The Standard Model Gauge Symmetry from Higher-Rank Unified Groups in Grand Gauge-Higgs Unification Models

We study grand unified models in the five-dimensional space-time where the extra dimension is compactified on $S^1/Z_2$. The spontaneous breaking of unified gauge symmetries is achieved via vacuum expectation values of the extra-dimensional components of gauge fields. We derive one-loop effective potentials for the zero modes of the gauge fields in SU(7), SU(8), SO(10), and $E_6$ models. In each model, the rank of the residual gauge symmetry that respects the boundary condition imposed at the orbifold fixed points is higher than that of the standard model. We verify that the residual symmetry is broken to the standard model gauge symmetry at the global minima of the effective potential for certain sets of bulk fermion fields in each model.


Introduction
For the past several decades, grand unification of the standard model gauge symmetry at a highenergy regime has been considered to be an attractive idea as physics beyond the standard model, since the unification helps us to understand unrevealed features involved in the standard model such as the charge quantization and the anomaly cancellation. In addition to the minimal grand unified theory (GUT) based on SU (5) [1], there are well known GUT models based on, for instance, SU (4) × SU (2) L × SU (2) R [2], SO(10) [3], and E 6 [4]. A common feature shared among various GUT models is that some symmetry breaking mechanism is required to obtain the standard model gauge symmetry G SM = SU (3) C × SU (2) L × U (1) Y at a low-energy regime. A standard prescription for the symmetry breaking in GUT models is to involve elementary Higgs scalars that develop vacuum expectation values (VEVs) and the VEVs lead to desired breaking patterns of the unified symmetries. This mechanism is an analogous to the electroweak symmetry breaking by the Higgs scalar in the standard model.
Besides the Higgs mechanism, if compactified extra dimensions are concealed in our Universe, another way of the spontaneous symmetry breaking becomes possible, namely the Hosotani mechanism [5][6][7]. In models with the Hosotani mechanism, the extra-dimensional components of the gauge fields effectively behave as "Higgs" scalars at low energy, and dynamics of the gauge fields reflects degrees of freedom of Wilson line phases. Although gauge invariance forbids the tree-level potential for the phases at the classical level, non-trivial VEVs of the phases are naturally emerged and spontaneous symmetry breaking is achieved when quantum corrections are involved [7]. Advantages of the Hosotani mechanism are predictivity and finiteness of the "Higgs" potential and masses [8], even though the potential arises from loop corrections. Hence, as a solution to the hierarchy problem in the standard model, the gauge-Higgs unification models has been widely investigated [9,10]. In these models, the zero modes of the extra-dimensional component of gauge fields are identified to the Higgs doublet in the standard model.
Recently, we have been focusing on application of the Hosotani mechanism to unified gauge symmetry breaking on an orbifold compactification S 1 /Z 2 , which enable us to incorporate chiral fermions in five-dimensional models [11,12]. In this case, the zero mode of the extra-dimensional gauge field, which have even parities under a boundary condition defined at the boundaries of the orbifold, plays a role of the Higgs field whose VEV breaks unified gauge symmetries into the standard model one. We refer to this scenario as grand gauge-Higgs unification (gGHU). Note that gGHU is different from the orbifold GUT models where boundary conditions directly break GUT symmetry into the standard model gauge symmetry [13].
In models with S 1 /Z 2 , the orbifold parities of the extra-dimensional component of the gauge field is opposite to those of the four-dimensional vector counterpart. Consequently, massless zero modes appearing from the extra-dimensional components tend not to belong to the adjoint representation of unbroken symmetries, though adjoint Higgs fields are often utilized in ordinary four-dimensional GUT models [1]. This situation leads to severe constraints on construction of gGHU models. We have shown that the difficulty is evaded and the adjoint "Higgs" field in the gGHU model is obtained [11] with the diagonal embedding method [14], which is known in the context of string theory. The doublettriplet splitting problem and several phenomenological aspects have been studied in the supersymmetric version of the model [12].
In this work, we focus on another way to construct phenomenologically viable gGHU models where "Higgs" fields originated from the extra-dimensional gauge field transform as non-adjoint representations of unbroken symmetries. In this case, spontaneous breaking of unified symmetries triggered by the "Higgs" fields generally involves rank reductions. As concrete examples, we examine the four models based on the unified symmetries SU (7), SU (8), SO(10), and E 6 . In each model, we derive the one-loop effective potential, which depends on the matter content, for the zero mode of the gauge field. In these models, accordingly, vacuum structure of the potential and symmetry breaking pattern are determined by bulk field contents. We show that the standard model gauge symmetry is achieved at a low-energy regime for certain sets of bulk fermion fields in each model. The paper is organized as follows. In Sec. 2, we give a general setup of five-dimensional gauge theories with a compactified dimension on S 1 /Z 2 . A calculation method of the one-loop effective potential for Wilson line phases is briefly summarized. In Sec. 3, as illustrative examples of gGHU, we discuss three models where each unified symmetry is SU (7), SU (8), or SO (10). The one-loop effective potential in each model is derived. In Sec. 4, the E 6 gGHU model is studied and the one-loop effective potential is examined. Vacuum structure of each effective potential is studied in Sec. 5. We finally summarize our discussions in Sec. 6. The appendices are devoted to show detailed calculations required for the discussion in Sec. 4.

