Diagonal Kaluza-Klein expansion under brane localized potential
aa r X i v : . [ h e p - ph ] M a y Diagonal Kaluza-Klein expansion under brane localizedpotential
Naoyuki Haba ∗ , Kin-ya Oda † , and Ryo Takahashi a,b, ‡ Department of Physics, Osaka University,Osaka 560-0043, Japan a Yukawa Institute for Theoretical Physics, Kyoto University,Kyoto 606-8502, Japan b Max-Planck-Institute f ¨ u r Kernphysik, Postfach 10 39 80, 69029Heidelberg, Germany October 8, 2018
Abstract
We clarify and study our previous observation that, under a compactification with bound-aries or orbifolding, vacuum expectation value of a bulk scalar field can have differentextra-dimensional wave-function profile from that of the lowest Kaluza-Klein mode ofits quantum fluctuation, under presence of boundary-localized potentials which would benecessarily generated through renormalization group running. For concreteness, we ana-lyze the Universal Extra Dimension model compactified on orbifold S /Z , with brane-localized Higgs potentials at the orbifold fixed points. We compute the Kaluza-Kleinexpansion of the Higgs and gauge bosons in an R ξ -like gauge by treating the brane-localized potential as a small perturbation. We also check that the ρ parameter is notaltered by the brane localized potential. OU-HET-643 ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail:
[email protected] Introduction
The five dimensional Quantum Field Theory (QFT), compactified on the orbifold S /Z ,has been paid much attention as the basis for the extra dimensional standard model withbulk gauge bosons [1, 2, 3, 4, 5], Universal Extra Dimension (UED) model [6, 7], Higgslessmodel [8], gauge-Higgs unification models (see e.g. [9] and references therein), and also thesupergravity models [10, 11, 12, 13]. The five dimensional QFT on S /Z is also the startingpoint for the QFT in the warped space, which is again utilized in the warped version of thebulk standard [16, 17], Higgsless [18, 19, 20, 21], gauge-Higgs unification [22, 23, 24, 25, 26],and supergravity [27] models.A five dimensional gauge theory is not renormalizable and must be seen as an effectivefield theory. We must take into account all the higher dimensional operators that are allowedby symmetries of a given theory, with appropriate suppression by a cutoff scale Λ. Especially,when there is a bulk scalar field, no symmetry prohibits the existence of the same type ofpotentials at the orbifold fixed-points as that of the bulk potential (with appropriate rescalingby the cutoff Λ to match its mass dimension). To repeat, the five dimensional QFT with abulk scalar, given as an effective theory, inevitably has the brane potentials.In [30] we stressed the importance of the brane-localized potential and considered anextreme case where the electroweak symmetry breaking is solely due to the brane-localizedpotential. In this paper, we concentrate on the opposite extreme where electroweak symmetrybreaking is mainly due to the bulk potential, as in the UED model, and take into account thebrane localized potentials as small perturbation. One of the main subjects of the currentstudy is to perform diagonalization of eigenmodes in order to present their profiles that evenleads to a difference between the vev and lowest mode profiles. Note that this diagonalizationhas never been achieved in any kind of models, except for our previous study [30].The organization of the paper is as follows. In the next section, we present our idea bythe simplest toy model with a single real scalar field in the bulk, under the presence of thebrane-localized potentials. In Section 3, we compute the Kaluza-Klein (KK) expansions forHiggs fields in the UED model with brane potentials, by taking it as a small perturbation.In Section 4, we compute the KK expansions for gauge fields similarly. We show that eventhough the KK masses are distorted by the brane potential, ρ parameter remains the sameas the standard model at the tree level. In Section 5, we summarize our result and showpossible future directions. In Appendix, we give our gauge fixing procedure and show thatextra-dimensional component of the gauge field and the would-be Nambu-Goldstone (NG)modes mix each other because of the position dependent vacuum expectation value (vev)while the four dimensional component of the gauge field does not receive such contribution. Originally Randall and Sundrum proposed it without any bulk field other than graviton [14]. See also [15]for a possible regularization of the negative tension brane. See Refs. [28, 29] for related works that also take into account the brane-localzed potential. In