Stabilization of moduli in spacetime with nested warping and the UED

The absence, so far, of any graviton signatures at the LHC imposes severe constraints on the Randall-Sundrum scenario. Although a generalization to higher dimensions with nested warpings has been shown to avoid these constraints, apart from incorporating several other phenomenologically interesting features, moduli stabilization in such models has been an open question. We demonstrate here how both the moduli involved can be stabilized, employing slightly different mechanisms for the two branches of the theory. This also offers a dynamical mechanism to generate and stabilize the scale for the Universal Extra Dimensions, another long-standing issue.

to be, at best, a few times heavier than the Higgs boson. Furthermore, the very warping that explains the hierarchy also concentrates the KK-modes (though, not the lowest and massless mode) near the TeV brane, thereby enhancing their couplings to the SM fields. Consequently, several search strategies at the LHC were designed [11,12,13,14] to detect their signatures in a multitude of channels. Negative results from the same viz. m 1 > ∼ 2.66 TeV (at 95% C.L.) [15,16], thus, impose severe constraints on the model. It can be argued that, unless we allow a little hierarchy in the ratioṽ/M 5 , the RS scenario can be ruled out as a solution to the SM hierarchy problem.
The situation improves significantly if one were to consider an extension of the scenario to two extra dimensions with nested warpings [17]. The graviton spectrum, while now being enlarged to a tower of towers, is different from that in the five-dimensional case in two crucial aspects. For one, the change wrought in the graviton wave function results in the mass of the first KK-mode being significantly higher than that of the corresponding mode in the (five-dimensional) RS case 1 . As this happens for natural values of the parameters, and does not need any fine-tuning, this feature, on its own, would imply a weakening of the aforementioned "little hierarchy" that the original RS scenario needed so as to explain the nonobservation of gravitons at the LHC [18]. More importantly, the large coupling (to the SM fields) enhancement that allowed for the graviton KK-modes to be extensively produced at the LHC, is now tempered to a great degree [18], a consequence, once again, of the double warping.
Consequently, the graviton production rates are further suppressed and the scenario easily survives the current bounds from the LHC [18,19]. On the other hand, while the allowed parameter space of the model is still quite extensive, it can be probed well in the current run of LHC.
This, along with the fact that formulating the theory in a six-dimensional world has many other benefits, especially when the SM fields are also allowed into the bulk [20,21], renders this construction rather interesting.
In particular, with the four-dimensional theory getting supplanted by a five-dimensional one at the lower of the two compactification scales, the infrared is effectively screened from modes traversing the far ultraviolet. This is also reflected by the explicit computations of the electroweak precision variables [21], which demonstrated that the little hierarchy is no longer a major issue. However, the very issue of stabilizing the moduli (two in the current case, as opposed to a single one in the 5-dimensional one) has not been addressed so far. This assumes particular significance in that the structure formulated in Ref. [17] does not boast of a conformally flat geometry. Furthermore, the branes are not necessarily flat and this introduces its own set of complications.
In this paper, we aim to rectify this situation and develop two related, but distinct, stabilization mechanisms, somewhat analogous to those in Refs. [9,22,23,24]. This would also be seen to offer a stabilization mechanism for the modulus in a UED theory, thereby addressing a long-standing general lacuna in this otherwise attractive scenario.
Before we venture into the actual stabilization mechanism or even a detailed discussion of the scenario, we wish to clarify certain issues. Naively, it might be argued that having the gravitons to be heavier than in the RS would result in a worse fine tuning for the electroweak scale. As we have already mentioned, this moderate heaviness is but a consequence of there being two extra dimensions. To appreciate this, let us consider a sequence of unrelated scenarios. The first example would be an ADD [4]-like scenario with two extra-dimensions being compactified toroidally, with radii (possibly different) only somewhat larger than M −1 In the analogous five-dimensional theory, one would have, instead, M 2 Pl = M 2 5 θ 1 . Thus, for the theory with the larger number of extra dimensions, one would have a smaller hierarchy between the fundamental M 5 , M 6 etc. as the case may be and the electroweak scale. Consider, as the next example, a simplistic generalization of the RS scenario to a slice of AdS 6 bounded by two 4-branes, such that the apparent scale on the IR-brane is a few TeVs. This particular (multi-TeV) scale would, then, be protected with the graviton KK-modes now lying at at the same scale. If the 5-dimensional world be further compactified (or, even, orbifolded) over a circle of small radius, there would extend an additional factor in the relation between M 6 and M Pl , thereby further ameliorating the hierarchy problem. As we shall see, much the same happens in the present case.
