Bulk gauge and matter fields in nested warping: I. the formalism

The lack of evidence for a TeV-mass graviton has been construed as constricting the Randall-Sundrum model. However, a doubly-warped generalization naturally avoids such restrictions. We develop, here, the formalism for extension of the Standard Model gauge bosons and fermions into such a six-dimensional bulk. Apart from ameliorating the usual problems such as flavour-changing neutral currents, this model admits two very distinct phases, with their own unique phenomenologies.


I. INTRODUCTION
The spectacularly successful Standard Model (SM) has received a recent fillip with the discovery of the long-sought for Higgs boson [1,2]. Yet, certain questions remain unanswered. These pertain to the existence of Dark Matter, the origin of baryon asymmetry in the Universe, the existence of multiple generations of fermions, the hierarchy in fermion masses and mixing, and, last but not the least, the stability of the Higgs sector under quantum corrections.
Over the years, several attempts have been made to answer these questions, albeit only with partial success. One such stream of thought envisages a world in more than three space dimensions as a possible panacea to some of the ills of the SM, and, in this paper, we concentrate on this possibility.
While such theories were first proposed nearly a century ago [3][4][5] in the quest to unify electromagnetism and gravity, the early efforts were quickly shown to lead to a dead end and were abandoned. The situation changed with the introduction of String Theory as a quantum theory of gravity as well as a possible ultraviolet completion of the SM. With the theory defined, of necessity, in at least ten dimensions, compactification of the extra dimensions is paramount before it can deemed a theory of the observed world. With the compactification scale being close to the Planck scale in most early constructions, understandably the new dimensions played virtually no role in low-energy physics. However, warped compactification [6,7], wherein the SM fields were confined to the usual (3 + 1) dimensions with only gravity being allowed to propagate in the (five-dimensional) bulk, led not only to a "resolution" of the hierarchy problem, but also to interesting consequences at colliders owing to distinct O( TeV) resonances in the form a Kaluza-Klein (KK) excitations of the graviton.
The last feature moved both the ATLAS [8] and CMS [9] experiments to investigate the existence of such Randall-Sundrum (RS) graviton resonances, especially through the dilepton and diphoton decay channels. In particular, the ATLAS experiment ruled out graviton masses below 1.03 (2.23) TeV at 95% C.L. with the lower bound being dependent on the ratio of the five-dimensional curvature and the fundamental mass scale. This ratio is constrained, on the upper side, by the applicability of a semiclassical treatment (only recourse available in the absence of a full quantum theory of gravity) and, on the lower, by the undesirability of fine-tuning. For reasonable values of this ratio, the mass of the first graviton excitation should, preferably, be a few times that of the Higgs boson; and certainly no higher than a few TeVs. Thus, the continued absence of any such resonance at the TeV scale begins to call into question the validity of this scenario as a cure for the Higgs mass stabilization problem. However, it should be realized that the RS model is only the simplest of possible warped world scenarios. In particular, there is no reason that there should be only one such extra dimension [10][11][12][13][14][15][16]. In particular, for a scenario with double (or more) warping [16], it was shown recently that the aforementioned ATLAS bounds are naturally evaded [17].
Although the mechanism for the formation of branes and the localization of fermions thereon is well understood, it is interesting to consider allowing them to propagate in the full six dimensions. The corresponding flat space theories have several interesting consequences. For example, the analogue of Witten anomaly cancellation leads to a prediction of the number of chiral generations [18], while suppressing the proton decay rate to below the current constraints [19]. Furthermore, some of these constructions [20] naturally lead to a small cosmological constant.
The rich collider phenomenology [21][22][23][24], apart from the existence of a viable cold dark matter candidate [25,26] renders these scenarios phenomenologically attractive. On the other hand, with a double tower of KK excitations, the quantum corrections to the SM amplitudes as well as to electroweak precision tests are potentially large, calling into question the consistency with low-energy phenomenology. However, as Ref. [27] demonstrated for a five-dimensional theory, it is possible to suppress the coupling between zero modes of the SM fields and the KK towers. This is of particular importance in the context of the aforementioned quantum corrections. Thus, it is of interest to investigate whether considering a warped space would allow us to preserve some of the advantages of going into six dimensions while simultaneously protecting us from the pitfalls. This paper is the first step towards this goal, and we set up the entire formulation here while commenting on some of the consequences. While Ref. [28] did consider bulk SM fields in such a geometry, the analysis therein had taken recourse to an approximation of the metric, thereby leading to a significant simplification of the equations of motion. However, as was demonstrated in Ref. [17], the said approximation, apart from being untenable close to the brane we live on, led to a drastic change in the form of the graviton wavefunctions (and, hence, their couplings). As we shall show, much the same happens for bulk gauge bosons as well, leading to very interesting phenomenological consequences. The detailed phenomenology would be presented in subsequent papers.
The rest of the article is constructed as follows. To begin with, we present a very brief review of the doubly warped space. Sections III&IV discuss, respectively, the bulk fermions and gauge bosons in this theory, without taking into consideration the spontaneous breaking of the gauge symmetry which, in turn, is discussed in Sec.V. The interactions are delineated in Sec.VI and the Feynman rules listed. Finally, we conclude in Sec.VII.

