Fermion Mass Hierarchy in Grand Gauge-Higgs Unification
OOCU-PHYS 499NITEP 10
Fermion Mass Hierarchyin Grand Gauge-Higgs Unification
Nobuhito Maru a,b and
Yoshiki Yatagai a , a Department of Mathematics and Physics, Osaka City University,Osaka 558-8585, Japan b Nambu Yoichiro Institute of Theoretical and Experimental Physics (NITEP),Osaka City University, Osaka 558-8585, Japan
Abstract
Grand gauge-Higgs unification of five dimensional SU (6) gauge theory on anorbifold S /Z is discussed. The Standard model (SM) fermions are introduced onone of the boundaries and some massive bulk fields are also introduced so that theycouple to the SM fermions through the mass terms on the boundary. Integratingout the bulk fields generates the SM fermion masses with exponentially small bulkmass dependences. The SM fermion masses except for top quark are shown to bereproduced by mild tuning the bulk masses. One-loop Higgs potential is calculatedand it is shown that the electroweak symmetry breaking occurs by introducingadditional bulk fields. Higgs boson mass is also computed. a r X i v : . [ h e p - ph ] M a r Introduction
Gauge-Higgs unification (GHU) [1] is one of the attractive scenarios among the physicsbeyond the Standard Model (SM), which solves the hierarchy problem by identifying theSM Higgs field with one of the extra spatial component of the higher dimensional gaugefield. In this scenario, the most appealing feature is that physical observables in Higgssector are calculable and predictable regardless of the non-renormalizable theory. Forinstance, the radiative corrections to Higgs mass and Higgs potential are known to befinite at one-loop [2] and two-loop [3] thanks to the higher dimensional gauge symmetry.Rich structures of the theory and its phenomenology have been investigated [4–12].The hierarchy problem was originally addressed in grand unified theory (GUT) as aproblem how the discrepancy between the GUT scale and the weak scale are kept. There-fore, the extension of GHU to grand unification is an interesting direction to explore. Thescenario of grand gauge-Higgs unification was discussed by one of the present authors [13], where the five dimensional SU (6) grand gauge-Higgs unification was considered and theStandard Model (SM) fermions were embedded in zero modes of some SU (6) multipletsin the bulk. This embedding was very elegant in that it was a minimal matter contentwithout massless exotic fermions which is not included in the SM. That immediatelymeans a minimal anomaly-free matter content. However, a crucial drawback was foundthat the down-type Yukawa couplings and the charged lepton Yukawa couplings are notallowed. This is because the left-handed quark (lepton) SU (2) L doublets and the right-handed down quark (charged lepton) SU (2) L singlets are embedded into different SU (6)multiplets. As a result, Yukawa coupling in GHU originated from the gauge couplingcannot be allowed. This feature seems to be generic in GHU, therefore we have to give upembedding all the SM fermions into the SU (6) multiplets in the bulk to obtain the SMYukawa couplings. Fortunately, we know another approach to generate Yukawa couplingin a context of GHU [15, 16]. In this approach, the SM fermions are introduced on theboundaries (i.e. fixed point in an orbifold compactification). We also introduce massivebulk fermions, which couple to the SM fermions through the mass terms on the boundary.Integrating out these massive fermions generates non-local SM fermion masses, which areproportional to the bulk to boundary couplings and exponentially sensitive to their bulkmasses. Then, the SM fermion mass hierarchy can be obtained by very mild tuning ofbulk masses. For earlier attempts and related recent works, see [14]
1n this paper, we propose an improved SU (6) grand GHU model [13], where theSM fermion mass hierarchy is obtained by following the approach mentioned in the lastparagraph. The SM fermions are introduced on the boundary as SU (5) multiplets, thefour types of massive bulk fermions in SU (6) multiplets coupling to the SM fermions areintroduced. We obtain the quark and lepton masses except for top quark by integratingout the massive bulk fermions and tuning of the bulk masses. We also calculate one-loop Higgs potential and study whether the electroweak symmetry breaking happens andHiggs mass can be obtained. This issue is very nontrivial in GHU since the potential isgenerated at one-loop and strongly depends on matter fermion content. We find that it isnot possible to break the electroweak symmetry by only the four types of bulk fermions.Then, we show that the electroweak symmetry breaking and a viable Higgs mass can berealized by introducing additional bulk fermions with large dimensional representation.This paper is organized as follows. In the next section, we describe our model indetail. In section 3, the mechanism of the SM fermion mass generation is explained.It is shown that the SM fermion masses except top quark can be reproduced by mildtuning of bulk masses. One-loop Higgs potential is calculated in section 3, where theelectroweak symmetry breaking and Higgs mass are analyzed. Section 4 is devoted to ourconclusions and discussions. The details of calculations are summarized in Appendices.The branching rules of the representations relevant to our model are shown in AppendixA. In Appendix B, calculations of the Kaluza-Klein (KK) mass spectrum of bulk fieldsare explained in some detail. In this section, we briefly explain an SU (6) GHU model [13]. We consider a five dimen-sional (5D) SU (6) gauge theory with an extra space compactified on an orbifold S /Z ,whose radius and coordinate are denoted by R and y , respectively. Z parities at eachfixed points are given as follows. P = diag(+ , + , + , + , + , − ) at y = 0 ,P (cid:48) = diag(+ , + , − , − , − , − ) at y = πR. (1)2e assign the Z parity for the gauge field and the scalar field as A µ ( − y ) = P A µ ( y ) P † , A y ( − y ) = − P A y ( y ) P † . Then, their fields have the following parities in components, A µ = (+ , +) (+ , +) (+ , − ) (+ , − ) (+ , − ) ( − , − )(+ , +) (+ , +) (+ , − ) (+ , − ) (+ , − ) ( − , − )(+ , − ) (+ , − ) (+ , +) (+ , +) (+ , +) ( − , +)(+ , − ) (+ , − ) (+ , +) (+ , +) (+ , +) ( − , +)(+ , − ) (+ , − ) (+ , +) (+ , +) (+ , +) ( − , +)( − , − ) ( − , − ) ( − , +) ( − , +) ( − , +) (+ , +) , (2) A y = ( − , − ) ( − , − ) ( − , +) ( − , +) ( − , +) (+ , +)( − , − ) ( − , − ) ( − , +) ( − , +) ( − , +) (+ , +)( − , +) ( − , +) ( − , − ) ( − , − ) ( − , − ) (+ , − )( − , +) ( − , +) ( − , − ) ( − , − ) ( − , − ) (+ , − )( − , +) ( − , +) ( − , − ) ( − , − ) ( − , − ) (+ , − )(+ , +) (+ , +) (+ , − ) (+ , − ) (+ , − ) ( − , − ) , (3)where (+ , − ) means that Z parity is even (odd) at y = 0 ( y = πR ) boundary, forinstance. We note that only the field with (+ , +) parity has a 4D massless zero mode( n = 0) as can