On a related note, the cutoff of the effective four-dimensional theory needs to be identified too. For an effective theory, this is often described as the scale at which the loop contributions (often very large) are to be cut off, for the new physics beyond this scale would naturally regulate such contributions. Nonetheless, with the ultraviolet completion of the present theory being unknown (in the absence of any quantum theory of gravity), this cancellation cannot be demonstrated exactly. However, within the five-dimensional context, it has been argued that the addition of the Planck-brane and/or the TeV-brane allows for a holographic interpretation, with the former acting as a regulator leading to a UV cutoff, of the order of the inverse of the modulus, on the corresponding conformal field theory [25,26,27]. A similar conclusion also holds for theories with gauge fields extended in to the warped bulk [10,28,29]. While no such duality has been explicitly constructed for the sixdimensional case, one such would obviously exist, for, in a certain limit, the bulk is indeed AdS 6 -like. Thus, the branes would provide a regulator, albeit in a deformed CFT. In particular, the cutoff for the four-dimensional quantum field theory is set not by M 6 , but the inverse of the larger of the two moduli. At such a scale, the higher-dimensional nature of the theory becomes quite apparent, and the four-dimensional effective theory (including the graviton KK-modes) is no longer an apt language. And while the compactification mechanism is not specified here (or within the RS theory), the physics responsible for it must be incorporated in any description that reaches beyond this scale.

The 6D warped model
The space-time of interest is a six-dimensional one with successive (nested) warpings along the two compactified dimensions. The uncompactified directions support four-dimensional (x µ ) Lorentz symmetry while the com-pactified directions are individually Z 2 -orbifolded. In other words, we have Representing the compact directions by the angular coordinates x 4,5 ∈ [0, π] with R y and r z being the corresponding moduli, the line element is, thus, given by [17] where η µν is the flat metric on the four-dimensional slice of spacetime. As in the RS case, orbifolding, in the presence of nontrivial warp factors, 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 four end-of-the-world 4-branes.
Denoting the natural (quantum gravity) scale in six dimensions by M 6 and the negative (six dimensional) bulk cosmological constant by Λ 6 , the total bulk-brane action is, thus, The five-dimensional metrics in S 5 are those induced on the appropriate 4-branes which accord a rectangular box shape to the space. Furthermore, the SM (and other) fields may be localized on additional 3-branes located at the four corners of the box, viz.
Since S 4 is not relevant to the discussions of this paper, we shall not discuss it any further.
Rather than limit ourselves to the solutions to the Einstein equations presented in Ref. [17], we consider, here, a more general class. To motivate it, let us recollect that, in such models, the presence of bent branes is due to a "lower-dimensional cosmological constant" induced on the brane. For example, the four dimensional components of the Einstein equations, in the presence of such a term Ω would read where primes(dots) denote derivatives with respect to x 4 (x 5 ). Introducing a constant of separation Ω, we have The first equation has the solution b( assuming 2 Ω < 0. While Ref. [17] had considered only the special case of b 2 = 0, we shall admit the more general solution. As we shall see below, a nonzero b 2 would have very important consequences. Physically, Ω (or equivalently b 2 ) is related to the induced cosmological constant on a five dimensional hypersurface 3 along the constant x 5 direction. Differing values of Ω, thus, correspond to inequivalent extent of bending of the fourbrane, and, hence, lead to different physical outcomes. We will demonstrate this shortly using widely different (in essence, limiting) values for Ω. However, while the quantitative results do differ, qualitatively they turn out to be quite similar, with certain aspects essentially not changing at all. This was to be expected as many of For future convenience, we have also introduced the dimensionless combination ǫ. Clearly, for a semi-classical approach to be valid, the curvature must be significantly smaller than the mass scale of the theory. In other words, ǫ must be small 4 , namely ǫ < ∼ 0.15. On the other hand, as we shall see below, too small an ǫ would either invalidate the resolution of the hierarchy problem, or, in the process, introduce a new (but smaller) hierarchy.