II. BRIEF REVIEW: DOUBLE WARPING IN SIX DIMENSIONS
We consider a compactified six-dimensional space-time with successive warpings and Z 2 orbifolding in each of the two extra dimensions, viz.
Demanding that a four-dimensional (x µ ) Lorentz symmetry be exhibited requires the line element to be given by [16] where the compact directions are represented by the dimensionless coordinates x 4,5 ∈ [0, π] with R y and r z being the corresponding moduli. It is interesting to examine the rationale for the two orbifoldings. A nontrivial a(x 4 ), when accompanied by compactification, demands (as in the RS case), the orbifolding in the x 4 -direction. Furthermore, it necessitates the presence of localized energy densities at the orbifold fixed points, and in the present case, these appear in the form of tensions associated with the two end-of-the-world 4-branes at x 4 = 0, π. On the other hand, even without any orbifolding in the x 5 -direction, a nontrivial b(x 5 ) for a compactified x 5 automatically requires that a 4-brane should exist at x 5 = π, whereas none needs to exist at x 5 = 0. The situation changes though if one wishes to introduce such a brane. While the latter could exist even in the absence of such an orbifolding, it would be free to traverse in the x 5 -direction in the absence of a constraining potential. Thus, if such branes are to be introduced, it is easiest to do so if the second S 1 is orbifolded too.
The total bulk-brane action for the six dimensional space time is, then, given by where Λ is the (six dimensional) bulk cosmological constant and M 6 is the fundamental scale (quantum gravity scale) in six dimensions. The five-dimensional metrics in S 5 are those induced on the appropriate 4-branes which lend a rectangular box shape to the space.
If the bulk cosmological constant Λ is negative, the solutions for the 6-dimensional Einstein field equations are given by [16] a( with the junction conditions specifying the brane tensions. The smoothness of the warp factor at x 5 = 0 obviates the necessity for a V 3 (x 4 ), while the fixed point at x 5 = π requires a negative tension, viz.
In contrast, the two 4-branes sitting at x 4 = 0 and x 4 = π require equal and opposite energy densities, just as in the RS case. However, the x 5 -warping dictates that, rather than being constants, these energy densities must be Such tensions could originate from different kinds of physics [16] , and we shall briefly touch upon this issue in Sec.V.
The (derived) 4-dimensional Planck scale can be related to the fundamental scale M through The point in this 2-dimensional (x 4 -x 5 ) plane associated with the lowest energy scale is given by x 4 = π, x 5 = 0.
Assuming us to be located at this juncture immediately gives the required hierarchy factor (i.e., the mass rescaling due to warping) to be w = e −cπ cosh kπ .
For the large w that we need, this equation, alongwith the relation between c and k (eqn.3) demands that, unless there is a very large hierarchy between the moduli, the warping is substantial in only one of the two directions, and rather subdominant in the other. In other words, we can have either (i) a large (∼ 10) value for k accompanied by an infinitesimally small c or (ii) a large (∼ 10) value for c with a moderately small ( < ∼ 0.3) k.
In summary, we are dealing with a brane world which is doubly warped, with the warping being large along one direction and small in the other. The very structure of the theory typically requires a small hierarchy between the two moduli, both of which remain comparable to the fundamental length scale in the theory. It should be realized that the two branches, namely (i) a large k and a small c, or (ii) a large c and a small k, are fundamentally different.

III. THE FERMIONS
As is well-known, fermions, of necessity, are defined as representations of the Lorentz algebra, as applicable to the tangent space. Hence, we begin by briefly reviewing the construction in flat space before embarking on the more germane issue of the warped space.