General setup
We consider five-dimensional gauge theories on M 4 × S 1 /Z 2 , where one of the spatial dimensions is compactified on the orbifold S 1 /Z 2 and M 4 is the four-dimensional Minkowski space-time. On the compact space S 1 , which has the radius R, the fifth-dimensional coordinate denoted by y is identified with y + 2πR by the translation. On the orbifold S 1 /Z 2 , in addition to the translation, there is another identification y ∼ −y, which is induced by the orbifold parity transformation. In the S 1 /Z 2 orbifold theories, the combination of the translation and the parity defines another orbifold parity transformation that leads to the identification (πR + y) ∼ (πR − y). There exists the gauge field A M (x, y) that has the four-dimensional part A µ (µ = 0, 1,2,3) and the extra-dimensional part A 5 . The gauge field is also denoted by A M = A a M T a , where T a is the generator of the gauge symmetry G of the theory. The above two orbifold parity transformations act on the gauge field in such a way that the Lagrangian is invariant. Let us define the parity operatorsP 0,1 around y = 0 and around y = πR where P 0 and P 1 are matrices in a representation space of G. The orbifold parities and the matrices are called boundary conditions. Note that the translation from y to y + 2πR is induced by the operator P 1P0 , whose eigenvalue corresponds to the periodicity, for each field.
One can introduce Dirac fermions ψ(x, y); they obey the parity transformations as (2.6) where the matrix T ψ [P 0,1 ] acts on the field in its representation space and corresponds to P 0,1 in Eqs. (2.1)- (2.4). The parameters η 0 and η 1 can be chosen as 1 or −1 for each fermion. We also use the notationη = η 0 η 1 , which is related to the periodicity of each field.
At an energy regime below 1/R, the theories are well described by four-dimensional effective pictures where A µ and A 5 behave as a four-dimensional vector field and a scalar field, respectively. Once the boundary condition in Eqs. (2.1)-(2.4) is specified, we readily see that there exist sets of the generators respectively. The corresponding component fields A α µ and Aα 5 have even parities of bothP 0 andP 1 . Hence, the Kaluza-Klein (KK) decompositions of A α µ and Aα 5 have massless zero modes. The set of generators {T α } corresponds to a subgroup of G, which we call the residual gauge symmetry H. The massless "scalar" zero modes of Aα 5 parametrize Wilson line phase degrees of freedom and can develop non-trivial VEVs Aα 5 . As a result, the residual gauge symmetry H is further broken by the VEVs.
The gauge symmetry forbids tree-level potential for the zero mode of A 5 . Therefore, a minimum of the potential, namely vacuum structure of the theory, is determined by quantum effects. The effective potential can be derived by the functional integral over fields in the theory. One can treat Aα 5 as classical backgrounds and substitute Aα 5 → Aα 5 + Aα 5 in the Lagrangian. Then quadratic terms of the gauge field in the Lagrangian are written as where we adopt the Feynman gauge and use η MN = diag(1, −1, −1, −1, −1) and = ∂ µ ∂ µ . We denote the five-dimensional gauge coupling constant by g and the generators in the adjoint representation by ad(Tα) ab . The functional integral over the fluctuations A a M leads to contributions to the effective potential for Aα 5 .
Notice that the contributions to the effective potential are determined by eigenvalues of the generators {Tα}, which are accompanied by the zero modes Aα 5 , in the covariant derivative D (0) y . The generators are regarded as the charge operators of U (1) subgroups of G. Thus once the U (1) charges of the fields in the functional integral are known, then the contributions to the effective potential are obtained. In addition, each U (1) generator is also regarded as the Cartan generator of an SU (2) subgroup of G. Therefore, we can easily understand the U (1) charges of the fields from the spin eigenvalues of the SU (2) subgroups accompanied by the zero modes Aα 5 [15]. We use this procedure for deriving the contributions to the effective potential.
In the following sections, we study several gGHU models that lead to the standard model gauge symmetry G SM at a low-energy regime. The gauge symmetry G in gGHU models is broken via boundary conditions and non-vanishing VEVs A 5 . We also study a model with gauge symmetry breaking induced by localized anomalies at a boundary.

Simple examples of gGHU models
In this section, we examine three models based on the unified symmetries SU (7), SU (8), and SO (10).
These models are simple and intuitive examples of the gGHU models that lead to G SM as a result of boundary conditions and the Hosotani mechanism. The study in this section helps us to understand the E 6 model, which will be studied in the next section.
We first give a brief explanation for symmetry breaking patterns in the models studied in this section. In the SU (N ) (N = 7, 8) model, we adopt the boundary condition that leads to the residual (1), under which the zero mode of A 5 transforms as the bifundamental representation. Non-zero VEVs of the Wilson line phases can lead to the spontaneous symmetry breaking H → G SM . The gauge symmetry breaking of H is similar to that in the productgroup unification [16], though in which the construction of the U (1) Y hypercharge generator is different from our model. In the SO(10) model, the residual symmetry is H = SU (5) × U (1) and the zero mode of A 5 behaves as the 10-dimensional anti-symmetric representation of SU (5). A non-zero VEV of the 10-dimensional field induces the spontaneous symmetry breaking SU (5)× U (1) → G SM . The symmetry breaking pattern is similar to the flipped SU (5) models [17], although there is a difference between the constructions of the U (1) Y hypercharge generator in our model and the flipped SU (5) models.
In the following discussions, we give explicit forms of the boundary condition and the Wilson line phase in each model. The effective potential for the zero mode of A 5 is derived with the calculation methods in Ref. [15]. The vacuum structures of the effective potentials will be analyzed in Sec. 5.