The solution to eqn.(4) for a nonzero Ω is given in terms of hyperbolic functions. While it is possible to work with the general solution, the consequent algebra is exceedingly complicated and the exercise does not proffer any extra insight that a simplifying choice does not. As a nonzero Ω results in a nonzero cosmological constant in the four-dimensional world, and as the observed cosmological constant in our world is infinitesimally small, we disregard it altogether and consider only 5 Ω = 0. We do not claim to offer any rationale for this choice but for the fact that it simplifies the algebra for the rest of the article without losing any of the essence. In this limit, the solution can be expressed as 6 Normalizing the warp factors, at their maximum values, through a(0) = 1 and b(π) = 1, and imposing the orbifolding conditions, we have The brane potentials are determined by the junction conditions. The ones at x 5 = 0, π are simple and are given by whereas the ones at x 4 = 0, π have x 5 -dependent tensions It should be noted that the Israel junction condition V 1 = −V 2 is necessitated only by our focus on Ω = 0, or, in other words, a configuration wherein the four-dimensional cosmological constant vanishes exactly. Had we admitted Ω = 0, this equality of magnitude would neither have been necessary nor would it have held. This, of course, is exactly as in the RS case. The dependence of V 1,2 on x 5 is easy to understand. Each slice of x 5 could, potentially, host a 4-brane, with distinct (3 + 1)-dimensional worlds at the ends. Only if the potentials localized at the end of the branes are equal and opposite and related to the "overall size" of the 5-dimensional metric in that slice (just as in the RS case), would these hypothetical (3 + 1)-dimensional worlds be associated with a vanishing cosmological constant. As has been demonstrated in Refs. [17,21], such a x 5 -dependent potential could be occasioned by a brane-localized scalar field, such as a kink solution corresponding to a quartic potential, or in a theory with a non-trivial kinetic term.
It should be noted, though, that with these particular forms for V 1,2 are not strict requirements for the model.
Such a choice only helps to reduce the algebra. Indeed, as long as eqn. (9) holds at x 5 = 0 (with no restrictions for x 5 = 0), the vanishing of the four-dimensional cosmological constant is guaranteed. However, the relaxation of eqn.(9) does not add anything qualitatively different to either the phenomenology (whether in the graviton sector [18,19] or in the SM sector [20,21]) or to the main thrust of this paper, namely the stability of the scenario.
We now turn to the consequences of choosing a particular value for b 2 (this choice, as we shall see later, also serves to determine c). Rather than discuss the generic case (which does not afford closed-form analytical solutions), we illustrate the situations for two extreme limits. Physically, one of the limits corresponds to a vanishing five-dimensional cosmological constant (equivalently, straight, or unbent, four-branes at the ends of the world). The opposite limit corresponds to the case wherein the four-branes suffer the maximum possible 6 Once again, we omit the second solution, viz. e cx 4 for reasons analogous to those operative for b(x 5 ).
bending commensurate with a semiclassical analysis (or, in other words, a five-dimensional cosmological constant comparable to the fundamental scale). The low energy phenomenology, naturally, would turn out to be quite different in the two cases. Clearly, any intermediate value of b 2 would correspond to an intermediate value of the five-dimensional cosmological constant and, similarly, for the low-energy phenomenology.