be seen from the KK expansion in terms of mode function described inAppendix B. The Z parity for A µ indicates that SU (6) gauge symmetry is broken to SU (3) C × SU (2) L × U (1) Y × U (1) X by the combination of the symmetry breaking patternat each boundary, SU (6) → SU (5) × U (1) X at y = 0 , (4) SU (6) → SU (2) × SU (4) at y = πR. (5)The hypercharge U (1) Y is contained in Georgi-Glashow SU (5) GUT, which is an upper-left 5 × × g = g = (cid:114) g Y (6)at the unification scale, which will not be so far from the compactification scale. g , ,Y are the gauge coupling constants for SU (3) C , SU (2) L , U (1) Y , respectively. This couplingrelation implies that the weak mixing angle is the same as that of Georgi-Glashow SU (5)GUT model, sin θ W = 3 / θ W :weak mixing angle). This result can be explicitly checkedfor the bulk fermion in representation of SU (6) (see the next section).sin θ W = Tr I Tr Q = (( ) + ( − ) ) × ) + ( − ) + ( ) + ( ) ) × + 1 = 38 , (7)where I is the third component of SU (2) L isospin and Q is an electric charge.3 U (2) L Higgs doublet field is identified with a part of an extra component of gaugefield A y . A y = 1 √ HH † . (8)We suppose that a vacuum expectation value (VEV) of the Higgs field is taken to be inthe 28-th generator of SU (6), (cid:104) A ay (cid:105) = αRg δ a . g is a 5D SU (6) gauge coupling and α isa dimensionless constant. The VEV of Higgs field is given by (cid:104) H (cid:105) = √ αRg . We note thatthe doublet-triplet splitting problem is solved by the orbifolding since the Z parity of thecolored Higgs field is (+ , − ) and it become massive [17].Here we give some comments on U (1) X gauge symmetry which remains unbroken byorbifolding. We first note that the U (1) X is anomalous as it stands since the masslessfermions are only the SM fermions and their U (1) X charge assignments are not anomaly-free (see Table 1 in the next section.). However, it is easy to cancel the anomaly by addingappropriate number of the SM singlet fermions with U (1) X charge only. In order to breakthe U (1) X spontaneously, U (1) X charged scalars are introduced on the y = 0 boundaryfor instance, and we write down the potential of quadratic and quartic terms like the SMHiggs potential. Then, U (1) X is spontaneously broken by having the VEV for the scalars. As mentioned in the introduction, we have to give up embedding all the SM fermions intothe SU (6) multiplets in the bulk to generate the fermion masses. The SM quarks andleptons are embedded into SU (5) multiplets localized at y = 0 boundary, three sets of Ψ ,Ψ ∗ , Ψ along the sprit of GUT as much as possible. We also introduce various pair of bulkfermions Ψ and ˜Ψ with opposite Z parities each other and constant mass term like M ¯Ψ ˜Ψin the bulk to avoid exotic massless fermions from them. ˜Ψ is referred as “mirror fermions”in this paper. In this setup, we have no massless chiral fermions from the bulk and itsmirror fermions. The massless fermions are the SM fermions only and the gauge anomaliesfor the SM gauge groups are trivially canceled. In order to realize the SM fermion masses,the boundary localized mass terms between the SM fermions localized at y = 0 and thebulk fermions are necessary. To this end, we have to choose appropriate representationsof SU (6) for bulk fermions so that the left(right)-handed fermion components in the bulkfermions couple to the right(left)-handed SM fermions after the decomposition into the4M model gauge group representations. Note that the mirror fermions have no couplingto the SM fermions. Table 1 shows various representations for bulk and mirror fermionsin our model in addition to the SM fermions, which corresponds to the matter contentfor one generation. Totally, three copies of them are present in our model.bulk fermion mirror fermion SM fermion coupling to bulk20 ∗ ( − , − ) ⊃ Q ∗ (3 ∗ , ( − , − ) − / , , U ∗ (3 ∗ , (+ , +) − / , − ∗ (+ , +) q ∗ L (3 ∗ , − / , , u ∗ R (3 ∗ , − / , − ( − , +) ⊃ Q (3 , (+ , +)1 / , − , D (3 , ( − , − ) − / , − (+ , − ) q L (3 , / , − , d R (3 , − / , − (+ , +) ⊃ L ∗ (1 , ( − , − )1 / , − , E ∗ (1 , (+ , +)1 , ( − , − ) l ∗ L (1 , / , − , e ∗ R (1 , , (+ , +) ⊃ L ∗ (1 , ( − , − )1 / , − , N ∗ (1 , (+ , +)0 , − ( − , − ) l ∗ L (1 , / , − , ν ∗ R (1 , , − Table 1: Representation of bulk fermions, the corresponding mirror fermions and SMfermions per a generation. R in R (+ , +) means an SU (6) representation of the bulkfermion. r , in ( r , r ) a,b are SU (3) , SU (2) representations in the SM, respectively. a, b are U (1) Y , U (1) X charges.Lagrangian for the fermions is L matter = (cid:88) a =20 , , , (cid:20) Ψ a i Γ M D M Ψ a + ˜Ψ a i Γ M D M ˜Ψ a + (cid:18) λ a πR Ψ a ˜Ψ a + h . c . (cid:19)(cid:21) + δ ( y ) (cid:2) Ψ i Γ µ D µ Ψ + Ψ ∗ i Γ µ D µ Ψ ∗ + Ψ i Γ µ D µ Ψ + (cid:114) πR (cid:0) q ∗ L Q ∗ + q L Q + u ∗ R U ∗ + d R D + l ∗ L ( L ∗ + L ∗ ) + e ∗ R E ∗ + ν ∗ R N ∗ + h . c . (cid:17)(cid:105) . (9)The first line is lagrangian for the bulk and the corresponding mirror fermions, and theremaining terms are lagrangian localized on y = 0 boundary. Note that the subscript “ a ”denotes the representations of the bulk and mirror fermions. The bulk masses between thebulk and the mirror fermions are normalized by πR and expressed by the dimensionlessparameter λ a . The last two lines are mixing mass terms between the bulk fermions and theSM fermions. In general, these mixing masses can be free parameters, but we set them tobe a common value (cid:112) /πR since we would like to avoid unnecessary arbitrary parametersin fitting the data of SM fermion masses. The five-dimensional gamma matrices Γ M isgiven by (Γ µ , Γ y ) = ( γ µ , iγ ). By integrating out y -direction, 4D effective Lagrangianfrom the bulk lagrangian is obtained. L ⊃ ∞ (cid:88) n = −∞ (cid:104) Ψ ( n ) ( i / ∂ − m ( qα ))Ψ ( n ) + ˜Ψ ( n ) ( i / ∂ + m ( qα )) ˜Ψ ( n ) + (cid:18) λπR Ψ ( n ) ˜Ψ ( n ) + ψ SM κ L P L + κ R P R πR Ψ ( n ) + h.c. (cid:19)(cid:21) , (10)5here Ψ ( n ) ( ˜Ψ ( n ) ) represents a n -th KK mode of bulk (mirror) fermion, and ψ SM is a SMfermion. P L,R are chiral projection operators and κ L,R are some constants. m ( qα ) = n + qαR denotes the sum of the ordinary KK mass and the electroweak symmetry breaking massproportional to the Higgs VEV. The factor q determined by the representation which thefermion under consideration belongs to. The mass spectrum of bulk and mirror fermionsis totally given by m n = (cid:0) λπR (cid:1) + m ( qα ) . Note that the Lagrangian (10) is illustrated forparticular bulk and mirror fermions as an example.In order to derive the SM fermion masses, we need the quadratic terms in the effectiveLagrangian for the SM fermion. L SM ⊃ ψ SM Kψ SM (11)with K ≡ / p (cid:18) κ L P L + κ R P R √ x + λ (cid:19) Re f ( √ x + λ , qα ) + iπR Im f ( √ x + λ , qα ) (12)where x ≡ πRp and f ( √ x + λ , qα ) ≡ ∞ (cid:88) n = −∞ √ x + λ + iπ ( n + qα ) = coth( √ x + λ + iπα ) . (13)In deriving L SM , we simply took the large bulk mass limit λ ( πR ) (cid:29) p so that the mixingsof the SM fermions with non-zero KK modes in the mass eigenstate become negligiblysmall.Integrating out all massive bulk fermions and normalizing the kinetic term to becanonical, we obtain the physical mass for the SM fermions. m a phys = m a (cid:112) Z aL Z aR (cid:39) m W e − λ ( a = u, d, e, ν ) (14)where the bare mass and the wave function renormalization factors are m a = 1 πR Im f ( √ x + λ , qα ) , (15) Z aL,R = 1 + (cid:88) i κ iL,R (cid:112) x + λ i Re f ( (cid:113) x + λ i , q i α ) (16)where the summation in Z aL,R means that it takes a summation for all the bulk fieldscontributing to mass m a . The explicit expressions are shown below.6e consider here ratios of the physical SM fermion mass and the weak boson mass m W to fit the experimental data. m u phys m W = (cid:0) − coth ( λ ) (cid:1)(cid:114)(cid:16) λ coth( λ ) + λ coth( λ ) (cid:17) (cid:16) λ coth( λ ) (cid:17) , (17) m d phys m W = √ (cid:0) − coth ( λ ) (cid:1)(cid:114)(cid:16) (cid:15) λ coth( λ ) (cid:17) (cid:16) (cid:15) λ coth( λ ) + (cid:15) λ coth( λ ) + (cid:15) λ coth( λ ) (cid:17) , (18) m e phys m W = (cid:0) − coth ( λ ) (cid:1)(cid:114)(cid:16) λ coth( λ ) (cid:17) (cid:16) λ coth( λ ) + λ coth( λ ) (cid:17) , (19) m ν phys m W = √ (cid:0) − coth ( λ ) (cid:1)(cid:114)(cid:16) λ coth( λ ) + λ coth( λ ) (cid:17) (cid:16) λ coth( λ ) + λ coth( λ ) (cid:17) , (20)where m u,d,e,ν denote up-type quark, down-type quark, charged lepton, and neutrinomasses, respectively. All these ratios depend on two kinds of bulk mass parameters,but one of them is always dominant to the other one. Fig. 1 shows the dependence onbulk mass parameter for various mass ratios. Note that λ in the horizontal axis of thefigure means a larger bulk mass parameter of the two kinds: λ = λ , λ , λ , λ forup-type quarks, down-type quarks, charged leptons, neutrinos, respectively. As can beseen from the Figure 1, the masses up to order of the weak boson mass can be realizedby choosing an appropriate bulk mass parameter. Table 2 summarizes the values of bulkmass parameters reproducing the SM fermion masses except for the top quark. It is a verynice feature of models in extra dimensions that the SM fermion mass hierarchy can be ob-tained by the mild tuning of bulk mass parameters. This is because the physical fermionmass has an exponential dependence on the bulk mass parameter as seen from (14). As (cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96)(cid:96) parameter generation 1 2 3 λ (up-type quark) 5.9 2.55 0.1 λ (down-type quark) 5.65 4.1 1.1 λ (charged lepton) 6.58 3.87 2.4 λ (neutrino) 13 10 10Table 2: Bulk masses fitted by the SM fermion masses except for the top quark mass.for the top quark, even if the vanishing bulk mass parameter is taken, the ratio between7 ,e d, ν λ m phys m W Figure 1: Ratios of the SM fermion mass and the weak boson mass as a function of thebulk mass parameter. The yellow (blue) curve denotes the down-type quark and theneutrino masses (up-type quark and the charged lepton masses).top and W-boson masses m t /m W is at most unity. In order to avoid this situation, thefermion components coupling to top quark on the boundary should be embedded intohigher rank representation as in [18]. We have investigated whether fermions included inthree and four rank tensor of SU (6) representations couple to the SM fermions on the y = 0 boundary, but we could not succeed in finding. It might be possible to considerrepresentations on other gauge groups. In this section, we calculate the effective potential for the Higgs field and study whetherthe electroweak symmetry breaking correctly occurs. Since the Higgs field is originally agauge field, the potential is generated at one-loop by Coleman-Weinberg mechanism. Thepotential from the bulk fields is given by V ( α ) = (cid:88) n ± g (cid:90) d p E (2 π ) log[ p E + m n ] ≡ g F ± ( qα ) (21)with F ± ( qα ) = ± (cid:88) n (cid:90) d p E (2 π ) log[ p E + m n ] , (22)8here overall signs +( − ) stand for fermion (boson), respectively. g means the spin degreesof freedom of the field running in the loop. The loop momentum p E is taken to beEuclidean.For the gauge bosons, bulk fermions and mirror fermions, the mass spectrum is calcu-lated as the following four types depending on the Z parity and the bulk mass. m n = ( n + qα ) R ,m n = ( n + 1 / qα ) R ,m n = ( n + qα ) R + (cid:18) λπR (cid:19) ,m n = ( n + 1 / qα ) R + (cid:18) λπR (cid:19) . (23)Using this information, we obtain the corresponding potentials [18]. F ± ( qα ) = ∓ π R ∞ (cid:88) k =1 cos(2 πqαk ) k , F ± / ( qα ) = ∓ π R ∞ (cid:88) k =1 ( − k cos(2 πqαk ) k , F ± λ ( qα ) = ∓ π R ∞ (cid:88) k =1 cos(2 πqαk ) e − kλ k (cid:20) (2 λ ) λk + 1 k (cid:21) , F ± / λ ( qα ) = ∓ π R ∞ (cid:88) k =1 ( − k cos(2 πqαk ) e − kλ k (cid:20) (2 λ ) λk + 1 k (cid:21) . (24)Table 3 lists the various potentials from the gauge field, bulk fermion and mirror fermioncontributions. The coefficients in the potential can be read from the branching rules inthe decomposition of the SU (6) representation into SU (3) C × SU (2) L × U (1) Y × U (1) X representations listed in Appendix A.bulk+mirror g = 820 ∗ ( − , − ) + 20 ∗ (+ , +) F − λ ( α ) + 3 F − / λ ( α )56 ( − , +) + 56 (+ , − ) F − λ ( α ) + 3 F − λ (2 α ) + 7 F − / λ ( α ) + F − / λ (2 α ) + F − / λ (3 α )15 (+ , +) + 15 ( − , − ) F − λ ( α ) + 3 F − / λ ( α )21 (+ , +) + 21 ( − , − ) F − λ ( α ) + F − λ (2 α ) + 3 F − / λ ( α )gauge g = 335 (+ , +) F + ( α ) + F + (2 α ) + 6 F + ( α )Table 3: Bulk fermion, mirror fermion and gauge field contributions to Higgs potential.9ext, we have to calculate the Higgs potential from the SM fermion contributionslocalized at y = 0 using K in eq.(12). The results are as follows. V u = − π R (cid:90) dx x × log (cid:34)(cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) + 1 (cid:112) x + λ Re f ( (cid:113) x + λ , α ) (cid:33) , × (cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) (cid:33) + (cid:18) x Im f ( (cid:113) x + λ , α ) (cid:19) (cid:35) ,V d = − π R (cid:90) dx x log (cid:34)(cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) (cid:33) × (cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) + 12 (cid:112) x + λ Re f ( (cid:113) x + λ , (cid:112) x + λ Re f ( (cid:113) x + λ , (cid:33) + (cid:18) √ x Im f ( (cid:113) x + λ , α ) (cid:19) (cid:35) ,V e = − π R (cid:90) dx x log (cid:34)(cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α )) (cid:33) × (cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) + 1 (cid:112) x + λ Re f ( (cid:113) x + λ , α ) (cid:33) + (cid:18) x Im f ( (cid:113) x + λ , α ) (cid:19) (cid:35) ,V ν = − π R (cid:90) dx x × log (cid:34)(cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) + 12 (cid:112) x + λ Re f ( (cid:113) x + λ , (cid:33) × (cid:32) (cid:112) x + λ Re f ( (cid:113) x + λ , α ) + 1 (cid:112) x + λ Re f ( (cid:113) x + λ , (cid:33) + (cid:18) √ x Im f ( (cid:113) x + λ , α ) (cid:19) (cid:35) . (25)In calculation of the potential from the both bulk and boundary contributions, we havesubtracted the α independent part of the potential since it corresponds to the divergentvacuum energy and is irrelevant to the electroweak symmetry breaking.Total potential is V ( α ) = V gauge + V bulk + V boundary , where V gauge , V bulk and V boundary denote the contributions from the gauge field, the bulk fermions and mirror fermions10espectively. The plots of the potentials are shown in Fig. 2. As we can see from Fig. 2,the electroweak symmetry breaking does not occur since the potential minimum is origin. V bulk V boundary V gauge α - - V V ( α ) α - - V Figure 2: Left: Each contribution of the effective potential; the blue line, the yellowline and green line corresponds to the gauge field, bulk fermion and boundary fermioncontributions, respectively. Right: Total Higgs potential.Therefore, we must add some extra fields to obtain the electroweak symmetry breaking.In this paper, we introduce a set of bulk and mirror fermion in representation of SU (6), which is the fourth rank symmetric tensor. The reason why such a bulk fermionwith a large dimensional representation is considered is as follows. As a generic feature ofHiggs potential in GHU, the curvature at the origin of the potential from the gauge field(bulk fermion) contribution is positive (negative) and is likely to make the electroweaksymmetry unbroken (broken). Furthermore, to realize the electroweak symmetry breaking SU (2) L × U (1) Y → U (1) em in GHU, the Higgs VEV (more precisely, the dimensionlessconstant Higgs VEV) must be smaller than one, 0 < α <
1. In order to obtain such asmall VEV, the field with larger representation is preferable since the periodicity of thepotential becomes smaller. The additional contribution to Higgs potential is shown inTable 4. extra g = 8126 (+ , +) + 126 ( − , − ) F − M ( α ) + 7 F − M (2 α ) + F − M (3 α ) + F − M (4 α )+13 F − / M ( α ) + 3 F − / M (2 α ) + 3 F − / M (3 α )Table 4: The extra bulk fermion contribution to Higgs potential.The corrected total potential by adding the contribution from a pair of fermions in representation is displayed in Fig. 3, where the bulk mass parameter λ is taken tobe 0.5. As Fig. 3 shows, the realistic electroweak symmetry breaking is realized. In fact,Higgs VEV α ∼ .
01 is found from the minimization of the potential. Higgs mass canbe obtained as a function of the compactification scale. We find Higgs mass m H ∼ g GeV at the compactification scale 1 /R ∼ . g is a four-dimensional SU (2) L gauge11oupling obtained from the five-dimensional one g = 2 πR g . V ( α ) α - V V ( α ) α - - - - - - - V Figure 3: Total potential corrected by adding extra fermions in the range 0 ≤ α ≤ . ≤ α ≤ .
12 (right). The bulk mass parameter λ is taken to be 0.5. In this paper, we have considered the fermion mass hierarchy in grand GHU. In thegrand GHU previously discussed [13], a 5D SU (6) GHU with an orbifold S /Z wasconsidered and all the SM fermions were elegantly embedded into a minimal set of SU (6)bulk multiplets without massless exotic fermions, namely anomaly-free matter content.However, the down-type Yukawa couplings and the charged lepton Yukawa couplingswere not allowed since the left-handed quark (lepton) doublets and the right-handeddown quark (charged lepton) singlets were embedded into different SU (6) multiplets andYukawa couplings in GHU is generated by the gauge interactions. From this observation,the SM fermions were introduced in the SU (5) multiplets on the boundary at y = 0in this paper. We have also introduced some massive bulk fermions in four types of SU (6) representations and couplings between the SM fermions on the boundary and thebulk fermions. By integrating out the massive bulk fermions, the SM fermion massesare generated. We have shown that the SM fermion masses except for top quark canbe reproduced by mild tuning of bulk masses. Furthermore, we have calculated one-loop Higgs potential and found that the electroweak symmetry breaking does not occurunfortunately for the fermion matter content mentioned above. To resolve this issue, wehave clarified that the electroweak symmetry breaking happened by introducing additionalbulk fermions in representation. The SM Higgs boson mass was also obtained.In our analysis, Higgs boson mass and the compactification scale are slightly small. Itmight be possible to solve these problems by introducing the localized gauge kinetic termson the boundary, as discussed in [16]. These terms are not forbidden by symmetry. If weconsider the localized gauge kinetic term on the boundary at y = 0 where the top quark12s present, the effects of the localized gauge kinetic term enhance the magnitude of thecompactification scale. This leads to the enhancement of the Higgs mass. Furthermore,top quark mass also enhanced as explained in [16]. To confirm this expectations, we haveto reanalyze the mass spectrum and the mode functions for the gauge fields since theyare corrected by the presence of the localized gauge kinetic terms. This direction is veryinteresting, but remained for our future study.There are issues to be explored in a context of GUT scenario, which are not discussedin this paper. First, it is important to study the gauge coupling unification. It is wellknown that the gauge coupling running in (flat) extra dimensions is not logarithmic butpower dependence on energy scale [19]. Therefore, the GUT scale is expected to be verylow comparing to the conventional 4D GUT, namely, not far from the compactificationscale. In this analysis, it is very nontrivial whether the unified SU (6) gauge coupling atthe GUT scale is perturbative. This is because we have introduced relatively many bulkfields in our model, which might lead to Landau pole below the GUT scale. For our modelto be a physically meaningful GUT model, this issue must be clarified. Second issue to beaddressed is proton decay. The masses of so-called X, Y gauge bosons are also extremelylight comparing to the conventional GUT scale. Therefore, proton decays very rapidlyand our model is immediately excluded by the constraints from the Super Kamiokandedata as it stands. Dangerous baryon number violating operators have to be forbidden attree level by imposing symmetry (see [20] for UED case) in order to ensure the protonstability. If U (1) X is broken to some discrete symmetry and this symmetry plays an rolefor it, it would be very interesting. Then, it is desirable to predict the main decay modeat quantum level.These issues are beyond the scope of this paper and also remained for our future work. Acknowledgments
This work is supported in part by JSPS KAKENHI Grant Number JP17K05420 (N.M.).
