The Standard Model Gauge Symmetry from Higher-Rank Unified Groups in Grand Gauge-Higgs Unification Models
aa r X i v : . [ h e p - ph ] J un KYUSHU-HET-176
The Standard Model Gauge Symmetry from Higher-Rank Unified Groupsin Grand Gauge-Higgs Unification Models
Kentaro Kojima a, ∗ , Kazunori Takenaga b, † , and Toshifumi Yamashita c, ‡ a Faculty of Arts and Science, Kyushu University, Fukuoka 819-0395, Japan b Faculty of Health Science, Kumamoto Health Science University, Izumi-machi, Kumamoto861-5598, Japan c Department of Physics, Aichi Medical University, Nagakute, 480-1195, Japan
Abstract
We study grand unified models in the five-dimensional space-time where the extra dimension iscompactified on S / Z . The spontaneous breaking of unified gauge symmetries is achieved viavacuum 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 models. In each model, the rank of the residual gauge symmetry that respects the boundarycondition imposed at the orbifold fixed points is higher than that of the standard model. We verifythat the residual symmetry is broken to the standard model gauge symmetry at the global minimaof the effective potential for certain sets of bulk fermion fields in each model. ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] ontents SU (7) and SU (8) models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53.2 The SO (10) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 E gGHU model 11 E model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2 The boundary condition and the A zero mode . . . . . . . . . . . . . . . . . . . . . . . 134.3 Contributions to the effective potential from bulk fermion fields . . . . . . . . . . . . . . 144.4 Contributions to the effective potential from the gauge field . . . . . . . . . . . . . . . . 16 E model 21 A.1 Contributions from bulk fermion fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21A.2 Contributions from the gauge field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
B Equivalence of vacua in the E model 27C Effective modification of the boundary condition of A with boundary breaking 29 For the past several decades, grand unification of the standard model gauge symmetry at a high-energy 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 suchas the charge quantization and the anomaly cancellation. In addition to the minimal grand unifiedtheory (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 [4]. A common feature shared among various GUT modelsis 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 symmetrybreaking in GUT models is to involve elementary Higgs scalars that develop vacuum expectation values1VEVs) and the VEVs lead to desired breaking patterns of the unified symmetries. This mechanism isan 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 mecha-nism [5–7]. In models with the Hosotani mechanism, the extra-dimensional components of the gaugefields effectively behave as “Higgs” scalars at low energy, and dynamics of the gauge fields reflectsdegrees of freedom of Wilson line phases. Although gauge invariance forbids the tree-level potential forthe phases at the classical level, non-trivial VEVs of the phases are naturally emerged and spontaneoussymmetry breaking is achieved when quantum corrections are involved [7]. Advantages of the Hosotanimechanism are predictivity and finiteness of the “Higgs” potential and masses [8], even though thepotential arises from loop corrections. Hence, as a solution to the hierarchy problem in the standardmodel, the gauge-Higgs unification models has been widely investigated [9, 10]. In these models, thezero modes of the extra-dimensional component of gauge fields are identified to the Higgs doublet inthe standard model.Recently, we have been focusing on application of the Hosotani mechanism to unified gauge symme-try breaking on an orbifold compactification S / Z , which enable us to incorporate chiral fermions infive-dimensional models [11,12]. In this case, the zero mode of the extra-dimensional gauge field, whichhave even parities under a boundary condition defined at the boundaries of the orbifold, plays a role ofthe Higgs field whose VEV breaks unified gauge symmetries into the standard model one. We refer tothis scenario as grand gauge-Higgs unification (gGHU). Note that gGHU is different from the orbifoldGUT models where boundary conditions directly break GUT symmetry into the standard model gaugesymmetry [13].In models with S / Z , the orbifold parities of the extra-dimensional component of the gauge fieldis opposite to those of the four-dimensional vector counterpart. Consequently, massless zero modesappearing from the extra-dimensional components tend not to belong to the adjoint representationof unbroken symmetries, though adjoint Higgs fields are often utilized in ordinary four-dimensionalGUT models [1]. This situation leads to severe constraints on construction of gGHU models. We haveshown 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 doublet-triplet splitting problem and several phenomenological aspects have been studied in the supersymmetricversion 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 representa-tions 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 modelsbased on the unified symmetries SU (7), SU (8), SO (10), and E . In each model, we derive the one-loop2ffective potential, which depends on the matter content, for the zero mode of the gauge field. In thesemodels, accordingly, vacuum structure of the potential and symmetry breaking pattern are determinedby bulk field contents. We show that the standard model gauge symmetry is achieved at a low-energyregime 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 gaugetheories with a compactified dimension on S / Z . A calculation method of the one-loop effectivepotential for Wilson line phases is briefly summarized. In Sec. 3, as illustrative examples of gGHU, wediscuss three models where each unified symmetry is SU (7), SU (8), or SO (10). The one-loop effectivepotential in each model is derived. In Sec. 4, the E gGHU model is studied and the one-loop effectivepotential is examined. Vacuum structure of each effective potential is studied in Sec. 5. We finallysummarize our discussions in Sec. 6. The appendices are devoted to show detailed calculations requiredfor the discussion in Sec. 4. We consider five-dimensional gauge theories on M × S / Z , where one of the spatial dimensions iscompactified on the orbifold S / Z and M is the four-dimensional Minkowski space-time. On thecompact space S , which has the radius R , the fifth-dimensional coordinate denoted by y is identifiedwith y + 2 πR by the translation. On the orbifold S / Z , in addition to the translation, there is anotheridentification y ∼ − y , which is induced by the orbifold parity transformation. In the S / Z orbifoldtheories, the combination of the translation and the parity defines another orbifold parity transformationthat leads to the identification ( πR + y ) ∼ ( πR − y ). There exists the gauge field A M ( x, y ) that hasthe four-dimensional part A µ ( µ = 0 , , ,
3) and the extra-dimensional part A . The gauge field is alsodenoted by A M = A aM 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 La-grangian is invariant. Let us define the parity operators ˆ P , around y = 0 A µ ( x, − y ) = ˆ P A µ ( x, y ) = P A µ ( x, y ) P † , (2.1) A ( x, − y ) = ˆ P A ( x, y ) = − P A ( x, y ) P † , (2.2)and around y = πR A µ ( x, πR − y ) = ˆ P A µ ( x, πR + y ) = P A µ ( x, πR + y ) P † , (2.3) A ( x, πR − y ) = ˆ P A ( x, πR + y ) = − P A ( x, πR + y ) P † , (2.4)where P and P are matrices in a representation space of G . The orbifold parities and the matricesare called boundary conditions. Note that the translation from y to y + 2 πR is induced by the operatorˆ P ˆ P , whose eigenvalue corresponds to the periodicity, for each field.3ne can introduce Dirac fermions ψ ( x, y ); they obey the parity transformations as ψ ( x, − y ) = ˆ P ψ ( x, y ) = η T ψ [ P ] γ ψ ( x, y ) , (2.5) ψ ( x, πR − y ) = ˆ P ψ ( x, πR + y ) = η T ψ [ P ] γ ψ ( x, πR + y ) , (2.6)where the matrix T ψ [ P , ] acts on the field in its representation space and corresponds to P , inEqs. (2.1)–(2.4). The parameters η and η can be chosen as 1 or − η = η η , 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 pictureswhere A µ and A behave as a four-dimensional vector field and a scalar field, respectively. Once theboundary condition in Eqs. (2.1)–(2.4) is specified, we readily see that there exist sets of the generators { T α } ⊂ { T a } and { T ˜ α } ⊂ { T a } that satisfy [ T α , P ] = [ T α , P ] = 0 and { T ˜ α , P } = { T ˜ α , P } = 0,respectively. The corresponding component fields A αµ and A ˜ α have even parities of both ˆ P and ˆ P .Hence, the Kaluza-Klein (KK) decompositions of A αµ and A ˜ α have massless zero modes. The set ofgenerators { T α } corresponds to a subgroup of G , which we call the residual gauge symmetry H . Themassless “scalar” zero modes of A ˜ α parametrize Wilson line phase degrees of freedom and can developnon-trivial VEVs h A ˜ α i . 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 . Therefore, a minimum ofthe potential, namely vacuum structure of the theory, is determined by quantum effects. The effectivepotential can be derived by the functional integral over fields in the theory. One can treat h A ˜ α i asclassical backgrounds and substitute A ˜ α → A ˜ α + h A ˜ α i in the Lagrangian. Then quadratic terms of thegauge field in the Lagrangian are written as L gauge = 12 η MN A aM h (cid:3) δ ac − ( D (0) y ) ab ( D (0) y ) bc i A cN , ( D (0) y ) ab = [ ∂ y δ ab − ig h A ˜ α i ad( T ˜ α ) ab ] , (2.7)where we adopt the Feynman gauge and use η MN = diag(1 , − , − , − , −