Case 1: The situation of b 2 = 0 recovers the results of Ref. [17] and we have This, obviously, corresponds to a bent brane scenario with nonvanishing induced five-dimensional cosmological constants on the hypersurfaces at x 5 = 0, π. This could easily be seen by observing that the induced metric on the x 5 = 0 surface, apart from an overall b(0) factor, is given by or, in other words, the induced geometry is AdS 5 -like.
and, hence, the normalization of the warp factor would imply b 1 ≈ 2 exp(−kπ − b 2 ) → 0. Consequently, one is forced to c ≪ k, unless one were to admit a large, and unpleasant, hierarchy between R y and r z . This situation should be contrasted to the previous case, where the limit was realizable for both branches of the theory, viz. c ≪ k as well as a moderate c > k.
With c → 0 the brane potentials now read The fact of V 3 ≈ −V 4 reveals the near vanishing of brane-induced cosmological constant. As for the line element, in this limit, or that the metric is nearly conformally flat. It should be realized, though, that the approximate conformal flatness would have followed as long as c ≪ k (i.e., for k > ∼ 10) and did not need b 2 → ∞. However, a finite value of b 2 would have translated to unequal brane tensions and, consequently, nonvanishing induced cosmological constants.
The two opposing limits of b 2 are not special, but only serve to simplify the algebra. Any intermediate value of b 2 would only lead to phenomenological situations that interpolate between those listed above. In the following, we shall detail not only the stabilization of the radii, but also that of b 2 .

Radii Stabilization
While it may seem, at first sight, that moduli stabilization in this (6D) framework can proceed in a fashion identical to that in the RS paradigm, there are certain crucial differences. In particular (and, as we shall see below), if we attempt a naive GW [9]-like mechanism, only one combination of the two moduli can be stabilized.
This is but a reflection of the well-known fact that, for a multidimensional hidden compact space, it is easier to stabilize the shape rather than the volume. It should be realized, though, that had we been interested in a different compactification (such as, for example, M (1,3) × S 2 with an appropriate orbifolding), a single-field GW-like mechanism would indeed be enough. This is as expected, for in such a case there would, but, be only one modulus to stabilize. However, such a compactification is not favoured phenomenologically as, on the one hand, it requires extra fields to counterbalance the curvature of S 2 , while, on the other, if the SM fields are extended in to the bulk (so as to fully exploit the advantages of the 6D construction), the resultant spectrum cannot, easily, be made consonant with low energy observations.
While the same mechanism would work irrespective of the choice for induced cosmological constant, the algebraic simplification is significant in the two limits discussed in the preceding section. Similarly, treating the two distinct regimes (viz small k and large k) separately brings forth an appreciation of both the overall mechanism, as well as the subtle differences in the implementation thereof.
Before we do this, though, let us reexamine some potentially confusing features of this scenario, in particular the roles of the brane localized potentials V i , the separation constant Ω and the constant b 2 . At first glance, the "choices" might seem to associated with fine-tunings. We begin by showing that not all of them are independent and, then, explore the stabilization of the truly independent.
To begin with, it should be realized that the special case of Ω = 0 would have led to a generic solution of the there is no severe fine-tuning associated with Ω.
Note further that eqn.(5) also implies whereas eqn.(8) In other words, there is a one-to-one relation between (V 3 , V 4 ) and (k, b 2 ) or, equivalently, (k, Ω). Stabilizing one set automatically stabilizes the others. While we propose below a mechanism to stabilize the last (or, equivalently, the first) set, note that we have already seen that the dependence of physical observables on Ω is a suppressed one. Thus, stabilizing k would be enough.

Small k and large c
As we have discussed in Sec.2, in this regime, the metric cannot be approximated by a conformally flat one, and, of the two limits discussed therein, only Case I can be applicable. Rather than work with the general solution, we shall work in this limit, for it simplifies the algebra considerably without altering the physical essence.
As we have also explained earlier, starting with a single canonically quantized scalar field, it is not possible to stabilize both the moduli. Consequently, we postulate two such scalar fields. In order to minimize the number of effective four-dimensional fields (on KK reduction), we incorporate one scalar field φ 1 (x µ , x 4 , x 5 ), permeating the entire bulk, that would serve to stabilize r z (or, equivalently, the dimensionless quantity k). A second field φ 2 (x µ , x 4 ), introduced (localized) only on the x 5 = 0 brane, would, similarly, stabilize the length (R y ) of the brane. Given the box structure and the orbifolding, together, they stabilize both the moduli.