The SU(7) and SU(8) models
At first we study the gGHU model with the SU (7) unified symmetry. We choose the boundary condition that is defined by the parity matrices: (1, 1, 1, 1, 1, −1, −1). (3.1) This boundary condition implies that the gauge field has the following eigenvalues of the parity operators: where the subscripts in the right-hand sides imply n × m submatrix in the SU (7) representation space. The zero mode of A µ appears as the adjoint representation of the residual gauge symmetry H = SU (5) × SU (2) × U (1). Meanwhile, A 5 has the zero mode that transforms as the bi-fundamental representation under the SU (5) × SU (2) symmetry.
Without loss of generality, the residual symmetry SU (5) × SU (2) × U (1) allows us to simplify the form of the VEVs of the A 5 zero mode as where a 1 and a 2 are real parameters. The gauge symmetry forbids the tree-level potential for a 1 and a 2 . If the effective potential, which is induced by quantum corrections, has the global minima where a 1 = a 2 = 0 (mod 1) is realized, then SU (5) × SU (2) × U (1) is spontaneously broken down to G SM at a vacuum. § We start to derive the effective potential. The parametrization of VEVs in Eq. (3.4) suggests that the SU (7) generator that corresponds to a 1 (a 2 ) can be seen as the Cartan generator of SU (2) 16 where SU (2) ij induces mixing between the i-th and j-th components of the SU (7) fundamental representation. This is an important point to understand eigenvalues of D (0) y in the Lagrangian (2.7) and contributions to the effective potential, as discussed in Sec. 2.
One can see that the SU (7) fundamental representation involves two doublets under SU (2) 16 × SU (2) 27 where the right-hand side indicates representations of (SU (2) 16 , SU (2) 27 ). Similarly, the anti-symmetric 21-dimensional, the symmetric 28-dimensional, and the adjoint representations are decomposed as § Since the parameter a i (i = 1, 2) has the phase property, the cases with a i = 0 and a i = 2 are physically equivalent. Among the vacua a 1 = a 2 = 0 (mod 2), there is a special one with a 1 = a 2 = 1, where the rank is preserved under the spontaneous breaking of the symmetry and SU (3) × SU (2) × SU (2) × U (1) × U (1) appears as the low-energy symmetry.
From the above decompositions, one can understand that how the SU (7) representations transform under U (1) generators accompanied by the parameters a 1 and a 2 . Then the eigenvalues of the covariant derivative in Eq. (2.7) and the contributions to the effective potential are easily evaluated. We denote the contribution to the effective potential from a bosonic degree of freedom of the R-dimensional SU (7) representation as F R 7 (a i , δ), where δ = 0 (−1) for the bulk fermion fields ofη = 1 (−1). For R = 7, 21, 28, and 48, the contributions are written as follows: 14) (3.16) where C = 3/(64π 7 R 5 ) and we usef cos (πw(x + δ)) w 5 . (3.17) In the above expressions, we denote a i -independent terms as constants, which have no effect on symmetry breaking patterns and are discarded in the following discussions. One can confirm that the results in Eqs. (3.13)-(3.16) are coincide with the potential in Ref. [15].
We specify the matter content for fermion fields in the model as where n (±) R stands for the number of bulk fermion fields that belong to the R-dimensional representations and haveη = ±1. The one-loop effective potential is written as follows: Next, let us start to study the gGHU model with the SU (8) unified symmetry. We assume the following boundary condition: The subscripts in the right-hand sides imply n × m submatrix in the SU (8) representation space. This boundary condition leads to the residual symmetry SU (5) × SU (3) × U (1).
Using the residual symmetry, we can simplify the VEVs of the zero mode of A 5 . In this case the Wilson line phase degrees of freedom are parametrized by the three real parameters a i (i = 1, 2, 3) as If the parameters evolve non-zero VEVs of a 1 = a 2 = a 3 = 0 (mod 1) at a vacuum, then the spontaneous From the parametrization in Eq. (3.22), we can see that a 1 , a 2 , and a 3 correspond to generators involved in SU (2) 16 , SU (2) 27 , and SU (2) 38 , respectively. In order to derive the effective potential for the Wilson line phases, we decompose SU ( representations. The 8, 28, 36, and 63-dimensional representations of SU (8) are decomposed as (3.23) 28 ∋ 2 × [(2, 1, 1) + (1, 2, 1) + (1, 1, 2)] + 1 × [(2, 2, 1) + (1, 2, 2) + (2, 1, 2)], (3.24) 36 (3.26) where the right-hand sides indicate the (SU (2) 16 , SU (2) 27 , SU (2) 38 ) irreducible representations. From the expressions, as in the SU (7) case, one can readily derive contributions to the effective potential for We denote the contributions to the effective potential from a bosonic degree of freedom of the , which is written as follows: We specify the numbers of the bulk fermion fields in this model by where we consider R = 8, 28, 36, 63 fermion fields having even (+) or odd (−) periodicities. We obtain the one-loop effective potential in the SU (8) model as Vacuum configurations determined by the effective potentials in the SU (7) model in Eq. (3.19) and the SU (8) model in Eq. (3.32) depend on numbers of bulk fermion fields in each model. We will discuss the vacuum structures in the SU (7) and SU (8) models in Sec. 5.