A. Flat six-dimensions
In the current case, the spin-1/2 representation is defined by six 8 × 8 matrices Γ a , satisfying a Clifford algebra where η ab = diag(−1, +1, +1, +1, +1, +1). We choose to work with a particular representation of the algebra defined by Γμ = γμ ⊗ σ 3 Γ 4 = 1 ⊗ σ 1 where γμ (withμ denoting the subspace of the flat space) are the four-dimensional Dirac matrices (in the Weyl representation) and γ 5 (Γ 7 ) is the parity operator in four (six) dimensions. The direct product is defined in a trivial sense and, for example, It is straightforward to construct the Dirac spinor Ψ(xμ, x 4 , x 5 ) satisfying the flat space Dirac equation The representation of the Lorentz generators given by Σ ab = i 2 [Γ a , Γ b ], viz.
is, of course, reducible, as is the case for all even dimensions. In other words, this space admits chiral representations Ψ ± and thus, we may directly export the SM quantum number assignments. This is quite unlike the five-dimensional case. Of course, on compactification, each such six-dimensional chiral representation would, in general, yield low-lying states carrying either value for the four-dimensional chirality unless boundary conditions (such as those pertaining to the orbifold fixed points) prevent this. Before we consider such details, we need to set up the Dirac equation in the six-dimensional space which we do next.

B. Fermions in the warped space
To define these, we need to consider the sechsbeins (namely, the transformations to the tangent space) e a M which satisfy the conditions Denoting the inverse sechsbeins by E M a , we define the spin connections ω bcM through the covariant derivatives of E M a , viz., For the metric of eqn.(1), the only nontrivial components of the spin connections are given by where primes (dots) denote derivatives with respect to x 4 (x 5 ). The Dirac Lagrangian in the warped geometry is, then, given by for the positive chirality field Ψ + and, analogously, for Ψ − as well. The corresponding equation of motion is Anticipating Kaluza-Klein reduction, we write the positive chirality Weyl spinor as with Here, F n,p r (x 4 , x 5 ) encapsulate the wavefunction dependences on the extra dimensions, with the subscripts (l, r) referring to the (four-dimensional) chirality of the putative four-dimensional fields ψ n,p l,r whereas the factor R y r z (ensuring the correct mass dimension) would have arisen if the compactified directions were flat instead. The Dirac equation then reduces to Expectedly, the two Weyl fields ψ n,p l/r (at each level) combine to give a Dirac fermion, and this results in Effecting a separation of variables and writing F n,p l/r (x 4 , where E p is the eigenvalue of the differential operator. The equations for the x 4 -modes differ slightly for the two chiralities, and actually couple them, viz.
To solve the x 4 equations, it is useful to define the operators and, then, we have Solving these, we get f n,p l/r (x 4 ) in terms of Bessel functions, namely where c i s are constants of integration. To obtain the physical fields, we need to impose the chiral projections and the boundary conditions and the latter we choose to be a mixture of Neumann and Dirichlet boundary conditions, viz. ∂ 5 Ψ +l | x5=0 = 0 and Ψ +r | x5=0 = 0. Thus, the six dimensional chiral fermion Ψ + could be written as This, obviously, ensures that only the component Ψ +l possesses the lowest (massless) mode on the 3-brane at (x 4 , x 5 ) = (0, π), a feature that is a phenomenological necessity once we identify Ψ + as the SU (2) doublet field.

C. KK masses for the fermions
As we shall see in the next section, the issue of spontaneous symmetry breaking is a tricky one, as the inclusion of a bulk Higgs has been shown [27] to resuscitate the hierarchy problem. Localizing the higgs onto a brane obviates the problem, though. Given this, we continue under the assumption of the six-dimensional fields to be strictly massless and, thus, the only contributions to the masses of the four-dimensional components would be that due to the compactification. Incorporating the effect of the Higgs field is a straightforward exercise, and would be undertaken in Sec.V. In any case, with the compactification scale being much larger than the electroweak scale, the Higgs contribution would be of little importance to any but the lowest mode and can be treated as a perturbation.
Determining the spectrum, as usual, needs the imposition of the boundary conditions. While we already have done so for the x 5 -modes, the x 4 -component is still unrestricted. Noting that the orbifolding demands that the fermion wavefunctions be even in the x 4 -direction, these must satisfy Rather than attempt to solve for the above for the most general choice of c and k, we restrict ourselves to the two cases that are of relevance in resolving the hierarchy problem, viz. small k and large c or large k and small c.