The Lagrangians for these scalars are given by and respectively. Here, g = det(g MN ) = −a 4 b 5 R y r z , whereas, for the induced metrics, we have g 5 = det(gMN ) = −R y a 4 b 5 and g 4 = det(g µν ) = −a 4 b 4 .
In particular, the 5d metric, apart from the constant b 2 (0), induced on the x 5 = 0 brane is Given this AdS 5 geometry and the form of L 5 , it is clear that the stabilization of R y can proceed exactly as in the GW mechanism [9] or its variants [22] using the classical configuration of φ 2 . Since this technique is well-known, we, for the sake of brevity, will eschew any details here, assuming that R y can be stabilized. Indeed, with an appropriately modified five-dimensional potential for φ 2 , it is also possible to take into account the back reaction and achieve an exact solution [23]. We will come back to a generalized version of this.
Unlike the 4-brane at x 5 = 0 (hereafter called the 4 0 brane) itself, the x 5 −direction possesses a non-zero induced cosmological constant, as shown in section 2. Stabilization of this direction, thus, requires a more careful analysis which we proceed to now.
The effective potential for r z (equivalently, k) can be obtained, starting from the Lagrangian of eq. 12.
While the classical configuration of φ could, in principle, have nontrivial dependences on both x 4 and x 5 (and, yet, maintain the requisite Lorentz symmetry), such a general consequence would have required complicating the boundary-localized terms and does not add anything qualitatively different to the system. Since we are primarily interested in the effective potential in the x 5 -direction, for brevity's sake, we restrict our discussion to the case where φ has a nontrivial dependence only along x 5 and denoting it the effective one-dimensional Lagrangian for φ(x 5 ) is given by Understandably, a(x 4 ) appears only as an overall multiplicative factor and plays no dynamical role. The corresponding equation of motion is The solution, in the bulk, is given in terms of associated Legendre functions, viz. φ = sech 5/2 (kx 5 ) c 1 P ν 3/2 (tanh(kx 5 )) + c 2 Q ν 3/2 (tanh(kx 5 )) , where c 1,2 are the constants of integration and The constants c 1,2 can be determined once boundary conditions are imposed. To do this, we turn to the brane-localized potentials U 1,2 (φ 1 ) which, until now, were unspecified. We are not sensitive to the exact form of U 1,2 (φ 1 ) as long as they admit nonzero minima at φ = v 1,2 respectively. Noting that the cutoff scale on this brane is given by R −1 y , such minima, for example, can be easily achieved if one were to consider with λ 1,2 being dimensionless constants and V 3,4 being defined as in Putting the solution back in the effective Lagrangian 14, we have Eliminating the irrelevant factor a 4 (x 4 ) and integrating L 6 over x 5 , we would obtain an effective potential for k, defined, in dimensionless form, as .
Since k = r z M 6 ǫ, with the last two quantities being fixed parameters of the theory, V eff is, thus, equivalently, a potential for r z . As a closed form expression for V eff is not possible, and even a good approximate form complicated enough, we present it, instead, only in a graphical form.
In Fig.1, we display V eff (k) for a fixed value of the mass parameter µ (equivalently, m). As is obvious, depending on the ratio v 2 /v 1 , minima exist for 0.1 < ∼ k < ∼ 0.6, the range that is of particular interest not only to explain the non-observation (so far) of the KK-graviton at the LHC [18,19], but also for scenarios wherein the SM fields are extended in to the bulk [20,21]. What is particularly encouraging is that such minima arise for very natural values of the parameters and are not overly sensitive to their precise values. Indeed, the strongest dependence, of the stabilized value of the modulus r z , is on the ratio v 1 /v 2 of the classical values.
It is also instructive to examine the dependence of V eff (k) on ǫ (as depicted in the two panels of Fig.1) and µ (as shown in Fig.2). As can be readily ascertained, while the size of the potential has a strong dependence on µ (understandable, since it is µ that allows for a nontrivial V eff ), the position of the minimum has only a muted dependence.