1) and (cid:3) = ∂ µ ∂ µ . We denotethe five-dimensional gauge coupling constant by g and the generators in the adjoint representation byad( T ˜ α ) ab . The functional integral over the fluctuations A aM leads to contributions to the effectivepotential for h A ˜ α i .Notice that the contributions to the effective potential are determined by eigenvalues of the gen-erators { T ˜ α } , which are accompanied by the zero modes h A ˜ α i , in the covariant derivative D (0) y . Thegenerators are regarded as the charge operators of U (1) subgroups of G . Thus once the U (1) chargesof the fields in the functional integral are known, then the contributions to the effective potential areobtained. In addition, each U (1) generator is also regarded as the Cartan generator of an SU (2) sub-group of G . Therefore, we can easily understand the U (1) charges of the fields from the spin eigenvaluesof the SU (2) subgroups accompanied by the zero modes h A ˜ α i [15]. We use this procedure for derivingthe contributions to the effective potential. 4n the following sections, we study several gGHU models that lead to the standard model gaugesymmetry G SM at a low-energy regime. The gauge symmetry G in gGHU models is broken via boundaryconditions and non-vanishing VEVs h A i . We also study a model with gauge symmetry breakinginduced by localized anomalies at a boundary. 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 ofboundary conditions and the Hosotani mechanism. The study in this section helps us to understandthe E model, which will be studied in the next section.We first give a brief explanation for symmetry breaking patterns in the models studied in thissection. In the SU ( N ) ( N = 7 ,
8) model, we adopt the boundary condition that leads to the residualsymmetry H = SU (5) × SU ( N − × U (1), under which the zero mode of A transforms as the bi-fundamental representation. Non-zero VEVs of the Wilson line phases can lead to the spontaneoussymmetry breaking H → G SM . The gauge symmetry breaking of H is similar to that in the product-group unification [16], though in which the construction of the U (1) Y hypercharge generator is differentfrom our model. In the SO (10) model, the residual symmetry is H = SU (5) × U (1) and the zero modeof A behaves as the 10-dimensional anti-symmetric representation of SU (5). A non-zero VEV of the10-dimensional field induces the spontaneous symmetry breaking SU (5) × U (1) → G SM . The symmetrybreaking pattern is similar to the flipped SU (5) models [17], although there is a difference between theconstructions 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 linephase in each model. The effective potential for the zero mode of A is derived with the calculationmethods in Ref. [15]. The vacuum structures of the effective potentials will be analyzed in Sec. 5. SU (7) and SU (8) models At first we study the gGHU model with the SU (7) unified symmetry. We choose the boundary conditionthat is defined by the parity matrices: P = P = diag(1 , , , , , − , − . (3.1)This boundary condition implies that the gauge field has the following eigenvalues of the parity opera-tors: ( ˆ P , ˆ P ) · ( A µ ) = (cid:18) (+ , +) · ( A µ ) × ( − , − ) · ( A µ ) × ( − , − ) · ( A µ ) × (+ , +) · ( A µ ) × (cid:19) , (3.2)( ˆ P , ˆ P ) · ( A ) = (cid:18) ( − , − ) · ( A ) × (+ , +) · ( A ) × (+ , +) · ( A ) × ( − , − ) · ( A ) × (cid:19) , (3.3)5here the subscripts in the right-hand sides imply n × m submatrix in the SU (7) representationspace. 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 has the zero mode that transforms as the bi-fundamentalrepresentation under the SU (5) × SU (2) symmetry.Without loss of generality, the residual symmetry SU (5) × SU (2) × U (1) allows us to simplify theform of the VEVs of the A zero mode as h A i = 12 gR (cid:18) × Θ a (Θ a ) † × (cid:19) , Θ a = a a , (3.4)where a and a are real parameters. The gauge symmetry forbids the tree-level potential for a and a . If the effective potential, which is induced by quantum corrections, has the global minima where a = a = 0 (mod 1) is realized, then SU (5) × SU (2) × U (1) is spontaneously broken down to G SM ata 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 ( a ) can be seen as the Cartan generator of SU (2) ( SU (2) ),where SU (2) ij induces mixing between the i -th and j -th components of the SU (7) fundamental repre-sentation. This is an important point to understand eigenvalues of D (0) y in the Lagrangian (2.7) andcontributions to the effective potential, as discussed in Sec. 2.In order to derive the effective potential, we consider the decomposition of the fundamental repre-sentation of SU (7) into representations of ( SU (3) × SU (2) × SU (2) ) as7 → { (3 , ,
1) + (1 , , } + { (1 , , } , (3.5)where representations in each curly bracket compose an irreducible representation of SU (5) × SU (2) .One can see that the SU (7) fundamental representation involves two doublets under SU (2) × SU (2) as 7 ∋ × [(2 ,
1) + (1 , , (3.6)where the right-hand side indicates representations of ( SU (2) , SU (2) ). Similarly, the anti-symmetric21-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 = a = 0 (mod 2), there is a special one with a = a = 1, where the rank is preserved under thespontaneous breaking of the symmetry and SU (3) × SU (2) × SU (2) × U (1) × U (1) appears as the low-energy symmetry. →{ (¯3 , ,
1) + (3 , ,
1) + (1 , , } + { (3 , ,
2) + (1 , , } + { (1 , , } , (3.7)28 →{ (6 , ,
1) + (3 , ,
1) + (1 , , } + { (3 , ,
2) + (1 , , } + { (1 , , } , (3.8)48 →{ (8 , ,
1) + (1 , ,
1) + (1 , ,
1) + (3 , ,
1) + (¯3 , , } + { (1 , , } + { (1 , , } + { (3 , ,
2) + (1 , , } + { (¯3 , ,
2) + (1 , , } , (3.9)in terms of ( SU (3) , SU (2) , SU (2) ). One finds that the irreducible representations of SU (7) involvethe following ( SU (2) , SU (2) ) representations:21 ∋ × [(2 ,
1) + (1 , × (2 , , (3.10)28 ∋ × [(2 ,
1) + (1 , × (2 ,
2) + 1 × [(3 ,
1) + (1 , , (3.11)48 ∋ × [(2 ,
1) + (1 , × (2 ,
2) + 1 × [(3 ,
1) + (1 , , (3.12)and the others are singlets.From the above decompositions, one can understand that how the SU (7) representations transformunder U (1) generators accompanied by the parameters a and a . Then the eigenvalues of the covariantderivative in Eq. (2.7) and the contributions to the effective potential are easily evaluated. We denotethe contribution to the effective potential from a bosonic degree of freedom of the R -dimensional SU (7)representation as F R ( a i , δ ), where δ = 0 ( −
