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Warping, Extra Dimensions and a Slice of AdS_d

Inspired by the Randall-Sundrum (RS) framework we consider a number of phenomenologically relevant model building questions on a slice of compactified AdS_d for d>5. Such spaces are interesting as they enable one to realize the weak scale via warping. We perform the Kaluza-Klein (KK) reduction for gravitons and bulk vectors in these spaces and for the case of AdS_6 consider the KK spectrum of gauge-scalars. We further obtain the KK towers for bulk fermions on a slice of AdS_7 and AdS_9 and show that the RS approach to flavor generalizes to these spaces with the localization of chiral zero mode fermions controlled by their bulk Dirac mass parameters. However for the phenomenologically interesting case where the transverse radius is R^{-1}\sim TeV we show that bulk Standard Model fields are not viable due to a resulting volume suppression of the gauge coupling constants. A similar suppression occurs for the case of UV localization. Thus it seems that the Standard Model fields should be confined to the infrared brane in such spaces. Sterile fields and extended gauge sectors may propagate in the bulk with the gauge-coupling volume suppression experienced by the latter motivating a weak coupling to Standard Model fields. We also discuss some issues regarding the effective 4D theory description in these spaces.


Introduction
Though a remarkably successful theory, the Standard Model (SM) of particle physics is almost certainly incomplete. There are two main reasons, one theoretical the other experimental, that lead us to suspect that new physics will appear at the TeV scale. The direct sensitivity of the Higgs mass to ultraviolet (UV) effects (the hierarchy problem) makes it difficult to take the SM seriously as a successful theory beyond the TeV scale. A likely scenario is that some mechanism is responsible for stabilizing the weak scale and the expectation is that this mechanism will manifest itself in the form of new particles with ∼ TeV masses. On the experimental side there is now a growing body of evidence suggesting that the matter density of the universe is dominated by an unknown particle or particles, referred to as dark matter (DM). Curiously the requisite behaviour of the DM can be obtained by a ∼ 10 2 GeV particle which interacts with weak scale strength with the SM fields.
One promising possibility is that Nature is supersymmetric, in which case there should exist ∼ TeV scale particles whose UV sensitive contribution to the Higgs mass via loop effects approximately cancel the UV sensitive contributions of SM particles. Supersymmetric extensions of the SM can also motivate the unknown DM density as the imposition of an extended symmetry (R-parity) on supersymmetric models renders the lightest new particle absolutely stable. Furthermore the coupling constant relations dictated by supersymmetry mandate weak scale interaction strengths for some of the supersymmetric particles.
An alternative solution to the hierarchy problem can occur if Nature possesses extra spatial dimensions. In particular if nature admits a non-factorizable geometry the weak scale may be realized as a red-shifted, or warped, incarnation of Planck scale sized input parameters [1]. In this case the break down of the SM at the TeV scale would be manifest by the existence of TeV scale Kaluza-Klein (KK) excitations of the graviton, and of the SM fields, if the latter propagate in the bulk. Interestingly if SM fermions propagate in the bulk theories of flavor can also be constructed by employing the wave function overlap of the SM fermions in the extra space [2,3]. The existence of extra spatial dimensions can also motivate a DM candidate if a subgroup of an isometry of the extra space is conserved in the low energy theory. This is precisely what happens in models with universal extra dimensions (UED models) [4] where all the SM fields propagate in the bulk of an extended spacetime and a remnant discrete symmetry, known as KK parity 1 , renders the lightest KK particle a good DM candidate [5].
If the RS scenario is realized in nature it is possible that additional spatial dimensions exist beyond the warped extra dimension. As discussed in [7], from a string theoretic perspective one may obtain the AdS 5 RS model from a stack of parallel D3 branes in type-IIB string theory [8], though additional compact dimensions will be present. Interestingly one may realize AdS 7 with additional compact dimensions from a stack of parallel M5 branes in M theory [8]. It has also been noted that AdS 6 (with additional compact dimensions) is the near-horizon limit of the Type I' D4-D8 brane system [9]. It is important to ask how these extra dimensions, if present, may modify our understanding of the RS model and what new features may emerge.
In a recent work we have considered the generalization of the RS model to the higher dimensional space AdS 5 × T 2 [11]. In that work we were primarily motivated by the observation that UED models and RS models are, in some sense, complementary. RS models motivate the weak/Planck hierarchy, the existence of TeV scaled particles (in the form of KK excitations) and can shed light on the flavor puzzle. However the warped geometry breaks translational invariance along the extra dimension in a maximal fashion so that the KK particles are not stable and do not admit a good DM candidate. UED models on the other hand motivate a stable DM candidate but do not shed any light on the weak/Planck hierarchy, nor do they provide any insight into the flavor structure of the SM 2 . The presence of the UED DM candidate at the weak scale is also not motivated within UED models as one obtains the TeV scale DM particle simply by assuming that the weak scale is similar to the KK scale; two scales which are otherwise independent. In [11] we showed that the extended space AdS 5 × T 2 permits the complementary features of the RS and UED frameworks to be unified, with the warped direction motivating the weak/Planck hierarchy and admitting a description of flavor whilst KK parity emerges as a remnant symmetry of the extra toroidal dimensions. Interestingly the AdS 5 warping also motivates the connection between the weak scale and the UED KK scale with the warping inducing an effective KK scale on the torus of order ∼ TeV, even if the toroidal scale is R −1 ∼ M P l . This motivates the connection between the weak scale and the DM scale usually assumed in UED models.