Small k and large c
The aforementioned boundary conditions, respectively, give and −c 2 As e −cπ is negligibly small, the corresponding Y ν l p (x np e −cπ ) is very large. Consequently, in this regime, the two equations above are simultaneously satisfied only if thereby determining the quantized values for x np (for a given ν l p determined, in turn, by eqns.(22&24)). For the right-chiral fields, relations analogous to eqns.(27&28) would obtain, but with ν r p instead. It is easy to see that the corresponding set of x np s identically match those in eqn. (29), but for the zero mode.

Large k and small c
Since c is now almost infinitesimally small, a(x 4 ) ≈ 1 is a very good approximation and eq.(23) can be simplifield down to This, of course, yields plane wave solutions, which is as expected since such a small warping means that the space is essentially flat. Hence, the spectrum is given by As R y is very small, we could as well neglect the n = 0 modes in any discussion of TeV scale physics. In other words, for all practical purposes, the fermions act as if they are confined onto a brane.

IV. THE GAUGE BOSONS
To begin with, we examine the mass spectrum of the gauge bosons postponing discussion of all interactions until later. Hence, it is convenient to consider only a U (1) theory in the six dimensional bulk. Similarly, rather than concerning ourselves with the issue of symmetry breaking in the bulk, we would introduce an explicit mass term so as to understand the consequences of a bulk mass term.
The gauge boson Lagrangian is given by While the structure of the gauge-fixing term is, a priori, undetermined, it is often useful to restrict ourselves to a covariant choice, so as to manifestly respect the symmetries of the theory, which, at first sight, would seem to be diffeomorphism invariance. Note, however, that the presence of the boundary branes (and the orbifolding), actually reduces the symmetry to only a four-dimensional general coordinate invariance. Hence, we introduce a curved-space analog of the oft-used generalized R ζ gauge (at least in the case of flat extra dimensions [29]) which, while respecting only the four-dimensional general coordinate invariance, also serves to eliminate the cumbersome kinetic mixing terms between A µ and A 4,5 . To be specific, we have This does not exhaust all of the gauge symmetries. On compactification down to four dimensions, the components thereby eliminating the kinetic mixing between A µ and A 5 . Adopting, for the time, the Minkowski metric, the quadratic term for the vector field is now given by while for the adjoint scalar, with a field redefinitionÃ 5 ≡ A 5 Ry rz , it is and reminiscent of the action for a scalar field. To this, we may add a mass term (presumably originating from spontaneous symmetry breaking in the bulk), viz.
We may now rexpress the field in terms of the eigenstates of the extra-dimensional parts of the differential operator corresponding to the equation of motion. In this case, these are the analogues of the "plane wave" solutions and given by with the components satisfying the orthogonality relations The corresponding equations of motion for the vector modes are whose solutions could be written as where c 1,2 (we explicitly retain both as this is useful in studying the boundary conditions) and c np are arbitrary constants whileB p and B np provide the normalization. The associated Legendre functions, appearing in the description of the fermions, or at any stage of the five-dimensional theory, is reminiscent of the x 5 -dependence of the graviton wavefunctions [17] and are a feature of the multiply warped theory.
Similarly, the equations of motion for the adjoint scalars are seen to be leading to Once again, d 1,2 and d np are constants of integration, whileD p and D np provide normalizations. With these conditions in place, the quadratic part of the Lagrangian for the vector fields can be expressed in terms of the KK-towers as while, for the adjoint scalars, we have

A. Gauge and Adjoint Scalar Masses
Our aim, now, is to compute the allowed values for m np andm np . In each case, one must first find the x 5 -equation eigenvalues (m p andm p respectively) using the boundary conditions for the corresponding wavefunctions and, then, find the desired spectrum in terms of these. For ease of appreciation, we perform the exercise in the reverse order.
We first establish the general conditions and then examine the situation for the two particular cases of interest (in terms of the relative sizes of k and c).
1. Boundary conditions for ηn,p(x4) and η S n,p (x4) As of now, we have not considered the existence of any localized fields 1 . Consequently, the wavefunctions η n,p (x 4 ) must be differentiable, and especially at the ends of the world. This implies that for each of x 4 = 0, π. Here, we have defined Once ν p is known, the two conditions of eqn.(42), together, determine η n,p (x 4 ) as well as serve to quantize α np (and, hence, m np ).
Similarly, since η S np (x 4 ) is an even function, its derivative has to vanish at both the fixed points (x 4 = 0, π) and this gives whereᾱ Again, onceν n is known, the two conditions in eqn.(44) together determine η S n,p (x 4 ) as well as quantize the mass.
or, in other words, An analogous condition would be obtained for the derivative at x 5 = π, but that is best analyzed separately for small and large k, which we come to later.
We have already seen that since η S n,p (x 4 ) are even functions, they need to satisfy Neumann boundary conditions at x 4 = 0, π. Thus, the absence of massless adjoint scalars necessitates that we impose Dirichlet boundary condition on χ S p (x 5 ), namely χ S p (0) = 0 = χ S p (π). This, then, implies that .
In other words, the eigenvalue spectrum is given by a condition that, again, is best analyzed separately for large and small k.