Until now, we have neglected the back reaction on the metric due to the scalar field. While we could, in principle, attempt this, as we shall indeed do for the other regime (namely, large k) in the next section. However, 7 On the boundary, once the scalar field φ settles down to the vacuum v 1,2 , the brane localized potential becomes U 1,2 = V 3,4 and one recovers the action given in eq. (2). While this unifies the explanation of the brane-tensions in the current context, the presence of a non-negligible induced cosmological constant Λ ind queers the pitch. In the absence of Λ ind , the change in the warp factor due to back-reaction can be computed exactly by integrating three first order equations (namely, the one for the scalar field, the warp factor and the bulk potential). For a non-zero Λ ind , additional nonlinearities , that couple these three equations in a non-trivial manner, emerge [23], and a closed-form analytic solution is not possible. Of course, the equations can still be solved numerically.
However, in view of the fact that such solutions can be easily obtained by deforming the solutions presented above, we eschew a detailed discussion for the sake of brevity. The neglect of the back-reaction is eminently justified, for the largest value of the scalar field mass that we have used is sufficiently smaller than Λ 1/6 6 (and, certainly M 6 ). Consequently, the energy content in the φ-field is rather subdominant to that due to the bulk cosmological constant and the back-reaction in the bulk is not much of a worry. Furthermore, with the hierarchy, for the most part, being dictated by the warp factor in the x 4 −direction, even moderate changes in b(x 5 ), as would be introduced by the back-reaction, have relatively little bearing on the phenomenology.

Large k and small c
In this regime, the metric is nearly conformally flat. With c being infinitesimally small, neglecting the c−dependence of the metric would not introduce a decipherable difference in the analysis. To simplify the algebra, we will take recourse 8 to Case 2 of section 2, whence the metric reduces to where in the absence of backreaction A(x 5 ) = k|x 5 |. With the 4 0 -brane-localized induced cosmological constant being infinitesimally small, it is possible to obtain an almost exact solution incorporating the back reaction as well and we now attempt this. Introducing a scalar field φ in the bulk, the entire action is given by 9 The corresponding equations of motion arë For a scalar with a localized potential on the x 5 −constant 4-branes V (φ) could be written as where V bulk (φ) is the bulk potential and f 0,π (φ(x 5 )) are some as-yet undetermined functions of the scalar field.
Integrating eqn.(21) across the 4-brane locations (α ≡ x 5 = 0, π), we havė which provide the junction conditions. An exact closed-form solution to eqns. (21) can be obtained only for particular bulk potentials. Borrowing from techniques of supersymmetric quantum mechanics, we assume the bulk potential can be expressed as where W (φ) can be thought of as a superpotential. This, immediately leads tȯ as long as W (φ) satisfies the junction conditions Each choice for W (φ) gives a different V bulk , but an analytic closed-form solution can be found for only some.
An explicit example is afforded by a quadratic superpotential [23,24], namely where u < ∼ 0.1 is a constant, parameterizing not only the mass of φ, but its (quartic) self-interaction as well.
The corresponding brane localized potentials read where γ π,0 are arbitrary positive constants that ensure that φ(x 5 ) assumes values v π,0 on the Planck (TeV) branes. The solutions to eqns.(23) are given by Note that the warp factor has changed from the simple exponential form that it had in the absence of φ.
It is worthwhile to reflect on the difference between this analysis and that presented in the preceding subsection. While we could have adopted the same procedure, namely substitute eqn. (24) in eqn. (20) and integrate over x 5 to yield an effective potential V eff (r z ), it is not necessary to do so. Rather, note that the very structure of the solution (eqn.24), along with the boundary-localized potential ensures that nonlinear differential equations that we are endowed with. It is also worthwhile to note that a natural set of values for v π /v 0 and u can reproduce the required r z (and, hence, the correct warping) without any fine-tuning being needed.