Ref. [28],the equivalence theorem is studied in a two Higgs doublet model with a brane-localized potential. In Ref. [29],it has been shown that the vev profile can be non-flat under the presence of a brane-localized potential. Inboth papers, the KK expansion of the Higgs field is not performed in a diagonal basis and the wave functionprofile of a KK mass eigenstate was hardly observable. In [31], Flacke, Menon and Phalen have emphasized the importance of the brane-localized interactions inthe context of the UED model and especially have analyzed the effect from the existence of the brane-localizedkinetic (quadratic) term upon the extra dimensional wave-function profile. The brane-localized potential waswritten but not taken into account in the calculation of the wave function profile. In this paper we continueto concentrate on the effect of the brane localized potential. VEV and Physical Fields under Brane Potentials
To clarify our previous observation [30], let us first consider a five dimensional theory with areal bulk scalar field Φ, compactified on a line segment y ∈ [0 , L ]. The action is given by S = Z d x Z L dy (cid:20) −
12 ( ∂ M Φ)( ∂ M Φ) − V (Φ) − δ ( y ) V (Φ) − δ ( y − L ) V L (Φ) (cid:21) , (1)where M, N, . . . run for 0 , . . . ,
3; 5, our metric convention is η MN = diag( − , , , , MN .Mass dimensions are [Φ] = 3 /
2, [ V ] = 5, and [ V ] = [ V L ] = 4. The variation of the action is δS = Z d x Z L dy δ Φ (cid:20) ✷ Φ + ∂ y Φ − ∂ V ∂ Φ − δ ( y ) ∂V ∂ Φ − δ ( y − L ) ∂V L ∂ Φ (cid:21) + Z d x [ − δ Φ ∂ y Φ] y = Ly =0 (2)where we have performed the partial integration and we define ✷ ≡ ∂ µ ∂ µ = − ∂ + ∇ with µ, ν, . . . running for 0 to 3. Resultant bulk equation of motion from the variation (2) is ✷ Φ + ∂ y Φ − ∂ V ∂ Φ = 0 , (3)while the boundary condition at y = 0 , L reads either Dirichlet δ Φ | y = η = 0 (4)or Neumann (cid:18) ∓ ∂ y Φ − ∂V η ∂ Φ (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y = η = 0 , (5)where signs above and below are for η = L and 0, respectively, throughout this paper. Wehave four choices of combination of Dirichlet and Neumann boundary conditions at y = 0and L , namely ( D, D ) , ( D, N ) , ( N, D ) , and ( N, N ) . (6)Difference choice of boundary condition corresponds to different choice of the theory. Thetheory is fixed once one chooses one of the four conditions.We comment on the relation between the above “downstairs” line-segment picture andthe orbifold picture. Sometimes it is convenient to first define fields on a circle y ∈ ( − L, L ],or even in the “upstairs” picture y ∈ ( −∞ , ∞ ). A special Dirichlet condition Φ | y = η = 0corresponds to the Z odd condition Φ( x, η + y ) = − Φ( x, η − y ) in the orbifolding, whilethe Neumann condition (5) corresponds to the Z even one Φ( x, η + y ) = Φ( x, η − y ) (withthe appropriate redefinition of the brane potential by factor two). The even ( N, N ) andodd (
D, D ) fields in the orbifold picture are given as (see e.g. [32])Φ even ( x, y ) = Φ( x, | y | ) , (7)Φ odd ( x, y ) = ǫ ( y )Φ( x, | y | ) , (8) An orbifold theory on S /Z can be obtained by identifying its brane-localized potentials with twice thecorresponding boundary-localized potentials in the line-segment theory. Note that there can be brane localized kinetic terms too [31] ∝ δ ( y − η )( ∂ M Φ)( ∂ M Φ) with η being 0 or L ,which we neglect for simplicity in this paper. ǫ ( y ) = ± ± y > < y < L subject to the boundary conditions (4) or (5).We utilize the background field method, separating the field into vev and quantum-fluctuation parts: Φ( x, y ) = Φ c ( x, y ) + φ q ( x, y ) . (9)In order to determine the vev profile, we need to solve the bulk equation of motion ✷ Φ c + ∂ y Φ c − ∂ V ∂ Φ c = 0 , (10)with either the Dirichlet boundary condition δ Φ c | y = η = 0 (11)or the Neumann boundary condition (cid:18) ∓ ∂ y Φ c − ∂V η ∂ Φ c (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y = η = 0 (12)at each brane. Here and hereafter, we utilize the following shorthand notation: ∂V∂ Φ c ( x, y ) ≡ ∂V∂ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ=Φ c ( x,y ) , ∂ V∂ Φ c ( x, y ) ≡ ∂ V∂ Φ (cid:12)(cid:12)(cid:12)(cid:12) Φ=Φ c ( x,y ) , (13)etc.We put the separation (9) into the action (1) and expand up to the quadratic terms ofthe fields φ q . Note that the Dirichlet boundary condition on the quantum fluctuation reads φ q | y = η = 0. After several partial integrations, utilizing the equation of motion (10) witheither the Dirichlet φ q | y = η = 0 or Neumann (12) boundary condition, we obtain the free fieldaction up to the quadratic terms in φ q S free = Z d x Z L dy (cid:18) φ q (cid:20) ✷ + ∂ y − ∂ V ∂ Φ c (cid:21) φ q + δ ( y )2 φ q (cid:20) ∂ y − ∂ V ∂ Φ c (cid:21) φ q + δ ( y − L )2 φ q (cid:20) − ∂ y − ∂ V L ∂ Φ c (cid:21) φ q (cid:19) . (14)A few comments are in order: • The free field action (14) is obtained by the expansion up to quadratic orders. Higherorder terms ∝ φ n with n > φ with the action (14). • The boundary conditions (4) and/or (5) is put on the whole field (9) when the theory isdefined. That is, when the vev Φ c obeys Dirichlet condition Φ c = const. at a boundary,the quantum fluctuation also obeys the Dirichilet one δ Φ = φ q = 0. When Φ c obeysNeumann condition (12) at a boundary, the quantum part φ q obeys (cid:18) ∓ ∂ y φ q − ∂ V η ∂ Φ c φ q (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) y = η = 0 , (15)where above (below) sign is for y = L (0).4 The Neumann boundary condition for vev (12) and for quantum fluctuation (15) aregenerically different. Therefore in general, the wave function profile for the vev andquantum fluctuation are different from each other . We will see it more in detail below. • The boundary condition (5) on the whole field (9) contains terms quadratic and higherorder in φ q , such as δ ( y )2 (cid:18) ∂ V ∂ Φ (cid:19) c ( φ q ) . (16)These terms are coming from the cubic and higher order brane-localized interactions,which are dropped to obtain the free field action (14). Note that exactly these termsaccount for the difference between the boundary conditions for vev and fluctuation. Forexample, the brane-localized term corresponding to the condition (16) is δ ( y )3! (cid:18) ∂ V ∂ Φ (cid:19) c ( φ q ) . (17)These dropped terms will be treated as boundary-localized interactions that genericallymix different KK modes.Now let us go on to the KK expansion. On physical ground, we assume that the vev doesnot depend on the flat four dimensional coordinates x µ : Φ c = Φ c ( y ). The equation of motionare then d Φ c dy ( y ) − ∂ V ∂ Φ c ( y ) = 0 . (18)Following the Sturm-Liouville theory, we can always expand any function of y , subject to oneof the four choices of boundary conditions (6), in terms of the orthonormal basis φ q ( x, y ) = X n φ qn ( x ) f n ( y ) , (19)where f n ( y ) are eigenfunctions of the Hermitian differential operator in the free action (14): (cid:18) d dy − ∂ V ∂ Φ c ( y ) (cid:19) f n ( y ) = − µ n f n ( y ) . (20)The eigenvalues − µ n are real but are not necessarily negative at the moment. For each n th mode, there are totally three unknown constants: two integration constantsof the second order differential equation (20) and the eigenvalue − µ n . Two of the three arefixed by the two boundary conditions at y = 0 and L , while the last one is fixed by thenormalization Z L dy f n ( y ) f m ( y ) = δ nm . (21)Consequent mass dimension is [ f n ] = 1 /
2. Eventually the free field action (14) is renderedinto S free = X n Z d x φ qn ( x ) (cid:0) ✷ − µ n (cid:1) φ qn ( x ) . (22) Recall also that they are not degenerate, that is, − µ n = − µ m if n = m . Boundary Potential on Universal Extra Dimension
In this section, we study the effect of the brane-localized potentials on the UED model [6, 7].In the UED model, the KK parity L − y → L + y plays a crucial role to make the LightestKK Particle (LKP) stable so that it can serve as a dark matter candidate. In this setup, itis convenient to utilize the new coordinate z ≡ y − L . The KK parity is realized as z → − z .Hereafter, we rewrite the labels η = L and 0, respectively by + and − . The action for the SU (2) L doublet Higgs field H is now S H = Z d x Z L/ − L/ dz h − ( D M H ) † ( D M H ) − V ( H ) − δ ( z − L/ V + ( H ) − δ ( z + L/ V − ( H ) i , (23)where D M is the gauge covariant derivative D M = ∂ M + ig T a W aM + ig ′ Y B M , (24)with Y = 1 / T a = σ a / H . (As usual, σ a are the Pauli matrices.) Mass dimensionsare [ H ] = [ W aM ] = [ B M ] = 3 / g ] = [ g ′ ] = − /
2. In the UED model, extra dimensionalcomponents of the gauge fields W ± , Z and A are odd under orbifold projection, taking( D, D ) boundary conditions, while all the other fields are even, taking (
N, N ) ones. An important point is that, as a non-renormalizable effective field theory in five dimen-sions, the bulk and brane potentials should contain all the higher dimensional operators,suppressed by a cutoff scale of the five dimensional theory Λ: V ( H ) = m | H | + ˆ λ Λ | H | + O (Λ − ) , (25) V ± ( H ) = m ± | H | + ˆ λ ± Λ | H | + O (Λ − ) , (26)where ˆ λ and ˆ λ ± are dimensionless constants. (Recall the mass dimensions: [ V ] = 5, [ V ± ] = 4,and [ H ] = 3 / inevitable since no symmetry can prohibit the existence of (26) when oneallows the bulk potential (25).Note that we have chosen the following basis H = (cid:18) ϕ + ϕ (cid:19) = ϕ +1 √ ( φ + iχ ) ! , (27)in which the real part φ (of the electrically neutral scalar ϕ ) takes a vev and plays the roleof the real scalar Φ in the previous section. Using | H | = φ + χ + ϕ + ϕ − with ϕ − ≡ ( ϕ + ) † , In the UED model, (