1) for the bulk fermion fields of ˜ η = 1 ( − R = 7, 21,28, and 48, the contributions are written as follows: F ( a i , δ ) = − C h ˆ f ( a + δ ) + ˆ f ( a + δ ) i = − C X i =1 ˆ f ( a i + δ ) , (3.13) F ( a i , δ ) = − C h f ( a + δ ) + 3 ˆ f ( a + δ ) + ˆ f ( a + a + δ ) + ˆ f ( a − a + δ ) i = − C X i =1 ˆ f ( a i + δ ) + X ≤ i 1. The one-loop effective potential is written as follows: V ( a i , N ) =3 F ( a i , − (cid:20) n (+)7 F ( a i , 0) + n ( − )7 F ( a i , 1) + n (+)21 F ( a i , 0) + n ( − )21 F ( a i , n (+)28 F ( a i , 0) + n ( − )28 F ( a i , 1) + n (+)48 F ( a i , 0) + n ( − )48 F ( a i , (cid:21) . (3.19)Next, let us start to study the gGHU model with the SU (8) unified symmetry. We assume thefollowing boundary condition:( ˆ P , ˆ P ) · ( A µ ) = (cid:18) (+ , +) · ( A µ ) × ( − , − ) · ( A µ ) × ( − , − ) · ( A µ ) × (+ , +) · ( A µ ) × (cid:19) , (3.20)( ˆ P , ˆ P ) · ( A ) = (cid:18) ( − , − ) · ( A ) × (+ , +) · ( A ) × (+ , +) · ( A ) × ( − , − ) · ( A ) × (cid:19) . (3.21)The subscripts in the right-hand sides imply n × m submatrix in the SU (8) representation space. Thisboundary 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 . In this case theWilson line phase degrees of freedom are parametrized by the three real parameters a i ( i = 1 , , 3) as h A i = 12 gR (cid:18) × Θ b (Θ b ) † × (cid:19) , Θ b = a a 00 0 a . (3.22)If the parameters evolve non-zero VEVs of a = a = a = 0 (mod 1) at a vacuum, then the spontaneoussymmetry breaking SU (5) × SU (3) × U (1) → G SM is realized.From the parametrization in Eq. (3.22), we can see that a , a , and a correspond to generatorsinvolved in SU (2) , SU (2) , and SU (2) , respectively. In order to derive the effective potentialfor the Wilson line phases, we decompose SU (8) representations into SU (2) × SU (2) × SU (2) SU (8) are decomposed as8 ∋ × [(2 , , 1) + (1 , , 1) + (1 , , , (3.23)28 ∋ × [(2 , , 1) + (1 , , 1) + (1 , , × [(2 , , 1) + (1 , , 2) + (2 , , , (3.24)36 ∋ × [(2 , , 1) + (1 , , 1) + (1 , , × [(2 , , 1) + (1 , , 2) + (2 , , × [(3 , , 1) + (1 , , 1) + (1 , , , (3.25)63 ∋ × [(2 , , 1) + (1 , , 1) + (1 , , × [(2 , , 1) + (1 , , 2) + (2 , , × [(3 , , 1) + (1 , , 1) + (1 , , , (3.26)where the right-hand sides indicate the ( SU (2) , SU (2) , SU (2) ) irreducible representations. Fromthe expressions, as in the SU (7) case, one can readily derive contributions to the effective potential for a i . We denote the contributions to the effective potential from a bosonic degree of freedom of the R -dimensional representation by F R ( a i , δ ), which is written as follows: F ( a i , δ ) = − C X i =1 ˆ f ( a i + δ ) , (3.27) F ( a i , δ ) = − C X i =1 ˆ f ( a i + δ ) + X ≤ i 63 fermion fields having even (+) or odd ( − ) periodicities. We obtainthe one-loop effective potential in the SU (8) model as V ( a i , N ) =3 F ( a i , − (cid:20) n (+)8 F ( a i , 0) + n ( − )8 F ( a i , 1) + n (+)28 F ( a i , 0) + n ( − )28 F ( a i , n (+)36 F ( a i , 0) + n ( − )36 F ( a i , 1) + n (+)63 F ( a i , 0) + n ( − )63 F ( a i , (cid:21) . (3.32)Vacuum configurations determined by the effective potentials in the SU (7) model in Eq. (3.19) andthe SU (8) model in Eq. (3.32) depend on numbers of bulk fermion fields in each model. We will discussthe vacuum structures in the SU (7) and SU (8) models in Sec. 5.9 .2 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 A M = A M (24) + A M (1) + A M (10) + A M (10) , (3.33)where A M ( R ) transforms as the R -dimensional representation of SU (5). The boundary condition istaken as follows:( ˆ P , ˆ P ) · A µ (24) = (+ , +) · A µ (24) , ( ˆ P , ˆ P ) · A µ (1) = (+ , +) · A µ (1) , (3.34)( ˆ P , ˆ P ) · A µ (10) = ( − , − ) · A µ (10) , ( ˆ P , ˆ P ) · A µ (10) = ( − , − ) · A µ (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 in A (10) and A (10). Using the residual SU (5) × U (1) symmetry, we can parametrize them by two real parameters ˜ a and ˜ b as h A (10) i ≡ ( H ij ) = H H H H H H H H H H → gR a b , (3.36)and (cid:10) A (10) (cid:11) = h A (10) i † . At a vacuum, if one of the parameters takes a non-zero VEV (mod 1) andthe other remains zero (mod 2), then the spontaneous gauge symmetry breaking SU (5) × U (1) → G SM is realized.We start to discuss the effective potential for ˜ a and ˜ b . In order to clarify the group structure, itis useful to consider the decomposition SU (5) → SU (3) × SU (2) × U (1) ′ , where ˜ a is a member of thetriplet of the SU (3) symmetry and ˜ b is a singlet under the SU (3) × SU (2) symmetry. Then one canintroduce a decomposition SO (10) → SU (4) × SU (2) × SU (2) ′ where the above SU (3) is involved in SU (4). In this basis, A is written as A = A ((15 , , A ((1 , , A ((1 , , A ((6 , , , (3.37)where each term in the right-hand side transforms as ( SU (4) , SU (2) , SU (2) ′ ) irreducible representa-tions. Since ˜ a is involved in h A ((15 , , i , one can find an SU (2) subgroup of SU (4) such that theparameter ˜ a corresponds to a generator of the SU (2) subgroup, which is referred to as SU (2) a in thefollowing. The other parameter ˜ b is involved in h A ((1 , , i . It is clear that ˜ b corresponds to agenerator of SU (2) ′ of SU (4) × SU (2) × SU (2) ′ . Thus we denote SU (2) ′ = SU (2) b .10s in the previous subsection, we will derive the effective potential for ˜ a and ˜ b focusing on SU (2) a × SU (2) b charges of fields in the SO (10) model. Under the SO (10) → SU (4) × SU (2) × SU (2) ′ decom-position, the 10 and 16-dimensional representations are written by10 → (6 , , 1) + (1 , , , → (4 , , 1) + (4 , , . (3.38)From Eqs. (3.37) and (3.38), one can see that the 10, 16, and adjoint representations involve thefollowing ( SU (2) a , SU (2) b ) representations:10 ∋ × [(2 , 1) + (1 , , (3.39)16 ∋ × [(2 , 1) + (1 , × (2 , , (3.40)45 ∋ × [(2 , 1) + (1 , × (2 , 2) + 1 × [(3 , 1) + (1 , . (3.41)The contributions to the effective potential for ˜ a and ˜ b from a bosonic degree of freedom of the R -dimensional representation is denoted by F R (˜ a, ˜ b, δ ); it is written as follows: F (˜ a, ˜ b, δ ) = − C h f (˜ a + δ ) + 2 ˆ f (˜ b + δ ) i , (3.42) F (˜ a, ˜ b, δ ) = − C h f (˜ a + δ ) + 2 ˆ f (˜ b + δ ) + ˆ f (˜ a + ˜ b + δ ) + ˆ f (˜ a − ˜ b + δ ) i , (3.43) F (˜ a, ˜ b, δ ) = − C h f (˜ a + δ ) + 4 ˆ f (˜ b + δ ) + 4 ˆ f (˜ a + ˜ b + δ ) + 4 ˆ f (˜ a − ˜ b + δ ) + ˆ f (2˜ a + δ ) + ˆ f (2˜ b + δ ) i . (3.44)We use N = ( n (+)10 , n ( − )10 , n (+)16 , n ( − )16 , n (+)45 , n ( − )45 ) as the numbers of R -dimensional ( R = 10 , , η = ± V (˜ a, ˜ b, N ) = 3 F (˜ a, ˜ b, − (cid:20) n (+)10 F (˜ a, ˜ b, 0) + n ( − )10 F (˜ a, ˜ b, n (+)16 F (˜ a, ˜ b, 0) + n ( − )16 F (˜ a, ˜ b, 1) + n (+)45 F (˜ a, ˜ b, 0) + n ( − )45 F (˜ a, ˜ b, (cid:21) . (3.45)We will discuss the vacuum structure of the potential in Sec. 5. E gGHU model E model In this section, we study the gGHU model based on the E unified symmetry. This model leads tothe gauge symmetry breaking E → G SM . We give an overview of the model in this subsection. Thedetailed structure of the model and the derivation of the effective potential for the zero mode of A isstudied in the following subsections.We first summarize the group structure of E , which has three maximal regular subgroups SO (10) × U (1), SU (6) × SU (2), and ( SU (3)) [18]. We denote the subgroup ( SU (3)) as SU (3) C × SU (3) L × U (3) R where SU (3) C is identified to the color gauge symmetry and SU (3) L involves the weak isospin SU (2) L . An SU (2) subgroup of SU (3) R is identified to SU (2) R that is found in the Pati-Salam unifiedsymmetry [2]. We take SU (2) R = SU (2) , where SU (2) ij induces mixing between the i -th and j -thcomponents of the SU (3) R fundamental representation. In addition, we refer to SU (2) and SU (2) as SU (2) E and SU (2) E F , respectively. Among the SU (2) symmetries in SU (3) R , only SU (2) E isorthogonal to G SM .In the rest of the paper, we use the notation E ⊃ ( SU (6) × SU (2) E ⊃ SU (5) × U (1) K × SU (2) E ,SU (6) F × SU (2) E F ⊃ SU (5) F × U (1) K F × SU (2) E F , (4.1)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) E F . Accordingly, SU (5) F is identified to the symmetry found in the flipped SU (5)models [17]. We also use E ⊃ SO (10) × U (1) V ′ ⊃ ( SU (5) × U (1) V × U (1) V ′ ,SU (5) F × U (1) V F × U (1) V ′ . (4.2)As in Eq. (4.1), the SU (2) R flip leads to the exchange of the above two different bases in the right-handsides.In this model, the symmetry breaking is induced by a boundary condition, the VEVs of Wilsonline phases, and an anomaly. As shown below, the boundary condition leads to the symmetry breaking E → SO (10) × U (1) V ′ at y = 0 and E → SU (6) F × SU (2) E F at y = πR . As a result, the residualsymmetry is SU (5) F × U (1) V F × U (1) V ′ . The zero mode of A can develop VEVs, which lead to thesymmetry breaking SU (5) F × U (1) V F → G SM .We assume that the U (1) V ′ is broken by localized anomalies. The orbifold allows us to obtainchiral fermions, and the fermion fields generally contribute to anomalies at boundaries [19]. In ourmodel, U (1) V ′ charges of the fermion fields that have the Neumann boundary condition at y = 0tend to become anomalous. Localized anomalies are assumed to be cancelled by the Green-Schwarzmechanism [20]. Namely, a pseudo-scalar field that transforms non-linearly under the U (1) V ′ symmetryand has the Wess-Zumino couplings is introduced on this boundary to cancel the anomaly. The scalarfield allows a mass term [21] for the U (1) V ′ gauge field on the boundary. Thus we also assume thatthere appears a localized heavy mass term for the U (1) V ′ gauge field at the boundary y = 0 due to the U (1) V ′ breaking.In the following subsections, we will show the explicit formulations of the E model. Then we willderive the contributions to the effective potential for A from bulk fermion fields and the gauge fieldtaking into account