In the present work we extend the program undertaken in [11] and consider the promotion of the RS model to a higher dimensional slice of compactified AdS d for d > 5. Our motivations are ultimately phenomenological and we seek to determine the extent to which the complementary features of UED and RS models can be combined in these higher dimensional warped spaces. However there is also a theoretical aspect to our work as we generalize many familiar RS expressions to these higher dimensional warped spaces. We find that, as one would expect, the RS realization of the weak scale via spacetime warping carries over to a slice of AdS d for d > 5 when the Higgs boson is a (d − 1) dimensional field localized on the IR brane. We further find that the RS approach to flavour also carries over to AdS 7 and AdS 9 , with the coupling between two chiral zero mode fermions and a brane localized scalar being exponentially sensitive to the fermion bulk mass parameters such that hierarchical Yukawas are expected in the 4D theory. However for the phenomenologically interesting case of R −1 ∼ 1 TeV the effective 4D gauge coupling between a chiral zero mode fermion and the zero mode of a bulk gauge field experiences volume suppression and, if the IR brane scale is ∼ TeV, the effective gauge coupling in the 4D theory is significantly suppressed. The severity of this suppression increases with d, though already for d = 7 it is of order ∼ 10 −15/2 . Consequently bulk SM fermions and gauge fields are not viable for both AdS 7 and AdS 9 . The appealing RS approach to flavour is therefore viable only on a slice of AdS 5 or, as shown in [11] for the case of AdS 5 × T 2 , for certain spaces of the form If the SM matter fields propagate in the transverse dimensions of AdS d one therefore expects them to be localized at either the UV or infrared (IR) brane. We shall show that in the former case a similar suppression of the effective 4D couplings is found for R −1 ∼ TeV so that only IR localization is viable. The main model building feature of the AdS d spaces seems to be their ability to combine the warped explanation for the weak/Planck hierarchy with the KK parity found in UED models, so these spaces admit only a partial unification of the appealing complimentary features of UED and RS models. The main experimental signature for the AdS d spaces in this instance is the observation of warped KK gravitons in addition to UED KK modes. Such a signature also occurs when the (d − 1) dimensional UED model is realized by embedding the SM fields on the IR brane of AdS 5 ×T d−5 , as discussed in [11]. However, as we shall show, the graviton KK towers on AdS d 2 Flavor structures may be viable in UED variants like split-UED [10]. and AdS 5 × T d−5 differ so that if a (d − 1) dimensional UED scenario is discovered one would be able to experimentally determine if the UED model in is embedded in either of these distinct warped spaces by carefully studying the graviton KK spectrum.
Fermions which are sterile with respect to the SM gauge group may propagate in the bulk, with such a scenario considered already for AdS 7 in the context of a brane localized UED model in [12]. Extended gauge sectors may also propagate in the bulk and the resulting volume suppression of the gauge coupling can motivate a very weak coupling for such sectors with SM fields. For example, if the SM is localized on the IR brane of AdS 6 , to realize an embedding of the minimal UED model on the IR brane and simultaneously motivate the weak/Planck hierarchy, the gauge group extension G SM × G X with G X in the bulk permits the G X -symmetry breaking to occur on the IR brane at the weak scale and yet remain experimentally viable. Such a scenario may offer an interesting way to employ, for example, a weakly coupled symmetry which plays a custodial role and is broken at the weak scale.
Before proceeding we note that works based on higher dimensional warped spaces exist already in the literature; see for example [13,14,15,16]. It is known, for example, that in AdS 7 the cancellation of boundary anomalies [16] necessarily constrains the boundary symmetries and field content. The combination of warped and universal extra dimensions has been previously considered on a slice of AdS 7 [12] and the graviton KK tower for AdS 7 was also studied in [7]. Some matters regarding moduli stabilization via bulk scalar fields in higher dimensional warped spaces were considered in [17] and the Casimir force was studied in [18] and [19], where, in the latter, it was noted that the contribution from the transverse extra dimensions resembles that of UED models. A study of DM candidates that result from approximate isometries of warped throats in compactified string models has also appeared [20].
The layout of the present work is as follows. In Section 2 we consider the Einstein equations and graviton KK tower for AdS d and in Section 3 we obtain the KK tower for bulk vectors in said spaces. Relative to AdS 5 the spaces AdS d for d > 5 admit additional modes in the form of metric and gauge boson polarizations in the transverse space. As an example of these modes we detail the KK spectrum for gauge-scalars in Section 3 for the d = 6 case of AdS 6 ; six being the lowest dimensionality which admits such modes. We derive the KK spectra for bulk fermions on a slice of AdS 7 and AdS 9 in Section 4 and show that in each case a single localizable chiral zero mode appears in the spectrum. In Section 5 we combine a number of these ingredients and consider the realization of the weak scale via warping with an IR brane localized Higgs boson, the mechanism of 4D flavor via fermion wavefunction overlap with an IR brane Higgs for AdS 7 and AdS 9 and the coupling of bulk vectors to bulk fermions in these spaces. Finally we comment on the range of validity of the effective 4D theory description and the case of UV localization in Section 6 before concluding in Section 7. In four Appendices we provide additional information which complements the analysis, including our conventions for bulk fermions in 7D and 9D.

