Collider signals of W' and Z' bosons in the gauge-Higgs unification
Shuichiro Funatsu, Hisaki Hatanaka, Yutaka Hosotani, Yuta Orikasa
aa r X i v : . [ h e p - ph ] M a r KEK-TH-1944, OU-HET/915
Collider signals of W ′ and Z ′ bosonsin the gauge-Higgs unification Shuichiro Funatsu a , Hisaki Hatanaka b , Yutaka Hosotani b , and Yuta Orikasa ca KEK Theory Center, Tsukuba, Ibaraki 305-0801, Japan b Department of Physics, Osaka University,Toyonaka, Osaka 560-0043, Japan and c Institute of Experimental and Applied Physics,Czech Technical University, Prague 12800, Czech Republic (Dated: February 16, 2016) bstract In the SO (5) × U (1) gauge-Higgs unification (GHU), Kaluza-Klein (KK) excited states of chargedand neutral vector bosons, W (1) , W (1) R , Z (1) , γ (1) and Z (1) R , can be observed as W ′ and Z ′ signalsin collider experiments. In this paper we evaluate the decay rates of the W ′ and Z ′ , and s -channel cross sections mediated by W ′ and Z ′ bosons with final states involving the standardmodel (SM) fermion pair ( ℓν , ℓ ¯ ℓ , q ¯ q ′ ), W H , ZH , W W and
W Z . W ′ and Z ′ resonances appeararound 6 . . θ H = 0 .
115 (0.0737) where θ H is the Aharonov-Bohm phase in the fifthdimension in GHU. For decay rates we find Γ( W ′ → W H ) ≃ Γ( W ′ → W Z ) ( W ′ = W (1) , W (1) R ),Γ( W (1) → W H, W Z ) ∼ Γ( W (1) R → W H, W Z ), Γ( Z (1) → ZH ) ≃ P Z ′ = Z (1) ,γ (1) Γ( Z ′ → W W ),and Γ( Z (1) R → ZH ) ≃ Γ( Z (1) R → W W ). W ′ and Z ′ signals of GHU can be best found at theLHC experiment in the processes pp → W ′ ( Z ′ ) + X followed by W ′ → t ¯ b , W H , and Z ′ → e + e − , µ + µ − , ZH near the W ′ and Z ′ resonances. For the lighter Z ′ ( θ H = 0 . − data of the 13 TeV LHC experiment we expect about ten µ + µ − events for the invariant massrange 3000 to 7000 GeV, though the number of the events becomes much smaller when θ H = 0 . W Z in the final state, it is confirmed that the leading contributions in theamplitude from the longitudinal polarizations of W and Z in the s -, t - and u -channels cancel witheach other so that the unitarity is preserved, provided that all KK excited states in the intermediatestates are taken into account. Deviation of the W W Z coupling from the SM is very tiny. Exoticpartners t (1) T and b (1) Y of the top and bottom quarks with electric charge +5 / − / M t (1) T ,b (1) Y = 4 . . θ H = 0 .
115 (0 . . INTRODUCTION At the LHC, W ′ and Z ′ are searched with various decay modes: decay to lepton pairs ( ℓ ¯ ℓ , ℓν , ν ¯ ν ) [1–6], a pair of top and bottom [7–9], q ¯ q -dijet [10, 11], W H ( → ℓνb ¯ b ) and ZH ( → ℓ ¯ ℓb ¯ b )[12–15] and W W and
W Z [16–22]. In the models with extra dimensions, W ′ and Z ′ appearas Kaluza-Klein (KK) excited states of charged and neutral vector bosons. The couplingsof W ′ and Z ′ to fields in the standard model (SM) depend on the details of the models. Forexample, in the minimal universal extra dimension (mUED) model [23], the conservationof KK number forbids tree-level couplings of KK-excited states to the SM fields so thatproduction of W ′ or Z ′ in the collider experiment is highly suppressed.Non-vanishing couplings of W ′ and Z ′ to the SM fields in models with extra-dimensionsoriginate from the violation of the KK number conservation which reflects the translationalinvariance in the direction of the extra dimensional space. The KK number conservation isbroken by, for instance, domain-wall like bulk mass terms, brane localized mass terms tothe fermion, or warped extra dimensions. It is well known that in the models in which SMfields live in the warped bulk space, couplings among fermions and KK-excited gauge bosonsare non-vanishing and can be large. In the minimal SU (3) gauge-Higgs unification (GHU)model [24] there are no W ′ or Z ′ couplings to the SM fields. In a custodially-protectedwarped extra dimensional model [25], W W and
W H diboson signals have been studied.The GHU provides a natural scenario for solving the gauge hierarchy problem. Manyadvances have been made in this direction recently [24, 26–71]. In the present paper, weevaluate couplings of KK excited gauge bosons to the SM gauge boson, Higgs boson, andfermions, and study collider signals of W ′ and Z ′ in the SO (5) × U (1) GHU model whichnaturally incorporates the Higgs boson of mass m H = 125 GeV and gives almost the samephenomenology as the SM at low energies [46–49]. In the previous work [47], we reportedthat in hadron collider experiments large Z ′ signals are expected due to the large couplingsof right-handed fermions to the KK-excited gauge bosons [72]. In the present paper weevaluate cross sections not only with fermionic final states but also with bosonic W H , ZH , W W and
W Z final states.In the warped space, in general, it is difficult to evaluate couplings among various fieldsas they should be calculated in their mass eigenstates. One remarkable feature of the GHUis that the Higgs VEV can be eliminated by a large gauge-transformation and its effect is3ransmitted to the change in the boundary conditions so that one can obtain mass eigenstateseasily. In the previous work [49], it is found that KK non-conserving couplings HW ( m ) W ( n ) and ZW ( m ) W ( n ) ( m = n ), including HW W ( n ) and ZW W ( n ) , are non-vanishing, and weused them in the calculation of the H → Zγ decay rate .This paper is organized as follows. In Sec. 2, the model is introduced . In Sec. 3 decaywidth and cross sections formulas are given. In Sec. 4 we introduce effective theories toqualitatively describe salient relations among various decay widths. In Sec. 5, couplings anddecay widths of W ′ and Z ′ are evaluated in GHU. In Sec. 6 cross sections are evaluated. W ′ and Z ′ signals in pp collision experiment at LHC are explored. We also show how theunitarity in the process d ¯ u → W Z is ensured by including contributions from KK states ofvector bosons and fermions in the intermediate states. Sec. 7 is devoted to summary. InAppendix A SO (5) generators and basis functions in the analysis are summarized. In Ap-pendices B and C masses and wave functions for bosonic and fermionic KK states are given,respectively. In Appendix D fermion couplings to vector bosons and the Higgs boson aresummarized, whereas cubic vector couplings and Higgs couplings are given in Appendix E.In Appendices F and G, formulae for decay widths and scattering cross sections are sum-marized, respectively.
2. MODEL
We consider five-dimensional (5D) gauge theory in the Randall-Sundrum space-time,whose metric is ds = e − σ ( y ) η µν dx µ dx ν + dy = G MN dx M dx N , (2.1)where η µν = diag( − , , , σ ( y ) = k | y | for − L ≤ y ≤ L and σ ( y + 2 L ) = σ ( y ) issatisfied. k is the AdS curvature. y = 0 and y = L boundaries are referred to as the UV(ultraviolet) and IR (infrared) branes, respectively. The Kaluza-Klein (KK) mass scale isgiven by m KK ≡ πke kL − , (2.2)and when kL &
5, it is followed by m KK ≪ k , L − .This space-time has symmetric under the Z reflection y → − y , and fundamental regionof the extra dimension is given by 0 ≤ y ≤ L . In this region we introduce a new coordinate4 = e ky , with which the metric becomes ds = 1 z (cid:18) η µν dx µ dx ν + dz k (cid:19) . (2.3)Note that ∂ y = kz∂ z and V y = kzV z for a vector field V M .In the 5D bulk space there are SO (5) and U (1) X gauge fields, four SO (5)-vector fermionsper generation Ψ ga ( a = 1 , , , g = 1 , , N F SO (5)-spinor fermions Ψ F i ( i =1 , · · · , N F ). We note that each of Ψ and Ψ is an SU (3)-color triplet.The bulk part of the action is given by S bulk = Z d x √− G (cid:26) −
12 tr G MR G NS F ( A ) MN F ( A ) RS − G MR G NS F ( B ) MN F ( B ) RS + 12 ξ ( A ) ( f ( A )gf ) + 12 ξ ( B ) ( f ( B )gf ) + L ( A ) GH + L ( B ) GH + X g =1 4 X a =1 ¯Ψ ga D ( c ga )Ψ ga + N F X i =1 ¯Ψ F i D ( c F i )Ψ F i (cid:27) , (2.4) D ( c a ) ≡ Γ A e AM (cid:18) ∂ M + 18 Ω MBC [Γ B , Γ C ] − ig A A M − ig B Q X,a − c a kǫ ( y ) (cid:19) , (2.5)where ǫ ( y ) ≡ σ ′ /k is a sign function. Γ M denotes gamma matrices which is defined by { Γ M , Γ N } = 2 η MN ( η = +1). e AM is an inverse fielbein, and Ω MBC is the spin connection. F ( A ) MN = ∂ M A N − ∂ N A M − ig A [ A M , A N ] and F ( B ) MN = ∂ M B N − ∂ N B M . g A and g B are 5D gaugecouplings of SO (5) and U (1) X , respectively. g w ≡ g A / √ L is the four-dimensional (4D) SO (5) coupling. f ( A )gf and f ( B )gf are gauge-fixing functions, and ξ ( A ) and ξ ( B ) are correspondinggauge parameters. L ( A ) GH and L ( B ) GH denote ghost Lagrangians.Bulk fermions are SO (5)-vectors. For the third generation, they are given byΨ = T tB b , t ′ = (( Q , q ) , t ′ ) = ( ˇΨ q , t ′ ) , Ψ = U XD Y , b ′ = (( Q , Q ) , b ′ ) = ( ˇΨ q , b ′ ) , Ψ = ν τ L X τ L Y , τ ′ = (( ℓ, L ) , τ ′ ) , = ( ˇΨ ℓ , τ ′ ) , Ψ = L X L X L Y L Y , ν ′ τ = (( L , L ) , ν ′ τ ) , = ( ˇΨ ℓ , ν ′ τ ) , (2.6)5here SO (4) vector is embedded in ( , )-representation of SU (2) L × SU (2) R by ψ ψ ψ ψ = 1 √ ψ + i~σ · ~ψ ) iσ = − iψ − ψ iψ + ψ iψ − ψ iψ − ψ . (2.7)The brane part of the action consists of scalar part S Φbrane and brane-fermion part S χ brane .The scalar part is given by S Φbrane = Z d x √− Gδ ( y )[ − ( D µ Φ) † ( D µ Φ) − λ Φ ( | Φ | − w ) ] ,D µ Φ = ∂ µ − i ( g A X a R =1 A a R µ T a R + 12 g B B µ )! Φ . (2.8)The fermion part of the brane action is S χ brane = Z d x √− Gδ ( y ) {L q + L ℓ } , L q ≡ X g =1 3 X α =1 ( ˆ χ q,g † αR i ¯ σ µ D µ ˆ χ q,gαR ) − i X g,g ′ =1 (cid:2) κ q,gg ′ ˆ χ q,g † R ˇΨ q,g ′ L ˜Φ + ˜ κ q,gg ′ ˆ χ q,g † R ˇΨ q,g ′ L Φ+ κ q,gg ′ ˆ χ q,g † R ˇΨ g ′ L ˜Φ + κ q,gg ′ ˆ χ q,g † R ˇΨ g ′ L Φ − (H.c.) (cid:3) , L ℓ ≡ X g =1 3 X α =1 ( ˆ χ ℓ,g † αR i ¯ σ µ D µ ˆ χ ℓ,gαR ) − i X g,g ′ =1 [˜ κ ℓ,gg ′ ˆ χ ℓ,g † R ˇΨ ℓ,g ′ L ˜Φ + κ ℓ,gg ′ ˆ χ ℓ,g † R ˇΨ ℓ,g ′ L Φ+ κ ℓ,gg ′ ˆ χ ℓ,g † R ˇΨ ℓ,g ′ L ˜Φ + κ ℓ,gg ′ ˆ χ ℓ,g † R ˇΨ ℓ,g ′ L Φ − (H.c.) , (2.9) D µ ˆ χ = ∂ µ − ig A X a L =1 A a L µ T a L − iQ X g B B µ ! ˆ χ, ˜Φ ≡ iσ Φ ∗ , (2.10)where ˆ χ q R = ˆ T R ˆ B R / , ˆ χ q R = ˆ U R ˆ D R / , ˆ χ q R = ˆ X R ˆ Y R − / , (2.11)ˆ χ ℓ R = ˆ L XR ˆ L Y R − / , ˆ χ ℓ R = ˆ L XR ˆ L Y R / , ˆ χ ℓ R = ˆ L XR ˆ L Y R − / , (2.12)are right-handed brane fermions. We note that each of ˆ χ q R , ˆ χ q R and ˆ χ q R is an SU (3)-colortriplet. We also have introduced 3 × κ q , , , κ ℓ , , , ˜ κ q and ˜ κ ℓ .6 . Orbifold symmetry breaking We impose Z boundary conditions at boundaries y = y i , y ≡ y ≡ L . A µ A y ( x, y i − y ) = P i A µ − A y ( x, y i + y ) P − i , (2.13) B µ B y ( x, y i − y ) = B µ − B y ( x, y i + y ) , (2.14)Ψ a ( x, y i − y ) = γ P i Ψ( x, y i + y ) a , (2.15)where P vec0 = P vec1 = diag( − , − , − , − , +1) , P sp0 = P sp1 = − (2.16)in the vector and spinor representations, respectively. These boundary conditions break SO (5) to SO (4) ≃ SU (2) L × SU (2) R . A a L ,a R µ and A ˆ ay are even function against reflections at y = y i and can have their zero-modes. Zero modes of A a L ,a R µ are the gauge fields of unbroken SO (4) symmetry. B. Symmetry breaking by brane scalar
Once Φ develops a VEV h Φ i = w , (2.17) SU (2) R × U (1) X symmetry is broken to U (1) Y After Φ develops a VEV, the boundaryconditions in the original gauge are given byat z = 1 : ∂ z A a L µ = (cid:16) ∂ z − κ k (cid:17) A R , R µ = (cid:18) ∂ z − κ ′ k (cid:19) A ′ R µ = ∂ z B Y ′ µ = 0 ,A ˆ aµ = A ˆ4 µ = 0 ,A a L z = A a R z + B z = 0 , ∂ z (cid:18) z A ˆ az (cid:19) = ∂ z (cid:18) z A ˆ4 z (cid:19) = 0 , at z = z L : ∂ z A a L µ = ∂ z A a R µ = ∂ z B Xµ = 0 ,A ˆ aµ = A ˆ4 µ = 0 , a L z = A a R z = B z = 0 , ∂ z (cid:18) z A ˆ az (cid:19) = ∂ z (cid:18) z A ˆ4 z (cid:19) = 0 . (2.18)Here we have defined κ ≡ g A w g w Lw , κ ′ ≡ ( g A + g B ) w , (2.19)and A ′ R M B Y ′ M ≡ c φ − s φ s φ c φ A R M B M ,c φ ≡ cos φ = g A p g A + g B , s φ ≡ sin φ = g B p g A + g B . (2.20)where a mixing angle φ is defined by tan φ ≡ g B /g A . KK modes of A , R µ and A ′ R µ with m n ≪ w obey effectively Dirichlet boundary conditions on the UV brane : A R , R µ = A ′ R µ = 0at z = 1.For fermions, non-vanishing h Φ i also induces brane mass terms given by S massbrane = Z d x √− Gδ ( y ) (cid:8) L massquark + L masslepton (cid:9) , (2.21) L massquark = X g,g ′ =1 (cid:20) − X α =1 iµ q,gg ′ α ( ˆ χ q,g † αR Q g ′ αL − Q g ′ † αL ˆ χ q,gαR ) − i ˜ µ q,gg ′ ( ˆ χ q,g † R q g ′ L − q g ′ † L ˆ χ q,g R ) (cid:21) , (2.22) L masslepton = X g,g ′ =1 (cid:20) − X α =1 iµ ℓ,gg ′ α ( ˆ χ ℓ,g † αL L g ′ αL − L g ′ † αL ˆ χ ℓ,gαR ) − i ˜ µ ℓ,gg ′ ( ˆ χ ℓ,g † R ℓ g ′ L − ℓ g ′ † L ˆ χ ℓ,g R ) (cid:21) . (2.23)where µ q,gg ′ α κ q,gg ′ α = ˜ µ q,gg ′ ˜ κ q,gg ′ = µ ℓ,gg ′ α κ ℓ,gg ′ α = ˜ µ ℓ,gg ′ ˜ κ ℓ,gg ′ = w. (2.24)For w ≫ m KK , and µ α , ˜ µ ≫ √ m KK , exotic fermions couple to brane fermions to becomevery heavy, so that only quark and leptons remain at low energy.Since the gauge field A y plays the role of the Higgs boson, the Yukawa couplings ofquarks and leptons are diagonal in the flavor space, and flavor mixing can be induced bynon-diagonal brane mass terms. For simplicity we assume that all brane mass terms areflavor-diagonal: µ q,gg ′ α = δ gg ′ µ qα , µ ℓ,gg ′ α = δ gg ′ µ ℓα , α = 1 , , , ˜ µ q,gg ′ = δ gg ′ ˜ µ q , ˜ µ ℓ,gg ′ = δ gg ′ ˜ µ ℓ . (2.25)8 . Electroweak symmetry breaking A ˆ az ( a = 1 , , A ˆ az ( x, z ) = φ a ( x ) s k ( z L − · z + · · · , (2.26)(where “ · · · ” includes higher-KK modes) and can develop a VEV. We assume that A z develops a VEV in the direction of T ˆ4 and we parameterize it by h φ a i = v W δ a . Then wedefine the Wilson-line phase parameter θ H byexp (cid:20) i θ H (2 √ T ˆ4 ) (cid:21) = exp (cid:20) ig A Z z L h A z i dz (cid:21) , (2.27)so that we obtain θ H = 12 g A v W r z L − k ∼ g v W π √ kLm KK , (2.28)where g w ≡ g A / √ L is the 4D SO (4) ≃ SU (2) L × SU (2) R gauge coupling constant. We alsohave a formula of W -boson mass. m W ≃ m KK π √ kL | sin θ H | , (2.29)and for θ H ≪ m W = g w v W is obtained. This may be compared with the SM formula m W = g w v H , v H = 246 GeV.To solve the equations of motion, we move to the twisted gauge in which h ˜ A z i = 0. Thisis achieved by ˜ A M = Ω A M Ω − + ig A Ω( ∂ M Ω − ) , ˜Ψ = ΩΨ , (2.30)Ω = exp h iθ ( z ) √ T ˆ4 i , θ ( z ) = θ H z L − z z L − . (2.31)Using Ω and making use of SO (5) algebra, we find gauge transformation as A a L M = 1 √ n ˜ A a + M + ˜ A a − M cos θ ( z ) − ˜ A ˆ aM sin θ ( z ) o ,A a R M = 1 √ n ˜ A a + M − ˜ A a − M cos θ ( z ) + ˜ A ˆ aM sin θ ( z ) o ,A ˆ aM = ˜ A a − M sin θ ( z ) + ˜ A ˆ aM cos θ ( z ) , a = 1 , , ,A ˆ4 µ = ˜ A ˆ4 µ , A ˆ4 z = ˜ A ˆ4 z − √ g A θ ′ ( z ) , (2.32)where ˜ A a ± M ≡ ( ˜ A a L M ± ˜ A a R M ) / √
2. 9 . Kaluza-Klein towers
1. Gauge bosons SO (5) × U (1) X gauge fields are decomposed into Kaluza-Klein towers given by A µ ( x µ , z ) + B µ ( x µ , z ) T B = ˆ W µ + ˆ W † µ + ˆ W Rµ + ˆ W † Rµ + ˆ Z µ + ˆ Z Rµ + ˆ A γµ + ˆ A ˆ4 µ , (2.33)where T B is a U (1) X generator. ˆ W , ˆ Z and ˆ A γ are KK towers for W , Z bosons and photonsin the SM, respectively. We note that each of ˆ W and ˆ Z towers contains two KK towers sothat there are eleven KK towers in (2.33).Each tower has an expansion of the formˆ A Cµ = X n A ( n ) µ ( x µ ) n h LA ( n ) ( z ) T − L + h RA ( n ) ( z ) T − R + ˆ h A ( n ) ( z ) T ˆ − o , ˆ A Nµ = X n A ( n ) µ ( x µ ) (cid:26) h LA ( n ) ( z ) T L + h RA ( n ) ( z ) T R + ˆ h A ( n ) ( z ) T ˆ3 + h BA ( n ) ( z ) T B (cid:27) , ˆ A ˆ4 µ = X n A ( n )ˆ4 µ ( x ) h A ( n )ˆ4 ( z ) T ˆ4 , (2.34)for ˆ A C = ˆ W , ˆ W R and ˆ A N = ˆ Z , ˆ Z R and ˆ A γ . T ± = ( T ± iT ) / √
2. Explicit forms of KKtowers of gauge fields are summarized in Appendix B and also found in [47]. A z and B z A z and B z are expanded, in the twisted gauge, as˜ A z ( x, z ) = X a =1 ˆ G a + X a =1 ˆ D a + ˆ H,B z ( x, z ) = ˆ B = ∞ X n =1 B ( n ) ( x ) u B ( n ) ( z ) T B . (2.35)ˆ D and ˆ G towers are expanded asˆ S − = X n S − ( n ) ( x ) n u LS ( n ) ( z ) T − L + u RS ( n ) ( z ) T − R + ˆ u S ( n ) ( z ) T ˆ − o , ˆ S = X n S n ) ( x ) n u LS ( n ) ( z ) T L + u RS ( n ) ( z ) T R + ˆ u S ( n ) T ˆ3 o , S = D, G, (2.36)10hereas ˆ H is expanded as ˆ H = ∞ X n =0 H ( n ) ( x ) u H ( n ) ( z ) T ˆ4 . (2.37) H (0) corresponds to the SM Higgs boson. ˆ D contain two KK towers so that ˜ A z contain tenKK towers. We note that KK modes other than H (0) are Nambu-Goldstone bosons andeaten by KK excited states of corresponding vector bosons.
3. Fermions
Bulk fermions are also decomposed into KK towers as follows. For example, quark bulkfermions Ψ and Ψ in the third generation are decomposed into(Ψ + Ψ ) = ˆ t T (5 / + ˆ t (2 / + ˆ t B (2 / + ˆ t U (2 / +ˆ b ( − / + ˆ b D ( − / + ˆ b X ( − / + ˆ b Y ( − / , (2.38)where numbers in subscripts denote the electric charge of fields. ˆ t and ˆ b are towers for topand bottom quarks, respectively, whereas others are towers for non-SM exotic partners ofquarks. We note that each of ˆ t and ˆ b contains two KK towers. In all Ψ and Ψ containten KK towers of fermions. In the same way KK towers of quarks of the first and secondgenerations are expressed as ˆ u T + ˆ u + · · · , ˆ d + ˆ d D + · · · , and so on.Lepton bulk fermions Ψ and Ψ (in the third generation) are decomposed as(Ψ + Ψ ) = ˆ ν τ (0) + ˆ τ ( − + ˆ τ X ( − + ˆ τ Y ( − +ˆ ν τ X (+1) + ˆ ν τ Y (0) + ˆ ν τ X (0) + ˆ τ Y ( − , (2.39)where ˆ τ and ˆ ν τ are towers for tau and tau-neutrino, respectively, and others are towers fornon-SM exotic lepton partners.Hereafter we work with rescaled bulk fermions ˘Ψ ≡ z Ψ. One can find mass spectraof KK towers corresponding to ˆ t T , ˆ b Y , ˆ t and ˆ b in references [39, 42]. For the fermionswith Q EM = +2 /
3, integrating brane fermions and utilizing orbifold boundary conditions,boundary conditions at z = 1 + in the original gauge are given by µ k [ µ ˘ U L + ˜ µ ˘ t L ] − D (2)+ ˘ U L = 0 , k ˘ B L − D (1)+ ˘ B L = 0 , ˜ µ k [ µ ˘ U L + ˜ µ ˘ t L ] − D (1)+ ˘ t L = 0 , ˘ t ′ L = 0 , (2.40)where D ( a ) ± ≡ ± ( d/dz ) + ( c a /z ). Bulk fermions in the twisted gauge, ˜ U , ˜ t , ˜ B and ˜ t ′ areobtained by gauge transformation (˜ t − ˜ B ) / √ t ′ = cos θ ( z ) sin θ ( z ) − sin θ ( z ) cos θ ( z ) ( t − B ) / √ t ′ , ˜ U = U, ˜ t + ˜ B = t + B. (2.41)where θ ( z ) = θ H ( z L − z ) / ( z L − ˘˜ U ˘˜ t ˘˜ B ˘˜ t ′ = √ k X n ψ ( n ) L ( x ) a U C (2) L a t C (1) L a B C (1) L a t ′ S (1) L ( z, λ n ) + √ k X n ψ ( n ) R ( x ) a U S (2) R a t S (1) R a B S (1) R a t ′ C (1) R ( z, λ n ) , (2.42)where C ( a ) L ( z, λ n ) = C L ( z ; λ n , c a ) and so on. ψ ( n ) L and ψ ( n ) R are 4D left- and right-handedfermions with the mass m n = kλ n , respectively. When we assume that µ , µ , ˜ µ , ˜ µµ ≫ kλ n , then the boundary conditions at z = 1 are simplified as follows˘ B L = 0 , µ ˘ U L + ˜ µ ˘ t L = 0 , ˜ µD (2)+ ˘ U L − µ D (1)+ ˘ t L = 0 , ˘ t ′ L = 0 . (2.43)Substituting left-handed KK modes in (2.42) into the above conditions, we can obtain KKmasses and corresponding eigenstates. KK fermions with Q EM = − / t (1) T and b (1) Y , which are exotic partners of the top andbottom quarks with electric charge +5 / − /
3, respectively, have mass M t (1) T ,b (1) Y = 4 . . θ H = 0 .
115 (0 . t (1) T and b (1) Y are the lightest non-SM states in GHUwhich can be singly produced in the colliders.12 . DECAY WIDTH AND CROSS SECTIONS The relevant parts of the Lagrangian for the present study consist of cubic interactionsamong vector and Higgs bosons L bosoneff = L W ′ W Z + L W ′ W H + L Z ′ W W + L Z ′ ZH , L W ′ W Z = X W ′ = W (1) ,W (1) R ig W ′ W Z ( η αγ η βδ − η αδ η βγ )( W ′− γ Z δ ∂ α W + β + W − γ Z δ ∂ α W ′ + β + Z γ W ′ + δ ∂ α W − β + Z γ W + δ ∂ α W ′− β + W ′ + γ W − δ ∂ α Z β + W + γ W ′− δ ∂ α Z β ) , L W ′ W H = g W (1) W H [ HW + µ W − (1) µ + HW − µ W +(1) µ ] , L Z ′ W W = X Z ′ = Z (1) ,γ (1) ,Z (1) R ig Z ′ W W ( η αγ η βδ − η αδ η βγ )( W − γ Z ′ δ ∂ α W + β + W − γ Z ′ δ ∂ α W + β + Z ′ γ W + δ ∂ α W − β + Z ′ γ W + δ ∂ α W − β + W + γ W − δ ∂ α Z ′ β + W + γ W − δ ∂ α Z ′ β ) , L Z ′ ZH = X Z ′ = Z (1) ,γ (1) ,Z (1) R g Z ′ ZH [ HZ µ Z ′ µ ] , (3.1)and the W ′ and Z ′ couplings to fermions L fermioneff = L W ′ f ¯ f + L Z ′ f ¯ f , L W ′ f ¯ f = X W ′ = W (1) ,W (1) R (cid:26) X ( ℓ,ν ℓ ) (cid:20) g LW ′ ℓν W ′− µ ¯ ℓγ µ − γ ν ℓ + (H.c.) (cid:21) + X color X ( U,D ) (cid:20) g LW ′ UD W ′− µ ¯ Dγ µ − γ U + (H.c.) (cid:21) + ( L → R, γ → − γ ) (cid:27) , L Z ′ f ¯ f = X Z ′ = Z (1) ,γ (1) ,Z (1) R (cid:26) X l = ℓ,ν ℓ (cid:20) g LZ ′ l Z ′− µ ¯ lγ µ − γ ν l + (H.c.) (cid:21) + X color X q = u,d,s,c,b,t (cid:20) g LZ ′ q Z ′− µ ¯ qγ µ − γ q + (H.c.) (cid:21) , +( L → R, γ → − γ ) (cid:27) . (3.2)The couplings L bosonseff and L fermionseff are summarized in Appendix E and Appendix D, respec-tively. Z ′ couplings to fermions are also given in Ref [47].Formulas for decay widths of W ′ and Z ′ are summarized in Appendix F. When M W ′ , M Z ′ ≫ M W , M Z , M H , partial decay widths of W ′ = W (1) and Z ′ = Z (1) , γ (1) , Z (1) R are approximately given byΓ( W ′ → W H ) ≃ M W ′ π (cid:18) g HW ′ W M W (cid:19) , Z ′ → ZH ) ≃ M Z ′ π (cid:18) g HZ ′ Z M Z (cid:19) , Γ( W ′ → W Z ) ≃ M W ′ π g W ′ W Z M W ′ M W M Z , Γ( Z ′ → W + W − ) ≃ M Z ′ π g Z ′ W W M Z ′ M W , (3.3)and Γ( W ′ → f ¯ f ′ ) ≃ N c M W ′ π ( | g LW ′ ff ′ | + | g RW ′ ff ′ | ) , Γ( Z ′ → f ¯ f ) ≃ N c M Z ′ π ( | g LZ ′ f | + | g RZ ′ f | ) , (3.4)where N c is the number of color of fermions f, f ′ . For later use, we define ratios of partialdecay widths as follows.Γ( W ′ → W H )Γ( W ′ → W Z ) ≃ g HW ′ W M W g W ′ W Z M W ′ M W M Z = g HW ′ W g W ′ W Z M Z M W ′ ≡ η W ′ , (3.5)Γ( Z ′ → ZH )Γ( Z ′ → W W ) ≃ g HZ ′ Z M Z g Z ′ W W M Z ′ M W = g HZ ′ Z g Z ′ W W M W M Z ′ M Z ≡ η Z ′ . (3.6)Formulas of s -channel cross sections mediated by W ′ or Z ′ are summarized in Appendix G.When √ s ≫ M W , M Z , M H , cross sections for the processes f ¯ f ′ → { W, W (1) } →
W H and f ¯ f → { Z, Z (1) , Z (1) R } → ZH are approximately given by σ ( f ¯ f ′ → { W, W (1) } →
W H ) ≃ N ic π sM W (cid:26) X V = W,W (1) g HV W [ | g LV ff ′ | + | g RV ff ′ | ]( s − M V ) + M V Γ V +2 Re " g HW W g HW (1) W [( g LW ff ′ )( g LW (1) ff ′ ) ∗ + ( g RW ff ′ )( g RW (1) ff ′ ) ∗ ][( s − M W ) + iM W Γ W ][( s − M W (1) ) − iM W (1) Γ W (1) ] , (3.7) σ ( f ¯ f → { Z, Z (1) , Z (1) R } → ZH ) ≃ N ic π sM Z (cid:26) X V = Z,Z (1) ,Z (1) R g HV Z [ | g LV f | + | g RV f | ]( s − M V ) + M V Γ V + X V ,V = Z,Z (1) ,Z (1) R V = V Re (cid:20) g HV Z g HV Z [( g LV f )( g LV f ) ∗ + ( g RV f )( g RV f ) ∗ ][( s − M V ) + iM V Γ V ][( s − M V ) − iM V Γ V ] (cid:21)(cid:27) . (3.8)For processes f ¯ f → W W and f ¯ f ′ → W Z , careful treatments are necessary. Each ampli-tude contains not only s - channel diagrams but also t - and u - channel diagrams. Therefore14he total amplitude M will be given by M = M SM + M NP , M SM = M SMs + M SMt + M SMu , M NP = M NPs + M NPt + M NPu , (3.9)where M SMs,t,u and M NPs,t,u are s -, t - and u -channel amplitudes for the SM fields and newphysics parts, respectively. The square of the total amplitude is given by |M| = |M SM | + |M NP | + (interference) , (3.10)where the interference terms contain products of M SM ( ∗ ) and M NP ( ∗ ) . When the energyof the initial state √ s is much larger than the EW scale, each of M SMs,t,u grows due to thelongitudinal part of the vector bosons in the final state. In the SM it is known that thegrowing contributions from longitudinal parts cancels with each other precisely, and thatthe unitarity of the amplitude M SM is protected. In our model, couplings among SM fieldsare very close to those of the SM values, so that M SM is well-behaved. In the vicinity of M W ′ and M Z ′ productions, |M NPs | dominates over the interference terms. The relevantformulas for cross sections for the SM part |M SM | are found in [73, 74]. For the NP partnear the resonance √ s ∼ M W ′ , M Z ′ ≫ M W , M Z , M H , one can neglect small M NPt,u . In theprecesses f ¯ f → W W and f ¯ f ′ → W Z , we approximate |M| by |M SM | + |M NPs | , thoughthe interference term need to be included for more rigorous treatment. Cross sections of s -channel processes f ¯ f ′ → { W ( n ) } → W Z and f ¯ f → { γ ( n ) , Z ( n ) , Z ( n ) R } → W + W − are givenin Appendix G. Hence the NP part of the cross sections are approximately given by σ ( f ¯ f ′ → { W ( n ) } → W Z ) ≃ N ic π s M W M Z (cid:26) X V = W ( n ) n ≥ [ | g LV ff ′ | + | g RV ff ′ | ] g V W Z ( s − M V ) + M V Γ V + X ( V ,V )=( W ( n ) ,W ( m ) )1 ≤ n 4. EFFECTIVE THEORIES Before jumping to calculate the couplings numerically, we formulate effective theorieswhich yield qualitative understandings of various couplings and decay widths.16 . 4D SO (4) × U (1) model Let us consider a 4D SO (4) × U (1) X model with two scalars Φ R and Φ H . Gauge couplingsof SO (4) ≃ SU (2) L × SU (2) R and U (1) X are denoted by g w and g b , respectively. Φ R is( , ) − / in SU (2) L × SU (2) R × U (1) X and Φ H is a SO (4)-vector ( , ¯ ) correspondingto the Higgs boson. When Φ R develops a VEV h Φ R i = (0 , µ ) / √ 2, the SU (2) R × U (1) X symmetry is broken to U (1) Y . We take µ = O ( m KK ). On the other hand non vanishing Φ H breaks SU (2) L × SU (2) R to SU (2) V whose generators are ( T a L + T a R ) / √ 2. With both h Φ R i and h Φ H i non-vanishing, SO (4) × U (1) X symmetry is broken to U (1) em . The mass matricesof gauge bosons in ( A a L µ , A a R µ ) ( a = 1 , 2) and ( A L µ , A R µ , B µ ) are given by M C = M LL − M LL − M LL M LL + M RR , M N = M LL − M LL − M LL M LL + M RR − M RB − M RB M BB ,M LL = g w v , M RR = g w µ , M RB = g w g b µ , M BB = g b µ , (4.1)respectively. M C has two eigenvalues corresponding to the mass-squared of W and W R bosons, respectively, which are given by M W = ¯ M W (1 + O ( v /µ )) , ¯ M W ≡ g w v ,M W R = g w µ O ( v /µ )) . (4.2) M N has three eigenvalues, which correspond to mass-squared of the photon, Z -boson, and Z R -boson, respectively. Here M γ = 0 ,M Z = g w v · g w + 2 g b g w + g b (1 + O ( v /µ ))= ¯ M Z (1 + O ( v /µ )) , ¯ M Z ≡ g w v θ W ,M Z R = ( g w + g b ) µ O ( v /µ ))= g w µ θ W cos θ W − sin θ W (1 + O ( v /µ )) . (4.3)Diagonalizing these mass matrices we obtain mass eigenstates W, W ′ and Z, Z ′ , γ . Mixing17atrices are given by A a L µ A a R µ = cos θ C sin θ C − sin θ C cos θ C W aµ W aRµ , a = 1 , , A L µ A R µ B µ = (cid:16) ~v γ , ~v − cos θ N + ~v + sin θ N , ~v + cos θ N − ~v − sin θ N (cid:17) γ µ Z µ Z Rµ ,~v γ = sin θ W sin θ W √ cos 2 θ W , ~v − = cos θ W − sin θ W cos θ W − tan θ W √ cos 2 θ W , ~v + = √ cos 2 θ W cos θ W − tan θ W , (4.4)where mixing angles are determined so as to mixing matrices properly diagonalize massmatrices: tan(2 θ C ) = 2 v µ , tan(2 θ N ) = 2 g w v p g w + 2 g b w ( g w + g b ) − g g b v . (4.5)The weak mixing angle is given bysin θ W = g b p g w + 2 g b , (4.6)so that γ µ couples to W ± bosons with e = g w sin θ W .Now we can calculate couplings among these mass eigenstates. For vector-boson trilinear V V V couplings and V V H couplings, we obtain g W W γ = g w sin θ W = e,g W W Z = g w cos θ W + O ( v /µ ) ,g Z R W W = − g w (cos θ W − sin θ W ) / cos θ W v µ ,g ZW R W = − g w θ W v µ , (4.7)and g HW W = g w ¯ M W (1 + O ( v /µ )) , HZZ = g w cos θ W ¯ M Z (1 + O ( v /µ )) ,g HW R W = − g w ¯ M W (1 + O ( v /µ )) ,g HZ R Z = − g w ¯ M Z p cos θ W − sin θ W cos θ W (1 + O ( v /µ )) , (4.8)where ¯ M W = g w v/ M Z = ¯ M W / cos θ W are used.Hence we see that (a) g ZW W is very close to its SM-value g w cos θ W and the deviation fromthe SM-value will be suppressed by a factor O ( v /m KK ), (b) Deviations of g HZZ and g HW W couplings from their SM values are both suppressed by v /m KK , (c) g Z ′ W W and g ZW R W aresuppressed by a factor O ( v /µ ). g W R W Z , g Z R W W ∼ g W W Z · v µ , (4.9)(d) In contrast, values of HW W R and HZZ R approximately equal to HW W and HZZ couplings in magnitudes but opposite in signs, respectively. g HW R W ∼ − g HW W , g HZ ′ Z ∼ − g HZZ . (4.10)(e) For the decay widths of W R and Z R one findsΓ( W R → ZW )Γ( W R → W H ) ≃ (cid:18) g ZW R W g HW R W M W R ¯ M Z (cid:19) = 1 + O ( v /µ ) , (4.11)Γ( Z R → W W )Γ( Z R → ZH ) ≃ (cid:18) g Z R W W g HZ ′ Z M Z R ¯ M Z ¯ M W (cid:19) = 1 + O ( v /µ ) . (4.12)We will find in the following section that the relation (4.11) will be satisfied, and that (4.12)needs to be generalized to incorporate KK- γ and KK- Z bosons. 2. 5D SO (5) × U (1) X GHU in the flat-space limit We also explore the flat-space limit of the warped SO (5) × U (1) X GHU by taking kL → L = πR finite [71]. A similar model is seen in [70]. In this limit m KK = 1 /R . The W -boson mass and the AB phase θ H are related with each other by √ m W πR ) = sin θ H . The couplings among vector bosons and Higgs are summarized asfollows. Vector boson trilinear couplings are given by g γ (0) W ( n ) W ( m ) = δ mn e, g γ (1) W ( n ) W ( m ) = δ mn √ e, Z (0) W (0) W (0) = g w cos θ W + O ( m W R ) ,g Z (1) W (0) W (0) , g Z (0) W (1) W (0) = O ( m W R ) ,g Z (1) R W (0) W (0) = 8 √ √ cos 2 θ W π cos θ W g w m W R + O ( m W R ) ,g Z (0) W (1) R W (0) = 8 √ π g w m W m Z R + O ( m W R ) . (4.13)Higgs vector-boson couplings are given by g HW ( m ) W ( n ) = δ mn ( − n g w m W ( n ) ,g HZ ( m ) Z ( n ) = δ mn ( − n g w cos θ W m Z ( n ) ,g HZ (1) R Z (0) = − √ √ cos 2 θ W π cos θ W g w m Z [1 + O ( m W R )] ,g HW (1) R W (0) = − √ g w π m W [1 + O ( m W R )] ,g Hγ ( n ) Z ( m ) , g Hγ ( n ) Z ( m ) R = 0 . (4.14)Here we note that m W (1) R = m Z (1) R = 1 / (2 R ).It seems that W and W R , Z and Z R are mixed in an ordinary manner as seen inthe 4D model, whereas W and W ( n ) , Z and Z ( n ) are very weakly mixed. This will beunderstood by mass-mixings in the original gauge. Gauge fields for charged bosons are A ( n ) ± L µ , A ( m ) ± R µ , A ( m ) ˆ ± µ , ( a = 1 , n = 0 , , , · · · and m = 1 , , · · · ). For up to first KKexcited states, the mass matrix in the ( A (0) ± L µ , A (1) ± R µ , A (1) ± L µ , A (1) ˆ ± µ ) basis is given by M v − M v − M v M v + M R − M ′ v − M ′ v M v + M L 00 0 0 2 M v + M X , (4.15)where M v = 14 g w v , M R = 14 R , M L = M X = 1 R ,M ′ v = 14 g w v πR Z πR cos (cid:18) πR − y R (cid:19) cos (cid:18) πR − yR (cid:19) dy. (4.16) A (0) ± L µ and A (1) ± R µ have a mixing term, whereas there is no mixing between A (0) ± L µ and A (1) ± L µ in the mass matrix due to the KK-number conservation. Therefore mixing between A (0) ± L µ and A (1) ± L µ is induced only through both A (0) ± L µ - A (1) ± R µ and A (1) ± L µ - A (1) ± R µ mixing terms sothat the mixing angle is suppressed. 