the effect of the localized mass term on the effective potential.12 .2 The boundary condition and the A zero mode In the E model, in order to show the boundary condition, we decompose the gauge field A M , whichbelongs to the 78-dimensional adjoint representation of E , into SO (10) and SU (6) F × SU (2) E F rep-resentations as A M = A M (45) + A M (1) + A M (16) + A M (16) (4.3)= A M ((35 , A M ((1 , A M ((20 , , (4.4)where each term in the right-hand sides transforms as the SO (10) or SU (6) F × SU (2) E F irreduciblerepresentations.We introduce the following boundary condition:ˆ P · A µ = + A µ (45) + A µ (1) − A µ (16) − A µ (16) , (4.5)ˆ P · A µ = + A µ ((35 , A µ ((1 , − A µ ((20 , . (4.6)The residual symmetry is SU (5) F × U (1) V F × U (1) V ′ .It is useful to decompose the fields in Eq. (4.3) into SU (5) F × U (1) V F representations as A µ (45) = A µ (24 ) (+ , +) + A µ (1 V F ) (+ , +) + A µ (10 − ) (+ , − ) + A µ (10 ) (+ , − ) , (4.7) A µ (1) = A µ (1 V ′ ) (+ , +) , (4.8) A µ (16) = A µ (10 ) ( − , − ) + A µ (5 − ) ( − , +) + A µ (1 ) ( − , +) , (4.9) A µ (16) = A µ (10 − ) ( − , − ) + A µ (5 ) ( − , +) + A µ (1 − ) ( − , +) , (4.10)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) V F charge Q . The superscript indicates orbifold parity ( ˆ P , ˆ P )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 obeya mixed boundary condition [22]; the effect of the modification will be discussed in Sec. 4.4. The above SU (5) F × U (1) V F representations are related to SU (6) F × SU (2) E F representations in Eq. (4.4) asfollows: A µ ((35 , A µ (24 ) (+ , +) + A µ (5 ) ( − , +) + A µ (5 − ) ( − , +) + A µ (1 K F ) (+ , +) , (4.11) A µ ((1 , A µ (1 ) ( − , +) + A µ (1 − ) ( − , +) + A µ (1 E F ) (+ , +) , (4.12) A µ ((20 , A µ (10 ) ( − , − ) + A µ (10 − ) ( − , − ) + A µ (10 − ) (+ , − ) + A µ (10 ) (+ , − ) , (4.13)where A µ (1 K F ) (+ , +) and A µ (1 E F ) (+ , +) are linear combinations of A µ (1 V F ) (+ , +) and A µ (1 V ′ ) (+ , +) .Note that A ( R Q ) has opposite orbifold parities to those of A µ ( R Q ). Thus the zero mode of A appears in A (10 ) (+ , +) and A (10 − ) (+ , +) . With the help of the residual SU (5) F × U (1) V F symmetry,13he VEVs of the zero mode are simplified as follows: D A (10 ) (+ , +) E → gR d n , (4.14)where ˜ d and ˜ n are real parameters, and (cid:10) A (10 − ) (+ , +) (cid:11) = (cid:10) A (10 ) (+ , +) (cid:11) † . The parameter ˜ d (˜ n )corresponds to a generator of an SU (2) subgroup of E , which is referred to as SU (2) d ( SU (2) n ) in thefollowing. If one of the parameters takes a non-trivial VEV (mod 1) and the other remains zero (mod2), then the symmetry breaking SU (5) F × U (1) V F → G SM is realized similarly to the previous SO (10)model. We start to derive the effective potential for the zero mode of A , namely parameters ˜ d and ˜ n inEq. (4.14). The effective potential is generated by quantum corrections from matter and the gaugefields in the model. We here focus on the contributions to the effective potential from bulk fermionfields; the contributions from the gauge field are studied in the next subsection.The contributions in the E 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 where the parameters ˜ d and ˜ n in Eq. (4.14)belongs to the 45-dimensional adjoint representation. For this purpose, we consider maximal SU (2) E F rotation of the SO (10) × U (1) V ′ decompositions in Eqs. (4.7)–(4.10). This leads to a new basis E ⊃ SO (10) ′ × U (1) V ′′ , where Eqs. (4.7)–(4.10) are changed to A µ (45 ′ ) = A µ (24 ) (+ , +) + A µ (1 V ′ F ) (+ , +) + A µ (10 ) ( − , − ) + A µ (10 − ) ( − , − ) , (4.15) A µ (1 ′ ) = A µ (1 V ′′ ) (+ , +) , (4.16) A µ (16 ′ ) = A µ (10 − ) (+ , − ) + A µ (5 − ) ( − , +) + A µ (1 ) ( − , +) , (4.17) A µ (16 ′ ) = A µ (10 ) (+ , − ) + A µ (5 ) ( − , +) + A µ (1 − ) ( − , +) . (4.18)In this expression, the left-hand sides transform as the irreducible representations of SO (10) ′ , and A µ (1 V ′ F ) (+ , +) and A µ (1 V ′′ ) (+ , +) are linear combinations of A µ (1 V F ) (+ , +) and A µ (1 V ′ ) (+ , +) . Note thatthe fields in A µ (45 ′ ) have the same orbifold parities as those in Eqs. (3.34) and (3.35). This coincideswith the fact that the zero mode of A parametrized by ˜ d and ˜ n appears in the adjoint representation A (45 ′ ) of SO (10) ′ . In addition, since the gauge interactions respect the SO (10) ′ symmetry and theorbifold parities, other SO (10) ′ representations in the E model should have the same orbifold paritiesas in the previous SO (10) model in Sec. 3.2 up to overall signs. These observations imply that thecontributions to the effective potential in the E model can be written by using the contributions inthe SO (10) model. 14et us consider the contribution from a bulk adjoint fermion Φ (˜ η ) A having ˜ η = η η , which is decom-posed into the SO (10) ′ multiplets as Φ (˜ η ) A = Φ (˜ η ) A (45 ′ ) + Φ (˜ η ) A (1 ′ ) + Φ (˜ η ) A (16 ′ ) + Φ (˜ η ) A (16 ′ ). One can seethat, for instance, the orbifold parity of Φ (+1) A (45 ′ ) (Φ (+1) A (16 ′ )) corresponds to one of the adjoint fieldwith ˜ η = +1 (16-plet with ˜ η = − 1) in the previous SO (10) model. As in the previous section, we denotethe contribution from a field that has a bosonic degree of freedom of the R -dimensional representation( R = 27 , 78) by F R ( ˜ d, ˜ n, δ ). From the above discussion, the contribution F ( ˜ d, ˜ n, δ ) is shown by theterms in Eqs. (3.42)–(3.44) as F ( ˜ d, ˜ n, 0) = F ( ˜ d, ˜ n, 0) + 2 F ( ˜ d, ˜ n, , (4.19) F ( ˜ d, ˜ n, 1) = F ( ˜ d, ˜ n, 1) + 2 F ( ˜ d, ˜ n, . (4.20)The explicit form of the contribution is F ( ˜ d, ˜ n, δ ) = − C (cid:20) f ( ˜ d ) + 4 ˆ f ( ˜ d + 1) + 4 ˆ f (˜ n ) + 4 ˆ f (˜ n + 1) + 2 ˆ f ( ˜ d + ˜ n ) + 2 ˆ f ( ˜ d + ˜ n + 1) + 2 ˆ f ( ˜ d − ˜ n )+ 2 ˆ f ( ˜ d − ˜ n + 1) + 2 ˆ f ( ˜ d + ˜ n + δ ) + 2 ˆ f ( ˜ d − ˜ n + δ ) + ˆ f (2 ˜ d + δ ) + ˆ f (2˜ n + δ ) (cid:21) . (4.21)Similar discussion holds for the contributions from a bulk 27-plet fermion Φ (˜ η ) F having ˜ η = η η . Inthis model, the orbifold parities of the 27-plet areˆ P · Φ (˜ η ) F = η · n − Φ (˜ η ) F (16) + Φ (˜ η ) F (10) + Φ (˜ η ) F (1) o , (4.22)ˆ P · Φ (˜ η ) F = η · n +Φ (˜ η ) F ((15 , − Φ (˜ η ) F ((6 , o , (4.23)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) E F . One can find the SO (10) → SU (5) F × U (1) V F decomposition:Φ (˜ η ) F (16) = Φ F (10 ) ( − η , + η ) + Φ F (5 − ) ( − η , − η ) + Φ F (1 ) ( − η , − η ) , (4.24)Φ (˜ η ) F (10) = Φ F (5 − ) (+ η , + η ) + Φ F (5 ) (+ η , − η ) , (4.25)Φ (˜ η ) F (1) = Φ F (1 ) (+ η , − η ) , (4.26)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) V F charge Q . The superscript in the right-hand sides indicates the eigenvaluesof the parity ( ˆ P , ˆ P ) of each field. Using the SU (2) E F rotation, one can obtain the SO (10) ′ × U (1) V ′′ decomposition: Φ (˜ η ) F (16 ′ ) = Φ F (10 ) ( − η , + η ) + Φ F (5 ) (+ η , − η ) + Φ F (1 ) (+ η , − η ) , (4.27)Φ (˜ η ) F (10 ′ ) = Φ F (5 − ) (+ η , + η ) + Φ F (5 − ) ( − η , − η ) , (4.28)Φ (˜ η ) F (1 ′ ) = Φ F (1 ) ( − η , − η ) . (4.29)15rom the expressions, one can obtain the contributions to the effective potential from the 27-plet as F ( ˜ d, ˜ n, 0) = F ( ˜ d, ˜ n, 1) + F ( ˜ d, ˜ n, , (4.30) F ( ˜ d, ˜ n, 1) = F ( ˜ d, ˜ n, 0) + F ( ˜ d, ˜ n, , (4.31)where terms in the right-hand sides are found in Eqs. (3.42)–(3.44). More explicitly, we obtain F ( ˜ d, ˜ n, δ ) = − C (cid:20) f ( ˜ d ) + 2 ˆ f ( ˜ d + 1) + 2 ˆ f (˜ n ) + 2 ˆ f (˜ n + 1) + ˆ f ( ˜ d + ˜ n + 1 − δ ) + ˆ f ( ˜ d − ˜ n + 1 − δ ) (cid:21) . (4.32)In Appendix A, we also derive the effective potential and confirm the result in Eqs. (4.21) and (4.32)by using another explicit formulation.Several comments are in order. In our setup, ˜ d and ˜ n parametrize the Wilson line phase degreesof freedom. Thus the contributions to the effective potential in Eqs. (4.21) and (4.32) are invariantunder discrete shift of the parameters as ˜ n → ˜ n + 2 or ˜ d → ˜ d + 2. In addition, the residual SU (5)symmetry of the model ensures that the contributions are invariant under the changes of signs of ˜ n and˜ d and the exchange of values of ˜ n and ˜ d . Similar invariance is found in the SU (7), SU (8), and SO (10)models. In addition, in this E model, the contributions are also unchanged under the transformation(˜ n, ˜ d ) → (˜ n + 1 , ˜ d + 1). This invariance is not accidental at the one-loop level but guaranteed by the E symmetry in this model as shown in Appendix B. 