D, D ) condition is set such that the fields W ± , Z and A are vanishing at theboundary. Generically one can consider fixed but non-vanishing value for ( D, D ) boundary condition. Thistype of boundary condition for the Higgs field is utilized in [33]. Generically one would also expect that m ∼ m ± ∼ Λ as an effective theory. Here we do not pursue thisso-called “naturalness problem” and take m and m ± , being either positive or negative, as free dimensionfulparameters. V = λ (cid:0) φ + χ + 2 ϕ + ϕ − − v (cid:1) + O (Λ − ) , (28) V ± = λ ± (cid:0) φ + χ + 2 ϕ + ϕ − − v ± (cid:1) + O (Λ − ) , (29)where we defined λ ≡ ˆ λ/ Λ and λ ± ≡ ˆ λ ± / Λ . The mass dimensions of the new parametersare [ λ ] = −
1, [ λ ± ] = −
2, and [ v ] = [ v ± ] = 3. Note that the parameters v ≡ m /λ and v ± ≡ m ± /λ ± can be either positive or negative. In this notation, the vev φ c ( z ) is determined by the bulk equation of motion d φ c ( z ) dz − λ (cid:16) φ c ( z ) − v (cid:17) φ c ( z ) = 0 , (30)with either the Neumann ∓ dφ c ( z ) dz − λ ± (cid:16) φ c ( z ) − v ± (cid:17) φ c ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = 0 (31)or Dirichlet φ c ( z ) | z = ± L/ = const. (32)boundary condition at each end.Hereafter, we rewrite h ( x, z ) ≡ φ q ( x, z ) and drop the label “ q ” from other quantumfluctuations: H ( x, z ) = ϕ + ( x, z ) √ [ φ c ( z ) + h ( x, z ) + iχ ( x, z )] ! . (33)For reader’s ease, we write down the potential quadratic in quantum fluctuation V ( x, z ) = λ (cid:16) φ c ( z ) − v (cid:17) (cid:16) χ ( x, z ) + 2 (cid:12)(cid:12) ϕ + ( x, z ) (cid:12)(cid:12) (cid:17) + λ (cid:16) φ c ( z ) − v (cid:17) h ( x, z ) , (34) V ± ( x ) = λ ± (cid:16) φ c ( z ) − v (cid:17) (cid:16) χ ( x, z ) + 2 (cid:12)(cid:12) ϕ + ( x, z ) (cid:12)(cid:12) (cid:17) + λ ± (cid:16) φ c ( z ) − v (cid:17) h ( x, z ) (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ . (35)Note that linear terms necessarily drop out, due to the equation of motion for the vev (cor-responding to Eq. (18)). The KK expansion for the quantum fluctuations is given as h ( x, z ) = X n h n ( x ) f hn ( z ) , χ ( x, z ) = X n χ n ( x ) f χn ( z ) , ϕ + ( x, z ) = X n ϕ + n ( x ) f ϕn ( z ) , (36) In this paper, we neglect all the back-reactions to the background spacetime geometry and shift zero ofthe potentials freely. As stated in footnote 8, the bulk mass squared and the brane mass, which can be positive and/or negative,are taken as free dimensionful parameters and hence v and v ± are also free parameters. h n ] = [ χ n ] = [ ϕ + n ] = 1. Here f n are eigenfunctions of the KK equations (cid:20) d dz − λ (cid:16) φ c ( z ) − v (cid:17)(cid:21) f hn ( z ) = − µ hn f hn ( z ) , (37) (cid:20) d dz − λ (cid:16) φ c ( z ) − v (cid:17)(cid:21) f Xn ( z ) = − µ Xn f Xn ( z ) , (38)subjecting to the boundary conditions (cid:20) ∓ ddz − λ ± (cid:16) φ c ( z ) − v ± (cid:17)(cid:21) f hn ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = 0 , (39) (cid:20) ∓ ddz − λ ± (cid:16) φ c ( z ) − v ± (cid:17)(cid:21) f Xn ( z ) (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = 0 , (40)where X stands for the labels χ and ϕ , both giving the same KK expansions in this casewithout boundary potential. Results presented in this section correspond to ξ = 1 in the R ξ gauge, see Appendix. Let us first review the case without any brane potential V ± ( H ) = 0, as in the original UEDmodel [6, 7]. In the model, there is only bulk potential (28), with O (Λ − ) terms beingneglected. The solution to the equation of motion (30) is φ c ( z ) = v . (41)Note that obviously χ c ( z ) = ( ϕ + ) c ( z ) = 0 is the solution for other modes. In the originalUED model, all the bulk fields are put the ( N, N ) boundary condition with V ± = 0: ∓ dHdz ( ± L/
2) = 0 , (42)which is trivially satisfied by the constant profile (41).The KK equation corresponding to (20) is now d f hn ( z ) dz − λv f hn ( z ) = − µ hn f hn ( z ) , (43) d f Xn ( z ) dz = − µ Xn f Xn ( z ) . (44)The ( N, N ) boundary condition (42) simply reads ∓ df n dz ( ± L/
2) = 0 , (45)for all h , χ and ϕ ± .There are three possible cases:1. When µ hn < λv or µ Xn <