The SO(10) model
In this subsection, we study another example of the gGHU model that has the SO(10) unified symmetry.
In the SO(10) model, the gauge field A M , which belongs to the 45-dimensional adjoint representation, can be decomposed into the representations of the subgroup SU (5) as (10), (3.33) where A M (R) transforms as the R-dimensional representation of SU (5). The boundary condition is taken as follows: (10). (3.35) This leads to SU (5) × U (1) as the residual symmetry.
Since the extra-dimensional component of the gauge field has the opposite parities to those of A µ in Eqs. (3.34) and (3.35), there appears zero mode of A 5 in A 5 (10) and A 5 (10). Using the residual SU (5) × U (1) symmetry, we can parametrize them by two real parametersã andb as and A 5 (10) = A 5 (10) † . At a vacuum, if one of the parameters takes a non-zero VEV (mod 1) and the other remains zero (mod 2), then the spontaneous gauge symmetry breaking SU ( We start to discuss the effective potential forã andb. In order to clarify the group structure, it is useful to consider the decomposition SU ( 4). In this basis, A 5 is written as (3.37) where each term in the right-hand side transforms as (SU (4), SU (2), SU (2) ′ ) irreducible representations. Sinceã is involved in A 5 ((15, 1, 1)) , one can find an SU (2) subgroup of SU (4) such that the parameterã corresponds to a generator of the SU (2) subgroup, which is referred to as SU (2) a in the following. The other parameterb is involved in A 5 ((1, 1, 3)) . It is clear thatb corresponds to a As in the previous subsection, we will derive the effective potential forã andb focusing on SU (2) The contributions to the effective potential forã andb from a bosonic degree of freedom of the Rdimensional representation is denoted by F R 10 (ã,b, δ); it is written as follows: We will discuss the vacuum structure of the potential in Sec. 5. 4 The E 6 gGHU model In this section, we study the gGHU model based on the E 6 unified symmetry. This model leads to the gauge symmetry breaking E 6 → G SM . We give an overview of the model in this subsection. The detailed structure of the model and the derivation of the effective potential for the zero mode of A 5 is studied in the following subsections.
We first summarize the group structure of E 6 , which has three maximal regular subgroups SO(10)× U (1), SU (6) × SU (2), and (SU (3)) 3 [18]. We denote the subgroup (SU (3)) 3 as SU (3) (3) C is identified to the color gauge symmetry and SU (3) L involves the weak isospin symmetry [2]. We take SU (2) R = SU (2) 12 , where SU (2) ij induces mixing between the i-th and j-th components of the SU (3) R fundamental representation. In addition, we refer to SU (2) 23 and SU (2) 31 as SU (2) E and SU (2) EF , respectively. Among the SU (2) symmetries in SU (3) In the rest of the paper, we use the notation where SU (5) is the Georgi-Glashow unified symmetry that involves G SM . Note that the maximal SU (2) R rotation, which we call SU (2) R flip, corresponds to the exchange of the bases of SU (6)×SU (2) E and SU (6) F × SU (2) EF . Accordingly, SU (5) F is identified to the symmetry found in the flipped SU (5) models [17]. We also use As in Eq. (4.1), the SU (2) R flip leads to the exchange of the above two different bases in the right-hand sides.
In this model, the symmetry breaking is induced by a boundary condition, the VEVs of Wilson line phases, and an anomaly. As shown below, the boundary condition leads to the symmetry breaking The zero mode of A 5 can develop VEVs, which lead to the symmetry breaking SU (5) We assume that the U (1) V ′ is broken by localized anomalies. The orbifold allows us to obtain chiral fermions, and the fermion fields generally contribute to anomalies at boundaries [19]. In our model, U (1) V ′ charges of the fermion fields that have the Neumann boundary condition at y = 0 tend to become anomalous. Localized anomalies are assumed to be cancelled by the Green-Schwarz mechanism [20]. Namely, a pseudo-scalar field that transforms non-linearly under the U (1) V ′ symmetry and has the Wess-Zumino couplings is introduced on this boundary to cancel the anomaly. The scalar field allows a mass term [21] for the U (1) V ′ gauge field on the boundary. Thus we also assume that there appears a localized heavy mass term for the U (1) V ′ gauge field at the boundary y = 0 due to the In the following subsections, we will show the explicit formulations of the E 6 model. Then we will derive the contributions to the effective potential for A 5 from bulk fermion fields and the gauge field taking into account the effect of the localized mass term on the effective potential.

The boundary condition and the A 5 zero mode
In the E 6 model, in order to show the boundary condition, we decompose the gauge field A M , which belongs to the 78-dimensional adjoint representation of E 6 , into SO (10) and SU (6) (20, 2)), (4.4) where each term in the right-hand sides transforms as the SO (10) or SU (6) We introduce the following boundary condition: (20, 2)).
( 4.6) The residual symmetry is SU ( It is useful to decompose the fields in Eq. where A µ (R Q ) in the right-hand sides corresponds to the field that transforms as the R-dimensional SU (5) F representation and has the U (1) VF charge Q. The superscript indicates orbifold parity (P 0 ,P 1 ) of each field. Note that due to the localized mass term for the U (1) V ′ gauge field at the boundary y = 0, the orbifold parities of the gauge field are effectively modified. As a result, they generally obey a mixed boundary condition [22]; the effect of the modification will be discussed in Sec. 4.4. The above where A µ (1 KF 0 ) (+,+) and A µ (1 EF 0 ) (+,+) are linear combinations of A µ (1 VF 0 ) (+,+) and A µ (1 V ′ 0 ) (+,+) . Note that A 5 (R Q ) has opposite orbifold parities to those of A µ (R Q ). Thus the zero mode of A 5 appears in A 5 (10 1 ) (+,+) and A 5 (10 −1 ) (+,+) . With the help of the residual SU (5) F × U (1) VF symmetry, the VEVs of the zero mode are simplified as follows: 14) whered andñ are real parameters, and A 5 (10 −1 ) (+,+) = A 5 (10 1 ) (+,+) † . The parameterd (ñ) corresponds to a generator of an SU (2) subgroup of E 6 , which is referred to as SU (2) d (SU (2) n ) in the following. If one of the parameters takes a non-trivial VEV (mod 1) and the other remains zero (mod 2), then the symmetry breaking SU (5) F × U (1) VF → G SM is realized similarly to the previous SO (10) model.