Gravity on AdS d
We consider the metric defined by the d-dimensional spacetime interval where M, N = 0, 1, 2, 3, 5, 6, .., d label the full d-dimensional space, µ, ν = 0, 1, 2, 3 label the 4D subspace and the extra dimensions are labeled by x a , with a, b = 5, 6, .., (d − 1), and x d = y (the latter being the warped direction). The extra dimensions are compact with x a ∈ [−πR, πR], y ∈ [−πr c , πr c ], and the points x a = ±πR (y = ±πr c ) identified. For simplicity we take equal radii in the x a directions and we shall, at times, refer to these as the 'transverse' extra dimensions.
As in the RS model the warped direction is orbifolded as S 1 /Z 2 with the Z 2 action defined by the identification Z 2 : y → −y. The transverse directions must also be orbifolded to ensure the absence of massless gravi-vectors. For much of what follows we need not specify this orbifolding, though for completeness we note that for odd (d − 5) we shall use where there are (d − 6)/2 factors of T 2 /Z 2 in the brackets. For even (d − 5) the last factor of S 1 /Z 2 in (2) is not present and there are (d − 5)/2 factors of T 2 /Z 2 . We provide additional details regarding this orbifolding as appropriate in the text. We take as sources a cosmological constant Λ and two codimension one branes with tensions V 0,L ; the resulting Einstein equations being Here M * (R (d) ) is the d-dimensional Planck scale (Ricci scalar),ḠMN denotes the induced five dimensional metric at the brane locations with brane Lorentz indicesM ,N = 0, 1, 2, 3, 5, 6, .., (d− 1), and G = |det(G M N )| (similarly forḠ). The Einstein equations give with solution so the warp factor may be written as e −σ = e −k|y| . Calculating the second derivative of σ and comparing with (5) requires the tunings and the effective 4D Planck scale is given by We note that the solution (6) requires Λ < 0 and also define a new (conformal) variable by kz = e ky to write (1) as We refer to this metric, sourced by a negative cosmological constant, as AdS d . Strictly speaking the compactification of the transverse dimensions breaks some of the isometries usually present in AdS d (see [21]) and our space is the compactification of AdS d by the action of the discrete translation isometries x a ∼ x a + 2πR. For brevity we refer to this simply as AdS d with the implied compactification understood. We display the approximate value of the 7D gravity scale M * in Table 1 for 3 d ∈ [5,9]. Throughout this work we take R −1 ∼ TeV as this is the interesting region to be explored by the LHC and is also the compactification scale for UED models which permits the lightest KK particle to be a suitable DM candidate. We also take the IR brane scale as ∼ TeV so the hierarchy between the fundamental gravity scale M * and the weak scale results from warping. For the phenomenologically interesting case of R −1 ∼ TeV with k ∼ M * equation (8) gives (M * /TeV) ∼ 10 30/(d−3) . As can be seen in the table, M * decreases with increasing d for R −1 ∼ TeV due to the relatively large transverse volume. The IR brane scale is e −kπrc M * so that the value of kr c required to realize the weak scale on the IR brane also decreases with d. We also show this in the Table. One observes that, as opposed to the RS value of kr c ∼ O(10), no hierarchy is required for larger values of d with kr c ∼ O(1) readily obtained.

Graviton KK spectrum
The masses and wave functions of the KK gravitons are found by making the metric replacement The KK expansion for h µν is: where 4 n = (n a , n) = (n 5 , n 6 , .., n d−1 , n) and g (na) + (x a ) are even parity wave functions on the transverse space. Working in the gauge ∂ µ h µν = h µ µ = 0 the expansion (10) leads to where we write the KK masses as m h, n and use a ∂ 2 a g + . The profiles obey the orthogonality conditions and the solutions to (11) are where N n is a normalization factor, β h is a constant and the order of the Bessel functions is Equation (13) is the generalization of the RS (AdS 5 ) result and as such reduces to known expressions in the literature; the d = 5, 6, 7 cases reproduce the AdS 5,6,7 results found in references [23], [13] and [7]  h | z * = 0, where z * = z 0,L = k −1 , e kπrc k −1 , and the KK masses m h, n follow from β n (z 0 ) = β n (z L ), which to good approximation gives For Considering the purely warped KK modes (n a = 0), one observes that as d increases the mass of the KK modes increases whilst the relative KK spacing, (m h,n+1 − m h,n )/m h,n , decreases. 4 We emphasize that n a (n) labels the quantized momenta in the compact x a (z) directions. We shall on occasion 5 For string theoretic realizations there may be additional winding modes present in the spectrum. These can be phenomenologically important (see [7]), though we do not consider them here.
Thus the purely warped KK gravitons on a slice of AdS d are discernible from those of the RS model. The AdS d KK gravitons also differ from those found in spaces of the form AdS 5 × M d−5 , given that purely warped gravitons in the latter match those of the RS model. The purely warped AdS d gravitons with n > 0 couple to IR brane localized stress-energy sources with coupling Λ −1 π ≫ M −1 P l , as in the RS model. In particular one finds Λ π ≃ e −(d−3)kπrc/2 M P l , which is ∼ TeV for the parameters we consider.