A Branching rules of bulk fields
In this appendix, several branching rules under the symmetry breaking SU (6) → SU (3) C × SU (2) L × U (1) Y × U (1) X y = 0. They are also useful to compute the one-loopeffective Higgs potential.gauge field (+ , +) = (8 , (+ , +)0 , ⊕ (1 , (+ , +)0 , ⊕ (1 , (+ , +)0 , ⊕ (1 , (+ , +)0 , ⊕ (1 , (+ , +)0 , ⊕ (3 , ( − , +) − / , ⊕ (3 ∗ , ( − , +)1 / , − ⊕ (2 , ( − , − )1 / , ⊕ (2 , ( − , − ) − / , − ⊕ (3 , (+ , − )0 , − / ⊕ (3 ∗ , (+ , − )0 , / , ( − , − ) = (8 , ( − , − )0 , ⊕ (1 , ( − , − )0 , ⊕ (1 , ( − , − )0 , ⊕ (1 , ( − , − )0 , ⊕ (1 , ( − , − )0 , ⊕ (3 , (+ , − ) − / , ⊕ (3 ∗ , (+ , − )1 / , − ⊕ (2 , (+ , +)1 / , ⊕ (2 , (+ , +) − / , − ⊕ (3 , ( − , +)0 , − / ⊕ (3 ∗ , ( − , +)0 , / . (26)bulk fermions (+ , +) = (3 ∗ , (+ , +) − / , ⊕ (3 , (+ , − )1 / , ⊕ (1 , (+ , +)1 , ⊕ (3 , ( − , +) − / , − ⊕ (1 , ( − , − )1 / , − , ∗ ( − , − ) = (3 , (+ , − )1 / , − ⊕ (3 ∗ , (+ , +) − / , − ⊕ (1 , (+ , +)1 , − ⊕ (3 ∗ , ( − , − ) − / , ⊕ (3 , ( − , +)2 / , ⊕ (1 , ( − , +) − , , (+ , +) = (6 , (+ , +) − / , ⊕ (3 , (+ , − )1 / , ⊕ (1 , ∗ ) (+ , +)1 , ⊕ (3 , ( − , +) − / , − ⊕ (1 , ( − , − )1 / , − ⊕ (1 , (+ , +)0 , − , ( − , +) = (10 , ( − , − ) − , ⊕ (6 , ( − , +) − / , ⊕ (3 , ( − , − )2 / , ⊕ (1 , ( − , +)3 / , − ⊕ (6 , (+ , − ) − / , − ⊕ (3 , (+ , +)1 / , − ⊕ (1 , (+ , − )1 , − ⊕ (3 , ( − , − ) − / , − ⊕ (1 , ( − , +)1 / , − ⊕ (1 , (+ , − )0 , − . (27) B Mass spectrum of Bulk fermions
In this appendix, the calculations of mass spectrum of bulk fermions are described indetail.
B.1 Mode expansion and reflection
Because of two Z parity conditions, the five-dimensional field can be decomposed intofour types of KK-modes classified by combination of Z eigenvalues ( P, P (cid:48) ), where the left14right) parity is with respect to y = 0( πR ). The mode functions are listed below. f ( n )(+ , +) ( y ) = 1 √ πR cos (cid:16) nR y (cid:17) ,f ( n )( − , − ) ( y ) = 1 √ πR sin (cid:16) nR y (cid:17) ,f ( n )(+ , − ) ( y ) = 1 √ πR cos (cid:18) n + 1 / R y (cid:19) ,f ( n )( − , +) ( y ) = 1 √ πR sin (cid:18) n + 1 / R y (cid:19) . (28)It is convenient to define the following reflection properties for KK-modes. ψ ( − n ) = ψ ( n ) for (+ , +) ,ψ ( − n ) = − ψ ( n ) for ( − , − ) ,ψ ( − n − = ψ ( n ) for (+ , − ) ,ψ ( − n − = − ψ ( n ) for ( − , +) . (29)Utilizing these reflection properties, the five-dimensional field Ψ( x, y ) is expanded in termsof mode function f ( y ) and four-dimensional field ψ ( x ) as follows.As an example, the KK decomposition of the field with (+ , +) parity is discussed indetail.Ψ( x, y ) (+ , +) = 1 √ πR (cid:34) ψ (0) ( x ) + √ ∞ (cid:88) n =1 cos (cid:16) nR y (cid:17) ψ ( n ) ( x ) (cid:35) = 1 √ πR ψ (0) ( x ) + ∞ (cid:88) n =1 √ f ( n )(+ , +) ( y ) ψ ( n ) ( x ) + ∞ (cid:88) n =1 √ f ( n )(+ , +) ( y ) ψ ( n ) ( x )= 1 √ πR ψ (0) ( x ) + ∞ (cid:88) n =1 √ f ( n )(+ , +) ψ ( n ) ( x ) + − (cid:88) n = −∞ √ f ( − n )(+ , +) ( y ) ψ ( − n ) ( x )= 1 √ πR ψ (0) ( x ) + ∞ (cid:88) n =1 √ f ( n )(+ , +) ( y ) ψ ( n ) ( x ) + − (cid:88) n = −∞ √ f ( n )(+ , +) ( y ) ψ ( n ) ( x )= (cid:88) n η n f ( n )(+ , +) ( y ) ψ ( n ) ( x ) (30)where η n = (cid:40) n = 0 , √ for n (cid:54) = 0 . Other types of fields can be also decomposed in a similar way and we obtain the results15s Ψ( x, y ) ( − , − ) = (cid:88) n η n f ( n )( − , − ) ( y ) ψ ( n ) ( x ) , Ψ( x, y ) (+ , − ) = (cid:88) n √ f ( n )(+ , − ) ( y ) ψ ( n ) ( x ) , Ψ( x, y ) ( − , +) = (cid:88) n √ f ( n )( − , +) ( y ) ψ ( n ) ( x ) . (31) B.2 Mass eigenvalues
We employed four representations of SU (6) as bulk fermion in our model; ∗ , , and . Since all of representations are higher rank representations, it is very nontrivialto find mass eigenvalues after the electroweak symmetry breaking. In this subsection, wedescribe how the mass eigenvalues are obtained for the above four bulk fields. In GHU,the electroweak symmetry breaking masses are generated from the gauge interaction sinceHiggs field is originated from the fifth component of the gauge field.TrΨ i Γ D Ψ = − Ψ ( − ) D Ψ (+) + Ψ (+) D Ψ ( − ) Turning on the Higgs VEV, we find that the KK masses and the symmetry breakingmasses take the following form depending on the tensor structure. ∓ (cid:0) Ψ ( ∓ ) D Ψ ( ± ) + Ψ ( ± ) D Ψ ( ∓ ) (cid:1) = ∓ Ψ ( − ) i ∂ Ψ (+) i ∓ αR (Ψ ( − )2 Ψ (+)6 − Ψ ( − )6 Ψ (+)2 ) (the first rank tensor) , ∓ Ψ ( − ) ji ∂ Ψ (+) ji ∓ αR (Ψ ( − ) j Ψ (+) j − Ψ ( − ) j Ψ (+) j ) (the second rank tensor) , ∓ Ψ ( − ) ikj ∂ Ψ (+) ijk ∓ αR (Ψ ( − )2 kj Ψ (+)6 jk − Ψ ( − )6 kj Ψ (+)2 jk ) (the third rank tensor) . (32)The point is that the coefficients of symmetry breaking mass α/R is determined by thenumber of rank of the field under consideration. Note that the only components 2 and 6appear in the symmetry breaking terms since the Higgs VEV is supposed to take in (2 , ,
2) components in A .In next subsubsections, we briefly discuss how the mass spectrum is derived for eachrepresentation. B.2.1 15: the second rank anti-symmetric tensor
The representation is the second rank anti-symmetric tensor of SU (6). The compo-nents after the decomposition into SU (3) C × SU (2) L × U (1) Y × U (1) X and the correspond-16ng parity and reflection are summarized in Table 5. The blanks in the matrix elementsmeans zero, hereafter.(+ , +) (+ , − )(1 , E ∗ ( − n )15( ± ) = ∓ E ∗ ( n )15( ± ) (3 , ζ ( − n − ± ) = ∓ ζ ( n )( ± ) (3 ∗ , ψ ( − n )( ± ) = ∓ ψ ( n )( ± ) ( − , − ) ( − , +)(1 , L ∗ ( − n )15( ± ) = ± L ∗ ( n )15( ± ) (3 , ω ( − n − ± ) = ± ω ( n )( ± ) Table 5: Parity and reflection for components of .Making use of the results in the previous subsection B.1., the KK expansion of isdescribed in a following matrix form.