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 massparameter and the background fields ( ˜ d, ˜ n ), of the gauge field. The detailed derivation of the massspectrum is shown in Appendix A.2. We here shortly summarize the calculation procedure and theresult of the calculation of the effective potential.The KK mass spectrum is obtained with the solution of equations of motion (EOM) of the gauge fieldin the bulk. The bulk EOM is simplified in the basis of SU (6) × SU (2) E , where the U (1) V ′ component A µ ( V ′ ) of the gauge field is written by a linear combination of the components A µ ( n (3) ), A µ ( d (3) ), and A µ ( X ), where n (3) ( d (3) ) implies the Cartan generator of SU (2) n ( SU (2) d ), which is defined belowEq. (4.14), and X is a generator involved in SU (6) /SU (2) d . Since the parameter ˜ n ( ˜ d ) appears in thebulk EOM, A µ ( n (3) ) ( A µ ( d (3) )) mixes with another SU (2) n ( SU (2) d ) gauge field component, which wecall A µ ( n (2) ) ( A µ ( d (2) )). Therefore, as discussed in sec. A.2, we should solve the EOM as simultaneousequations concerning the set of fields ( A µ ( n (3) ) , A µ ( n (2) ) , A µ ( d (3) ) , A µ ( d (2) ) , A µ ( X )).To obtain the solution of the EOM, the boundary condition should be imposed. The boundarycondition at the fixed points y = 0 and πR corresponds to ( ˆ P , ˆ P ) parities as in Eqs. (4.7)–(4.10),16here even (odd) parity means the Neumann (Dirichlet) boundary condition. In addition the boundarycondition for A µ ( V ′ ) is effectively modified from ( ˆ P , ˆ P ), since it has a localized mass term at theboundary y = 0 [22]. Note that if the localized mass scale is much larger than the compactificationscale 1 /R , the Neumann boundary condition of A µ ( V ′ ) at y = 0 is effectively modified to the Dirichletboundary 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: m ( n )2KK = (cid:18) n + ρ + R (cid:19) , ρ + = 1 π arcsin r S + S , n = 0 , ± , ± , · · · , (4.33) m ( n )2KK = (cid:18) n + ρ − R (cid:19) , ρ − = 1 π arcsin r S − S , n = 0 , ± , ± , · · · , (4.34) S = 52 h sin ( π ˜ d ) + sin ( π ˜ n ) i , S = 52 (cid:20)(cid:16) sin ( π ˜ d ) − sin ( π ˜ n ) (cid:17) + 3625 sin ( π ˜ d ) sin ( π ˜ n ) (cid:21) / , (4.35)in addition to a ˜ d , ˜ n -independent mass.For the extra-dimensional component A , there is no direct coupling to the localized mass parameter.However, the boundary condition of A could be modified in accordance with the modification of theboundary condition of A µ due to a gauge fixing term, which mixes A with A µ . We demonstrate howthe boundary condition of A 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 thecontributions to the effective potential from the gauge sector in this model. The contribution from abosonic degree of freedom is F A ( ˜ d, ˜ n ) = − C (cid:20) f ( ˜ d ) + 4 ˆ f ( ˜ d + 1) + 4 ˆ f (˜ n ) + 4 ˆ f (˜ n + 1) + 4 ˆ f ( ˜ d + ˜ n ) + 4 ˆ f ( ˜ d − ˜ n )+ 2 ˆ f ( ˜ d + ˜ n + 1) + 2 ˆ f ( ˜ d − ˜ n + 1) + ˆ f (2 ρ + ) + ˆ f (2 ρ − ) (cid:21) . (4.36)Using the contributions in Eqs. (4.21), (4.32), and (4.36), we can express the effective potential inthe E model. The matter content of the model is specified by n ( ± ) R ( R = 27 , 78) that is a numberof R -dimensional bulk fermion fields with ˜ η = ± 1, and we use N E = ( n (+)27 , n ( − )27 , n (+)78 , n ( − )78 ). Theone-loop effective potential for ˜ d and ˜ n is V E ( ˜ d, ˜ n, N E ) = 3 F A ( ˜ d, ˜ n ) − (cid:20) n (+)27 F ( ˜ d, ˜ n, 0) + n ( − )27 F ( ˜ d, ˜ n, 1) + n (+)78 F ( ˜ d, ˜ n, 0) + n ( − )78 F ( ˜ d, ˜ n, (cid:21) . (4.37) In this section, we study vacuum structure of the one-loop effective potentials in the SU (7), SU (8), SO (10), and E models discussed in the previous sections. In general, positions of the vacua of thepotentials depend on bulk matter contents of the models. Without bulk matter fields, for instance,17able 1: Examples of vacuum configurations in SU (7), SU (8), SO (10), and E models. Matter contents,VEVs at the global minimum of the one-loop effective potentials, and physical squared mass eigenvaluesof A zero modes normalized by the typical mass parameter m in Eq. (5.1) are shown.model matter contents VEVs normalized squared masses SU (7) N = (0 , , , , , , , a = a = 0 . . , . SU (7) N = (2 , , , , , , , a = a = 0 . . , . SU (8) N = (0 , , , , , , , a = a = a = 0 . . , . , . SU (8) N = (2 , , , , , , , a = a = a = 0 . . , . , . SO (10) N = (0 , , , , , 0) ˜ a = 0, ˜ b = 0 . . , . SO (10) N = (0 , , , , , 0) ˜ a = 0, ˜ b = 0 . . , . E N E = (0 , , , 2) ˜ d = 0, ˜ n = 0 . . , . E N E = (0 , , , 2) ˜ d = 0, ˜ n = 0 . . , . /R . On the other hand, if there arebulk fields that lead to non-zero VEVs of the phases, then the residual symmetries are further brokenaround the compactification scale. In view of the gGHU scenario, we are interested in the case wherelow-energy symmetries are compatible with the standard model gauge group.In each model, we found matter contents for bulk fermion fields that lead to the symmetry breakinginto G SM . Examples are summarized in Table 1, where the matter contents and the VEVs at vacua areshown. In addition, we show the physical mass spectrum of the zero mode of A in a normalization,whose definition is expressed in Eq. (5.1). In all the models, bulk fermions that have periodic boundaryconditions (˜ η = 0) are required to obtain non-zero VEVs.In the SU (7) model, non-zero VEVs a = a = 0 (mod 1) are obtained at vacua, where the residualsymmetry SU (5) × SU (2) × U (1) is broken to G SM . Figure 1 left shows the contour plot of the one-loopeffective potential V ( a i , N ) for the case with N = (0 , , , , , , , a i → a i + 2 for each i , thepotential has invariance, which reflects the phase property of each a i . Also one can see the potentialis invariant under a i → − a i or ( a , a ) → ( a , a ). In this figure, one can see that there appear fourdegenerate vacua, which are physically equivalent. Around the vacua, the physical zero mode of A becomes massive and the mass matrix is evaluated as( M ) jk = m ∂ ∂a k ∂a j V ( a i , N ) C , m = 3 g D π R , g D = g √ πR , (5.1)where we introduce m as a typical (squared) mass scale and the effective four-dimensional gaugecoupling g D . In the present case, at one of the vacua, the VEVs take a = a = 0 . . m and 65 . m . The values in Table 1 show the eigenvalues ofsquared masses normalized by m . For the case with N = (2 , , , , , , , m in Table 1.18igure 1: The color contour plots of the one-loop effective potentials as functions of the parametrizedzero modes of A in SU (7) model with N = (0 , , , , , , , 0) (left), SO (10) model with N =(0 , , , , , 0) (center), and E model with N E = (0 , , , 2) (right). From light orange region to darkblue region, values of the potential decrease. In each figure, we show positions of the global minima ofthe potential by the square symbols.In the SU (8) model, the residual symmetry is SU (5) × SU (3) × U (1), which is broken to G SM by VEVs a = a = a = 0 (mod 1). For instance, the symmetry breaking is achieved for the caseswith N = (0 , , , , , , , 0) and N = (2 , , , , , , , A is evaluated from V ( a i , N ) similarly to Eq. (5.1). InTable 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 , , 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 theparameters ˜ a and ˜ b 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 correspondingVEVs that lead to G SM . For the case with N = (0 , , , , , A using the potential V (˜ a, ˜ b, N ) similarly to Eq. (5.1). The eigenvalues of the squared massesare also shown in Table 1.Finally we discuss the E model, where ˜ d and ˜ n in Eq. (4.14) parametrize the VEVs. The residualsymmetry of the model is SU (5) F × U (1) V F . We are particularly interested in the vacua that lead to G SM , where VEVs are ˜ d = 0 (mod 2) and ˜ n = 0 (mod 1). We show two cases with N E = (0 , , , N E = (0 , , , 2) in Table 1. For the latter case, the contour plot of V E ( ˜ d, ˜ n, N E ) is also shown inFigure 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 become massive. TheVEVs and the squared mass eigenvalues are shown in Table 1.19 Summary and discussion We have studied the five-dimensional gGHU models, namely the applications of the Hosotani mechanismto the unified gauge symmetry breaking, on the orbifold compactification S / Z . In these models, VEVsof the zero modes of the extra-dimensional gauge fields, whose dynamics reflects degrees of freedom ofWilson line phases, are available to break the residual symmetry. The effective potential of the zeromode 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 gauge symmetries. In eachmodel, the standard model gauge symmetry is achieved at a low-energy regime when the suitable bulkfermion fields are contained. We have derived the one-loop effective potentials for the zero modes ofthe extra-dimensional gauge fields in all the models. We have also studied the effects of the