Bulk Vectors on AdS d
In this section we consider a bulk U(1) gauge field in the AdS d background. As our ultimate purpose is to determine the viability of modeling a SM gauge boson by such a state the vector modes M = µ should have even parity to ensure a zero mode. The action for A M is and we work with the conformal coordinates defined by (9). The mixing between the vector mode and the gauge-scalar modes may be decoupled by introducing a bulk gauge fixing term, where we use the indexā to denote a, z so that ā = ā=a,z and we define the quantity K = K(z) by Varying the action S A + S GF gives the bulk equations of motion, where the first equation describes the vector modes and the remaining (d − 4) equations are mixed and describe the gauge-scalars. Taking suitable combinations of (21) and (22) gives where G A = (kz) d−4 b ∂b(KAb). Note that the states described by equation (23) have decoupled and are, in fact, the Goldstone modes. The (d − 5) equations (24) remain mixed and describe the physical gauge-scalars.

KK decomposition of the vector mode
We expand the vector modes A µ as and the profiles f ( n) A (z) must satisfy the following orthogonality relations: and the equation of motion: The solution to (27) is where N ( n) A is a normalization constant and the order of the Bessel functions is Equation (28) generalizes the wavefunction for a bulk vector in the AdS 5 RS background to a higher dimensional slice of AdS d . As such the d = 5 case reduces to that of [24]. The constants β ( n) A are determined by the boundary conditions ∂ z f ( n) A | z * = 0 and are found to be with the KK masses m n determined by solving β As with the KK gravitons, the wavefunction along the warped direction for a bulk vector differs from the RS result for d > 5 with both the order of the Bessel functions and the power of the prefactor (kz) increasing with d. The mass of the purely warped KK vectors (n a = 0) also increases with d whilst the relative spacing of the KK modes (m n+1 − m n )/m n decreases. Thus the vector KK tower for d > 5 is discernible from its RS counterpart. The KK action for the vector modes is finally given by which reduces to the usual RS expression for d = 5. We note that for d > 5 the massless zero mode gauge boson has wave function which remains finite for r c → ∞ and differs from the d = 5 case, for which f . This difference has been noted already in [15].

Gauge-scalar modes in AdS 6
For d > 5 a bulk vector has (d−5) additional degrees of freedom in the form of polarizations along the transverse directions. These, combined with the polarization in the warped direction, give rise to (d − 5) KK towers of physical gauge-scalars and a single KK tower of Goldstone modes; see e.g. [25] for studies of gauge-scalars in UED models. We shall not determine the KK towers of gauge-scalars for arbitrary d in what follows but instead, as an example, provide the KK spectrum of scalar modes for AdS 6 . The value d = 6 is the smallest number of spacetime dimensions which admits a KK tower of physical gauge-scalars. For AdS 6 we take the transverse direction x 5 to be orbifolded as S 1 /Z ′ 2 , where the action of the orbifold symmetry is Z ′ 2 : The parities for the scalar modes are fixed by the demand that A µ be even and are such that A 5,z do not posses zero modes. For AdS 6 the equations of motion (23) and (24) reduce to where We KK expand the scalar modes as where m G(S), n is the mass for the n-th KK mode and g (n 5 ) ± are the usual even/odd parity wave functions for the S 1 /Z 2 orbifold. The 4D fields satisfy and the orthogonality relations are: Using and equation (37) in the equations of motion gives: which have solutions From (33) one obtains the boundary conditions along the warped direction for A 5,z as which lead to and the KK masses follow from enforcing β one finds that f ( n) Combining the above gives the effective 4D action for the gauge-scalars, and by adding (49) to the d = 6 case of the vector KK action (31) one obtains the complete KK action for a bulk vector in AdS 6 . Observe that in the unitary gauge ξ → ∞ the modes A ( n) G become infinitely heavy and disappear from the spectrum so that, as advertised, these are the Goldstone modes which are 'eaten' by the massive KK vectors A

Bulk Fermions
In RS models the KK decomposition of a bulk 5D vectorial fermion produces a single chiral massless mode [2]. Being vectorial, an RS fermion may posses a bulk mass 6 and by varying this mass over order one values (in units of k) the chiral mode is readily localized towards either the UV or IR brane [2,3]. By localizing the lighter (heavier) SM fermions towards the Planck (TeV) brane one may generate the observed SM fermion mass hierarchies with order one Yukawa couplings [3,26] and thus the RS framework provides a mechanism by which to construct theories of flavor. We are interested in considering the generalization of the RS approach to flavor for AdS d with d > 5. The vectorial nature of bulk RS fermions persists only in spacetimes with an odd number of dimensions; for even d the minimal fermion is chiral with respect to the d-dimensional chiral projection operators and therefore does not admit a bulk mass. The RS approach to flavor is thus expected to generalize only for odd values of d and we consider the simplest cases of odd d > 5 in what follows and we obtain the KK spectrum for a bulk fermions on a slice of AdS 7 and AdS 9 . We then consider the coupling of these bulk fermions, including the localizable chiral modes, to an IR brane scalar and a bulk vector in Section 5.
We point out that a bulk fermion on a slice of AdS 7 was considered already in [12]. In that work the bulk fermion acquired an effective bulk mass by coupling to a bulk scalar with a non-vanishing background profile. The Yukawa coupling of the chiral zero-mode fermion to brane fields was then considered. Our analysis differs as we admit a mass for the bulk fermion and obtain the entire fermion KK spectrum; not just the zero mode profile as was done in [12]. We then consider the Yukawa coupling of two such bulk fields to a brane scalar and the coupling of a bulk fermion to a bulk gauge boson. Our results and notation provide a transparent generalization of the familiar RS expressions.