Ψ ( ± ) = 1 √ ∞ (cid:88) n = −∞ η n f ( n )( ∓ , ∓ ) E ( n )15( ± ) − η n f ( n )( ∓ , ∓ ) E ( n )15( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )1( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )2( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )3( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )4( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )5( ± ) − √ f ( n )( ∓ , ± ) ζ ( n )6( ± ) ∓ η n f ( n )( ± , ± ) L ( n )115( ± ) ∓ η n f ( n )( ± , ± ) L ( n )215( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )5( ± ) ± η n f ( n )( ± , ± ) L ( n )115( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )2( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )4( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )6( ± ) ± η n f ( n )( ± , ± ) L ( n )215( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )1( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) − η n f ( n )( ∓ , ∓ ) ψ ( n )1( ± ) √ η n f ( n )( ∓ , ∓ ) ψ ( n )3( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ∓ ) − η n f ( n )( ∓ , ∓ ) ψ ( n )2( ± ) − ± η n f ( n )( ∓ , ∓ ) ψ ( n )3( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )3( ± ) . Substituting this expansion into the mass term and diagonalizing it, we find mass spectrum L ⊃ − ∞ (cid:88) n = −∞ (cid:34) n + αR Ψ ( n )1( ± ) Ψ ( n )1( ± ) + (cid:88) i =2 n + 1 / αR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) (cid:35) − ∞ (cid:88) n =1 (cid:34) (cid:88) i =5 nR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + (cid:88) i =9 n + 1 / R Ψ ( n ) i ( ± ) Ψ i ( n )( ± ) (cid:35) (33)and the corresponding mass eigenstates are given byΨ ( n )1( ± ) = η n (cid:16) L ( n )115( ± ) + E ( n )15( ± ) (cid:17) , Ψ ( n ) { , , } ( ± ) = 1 √ (cid:16) ω ( n ) { , , } ( ± ) − ζ ( n ) { , , } ( ± ) (cid:17) , Ψ ( n ) { , , } ( ± ) = ψ ( n ) { , , } ( ± ) , Ψ ( n )8( ± ) = L ( n )215( ± ) , Ψ ( n ) { , , } ( ± ) = ζ ( n ) { , , } ( ± ) . (34)17 .2.2 21: the second rank symmetric tensor The representation is the second rank symmetric tensor of SU (6). The componentsafter the decomposition into SU (3) C × SU (2) L × U (1) Y × U (1) X and the correspondingparity and reflection are summarized in Table 6.(+ , +) (+ , − )(1 , φ ( − n )( ± ) = ∓ φ ( n )( ± ) (3 , ζ ( − n − ± ) = ∓ ζ ( n )( ± ) (6 , ψ ( − n )( ± ) = ∓ ψ ( n )( ± ) (1 , N ∗ ( − n )21( ± ) = ∓ N ∗ ( n )21( ± ) ( − , − ) ( − , +)(1 , L ∗ ( − n )21( ± ) = ± L ∗ ( n )21( ± ) (3 , ω ( − n − ± ) = ± ω ( n )( ± ) Table 6: Parity and reflection for components of .KK expansion and diagonalization of mass matrix can proceed similarly as repre-sentation in the previous subsubsection.Ψ ( ± ) = 1 √ ∞ (cid:88) n = −∞ √ η n f ( n )( ∓ , ∓ ) φ ( n )1( ± ) η n f ( n )( ∓ , ∓ ) φ ( n )2( ± ) η n f ( n )( ∓ , ∓ ) φ ( n )2( ± ) √ η n f ( n )( ∓ , ∓ ) φ ( n )3( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )2( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )4( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )5( ∓ ) 1 √ f ( n )( ∓ , ± ) ζ ( n )6( ∓ ) ± η n f ( n )( ± , ± ) L ( n )121( ± ) ± η n f ( n )( ± , ± ) L ( n )221( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )5( ± ) ± η n f ( n )( ± , ± ) L ( n )121( ± )1 √ f ( n )( ∓ , ± ) ζ ( n )2( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )4( ± ) 1 √ f ( n )( ∓ , ± ) ζ ( n )6( ± ) ± η n f ( n )( ± , ± ) L ( n )221( ± ) √ η n f ( n )( ∓ , ∓ ) ψ ( n )1( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )2( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )2( ± ) √ η n f ( n )( ∓ , ∓ ) ψ ( n )3( ± ) √ η n f ( n )( ∓ , ∓ ) ψ ( n )5( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )2( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )4( ± ) η n f ( n )( ∓ , ∓ ) ψ ( n )5( ± ) √ η n f ( n )( ∓ , ∓ ) ψ ( n )6( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) √ η n f ( ∓ )( ∓ , ∓ ) N ( n )21( ± ) . The diagonalized mass terms are L ⊃ − ∞ (cid:88) n = −∞ (cid:34) n + αR Ψ ( n )1( ± ) Ψ ( n )1( ± ) + n + 2 αR Ψ ( n )2( ± ) Ψ n )( ± ) + (cid:88) i =3 , , n + 1 / αR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) (cid:35) − ∞ (cid:88) n =1 (cid:34) (cid:88) i =6 nR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + (cid:88) i =14 n + 1 / R Ψ ( n ) i ( ± ) Ψ i ( n )( ± ) (cid:35) (35)18nd the corresponding mass eigenstates are givenΨ ( n )1( ± ) = η n (cid:16) φ ( n )2( ± ) − L ( n )121( ± ) (cid:17) , Ψ ( n )2( ± ) = η n (cid:18) L ( n )221( ± ) − √ (cid:16) φ ( n )3( ± ) − N ( n )21( ± ) (cid:17)(cid:19) , Ψ ( n ) { , , } ( ± ) = 1 √ (cid:16) ζ ( n ) { , , } ( ± ) − ω ( n ) { , , } ( ± ) (cid:17) , Ψ ( n )6( ± ) = 1 √ (cid:16) φ ( n )3( ± ) + N ( n )21( ± ) (cid:17) , Ψ ( n )7( ± ) = φ ( n )1( ± ) , Ψ ( n ) { , , , , , } ( ± ) = ψ ( n ) { , , , , , } , Ψ ( n ) { , , , } ( ± ) = ζ ( n ) { , , } ( ± ) . (36) B.2.3 20 ∗ : the third rank anti-symmetric tensor The ∗ representation is the third rank anti-symmetric tensor of SU (6). The componentsafter the decomposition into SU (3) C × SU (2) L × U (1) Y × U (1) X and the correspondingparity and reflection are summarized in Table 7.(+ , +) (+ , − )(3 ∗ , U ∗ ( − n )20( ± ) = ∓ U ∗ ( n )20( ± ) (3 , σ ( − n − ± ) = ∓ σ ( n )( ± ) (1 , τ ( − n )( ± ) = ∓ τ ( n )( ± ) ( − , − ) ( − , +)(3 ∗ , Q ∗ ( − n )20( ± ) = ± Q ∗ ( n )20( ± ) (3 , ω ( − n − ± ) = ± ω ( n )( ± ) (1 , ζ ( − n − ± ) = ± ζ ( n )( ± ) Table 7: Parity and reflection for components of ∗ .It is straightforward to extend the KK expansion to the third rank tensor case, buttakes a more complicated form.(Ψ ( ± )1 ) jk = Ψ ( ± )1 jk = 1 √ ∞ (cid:88) n = −∞ ∓ √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )3( ± ) − η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) ± η n f ( n )( ± , ± ) Q ( n )120( ± ) ± η n f ( n )( ± , ± ) Q ( n )220( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )1( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )120( ± ) ± η n f ( n )( ± , ± ) Q ( n )320( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )3( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )220( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )320( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )5( ± ) − √ f ( n )( ∓ , ± ) σ ( n )1( ± ) − √ f ( n )( ∓ , ± ) σ ( n )3( ± ) − √ f ( n )( ∓ , ± ) σ ( n )5( ± ) , ( ± )2 ) jk = Ψ ( ± )2 jk = 1 √ ∞ (cid:88) n = −∞ ∓ √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )3( ± ) − η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) ± η n f ( n )( ± , ± ) Q ( n )420( ± ) ± η n f ( n )( ± , ± ) Q ( n )520( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )2( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )420( ± ) ± η n f ( n )( ± , ± ) Q ( n )620( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )4( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )520( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )620( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )6( ± ) − √ f ( n )( ∓ , ± ) σ ( n )2( ± ) − √ f ( n )( ∓ , ± ) σ ( n )4( ± ) − √ f ( n )( ∓ , ± ) σ ( n )6( ± ) , (Ψ ( ± )3 ) jk = Ψ ( ± )3 jk = 1 √ ∞ (cid:88) n = −∞ ± √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )1( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )120( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )420( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )220( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )520( ± ) − √ f ( n )( ∓ , ± ) σ ( n )1( ± ) − √ f ( n )( ∓ , ± ) σ ( n )2( ± ) ± η n f ( n )( ± , ± ) Q ( n )120( ± ) ± η n f ( n )( ± , ± ) Q ( n )220( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )1( ± ) ± η n f ( n )( ± , ± ) Q ( n )420( ± ) ± η n f ( n )( ± , ± ) Q ( n )520( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) ∓ √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) , ( ± )4 ) jk = Ψ ( ± )4 jk = 1 √ ∞ (cid:88) n = −∞ ± √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )2( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )120( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )420( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )320( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )620( ± ) − √ f ( n )( ∓ , ± ) σ ( n )3( ± ) − √ f ( n )( ∓ , ± ) σ ( n )4( ± ) ± η n f ( n )( ± , ± ) Q ( n )120( ± ) ± η n f ( n )( ± , ± ) Q ( n )320( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )3( ± ) ± η n f ( n )( ± , ± ) Q ( n )420( ± ) ± η n f ( n )( ± , ± ) Q ( n )620( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) ∓ √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )320( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )320( ± ) , (Ψ ( ± )5 ) jk = Ψ ( ± )5 jk = 1 √ ∞ (cid:88) n = −∞ ± √ f ( n )( ± , ∓ ) ω ( n )3( ± ) ∓ √ f ( n )( ± , ∓ ) ω ( n )3( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )220( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )520( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )320( ± ) ∓ η n f ( n )( ± , ± ) Q ( n )620( ± ) − √ f ( n )( ∓ , ± ) σ ( n )5( ± ) − √ f ( n )( ∓ , ± ) σ ( n )6( ± ) ± η n f ( n )( ± , ± ) Q ( n )220( ± ) ± η n f ( n )( ± , ± ) Q ( n )320( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )5( ± ) ± η n f ( n )( ± , ± ) Q ( n )520( ± ) ± η n f ( n )( ± , ± ) Q ( n )620( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )6( ± ) ± √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) ∓ √ f ( n )( ± , ∓ ) ζ ( n )( ± ) η n f ( n )( ∓ , ∓ ) U ( n )320( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )320( ± ) , ( ± )6 ) jk = Ψ ( ± )6 jk = 1 √ ∞ (cid:88) n = −∞ η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) − η n f ( n )( ∓ , ∓ ) τ ( n )( ± ) − √ f ( n )( ∓ , ± ) σ ( n )1( ± ) − √ f ( n )( ∓ , ± ) σ ( n )2( ± ) − √ f ( n )( ∓ , ± ) σ ( n )3( ± ) − √ f ( n )( ∓ , ± ) σ ( n )4( ± ) − √ f ( n )( ∓ , ± ) σ ( n )4( ± ) − √ f ( n )( ∓ , ± ) σ ( n )6( ± )1 √ f ( n )( ∓ , ± ) σ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )5( ± )1 √ f ( n )( ∓ , ± ) σ ( n )2( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )4( ± ) 1 √ f ( n )( ∓ , ± ) σ ( n )6( ± ) η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )120( ± ) η n f ( n )( ∓ , ∓ ) U ( n )320( ± ) − η n f ( n )( ∓ , ∓ ) U ( n )220( ± ) − η n f ( n )( ∓ , ± ) U ( n )320( ± ) . The diagonalized mass terms are derived as L ⊃ − ∞ (cid:88) n = −∞ (cid:34) (cid:88) i =1 n + αR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + (cid:88) i =4 n + α + 1 / R Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) (cid:35) − ∞ (cid:88) n =0 (cid:34) (cid:88) i =7 nR Ψ ( n ) i ( ± ) Ψ i ( n )( ± ) + (cid:88) i =11 n + 1 / R Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) (cid:35) (37)and the corresponding mass eigenstates are foundΨ ( n ) { , , } ( ± ) = η n (cid:16) Q ( n ) { , , } ± ) − U ( n ) { , , } ± ) (cid:17) , Ψ ( n ) { , , } ( ± ) = 1 √ (cid:16) ω ( n ) { , , } ( ± ) − σ ( n ) { , , } ( ± ) (cid:17) Ψ ( n )7( ± ) = τ ( n )( ± ) , Ψ ( n ) { , , } ( ± ) = Q ( n ) { , , } ± ) , Ψ ( n ) { , , } ( ± ) = σ ( n ) { , , } ( ± ) , Ψ ( n )14( ± ) = ζ ( n )( ± ) . (38) B.2.4 56: the third rank symmetric tensor
The representation is the third rank symmetric tensor of SU (6). The componentsafter the decomposition into SU (3) C × SU (2) L × U (1) Y × U (1) X and the correspondingparity and reflection are summarized in Table 8.KK expansion and the diagonalization of mass matrix can be done similarly as inB.2.3. 22+ , +) (+ , − )(3 , Q ( − n )56( ± ) = ∓ Q ( n )56( ± ) (1 , χ ( − n − ± ) = ∓ χ ( n )( ± ) (6 , ρ ( − n − ± ) = ∓ ρ ( n )( ± ) (1 , τ ( − n − ± ) = ∓ τ ( n )( ± ) ( − , − ) ( − , +)(3 , ω ( − n )( ± ) = ± ω ( n )( ± ) (6 , θ ( − n − ± ) = ± θ ( n )( ± ) (10 , ζ ( − n )( ± ) = ± ζ ( n )( ± ) (1 , φ ( − n − ± ) = ± φ ( n )( ± ) (3 , D ( − n )56( ± ) = ± D ( n )56( ± ) (1 , ν ( − n − ± ) = ± ν ( n )( ± ) Table 8: Parity and reflection for inner component of .(Ψ ( ± )1 ) jk = Ψ ( ± )1 jk = 1 √ ∞ (cid:88) n = −∞ ±√ √ f ( n )( ± , ∓ ) φ ( n )1( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )2( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )2( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )3( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )1( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )3( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )5( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) √ √ f ( n )( ∓ , ± ) τ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) τ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )1( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )3( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )5( ± ) √ √ f ( n )( ∓ , ± ) τ ( n )1( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) 1 √ f ( n )( ∓ , ± ) τ ( n )2( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )1( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )3( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )5( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )5( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )6( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )556( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )556( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )1( ± ) , (Ψ ( ± )2 ) jk = Ψ ( ± )2 jk = 1 √ ∞ (cid:88) n = −∞ ±√ √ f ( n )( ± , ∓ ) φ ( n )2( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )3( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )3( ± ) ±√ √ f ( n )( ± , ∓ ) φ ( n )4( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )7( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )8( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )9( ± )1 √ f ( n )( ∓ , ± ) τ ( n )2( ± ) √ √ f ( n )( ∓ , ± ) τ ( n )3( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) 1 √ f ( n )( ∓ , ± ) τ ( n )2( ± ) ± η n f ( n )( ± , ± ) ω ( n )7( ± ) ± η n f ( n )( ± , ± ) ω ( n )8( ± ) ± η n f ( n )( ± , ± ) ω ( n )9( ± ) √ √ f ( n )( ∓ , ± ) τ ( n )3( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )7( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )9( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )11( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )11( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )12( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )2( ± ) , ( ± )3 ) jk = Ψ ( ± )3 jk = 1 √ ∞ (cid:88) n = −∞ ±√ η n f ( n )( ± , ± ) ω ( n )1( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ± η