Contributions to the effective potential from bulk fermion fields
We start to derive the effective potential for the zero mode of A 5 , namely parametersd andñ in Eq. (4.14). The effective potential is generated by quantum corrections from matter and the gauge fields in the model. We here focus on the contributions to the effective potential from bulk fermion fields; the contributions from the gauge field are studied in the next subsection.
The contributions in the E 6 model can be easily obtained from the result in Sec. 3.2. To see this, it is required to find another SO (10) subgroup of E 6 where the parametersd andñ in Eq. (4.14) belongs to the 45-dimensional adjoint representation. For this purpose, we consider maximal SU (2) 18) In this expression, the left-hand sides transform as the irreducible representations of SO(10) ′ , and +) . Note that the fields in A µ (45 ′ ) have the same orbifold parities as those in Eqs. (3.34) and (3.35). This coincides with the fact that the zero mode of A 5 parametrized byd andñ appears in the adjoint representation A havingη = η 0 η 1 , which is decomposed into the SO(10) ′ multiplets as Φ (η) F 78 (d,ñ, 1) = F 45 10 (d,ñ, 1) + 2F 16 10 (d,ñ, 0). (4.20) The explicit form of the contribution is

15)
Similar discussion holds for the contributions from a bulk 27-plet fermion Φ (η) F havingη = η 0 η 1 . In this model, the orbifold parities of the 27-plet arê where each of the terms in the right-hand sides is the irreducible representation of SO(10) × U (1) V ′ or SU (6) F × SU (2) EF . One can find the SO(10) → SU (5) F × U (1) VF decomposition: where Φ F (R Q ) in the right-hand sides means the field transforms as the R-dimensional representation of SU (5) F and has the U (1) VF charge Q. The superscript in the right-hand sides indicates the eigenvalues of the parity (P 0 ,P 1 ) of each field. Using the SU (2) EF rotation, one can obtain the SO (10) From the expressions, one can obtain the contributions to the effective potential from the 27-plet as F 27 (d,ñ, 0) = F 16 10 (d,ñ, 1) + F 10 10 (d,ñ, 0), (4.30) F 27 (d,ñ, 1) = F 16 10 (d,ñ, 0) + F 10 10 (d,ñ, 1), (4.31) where terms in the right-hand sides are found in Eqs. (3.42)- (3.44). More explicitly, we obtain (4.32) In Appendix A, we also derive the effective potential and confirm the result in Eqs. (4.21) and ( (7), SU (8), and SO (10) models. In addition, in this E 6 model, the contributions are also unchanged under the transformation (ñ,d) → (ñ + 1,d + 1). This invariance is not accidental at the one-loop level but guaranteed by the E 6 symmetry in this model as shown in Appendix B.

Contributions to the effective potential from the gauge field
In this subsection, we will discuss the contributions to the effective potential from the gauge field, whose U (1) V ′ component has a localized mass term at the y = 0 boundary due to the anomaly cancellation.
In order to do this, we need to show the KK mass spectrum, which depends on the boundary mass parameter and the background fields (d,ñ), of the gauge field. The detailed derivation of the mass spectrum is shown in Appendix A.2. We here shortly summarize the calculation procedure and the result of the calculation of the effective potential.
The KK mass spectrum is obtained with the solution of equations of motion (EOM) of the gauge field in the bulk. The bulk EOM is simplified in the basis of SU (6)
To obtain the solution of the EOM, the boundary condition should be imposed. The boundary condition at the fixed points y = 0 and πR corresponds to (P 0 ,P 1 ) parities as in Eqs. (4.7)-(4.10), where even (odd) parity means the Neumann (Dirichlet) boundary condition. In addition the boundary condition for A µ (V ′ ) is effectively modified from (P 0 ,P 1 ), since it has a localized mass term at the boundary y = 0 [22]. Note that if the localized mass scale is much larger than the compactification scale 1/R, the Neumann boundary condition of A µ (V ′ ) at y = 0 is effectively modified to the Dirichlet boundary condition. With this heavy mass limit, the mass spectrum of the n-th KK mode of the fields (A µ (n (3) ), A µ (n (2) ), A µ (d (3) ), A µ (d (2) ), A µ (X)) has the following form: For the extra-dimensional component A 5 , there is no direct coupling to the localized mass parameter.
However, the boundary condition of A 5 could be modified in accordance with the modification of the boundary condition of A µ due to a gauge fixing term, which mixes A 5 with A µ . We demonstrate how the boundary condition of A 5 is modified in a simple setup in Appendix C.
With the KK mass spectrum of the gauge fields in Eqs. (4.33) and (4.34), we can easily derive the contributions to the effective potential from the gauge sector in this model. The contribution from a bosonic degree of freedom is (4.37)