localizedmass term for the gauge field induced by the anomaly in the E model.We have analyzed vacuum structures of the effective potentials and shown examples of the mattercontents for bulk fermion fields that lead to the non-trivial VEVs of the zero modes and the standardmodel gauge symmetry at the vacua. Our discussions have shown that not so many bulk fermionsare required to achieve the desired vacua; it implies that the unified symmetry is naturally broken bydynamics of the Wilson line phases, owing to extra dimensions in our Universe.To make our discussion more concrete, the mass spectrum of the bulk fermion fields and the standardmodel matter sector should be explicitly treated. In this case, one can examine the renormalizationgroup evolution of the gauge coupling constants, which has large dependence on the bulk fermion massspectrum. Since the mass spectrum is not severely constrained, we tend to lose precise predictions forthe values of the gauge couplings in this setup. For instance, if we introduce bulk masses for bulk fermionfields slightly smaller than the compactification scale, which do not change the present analysis of theeffective potentials approximately, their contributions to the evolution are suppressed. The standardmodel fermions and the Higgs scalar can be introduced into the models as bulk fields or localized fieldsin the present setup. ¶ As mentioned, the models based on SU ( N ) ( N = 7 , 8) and SO (10) share a partof 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 fromthe known models. This implies that the standard model fields are realized as boundary localizedfields in our SU ( N ) and SO (10) models. On the other hand, the standard model fields are naturallyincorporated into bulk fields in the E model k . In addition, the supersymmetric extension of the E model can supply the doublet-triplet splitting via an analogues of the missing partner mechanism thatis often discussed in the flipped SU (5) models. These subjects are left to our future studies. ¶ This means that the hierarchy problem is not addressed in the present study, as in usual four-dimensional non-supersymmetric GUT models. k Recently, the classification of the standard model fermions from bulk 27-plet fields in models with orbifold breakingsof the E unified symmetry is studied in Ref. [23]. cknowledgments The authors would like to thank N. Yamatsu for valuable discussions. A Calculation of the effective potential in the E 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 model isshown, with the help of the effective potential in the SO (10) model. In this subsection, we derive thecontribution by using another explicit formulation.For deriving the contribution, a key point is that the U (1) directions accompanied by ˜ d and ˜ n inEq. (4.14) are identified to generators in SU (6) and SU (2) E , respectively. This can be realized becausethe zero mode of A appears in the adjoint representations of SU (6) and SU (2) E . To see this, we firstfocus on the decomposition of A µ in Eqs. (4.7)–(4.10). It is useful to consider the U (1) Y F subgroup thatappears in SU (5) F ⊃ SU (3) C × SU (2) L × U (1) Y F . One can further decompose the SU (5) F × U (1) V F representations of A µ as A µ (24 ) (+ , +) = A µ ( G ) (+ , +) + A µ ( W ) (+ , +) + A µ ( n Y F ) (+ , +) + A µ ( Q ) (+ , +) + A µ ( Q ) (+ , +) , (A.1) A µ (1 V F ) (+ , +) = A µ ( n V F ) (+ , +) , (A.2) A µ (10 − ) (+ , − ) = A µ ( X ) (+ , − ) + A µ ( U ) (+ , − ) + A µ ( E ) (+ , − ) , (A.3) A µ (10 ) (+ , − ) = A µ ( X ) (+ , − ) + A µ ( U ) (+ , − ) + A µ ( E ) (+ , − ) , (A.4) A µ (1 V ′ ) (+ , +) = A µ ( n V ′ ) (+ , +) , (A.5) A µ (10 ) ( − , − ) = A µ ( q ) ( − , − ) + A µ ( d ) ( − , − ) + A µ ( n ) ( − , − ) , (A.6) A µ (5 − ) ( − , +) = A µ ( u ) ( − , +) + A µ ( ℓ ) ( − , +) , (A.7) A µ (1 ) ( − , +) = A µ ( e ) ( − , +) , (A.8) A µ (10 − ) ( − , − ) = A µ ( q ) ( − , − ) + A µ ( d ) ( − , − ) + A µ ( n ) ( − , − ) , (A.9) A µ (5 ) ( − , +) = A µ ( u ) ( − , +) + A µ ( ℓ ) ( − , +) , (A.10) A µ (1 − ) ( − , +) = A µ ( e ) ( − , +) , (A.11)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) Y F × U (1) V F symmetry. We denote the complex conjugate of R by R . Note that the U (1) Y hyperchargeand the charge of the Cartan generator of SU (2) R , which we denote by T R , are linear combinationsof U (1) Y F and U (1) V F 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 isa linear combination of the zero modes of A µ ( n Y F ) (+ , +) and A µ ( n V F ) (+ , +) . If the symmetry breaking SU (5) F × U (1) V F → G SM is realized, then the zero modes A µ ( Q ) (+ , +) , A µ ( Q ) (+ , +) , and a linear21able 2: Representations and charges under SU (3) C , SU (2) L , U (1) Y F , U (1) V F , U (1) Y , and T R . G W n Y F Q n V F U X E n V ′ d q n u ℓ eSU (3) C SU (2) L U (1) Y F − 23 16 23 16 − − − U (1) V F − − − − − − U (1) Y − − − 13 16 − − T R − − − combination of A µ ( n Y F ) (+ , +) and A µ ( n V F ) (+ , +) become massive with would-be NG bosons that belongto the zero mode of A .We next focus on the SU (6) F × SU (2) E F decomposition of A . Similarly to A µ in Eqs. (4.11)–(4.13),we can obtain A ((35 , A (24 ) ( − , − ) + A (5 ) (+ , − ) + A (5 − ) (+ , − ) + A (1 K F ) ( − , − ) , (A.12) A ((1 , A (1 ) (+ , − ) + A (1 − ) (+ , − ) + A (1 E F ) ( − , − ) , (A.13) A ((20 , A (10 ) (+ , +) + A (10 − ) (+ , +) + A (10 − ) ( − , +) + A (10 ) ( − , +) . (A.14)These equations can be rewritten by the representations in Table 2 as A ((35 , A ( G ) ( − , − ) + A ( W ) ( − , − ) + A ( n Y F ) ( − , − ) + A ( Q ) ( − , − ) + A ( Q ) ( − , − ) + A ( u ) (+ , − ) + A ( ℓ ) (+ , − ) + A ( u ) (+ , − ) + A ( ℓ ) (+ , − ) + A ( n K F ) ( − , − ) , (A.15) A ((1 , A ( e ) (+ , − ) + A ( e ) (+ , − ) + A ( n E F ) ( − , − ) , (A.16) A ((20 , A ( q ) (+ , +) + A ( d ) (+ , +) + A ( n ) (+ , +) + A ( q ) (+ , +) + A ( d ) (+ , +) + A ( n ) (+ , +) + A ( X ) ( − , +) + A ( U ) ( − , +) + A ( E ) ( − , +) + A ( X ) ( − , +) + A ( U ) ( − , +) + A ( E ) ( − , +) , (A.17)where A ( n K F ) ( − , − ) and A ( n E F ) ( − , − ) are linear combinations of A ( n V F ) ( − , − ) and A ( n V ′ ) ( − , − ) .From the expression, using the SU (2) R flip, we can obtain the E ⊃ SU (6) × SU (2) E decompositionof A : A = A ((35 , ′ ) + A ((1 , ′ ) + A ((20 , ′ ) , (A.18)where the terms in the right-hand side transform as the irreducible representations of SU (6) × SU (2) E .22hey are rewritten as follows: A ((35 , ′ ) = A ( G ) ( − , − ) + A ( W ) ( − , − ) + A ( n Y ) ( − , − ) + A ( X ) ( − , +) + A ( X ) ( − , +) + A ( d ) (+ , +) + A ( ℓ ) (+ , − ) + A ( d ) (+ , +) + A ( ℓ ) (+ , − ) + A ( n K ) ( − , − ) , (A.19) A ((1 , ′ ) = A ( n ) (+ , +) + A ( n ) (+ , +) + A ( n E ) ( − , − ) , (A.20) A ((20 , ′ ) = A ( q ) (+ , +) + A ( u ) (+ , − ) + A ( e ) (+ , − ) + A ( q ) (+ , +) + A ( u ) (+ , − ) + A ( e ) (+ , − ) + A ( Q ) ( − , − ) + A ( U ) ( − , +) + A ( E ) ( − , +) + A ( Q ) ( − , − ) + A ( U ) ( − , +) + A ( E ) ( − , +) . (A.21)In this expression, A ( n Y F ) ( − , − ) , A ( n K F ) ( − , − ) , and A ( n E F ) ( − , − ) are rearranged as A ( n Y ) ( − , − ) , A ( n K ) ( − , − ) , and A ( n E ) ( − , − ) . One can now clearly see that ˜ d and ˜ n in Eq. (4.14), which parametrizethe zero mode of A and thus are involved in the parity (+ , +) fields in Eqs. (A.19)–(A.21), belong tothe gauge field of an SU (2) subgroup of SU (6) and SU (2) E , respectively; these SU (2) symmetries arejust what we called SU (2) d and SU (2) n below Eq. (4.14).Let us derive the contributions to the effective potential for ˜ d and ˜ n from a bulk adjoint fieldΦ (˜ η ) A , where ˜ η is a parameter related to the periodicity of the field. For this purpose, we show thetransformation 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) × SU (2) E representations asΦ (˜ η ) A = Φ (˜ η ) A ((35 , ′ ) + Φ (˜ η ) A ((1 , ′ ) + Φ (˜ η ) A ((20 , ′ ) . (A.22)One can further decompose the fields asΦ (˜ η ) A ((35 , ′ ) =Φ A ( G ) (+˜ η ) + Φ A ( W ) (+˜ η ) + Φ A ( n Y ) (+˜ η ) + Φ A ( X ) ( − ˜ η ) + Φ A ( X ) ( − ˜ η ) + Φ A ( d ) (+˜ η ) + Φ A ( ℓ ) ( − ˜ η ) + Φ A ( d ) (+˜ η ) + Φ A ( ℓ ) ( − ˜ η ) + Φ A ( n K ) (+˜ η ) , (A.23)Φ (˜ η ) A ((1 , ′ ) =Φ A ( n ) (+˜ η ) + Φ A ( n ) (+˜ η ) + Φ A ( n E ) (+˜ η ) , (A.24)Φ (˜ η ) A ((20 , ′ ) =Φ A ( q ) (+˜ η ) + Φ A ( u ) ( − ˜ η ) + Φ A ( e ) ( − ˜ η ) + Φ A ( q ) (+˜ η ) + Φ A ( u ) ( − ˜ η ) + Φ A ( e ) ( − ˜ η ) + Φ A ( Q ) (+˜ η ) + Φ A ( U ) ( − ˜ η ) + Φ A ( E ) ( − ˜ η ) + Φ A ( Q ) (+˜ η ) + Φ A ( U ) ( − ˜ η ) + Φ A ( E ) ( − ˜ η ) , (A.25)where in the right-hand sides ± ˜ η denotes the periodicity that coincides with the eigenvalue of thetranslation operator ˆ P ˆ P .From