ=±(1/2)Ψ, where this condition is given in (2.43).

For the 4D gauge field 𝐴_𝜇, its kinetic term, and KK mass term are obtained from the terms: −1𝐿=𝑑ΩTr4𝐹_𝜇𝜈+12𝑅²𝐹_𝜇𝜃𝐹^𝜇_𝜃+12𝑅²sin²𝜃𝐹_𝜇𝜙𝐹^𝜇_𝜙. (3.59) Taking terms quadratic in 𝐴_𝜇, we get 𝐿_quad=−1𝑑ΩTr4𝜕_𝜇𝐴_𝜈−𝜕_𝜈𝐴_𝜇𝜕^𝜇𝐴^𝜈−𝜕^𝜈𝐴^𝜇+12𝑅²𝜕_𝜃𝐴_𝜇𝜕_𝜃𝐴^𝜇+12𝑅²sin²𝜃𝜕_𝜙𝐴_𝜇𝜕_𝜙𝐴^𝜇−12𝑅²𝐴_𝜇,𝐴^𝐵_𝜙𝐴^𝜇,𝐴^𝐵_𝜙, (3.60) where 𝐴^𝐵_𝜙 is the background gauge field given in (2.35). The KK expansion of 𝐴_𝜇 is 𝐴_𝜇=_ℓ𝑚𝐴_𝜇^ℓ𝑚(𝑥)𝑌^±_ℓ𝑚𝜃,𝜙, (3.61) where 𝑌^±_ℓ𝑚(𝜃,𝜙) are the linear combinations of spherical harmonics satisfying the boundary condition 𝑌^±_ℓ𝑚(𝜋−𝜃,−𝜙)=±𝑌^±_ℓ𝑚(𝜃,𝜙). Their explicit forms are [35] 𝑌⁺_ℓ𝑚≡(𝑖)𝜃,𝜙^ℓ+𝑚√2𝑌_ℓ𝑚𝜃,𝜙+(−1)^ℓ𝑌_ℓ−𝑚𝜃,𝜙for𝑌𝑚≠0,_−ℓ𝑚≡(𝑖)𝜃,𝜙^ℓ+𝑚+1√2𝑌_ℓ𝑚𝜃,𝜙−(−1)^ℓ𝑌_ℓ−𝑚𝜃,𝜙for𝑌𝑚≠0,_+(−)ℓ0(𝜃)≡𝑌_ℓ0(𝜃)for𝑚=0,ℓ=even(odd)0for𝑚=0,ℓ=odd(even). (3.62) Note that we do not have KK mode functions that are odd under 𝜙→𝜙+2𝜋 since the KK modes are specified by the integer angular momentum quantum numbers ℓ and 𝑚 of gauge field 𝐴_𝑀 on the two spheres. Thus, the components of 𝐴_𝜇 and 𝐴_𝜃,𝜙 with (+,−) or (−,+) parities do not have corresponding KK modes. Applying the KK expansion and integrating about 𝑑Ω, we obtain the kinetic and KK mass terms for the KK modes of 𝐴_𝜇𝐿_𝑀1=−2𝜕_𝜇𝐴_𝜈^ℓ𝑚(𝑥)−𝜕_𝜈𝐴_𝜇^ℓ𝑚(𝑥)𝜕^𝜇𝐴^ℓ𝑚𝜈(𝑥)−𝜕^𝜈𝐴^ℓ𝑚𝜇(𝑥+ℓ())ℓ+1𝑅²𝐴_𝜇^ℓ𝑚(𝑥)𝐴^ℓ𝑚𝜇(𝑥)+9𝑞²_𝐼𝑅²()𝑑Ωcos𝜃±1²sin²𝜃𝑌^∓_ℓ𝑚²𝐴_𝜇^ℓ𝑚(𝑥)𝐴^ℓ𝑚𝜇(𝑥), (3.63) where we have used Tr[𝑄_𝑖𝑄^𝑖]=2 and [𝐴_𝜇(𝑥),𝑄_𝐼]=𝑞_𝐼(𝐴^𝑖_𝜇(𝑥)𝑄_𝑖−𝐴_𝑖𝜇(𝑥)𝑄^𝑖). Therefore, the KK masses of 𝐴_𝜇 are 𝑀^KK_ℓ𝑚𝐴_𝜇=1𝑅ℓ(ℓ+1)+𝑚^𝐵_ℓ𝑚²,𝑚^𝐵_ℓ𝑚²=9𝑞²_𝐼()𝑑Ωcos𝜃±1²sin²𝜃𝑌^∓_ℓ𝑚², (3.64)(3.65) where 𝑚^𝐵_ℓ𝑚 corresponds to the contribution from the background gauge field. Note that (3.64) agrees with (2.41) when ℓ=0. Also, since the SM gauge bosons have 𝑞_𝐼=0, their KK masses are simply ℓ(ℓ+1)/𝑅 at the tree level.