Analysis of vacuum structure
In this section, we study vacuum structure of the one-loop effective potentials in the SU (7), SU (8), SO (10), and E 6 models discussed in the previous sections. In general, positions of the vacua of the potentials depend on bulk matter contents of the models. Without bulk matter fields, for instance,  (7), SU (8), SO (10), and E 6 models. Matter contents, VEVs at the global minimum of the one-loop effective potentials, and physical squared mass eigenvalues of A 5 zero modes normalized by the typical mass parameter m 2 0 in Eq. In each model, we found matter contents for bulk fermion fields that lead to the symmetry breaking into G SM . Examples are summarized in Table 1, where the matter contents and the VEVs at vacua are shown. In addition, we show the physical mass spectrum of the zero mode of A 5 in a normalization, whose definition is expressed in Eq. (5.1). In all the models, bulk fermions that have periodic boundary conditions (η = 0) are required to obtain non-zero VEVs.
In the SU (7) model, non-zero VEVs a 1 = a 2 = 0 (mod 1) are obtained at vacua, where the residual symmetry SU (5) × SU (2) × U (1) is broken to G SM . Figure 1 left shows the contour plot of the one-loop effective potential V 7 (a i , N 7 ) for the case with N 7 = (0, 0, 0, 0, 2, 0, 0, 0). The positions of the vacua are denoted by the square symbols in the contour plot. Under the shift a i → a i + 2 for each i, the potential has invariance, which reflects the phase property of each a i . Also one can see the potential is invariant under a i → −a i or (a 1 , a 2 ) → (a 2 , a 1 ). In this figure, one can see that there appear four degenerate vacua, which are physically equivalent. Around the vacua, the physical zero mode of A 5 becomes massive and the mass matrix is evaluated as where we introduce m 2 0 as a typical (squared) mass scale and the effective four-dimensional gauge coupling g 4D . In the present case, at one of the vacua, the VEVs take a 1 = a 2 = 0.5849 and the eigenvalues of the mass matrix are 105.4m 2 0 and 65.76m 2 0 . The values in Table 1 show the eigenvalues of squared masses normalized by m 2 0 . For the case with N 7 = (2, 0, 0, 2, 0, 0, 2, 0), we also show the VEVs and squared mass eigenvalues normalized by m 2 0 in Table 1. In the SU (8) model, the residual symmetry is SU (5) × SU (3) × U (1), which is broken to G SM by VEVs a 1 = a 2 = a 3 = 0 (mod 1). For instance, the symmetry breaking is achieved for the cases with N 8 = (0, 0, 0, 0, 2, 0, 0, 0) and N 8 = (2, 0, 0, 2, 0, 0, 2, 0). Around the vacua, the mass matrix of the physical mass spectrum of the zero mode of A 5 is evaluated from V 8 (a i , N 8 ) similarly to Eq. (5.1). In Table 1, the VEVs and squared mass eigenvalues are shown for the above two cases. In the eigenvalues, there appears degeneracy, which reflects that two linear combinations of the parametrized VEVs a 1,2,3 belong to the adjoint representation of SU (3) C in G SM .
The SO(10) model has the residual symmetry SU (5) × U (1). For the cases where one of the parametersã andb in Eq. (3.36) has a non-zero VEV (mod 1) while the other remains zero (mod 2), then G SM is obtained. In Table 1, we show the examples of the matter contents and corresponding VEVs that lead to G SM . For the case with N 10 = (0, 1, 0, 0, 1, 0), we also show the contour plot of the effective potential in Figure 1 center, where the square symbols indicate the positions of degenerate vacua, which are physically equivalent. Around the vacua, one can evaluate the physical mass spectrum of A 5 using the potential V 10 (ã,b, N 10 ) similarly to Eq. (5.1). The eigenvalues of the squared masses are also shown in Table 1.
Finally we discuss the E 6 model, whered andñ in Eq. (4.14) parametrize the VEVs. The residual symmetry of the model is SU (5) F × U (1) VF . We are particularly interested in the vacua that lead to G SM , where VEVs ared = 0 (mod 2) andñ = 0 (mod 1). We show two cases with N E6 = (0, 2,1,2) and N E6 = (0, 4, 1, 2) in Table 1. For the latter case, the contour plot of V E6 (d,ñ, N E6 ) is also shown in Figure 1 right. One can confirm that the potential has the invariance that was mentioned in Sec. 4.3.
Around the vacua of the potential, physical components of the zero mode of A 5 become massive. The VEVs and the squared mass eigenvalues are shown in Table 1.

Summary and discussion
We have studied the five-dimensional gGHU models, namely the applications of the Hosotani mechanism to the unified gauge symmetry breaking, on the orbifold compactification S 1 /Z 2 . In these models, VEVs of the zero modes of the extra-dimensional gauge fields, whose dynamics reflects degrees of freedom of Wilson line phases, are available to break the residual symmetry. The effective potential of the zero mode is generated by quantum corrections that depend on matter contents in the models.
We have discussed the models based on SU (7), SU (8), SO (10), and E 6 gauge symmetries. In each model, the standard model gauge symmetry is achieved at a low-energy regime when the suitable bulk fermion fields are contained. We have derived the one-loop effective potentials for the zero modes of the extra-dimensional gauge fields in all the models. We have also studied the effects of the localized mass term for the gauge field induced by the anomaly in the E 6 model. To make our discussion more concrete, the mass spectrum of the bulk fermion fields and the standard model matter sector should be explicitly treated. In this case, one can examine the renormalization group evolution of the gauge coupling constants, which has large dependence on the bulk fermion mass spectrum. Since the mass spectrum is not severely constrained, we tend to lose precise predictions for the values of the gauge couplings in this setup. For instance, if we introduce bulk masses for bulk fermion fields slightly smaller than the compactification scale, which do not change the present analysis of the effective potentials approximately, their contributions to the evolution are suppressed. The standard model fermions and the Higgs scalar can be introduced into the models as bulk fields or localized fields in the present setup. ¶ As mentioned, the models based on SU (N ) (N = 7, 8) and SO(10) share a part of symmetry breaking pattern with the product GUT models [16] and the flipped SU (5) models [17], respectively, although the construction of the hypercharge generator in our models is different from the known models. This implies that the standard model fields are realized as boundary localized fields in our SU (N ) and SO(10) models. On the other hand, the standard model fields are naturally incorporated into bulk fields in the E 6 model . In addition, the supersymmetric extension of the E 6 model can supply the doublet-triplet splitting via an analogues of the missing partner mechanism that is often discussed in the flipped SU (5) models. These subjects are left to our future studies.