the above expression, we can easily see the SU (2) n transformation law of Φ (˜ η ) A with theperiodicity. In Φ (˜ η ) A ((1 , ′ ), there is one triplet of the periodicity +˜ η . In Φ (˜ η ) A ((20 , ′ ), there are twentydoublets, of which the twelve have +˜ η and the rest have − ˜ η .For SU (2) d , a little difficulty remains; the transformation law is not manifest in Eqs. (A.23)–(A.25)since SU (2) d does not commute with SU (3) C . Fortunately, we can find another SU (2) that helps usto see the SU (2) d transformation law of each SU (6) × SU (2) E representation in Eqs. (A.23)–(A.25)taking account of the periodicity ± ˜ η . Note that the SU (4) subgroup in SU (6) /SU (2) L commutes23ith the translation operator ˆ P ˆ P and involves both SU (3) C and SU (2) d . This implies that usingany SU (2) subgroup of SU (3) C instead of SU (2) d , we can derive the SU (2) d transformation law ofeach SU (6) × SU (2) E representation with the periodicity. Accordingly, under SU (2) d , it is realizedthat there are one triplet of +˜ η , four doublets of +˜ η , and four doublets of − ˜ η in Φ (˜ η ) A ((35 , ′ ). ForΦ (˜ η ) A ((20 , ′ ), eight doublets of +˜ η and four doublets of − ˜ η are contained. Among the doublets, sinceΦ (˜ η ) A ((20 , ′ ) is also SU (2) n doublet, there are bi-doublets under SU (2) d × SU (2) n ; four bi-doublets of+˜ η and two bi-doublets of − ˜ η are contained.In this way, the decompositions in Eqs. (A.23)–(A.25) tell us the transformations of the fields under SU (2) d × SU (2) n . The result isΦ (˜ η ) A ((35 , ′ ) ∋ × Φ A (2 , (+˜ η ) + 4 × Φ A (2 , ( − ˜ η ) + 1 × Φ A (3 , (+˜ η ) , (A.26)Φ (˜ η ) A ((1 , ′ ) ∋ × Φ A (1 , (+˜ η ) , (A.27)Φ (˜ η ) A ((20 , ′ ) ∋ × Φ A (1 , (+˜ η ) + 4 × Φ A (1 , ( − ˜ η ) + 4 × Φ A (2 , (+˜ η ) + 2 × Φ A (2 , ( − ˜ η ) , (A.28)where in the right-hand sides Φ A ( R, R ′ ) transforms under SU (2) d ( SU (2) n ) as R ( R ′ ) and the super-script for each term denotes the periodicity. Hence one can obtainΦ ( ± ) A ∋ h Φ A (2 , ( ± ) + Φ A (2 , ( ∓ ) + Φ A (1 , ( ± ) + Φ A (1 , ( ∓ ) i + 4Φ A (2 , ( ± ) + 2Φ A (2 , ( ∓ ) + Φ A (3 , ( ± ) + Φ A (1 , ( ± ) . (A.29)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) E F multiplets as Φ (˜ η ) F = Φ (˜ η ) F ((15 , (˜ η ) F ((6 , . (A.30)Further decomposition leads toΦ (˜ η ) F ((15 , F ( q ) ( − ˜ η ) + Φ F ( d ) ( − ˜ η ) + Φ F ( n ) ( − ˜ η ) + Φ F ( D ) (+˜ η ) + Φ F ( L ) (+˜ η ) , (A.31)Φ (˜ η ) F ((6 , F ( u ) (+˜ η ) + Φ F ( ℓ ) (+˜ η ) + Φ F ( s ) ( − ˜ η ) + Φ F ( D ) ( − ˜ η ) + Φ F ( L ) ( − ˜ η ) + Φ F ( e ) (+˜ η ) . (A.32)In the right-hand sides Φ F ( R ) transforms under SU (3) C × SU (2) L × U (1) Y F × U (1) V F symmetry asin Table 2 and 3 and the superscript for each term denotes the periodicity. Using the SU (2) R flip, wecan obtain the E ⊃ SU (6) × SU (2) E decomposition asΦ (˜ η ) F ((15 , ′ ) = Φ F ( q ) ( − ˜ η ) + Φ F ( u ) (+˜ η ) + Φ F ( e ) (+˜ η ) + Φ F ( D ) (+˜ η ) + Φ F ( L ) ( − ˜ η ) , (A.33)Φ (˜ η ) F ((6 , ′ ) = Φ F ( d ) ( − ˜ η ) + Φ F ( ℓ ) (+˜ η ) + Φ F ( s ) ( − ˜ η ) + Φ F ( D ) ( − ˜ η ) + Φ F ( L ) (+˜ η ) + Φ F ( n ) ( − ˜ η ) . (A.34)Therefore, a 27-plet involves the fields that transform under SU (2) d × SU (2) n asΦ (˜ η ) F ((15 , ′ ) ∋ × Φ F (2 , (+˜ η ) + 2 × Φ F (2 , ( − ˜ η ) , (A.35)Φ (˜ η ) F ((6 , ′ ) ∋ × Φ F (1 , (+˜ η ) + 2 × Φ F (1 , ( − ˜ η ) + 1 × Φ F (2 , ( − ˜ η ) . (A.36)24able 3: Representations and charges under SU (3) C , SU (2) L , U (1) Y F , U (1) V F , U (1) Y , and T R . D L sSU (3) C SU (2) L U (1) Y F − 13 12 U (1) V F − − U (1) Y − − T R ( ± ) F ∋ h Φ F (2 , ( ± ) + Φ F (2 , ( ∓ ) + Φ F (1 , ( ± ) + Φ F (1 , ( ∓ ) i + Φ F (2 , ( ∓ ) . 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 inSec. 4.4. The contribution is generated by the gauge field, whose U (1) V ′ component is assumed to havea large mass term at the y = 0 boundary due to the anomaly cancellation. The mass term effectivelymodifies the boundary condition and the KK masses of some components of the gauge field. Withoutthe boundary mass term, the contribution takes the form of Eq. (4.21) with δ = 0, since the gauge fieldbelongs to the adjoint representation. The modification of the KK mass alters a part of the contributionin Eq. (4.21).As explained in Sec. 4.4, the KK mass spectrum that is affected by the localized mass term isobtained as a solution to the EOM of the following set of the fields: (cid:16) A µ ( n (3) ) , A µ ( n (2) ) , A µ ( d (3) ) , A µ ( d (2) ) , A µ ( X ) (cid:17) , (A.37)where n (3 , , d (3 , , and X imply generators of SU (2) n , SU (2) d , and U (1) X , respectively. For conve-nience we introduce a column vector Φ αµ ( α = 1–5) that consists of the fields:Φ αµ ≡ (cid:18) A µ ( d ) − iA µ ( d ) √ , A µ ( d ) − iA µ ( d ) √ , A µ ( X ) , A µ ( n ) − iA µ ( n ) √ , A µ ( n ) − iA µ ( n ) √ (cid:19) T . (A.38)The Lagrangian in Eq. (2.7) is diagonalized by the vector as L gauge ∋ η µν (Φ αµ ) † h (cid:3) δ αβ − ˜ D αβ i Φ βν , δ αβ = diag(1 , , , , , (A.39)˜ D αβ = diag (cid:16) ( ∂ − i ˜ d/R ) , ( ∂ + i ˜ d/R ) , ( ∂ ) , ( ∂ + i ˜ n/R ) , ( ∂ − i ˜ n/R ) (cid:17) . (A.40)25e introduce the KK mode expansion:Φ αµ ( x, y ) = ∞ X n = −∞ ψ ( n ) µ ( x ) φ α ( n ) ( y ) , ( (cid:3) + m ( n )2KK ) ψ ( n ) µ ( x ) = 0 . (A.41)The solution to the bulk EOM is obtained as follows: φ n ) ( y ) = e i ˜ dy/R h ξ cos( m ( n )KK y ) + ζ sin( m ( n )KK y ) i , (A.42) φ n ) ( y ) = e − i ˜ dy/R h ξ cos( m ( n )KK y ) + ζ sin( m ( n )KK y ) i , (A.43) φ n ) ( y ) = ξ cos( m ( n )KK y ) + ζ sin( m ( n )KK y ) , (A.44) φ n ) ( y ) = e − i ˜ ny/R h ξ cos( m ( n )KK y ) + ζ sin( m ( n )KK y ) i , (A.45) φ n ) ( y ) = e i ˜ ny/R h ξ cos( m ( n )KK y ) + ζ sin( m ( n )KK y ) i , (A.46)where ξ α and ζ α ( α = 1–5) are real constants and m ( n )KK is the mass of the n -th KK mode.In order to determine the KK mass m ( n )KK , the boundary condition at y = 0 and πR should beimposed to the solutions in Eqs. (A.42)–(A.46). In the present case, the condition is simplified in abasis where U (1) V ′ is manifest, since there is a boundary mass term for A µ ( V ′ ). We introduce the newbasis, Φ ′ αµ = U αβ Φ βµ , U αβ = 14 √ √ 10 2 i √ 10 0 0 04 i − √ i √ √ 15 2 √ √ i √ i √ − − i i √ 10 2 √ , (A.47)and we denote Φ ′ αµ = ( A µ ( d ) , A µ ( K ′ ) , A µ ( V ′ ) , A µ ( V ) , A µ ( n )) T , (A.48)where A µ ( K ′ ), A µ ( V ′ ), and A µ ( V ) are defined by (cid:18) A µ ( K ′ ) A µ ( K ) (cid:19) = 1 √ (cid:18) √ −√ √ √ (cid:19) (cid:18) A µ ( d ) A µ ( X ) (cid:19) , (cid:18) A µ ( V ′ ) A µ ( V ) (cid:19) = 14 (cid:18) √ √ √ −√ (cid:19) (cid:18) A µ ( K ) A µ ( n ) (cid:19) . (A.49)It is realized that A µ ( V ′ ) and A µ ( V ) correspond to the U (1) V ′ and U (1) V gauge field, respectively. Byusing the above fields, the boundary condition is simplified; at y = 0, the condition is written asΦ ′ µ = Φ ′ µ = 0 , ∂ y Φ ′ µ = ∂ y Φ ′ µ = 0 , (2 ∂ y − M )Φ ′ µ = 0 , (A.50)where M represents the boundary mass parameter. On the other hand, at y = πR , the condition iswritten as Φ ′ µ = Φ ′ µ = 0 , ∂ y Φ ′ µ = ∂ y Φ ′ µ = ∂ y Φ ′ µ = 0 . (A.51)Imposing the boundary condition to the solutions in Eqs. (A.42)–(A.46), we can determine the KKmass m ( n )KK . In our model, we are interested in the case where M is much larger than the compactification26cale 1 /R . In this case, we obtain sin ( m ( n )KK πR ) = S ± S , (A.52)where S and S are found in Eq. (4.35). The above equation leads to Eqs. (4.33) and (4.34).If there were no boundary mass term, the gauge field and a bulk adjoint field with δ = 0, whichgives the contribution F ( ˜ d, ˜ n, 0) in Eq. (4.21), have the same KK mass spectrum. In this case, the n -th KK masses of the fields in Eq. (A.37) are (cid:18) n ± ˜ dR (cid:19) , (cid:18) n ± ˜ nR (cid:19) , (A.53)and n /R . In the contribution F ( ˜ d, ˜ n, f (2 ˜ d ) and ˆ f (2˜ n ). The boundary mass term alter the KK masses in Eq. (A.53) into theform in Eqs. (4.33) and (4.34). Thus the contribution from the gauge field with a large boundary massis obtained by the replacement ˆ f (2 ˜ d ) and ˆ f (2˜ n ) with ˆ f (2 ρ + ) and ˆ f (2 ρ − ) in F ( ˜ d, ˜ n, A µ , we also incorporate the contribu-tions from A . Although the U (1) V ′ component of A has neither zero mode nor direct coupling to thelocalized mass, the boundary condition of the field is modified from the Dirichlet to the Neumann typeeffectively by a large boundary mass ( M ≫ /R ). This modification is induced by proper treatmentof gauge fixing terms. In ref. [24], a similar situation can be seen in terms of the four-dimensionaleffective description. In Appendix C, five-dimensional treatment in a simple U (1) case is shown as anillustrative example. ∗∗ B Equivalence of vacua