AdS 7 : Fermion orbifold parities
Before proceeding to discuss bulk AdS 7 fermions we specify the action of the orbifold symmetries acting in the extra dimensions. We write the index of the toroidal transverse dimensions as a, b = 5, 6, with the metric defined by The extra dimensions x a , y are orbifolded via with the action of Z 2 , ′ , Z 2 defined by A bulk field in the above background is in general specified by two parities (Z ′ 2 , Z 2 ) = (P ′ , P ), where P ′ , P = ±, and we note that the orbifolding (51) ensures there are no massless gravi-vectors in the spectrum. The action of the orbifold symmetries on a bulk fermion Ψ is and our conventions for the 7D gamma matrices Γ M may be found in Appendix B, where we also discuss some general properties of 7D fermions. We shall work with P = −1 and P ′ = +1 so that the (Z ′ 2 , Z 2 ) parities of the components of Ψ are and ψ +L is the only field which is even under both symmetries. Regardless of which values are used for P ′ , P there is always only one component of Ψ which is even under both Z ′ 2 and Z 2 ; the selection of different values for P ′ , P simply determines which component is even. The action of the orbifold symmetries on a Dirac mass bilinear is so that a bulk fermion may only have a Dirac mass if the mass is odd under the action of Z 2 , as in the RS model.

AdS 7 : Fermion KK spectrum
The action for a bulk fermion in the AdS 7 background is: where e M M = (kz)δ M M . We have already dropped the spin connection terms, which arise from the use of the covariant derivative D M = ∂ M + ω M , and cancel in the above. After rescaling the field Ψ → (kz) 3 Ψ and integrating by parts one has We define the four component spinors ψ + = (ψ +L , ψ +R ) T and ψ − = (ψ −L , ψ −R ) T in terms of the component fields, and KK expand these four component fields as where ψ ±L,R = P L,R ψ ± . The wave functions obey the following orthogonality relations, and the explicit form of the toroidal wave functions g (na) ±L,R is given in Appendix D.1. With m na given by the equations of motion for the wave functions along the warped direction may be written as where m n are the KK masses and the dimensionless mass c is defined by m D = ck. The equations of motion (64), (65) may be separated as and, noting the parities (55), one may use the equations of motion to obtain the boundary conditions, The solutions to the above are where the order of the Bessel functions is ν ± = |c ± 1 2 | and the equations of motion require that the normalization constants satisfy The KK masses are fixed by enforcing β ( n) and, similar to the RS case [3], they may be approximated as for large n. Putting the above together the bulk fermion action reduces to the canonical KK form Our primary interest is in the spectrum of massless modes as this will determine the viability of employing a bulk 7D fermion. Consider first the case with m na = n a = 0, for which equations (64), (65) have the solution f where for completeness we retain the factor of (kz) 3 previously scaled out. This is identical to the usual RS profile [2,3] modulo the replacement (kz) 2 → (kz) 3 for the factor scaled out in the above decomposition. One can easily show that no massless modes obtain when m na = 0 so the chiral mode (74) is the only massless mode in the spectrum.

AdS 9 : Fermion orbifold parities
As it will be helpful in what follows to be able to distinguish between the different transverse directions we use a slightly different notation for the transverse coordinate labels in this section and write the index of the toroidal dimensions as a, a ′ = 5, 6 and b, b ′ = 7, 8 with the metric defined by The extra dimensions x a,b , y are orbifolded via with the action of the orbifold symmetry defined by (51) and The action of the orbifold symmetries on a bulk fermion Ψ is where the three parities P, P ′ , P ′′ all take the values ±1. Here G M the 9D Dirac gamma matrices; our conventions for which may be found in Appendix C, where we also discuss some general properties of 9D fermions. We shall work with P = −1, P ′ = +1 and P ′′ = −1 so the (Z 2 , Z ′′ 2 , Z ′ 2 ) parities for the components of Ψ = (ψ ↓ , ψ ↑ ) T are and ψ 4L is the only completely even field. Regardless of which values are used for the parities P, P ′ , P ′′ only one component of Ψ is even under Z 2 , Z ′ 2 and Z ′′ 2 ; the selection of different parities simply determines which component is completely even. The action of the orbifold symmetries Z ′ 2 , Z 2 on a Dirac mass bilinear is again given by equations (56) and (57) whilst the action of Z ′′ 2 is Z ′′ 2 : ΨΨ → +ΨΨ.

AdS 9 : Fermion KK spectrum
The action for a bulk fermion in the AdS 9 background is where e M M = (kz)δ M M and we have already dropped the spin connection terms which cancel in the above. After rescaling the field Ψ → (kz) 4 Ψ and integrating by parts one has We define the four component spinors ψ α = (ψ αL , ψ αR ) T with α = 1, 2, 3, 4, in terms of the component fields, and the KK expansion for the four component fermions is The wave functions obey the following orthogonality relations, [Π a dx a ]g * (na) and similarly with the replacement L → R. The explicit form of g αL,R are given in Appendix D.2. In terms of the masses the equations of motion for the warped direction wave functions are where m n are the KK masses and the dimensionless mass c is again defined by m D = ck. The wave functions f Equations (90)-(93) may be separated as where we definem 2 n ≡ m 2 n − m 2 na − m 2 n b . The solutions are, where the order of the Bessel functions is ν ± = |c ± 1 2 | and we have used of the equations of motion. The normalization constants are not independent and may be expressed in terms of a single constant, as given in Appendix D.3. The KK masses m n are found by solving β The spectrum contains a single chiral massless mode with profile which matches the RS result modulo the replacement (kz) 2 → (kz) 4 for the scale factor. For large n the KK masses may be approximated as [3] m 2 n ≃ (n + and putting the above together the bulk fermion action reduces to the canonical KK form. Before proceeding to consider the coupling of bulk fermions in AdS 7,9 to bosons we note that, relative to the RS model, the order of the Bessel functions in the fermion profiles (98), (99) has not changed as a result of having increased d from the RS value of d = 5 to d = 7, 9. This differs from the explicit d dependence found earlier for bulk vectors and gravitons. As noted already for the zero modes, the power of the factor initially scaled out of the fermion wave functions does increase with d so the fermion profiles do display some d dependence.