Small k and large c
Since the Legendre functions are well-behaved in this domain, we could use the Neumann boundary conditions (namely, the vanishing of their derivatives) for χ p (x 5 ) at x 5 = π in a straightforward fashion, and this dictates where τ π ≡ tanh(k π). This equation has to be solved numerically to obtain the discrete set of values allowed to ν p .
For a given ν p , to solve for m np , we need to consider both of eqs.(42). Note that, for the (k, c) values under discussion, the combination R −1 y e −cπ roughly gives the electroweak scale [17]. Thus, if m np are to be important in low-energy phenomenology, α np < ∼ O(1). On the other hand, with e c π being very large, the argument of the Bessel functions essentially vanishes at x 4 = 0. This implies that c np ≈ 0, and considering the boundary condition at Solving this, for a given ν p , would lead to the gauge boson KK tower.
Let us, now, turn our attention to the adjoint scalars. Again, since the Legendre functions are well behaved, values ofm p can be found by solving the equation Since the ratio P with the consequence that the first nonzero component of the adjoint scalar field would be A (0,1) 5 (x µ ).

Large k and small c
Denoting τ = tanh(k x 5 ), we may reexpress the wavefunctions χ p (x 5 ) as .
For x 5 = π, we are very close to τ = 1. Defining f (τ ) ≡ dχ p /dτ , we then have which automatically vanishes for the zero mode as the corresponding ν p = 1/2. For the higher modes to satisfy the Dirichlet boundary condition, we require cot θ p = 0, or in other words ν p = 2 p − 1/2, where p ∈ Z + .
Since we are now in the very small c regime, the warping in the x 4 direction is virtually nonexistent, and the modes with n = 0 will have a mass ∼ n R −1 y and, hence, will decouple from the theory. This is exactly analogous to the case for the corresponding graviton modes [17]. For the large k case, even if M is of the order of the natural cutoff, the quantities u, v do not differ substantially from their M = 0 limit values of 3 and 1 respectively. Naively, this would lead one to presume that the existence of even a seemingly large M would not alter the low-lying part of the KK-tower spectrum to any significant degree. However, as has been shown in [27], this argument is fraught with danger, and much the same follows here. As can be appreciated easily, with M being the bulk mass term, the mass of any given mode m np > M . Indeed, in the absence of warping, we would expect m 00 = M and m 2 np = M 2 + n 2 /R 2 y + p 2 /r 2 z . With M −1 being of the same order as the compactification radii, the mass of the first excitation, in the absence of warping, would be of similar order as that of the zero mode. Thus, if warping is supposed to bring down the mass of the zero mode, as perceived in four-dimensions, from M to M Z (Z-boson mass), it would also, typically, bring down the first excitation to well below a TeV, resulting in severe phenomenological contradictions. In other words, a bulk M , if present, would need to be much smaller than the compactification scale, thereby bringing back the hierarchy problem in a new guise. A similar argument holds for the large c case too.

The boundary conditions for the adjoint scalars simplifies to
All of the analysis presented above carries through for a nonabelian theory as well. The additional features are the gauge boson self-interactions (a discussion of which we postpone until Sec.VI) and ghost fields. The latter, we consider next.

B. Ghosts
As is well-known, a covariant gauge fixing condition for a nonabelian theory always gives rise to ghost fields.
Accounting for the spontaneous symmetry breaking, the gauge-fixing term can be written as with T a ij being the Yang Mills group generators in the representation of the scalar fields φ i . The latter are decomposed into the vacuum values φ 0i and the perturbations χ i around it (φ i = φ 0i + χ i ). Evidently, the gauge boson mass matrix is given by Note that the scalars φ i need not be bulk fields, but may be confined to branes (as we shall argue for in Sec.V), with the appropriate delta-functions being included accordingly.