0, general solutions are f n ( z ) = α n cosh( κ n z ) + β n sinh( κ n z ) , (46)where κ n = q λv − µ hn or κ n = q | µ Xn | , respectively. This cannot satisfy the bound-ary condition (45). 8. When µ hn = 2 λv or µ Xn = 0, general solutions are f n ( z ) = α n + β n z. (47)This is conventionally called zero mode and is written with n = 0. With the boundarycondition (45) and the normalization (21), we get f ( z ) = r L . (48)3. When µ hn > λv or µ Xn >
0, general solutions with integration constants α n , β n are f n ( z ) = α n cos( k n z ) + β n sin( k n z ) (49)where k n = q µ hn − λv or k n = µ Xn >
0, respectively. With the boundary condi-tion (45) and the normalization (21), we obtain f n ( z ) = q L cos( k n z ) ( n : even), q L sin( k n z ) ( n : odd), (50)where k n = nπ/L . The cosine and sine modes are KK parity even and odd, respectively.To summarize, the Kaluza-Klein mass for n ≥ µ hn = q k n + 2 λv = s n + (cid:18) λv m KK (cid:19) m KK , (51) µ Xn = k n = nm KK , (52)where we defined the unit KK mass m KK ≡ π/L . In Ref. [30], we have considered an extreme case where electroweak symmetry breaking issolely due to the brane potential. Here we concentrate on the opposite limit where branepotential is put as a small perturbation on the above UED model.Let us start from the bulk potential (28) and treat the brane potential V ± in (29) as asmall perturbation of O ( ǫ ). Note that v ± can be negative here, corresponding to the positivemass term in the brane potential, while v is always positive by the starting assumption thatthe symmetry breaking in mainly generated by the bulk potential. We take v > λ ± → m ± = − λ ± v ± .Firstly the equation of motion (30) is not altered. We seek for a solution of the type φ c ( z ) = v + ǫφ c ( z ) , (53)where φ c ( z ) is a small perturbation and ǫ is the expansion parameter eventually set to beunity. We put Eq. (53) into Eq. (30) to get d φ c dz ( z ) − λv φ c ( z ) = 0 . (54)9he general solution is φ c ( z ) = A cosh( κz ) + B sinh( κz ) , (55)where we define κ ≡ √ λ v . Note the mass dimensions [ κ ] = 1 and [ A ] = [ B ] = 3 /
2. Wesometimes trade λ by κ in the following.Noting that the brane potential itself is treated as a perturbation of O ( ǫ ), the ( N, N )boundary condition (31) reads: ∓ dφ c ( z ) dz − λ ± (cid:0) v − v ± (cid:1) v (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = 0 , (56)that is, − κA sinh( κL/ ∓ κB cosh( κL/ − λ ± (cid:0) v − v ± (cid:1) v = 0 . (57)When we assume conserved KK parity on our setup, namely V + ( H ) = V − ( H ) and hence λ + = λ − and v = v − , the solution to Eq. (57) simplifies to A = − λ + v (cid:0) v − v (cid:1) κ sinh κL , B = 0 . (58)To summarize, when the brane potentials respect the KK parity V + = V − the vev becomesKK parity even: φ c ( z ) = v + ǫφ c ( z ) + O ( ǫ ) , with φ c ( z ) = − λ + v (cid:0) v − v (cid:1) κ sinh κL cosh( κz ) . (59)Recall that v ± in the perturbation potential ǫV ± can be negative while we take v > We treat the brane potential as a perturbation on the eigenvalue problem (37) with theboundary condition (39). Recall that we are regarding V ± as a small perturbation of O ( ǫ ): ǫV ± = ǫ λ ± (cid:2) ( v + h ) + χ + 2 ϕ + ϕ − − v ± (cid:3) + O ( ǫ ) , (60) V = λ (cid:0) v h + h + χ + 2 ϕ + ϕ − (cid:1) + ǫλ (cid:0) v h + h + χ + 2 ϕ + ϕ − (cid:1) ( v + h ) φ c + O ( ǫ ) . (61)We separate the KK wave function of the physical Higgs field into the unperturbed andperturbed parts f hn ( z ) = f (0) n ( z ) + ǫf (1) n ( z ) + O ( ǫ ) , (62)where f (0) n ( z ) are explicitly given as the r.h.s. of Eqs. (48) and (50) with the unperturbedeigenvalues − µ n given by (51). Let us write the new perturbed eigenvalues as − µ n − ǫ ∆ n , with10 n being given by r.h.s. of Eq. (51) and ∆ n being real constant of mass dimension [∆ n ] = 2.The first order KK equation from Eq. (37) becomes (cid:18) d dz − λv + µ n (cid:19) f (1) n ( z ) = (6 λv φ c ( z ) − ∆ n ) f (0) n ( z ) . (63)The boundary condition (39) is now, to the first order, ∓ df (1) n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = λ ± (cid:0) v − v ± (cid:1) f (0) n ( z ) (cid:12)(cid:12)(cid:12) z = ± L/ . (64) Let us first consider the zero mode KK equation from Eq. (63) d f (1)0 dz ( z ) = 6 λv φ c ( z ) − ∆ √ L = 6 λv ( A cosh( κz ) + B sinh( κz )) − ∆ √ L , (65)where constants A and B are given by Eq. (58) when there is the conserved KK parity.General solution is f (1)0 ( z ) = α + β z − ∆ √ L z + 3 v √ L ( A cosh( κz ) + B sinh( κz )) , (66)where α and β are integration constants of mass dimensions [ α ] = 1 / β ] = 3 / λ + = λ − and v = v − , for simplicity.The solution to the boundary condition (64) is∆ = 4 λ + v L , β = 0 . (67)The zero mode becomes KK parity even. The constant α can be fixed by the normalizationcondition (21), or to the first order, Z L/ − L/ dz f (0)0 ( z ) f (1)0 ( z ) = 0 , (68)so that α = λ + v √ L λ + v κ L / (cid:18) − v v (cid:19) . (69)Recall the mass dimensions [ v ] = [ v + ] = 3 /