Coupling to a Brane Scalar and a Bulk Gauge Field
We have seen that localizable chiral zero mode fermions, familiar from RS models, may also be obtained in AdS 7 and AdS 9 . In this section we consider an IR brane scalar to show that the RS warped realization of the weak scale also carries over to AdS d and, by coupling two bulk fermions to such a brane scalar, we show that for AdS 7,9 the RS approach to flavour also generalizes. We then consider the coupling of a bulk fermion to a bulk gauge field for AdS 7,9 and show that, in the phenomenologically interesting case of R −1 ∼ TeV, the effective 4D coupling for the zero modes experiences volume suppression. After these considerations we shall comment on the model building possibilities in AdS d and contrast these with spaces where the transverse directions are external to the warping, AdS 5 × M d−5 , with an emphasis on the AdS 5 × T 2 case [11].
Consider a (d − 1) dimensional scalar Φ localized on the IR brane of a slice of AdS d with the usual quartic potential: where where, in the case of electroweak symmetry breaking, v ∼ 246 GeV would be the electroweak scale. The d = 5 case of (104) reproduces the usual RS expression for the VEV of an IR brane scalar whilst for d > 5 it generalizes the RS expression to AdS d . We are working with the phenomenologically interesting case of R −1 ∼ TeV and as e −kπrc M * ∼ TeV the factor in brackets in (104) is O(1 − 10). In particular for e −kπrc M * 2πR ∼ 1 the weak scale is v ∼ e −kπrc v 0 so that, as in the RS model, the weak scale is realized via the warped suppression of the order ∼ k input parameter 7 v 0 . The RS approach to flavor also carries through to AdS 7 and AdS 9 , with the Yukawa Lagrangian between two bulk fermions Ψ 1,2 and an IR brane scalar being where the dots denote terms containing modes with n > 0 and the fermion mass is m 12 = λ y v, with the effective 4D Yukawa coupling between the zero modes defined as 8 For d = 5 this reproduces the familiar expression for the effective 4D Yukawa coupling in RS models [2,3] whilst the d = 7, 9 cases generalize the RS result and show that the RS approach to flavour holds for the warped spaces AdS 7 and AdS 9 . We may also consider the coupling between a bulk fermion and a bulk gauge boson in AdS 7,9 : where g d is a dimensionless bulk gauge coupling and for d = 7 (d = 9) the gamma matrices are the 7D (9D) Dirac matrices given in Appendix B (Appendix C). In the last line we have retained only the terms with the chiral mode and defined the 4D gauge coupling as Using the vector zero mode profile for d > 5 (32) gives 7 We note that equation (104) seems to indicate that for RM * ∼ O(1) the 4D Higgs VEV is φ (0) ∼ v 0 e −(d−3)kπrc/2 . However, as we show in Section 6, the effective 4D quartic coupling for the IR brane Higgs becomes non-perturbative in this region of parameter space so it is not clear that this deduction can be trusted. 8 The numerical subscripts here label the different fermion fields Ψ 1,2 and not different spinor components.
where we have used the leading order expression for the 4D Planck mass via (8). One readily observes a volume suppression of the effective 4D couplings. For example, with k ∼ M * and R −1 ∼ 1 TeV one has and provided d is not too large this gives M P l ∼ e (d−3)kπrc/2 TeV so that From the above considerations we may surmise the following. The RS realization of the weak scale via spacetime warping carries over to a slice of AdS d for d > 5 when the Higgs boson is a (d − 1) dimensional field localized on the IR brane. The RS approach to flavour also carries over to AdS 7 and AdS 9 , with the coupling between two chiral zero mode fermions and a brane localized scalar sensitive to the fermion bulk mass parameters such that hierarchical Yukawas are expected in the 4D theory. However for the phenomenologically interesting case of R −1 ∼ 1 TeV the effective 4D gauge coupling between a chiral zero mode fermion and the zero mode of a bulk gauge field experiences volume suppression. The severity of this suppression increases with d, though already for d = 7 it is of order ∼ 10 −15/2 . A similar volume suppression is known to occur for models with large extra dimensions [27].
Consequently bulk SM fermions and gauge fields are not viable for both AdS 7 and AdS 9 . The appealing RS approach to flavour is therefore successful only on a slice of AdS 5 or, as shown in [11] for the case of AdS 5 × T 2 , for certain spaces of the form AdS 5 × M d−5 . If SM matter fields propagate in the transverse dimensions of AdS d one expects them to be localized at either the UV or IR brane, with the weak scale realized via warping in the latter case. Actually, as we shall show in Section 6, for R −1 ∼ TeV the effective 4D couplings are highly suppressed for UV localization so that only IR brane localization of the SM is viable.
Note that fermions which are sterile with respect to the SM gauge group may propagate in the bulk, with such a scenario considered already for AdS 7 in the context of a brane localized UED model in [12]. We further note that extended gauge sectors can also propagate in the bulk and the gauge coupling volume suppression can motivate a very weak coupling for such sectors. As an example consider the localization of the SM on the IR brane of AdS 6 . This would realize an embedding of the minimal UED model on the IR brane and simultaneously motivate the weak/Planck hierarchy. With the gauge group extension G SM × G X , the G X -symmetry could be broken on the IR brane at a scale of ∼ TeV and yet remain experimentally viable if it propagates in the bulk. The effective couplings in the 4D theory would be of order g 6 M * /M P l ∼ g 6 e −kπrc/2 ∼ 10 −5 g 6 and are therefore automatically suppressed. Such a scenario may offer an interesting way to employ, for example, a weakly coupled symmetry which plays a custodial role and is broken at the weak scale.
The main model building feature of the AdS d spaces seems to be the ability to combine the warped explanation for the weak/Planck hierarchy with the KK parity found in UED models. In UED models KK parity is a residual from an underlying spacetime isometry. The transverse space in AdS d admits such an isometry so that KK parity may remain viable when the SM fields propagate in the transverse space. In particular if the SM fields are localized on the (d − 1) dimensional IR brane of AdS d one obtains a geometrical motivation for both the weak/Planck hierarchy and the existence of stable dark matter. The main experimental signature for the AdS d spaces in this instance would be the observation of warped KK gravitons in addition to the UED KK modes. Such a signature also occurs when the (d − 1) dimensional UED model is realized by embedding the SM fields on the IR brane of AdS 5 × T d−5 , as discussed in [11]. However the graviton KK towers on AdS d differ from AdS 5 ×T d−5 so that if a (d−1) dimensional UED scenario is discovered one would be able to experimentally determine if the UED model is embedded in either of these distinct warped spaces by carefully studying the graviton KK spectrum.
We note that for RM * ∼ O(1) one has M * ∼ M P l and the volume suppression observed in (109) disappears. Although the transverse KK modes disappear from the low energy spectrum in this limit, this case may be interesting unto itself. However, as we show in the next section, it is not clear at present what the correct description of the IR brane (including the localized Yukawa coupling) should be in this instance. It should also be stated that whilst the transverse KK (or dark matter) scale in AdS d is set by R −1 , there is no a priori connection between this scale and the weak scale. Thus the usual WIMP paradigm requires the transverse radius to be stabilized at R −1 ∼ TeV ≪ M * . As we will show below, such a condition is in any case necessary for the validity of the effective theory description we have employed. This situation is to be contrasted with AdS 5 × T 2 [11] where the underlying geometry also motivates the weak/Planck hierarchy (via warping) and dark matter (via KK parity as an isometry remnant). In that case the transverse KK scale is automatically warped to the IR brane scale, so that once the weak/Planck hierarchy is established via warping the dark matter scale is also ∼ TeV, even if the transverse radius is stabilized at R −1 ∼ M * .