Under a gauge transformation,
where f abc are the structure constants. Using the notations of Ref. [30], the corresponding Faddev-Popov (FP) where D ac M = ∂ M δ ac + gf abc A b M . Though they are important in their own right, since a study of the higgs interactions does not constitute the main aim of this paper, we neglect the O(χ) term. Writing the determinant as a gaussian integral over an anticommuting scalar (θ) in the adjoint representation of the gauge group yields the Lagrangian for the ghost field, namely The ghost kinetic term, then, is Decomposing, in anticipation, as it is easy to see that η n,p and and χ p satisfy the same equations of motion (and orthonormality property) as the corresponding wavefunctions for the vector modes (see eqns.(37& 38)). In other words, not only are their masses identical to those for the corresponding vector modes, the wavefunction modes are the same too. This, of course, is as expected.

V. HIGGS
In the preceding section, we had, for the sole purpose of determining the spectrum, included an explicit mass term for the gauge boson, without ascribing any dynamic origin to it. The simplest mechanism to generate masses while preserving a gauge symmetry, of course, is to introduce a Higgs field and effect a spontaneous breaking of the symmetry. The most straightforward, and seemingly natural way, to do so would be to consider a bulk Higgs field.
However, this immediately leads to a problem (also seen in the context of bulk SM fields in the RS scenario [27]) in that the consequent masses of the first excited modes of the gauge bosons typically turn out to be well below 2 1 TeV, in stark contradiction to the direct bounds from gauge boson (most typically, generic Z ′ ) searches.
The alternative, then, is to localize the Higgs on one of the 4-branes, or, perhaps even to a 3-brane at one of the junctions. A junction localized higgs would lead to a gauge field equation of motion analogous to a two dimensional Poisson equation with a point source term. With the rotational symmetry being absent, a solution of such an equation consistent with the boundary conditions, though, is a complicated one.

A. 3-brane localization
Notwithstanding the anticipated technical problems, we begin by considering a 3-brane localized Higgs with a Lagrangian given by Rewriting Φ(x µ ) = (ṽ + h(x µ )/ (a(π)b(0)) allows us to canonically quantize the Higgs field, with the Higgs mass being given by Instead of solving the consequent equations of motion for the gauge fields with the delta function sources included, we consider the latter to be localized perturbations to the system with the symmetry intact. This is a valid approximation as the effect of the spontaneous breaking of the gauge symetry is parametrized by v; since it is suppressed down to the electroweak scale, its contribution to the KK-gauge boson masses would be small compared to those due to compactification (even after the warping). To the first order, then, the mass spectrum that we computed in section( IV A) do not get disturbed in the bulk, but for a small correction 3 due to symmetry breaking in the brane.
Similarly, the wavefunction profile would change only close to the brane. With the new contribution to the gauge boson mass term being it could be expressed in terms of the component fields (see eqn.36) as It comes as no surprise that this term mixes the KK modes, for this is a generic feature of brane localized Higgs fields. To the zeroth order, the gauge wavefunctions remain unchanged and have no discontinuities on the branes.
Of course, to obtain the physical gauge boson states and their masses, one would need to diagaonalize the mass matrix including the terms of eqn.(56). While a closed form solution is not obvious, inspiration may be taken from the see-saw mechanism; although the hierarchy between the compactification and the Higgs contributions is not as large, it is still sufficiently large for such approximate solutions to be valid. And while the absolute contribution of the Higgs grows for the higher-p modes, primarily because χ p are localized near z = 0, this growth is relatively slow and consequently the ratio m s.b. /m np becomes smaller. Thus, neglecting the Higgs contribution for the higher modes is progressiely a better approximation, and obtaining good estimates of the gauge boson masses and eigenstates is a relatively straightforward task.
Going beyond the first order approximation renders the algebra to be rather cumbersome, without providing us any real insight. To gain the latter, we would need to take recourse to some approximation. One such could be a smearing of the delta-function localized Higgs field.

