Diagonal KaluzaKlein expansion under brane localized potential
Abstract
We clarify and study our previous observation that, under a compactification with boundaries or orbifolding, vacuum expectation value of a bulk scalar field can have different extradimensional wavefunction profile from that of the lowest KaluzaKlein mode of its quantum fluctuation, under presence of boundarylocalized potentials which would be necessarily generated through renormalization group running. For concreteness, we analyze the Universal Extra Dimension model compactified on orbifold , with branelocalized Higgs potentials at the orbifold fixed points. We compute the KaluzaKlein expansion of the Higgs and gauge bosons in an like gauge by treating the branelocalized potential as a small perturbation. We also check that the parameter is not altered by the brane localized potential.
OUHET643
1 Introduction
The five dimensional Quantum Field Theory (QFT), compactified on the orbifold , has been paid much attention as the basis for the extra dimensional standard model with bulk gauge bosons [1, 2, 3, 4, 5], Universal Extra Dimension (UED) model [6, 7], Higgsless model [8], gaugeHiggs unification models (see e.g. [9] and references therein), and also the supergravity models [10, 11, 12, 13]. The five dimensional QFT on is also the starting point for the QFT in the warped space,^{1}^{1}1 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. which is again utilized in the warped version of the bulk standard [16, 17], Higgsless [18, 19, 20, 21], gaugeHiggs unification [22, 23, 24, 25, 26], and supergravity [27] models.
A five dimensional gauge theory is not renormalizable and must be seen as an effective field theory. We must take into account all the higher dimensional operators that are allowed by symmetries of a given theory, with appropriate suppression by a cutoff scale . Especially, when there is a bulk scalar field, no symmetry prohibits the existence of the same type of potentials at the orbifold fixedpoints as that of the bulk potential (with appropriate rescaling by the cutoff to match its mass dimension). To repeat, the five dimensional QFT with a bulk scalar, given as an effective theory, inevitably has the brane potentials.
In [30] we stressed the importance of the branelocalized potential and considered an extreme case where the electroweak symmetry breaking is solely due to the branelocalized potential.^{2}^{2}2See Refs. [28, 29] for related works that also take into account the branelocalzed potential. In Ref. [28], the equivalence theorem is studied in a two Higgs doublet model with a branelocalized potential. In Ref. [29], it has been shown that the vev profile can be nonflat under the presence of a branelocalized potential. In both papers, the KK expansion of the Higgs field is not performed in a diagonal basis and the wave function profile of a KK mass eigenstate was hardly observable. In this paper, we concentrate on the opposite extreme where electroweak symmetry breaking is mainly due to the bulk potential, as in the UED model, and take into account the brane localized potentials as small perturbation.^{3}^{3}3In [31], Flacke, Menon and Phalen have emphasized the importance of the branelocalized interactions in the context of the UED model and especially have analyzed the effect from the existence of the branelocalized kinetic (quadratic) term upon the extra dimensional wavefunction profile. The branelocalized potential was written but not taken into account in the calculation of the wave function profile. In this paper we continue to concentrate on the effect of the brane localized potential. One of the main subjects of the current study is to perform diagonalization of eigenmodes in order to present their profiles that even leads to a difference between the vev and lowest mode profiles. Note that this diagonalization has 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 by the simplest toy model with a single real scalar field in the bulk, under the presence of the branelocalized potentials. In Section 3, we compute the KaluzaKlein (KK) expansions for Higgs 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 even though the KK masses are distorted by the brane potential, parameter remains the same as the standard model at the tree level. In Section 5, we summarize our result and show possible future directions. In Appendix, we give our gauge fixing procedure and show that extradimensional component of the gauge field and the wouldbe NambuGoldstone (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.
2 VEV and Physical Fields under Brane Potentials
To clarify our previous observation [30], let us first consider a five dimensional theory with a real bulk scalar field , compactified on a line segment .^{4}^{4}4An orbifold theory on can be obtained by identifying its branelocalized potentials with twice the corresponding boundarylocalized potentials in the linesegment theory. The action is given by
(1) 
where run for , our metric convention is . Mass dimensions are , , and .^{5}^{5}5Note that there can be brane localized kinetic terms too [31] with being 0 or , which we neglect for simplicity in this paper.
The variation of the action is
where we have performed the partial integration and we define with running for 0 to 3. Resultant bulk equation of motion from the variation (LABEL:S_variation) is
(3) 
while the boundary condition at reads either Dirichlet
(4) 
or Neumann
(5) 
where signs above and below are for and , respectively, throughout this paper. We have four choices of combination of Dirichlet and Neumann boundary conditions at and , namely
(6) 
Difference choice of boundary condition corresponds to different choice of the theory. The theory is fixed once one chooses one of the four conditions.