n f ( n )( ± , ± ) ω ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )7( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )1( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )7( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )1( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )7( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )1( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )4( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )1( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )3( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )4( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )6( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) ± η n √ f ( n )( ± , ± ) D n )1 ∗ ( ± ) , (Ψ ( ± )4 ) jk = Ψ ( ± )4 jk = 1 √ ∞ (cid:88) n = −∞ ±√ η n f ( n )( ± , ± ) ω ( n )3( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ± η n f ( n )( ± , ± ) ω ( n )4( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )8( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )3( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )9( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )5( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )11( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )2( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )3( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )5( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )8( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )9( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )11( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )3( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )3( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )7( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )8( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )3( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )8( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )9( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± )1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) D n )2 ∗ ( ± ) , ( ± )5 ) jk = Ψ ( ± )5 jk = 1 √ ∞ (cid:88) n = −∞ ±√ η n f ( n )( ± , ± ) ω ( n )5( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) ± η n f ( n )( ± , ± ) ω ( n )6( ± ) ±√ η n f ( n )( ± , ± ) ω ( n )9( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )5( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )11( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )6( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )12( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )556( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )4( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )5( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )6( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )556( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )10( ± ) ± √ f ( n )( ± , ∓ ) θ ( n )11( ± ) ±√ √ f ( n )( ± , ∓ ) θ ( n )12( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )4( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )6( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) ± η n f ( n )( ± , ± ) ζ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )8( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )9( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )6( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )9( ± ) ±√ η n f ( n )( ± , ± ) ζ ( n )10( ± ) ] √ √ f ( n )( ∓ , ± ) ρ ( n )6( ± )1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )6( ± ) ±√ η n f ( n )( ± , ± ) D n )3 ∗ ( ± ) , (Ψ ( ± )6 ) jk = Ψ ( ± )6 jk = 1 √ ∞ (cid:88) n = −∞ √ √ f ( n )( ∓ , ± ) τ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) τ ( n )2( ± )1 √ f ( n )( ∓ , ± ) τ ( n )2( ± ) √ √ f ( n )( ∓ , ± ) τ ( n )3( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )1( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )2( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )156( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )356( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )556( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )1( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )256( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )456( ± ) η n f ( n )( ∓ , ∓ ) Q ∗ ( n )656( ± ) ±√ √ f ( n )( ± , ∓ ) ν ( n )2( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )1( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )156( ± )1 √ f ( n )( ∓ , ± ) ρ ( n )2( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )3( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )256( ± )1 √ f ( n )( ∓ , ± ) ρ ( n )4( ± ) 1 √ f ( n )( ∓ , ± ) ρ ( n )5( ± ) √ √ f ( n )( ∓ , ± ) ρ ( n )6( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )356( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )156( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )256( ± ) ±√ η n f ( n )( ± , ± ) D ∗ ( n )356( ± ) √ √ f ( n )( ∓ , ± ) χ ( n )( ± ) . The diagonalized mass terms are L ⊃ − ∞ (cid:88) n = −∞ (cid:34) (cid:88) i =1 n + αR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + (cid:88) i =4 n + 2 αR Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + (cid:88) i =7 n + α + 1 / R Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) + n + 2 α + 1 / R Ψ ( n )15( ± ) Ψ ( n )15( ± ) + n + 3 α + 1 / R Ψ ( n )16( ± ) Ψ ( n )16( ± ) (cid:21) − ∞ (cid:88) n =0 (cid:34) (cid:88) i =17 nR Ψ ( n ) i ( ± ) Ψ i ( n )( ± ) + (cid:88) i =33 n + 1 / R Ψ ( n ) i ( ± ) Ψ ( n ) i ( ± ) (cid:35) (39)25nd the corresponding mass eigenstates are given byΨ ( n ) { , , } ( ± ) = η n (cid:16) ω ( n ) { , , } ( ± ) − Q ∗ ( n ) { , , } ± ) (cid:17) , Ψ ( n ) { , , } ( ± ) = η n (cid:18) Q ∗ ( n ) { , , } ± ) − √ (cid:16) ω ( n ) { , , } ( ± ) − D ∗ ( n ) { , , } ± ) (cid:17)(cid:19) , Ψ ( n ) { − } ( ± ) = 1 √ (cid:16) θ ( n ) { − } ( ± ) − ρ ( n ) { − } ( ± ) (cid:17) , Ψ ( n )13( ± ) = 1 √ (cid:16) φ ( n )2( ± ) − τ ( n )1( ± ) (cid:17) , Ψ ( n )14( ± ) = 12 √ (cid:16) √ (cid:16) φ ( n )4( ± ) − χ ( n )( ± ) (cid:17) − (cid:16) τ ( n )3( ± ) − ν ( n )2( ± ) (cid:17)(cid:17) , Ψ ( n )15( ± ) = 1 √ (cid:18) τ ( n )2 − √ (cid:16) φ ( n )3( ± ) − ν ( n )1( ± ) (cid:17)(cid:19) , Ψ ( n )16( ± ) = 12 √ (cid:16)(cid:16) φ ( n )4( ± ) − χ ( n )( ± ) (cid:17) − √ (cid:16) τ ( n )3( ± ) − ν ( n )2( ± ) (cid:17)(cid:17) , Ψ ( n ) { , , } ( ± ) = η n (cid:16) ω ( n ) { , , } ( ± ) + D ∗ ( n ) { , , } ± ) (cid:17) , Ψ ( n ) { , , } ( ± ) = ω ( n ) { , , } ( ± ) , Ψ ( n )23 − ± ) = ζ ( n )1 − ± ) , Ψ ( n )33( ± ) = 1 √ (cid:16) φ ( n )3( ± ) + ν ( n )1( ± ) (cid:17) , Ψ ( n ) { − } ( ± ) = θ ( n ) { − } ( ± ) , Ψ ( n )40( ± ) = φ ( n )1( ± ) . 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