B. 4-brane localization
Since we have chosen to work in the regime wherein the warping in the x 5 -direction is small, we consider a configuration wherein the factor δ(x 5 ) in eqn.(54) is smeared onto a nonsingular but localized function. This, of course, is equivalent to considering the Higgs-field to be localized onto the 4-brane at x 4 = π (rather than on the 3-brane at the junction), with a further concentration of its wavefunction close to x 5 = 0. The relevant part of the symmetry breaking Lagrangian can now be parametrized as with M (x 5 ) encapsulating the x 5 -dependent profile of the Higgs vacuum expectation value. Assuming, for the moment, that M = m/ b(x 5 ) where m is a constant (we shall comment later on the origin of such a profile), the equations of motion for the gauge boson modes are altered from those in eqn. (38) to Indeed, the particular form of M (x 5 ) was chosen so as to render the equation of motion separable and, hence, easily solvable. The form of the second equation above is evidently the same as that in eqn. (38) but for the bulk mass term M in the latter. As for the first, for a given m np it differs from its predecessor only as far as the boundary term is concerned. Consequently, the bulk solutions are exactly the same as in eqn. (39) but with u = 3. The boundary conditions on χ p (x 5 ) remain exactly the same as before (see Sec.IV A 2), namely and, thus, the m p -spectrum is unchanged. To obtain the full spectrum, we must consider the boundary conditions on η n,p , and one of these now changes to accommodate the brane localized term, to wit, Since m p=0 = 0, we have, for the modes η n0 , where, as before, α n0 ≡ m n0 R y /(c e cπ ). Since the lightest mass mode is to be identified with the W/Z bosons, we have x 00 ∼ m 00 /(c v) ≪ 1 (as c ∼ 10). Expanding the Bessel functions, we obtain Clearly, for the W boson, m 2 = g 2 2 /R 2 y , whereas for the Z boson, m 2 = (g 2 2 + g 2 1 )/R 2 y , with g 2,1 being the weak and hypercharge coupling constants respectively.
In essence, the new boundary condition on η n,p manifests itself in the change in spectral condition of gauge fields through α np J νp−1/2 (α np ) as distinct from eqn.(49). With there being no singularities in the x 5 -direction, all the χ p modes remain unchanged (in particular, χ 0 is flat), and all effects of the brane localized mass term for the gauge bosons manifest themselves in altering the quantized values of α np , and, hence, η n,p . Quite in parallel to the RS case [31], the change in the wavefunction for the zero modes (W ±(0,0) , Z (0,0) ) is concentrated close to the brane and, in magnitude, restricted to . Away from the brane, even this relative change falls off exponentially.
As can be gleaned from the discussions in this section, a generic brane localized higgs profile would lead to equations for the gauge bosons that do not admit simple closed form solutions. While the particular choice of φ = v/ b(x 5 ) may seem an ad hoc one, we end this section delineating a mechanism to achieve this. The effective potential for a scalar field φ localized on the 4-brane at x 4 = π could be written as where V (φ) is the potential term appearing in the (flat space) Lagrangian. To find what potential would give rise to the required form for φ , we may treat V eff as the Lagrangian for a one-dimensional particle. Varying with respect to φ, we get Demanding that the solution be of the form φ ∝ 1/ b(x 5 ), we get where d = sech(kπ) and Λ is an arbitrary mass parameter (presumably of the order of compactification scale) that sets the scale for φ .

VI. INTERACTIONS
A. Gauge-fermion The relevant piece of Dirac Lagrangian for the positive chirality six-dimensional field is given by where the group representation has been suppressed. (A similar account holds for the negative chirality field as well.) Writing the term above in its component form, we have with the four dimensional charges being given by It is evident that, for the gauge boson zero mode, we have This universal coupling of the zero mode is, of course, mandated by gauge invariance.
For the adjoint scalar, we start from Redefining (as before) where the four dimensional coupling constants are given by Concentrating on the small k scenario (Table I) universal. In particular, the coupling to the first excitation is enhanced compared to that for the zero mode, while those to the higher ones are suppressed. Furthermore, the enhancement to the (1, 0) mode is only slightly smaller than that for the corresponding five-dimensional theory, with this effect having only a marginal dependence on the value of k. This is not surprising since the leading dependence of the couplings on k is common to the ground and the (1, 0) state. On the other hand, if the extent of warping is kept constant, the value of c progressively decreases from the RS value as k increases from zero. Consequently, C np decreases, although only slowly. Although a large decrease in C 10 is possible (thereby making these bosons less accessible to collider searches etc.), a substantial decrease would require a relatively large k. However, as already mentioned, if c and k are to be of the same order, a very large hierarchy between R y and r z would be required, thereby bringing back the hierarchy problem in a different guise. In other words, the aesthetically pleasant region of the parameter space would lead to gauge bosons discoverable in the next run of the LHC.
The situation is very different in the large k regime. Since the warping in the x 4 -direction is very small, the n = 0 modes are all superheavy and decouple from the TeV scale physics. In other words, just like the situation for the gravitons [17], essentially only one tower of gauge bosons (or, fermions for that matter) remains. Moreover, the couplings of the KK-tower are enhanced to nonperturbative levels. This is demonstrated, for a moderate hierarchy, in further, with the only way out of this eventuality being the introduction of a big hierarchy between the moduli.
Indeed, one had seen the germ of this problem even in the graviton sector [17]. Thus, bulk fermions or gauge bosons in the large k regime is ill-suited for a perturbative treatment.