in the E model As mentioned in Sec. 4.3, the effective potential in the E model has the invariance and the periodicityunder the characteristic transformation (˜ n, ˜ d ) → (˜ n + 1 , ˜ d + 1). In this section, we show the invarianceof the potential is ensured by the E gauge transformation.In a five-dimensional orbifold model, the gauge transformation is generally written by A M ( x, y ) → A ′ M ( x, y ) = Λ( x, y ) A M ( x, y )Λ † ( x, y ) + ig Λ( x, y ) ∂ M Λ † ( x, y ) , (B.1)where Λ( x, y ) = exp ( iλ a ( x, y ) t a ) , (B.2) ∗∗ Also in ref. [25], discussion about five-dimensional treatment is found, while the form of the gauge fixing terms aredifferent from our example. λ a ( x, y ) is a five-dimensional gauge transformation function. When the gauge field satisfies theboundary condition given in Eqs. (2.1)–(2.4), then the gauge transformation implies A ′ µ ( x, − y ) = P ′ A ′ µ ( x, y ) P ′† + ig P ′ ∂ µ P ′† , A ′ µ ( x, πR − y ) = P ′ A ′ µ ( x, πR + y ) P ′† + ig P ′ ∂ µ P ′† , (B.3) A ′ ( x, − y ) = − P ′ A ′ ( x, y ) P ′† − ig P ′ ∂ P ′† , A ′ ( x, πR − y ) = − P ′ A ′ ( x, πR + y ) P ′† − ig P ′ ∂ P ′† , (B.4)where P ′ = Λ( x, − y ) P Λ † ( x, y ) , P ′ = Λ( x, πR − y ) P Λ † ( x, πR + y ) . (B.5)Although generally the gauge transformation changes the boundary conditions, one can find theparticular gauge transformation such that the relations P = P ′ and P = P ′ hold and the gauge fieldis shifted as A ′ M = A M . In this case, the non-linear terms in Eqs. (B.3) and (B.4) vanish and hencethe same boundary conditions are imposed on A M and A ′ M . For example, suppose that a vacuumconfiguration h A i is shifted to h A ′ i by the gauge transformation function that preserves the boundaryconditions, then the potential should have degenerate vacua around h A i and h A i ′ .We start to discuss our E model, where the VEV in Eq. (4.14) can be written as h A i = 1 gR ( ˜ dt ˜ d + ˜ nt ˜ n ) , (B.6)where t ˜ d and t ˜ n are generators of SU (2) d ⊂ SU (6) and SU (2) n = SU (2) E , respectively. In a definitebasis of the fundamental representation of SU (6) × SU (2) E , the parity matrix of the boundary conditioncan be written as follows: P = diag(+1 , +1 , +1 , +1 , +1 , − ⊗ diag(+1 , − , (B.7) P = diag(+1 , +1 , +1 , − , − , − ⊗ diag(+1 , − . (B.8)Here we take that t ˜ d generates mixing between 1st and 6th entries in the SU (6) fundamental represen-tation. As discussed in Sec. 2, the generators and the parity matrix satisfy { P , t ˜ d } = { P , t ˜ n } = { P , t ˜ d } = { P , t ˜ n } = 0 , [ t ˜ d , t ˜ n ] = 0 . (B.9)Let us now consider the gauge transformationΛ( x, y ) = exp (cid:16) i yR ( t ˜ d + t ˜ n ) (cid:17) . (B.10)From Eq. (B.1), one can see that the transformation shifts the parameters in Eq. (B.6) as( ˜ d, ˜ n ) → ( ˜ d ′ , ˜ n ′ ) = ( ˜ d + 1 , ˜ n + 1) . (B.11)The gauge-transformed field satisfies the boundary condition in Eq. (B.4) with the new parity matrices P ′ = P , P ′ = exp (cid:0) πit ˜ d (cid:1) exp (2 πit ˜ n ) P . (B.12)28s mentioned above, if P ′ = P is satisfied, then the effective potential should have invariance underthe shift in Eq. (B.11). This can be shown as follows. In the SU (6) × SU (2) E representation space inEqs. (B.7) and (B.8), the matrices exp(2 πit ˜ d ) and exp(2 πit ˜ n ) correspond to 2 π rotations of fundamentalrepresentation of SU (2) d and SU (2) E , respectively. Thus we obtainexp (cid:0) πit ˜ d (cid:1) = diag( − , +1 , +1 , +1 , +1 , − ⊗ diag(+1 , +1) , (B.13)exp (2 πit ˜ n ) = diag(+1 , +1 , +1 , +1 , +1 , +1) ⊗ diag( − , − . (B.14)The parity matrix P ′ explicitly written as follows: P ′ = diag( − , +1 , +1 , − , − , +1) ⊗ diag( − , +1)= diag(+1 , − , − , +1 , +1 , − ⊗ diag(+1 , − 1) (B.15) → diag(+1 , +1 , +1 , − , − , − ⊗ diag(+1 , − 1) = P . (B.16)In the last line, we use an SU (4) ⊂ SU (6) /SU (2) d rotation; this rotation does not change P and theVEV since both P and t ˜ d commute with the SU (4) subgroup. Although the boundary condition inthe E model is unchanged under the gauge transformation in Eq. (B.10), the transformation shifts theparameters as in Eq. (B.11). Therefore, the effective potential should be invariant against the shift inEq. (B.11). This leads to the periodicity in the potential. C Effective modification of the boundary condition of A withboundary breaking In Sec. A.2, we show that the boundary condition of A µ is effectively modified due to the existence ofthe boundary mass term. As mentioned, while the U (1) V ′ component of A does not have zero modes,one can see that the boundary condition of A is also modified by introducing proper gauge fixingterms in the theory. As a result, one can choose the specific gauge, namely ξ = 1 shown just below,where the KK masses of A coincide with those of A µ . Here, we consider a five-dimensional U (1) modelcompactified on an S / Z orbifold, and illustrate the essential feature of the modification in the simplesetup.In the U (1) model, the orbifold parity around the fixed point y = 0 is expressed as A µ ( − y ) = A µ ( y ) , A ( − y ) = − A ( y ) . (C.1)Then, as the E model discussed in Sec. 4, we study the effect of a mass term localized on this fixedpoint. Below, we consider an anomaly as the origin of the mass term, while it can be the Higgsmechanism. 29he 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 theSt¨uckelberg mass term [21] (on the boundary) L St = 12 M (cid:18) A µ + 1 √ M ∂ µ χ (cid:19) δ ( y ) , (C.2)which is U (1) invariant and thus the naive scale of the mass M is around the cutoff scale of the five-dimensional theory, much larger than the compactification scale. As well-known, such a huge massrepels the wave functions of the lower-laying KK modes of A µ to modify its boundary condition fromthe Neumann to the Dirichlet type effectively [22]. Below, we examine the effect on those of A , whichdoes not directly couple to the localized mass due to the orbifold parity. (See Ref. [24] for the sameanalysis in terms of the KK decomposed language.)For this purpose, as A is unphysical except for the zero mode, we should treat the gauge fixingterm properly. †† The mixing terms of the four-dimensional gauge field are L mix = − ∂ A µ ∂ µ A + √ M A µ ∂ µ χδ ( y ) , (C.3)and we adopt the usual gauge fixing term with a constant gauge parameter ξ , L GF = − ξ h ∂ µ A µ − ξ (cid:16) ∂ A + √ M χδ ( y ) (cid:17)i , (C.4)to remove the above mixing terms (up to surface terms). Then the quadratic terms of A and χ become L quad = 12 A (cid:0) − (cid:3) + ξ∂ (cid:1) A − δ ( y ) (cid:18) ξ √ M χ∂ A + 12 χ ( (cid:3) + ξM δ ( y )) χ (cid:19) . (C.5)Note that this quadratic part is essentially the same as the one in the case that the mass term originatesfrom the Higgs mechanism, and thus the derivation below is applied also for the case.Since there is an awkward term proportional to δ ( y ) in the quadratic part, we should regularizethe delta function. To be more concrete, we replace the delta function by a finite, sufficiently smoothfunction δ ǫ ( y ) that vanishes for | y | ≥ ǫ and is normalized as R ǫ − ǫ δ ǫ ( y ) dy ∼ ‡‡ The EOMs of A and χ are respectively (cid:0) − (cid:3) + ξ∂ (cid:1) A ( y ) + ξ ( ∂ δ ǫ ( y )) √ M χ = 0 , (C.6) − δ ǫ ( y ) (cid:16) ξ √ M ∂ A ( y ) + ( (cid:3) + ξM δ ǫ ( y )) χ (cid:17) = 0 , (C.7)where the y -dependences are explicitly shown. Due to the overall delta function in Eq. (C.7), we maynot 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 asin Ref. [26] to make the ”mass spectrum” of A unchanged. In such cases, however, the calculation of, for instance, theeffective potential would be complicated. ‡‡ One may impose the periodicity, for completeness if necessary. y = 0 as ξ √ M ∂ A (0) + ( (cid:3) + ξM δ ǫ (0)) χ = 0 . (C.8)As often done, we integrate Eq. (C.6), which is an odd function of y , over a tiny region 0 ≤ y ≤ z = O ( ǫ ).Then, the contribution of the regular function, − (cid:3) A is negligible and we get ξ h ∂ A ( y ) + δ ǫ ( y ) √ M χ i z = 0 . (C.9)Using Eq. (C.8) to remove the factor δ ǫ (0), we obtain ξ (cid:16) ∂ A ( z ) + δ ǫ ( z ) √ M χ (cid:17) + 1 √ M (cid:3) χ = 0 . (C.10)Its integration over again a tiny region, − ǫ ≤ z ≤ ǫ , leads to ξ (cid:16) A ( ǫ ) + √ M χ (cid:17) = 0 , (C.11)where we use A ( − ǫ ) = − A ( ǫ ). Operating the four-dimensional Laplacian (cid:3) on Eq. (C.11) andapplying Eq. (C.10) evaluated at y = ǫ where the delta function vanishes, we can derive an effectivemixed boundary condition 2 ξ (cid:3) A ( ǫ ) − ξ M ∂ A ( ǫ ) = 0 . (C.12)The result shows that A obeys the Dirichlet boundary condition in the limit M → 0; the conditionchanges to the Neumann boundary condition in the opposite limit M → ∞ . 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