The kinetic and KK mass terms of 𝐴_𝜃 and 𝐴_𝜙 are obtained from the terms in the higher dimensional gauge sector 1𝐿=2𝑔²𝑑ΩTr𝜕_𝜇𝐴_𝜃−𝑖𝐴_𝜇,𝐴_𝜃²+Tr𝜕_𝜇𝐴_𝜙−𝑖𝐴_𝜇,𝐴_𝜙²−1𝑅²1Tr𝜕sin𝜃^𝜃sin𝜃𝐴_𝜙+sin𝜃𝐴^𝐵_𝜙−1𝜕sin𝜃^𝜙𝐴_𝜃−𝑖𝐴_𝜃,𝐴_𝜙+𝐴^𝐵_𝜙². (3.66) The first line on the right-hand side of (3.66) corresponds to the kinetic terms, and the second line corresponds to the potential term. Applying the background gauge field (2.35), the potential becomes 𝐿_𝑉1=−2𝑔²𝑅²1𝑑ΩTr𝜕sin𝜃^𝜃sin𝜃𝐴_𝜙1+𝑄−𝜕sin𝜃^𝜙𝐴_𝜃−𝑖𝐴_𝜃,𝐴_𝜙+𝐴^𝐵_𝜙². (3.67) For 𝐴_𝜃 and 𝐴_𝜙, we use the following KK expansions to obtain the KK mass terms, 𝐴_𝜃𝑥,𝜃,𝜙=∑_{()ℓ𝑚≠0}−1ℓ()Φℓ+1^1ℓ𝑚(𝑥)𝜕_𝜃𝑌^±_ℓ𝑚𝜃,𝜙+Φ₂^ℓ𝑚(𝑥)1𝜕sin𝜃^𝜙𝑌^±_ℓ𝑚,𝐴𝜃,𝜙_𝜙𝑥,𝜃,𝜙=∑_{()ℓ𝑚≠0}1ℓ()Φℓ+1^2ℓ𝑚(𝑥)𝜕_𝜃𝑌^±_ℓ𝑚𝜃,𝜙−Φ₁^ℓ𝑚(𝑥)1𝜕sin𝜃^𝜙𝑌^±_ℓ𝑚𝜃,𝜙, (3.68) where the factor of 1/ℓ(ℓ+1) is needed for normalization. These particular forms are convenient in giving diagonalized KK mass terms [35]. Applying the KK expansions equations (3.68), we obtain the kinetic term 𝐿_𝐾=12𝑔²_{()ℓ𝑚≠0}Tr𝜕_𝜇Φ₁^ℓ𝑚(𝑥)𝜕^𝜇Φ₁^ℓ𝑚(𝑥)+𝜕_𝜇Φ₂^ℓ𝑚(𝑥)𝜕^𝜇Φ₂^ℓ𝑚(𝑥), (3.69) where only terms quadratic in 𝜕_𝜇Φ are retained. The potential term is 𝐿_𝑉1=−2𝑔²𝑅²_{{)ℓ𝑚}(≠0}Φ𝑑ΩTr^2ℓ𝑚ℓ()1ℓ+1𝜕sin𝜃^𝜃sin𝜃𝜕_𝜃𝑌^±_ℓ𝑚Φ+𝑄+^2ℓ𝑚ℓ()1ℓ+1sin²𝜃𝜕²_𝜙𝑌^±_ℓ𝑚−𝑖ℓ(ℓ+1)ℓ^′ℓ^′+1−Φ₁^ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚−Φ₂^ℓ𝑚1𝜕sin𝜃^𝜙𝑌^±_ℓ𝑚,Φ₂^{ℓ′𝑚′}𝜕_𝜃𝑌^±_{ℓ′𝑚′}−Φ₁^{ℓ′𝑚′}1𝜕sin𝜃^𝜙𝑌^±_{ℓ′𝑚′}+ℓ^′ℓ^′+1𝐴^𝐵_𝜙². (3.70) Note that these terms are not diagonal in (ℓ,𝑚) in general. Using the relation (1/sin𝜃)𝜕_𝜃(sin𝜃𝜕_𝜃𝑌_ℓ𝑚)+(1/sin²𝜃)𝜕²_𝜙𝑌_ℓ𝑚=−ℓ(ℓ+1)𝑌_ℓ𝑚, the potential term is simplified as 𝐿_𝑉1=−2𝑔²𝑅²×_{()ℓ𝑚≠0}𝑑ΩTr−ℓ(ℓ+1)Φ₂^ℓ𝑚𝑌^±_ℓ𝑚𝑖+𝑄+ℓ(ℓ+1)ℓ^′ℓ^′Φ+1^1ℓ𝑚,Φ^{ℓ′𝑚′}₂×𝜕_𝜃𝑌^±_ℓ𝑚𝜕_𝜃𝑌^±_{ℓ′𝑚′}+1sin²𝜃𝜕_𝜙𝑌^±_ℓ𝑚𝜕_𝜙𝑌^±_{ℓ′𝑚′}+𝑖ℓ()Φℓ+1^1ℓ𝑚,𝐴^𝐵_𝜙𝜕_𝜃𝑌^±_ℓ𝑚+𝑖ℓ()Φℓ+1^2ℓ𝑚,𝐴^𝐵_𝜙𝜕_𝜙𝑌^±_ℓ𝑚sin𝜃². (3.71) To obtain the mass term, we focus on terms quadratic in Φ_1,2: 𝐿_𝑀1=−2𝑔²𝑅²𝑑ΩTrℓ(ℓ+1)Φ₂^ℓ𝑚2𝑌^±_ℓ𝑚²+2𝑖𝑄ℓ()Φℓ+1^1ℓ𝑚,Φ₂^ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚+1sin²𝜃𝜕_𝜙𝑌^±_ℓ𝑚𝜕_𝜙𝑌^±_ℓ𝑚+2𝑖𝐴^𝐵_𝜙Φ₁^ℓ𝑚,Φ₂^ℓ𝑚𝑌^±_ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚−1ℓ()Φℓ+1^1ℓ𝑚,𝐴^𝐵_𝜙²𝜕_𝜃𝑌^±_ℓ𝑚²−1ℓ()Φℓ+1^2ℓ𝑚,𝐴^𝐵_𝜙²𝜕_𝜙𝑌^±_ℓ𝑚²sin²𝜃. (3.72) Here we take terms which are diagonal in (ℓ,𝑚) for simplicity. Note that we have dropped the term proportional to [Φ₁,𝐴^𝐵_𝜙][Φ₂,𝐴^𝐵_𝜙] because this term vanishes after turning the field into the linear combinations of Φ and Φ^†, (2.51) and (2.52):TrΦ₁,𝐴^𝐵_𝜙Φ₁,𝐴^𝐵_𝜙⟶TrΦ+Φ^†,𝑄Φ−Φ^†,𝑄∝TrΦ−Φ^†Φ+Φ^†∝TrΦΦ^†−TrΦ^†Φ=0. (3.73) Integrating the second term of (3.72) by part, we obtain𝐿_𝑀1=−2𝑔²𝑅²ℓ(ℓ+1)TrΦ₂^ℓ𝑚2+2𝑖Tr𝑄Φ₁^ℓ𝑚,Φ₂^ℓ𝑚−2𝑖Tr𝑄Φ₁^ℓ𝑚,Φ₂^ℓ𝑚𝑑Ωcos𝜃∓1𝑌sin𝜃^±ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚−1ℓ()Φℓ+1^1ℓ𝑚,𝑄²()𝑑Ωcos𝜃∓1²sin²𝜃𝜕_𝜃𝑌^±_ℓ𝑚²−1ℓ()Φℓ+1^2ℓ𝑚,𝑄²()𝑑Ωcos𝜃∓1sin²𝜃𝜕_𝜙𝑌^±_ℓ𝑚²sin²𝜃. (3.74) Therefore, the KK masses depend on the U(1)_𝐼 charges of the scalar fields. Note that terms in the second line to the last line of (3.74) are not diagonal in (ℓ,𝑚) in general.