B. Triple gauge boson
The trilinear self interaction term can be written as where f abc are the structure constants. The six-dimensional Ward identities ensure that the coupling g YM equals the six-dimensional gauge-fermion coupling even after quantum corrections are included. On redefining A 5 and rewriting in terms of the four dimensional fields, we have {ni,pi} η µα η νρ ∂ µ A a(n1,p1) {ni,pi} η µν A a(n1,p1) {ni,pi} η µν ∂ µÃ a(n1,p1) 5 where the coupling constants are defined through For the three vector vertex, clearly if one of them is a zero-mode, the other two must be identical (so as to preserve the six-dimensional momentum). Similarly, the vector zero-mode couples only to a pair of identical scalars. Finally, for either case, the coupling is the same as that in eqn.(63). All of the above are, of course, consequences of gauge invariance.

C. Quartic gauge interaction
The corresponding term in the Lagrangian is Once again, redefining A 5 , and reexpressing in terms of four dimensional fields, we have with the coupling constants being defined through Again, for the zero mode vectors, the Ward identity is satisfied.

VII. SUMMARY
While the negative results for graviton resonance searches by the ATLAS [8] and CMS [9] collaborations have caused a fair bit of tension for the Randall-Sundrum scenario, a six-dimensional analogue with a double warping [16] has been seen to be very consistent with the experimental results [17]. Such scenarios are of interest in their own right as they could, for example, be an intermediate step in the compactification down from a theory in higher dimensions.
Moreover, the fundamental scale in such theories are naturally lower than the Planck-scale, and this could play a significant role in the context of gauge unification. But, most interestingly, it provides a tunable parameter that smoothly takes one from a nearly-conformal theory to another that is a large departure from one, with the added feature that both the ends provide a resolution of the hierarchy problem (although this is not apparent in the interim regime).
Just as the RS model would, generically, admit operators that lead to unsuppressed flavour changing neutral currents and/or proton decay, so would the model considered in Refs. [16,17]. On a different vein, the exact cutoff scale of this theory (normally described as the scale at which the loop contributions are to be cutoff) needs to be identified too. It has been argued that, within the five-dimensional context, the addition of the Planck-brane and/or the TeV-brane allows a holographic interpretation [32], with the former acting as a regulator leading to an ultraviolet 4 cutoff ( < ∼ r −1 c ) on the corresponding CFT [33][34][35]. It has been demonstrated that, for RS-like theories with gauge fields extended in to the warped bulk, this is indeed so [36][37][38]. Thus, the obvious way to ameliorate flavour changing currents is to allow the fermions as well as the gauge fields to propagate into the six dimensional bulk, for now the higher dimensional operators get suppressed by the bulk cutoff of the theory, which is higher than the GUT scale.
Furthermore, a six-dimensional theory allows one to make predictions about the number of chiral generations in the theory.
In this paper, we have derived the wave profiles for these fields and computed the master formulae to calculate their spectra. It is seen that, of the two branches allowed to the theory by the resolution of the hierarchy problem, the one close to a conformally flat space leads to the collapse of the double-tower (for both the gauge fields and the fermions) to a single tower each (the other excitations are too heavy and decouple from the low-energy theory).
However, the remaining KK-excitations of the gauge bosons couple very strongly, thereby invalidating a perturbative treatment. The other branch, which cannot be thought of as a perturbation of the AdS 6 theory, leads to very interesting phenomenology. This is of paramount interest in the context of electroweak precision tests, rare decays and renormalization group evolutions [39].
As for the Higgs field, just as in the five-dimensional theory, putting the Higgs field into the bulk either brings back the hierarchy problem, or renders the masses of the gauge boson KK-tower unnacceptably low. Thus, it is wise to localize the Higgs on to a 3-brane. This, though, has the unfortunate consequence of making the combination of the equations of motion and the boundary condition too complicated to permit an easy understanding of the dynamics.
On the other hand, if we localize the Higgs onto a 4-brane (located at y = π and z = 0), and work in the large c regime, we see that the Higgs vacuum expectation value does get warped down to the electroweak scale. With the consequent brane localized contribution being small, it can be treated as a perturbation, and the consequent shifts in the spectrum as well as the wavefunction profiles can be calculated. It is interesting to note that, although small, the brane localized Higgs mixes the KK states with possible phenomenolgical ramifications. These issues are under