We comment on the relation between the above “downstairs” linesegment picture and the orbifold picture. Sometimes it is convenient to first define fields on a circle , or even in the “upstairs” picture . A special Dirichlet condition corresponds to the odd condition in the orbifolding, while the Neumann condition (5) corresponds to the even one (with the appropriate redefinition of the brane potential by factor two). The even () and odd () fields in the orbifold picture are given as (see e.g. [32])
(7)  
(8) 
where for and in the r.h.s. is the solution to the bulk equation (3) in subject to the boundary conditions (4) or (5).
We utilize the background field method, separating the field into vev and quantumfluctuation parts:
(9) 
In order to determine the vev profile, we need to solve the bulk equation of motion
(10) 
with either the Dirichlet boundary condition
(11) 
or the Neumann boundary condition
(12) 
at each brane. Here and hereafter, we utilize the following shorthand notation:
(13) 
etc.
We put the separation (9) into the action (1) and expand up to the quadratic terms of the fields . Note that the Dirichlet boundary condition on the quantum fluctuation reads . After several partial integrations, utilizing the equation of motion (10) with either the Dirichlet or Neumann (12) boundary condition, we obtain the free field action up to the quadratic terms in
(14) 
A few comments are in order:

The boundary conditions (4) and/or (5) is put on the whole field (9) when the theory is defined. That is, when the vev obeys Dirichlet condition at a boundary, the quantum fluctuation also obeys the Dirichilet one . When obeys Neumann condition (12) at a boundary, the quantum part obeys
(15) where above (below) sign is for (0).

The boundary condition (5) on the whole field (9) contains terms quadratic and higher order in , such as
(16) These terms are coming from the cubic and higher order branelocalized interactions, which are dropped to obtain the free field action (14). Note that exactly these terms account for the difference between the boundary conditions for vev and fluctuation. For example, the branelocalized term corresponding to the condition (16) is
(17) These dropped terms will be treated as boundarylocalized interactions that generically mix different KK modes.
Now let us go on to the KK expansion. On physical ground, we assume that the vev does not depend on the flat four dimensional coordinates : . The equation of motion are then
(18) 
Following the SturmLiouville theory, we can always expand any function of , subject to one of the four choices of boundary conditions (6), in terms of the orthonormal basis
(19) 
where are eigenfunctions of the Hermitian differential operator in the free action (14):
(20) 
The eigenvalues are real but are not necessarily negative at the moment.^{6}^{6}6Recall also that they are not degenerate, that is, if .
For each th mode, there are totally three unknown constants: two integration constants of the second order differential equation (20) and the eigenvalue . Two of the three are fixed by the two boundary conditions at and , while the last one is fixed by the normalization
(21) 
Consequent mass dimension is . Eventually the free field action (14) is rendered into
(22) 
3 Boundary Potential on Universal Extra Dimension
In this section, we study the effect of the branelocalized potentials on the UED model [6, 7]. In the UED model, the KK parity . The KK parity is realized as . Hereafter, we rewrite the labels and 0, respectively by and . The action for the doublet Higgs field is now plays a crucial role to make the Lightest KK Particle (LKP) stable so that it can serve as a dark matter candidate. In this setup, it is convenient to utilize the new coordinate
(23) 
where is the gauge covariant derivative
(24) 
with and on . (As usual, are the Pauli matrices.) Mass dimensions are and . In the UED model, extra dimensional components of the gauge fields , and are odd under orbifold projection, taking boundary conditions, while all the other fields are even, taking ones.^{7}^{7}7In the UED model, condition is set such that the fields , and are vanishing at the boundary. Generically one can consider fixed but nonvanishing value for boundary condition. This type of boundary condition for the Higgs field is utilized in [33].
An important point is that, as a nonrenormalizable effective field theory in five dimensions, the bulk and brane potentials should contain all the higher dimensional operators, suppressed by a cutoff scale of the five dimensional theory :
(25)  
(26) 
where and are dimensionless constants.^{8}^{8}8 Generically one would also expect that as an effective theory. Here we do not pursue this socalled “naturalness problem” and take and , being either positive or negative, as free dimensionful parameters. (Recall the mass dimensions: , , and .) We emphasize that the presence of the brane potential (26), which has been overlooked so far, is inevitable since no symmetry can prohibit the existence of (26) when one allows the bulk potential (25).
Note that we have chosen the following basis
(27) 
in which the real part (of the electrically neutral scalar ) takes a vev and plays the role of the real scalar in the previous section. Using with , let us rewrite the potentials^{9}^{9}9In this paper, we neglect all the backreactions to the background spacetime geometry and shift zero of the potentials freely.
(28)  
(29) 
where we defined and . The mass dimensions of the new parameters are , , and . Note that the parameters and can be either positive or negative.^{10}^{10}10As 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 and are also free parameters.