20 . Comment on the warped SO (5) × U (1) X GHU In the flat GHU case, A (0) a L µ - A (1) a L µ mass terms vanish. This is because the Higgs wavefunction is constant along the extra dimension. The mass term, which is written as anoverlap integral of wave functions of A (0) ± L µ , A (1) ± L µ and the Higgs boson, vanishes by theorthonormality conditions of wave functions. In the warped case, the Higgs wave function isnot constant along the direction of the extra dimension. Therefore A (0) a L µ - A (1) a L µ mass termsdoes not vanish. 5. COUPLINGS AND DECAY WIDTHS Input parameters used in the numerical study is summarized in Table I. The W bosonmass at the tree level becomes M tree W = 79 . N F = 4 and z L = 10 , 10 . With ( N F , z L ) given bulk mass parameterof dark fermions, c F , is determined such that the resultant Higgs mass becomes M H = 125GeV. This procedure determine the value of θ H and the bulk mass parameters of quarks andleptons.(See for the details [46, 47]) The resultant values of θ H and k , m KK and first-KKgauge boson masses are tabulated in Table II. We set fermion bulk mass parameters as c g = c g ≡ { c u , c c , c t } and c g = c g ≡ { c e , c µ , c τ } . These parameters are tuned so that fermionmasses coincide with values in Tables I. These are listed in listed in Table III, TABLE I: Input parameters. Masses of Z boson, leptons and quarks in the unit of GeV. M Z sin θ W m e ( M Z ) m µ ( M Z ) m τ ( M Z )91.1876 0.23126 0 . × − m u ( M Z ) m d ( M Z ) m s ( M Z ) m c ( M Z ) m b ( M Z ) m t ( M Z )1 . × − . × − A. Couplings The couplings among vector bosons V , Higgs H , and SM fermions f SM are evaluatedfrom overlap integrals where functions of V , H and f SM are inserted. The detailed formulasare given in Appendices E and D, and Refs. [47, 49].21 ABLE II: Aharonov-Bohm phase θ H , the bulk mass parameter of dark fermions c F , AdS cur-vature k , Kaluza-Klein scale m KK and masses of first KK gauge bosons are given in the unit ofGeV with respect to z L for N F = 4 are summarized. Z (1) , W (1) and γ (1) are almost degenerate.Their mass differences is 1 − z L θ H c F k m KK m Z (1) m W (1) m γ (1) m Z (1) R = m W (1) R [rad.] [GeV] [TeV] [TeV] [TeV] [TeV] [TeV]10 . × . × z L c u c c c t c e c µ c τ In Table IV, the left-handed couplings of SM fermions to the W boson and its KK excitedstates are tabulated. Similar results for θ H = π/ W R boson vanish.It is seen that couplings to W (0) (the W boson) are slightly larger than the SM value g w / √ t and b to W (1) are smaller thancouplings to W (0) and their signs are opposite. W (1) t ¯ b coupling is larger than the SM value.The difference between W (1) ud and W (1) tb couplings is understood as follows. The cou-plings among left-handed up- and down-sector fermions ( U, D ) = ( u, d ), ( t, b ) and W boson g LW (1) UD are given by the overlapping (D.4). The integration in (D.4) is dominated by theterm h LW ( n ) f DbL f UtL (( U, D ) = ( u (0) , d (0) ) , ( t (0) , b (0) )). In Fig. 1, leading part of bulk wave func-tions for fermions and gauge bosons are plotted. In Fig. 1-(a), we see that f u (0) tL decreasesmuch faster than f t (0) tL . In particular, near z = z L , f t (0) tL has small but sizable value whereasthe value of f u (0) tL almost vanishes. In Fig. 1-(b), h LW (0) is almost constant, whereas h LW (1) hasnegative values for small values of z and has large positive value near z = z L . In Fig. 2, weplot overlapping of wave functions of up- and down-type fermions and W (1) . In the Figure,overlapping of wave functions of light quarks and W (1) has large negative value only near22 ABLE IV: Masses and couplings of W ( n ) to left-handed SM fermions. N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 m W ( n ) [GeV] 79.9 6004 9034 13378 16538 g LW ( n ) ℓν / ( g w / √ ℓ, ν ) = ( e, ν e ) 1.00019 -0.3455 -0.02507 0.2510 0.01937( µ, ν µ ) 1.00019 -0.3455 -0.02507 0.2510 0.01937( τ, ν τ ) 1.00019 -0.3452 -0.02505 0.2507 0.01934 g LW ( n ) UD / ( g w / √ U, D ) = ( u, d ) 1.00019 -0.3455 -0.02507 0.2510 0.01937( c, s ) 1.00019 -0.3454 -0.02506 0.2510 0.01936( t, b ) 0.9993 1.2970 0.06527 -0.4342 -0.03110 N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 m W ( n ) [GeV] 79.9 8520 12624 18852 23112 g LW ( n ) ℓν / ( g w / √ ℓ, ν ) = ( e, ν e ) 1.00009 -0.3904 -0.01861 0.2901 0.01461( µ, ν µ ) 1.00009 -0.3904 -0.01861 0.2901 0.01461( τ, ν τ ) 1.00009 -0.3901 -0.01858 0.2896 0.01457 g LW ( n ) UD / ( g w / √ U, D ) = ( u, d ) 1.00009 -0.3904 -0.01861 0.2901 0.01461( c, s ) 1.00009 -0.3904 -0.01860 0.2900 0.01460( t, b ) 0.9995 1.7517 0.04516 -0.2925 -0.01490 the UV brane ( z = 1). On the other hand, the overlapping of heavy quarks and W (1) takesnegative value for small values of z but becomes positive for larger z . This difference inthe integrand results in the differences in the signs and magnitudes of g LW (1) tb , g LW (1) ud and g W (0) ud .In Table V, HW W ′ ( W ′ = W ( n ) , W ( n ) R ) and HZZ ′ ( Z ′ = Z ( n ) , Z ( n ) R ) couplings are tabu-23 a ) z f tLt ( ) f tLu ( ) ( b ) - z h W ( ) L h W ( ) L FIG. 1: Behavior of dominant component of the wave functions for fermions and W ( n ) bosons for N F = 4, z L = 10 ( θ H = 0 . f t (0) tL for the top and f u (0) tL for the up quark, respectively. (b) Red-solid and blue-dashed lines are h LW (0) for W (0) and h LW (1) for W (1) bosons, respectively. 60 000 80 000 100 000 - - z g W ( ) tbL g W ( ) udL FIG. 2: Integrand of the coupling g LW (1) UD in (D.4). Red-solid and blue-dashed lines are for( U, D ) = ( t, b ) and ( u, d ), respectively. Hγ ( n ) Z ( m ) and Hγ ( n ) Z ( m ) R couplings vanish. We find that g W W H and g ZZH couplings are given by the SM values multiplied by cos θ H . g W (1) W H and g W (1) R W H area few times larger than g W W H , and similar relations holds among g Z (1) R ZH , g Z (1) ZH and g ZZH .All g W ′ ( n ) W H and g Z ′ ( n ) ZH couplings become small as n becomes larger. TABLE V: Couplings of W ( n ) , Z ( n ) , γ ( n ) and Z ( n ) R to W H , ZH in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g W ( n ) W H / ( g w cos θ H ) 80.0 255 2.57 45.4 0.220 g Z ( n ) ZH / ( g w cos θ H / cos θ W ) 91.2 291 3.35 51.8 0.286 g W ( n ) R W H /g w — 266 50.5 20.6 11.1 g Z ( n ) R ZH / ( g w / cos θ W ) — 223 42.3 17.2 9.27 N F = 4 z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g W ( n ) W H / ( g w cos θ H ) 80.0 225 1.89 39.2 0.169 g Z ( n ) ZH / ( g w cos θ H / cos θ W ) 91.2 257 2.46 44.8 0.220 g W ( n ) R W H /g w — 238 45.1 18.4 9.89 g Z ( n ) R ZH / ( g w / cos θ W ) — 199 37.7 15.4 8.27 In Table VI, trilinear vector-boson couplings are tabulated. The g W W Z coupling is veryclose to its SM value g w cos θ W . The deviation is one part in 10 . g γW W is exactly e , whichreflects the unbroken U (1) em gauge symmetry. Couplings among the first KK and two SMvector bosons are suppressed by a factor of O (10 − ), which is close to the square of the ratioof the weak boson mass to the 1st KK boson mass. B. Decay width In Tables VII and VIII, decay widths of W and W R boson are tabulated, respectively.Since the W (1) boson couples equally to the light SM fermions except for b and t quarks,partial decay widths to light SM fermions are almost identical besides the QCD color factor.The W (1) coupling to t and b quarks is larger than the couplings to other fermion pairs. The25 ABLE VI: Trilinear vector-boson couplings of W ( n ) , Z ( n ) , γ ( n ) and Z ( n ) R to W + W − , W ZN F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g W ( n ) W Z / ( g w cos θ W ) 0.9999998 − . × − − . × − − . × − − . × − g Z ( n ) W W / ( g w cos θ W ) 0.9999998 − . × − . × − − . × − − . × − g γ ( n ) W W /e − . × − − . × − − . × − − . × − g Z ( n ) R W W /g w — 5 . × − . × − . × − . × − g W ( n ) R W Z /g w — 7 . × − . × − . × − . × − N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g W ( n ) W Z / ( g w cos θ W ) 0.99999995 − . × − − . × − − . × − − . × − g Z ( n ) W W / ( g w cos θ W ) 0.99999995 − . × − − . × − − . × − − . × − g γ ( n ) W W /e − . × − − . × − − . × − − . × − g Z ( n ) R W W /g w — 2 . × − . × − . × − . × − g W ( n ) R W Z /g w — 3 . × − . × − . × − . × − W (1) decay to tb dominates over decay to other fermion pairs. Partial decay widths of W (1) to W Z and W H are almost identical:Γ( W (1) → W H ) ≃ Γ( W (1) → W Z ) , ∴ η W (1) ≃ , Γ( W (1) R → W H ) ≃ Γ( W (1) R → W Z ) , ∴ η W (1) R ≃ . (5.1)We also findΓ( W (1) → W H ) ∼ Γ( W (1) R → W H ) , Γ( W (1) → W Z ) ∼ Γ( W (1) R → W Z ) . (5.2)Since the W ( n ) R boson does not couple to SM fermions, and W (1) R decay only to the SMbosons.In Tables IX, X and XI, decay widths of Z (1) , γ (1) and Z (1) R are tabulated, respectively.Compared with W (1) and W (1) R , γ (1) and Z (1) R have large total widthsΓ γ (1) /M γ (1) = . 151 for N F = 4, z L = 10 ( θ H = 0 . . 125 for N F = 4, z L = 10 ( θ H = 0 . Z (1) R /M Z (1) R = . 129 for N F = 4, z L = 10 ( θ H = 0 . . 133 for N F = 4, z L = 10 ( θ H = 0 . . (5.4)From the tables, one finds thatΓ( Z (1) → HZ ) ≃ X Z ′ = Z (1) ,γ (1) Γ( Z ′ → W + W − ) , (5.5)Γ( Z (1) R → HZ ) ≃ Γ( Z (1) R → W + W − ) . (5.6) γ (1) and Z (1) are almost degenerate. The relation (5.5) follows from the relation amongHiggs-vector boson and trilinear vector boson couplings g γ (1) ZH + g Z (1) ZH M Z ≃ ( g γ (1) W W + g Z (1) W W ) M Z ′ M Z , (5.7)where M γ (1) ≃ M Z (1) ≡ M Z ′ and g γ (1) ZH = 0. TABLE VII: Partial and total decay width of W − (1) in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . e − ¯ ν e µ − ¯ ν µ τ − ¯ ν τ d ¯ u s ¯ c b ¯ t W − Z W − H totalΓ 2.00 2.00 1.99 5.99 5.98 84.7 42.7 42.1 187 N F = 4, z L = 10 ( θ H = 0 . e − ¯ ν e µ − ¯ ν µ τ − ¯ ν τ d ¯ u s ¯ c b ¯ t W − Z W − H totalΓ 3.63 3.63 3.62 10.88 10.88 219 46.9 47.2 346TABLE VIII: Partial and total decay widths of W − (1) R in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . W − Z W − H total0 43.4 43.4 86.8 N F = 4, z L = 10 ( θ H = 0 . W − Z W − H total0 48.7 48.8 97.4 ABLE IX: Partial and total decay widths of Z (1) in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total40.4 35.7 32.1 1.30 1.30 1.30 53.3 46.1 48.5 15.7 13.8 45.7 16.1 54.8 406 N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total48.1 42.5 38.0 2.36 2.36 2.36 64.4 55.5 84.4 20.3 18.1 106.8 17.8 61.1 564TABLE X: Partial and total decay width of γ (1) in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total133.0 117.4 105.7 0 0 0 171.0 147.2 93.0 42.8 36.8 23.3 39.3 0 909 N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total158.9 140.2 125.2 0 0 0 204.6 175.0 101.0 51.1 43.8 25.3 43.6 0 1068 6. CROSS SECTION In this section we evaluate cross sections in pp -collisions for various final states. In thenumerical