2, [ κ ] = 1, [ α ] = 1 /
2, [ λ ] = [ L ] = −
1, and[ λ + ] = −
2. 11 .3.2 Even Modes
For even n , the KK equation (63) reads (cid:18) d dz + k n (cid:19) f (1) n ( z ) = (6 λv φ c ( z ) − ∆ n ) r L cos( k n z )= − κλ + ( v − v )sinh κL cosh( κz ) + ∆ n ! r L cos( k n z ) . (70)Recall k n = nπ/L . The solution is f (1) n ( z ) = α n cos( k n z ) + β n sin( k n z ) − ∆ n √ Lk n [cos( k n z ) + 2 k n z sin( k n z )]+ 3 √ v √ L (4 k n + κ ) (cid:2) κ φ c ( z ) cos( k n z ) + 2 k n φ c ′ ( z ) sin( k n z ) (cid:3) , (71)where φ c ( z ) is given in Eq. (59).The boundary condition (64) for even n mode is now, to the first order, ∓ df (1) n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = λ + (cid:0) v − v (cid:1) r L ( − n/ , (72)which gives β n = 0 and ∆ n = 8 λ + v L + 24 λ + k n ( v − v ) L (4 k n + κ ) . (73)For n ≫
1, we get ∆ n → λ + (3 v + v ) /L . As in the zero mode case, the constant α n canbe fixed by the normalization condition α n = 12 √ λ + (cid:0) v − v (cid:1) κ L / (4 k n + κ ) . (74) Finally we consider the odd n modes. The KK equation reads (cid:18) d dz + k n (cid:19) f (1) n ( z ) = (6 λv φ c ( z ) − ∆ n ) r L sin( k n z )= − κλ + ( v − v )sinh κL cosh( κz ) + ∆ n ! r L sin( k n z ) (75)and its general solution is f (1) n ( z ) = α n cos( k n z ) + β n sin( k n z ) − ∆ n √ Lk n [sin( k n z ) − k n z cos( k n z )]+ 3 √ v √ L (4 k n + κ ) (cid:2) κ φ c ( z ) sin( k n z ) − k n φ c ′ ( z ) cos( k n z ) (cid:3) . (76)12he boundary condition (64) for odd n mode is now, to the first order, ∓ df (1) n ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = ± λ + (cid:0) v − v (cid:1) r L ( − ( n − / , (77)which gives α n = 0 and ∆ n again as in Eq. (73). From the normalization, the last constant β n is obtained as β n = 12 √ λ + (cid:0) v − v (cid:1) κ L / (4 k n + κ ) , (78)which is equal to the value of even-mode’s α n . To summarize, under the presence of small brane-localized potential, the KK expansion isgiven by f h ( z ) = 1 √ L λ + v L λ + ( v − v ) v κ L − λ + v L z − λ + ( v − v ) κ sinh κL cosh( κz ) ! (79) f hn ( z ) = r L (cid:18) C n + 3 κ v (4 k n + κ ) φ c ( z ) (cid:19) (cid:26) cos( k n z )sin( k n z ) (cid:27) + ∆ n √ Lk n z − r L k n v (4 k n + κ ) φ c ′ ( z ) ! (cid:26) ( − sin( k n z ))cos( k n z ) (cid:27) for ( n : even positive, n : odd, (80)where ǫ = 1, and φ c ( z ) and ∆ n for n > C n = 1 + 12 λ + (cid:0) v − v (cid:1) κ L (4 k n + κ ) − ∆ n k n . (81)The perturbed KK mass becomes, respectively for n = 0 and n > µ h = κ + ∆ = 2 λv + 4 λ + v L , (82) µ hn = k n + κ + ∆ n = (cid:16) πnL (cid:17) + 2 λv + ∆ n . (83)The case where we have only positive mass term on the brane V + = m + | H | = m + φ + · · · can be obtained by taking limit λ + → m + = − λ + v > φ c ( z ) → − m + v κ sinh κL cosh( κz ) , (84)∆ → − m + L , (85)∆ n> → − m + L k n + κ k n + κ , (86) C n → m + κ L (4 k n + κ ) + 2 m + ( k n + κ ) Lk n (4 k n + κ ) . (87)13or a very high KK mode n ≫
1, the limit further simplifies to∆ n> → − m + L , C n → . (88) Under the presence of the brane potential, the vev of the Higgs field is distorted as in Eq. (59)so that it has non-trivial extra dimensional profile. Let us see how the gauge field wavefunction is modified in this case.As shown in Appendix, the position dependent vev v ( z ) ≡ φ c ( z ) generates the positiondependent bulk mass terms for the gauge fields W ± µ and Z µ . When KK-expanding as W ± µ ( x, z ) = X n f Wn ( z ) W ± nµ ( x ) , Z µ ( x, z ) = X n f Zn ( z ) Z nµ ( x ) , (89)resultant bulk KK equation becomes (cid:18) d dz − m V ( z ) (cid:19) f V ( z ) = − µ V n f V ( z ) , (90)where the label V stands for W and Z . In contrast, their boundary conditions are not modifiedfrom the ordinary ( N, N ) ones df V ( z ) dz (cid:12)(cid:12)(cid:12)(cid:12) z = ± L/ = 0 , (91)since we neglect the brane-localized Higgs kinetic terms in our analysis.Again let us solve the KK equation iteratively by taking the Higgs brane potential assmall perturbation. From Eq. (59), we see m V = m V + ǫg V m V φ c ( z ) + O ( ǫ ) , (92)where we define m W ≡ gv , m Z ≡ g Z v , (93)with g Z ≡ p g + g ′ . The zeroth order solution with the boundary condition (91) is, bothfor W and Z , f (0)0 ( z ) = 1 √ L , (94) f (0) n ( z ) = q L cos( k n z ) for n : even positive , q L sin( k n z ) for n : odd , (95)where again k n = πn/L and the zeroth order KK masses are given by µ V n = k n + m V . (96)14riting