Validity of the Effective 4D Description
Throughout the present work we have assumed a transverse compactification scale of R −1 ∼ TeV. There are two reasons for having restricted our attention to this case. The first reason is phenomenological as the new KK modes associated with the transverse space in AdS d will be accessible to colliders only for R −1 of order TeV and the lightest transverse KK mode may also be a good DM candidate for a TeV scale compactification. The second reason is theoretical as the effective 4D theory description on the IR brane breaks down for R −1 > TeV when the IR brane scale is ∼ TeV. We briefly demonstrate the latter point in what follows. To this end we use AdS 7 as an example and consider a non-interacting 6D scalar field localized on the IR brane: where the barred quantities denote brane restriction. To obtain the last line we have rescaled the field Φ → e 2kπrc Φ to bring the kinetic term in the x µ directions into a canonical form. With the KK expansion where the profiles obey ∂ 2 a g (na) = −m 2 na g (na) with m na ∼ R −1 , the action reduces to the standard KK form: The KK masses are where the bare mass is warped down as m Φ e −kπrc whilst the KK mass m na is not. As the effective transverse radius in the 4D theory is not warped it may lie below the cutoff of the 7D theory and yet exceed the warped down cutoff on the IR brane, that is R may lie in the range M * > R −1 > Λ IR . Let us add a series of higher order interaction terms for the scalar to consider this matter further: where λ 2q is a dimensionless coupling and we have performed the rescaling necessary to return the kinetic term to a canonical form to obtain the second line. The brane cutoff is warped down to Λ IR = e −kπrc M * , exactly as occurs in RS models. One may expect that for R −1 > Λ IR the IR brane theory could be trusted provided one neglects all KK modes whose mass exceeds the brane cutoff. However the effective description on the brane breaks down even when these states are discarded, as is seen by considering the interactions involving only the zero modes. In the effective 4D theory these are As an example consider M * ∼ [2πR] −1 , a relationship which, from the 7D perspective, appears within the range of validity of the effective theory description as R −1 < M * . However in the 4D theory the coupling of the zero mode quartic interaction φ (0)4 is ∼ λ 4 e 2kπrc and if the 7D couplings assume 'natural' values of order λ 2q ∈ [10 −2 , 1] this 4D coupling is severely non-perturbative. A similar enhancement is found for the higher order interaction terms. Thus it is not enough to simply discard the higher KK modes whose mass exceeds the IR brane cutoff; the effective theory description has broken down even for the zero modes. We deduce that the usual constraint R −1 < M * , required to ensure validity of the effective theory description, is inadequate to ensure that the effective 4D theory on the IR brane is sensible on a slice of AdS 7 . This result holds more generally on a slice of AdS d . The effective description of the 4D theory on the IR brane does make sense provided the transverse radius is less than the IR brane cutoff, R −1 < Λ IR = e −kπrc M * . This motivates the assumed value of R −1 e −kπrc M * ∼ TeV employed in the text. One may understand why the validity of the effective theory description requires R −1 < Λ IR rather than simply R −1 < M * as follows. Consider two points with separation ∆x a ∼ R in the transverse space. When localized on one of the branes this corresponds to a physical separation of: Within the present effective theory description one may only talk sensibly about proper distances satisfying ∆s −1 k, which translates into R −1 k for UV localization and R −1 e −kπrc k for IR localization. With k ∼ M * the latter relation gives R −1 Λ IR as promised. As the IR brane theory breaks down for R −1 > Λ IR the effective theory description employed in this work remains valid for the entire space only for R −1 < Λ IR .
The relation R −1 e −kπrc M * may seem strange as in the limit r c → ∞ the effective description breaks down for any finite R. However this behaviour is understood as in the r c → ∞ limit the spacetime has a conical singularity, which is observed by noting that at the horizon the proper radius in the transverse directions shrinks to zero as e −kπrc R. The resolution of this singularity requires knowledge of the UV completion; for example the slice of AdS d may emerge from a more fundamental string theory. The presence of this singularity is known already in the literature and has been discussed in [28], where a supergravity embedding of AdS 6 was considered to flush out possible ways to resolve it.
Before concluding we briefly consider the case of UV brane localization with R −1 ∼ TeV to show that the resulting effective 4D couplings can be highly suppressed. If the brane scalar Φ is instead localized on the UV brane of AdS 7 equation (117) becomes and with e −kπrc M * 2πR ∼ 1 and λ 2q ∼ 1 the effective zero mode coupling, for a given value of q, is: As expected, the UV brane cutoff is Λ U V = M * whilst the effective dimensionless coupling is of order e −(2q−2)kπrc , which, even for the zero mode quartic coupling case of q = 2, is highly suppressed with e −2kπrc ∼ 10 −15 . A similar suppression holds more generally for AdS d and thus the SM cannot be localized on the UV brane for the phenomenologically interesting case of R −1 ∼ TeV. We also note that the effective coupling for the interaction φ (0)2 φ (na)2 between two zero modes and two n a = 0 KK modes is ∼ e −2kπrc (∼ 1) in the case of UV (IR) localization when λ 4 ∼ O(1). It is easy to understand why the effective quartic couplings on the IR brane can be O(1) whilst those on the UV brane must be highly suppressed. The running of the quartic coupling will receive contributions from loops containing transverse KK mode scalars and, if the relevant effective 4D couplings are of order unity, will rapidly become non-perturbative. This is ok on the IR brane where the cutoff is warped down to e −kπrc M * ∼ TeV so that a rapid approach to the non-perturbative regime is consistent with the expectation that the IR brane theory will break down at the TeV scale. However on the UV brane the description is expected to be valid up to the fundamental scale M * . This requires the effective 4D couplings to be highly suppressed to ensure a slow running and to avoid a breakdown of the theory at scales E ≪ M * . In this way we observe that the theory automatically generates couplings that are appropriate for, and consistent with, the expected domain of validity of the effective theory description when R −1 e −kπrc M * .