evaluation we use CTEQ5 parton distribution functions [75]. A. W ′ → tb , µν and Z ′ → ℓ + ℓ − In Figure 3, the differential cross sections of processes pp → { W − , W (1) − } → b ¯ t, µ − ¯ ν areplotted. For light fermion doublet paris ℓ ¯ ν, d ¯ u, s ¯ c in the final state, due to the flipped-signs ofthe couplings to W (1) , a clear deficit of cross section just below the resonance M µν ∼ M W (1) is observed. For processes with b ¯ t final state, a deficit of cross section is observed above theresonance M tb > M W (1) , since the W (1) ¯ tb coupling has opposite sign relative to the W (1) ¯ ud coupling.When the final state contains a neutrino, the transverse momentum distribution dσ/dp T with respect to the transverse momentum of charged lepton, p T , gives information on the28 ABLE XI: Partial and total decay widths of Z (1) R in the unit of GeV. N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ O (10 − ) O (10 − ) O (10 − ) u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total30.5 30.7 729 N F = 4, z L = 10 ( θ H = 0 . e + e − µ + µ − τ + τ − ν e ¯ ν e ν µ ¯ ν µ ν τ ¯ ν τ O (10 − ) O (10 − ) O (10 − ) u ¯ u c ¯ c t ¯ t d ¯ d s ¯ s b ¯ b W + W − ZH total34.1 34.3 1058 mass of W ′ . The transverse-momentum distribution at parton-level is given in Appendix G.The p T distribution in pp collision is given by dσ ( pp → e − ¯ ν + X ) dp T ( p T ) = Z dτ (cid:26) dp T ( d ¯ u → e − ¯ ν ) dp T ( s pp τ, p T ) · dL d ¯ u dτ ( τ ) (cid:27) , (6.1)where the parton luminosity dL d ¯ u /dτ is given in terms of parton distribution functions f q ( x , Q ) by dL d ¯ u dτ ( τ ) = Z dx Z dx [ f d ( x , Q ) f ¯ u ( x , Q ) + f d ( x , Q ) f ¯ u ( x , Q )] δ ( τ − x x ) ,Q = s pp τ. (6.2)In Figure 4 the p T -distribution dσ ( pp → e − ¯ ν ) /dp T is plotted. In the figure, Jacobian peakat p T = M M (1) / ≃ pp ( u ¯ u, d ¯ d ) → Z ′ → ℓ + ℓ − , we show the plot of the differential crosssection dσ/dM µµ in Figure 5. In the plot the updated decay widths of Z ′ ( Z ′ = γ (1) , Z (1) and Z (1) R ) has been used, which takes bosonic final states ( W + W − and ZH ) into29 IG. 3: pp ( d ¯ u ) → { W − , W (1) − } → b ¯ t, µ − ¯ ν µ differential cross sections dσ/dM ff at √ s pp = 14 TeVfor N F = 4, z L = 10 ( θ H = 0 . M ff is the invariant mass of ( b, ¯ t ) or ( µ, ¯ ν ). Red-solid, black-dotted lines are for the b ¯ t and µ − ¯ ν final states in GHU. Blue-dashed line is the cross section inthe SM. Cross sections for the processes pp → W (1) − → f ¯ f ′ , ( f, f ′ ) = ( e − , ν e ) , ( τ − , ν τ ) are almostidentical with that of µ − ¯ ν , whereas cross section for ( f, f ′ ) = ( d, u ) , ( s, c ) final states is three timesas large as that of µ − ¯ ν due to the color factor.FIG. 4: dσ ( pp → e − ¯ ν e ) /dp T as a function of p T at √ s pp = 14 TeV for N F = 4, z L = 10 ( θ H = 0 . account, and is O (10%) bigger than that used in the previous paper [47]. We stressthat the rate of Z ′ production is rather large, and it is promising to see the Z ′ eventsat the current LHC Run 2. Since in this model Z ′ bosons have large widths, at theearly stage of LHC experiment, sporadic events of high-energy µ + µ − final states will30e seen. For θ H = 0 . 115 ( M Z (1) ,γ (1) ∼ . M Z (1) R ∼ . − and √ s pp = 13 TeV data, expected numbers of events in GHU N GHU and SM signal N SM , and significance S are N GHU /N SM ( S ) = 8 . / . . . / . 26 (3 . . / . 02 (4 . . / . 004 (6 . . / . . 0) and 0 . / × − (0 . 34) for bins (GeV) [2000 , , , , , , M µµ & θ H (heav-ier M Z ′ ), the signals becomes smaller and more data is required for confirming/rejectingthe model. For θ H = 0 . M Z (1) ,γ (1) ∼ . M Z (1) R ∼ . − and √ s pp = 14 TeV data, N GHU /N SM ( S ) = 140 / 155 (1 . / 12 (2 . . / . . . / . 14 (4 . . / . 006 (2 . 3) and 0 . / . . 7) for bins (GeV) [2000 , , , , , , FIG. 5: Differential cross section dσ/dM µµ of the process pp ( u ¯ u, d ¯ d ) → { γ, Z, Z (1) , γ (1) , Z (1) R } → µ + µ − at √ s = 14 TeV. M µµ is the invariant mass of µ + µ − . Solid [red], dotted [green] and dashed[blue] lines are the Z ′ resonance in the GHU for N F = 4, z L = 10 ( θ H = 0 . N F = 4, z L = 10 ( θ H = 0 . e + e − final state is identical tothe µ + µ − final state. B. W ′ → W H and Z ′ → ZH In Figures 6 and 7, differential cross sections of processes pp → { W − , W (1) − } → W − H and pp → { Z, Z (1) , Z (1) R } → ZH are plotted, respectively. Compared with the W H mode,cross section for ZH mode is bigger and the width is wider. It is due to the fact that Z (1) Z (1) R have large couplings to the right-handed quarks, and their widths are large. FIG. 6: Differential cross section dσ/M W h of the process pp ( d ¯ u ) → { W − , W (1) − } → W − H at √ s pp = 14 TeV for N F = 4, z L = 10 ( θ H = 0 . M W h is the invariant mass of W H . Red-solidand blue-dashed lines show cross sections in GHU and in the SM, respectively.FIG. 7: Differential cross section dσ/dM Zh of the process pp ( u ¯ u, d ¯ d ) → { Z, Z (1) , Z (1) R } → ZH cross section at √ s pp = 14 TeV for N F = 4, z L = 10 ( θ H = 0 . M Zh is the invariant mass of ZH . Red-solid and blue-dashed lines show cross sections in GHU and in the SM, respectively. C. W ′ → W Z and Z ′ → W W In Figures 8 and 9, differential cross sections of the processes pp →{ γ, Z, Z (1) , γ (1) , Z (1) R } → W + W − and pp → { W − , W (1) − } → W − Z are plotted, respec-tively. For the W Z final states, the signal of the resonance of W ′ is a few times larger32han that of the SM. For the W W final states, the contribution from Z ′ resonances is muchsmaller than the SM cross section so that the signal is hard to see.We comment that there are no s -channel processes with ZZ final states mediated byvector bosons. The process mediated by KK gravitons [76, 77] can be ignored, as thecouplings of KK gravitons to the SM fields are suppressed by k/M P l ≪ 1, where M P l is thePlanck mass. FIG. 8: Differential cross section dσ/dM W Z of the process pp ( d ¯ u ) → { W − , W (1) − } → W − Z at √ s pp = 14 TeV for N F = 4, z L = 10 ( θ H = 0 . M WZ is the invariant mass of W Z .Green-dotted line shows the s -channel W ′ signals in GHU model. Blue-dashed line shows the SMprediction including s -, t - and u -channels [74]. Red-solid line is the sum of SM and GHU signals. D. Unitarity in f ¯ f ′ → W Z It is important to see how the unitarity is ensured when vector bosons are involved inthe final states. The unitarity in the W boson scattering, W W → W W in the gauge-Higgsunification has been studied in [41] by using position-space propagators. In the presentpaper we are considering f ¯ f → V ′ → W W , W Z . In these processes, t - and u -channelamplitudes must be included to cancel the growing part of the the s -channel amplitude at √ s ≫ m V ′ (See Fig. 10).Let us consider the process d ( p )¯ u ( p ) → W − ( k ) Z ( k ). When the initial states are givenby d R u L , there are no s -channel contribution as W ( n ) do not couple to the right-handedquarks. For the final state bosons with longitudinal polarization, ε µ ( k ) ≃ k µ /M W and33 IG. 9: Differential cross section dσ/dM W W of the process pp ( u ¯ u, d ¯ d ) → { γ, Z, Z (1) , γ (1) , Z (1) R } → W + W − at √ s pp = 14 TeV for N F = 4, z L = 10 ( θ H = 0 . M W W is the invariant mass of W + W − . Green-dotted line shows the s-channel W ′ signals in GHU model. Blue-dashed line showsthe SM prediction including s -, t - and u -channels [73]. Red-solid line is the sum of SM and GHUsignals. d ¯ u W ( n ) − W − Z (a) W − Zu ( n ) , u ( n ) U , u ( n ) B d ¯ u (b) W − Zd ( n ) , d ( n ) D , d ( n ) X d ¯ u (c) FIG. 10: Diagrams of d ¯ u → W Z at a energy scale above M KK . (a), (b) and (c) represent s -, t -and u -channels, respectively. W ( n ) is the n -th KK state of W , whereas D ( n ) and U ( n ) are n -th KKstates of d and u , and partners d ( n ) D,X , u ( n ) U,B , respectively. ε ν ( k ) ≃ k ν /M Z , t - and u -channel amplitudes at very high-energy √ s ≫ m KK are expressedas M t ∼ M W M Z X U g RW d U g RZu U ¯ v ( p ) /k P R ( /p − /k )( p − k ) /k P R u ( p )= 12 M W M Z X U g RW d U g RZu U ¯ v ( p )( /k − /k ) P R u ( p ) , (6.3)34 u ∼ M W M Z X D g RZd D g RW u D ¯ v ( p ) /k P R ( /p − /k )( p − k ) /k P R u ( p )= 12 M W M Z X D g RZd D g RW u D ¯ v ( p )( /k − /k ) P R u ( p ) , (6.4)where P L/R ≡ (1 ∓ γ ) / 2, and U and D denote KK-excited states with Q EM = +2 / − / 3, respectively. Here we have retained contributions only from the first KK states offermions, as the W u D ( n ) , W d U ( n ) , Zu U ( n ) and Zd D ( n ) couplings ( n ≥ M t and M u to cancel with each other, the relation X U g RW U d g RZu U ≃ X D g RZ D d g RW u D (6.5)should be satisfied. With the values in Tables XIV-XVIII, one finds that in (6.5)(L.H.S) ≃ g RW du (1) g RZuu (1) + g RW du (1) B g RZuu (1) B = [( − . · ( − . 65) + (1 . · ( − . × − g w / √ , (6.6)(R.H.S) ≃ g Rdd (1) g RW ud (1) + g RZdd (1) X g RW ud (1) X = [(1 . · ( − . 3) + (0 . · (2 . × − g w / √ . (6.7)We observe that the relation (6.5) is well satisfied. We note that g RW du (1) = − g RW ud (1) X , g RZuu (1) = − g RZdd X (1) ,g RW du (1) B = − g RW ud (1) , g RZdd (1) = − g RZuu B (1) . (6.8)When the initial states are given by d L u R , there are contributions from s -channel ampli-tudes. The condition for the cancellations is given by (c.f. Chapter 21 of [78]) ∞ X n =0 g LW ( n ) ud g W ( n ) W Z ≃ X U ′ g LW d U ′ g LZu U ′ − X D ′ g LZd D ′ g LW u D ′ , (6.9)where U ′ and D ′ represent all SM and non-SM fermions in the first generation with Q EM =2 / − / 3, respectively. From Tables XIV, XV, XVII and XVIII in Appendix D, onefinds that W u D ( n ) , W d U ( n ) , Zu U ( n ) and Zd D ( n ) couplings ( n ≥ 1) are all small. Hence theright-hand-side of (6.9) will be approximately given by(R.H.S.) ≃ g W du ( g Zuu − g Zdd ) = 0 . · g w / √ , (6.10)35he left-hand side is approximately given, with use of Tables. IV and VI, by(L.H.S.) ≃ X n =0 g W ( n ) ud g W ( n ) W Z = 0 . · g w / √ . (6.11)It is recognized that (6.9) is also quantitatively well-satisfied.In an analogous way one can confirm the unitarity of the amplitude of f ¯ f → W + W − . Inthis case KK bosons of γ , Z and Z R are involved in the s -channel amplitudes. 7. SUMMARY In this paper we have studied the collider signals of W ′ and Z ′ in the SO (5) × U (1) X gauge-Higgs unification.First we evaluated the couplings of W ′ and Z ′ to the SM fields. We found that the W (1) couplings to light fermions and to top-bottom are different in signs, which is explained fromthe different behavior of wave functions of fields along the extra dimension.Next we evaluated the decay rates of neutral and charged KK vector bosons. The totaldecay widths of Z ′ are large. Γ Z ′ /M Z ′ = 15%, 6 . 