Acknowledgments
The authors would like to thank N. Yamatsu for valuable discussions.
A Calculation of the effective potential in the E 6 model

A.1 Contributions from bulk fermion fields
In Sec. 4.3, the contribution to the effective potential from bulk fermion fields in the E 6 model is shown, with the help of the effective potential in the SO(10) model. In this subsection, we derive the contribution by using another explicit formulation.
For deriving the contribution, a key point is that the U (1) directions accompanied byd andñ in Eq. (4.14) are identified to generators in SU (6) and SU (2) E , respectively. This can be realized because the zero mode of A 5 appears in the adjoint representations of SU (6) and SU (2) E . To see this, we first focus on the decomposition of A µ in Eqs. (4.7)-(4.10). It is useful to consider the U (1) YF subgroup that where A µ (R) in the right-hand sides transforms as R in Table 2 under the SU (3) C × SU (2) L × U (1) YF × U (1) VF symmetry. We denote the complex conjugate of R by R. Note that the U (1) Y hypercharge and the charge of the Cartan generator of SU (2) R , which we denote by T 3 R , are linear combinations of U (1) YF and U (1) VF charges; they are also shown in Table 2. One can see that SU (3) C × SU (2) L gauge fields correspond to the zero modes of A µ (G) (+,+) and A µ (W ) (+,+) , and the U (1) Y gauge field is a linear combination of the zero modes of A µ (n YF ) (+,+) and A µ (n VF ) (+,+) . If the symmetry breaking SU (5) F × U (1) VF → G SM is realized, then the zero modes A µ (Q) (+,+) , A µ (Q) (+,+) , and a linear  (3) C 8  1  1  3  1  3  3  1  1  3  3  1  3  1 combination of A µ (n YF ) (+,+) and A µ (n VF ) (+,+) become massive with would-be NG bosons that belong to the zero mode of A 5 .
Let us derive the contributions to the effective potential ford andñ from a bulk adjoint field Φ (η) A , whereη is a parameter related to the periodicity of the field. For this purpose, we show the transformation law of Φ

(η)
A under SU (2) d ×SU (2) n accompanied by the periodicity of the field, explicitly. The adjoint field is decomposed into SU (6) One can further decompose the fields as .25) where in the right-hand sides ±η denotes the periodicity that coincides with the eigenvalue of the translation operatorP 1P0 .
From the above expression, we can easily see the SU (2) n transformation law of Φ

(η)
A with the periodicity. In Φ where in the right-hand sides Φ A (R, R ′ ) transforms under SU (2) d (SU (2) n ) as R (R ′ ) and the superscript for each term denotes the periodicity. Hence one can obtain From the above, the contribution in Eq. (4.21) is obtained.
The contribution from a 27-plet Φ (η) F is obtained in a similar fashion. The field is decomposed into SU (6) F × SU (2) EF multiplets as (6, 2)). (A.30) Further decomposition leads to Table 2 and 3 and the superscript for each term denotes the periodicity. Using the SU (2) R flip, we Therefore, a 27-plet involves the fields that transform under SU (2) d × SU (2) n as Φ (η) From the above, we can easily lead to Eq. (4.32).