For components with zero U(1)_𝐼 charge, we write Φ₁₍₂₎(𝑥) as 𝜙₁₍₂₎(𝑥)𝑄 where 𝑄 is the corresponding generator of E₆ in (3.30) with zero U(1)_𝐼 charge. Taking the trace, we have the following kinetic and KK mass terms instead: 𝐿=_{()ℓ𝑚≠0}𝜕_𝜇𝜙₁^ℓ𝑚(𝑥)𝜕^𝜇𝜙₁^ℓ𝑚(𝑥)+𝜕_𝜇𝜙₂^ℓ𝑚(𝑥)𝜕^𝜇𝜙₂^ℓ𝑚(𝑥)+ℓ(ℓ+1)𝜙₂^ℓ𝑚(𝑥)𝜙₂^ℓ𝑚(𝑥), (3.75) where we have made the substitution 𝜙_𝑖→𝑔𝜙_𝑖. Note that 𝜙₁ is considered as a massless Nambu-Goldstone (NG) boson in this case. For components with nonzero U(1)_𝐼 charge, mass terms are not diagonal for Φ₁₍₂₎, and Φ₁ does not correspond to the NG boson. In this case, we need to diagonalize the mass terms and some linear combination of Φ₁₍₂₎ becomes the NG boson mode.