6% and 13% for Z ′ = γ (1) , Z (1) and Z (1) R , respectively. On the other hand, W (1) has a narrow total width: Γ W (1) /M W (1) ≃ W (1) and W (1) R can decay to W H and W Z . Decay width of W ′ to W H and W Z are all nearly equal with each other. For Z ′ it is found that Γ( Z (1) → ZH ) ≃ Γ( Z (1) → W W ) + Γ( γ (1) → W W ) and Γ( Z R → ZH ) ≃ Γ( Z R → W W ). These properties of W R and Z R are qualitatively understood in terms of the 4D SO (4) × U (1) model introducedin Section 4.Further we have numerically evaluated the s -channel cross sections of W ′ and Z ′ in theLHC. We studied not only processes with fermionic final states but also bosonic W H , ZH , W W and W Z final states. W ′ and Z ′ signals of GHU can be found at the LHC experimentin the processes pp → W ′ ( Z ′ ) + X , W ′ → tb, W H , and Z ′ → e + e − , µ + µ − , ZH near the W ′ and Z ′ resonances. For θ H = 0 . 115 ( M Z (1) ,γ (1) ≃ . M Z (1) R ≃ . − , √ s pp = 13 TeV at LHC, an excess of the events of µ + µ − with invariantmass is expected. (e.g. expected signal[background] is 3 . . 29] events for the bin (GeV)[3000 , W Z in the final state, it is found that in the amplitude the leadingcontributions from the longitudinal polarizations of W and Z in the s -, t - and u -channelscancel with each other so that the unitarity is preserved, provided that both KK vectorbosons and KK fermions in the intermediate states are taken into account. We have con-firmed numerically that this cancellation of the leading terms in the amplitude with 6 digitsof precision by taking into account contributions of up to the 4-th level of KK excited states.We also found that the non-SM 1st KK excited state of fermions can be much lighterthan other KK states. Especially the 1st KK excited top and bottom partners ( t (1) U,B,T and b (1) D,X,Y ) are the lightest non-SM particles and can be singly produced in colliders. It is seenin Tables XII and XIII that t (1) U,B,T and b (1) D,X,Y , which are exotic partners of the top andbottom quarks respectively, have mass M t (1) U,B,T ,b (1) D,X,Y = 4 . . θ H = 0 . . t (1) T and b (1) Y have electric charges +5 / − / t + W + → t T → t + W + and b + W − → b Y → b + W − in colliders [79–81].The gauge-Higgs unification scenario is promising. It gives many predictions to be testedat LHC and future colliders. The 4D Higgs boson appears as the gauge boson in the extradimension. The gauge hierarchy problem is solved. The AB phase θ H is the importantparameter in GHU. Many of the physical quantities are determined by θ H . The universalrelations among θ H and m KK , Higgs cubic and quartic couplings have been found. Correc-tions to the decay rates for H → γγ , Zγ due to infinitely many KK states turn out finiteand small. Z ′ and W ′ are predicted around 6 - 8 TeV. Discovery of Z ′ and W ′ is mostawaited. Acknowledgments This work was supported in part by the Japan Society for the Promotion of Science,Grants-in-Aid for Scientific Research No 15K05052 (HH and YH).37 ppendix A: Basic formulas1. SO (5) generators The SO (5) generators in the spinor-representation are given by T a L = 12 σ a 00 0 , T a R = 12 σ a ,T ˆ a = 12 √ iσ a − iσ a , T ˆ4 = , (A.1)and tr( T a T b ) = δ ab is satisfied. Here σ a ( a = 1 , , 3) are Pauli matrices. T a L and T a R aregenerators for SU (2) L and SU (2) R subgroups, respectively. 2. Bulk wave functions a. Gauge boson bulk functions Bulk functions of gauge bosons C = C ( z ; λ ) and S = S ( z ; λ ) are defined as solutions of (cid:18) d dz − z ddz + λ (cid:19) CS = 0 , (A.2)with boundary conditions C = z L , S = 0 , C ′ = 0 , S ′ = λ at z = z L . (A.3)Here C ′ ≡ ( d/dz ) C etc. The solutions are given by C ( z ; λ ) = + π λzz L F , ( λz, λz L ) ,C ′ ( z ; λ ) = + π λ zz L F , ( λz, λz L ) ,S ( z ; λ ) = − π λzF , ( λz, λz L ) ,S ′ ( z ; λ ) = − π λ zF , ( λz, λz L ) , (A.4)where F α,β ( u, v ) ≡ J α ( u ) Y β ( v ) − Y α ( u ) J β ( v ) and J α ( x ) and Y α ( x ) are Bessel functions of the1st and 2nd kind, respectively. C , S and C ′ , S ′ satisfy CS ′ − SC ′ = λ. (A.5)38 . Fermion bulk functions Fermion bulk functions C L/R ( z ; λ, c ), S L/R ( z ; λ, c ) are defined by C L ( z ; λ, c ) = + π λ √ zz L F c + ,c − ( λz, λz L ) ,S L ( z ; λ, c ) = − π λ √ zz L F c + ,c + ( λz, λz L ) ,C R ( z ; λ, c ) = + π λ √ zz L F c − ,c + ( λz, λz L ) ,S R ( z ; λ, c ) = + π λ √ zz L F c − ,c − ( λz, λz L ) . (A.6)These satisfy D + C L S L = λ S R C R , D − C R S R = λ S L C L ,D ± ( c ) ≡ ± ddz + cz ,C L C R − S L S R = 1 , (A.7)and C R = C L = 1 , S R = S L = 0 , at z = z L . (A.8)In particular, for c = 0 we have C L ( z ; λ, 0) = C R ( z ; λ, 0) = cos( λ ( z − z L )) ,S L ( z ; λ, 0) = − S R ( z ; λ, 0) = sin( λ ( z − z L )) . (A.9) Appendix B: Gauge boson wave functions Wave functions for a charged vector boson V C = W ( n ) , W ( m ) R ( n = 0 , , , · · · , m =1 , , · · · ) are given by h LV C h RV C ˆ h V C = v LV C C ( z ; λ V C ) v RV C C ( z ; λ V C )ˆ v V C ˆ S ( z ; λ V C ) , (B.1)39here v LV C v RV C ˆ v V C = 1 √ r V C c H √ − c H √ − s H V C = W ( n ) q c H +1 − c H √ − − c H √ V C = W ( m ) R (B.2)with C = C ( z ; λ V C ) etc. c H , s H ≡ cos θ H , sin θ H . We have definedˆ S ( z ; λ ) ≡ C (1; λ ) S (1; λ ) S ( z ; λ ) . (B.3)The mass spectrum { m V C = kλ V C } is determined by2 SC ′ (1 , λ W ( n ) ) + λ W ( n ) s H = 0 ,C (1; λ W ( m ) R ) = 0 , (B.4)and normalization factors are given by r W ( m ) R = Z z L dzkz C ( z ; λ W ( m ) R ) ,r W ( n ) = Z z L dzkz (cid:8) (1 + c H ) C ( z ; λ W ( n ) ) + s H ˆ S ( z ; λ W ( n ) ) (cid:9) . (B.5)Wave functions for the photon γ = γ (0) is given by h Lγ (0) h Rγ (0) h Bγ (0) = 1 q (1 + s φ ) L s φ s φ c φ , ˆ h γ (0) = 0 , (B.6)where s φ ≡ sin φ and c φ ≡ cos φ . Wave functions for a massive neutral vector boson V = Z ( n ) , γ ( m ) and Z ( m ) R ( n = 0 , , , · · · , m = 1 , , · · · ) are given by h LV h RV ˆ h V h BV = v LV C ( z ; λ V ) v RV C ( z ; λ V )ˆ v V ˆ S ( z ; λ V ) v BV C ( z ; λ V ) , (B.7)40here v LV v RV ˆ v V v BV = 1 √ r V q s φ (1 + s φ )(1 + c H ) − s φ √ s φ )(1 − c H ) − s φ √ − (1 + s φ ) s H −√ s φ c φ V = Z ( n ) q s φ s φ s φ c φ V = γ ( m ) , q t φ ) c H +1 − c H √ − − c H √ √ t φ c H V = Z ( m ) R , (B.8)where t φ = tan φ . The mass spectrum { m V = kλ V } is determined by C ′ (1; λ γ ( m ) ) = 0 , SC ′ (1; λ Z ( n ) ) + (1 + s φ ) λ Z ( n ) sin θ H = 0 ,C (1; λ Z ( m ) R ) = 0 , (B.9)and normalization factors are given by r V = R z L dzkz F V where F V = C ( z ; λ V ) V = Z ( m ) R , γ ( m ) ,c φ C ( z ; λ V ) + (1 + s φ )[ c H C ( z ; λ V ) + s H ˆ S ( z ; λ V ) ] V = Z ( n ) . (B.10)41 ppendix C: Masses and wave functions of SO (5) -vector fermions1. Quark sector a. Q em = +5 / quark partners ( t T ) ( T L , T R ) = (+ , + ) of Ψ q,g =31 state has an expansion T ( x, z ) = √ kz ∞ X n =1 (cid:26) t ( n ) T,L ( x ) 1 q r t ( n ) T,L C L ( z, λ t ( n ) T , c ) + t ( n ) T,R ( x ) 1 q r t ( n ) T,R S R ( z, λ t ( n ) T , c ) (cid:27) , (C.1)where λ t ( n ) T = m t ( n ) T /k . The KK mass m t ( n ) T is determined by C L (1 , λ t ( n ) T , c ) = 0 . (C.2)Normalization factors r t ( n ) T,L/R are determined so that they satisfy1 r t ( n ) T,L Z z L C L ( z, λ ( n ) T , c ) dz = 1 r t ( n ) T,R Z z L S R ( z, λ ( n ) T , c ) dz = 1 , (C.3)and one finds r t ( n ) T,L = r t ( n ) T,R . b. Q em = − / quark partners ( b Y ) ( T L , T R ) = ( − , − ) of Ψ q,g =32 state has an expansion Y ( x, z ) = √ kz ∞ X n =1 (cid:26) b ( n ) Y,L ( x ) 1 q r b ( n ) Y,L C L ( z, λ b ( n ) Y , c ) + b ( n ) Y,R ( x ) 1 q r b ( n ) Y,R S R ( z, λ b ( n ) Y , c ) (cid:27) , (C.4)where λ b ( n ) Y = m b ( n ) Y /k and m b ( n ) Y is the KK mass, which is determined by C L (1 , λ b ( n ) Y , c ) = 0 . (C.5)Factors r b ( n ) Y,L/R are normalized so that they satisfy1 r b ( n ) Y,L Z z L C L ( z, λ b ( n ) Y , c ) dz = 1 r b ( n ) Y,R Z z L S R ( z, λ b ( n ) Y , c ) dz = 1 , (C.6)and one finds that r b ( n ) Y,L = r b ( n ) Y,R . 42 . Q em = +2 / quark and its partners ( t , t B , t U ) ( T L , T R ) = (+ , − ), ( − , + ) and (0 , 0) of Ψ states B, t, t ′ together with (+ , + ) ofΨ state U have Q em = +2 / q,g =3 a =1 , contain ˆ t , ˆ t B and ˆ t U .We have an expansion as follows. UtBt ′ ( x, z ) = √ kz ∞ X n =0 (cid:26) t ( n ) L ( x ) 1 p r t ( n ) L a ( t ) U C ( t ( n ) ) L ( z ) a ( t ) t C ( t ( n ) ) L ( z ) a ( t ) B C ( t ( n ) ) L ( z ) a ( t ) t ′ ˆ S ( t ( n ) ) L ( z ) + t ( n ) R ( x ) 1 p r t ( n ) R a ( t ) U S ( t ( n ) ) R ( z ) a ( t ) t S ( t ( n ) ) R ( z ) a ( t ) B S ( t ( n ) ) R ( z ) a ( t ) t ′ ˆ C ( t ) R ( z ) (cid:27) + √ kz ∞ X n =1 (cid:26) t ( n ) B,L ( x ) 1 q r t ( n ) B,L a ( t B ) U C ( t ( n ) B ) L ( z ) a ( t B ) t C ( t ( n ) B ) L ( z ) a ( t B ) B C ( t ( n ) B ) L ( z ) a ( t B ) t ′ ˆ S ( t ( n ) B ) L ( z ) + t ( n ) B,R ( x ) 1 q r t ( n ) B,R a ( t B ) U S ( t ( n ) B ) R ( z ) a ( t B ) t S ( t ( n ) B ) R ( z ) a ( t B ) B S ( t ( n ) B ) R ( z ) a ( t B ) t ′ ˆ C ( t ( n ) B ) R ( z ) (cid:27) + √ kz ∞ X n =1 (cid:26) t ( n ) U,L ( x ) 1 q r t ( n ) U,L a ( t U ) U C ( t ( n ) U ) L ( z ) a ( t U ) t C ( t ( n ) U ) L ( z ) a ( t U ) B C ( t ( n ) U ) L ( z ) a ( t U ) t ′ ˆ S ( t ( n ) U ) L ( z ) + t ( n ) U,R ( x ) 1 q r t ( n ) U,R a ( t U ) U S ( t ( n ) U ) R ( z ) a ( t U ) t S ( t ( n ) U ) R ( z ) a ( t U ) B S ( t ( n ) U ) R ( z ) a ( t U ) t ′ ˆ C ( t ( n ) U ) R ( z ) (cid:27) , (C.7)where C ( t ( n ) B ) L ( z ) ≡ C L ( z, λ t ( n ) B , c ), λ t ( n ) B = m t ( n ) B /k etc. We have defined { ˆ S L ( z, λ, c ) , ˆ C R ( z, λ, c ) } ≡ C L (1 , λ, c ) S L (1 , λ, c ) { S L ( z, λ, c ) , C R ( z, λ, c ) } . (C.8)KK masses m t ( n ) , m t ( n ) B and m t ( n ) U are determined by s H ( µ q ) ( µ q ) + (˜ µ q ) + 2 S R S L ( z = 1; λ t ( n ) , c ) = 0 , c = c ≡ c, (C.9)and C L (1 , λ t ( n ) B , c ) = 0 ,C L (1 , λ t ( n ) U , c ) = 0 , (C.10)43espectively. Common coefficients are given by a ( t ) U a ( t ) t a ( t ) B a ( t ) t ′ = −√ µ/µ (1 + c H ) / √ − c H ) / √ − s H , a ( t B ) U a ( t B ) t a ( t B ) B a ( t B ) t ′ = c H − / √ c H + 1) / √ , a ( t U ) U a ( t U ) t a ( t U ) B a ( t U ) t ′ = c H (˜ µ/µ )(1 + c H )(˜ µ/µ )(1 − c H )0 , (C.11)Normalization factors r f ( n ) L , r f ( n ) R ( f = t, t B , t U ) are determined by1 r f ( n ) L Z z L { [( a ( f ) U ) + ( a ( f ) t ) + ( a ( f ) B ) ]( C ( f ( n ) ) L ) + ( a ( f ) t ′ ) ( ˆ S ( f ( n ) ) L ) ] } dz = 1 r f ( n ) R Z z L { [( a ( f ) U ) + ( a ( f ) t ) + ( a ( f ) B ) ]( S ( f ( n ) ) R ) + ( a ( f ) t ′ ) ( ˆ C ( f ( n ) ) R ) ] } dz = 1 , (C.12)and one finds that r f ( n ) L = r f ( n ) R are satisfied. d. Q em = − / quark and its partners ( b , b D , b X ) ( T L , T R ) = ( − , − ) of Ψ states b together with (+ , − ), ( − , + ) and (0 , 0) of Ψ states X, D, b ′ have Q em = − / b , ˆ b D and ˆ b X . Hence we have an expansion bXDb ′ ( x, z ) = √ kz ∞ X n =0 (cid:26) b ( n ) L ( x ) 1 q r b ( n ) X,L a ( b ) b C ( b ( n ) ) L ( z ) a ( b ) X C ( b ( n ) ) L ( z ) a ( b ) D C ( b ( n ) ) L ( z ) a ( b ) b ′ ˆ S ( b ( n ) ) L ( z ) + b ( n ) R ( x ) 1 q r b ( n ) X,R a ( b ) b S ( b ( n ) ) R ( z ) a ( b ) X S ( b ( n ) ) R ( z ) a ( b ) D S ( b ( n ) ) R ( z ) a ( b ) b ′ ˆ C ( b ( n ) ) R ( z ) (cid:27) + √ kz ∞ X n =1 (cid:26) b ( n ) X,L ( x ) 1 q r b ( n ) X,L a ( b X ) b C ( b ( n ) X ) L ( z ) a ( b X ) X C ( b ( n ) X ) L ( z ) a ( b X ) D C ( b ( n ) X ) L ( z ) a ( b X ) b ′ ˆ S ( b ( n ) X ) L ( z ) + b ( n ) X,R ( x ) 1 q