In this notation, the vev is determined by the bulk equation of motion
(30) 
with either the Neumann
(31) 
or Dirichlet
(32) 
boundary condition at each end.
Hereafter, we rewrite and drop the label “” from other quantum fluctuations:
(33) 
For reader’s ease, we write down the potential quadratic in quantum fluctuation
(34)  
(35) 
Note that linear terms necessarily drop out, due to the equation of motion for the vev (corresponding to Eq. (18)). The KK expansion for the quantum fluctuations is given as
(36) 
where . Here are eigenfunctions of the KK equations
(37)  
(38) 
subjecting to the boundary conditions
(39)  
(40) 
where stands for the labels and , both giving the same KK expansions in this case without boundary potential. Results presented in this section correspond to in the gauge, see Appendix.
3.1 No brane potential case
Let us first review the case without any brane potential , as in the original UED model [6, 7]. In the model, there is only bulk potential (28), with terms being neglected. The solution to the equation of motion (30) is
(41) 
Note that obviously is the solution for other modes. In the original UED model, all the bulk fields are put the boundary condition with :
(42) 
which is trivially satisfied by the constant profile (41).
The KK equation corresponding to (20) is now
(43)  
(44) 
The boundary condition (42) simply reads
(45) 
for all , and .
There are three possible cases:

When or , general solutions are
(46) where or , respectively. This cannot satisfy the boundary condition (45).
To summarize, the KaluzaKlein mass for is given by
(51)  
(52) 
where we defined the unit KK mass .
3.2 Brane Potential as Perturbation: VEV Part
In Ref. [30], we have considered an extreme case where electroweak symmetry breaking is solely due to the brane potential. Here we concentrate on the opposite limit where brane potential is put as a small perturbation on the above UED model.
Let us start from the bulk potential (28) and treat the brane potential in (29) as a small perturbation of . Note that can be negative here, corresponding to the positive mass term in the brane potential, while is always positive by the starting assumption that the symmetry breaking in mainly generated by the bulk potential. We take hereafter. When we are interested solely in the brane mass term, we can take limit with fixed .
Firstly the equation of motion (30) is not altered. We seek for a solution of the type
(53) 
where is a small perturbation and is the expansion parameter eventually set to be unity. We put Eq. (53) into Eq. (30) to get
(54) 
The general solution is
(55) 
where we define . Note the mass dimensions and . We sometimes trade by in the following.
Noting that the brane potential itself is treated as a perturbation of , the boundary condition (31) reads:
(56) 
that is,
(57) 
When we assume conserved KK parity on our setup, namely and hence and , the solution to Eq. (57) simplifies to
(58) 
To summarize, when the brane potentials respect the KK parity the vev becomes KK parity even:
(59) 
Recall that in the perturbation potential can be negative while we take by construction.
3.3 Brane Potential as Perturbation: Quantum Part
We treat the brane potential as a perturbation on the eigenvalue problem (37) with the boundary condition (39). Recall that we are regarding as a small perturbation of :
(60)  
(61) 
We separate the KK wave function of the physical Higgs field into the unperturbed and perturbed parts
(62) 
where are explicitly given as the r.h.s. of Eqs. (48) and (50) with the unperturbed eigenvalues given by (51). Let us write the new perturbed eigenvalues as , with being given by r.h.s. of Eq. (51) and being real constant of mass dimension . The first order KK equation from Eq. (37) becomes
(63) 
The boundary condition (39) is now, to the first order,
(64) 
3.3.1 Zero Mode
Let us first consider the zero mode KK equation from Eq. (63)
(65) 
where constants and are given by Eq. (58) when there is the conserved KK parity. General solution is
(66) 
where and are integration constants of mass dimensions and , respectively.
Hereafter, we assume the conserved KK parity: and , for simplicity. The solution to the boundary condition (64) is
(67) 
The zero mode becomes KK parity even. The constant can be fixed by the normalization condition (21), or to the first order,
(68) 
so that
(69) 
Recall the mass dimensions , , , , and .
3.3.2 Even Modes
For even , the KK equation (63) reads
(70) 
Recall . The solution is
(71) 
where is given in Eq. (59).
The boundary condition (64) for even mode is now, to the first order,
(72) 
which gives and
(73) 
For , we get . As in the zero mode case, the constant can be fixed by the normalization condition
(74) 
3.3.3 Odd Modes
Finally we consider the odd modes. The KK equation reads
(75) 
and its general solution is
(76) 
The boundary condition (64) for odd mode is now, to the first order,
(77) 
which gives and again as in Eq. (73). From the normalization, the last constant is obtained as
(78) 
which is equal to the value of evenmode’s .
3.4 KK expansion of physical Higgs
To summarize, under the presence of small branelocalized potential, the KK expansion is given by