For components with nonzero U(1)_𝐼 charge, we use (2.51) and (2.52) and write Φ(𝑥) as 𝜙^𝑖(𝑥)𝑄_𝑖 where 𝑄_𝑖 is the corresponding generator of E₆ in (3.30) with nonzero U(1)_𝐼 charge. The commutator between 𝑄 and Φ is[𝑄,Φ]=3𝑄_𝐼,𝑄_𝑖𝜙^𝑖=3𝑞_𝐼𝜙^𝑖, (3.76) where we have used 𝑄=3𝑄_𝐼 as required to obtain chiral fermions in Section 3.2.3, and that 𝑞_𝐼 is a constant determined by the U(1)_𝐼 charge of the corresponding component. Finally, the Lagrangian becomes𝐿=_{()ℓ𝑚≠0}𝜕_𝜇𝜙^†_ℓ𝑚𝜕^𝜇𝜙_ℓ𝑚−14𝑅²2ℓ(ℓ+1)𝜙^†_ℓ𝑚𝜙_ℓ𝑚−12𝑞_𝐼𝜙^†_ℓ𝑚𝜙_ℓ𝑚+12𝑞_𝐼𝜙^†_ℓ𝑚𝜙_ℓ𝑚𝑑Ωcos𝜃∓1𝑌sin𝜃^±ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚+18𝑞²_𝐼ℓ()𝜙ℓ+1^†ℓ𝑚𝜙_ℓ𝑚()𝑑Ωcos𝜃∓1²sin²𝜃𝜕_𝜃𝑌^±_ℓ𝑚²+𝜕_𝜙𝑌^±_ℓ𝑚²sin²𝜃, (3.77) where the subscript 𝑖 is omitted for simplicity. The KK masses of the complex scalar field 𝜙 are then 𝑀^KK_ℓ𝑚𝜙=1𝑅ℓ(ℓ+1)2+𝑚^𝐵_ℓ𝑚²,𝑚^𝐵_ℓ𝑚²=−3𝑞_𝐼+3𝑞_𝐼𝑑Ωcos𝜃∓1𝑌sin𝜃^±ℓ𝑚𝜕_𝜃𝑌^±_ℓ𝑚+9𝑞²_𝐼()()2ℓℓ+1𝑑Ωcos𝜃∓1²sin²𝜃𝜕_𝜃𝑌^±_ℓ𝑚²+9𝑞²_𝐼()()2ℓℓ+1𝑑Ωcos𝜃∓1²sin²𝜃𝜕_𝜙𝑌^±_ℓ𝑚²sin²𝜃. (3.78) The squared KK mass (𝑀^KK_ℓ𝑚)² is always positive except for the lowest mode (ℓ=1,𝑚=1). In fact, the squared KK mass of the (1,1) mode agrees with the coefficient of quadratic term in the Higgs potential (3.48).