the eigenvalues of the KK equation − µ V − ǫ ∆ Vn , the first order KK equation forthe eigenfunction f (0) n ( z ) + ǫf V (1) n ( z ) is (cid:18) d dz + k n (cid:19) f V (1) n ( z ) = (cid:0) g V m V φ c ( z ) − ∆ Vn (cid:1) f (0) n ( z ) . (97)The solution subjecting to the boundary condition (91) is obtained similarly to the Higgs case f V (1)0 ( z ) = 1 √ L α V − ∆ V z − m V λ + ( v − v ) κ sinh κL cosh( κz ) ! , (98) f V (1) n> ( z ) = r L (cid:18) α Vn − ∆ Vn k n + g V m V k n + κ φ c ( z ) (cid:19) (cid:26) cos( k n z )sin( k n z ) (cid:27) + r L (cid:18) ∆ Vn k n z − g V m V k n (4 k n + κ ) κ φ c ′ ( z ) (cid:19) (cid:26) ( − sin( k n z ))cos( k n z ) (cid:27) , (99)where ∆ V = − m V λ + ( v − v ) κ L , (100)∆ Vn = − m V λ + ( v − v )(2 k n + κ ) Lκ (4 k n + κ ) , (101)and α V = m V λ + ( v − v )(24 − κ L )6 κ L , (102) α Vn = 8 m V λ + (cid:0) v − v (cid:1) L (4 k n + κ ) . (103)When there is only positive mass term on the brane V + = m + | H | , the solution is obtainedby taking limit λ + → m + ≡ − λ + v ∆ V → − m V m + κ L , (104)∆ Vn → − m V m + (2 k n + κ ) Lκ (4 k n + κ ) , (105)and α V = m V m + (24 − κ L )6 κ L , (106) α Vn = 4 m V m + L (4 k n + κ ) . (107)For a very high KK modes n ≫
1, they further simplify to∆ Vn → − m V m + Lκ , (108) α Vn → . (109)15e note that the observed physical mass-squared for W ± and Z bosons correspond to m V + ∆ V . Since the correction to the gauge boson mass-squared ∆ V is proportional to m V ,the correction to the W and Z masses are proportial to the corresponding gauge coupling g and g Z , respectively, with the uniform coefficient − v λ + ( v − v ) κ L . Therefore, the ratio of the W and Z boson masses are still proportional to the ratio of the gauge coupling g/g Z . Thebrane localized Higgs potential does not change the ρ parameter of the model even though itdoes change the mass formula, as is expected from the fact that the introduction of the branepotential does not violate the custodial symmetry. We have further clarified our previous observation that the brane localized potential can makethe extra-dimensional profiles of the vev and lowest KK mode different from each other. Oneof the main subjects of this paper is to perform diagonalization of eigenmodes in order topresent their profiles that even leads to a difference between the vev and lowest mode profiles.We note that this diagonalization has never been achieved in any kind of models, except forour previous study [30]. Especially we have explained what makes the difference from theview point of free part of the Lagrangian.We have considered the UED model and obtained the KK expansion for the Higgs field,under the presence of the brane-localized potential. We find that small boundary potentialraises the KK masses when it is wine-bottle shape with negative mass-squared at its origin,while it lowers the KK masses when there is only positive mass term. KK parity is conservedin all the modes by introduction of the KK parity even potential V + = V − .We have also computed the KK expansion for the four dimensional components of thegauge fields W ± µ and Z µ . Contrary to the Higgs field case, gauge boson KK masses acquirenegative contribution for both the wine-bottle and positive-mass shapes of boundary poten-tial. Even though W ± µ and Z µ have different position-dependent bulk masses and hence theoscillation of their wave function is different in the extra dimension, the resultant ρ parameterremains the same. This reflects the fact that the custodial symmetry remains intact underthe presence of the boundary potential.It would be interesting to compute the KK expansions of extra dimensional componentof gauge fields and the would-be NG modes as well as the bulk fermions, whose masses aremodified by the position dependent vev too. It is also worth studying the brane-localizedHiggs kinetic term simultaneously in our setup. These subjects will be treated in a separatepublication. Acknowledgement
We would like to thank T. Yamashita for very helpful discussions, and also thank S. Mat-sumoto for useful discussions. This work is partially supported by Scientific Grant by Ministryof Education and Science, Nos. 20540272, 20039006, 20025004, 20244028, and 19740171. Thework of RT is supported by the GCOE Program, The Next Generation of Physics, Spun fromUniversality and Emergence. 16 ppendixGauge Fixing
Basically we follow the notation of Ref. [30], summarized in its Appendix C, except for thenormalization of the vev v which differs by a factor √