A.2 Contributions from the gauge field
In this subsection, we show the calculation of the contributions to the effective potential discussed in Sec. 4.4. The contribution is generated by the gauge field, whose U (1) V ′ component is assumed to have a large mass term at the y = 0 boundary due to the anomaly cancellation. The mass term effectively modifies the boundary condition and the KK masses of some components of the gauge field. Without the boundary mass term, the contribution takes the form of Eq. (4.21) with δ = 0, since the gauge field belongs to the adjoint representation. The modification of the KK mass alters a part of the contribution in Eq. (4.21).
As explained in Sec. 4.4, the KK mass spectrum that is affected by the localized mass term is obtained as a solution to the EOM of the following set of the fields: A µ (n (3) ), A µ (n (2) ), A µ (d (3) ), A µ (d (2) ), A µ (X) , (A.37) where n (3,2) , d (3,2) , and X imply generators of SU (2) n , SU (2) d , and U (1) X , respectively. For convenience we introduce a column vector Φ α µ (α = 1-5) that consists of the fields: The Lagrangian in Eq. (2.7) is diagonalized by the vector as (1,1,1,1,1), (A.39) We introduce the KK mode expansion: The solution to the bulk EOM is obtained as follows: φ 1(n) (y) = e idy/R ξ 1 cos(m In the present case, the condition is simplified in a basis where U (1) V ′ is manifest, since there is a boundary mass term for A µ (V ′ ). We introduce the new basis, and we denote It is realized that A µ (V ′ ) and A µ (V ) correspond to the U (1) V ′ and U (1) V gauge field, respectively. By using the above fields, the boundary condition is simplified; at y = 0, the condition is written as where M represents the boundary mass parameter. On the other hand, at y = πR, the condition is written as In a five-dimensional orbifold model, the gauge transformation is generally written by and λ a (x, y) is a five-dimensional gauge transformation function. When the gauge field satisfies the boundary condition given in Eqs. (2.1)- (2.4), then the gauge transformation implies Although generally the gauge transformation changes the boundary conditions, one can find the particular gauge transformation such that the relations P 0 = P ′ 0 and P 1 = P ′ 1 hold and the gauge field is shifted as We start to discuss our E 6 model, where the VEV in Eq. (4.14) can be written as where td and tñ are generators of SU (2) d ⊂ SU (6) and SU (2) n = SU (2) E , respectively. In a definite basis of the fundamental representation of SU (6)×SU (2) E , the parity matrix of the boundary condition can be written as follows: P 0 = diag(+1, +1, +1, +1, +1, −1) ⊗ diag(+1, −1), (B.7) Here we take that td generates mixing between 1st and 6th entries in the SU (6)  As mentioned above, if P ′ 1 = P 1 is satisfied, then the effective potential should have invariance under the shift in Eq. (B.11). This can be shown as follows. In the SU (6) × SU (2) E representation space in Eqs. (B.7) and (B.8), the matrices exp(2πitd) and exp(2πitñ) correspond to 2π rotations of fundamental representation of SU (2) d and SU (2) E , respectively. Thus we obtain exp 2πitd = diag(−1, +1, +1, +1, +1, −1) ⊗ diag(+1, +1), (B.13) exp (2πitñ) = diag(+1, +1, +1, +1, +1, +1) ⊗ diag(−1, −1). (B.14) The parity matrix P ′ 1 explicitly written as follows: C Effective modification of the boundary condition of A 5 with boundary breaking In Sec. A.2, we show that the boundary condition of A µ is effectively modified due to the existence of the boundary mass term. As mentioned, while the U (1) V ′ component of A 5 does not have zero modes, one can see that the boundary condition of A 5 is also modified by introducing proper gauge fixing terms in the theory. As a result, one can choose the specific gauge, namely ξ = 1 shown just below, where the KK masses of A 5 coincide with those of A µ . Here, we consider a five-dimensional U (1) model compactified on an S 1 /Z 2 orbifold, and illustrate the essential feature of the modification in the simple setup.
In the U (1) model, the orbifold parity around the fixed point y = 0 is expressed as A µ (−y) = A µ (y), A 5 (−y) = −A 5 (y). (C.1) Then, as the E 6 model discussed in Sec. 4, we study the effect of a mass term localized on this fixed point. Below, we consider an anomaly as the origin of the mass term, while it can be the Higgs mechanism.
The anomaly is assumed to be made harmless via the Green-Schwarz mechanism [20]. Namely, a pseudo-scalar field χ that transforms non-linearly under the U (1) symmetry and has the Wess-Zumino couplings is introduced on this boundary to cancel the anomaly. Such a scalar field allows the Stückelberg mass term [21] (on the boundary) which is U (1) invariant and thus the naive scale of the mass M is around the cutoff scale of the fivedimensional theory, much larger than the compactification scale. As well-known, such a huge mass repels the wave functions of the lower-laying KK modes of A µ to modify its boundary condition from the Neumann to the Dirichlet type effectively [22]. Below, we examine the effect on those of A 5 , which does not directly couple to the localized mass due to the orbifold parity. (See Ref. [24] for the same analysis in terms of the KK decomposed language.) For this purpose, as A 5 is unphysical except for the zero mode, we should treat the gauge fixing term properly. † † The mixing terms of the four-dimensional gauge field are and we adopt the usual gauge fixing term with a constant gauge parameter ξ, to remove the above mixing terms (up to surface terms). Then the quadratic terms of A 5 and χ become L quad = 1 2 A 5 − + ξ∂ 2 5 A 5 − δ(y) ξ √ M χ∂ 5 A 5 + 1 2 χ ( + ξM δ(y)) χ . (C.5) Note that this quadratic part is essentially the same as the one in the case that the mass term originates from the Higgs mechanism, and thus the derivation below is applied also for the case.
Since there is an awkward term proportional to δ(y) 2 in the quadratic part, we should regularize the delta function. To be more concrete, we replace the delta function by a finite, sufficiently smooth function δ ǫ (y) that vanishes for |y| ≥ ǫ and is normalized as ǫ −ǫ δ ǫ (y)dy ∼ 1. ‡ ‡ The EOMs of A 5 and χ are respectively − + ξ∂ 2 5 A 5 (y) + ξ (∂ 5 δ ǫ (y)) √ M χ = 0, (C.6) − δ ǫ (y) ξ √ M ∂ 5 A 5 (y) + ( + ξM δ ǫ (y)) χ = 0, (C.7) where the y-dependences are explicitly shown. Due to the overall delta function in Eq. (C.7), we may not suppose that the combination in the parenthesis there vanishes for |y| ≥ ǫ, while it does vanish, for † † This means that, of course, the effect is gauge dependent and thus an unusual gauge fixing term may be selected as in Ref. [26] to make the "mass spectrum" of A 5 unchanged. In such cases, however, the calculation of, for instance, the effective potential would be complicated. ‡ ‡ One may impose the periodicity, for completeness if necessary.
Then, the contribution of the regular function, − A 5 is negligible and we get ξ ∂ 5 A 5 (y) + δ ǫ (y) √ M χ The result shows that A 5 obeys the Dirichlet boundary condition in the limit M → 0; the condition changes to the Neumann boundary condition in the opposite limit M → ∞. The modification of the boundary condition of A 5 is in accordance with that of A µ .