4. Summary and Discussions

We have reviewed a gauge theory defined on 6D spacetime with the 𝑆²/𝑍₂ topology on the extra space. Two scenarios are considered to construct a 4D theory from the 6D model. One scenario based on the SO(12) gauge group requires a symmetry condition for the gauge field. The other involves the E₆ gauge group, but does not need the symmetry condition. Nontrivial boundary conditions on the extra space are imposed in both scenarios.

We explicitly give the prescriptions to identify the gauge field and the scalar field remaining in 4D spacetime after the dimensional reduction. We show that the SU(3)_𝐶×SU(2)_𝐿×U(1)_𝑌×U(1)_𝑋×U(1)_𝐼 gauge symmetry remains in 4D spacetime, and that the SM Higgs doublet with a suitable potential for electroweak symmetry breaking can be derived from the gauge sector in both models. The Higgs boson mass is also predicted in such models. Our tree-level prediction of the Higgs boson mass is 196 GeV for the SO(12) model and 152 GeV for the E₆ model. These mass values are in the range of 127–600 GeV already excluded at 95% CL by recent LHC data [44, 45]. However, the mass value will become different once quantum corrections to the Higgs potential are taken into account. We expect that the Higgs boson mass in our model will become smaller than the lower limit of the exclusion region by quantum corrections. In particular, the E₆ case gives a 152 GeV Higgs boson mass at tree level that is not far from the lower limit of the exclusion region at 95%CL. However, a full analysis of the quantum corrections is beyond the scope of this paper and left as a future work. Massless fermion modes are also successfully obtained as the SM fermions by introducing appropriate field contents in 6D spacetime, with suitable parity assignments on the 𝑆²/𝑍₂ extra dimension and incorporating the background gauge field. We also discuss about the massive KK modes of fermions for the scenario with the symmetry condition and the KK modes of all fields for the one without the symmetry condition. The lightest fermonic KK mode can serve as a dark matter candidate. In general, they may give rise to rich phenomena in collider experiments and implications in cosmological studies.

To make our models more realistic, there are several challenges such as eliminating the extra U(1) symmetries and constructing the realistic Yukawa couplings, which are the same as other gauge-Higgs unification models. We, however, can get Kaluza-Klein modes in our models. This suggests that we obtain the dark matter candidate in our model. Thus, it is very important to study these models further such as dark matter physics and collider physics.

Acknowledgments

This research was supported in part by the Grant-in-Aid for the Ministry of Education, Culture, Sports, Science, and Technology, Government of Japan, No. 20540251 (J. Sato), and the National Science Council of R.O.C. under Grant No. NSC-100-2628-M-008-003-MY4.