r b ( n ) X,R a ( b X ) b S ( b ( n ) X ) R ( z ) a ( b X ) X S ( b ( n ) X ) R ( z ) a ( b X ) D S ( b ( n ) X ) R ( z ) a ( b X ) b ′ ˆ C ( b ( n ) X ) R ( z ) (cid:27) √ kz ∞ X n =1 (cid:26) b ( n ) D,L ( x ) 1 q r b ( n ) D,L a ( b D ) b C ( b ( n ) D ) L ( z ) a ( b D ) X C ( b ( n ) D ) L ( z ) a ( b D ) D C ( b ( n ) D ) L ( z ) a ( b D ) b ′ ˆ S ( b ( n ) D ) L ( z ) + b ( n ) D,R ( x ) 1 q r b ( n ) D,R a ( b D ) b S ( b ( n ) D ) R ( z ) a ( b D ) X S ( b ( n ) D ) R ( z ) a ( b D ) D S ( b ( n ) D ) R ( z ) a ( b D ) b ′ ˆ C ( b ( n ) D ) R ( z ) (cid:27) , (C.13)where λ b ( n ) X ≡ m b ( n ) X /k etc. Mass spectra m b ( n ) , m b ( n ) X , m b ( n ) D are determined by s H (˜ µ q ) ( µ q ) + (˜ µ q ) + 2 S R S L (1; λ b ( n ) , c ) = 0 , c = c ≡ c (C.14)and C L (1; λ b ( n ) X , c ) = 0 ,C L (1; λ b ( n ) D , c ) = 0 . (C.15)Combining (C.9) and (C.14), one finds (cid:18) ˜ µ q µ q (cid:19) = − (cid:26) s H S L S R (1; λ t ( n ) , c ) (cid:27) = − (cid:26) s H S L S R (1; λ b ( n ) , c ) (cid:27) − , (C.16)and c and ˜ µ q /µ q are determined from the masses of top and bottom quarks. Commoncoefficients are given by a ( b ) b a ( b ) X a ( b ) D a ( b ) b ′ = − −√ µ / ˜ µ (1 − c H ) / √ c H ) / √ s H , a ( b D ) b a ( b D ) X a ( b D ) D a ( b D ) b ′ = (˜ µ/µ )(1 + c H )1 − c H c H , a ( b X ) b a ( b X ) X a ( b X ) D a ( b X ) b ′ = c H ) / √ − c H ) / √ , (C.17)Factors r f ( n ) L/R ( f = b, b X , b D ) are normalized so that1 r f ( n ) L Z z L { [( a ( f ) b ) + ( a ( f ) X ) + ( a ( f ) D ) ]( C ( f ) L ) + ( a ( f ) b ′ ) ( ˆ S ( f ) L ) ] } dz = 1 r f ( n ) R Z z L { [( a ( f ) b ) + ( a ( f ) X ) + ( a ( f ) D ) ]( S ( f ) R ) + ( a ( f ) b ′ ) ( ˆ C ( f ) R ) ] } dz = 1 , (C.18)45nd one finds r f ( n ) L = r f ( n ) R are satisfied.In Table XII and XIII, masses of KK fermions are tabulated. In tables masses of exoticpartners of up- and down-type quarks are M u ( n ) U = M u ( n ) B = M u ( n ) T ≡ M u ( n ) x ,M d ( n ) D = M d ( n ) X = M d ( n ) Y ≡ M d ( n ) x , (C.19)and M u ( n ) x = M d ( n ) x are satisfied.Here we note that KK masses of exotics largely depend on their bulk mass parameters.In particular, since the bulk mass parameter of top and bottom quarks approaching to zerofor smaller z L ( θ H ), the mass spectrum for exotic partners of top and bottom quarks areapproximately given by C L (1; λ f (1) , c t = 0) = cos(( z L − m f (1) /k ) = 0 so that m f (1) ≃ m KK , f = t T,U,B , b Y,X,D . (C.20) TABLE XII: Masses of KK fermions for N F = 4, z L = 10 ( θ H = 0 . M q ( n ) x is the mass of the n -th KK excited state of exotic partners of q -quark (see text). n M u ( n ) M d ( n ) M t ( n ) M b ( n ) M u ( n ) x = M d ( n ) x M t ( n ) x = M b ( n ) x 2. Lepton sector For charged lepton, neutrino and their exotic partners, KK states are given as follows.46 ABLE XIII: Same as Table XII but for N F = 4, z L = 10 ( θ H = 0 . n M u ( n ) M d ( n ) M t ( n ) M b ( n ) M u ( n ) x = M d ( n ) x M t ( n ) x = M b ( n ) x a. Q em = +1 and − lepton partners ( T L , T R ) = ( − , − ) of Ψ , L Y , and ( , ) of Ψ , L X , have Q em = − L X ( x, z ) = √ kz ∞ X n =1 (cid:26) ν ( n ) τ X,L ( x ) 1 q r ν ( n ) τ X,L C L ( z ; λ ν ( n ) τ X , c )+ ν ( n ) τ X,R ( x ) 1 q r ν ( n ) τ X,R S R ( z ; λ ν ( n ) τ X , c ) (cid:27) , (C.21) L Y ( x, z ) = √ kz ∞ X n =1 (cid:26) τ ( n )1 Y,L ( x ) 1 q r τ ( n )1 Y,L C L ( z ; λ τ ( n )1 Y , c )+ τ ( n )1 Y,R ( x ) 1 q r τ ( n )1 Y,R S R ( z ; λ τ ( n )1 Y , c ) (cid:27) , (C.22)where the KK masses are given by ( m ν ( n ) τ X , m τ ( n )1 Y ) = k ( λ ( n ) ν τ X , λ τ ( n )1 Y ) and determined by C L (1; λ ν ( n ) τ X , c ) = C L (1; λ τ ( n )1 Y , c ) = 0 . (C.23)Normalization factors are determined by1 r f ( n ) L Z z L C L ( z ; λ f ( n ) , c ) dz = 1 r f ( n ) R Z z L S R ( z ; λ f ( n ) , c ) dz = 1 , (C.24)for ( f, c ) = ( τ Y , c ), ( ν τ X , c ). 47 . Q em = − charged lepton and its partners ( T L , T R ) = ( − , ), ( , − ) and (0 , 0) of Ψ together with ( − , − ) of Ψ are Q em = − L Y L X ττ ′ ( x, z ) = √ kz X f X n (cid:26) f ( n ) L p r f ( n ) L a ( f )3 Y C f ( n ) L ( z ) a ( f )1 X C f ( n ) L ( z ) a ( f ) τ C f ( n ) L ( z ) a ( f ) τ ′ ˆ S f ( n ) L ( z ) + f ( n ) R p r f ( n ) R a ( f )3 Y S f ( n ) R ( z ) a ( f )1 X S f ( n ) R ( z ) a ( f ) τ S f ( n ) R ( z ) a ( f ) τ ′ ˆ C f ( n ) R ( z ) (cid:27) , (C.25)where f = τ , τ X and τ Y and C f ( n ) L ( z ) ≡ C L ( z ; λ f ( n ) , c ) etc. Common coefficients are givenby a ( f )3 Y a ( f )1 X a ( f ) τ a ( f ) τ ′ = √ µµ − c H √ c H √ s H , c H +1 √ c H − √ and c H (1 − c H ) ˜ µ ℓ µ ℓ (1 + c H ) ˜ µ ℓ µ ℓ , (C.26)for f = τ , τ X and τ Y , respectively. The mass of τ ( n ) is given by m τ ( n ) = kλ τ ( n ) where λ τ ( n ) are determined by s H ( µ ℓ ) ( µ ℓ ) + (˜ µ ℓ ) + 2 S L S R ( z = 1; λ τ ( n ) , c ℓ ) = 0 , (C.27)and τ (0) corresponds to the tau lepton. For τ ( n ) E ( E = 3 Y and 1 X , n = 1 , , · · · ), the KKmasses are determined by C L (1; λ τ ( n ) E , c ℓ ) = 0 , m τ ( n ) E ≡ kλ τ ( n ) E . (C.28)Normalization factors are determined by1 r f ( n ) L Z z L nh ( a ( f )3 Y ) + ( a ( f )1 X ) + ( a ( f ) τ ) ]( C ( f ) L ) + ( a ( f ) τ ′ ) ( ˆ S ( f ) L ) io dz = 1 r f ( n ) R Z z L nh ( a ( f )3 Y ) + ( a ( f )1 X ) + ( a ( f ) τ ) ]( S ( f ) R ) + ( a ( f ) τ ′ ) ( ˆ C ( f ) R ) io dz = 1 , (C.29)where f = τ , τ Y and τ τ X . One finds r f ( n ) L = r f ( n ) R are satisfied.48 . Q em = 0 neutrino and its partners ( T L , T R ) = ( , ) of Ψ , and ( , − ) and ( − , ) and (0 , 0) of Ψ are Q em = 0 states.They are expanded as νL X L Y ν ′ ( x, z ) = √ kz X f X n (cid:26) f ( n ) L p r f ( n ) L a ( f ) ν C f ( n ) L ( z ) a ( f )3 X C f ( n ) L ( z ) a ( f )2 Y C f ( n ) L ( z ) a ( f ) ν ′ ˆ S f ( n ) L ( z ) + f ( n ) R p r f ( n ) R a ( f ) ν S f ( n ) R ( z ) a ( f )3 X S f ( n ) R ( z ) a ( f )2 Y S f ( n ) R ( z ) a ( f ) ν ′ ˆ C f ( n ) R ( z ) (cid:27) , (C.30)where f = ν , ν τ X and ν τ Y . C f ( n ) L = C L ( z ; λ f ( n ) , c ), c = c ≡ c ℓ etc. Common coefficientsare given by a ( f ) ν a ( f )3 X a ( f )2 Y a ( f ) ν ′ = √ µ ℓ ˜ µ ℓ − c H √ − − c H √ s H , − c H √ c H √ , and (1 + c H ) ˜ µ ℓ µ ℓ c H − c H (C.31)for f = τ , ν τ Y and ν τ X , respectively. KK masses of ν ( n ) τ , m ν ( n ) τ ≡ kλ ν ( n ) τ ,( n = 0 , , , · · · )are determined by s H (˜ µ ℓ ) ( µ ℓ ) + (˜ µ ℓ ) + 2 S L S R ( z = 1; λ ν ( n ) , c ℓ ) = 0 , (C.32)and ν (0) τ corresponds to the tau neutrino. From (C.27) and (C.32), one finds (cid:18) µ ℓ ˜ µ ℓ (cid:19) = − ( s H S L S R (1; λ ν ( n ) τ , c ) ) = − (cid:26) s H S L S R (1; λ τ ( n ) , c ) (cid:27) − , (C.33)and c and ˜ µ ℓ /µ ℓ are determined from the masses of τ and ν τ . For ν ( n ) τ N ( N = 3 X and 1 Y , n = 1 , , · · · ), the KK masses are determined by C L (1; λ ( n ) ν τ N , c ℓ ) = 0 , m ν ( n ) τ N ≡ kλ ν ( n ) τ N . (C.34)Normalization factors are determined by1 r f ( n ) L Z z L nh ( a ( f ) ν ) + ( a ( f )3 X ) + ( a ( f )2 Y ) ]( C ( f ) L ) + ( a ( f ) ν ′ ) ( ˆ S ( f ) L ) io dz 49 1 r f ( n ) R Z z L nh ( a ( f ) ν ) + ( a ( f )3 X ) + ( a ( f )2 Y ) ]( S ( f ) R ) + ( a ( f ) ν ′ ) ( ˆ C ( f ) R ) io dz = 1 , (C.35)for f = ν , ν X and ν Y . One finds r f ( n ) L = r f ( n ) R are satisfied. Appendix D: Fermion couplings The KK expansions (C.1), (C.4), (C.7) (C.13), (C.21), (C.22), (C.25) and (C.30) arewritten in the form of T ( x, z ) = √ kz ∞ X n =1 [ t ( n ) T,L ( x ) f t ( n ) T T L ( z ) + t ( n ) T,R ( x ) f ( t ( n ) T ) T R ( z )] ,Y ( x, z ) = √ kz ∞ X n =1 [ b ( n ) Y,L ( x ) f b ( n ) Y Y L ( z ) + b ( n ) Y,R ( x ) f ( b ( n ) Y ) Y R ( z )] , (D.1) UtBt ′ ( x, z ) = √ kz X t u = t,t B ,t U X n (cid:26) t u,L ( x ) f t ( n ) u UL ( z ) f t ( n ) u tL ( z ) f t ( n ) u BL ( z ) f t ( n ) u t ′ L ( z ) + t u,R ( x ) f t ( n ) u UR ( z ) f t ( n ) u tR ( z ) f t ( n ) u BR ( z ) f t ( n ) u t ′ R ( z ) (cid:27) , (D.2) bXDb ′ ( x, z ) = √ kz X b d = b,b X ,b D X n (cid:26) b d,L ( x ) f b ( n ) d bL ( z ) f b ( n ) d XL ( z ) f b ( n ) d DL ( z ) f t ( n ) u b ′ L ( z ) + b d,R ( x ) f b ( n ) d bR ( z ) f b ( n ) d XR ( z ) f b ( n ) d DR ( z ) f b ( n ) d b ′ R ( z ) (cid:27) . (D.3)In terms of these wave functions we write gauge-boson couplings and Yukawa couplings asfollows. 1. Vector boson couplings a. ¯ ψ ( − / n V − ψ (2 / m and ¯ ψ ( − n V − ψ (0) m couplings For b d = b, b D , b X , t u = t, t U , t B and V − = W − , W − R we have Z dzkz ¯Ψ [ γ µ g A A µ + ¯Ψ γ µ g A A µ Ψ ] ⊃ ¯ b ( n ) dL V − µ t ( m ) uL · g w √ L Z z L dz √ (cid:26) h LV (cid:20) f b ( n ) b bL f t ( m ) u tL + f b ( n ) d DL f t ( m ) u UL (cid:21) + h RV (cid:20) f b ( n ) d bL f t ( m ) u BL + f b ( n ) d XL f t ( m ) u UL (cid:21) h V (cid:20) f b ( n ) d bL f t ( m ) u t ′ L − f b ( n ) d b ′ L f t ( m ) u UL (cid:21)(cid:27) + H.c. ≡ ¯ b ( n ) dL V − µ t ( m ) uL · g LV d ( n ) d t ( m ) u + H.c. (D.4)and right-handed couplings with replacements L → R in spinors and their wave functions.For leptons couplings we obtain ¯ ℓV − ν couplings from the above formula with replacements b d → τ E , t u → ν τ N , ( b, b D , b X , b ′ ) → ( τ, τ Y , τ X , τ ′ ) , ( t, t U , t B , t ′ ) → ( ν τ , ν τ Y , ν τ X , ν ′ τ ) . (D.5) b. ¯ ψ (2 / n V µ ψ (2 / m and ¯ ψ ( − / n V µ ψ ( − / m For up-type quarks and their exotic partners t u , t u ′ = t, t B , t U , down-type quarks andtheir exotic partners b d , b d ′ = b, b D , b X and neutral vector boson V = γ ( l ) , Z ( l ) , Z ( l ) R , we have¯ t u V t u ′ and ¯ b d V b d ′ couplings as g A ¯Ψ γ µ (cid:20) A µ + (cid:18) (cid:19) g B g A B µ (cid:21) Ψ + g A ¯Ψ γ µ (cid:20) A µ + (cid:18) − (cid:19) g B g A B µ (cid:21) Ψ ⊃ ¯ t ( n ) u ′ L γ µ V µ t ( m ) uL · g w √ L Z z L dz (cid:26) 12 ( h LV − h RV ) (cid:20) − f t ( n ) u ′ BL f t ( m ) u BL + f t ( n ) u ′ tL f t ( m ) u tL (cid:21) + 12 ( h LV + h RV ) f t ( n ) u ′ UL f t ( m ) u UL + 12 ˆ h V (cid:20) f t ( n ) u ′ t ′ L (cid:16) f t ( m ) u BL + f t ( m ) u tL (cid:17) + (cid:18) f t ( n ) u ′ BL + f t ( n ) u ′ tL (cid:19) f t ( m ) u t ′ L (cid:21) + g B g A h BV (cid:20) (cid:18) f t ( n ) u ′ tL f t ( m ) u tL + f t ( n ) u ′ BL f t ( m ) u BL + f t ( n ) u ′ t ′ L f t ( m ) u t ′ L (cid:19) − (cid:18) f t ( n ) u ′ UL f t ( m ) u UL (cid:19)(cid:21)(cid:27) + H.c.