2. In our basis H c = v ( z ) √ ! , H q = ϕ + h ( x,z )+ iχ ( x,z ) √ ! , (110)where we have rewritten the vev v ( z ) ≡ φ c ( z ). The covariant derivative on the Higgs field is D M H = ∂ M H + ig √ (cid:18) W + M W − M (cid:19) H + ie θ W Z M + A M − θ W Z M ! H = ∂ M ϕ + ∂ M v + ∂ M h + i∂ M χ √ ! + ig √ W + M v + h + iχ √ W − M ϕ + ! + ie (cid:16) θ W Z M + A M (cid:17) ϕ + − θ W Z M v + h + iχ √ , (111)where we have defined W ± M = W M ∓ iW M √ , (cid:18) Z M A M (cid:19) = (cid:18) c − ss c (cid:19) (cid:18) W M B M (cid:19) , (112)with c ≡ cos θ W = g p g + g ′ , s ≡ sin θ W = g ′ p g + g ′ , e ≡ gg ′ p g + g ′ . (113)Note that the bulk gauge boson masses m W and m Z are z dependent now m W ( z ) ≡ gv ( z )2 , m Z ( z ) ≡ p g + g ′ v ( z ) = e sin 2 θ W v ( z ) . (114)Mass dimensions are [ g ] = [ g ′ ] = [ e ] = − / v ] = [ W ± M ] = [ Z M ] = [ A M ] = 3 /
2. TheHiggs kinetic Lagrangian is L H = − | D M H | = − (cid:12)(cid:12)(cid:12)(cid:12) ∂ M ϕ + + im W W + M + ig W + M ( h + iχ ) + ie (cid:18) θ W Z M + A M (cid:19) ϕ + (cid:12)(cid:12)(cid:12)(cid:12) − (cid:12)(cid:12)(cid:12)(cid:12) ∂ M v + ∂ M h + i∂ M χ + igW − M ϕ + − im Z Z M − ie sin 2 θ W Z M ( h + iχ ) (cid:12)(cid:12)(cid:12)(cid:12) , (115)where the contraction of the Lorentz indices is understood. The quadratic terms are L quad H = − (cid:12)(cid:12) ∂ M ϕ + (cid:12)(cid:12) − ( ∂ M h ) + ( ∂ M χ ) − m W (cid:12)(cid:12) W + M (cid:12)(cid:12) − m Z Z M ) + im W (cid:0) W − M ∂ M ϕ + − W + M ∂ M ϕ − (cid:1) + m Z Z M ∂ M χ − ( ∂ v ) (cid:18) ∂ h + ig (cid:0) W − ϕ + − W +5 ϕ − (cid:1) + e sin 2 θ W Z χ (cid:19) . (116)17he terms in the last line are coming from the non-trivial profile of the vev in the extradimension.We employ the following R ξ -like gauge fixing L GF = − ξ X a =1 f a f a + f B f B ! , (117)where f a = ∂ M W aM + igξ (cid:16) H q † T a H c − H c † T a H q (cid:17) ,f B = ∂ M B M + ig ′ ξ (cid:16) H q † Y H c − H c † Y H q (cid:17) . (118)By the redefinition f ± ≡ f ∓ if √ ∂ M W ± M ∓ iξm W ϕ ± , (119) f Z ≡ cf − sf B = ∂ M Z M − ξm Z χ, (120) f A ≡ sf + cf B = ∂ M A M , (121)we can rewrite L GF = − ξ f + f − − ξ (cid:0) f Z f Z + f A f A (cid:1) = − ξ (cid:12)(cid:12) ∂ M W + M (cid:12)(cid:12) − ξ (cid:16)(cid:0) ∂ M Z M (cid:1) + (cid:0) ∂ M A M (cid:1) (cid:17) + im W (cid:0) ϕ + ∂ M W − M − ϕ − ∂ M W + M (cid:1) + m Z χ∂ M Z M − ξm W (cid:12)(cid:12) ϕ + (cid:12)(cid:12) − ξm Z χ . (122)The following gauge choices can be considered.1. For ξ = 1, the sum of quadratic terms simplifies to L quad H +GF ξ =1 = − (cid:12)(cid:12) ∂ M ϕ + (cid:12)(cid:12) − m W (cid:12)(cid:12) ϕ + (cid:12)(cid:12) −
12 ( ∂ M χ ) − m Z χ −
12 ( ∂ M h ) − (cid:12)(cid:12) ∂ N W + N (cid:12)(cid:12) − m W (cid:12)(cid:12) W + M (cid:12)(cid:12) − (cid:0) ∂ N Z N (cid:1) − m Z Z M ) − (cid:0) ∂ M A M (cid:1) + ∂ (cid:2) im W (cid:0) W − ϕ + − W +5 ϕ − (cid:1) + m Z Z χ (cid:3) − ( ∂ v ) ( ∂ h ) − ∂ v ) (cid:18) ig (cid:0) W − ϕ + − W +5 ϕ − (cid:1) + e sin 2 θ W Z χ (cid:19) . (123)The third (second last) line is a total derivative and potentially contributes as boundarylocalized mixing terms between gauge fields and the would-be NG modes when weintegrate out the extra dimension for the KK reduction where the vev is independent offour-dimensional spacetime coordinate. In the UED model of our current consideration, When we also introduce brane-localized Higgs kinetic terms, we need to add extra gauge fixing termslocalized on the branes. y → − y and take the following Dirichlet boundary conditions W ± (cid:12)(cid:12) z = ± L/ = Z | z = ± L/ = A | z = ± L/ = 0 . (124)Under this assumption, the third line can be safely neglected.The last line in Eq. (123) is due to the non-trivial wave function profile of the vev,which mixes the extra-dimensional component of the gauge fields and the would-be NGmodes. The first term in the last line − ( ∂ v )( ∂ h ) is treated properly in Secs. 2 and 3,while impact from the other mixing terms will be presented elsewhere.2. In the unitary gauge ξ → ∞ , the would-be NG bosons ϕ ± and χ become infinitelyheavy and decouple L quad H +GF → −
12 ( ∂ M h ) − m W (cid:12)(cid:12) W + M (cid:12)(cid:12) − m Z Z M ) . (125)Hereafter, we employ the ξ = 1 gauge.The gauge kinetic Lagrangian is L YM = − X a =1 F aMN F aMN + F BMN F B MN ! . (126)From the redefinition W ± M = W M ∓ iW M √ , (127) Z M = cW M − sB M , (128) A M = sW M + cB M , (129)we get F ± MN = ∂ M W ± N − ∂ N W ± M ± ig (cid:0) W M W ± N − W N W ± M (cid:1) , (130) F MN = ∂ M W N − ∂ N W M + 2 g (cid:0) W + M W − N − W + N W − M (cid:1) , (131)with W M = cZ M + sA M , and L YM = − F + MN F − MN − (cid:2) F MN F MN + F BMN F B MN (cid:3) . 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