Conclusion
In this work we have extended the program begun in [11] for AdS 5 ×T 2 and considered the promotion of the RS model to a higher dimensional slice of AdS d for d > 5. Such spaces are interesting as they admit a generalized version of the warped realization of the weak scale employed in the RS model. Our primary motivation was to determine the viability of combining the phenomenologically appealing features of RS and UED models in such spaces. We have performed the KK reduction for gravitons, bulk vectors and, for the case of AdS 6 , the gauge-scalars. We also obtained the KK spectra for bulk fermions on a slice of AdS 7,9 and showed that the RS approach to flavor generalizes to these spaces with the localization of chiral zero mode fermions controlled by their bulk Dirac mass parameters. However for the phenomenologically interesting case where the transverse radius is R −1 ∼ TeV we find that bulk standard model fields are not viable due to a resulting volume suppression of the gauge coupling constants. A similar suppression occurs for UV localization so that, when propagating in the transverse directions, the SM fields should be confined to the IR brane, consistent with the warped realization of the weak/Planck hierarchy. The main experimental signature of the AdS d spaces in this instance is the observation of warped KK gravitons in addition to the usual UED KK modes.
The above notation clearly labels the components of Ψ in terms of their various lower dimensional chiral properties. It is, however, somewhat cumbersome and we employ a simpler notation in the text; see (85).

D Fermion Wave functions D.1 Toroidal wave functions: AdS 7
The fermion wave functions on the toroidal dimensions may be written in terms of g where n a = (n 5 , n 6 ). For AdS 7 with toroidal compactification the T 2 profiles must satisfy (∂ 5 ± i∂ 6 )g The wave functions along x a must satisfy (∂ 5 ± i∂ 6 )g giving g (na) 1L (x a ) = g where n b = (n 7 , n 8 ). The solutions are with g (n b ) ± (x b ) given by (146), (147) with the replacement n a , x a → n b , x b .

D.3 Normalization factors in AdS 9
The equations of motion require the normalization factors for f ( n) αL,R to be related and one can show that they may be expressed in terms of a single normalization factor N