+¯ b ( n ) d ′ L γ µ V µ b ( m ) dL · g w √ L Z z L dz (cid:26) 12 ( h LV − h RV ) (cid:20) − f b ( n ) d ′ DL f b ( m ) d DL + f d ( n ) b ′ XL f d ( m ) b XL (cid:21) − 12 ( h LV + h RV ) f b ( n ) d ′ bL f b ( m ) d bL + 12 ˆ h V (cid:20) f b ( n ) d ′ b ′ L (cid:18) f b ( m ) d XL + f b ( m ) d DL (cid:19) + (cid:18) f b ( n ) d ′ XL + f b ( n ) d ′ DL (cid:19) f b ( m ) d b ′ L (cid:21) + g B g A h BV (cid:20) − (cid:18) f b ( n ) d ′ XL f b ( m ) d XL + f b ( n ) d ′ DL f b ( m ) d DL + f b ( n ) d ′ b ′ L f b ( m ) d b ′ L (cid:19) + 23 (cid:18) f b ( n ) d ′ bL f b ( m ) d bL (cid:19)(cid:21)(cid:27) + H.c. ≡ ¯ t ( n ) u ′ L γ µ V µ t ( m ) uL · g LV t ( n ) u ′ t ( m ) u + H.c. + ¯ b ( n ) d ′ L γ µ V µ b ( m ) dL · g LV b ( n ) d ′ b ( m ) d + H.c. (D.6)and right-handed couplings. Lepton couplings ¯ ψ ( − V µ ψ ( − and ¯ ψ (0) V µ ψ (0) are obtainedfrom the above formula with replacements (D.5).51e note that photon wave functions (B.6) which can be rewritten as h Lγ (0) = h Rγ (0) = g B g A h Bγ (0) = sin θ W √ L , ˆ h γ (0) = 0 (D.7)yield proper electromagnetic couplings Q em e ¯ ψγ µ A γµ ψ . c. ¯ ψ (2 / n V − µ ψ (5 / m and ¯ ψ (0) n V − µ ψ (+1) m For t T , t u = t, t U , t B and V − = W − , W − R we have Z dzkz ¯Ψ γ µ g A A µ Ψ ⊃ ¯ t ( n ) uL γ µ V − µ t ( m ) T L · g w √ L Z z L dz √ (cid:26) h LV (cid:20) f t ( n ) u BL f t ( m ) T T L (cid:21) + h RV (cid:20) f t ( n ) u tL f t ( m ) T T L (cid:21) − ˆ h V (cid:20) f t ( n ) u t ′ L f t ( m ) T T L (cid:21)(cid:27) + H.c. ≡ ¯ t ( n ) uL γ µ V − µ t ( m ) T L · g LV t ( n ) u t ( m ) T + H.c. (D.8)and corresponding right-handed couplings.Lepton couplings are obtained from the above formula with replacements (D.5) and t T → ν τ X , f T → f ν τ X . (D.9) d. ¯ ψ ( − / n V + µ ψ ( − / m and ¯ ψ ( − n V + µ ψ ( − m couplings For b Y , b d = b, b X , b D and V + = W + , W + R , we have Z dzkz ¯Ψ γ µ g A A µ Ψ ⊃ ¯ b ( n ) dL V + µ b ( m ) Y L · g w √ L Z z L dz √ (cid:26) h LV (cid:20) f b ( n ) d XL f b ( m ) Y Y L (cid:21) + h RV (cid:20) f b ( n ) d DL f b ( m ) Y Y L (cid:21) +ˆ h V (cid:20) f b ( n ) d b ′ L f b ( m ) Y Y L (cid:21)(cid:27) + H.c. ≡ ¯ b ( n ) dL V + µ b ( m ) Y L · g LV b ( n ) d b ( m ) Y + H.c. (D.10)and corresponding right-handed couplings.Lepton couplings are obtained from the above formula with replacements (D.5) and b Y → τ Y , f Y → f Y . (D.11)52 . Numerical values In Tables XIV - XVIII, KK fermions’ couplings to W and Z bosons for N F = 4, z L = 10 ( θ H = 0 . TABLE XIV: Couplings of the W boson to the up-quark and KK excited down-type states in unitof g w / √ N F = 4, z L = 10 ( θ H = 0 . n = 1 2 3 4 g LW u (0) d ( n ) / ( g w / √ − . × − − . × − − . × − . × − g RW u (0) d ( n ) / ( g w / √ − . × − − . × − − . × − − . × − g LW u (0) d ( n ) D / ( g w / √ 2) 1 . × − − . × − . × − − . × − g RW u (0) d ( n ) D / ( g w / √ 2) 2 . × − . × − . × − . × − g LW u (0) d ( n ) X / ( g w / √ 2) 8 . × − − . × − . × − − . × − g RW u (0) d ( n ) X / ( g w / √ 2) 1 . × − − . × − . × − − . × − TABLE XV: Couplings of the W boson to the down-quark and KK excited up-type states in unitof g w / √ N F = 4, z L = 10 ( θ H = 0 . n = 1 2 3 4 g LW d (0) u ( n ) / ( g w / √ − . × − − . × − . × − . × − g RW d (0) u ( n ) / ( g w / √ − . × − − . × − − . × − − . × − g LW d (0) u B ( n ) / ( g w √ 2) 1 . × − − . × − − . × − − . × − g RW d (0) u ( n ) B / ( g w / √ 2) 5 . × − − . × − − . × − − . × − g LW d (0) u ( n ) U / ( g w / √ 2) 1 . × − − . × − . × − − . × − g RW d (0) u ( n ) U / ( g w / √ 2) 1 . × − . × − . × − . × − 2. Higgs Yukawa couplings Yukawa couplings among Higgs H ( k ) and quark-sector fermions are read from X i =1 , Z z L √− ge z i Ψ i Γ m ( − ig A A z )Ψ i ABLE XVI: Couplings of the W boson to the up- or down-quark and KK excite states withnon-SM electric charges in unit of g w / √ N F = 4, z L = 10 ( θ H = 0 . n = 1 2 3 4 g LW u (0) u ( n ) T / ( g w / √ − . × − . × − − . × − . × − g RW u (0) u ( n ) T / ( g w / √ − . × − − . × − − . × − − . × − g LW d (0) d ( n ) Y / ( g w / √ − . × − . × − − . × − . × − g RW d (0) d ( n ) Y / ( g w / √ − . × − − . × − − . × − − . × − TABLE XVII: Couplings of the Z boson to the up quark and KK excited states in unit of g w / cos θ W for N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g LZu (0) u ( n ) / ( g w / cos θ W ) 0 . − . × − − . × − . × − . × − g RZu (0) u ( n ) / ( g w / cos θ W ) − . − . × − − . × − − . × − − . × − g LZu (0) u ( n ) B / ( g w / cos θ W ) – − . × − . × − − . × − . × − g RZu (0) u ( n ) B / ( g w / cos θ W ) – − . × − − . × − − . × − − . × − g LZu (0) u ( n ) U / ( g w / cos θ W ) – − . × − . × − − . × − . × − g RZu (0) u ( n ) U / ( g w / cos θ W ) – − . × − − . × − − . × − − . × − TABLE XVIII: Couplings of the Z boson to the down quark and KK excited states in unit of g w / cos θ W for N F = 4, z L = 10 ( θ H = 0 . n = 0 1 2 3 4 g LZd (0) d ( n ) / ( g w / cos θ W ) − . . × − − . × − − . × − . × − g RZd (0) d ( n ) / ( g w / cos θ W ) 0 . . × − . × − . × − . × − g LZd (0) d ( n ) X / ( g w / cos θ W ) – − . × − . × − − . × − . × − g RZd (0) d ( n ) X / ( g w / cos θ W ) – 1 . × − . × − . × − . × − g LZd (0) d ( n ) D / ( g w / cos θ W ) – 8 . × − − . × − . × − − . × − g RZd (0) d ( n ) D / ( g w / cos θ W ) – 6 . × − . × − . × − . × − H ( k ) (cid:16) ¯ t ( m ) uL t ( n ) u ′ R + ¯ t ( n ) u ′ R t ( m ) uL (cid:17) ( x ) × g w √ L Z z L u H ( k ) ( z ) (cid:20) f t ( m ) u t ′ L (cid:18) f t ( n ) u ′ BR − f t ( n ) u ′ tR (cid:19) − (cid:16) f t ( m ) u BL − f t ( m ) u tL (cid:17) f t ( n ) u ′ t ′ R (cid:21) dz − H ( k ) (cid:16) ¯ t ( m ) uR t ( n ) u ′ L + ¯ t ( n ) u ′ L t ( m ) uR (cid:17) ( x ) × g w √ L Z z L u H ( k ) ( z ) (cid:20) f t ( m ) u t ′ R (cid:18) f t ( n ) u ′ BL − f t ( n ) u ′ tL (cid:19) − (cid:16) f t ( m ) u BR − f t ( m ) u tR (cid:17) f t ( n ) u ′ t ′ L (cid:21) dz, (D.12)and when t ( m ) u = t ( n ) u ′ , one obtain H ( n ) ( x ) (cid:16) ¯ t ( m ) uL t ( m ) uR + ¯ t ( m ) uR t ( m ) uL (cid:17) × g w √ L Z z L u H ( n ) h f t ( m ) u t ′ L (cid:16) f t ( m ) u BR − f t ( m ) u tR (cid:17) − (cid:16) f t ( m ) u BL − f t ( m ) u tL (cid:17) f t ( m ) u t ′ R i dz. (D.13)For the Higgs boson H = H (0) , the wave function is given by u H ( z ) = u H (0) ( z ) = s k ( z L − z. (D.14) Appendix E: Boson couplings1. Vector boson trilinear couplings The V ( l ) W ( m ) W ( n ) couplings for V = γ, Z, Z R are contained in Z z L dzkz (cid:18) − (cid:19) Tr [ F µν F ρσ ] η µρ η νσ ⊃ ig A Z z L dzkz Tr h ( ∂ µ ˆ V ν − ∂ ν ˆ V µ )[ ˆ W + ρ , ˆ W − σ ]+ ( ∂ µ ˆ W − ν − ∂ ν ˆ W − µ )[ ˆ V ρ , ˆ W + σ ] + ( ∂ µ ˆ W + ν − ∂ ν ˆ W + µ )[ ˆ V ρ , ˆ W − σ ] i η µρ η νσ ⊃ i X m,n g V ( l ) W ( m ) W ( n ) η µρ η νσ n ( ∂ µ Z ( l ) ν − ∂ ν Z ( l ) µ ) W +( m ) ρ W − ( n ) σ − ( ∂ µ W +( m ) ν − ∂ ν W +( m ) µ ) Z ( l ) ρ W − ( n ) σ + ( ∂ µ W − ( n ) ν − ∂ ν W − ( n ) µ ) Z ( l ) ρ W +( m ) σ o (E.1)so that one finds that g V ( l ) W ( m ) W ( n ) = g w √ L Z z L dzkz × (cid:26) h LV ( l ) (cid:18) h LW ( m ) h LW ( n ) + ˆ h W ( m ) ˆ h W ( n ) (cid:19) + h RV ( l ) (cid:18) h RW ( m ) h RW ( n ) + ˆ h W ( m ) ˆ h W ( n ) (cid:19) + ˆ h V ( l ) (cid:18) h LW ( m ) ˆ h W ( n ) + h RW ( m ) ˆ h W ( n ) + ˆ h W ( m ) h LW ( n ) + ˆ h W ( m ) h RW ( n ) (cid:19)(cid:27) . (E.2)55ere C W ( m ) = C ( z ; λ W ( m ) ) etc. V ( l ) W ( m ) W ( n ) R and V ( l ) W ( m ) R W ( n ) R couplings are obtained from above expression with re-placements W ( n ) → W ( n ) R . HZZ and HZZ R couplings The Higgs coupling HZ ( m ) Z ( n ) is contained in the Tr F µz F µz term − ig A k Z z L dzkz Tr h(cid:0) ∂ z ˆ Z µ (cid:1)(cid:2) ˆ H, ˆ Z ν (cid:3)i η µν ⊃ − X n g HZ ( m ) Z ( n ) HZ ( m ) µ Z ( n ) ν η µν − X m HW W R couplings are seen in [49]. Appendix F: Decay width For a heavy charged vector boson W ′ , the W ′ → W H decay width is given byΓ( W ′ → W H ) = M W ′ π (cid:18) g W ′ W H M W (cid:19) (cid:18) M W − M H M W ′ + M W + M H − M W M H M W ′ (cid:19) , (F.1)and Γ( Z ′ → ZH ) is obtained from the above expression by replacements of W with Z . Thedecay width for W ′ → W Z is given byΓ( W ′ → ZW ) = M W ′ π g W ′ W Z M W ′ M W M Z × (cid:18) − ( M Z + M W ) M W ′ (cid:19) (cid:18) − ( M Z − M W ) M W ′ (cid:19) × (cid:18) M Z + M W ) M W ′ + M Z + M W + 10 M Z M W M W ′ (cid:19) , (F.2)and Γ( Z ′ → W + W − ) is obtained by replacements W ′ → Z ′ and Z → W .For the decay of a heavy fermion F (mass m F ) to a light fermion f (mass m f ) and avector boson V (mass m V ), the decay rate is given byΓ( F → f V ) = m F π q λ (1 , m f /m F , m V /m F ) (cid:26) ( g LV F f ) + ( g LV F f ) m V m F (cid:2) ( m F − m f ) + m V ( m F + m f ) − m V (cid:3) − g L g R m f m F (cid:27) , (F.3)where λ ( A, B, C ) ≡ A + B + C − A B + B C + C A ) , (F.4)where g L/RV F f are the left- and right- handed coupling of ¯ f F V .For decay widths of exotic fermions t ( n ) T and b ( n ) Y , we haveΓ( t ( n ) T → tW + ) = 132 π M t ( n ) T q λ (1 , M t /M t ( n ) T , M W /M T ( n ) ) × (cid:26) (cid:0) g L (cid:1) + (cid:0) g R (cid:1) M W M t ( n ) T h ( M t ( n ) T − M t ) + M W ( M t ( n ) T + M t ) − M W i − g L g R M t M t ( n ) T (cid:27) , (F.5)where g L,R = g L,RW t ( n ) T t , (F.6)are left- and right-hand couplings of t ( n ) T to tW + . Decay width for b ( n ) Y to bW − are obtainedby replacements ( t ( n ) T , t, W + ) → ( b ( n ) Y , W − , b ) . (F.7)57 ppendix G: Cross section Cross sections of processes f ¯ f ′ → W ′ → W H and f ¯ f → Z ′ → ZH in the center-of-massframe are given as follows. For the process f ( p ) ¯ f ( p ) → V ′ → V ( k ) H ( k ), the differentialcross section is given by dσd cos θ = 164 π | k | s √ s g HV ′ V ( | g LV ′ f | + | g RV ′ f | ) × M V − | k | (cos θ − M V s ( s − M V ′ ) + M V ′ Γ V ′ , (G.1)where θ is the angle between p and k . | k | ≡ | k | = | k | = p λ ( M V ′ , M V , M H )2 M V ′ , (G.2)is the momentum of a final state particle. Integrating with respect to θ , and taking inter-ferences among intermediate bosons into account we obtain σ ( f ¯ f ′ → W, W ′ → W H )= 1 N ic π | k | √ s M W + | k | M W (cid:26) X V = W,W (1) g HV W [ | g LV ff ′ | + | g RV ff ′ | ]( s − M V ) + M V Γ V +2 Re " g HW W g HW (1) W [( g LW ff ′ )( g LW (1) ff ′ ) ∗ + ( g RW ff ′ )( g RW (1) ff ′ ) ∗ ][( s − M W ) + iM W Γ W ][( s − M W (1) ) − iM W (1) Γ W (1) ] , (G.3) σ ( f ¯ f → γ, Z, Z ′ → ZH )= 1 N ic π | k | √ s M Z + | k | M Z (cid:26) X V = Z,Z (1) ,γ (1) ,Z (1) R g HV Z [ | g LV f | + | g RV f | ]( s − M V ) + M V Γ V + X V ,V = Z,Z (1) ,γ (1) ,Z (1) R V = V Re " g HV Z g HV Z [( g LV f )( g LV f ) ∗ + ( g RV f )( g RV f ) ∗ ][( s − M V ) + iM V Γ V ][( s − M V ) − iM V Γ V ] , (G.4)where N ic is the number of colors of initial-state fermions.For the process f ( p ) ¯ f ′ ( p ) → W ( k ) Z ( k ), we adopt the approximation in which theinterference term between the SM part and NP part is dropped. The cross section formulaein the SM are found in [74]. The differential cross section mediated by heavy charged vectorbosons W ′ in the center-of-mass frame is given by dσdt ( f ¯ f → W ′ → W Z ) 58 164 πs · s A ( t, u ) (cid:26) X W ′ ∈{ W ( n ) } W ′ = W g W ′ W Z ( | g LW ′ ff ′ | + | g RW ′ ff ′ | ) ( s − M W ′ ) + M W ′ Γ W ′ + X W ,W ∈{ W ( n ) } M W ≪ M W 2) with | k | = | k , | = p λ ( M W ′ , M W , M Z ) / M W ′ . Integrating dσ/dt with respect to t and using Z t max t min A ( t, u ) dt = s β M W M Z (cid:20) M W + M Z ) s + M W + M Z + 10 M W M Z s (cid:21) , (G.7)we obtain σ ( f f ′ → W ′ → W Z )= 1384 π s β M W M Z (cid:20) M W + M Z ) s + M W + M Z + 10 M W M Z s (cid:21) × (cid:26) X W ′ ∈{ W ( n ) } M W ′ ≫ M W g W ′ W Z ( | g LW ′ ff ′ | + | g RW ′ ff ′ | ) ( s − M W ′ ) + M W ′ Γ W ′ X W ,W ∈{ W ( n ) } M W ≪ M W 2) sin θ is obtained by dσdp T ( p T , s ) = dσd cos θ (cid:12)(cid:12)(cid:12)(cid:12) cos θ = √ − p T /s · p T s p − p T /s . (G.10) [1] M. Aaboud et al. 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