We now turn to the stabilization of R y . If the approximation of eqn. (19) were truly exact, R y cannot be stabilized. On the other hand, it need not be, at least in the context of hierarchy stabilization, for it really does not play a role in defining the overall warp-factor. Apparently, thus, the primary constraints would be those on the ADD scenario [4], such as deviation from Newton's law or the fast cooling of a supernova. And while it might be argued that such an extremely large value for R y reintroduces a hierarchy, it is not obvious that this is a problem (far less a serious one), given that R y plays only a subservient role in defining the gap between M 6 and the electroweak scale. Indeed, well before R y becomes so large (the sub-millimeter range), c becomes quite non-negligible. This not only invalidates the approximation of eqn. (19), but also carries the seed for the stabilization of R y .
The latter can proceed, for example, in a fashion exactly analogous to the GW mechanism as defined for the original RS scenario. Consider, for example, a second scalar φ 2 (of mass m 2 < ∼ M 6 ) confined to the 4-brane at x 5 = π. Assume that the only self-interactions are localized at the boundaries, viz. at (x 4 , x 5 ) = (0, π) and (π, π) which, in turn, force φ 2 (0, π) = v 3 and φ 2 (π, π) = v 4 . Clearly, this would lead to a stabilized , and, consequently, to a moderate R y /r z and a small c (as desired). A more interesting option would be to locate φ 2 on the 4 0 brane instead, with the boundaries now corresponding to (x 4 , x 5 ) = (0, 0) and (π, 0) respectively. With m 2 now suffering a large warping (due to b(x 5 )), the stabilized value for R −1 y would, naturally, be in the TeV range. This, immediately, raises the intriguing possibility that new physics at a few-TeV scale could indeed be stabilized by the SM Higgs itself (or a cousin of its). Even more intriguingly, if one allows the SM fields to percolate into the x 4 direction, the setup under discussion would provide a dynamical justification for the scale in a Universal Extra Dimension-like scenario [5].

Conclusion
The six dimensional warped scenario provides a cure for various ailments of Randall-Sundrum model. Nevertheless, the problem of modulus stabilization, which was quite simple for Randall-Sundrum scenario, had, until now, not been executed for either of the two moduli in the nested warped model, largely on account of the fact that the model's space-time is neither conformally flat, nor are the end-of-the world branes flat. Consequently, the stabilization mechanism presents a technical challenge, and this is the issue that we have addressed in this paper. To this end, we begin by exploring the metric for nested warping, showing that the solutions for each of the two regimes allowed to the theory can be generalized beyond what was considered earlier.
In the small k (equivalently, large c) regime of the theory, the induced geometries on the x 5 -constant 4-branes are AdS 5 -like, and hence a Goldberger-Wise mechanism (or even one incorporating back-reaction) involving a brane-localized scalar field trivially stabilizes the corresponding modulus R y . The second modulus r z cannot be stabilized by the same scalar field. It is intriguing to consider leaving it unstabilized, especially since the corresponding warping is minor, and a slow temporal variation would have very interesting cosmological ramifications. However, r z can also be stabilized by a six-dimensional analogue of the GW mechanism, as we have demonstrated here. Although the form of the effective potential for r z is much more complicated than that in the minimal RS scenario, a numerical analysis shows that a minimum does exist and reproduces the desired hierarchy without the need for any fine tuning. Indeed, the phenomenologically acceptable domain in the parameter space of the theory is more extensive than that in the RS model. As for the back-reaction, while it can be incorporated, closed-form analytic solutions are not possible owing to the non-zero induced cosmological constants on the constant-x 5 hypersurfaces. However, numerical solutions are indeed possible.
In the other regime of the theory, characterized by a vanishingly small induced cosmological constant, the scenario changes dramatically, with the bulk tending to become conformally flat (and the warp factor nearly exponential). With the induced cosmological constant on the branes being infinitesimally small 10 , a closed form solution can be found even on the inclusion of the backreaction. This allows us to stabilize r z without taking recourse to any unnatural values of the parameters. And while the aforementioned exact solution is achievable for only certain specific potentials, deviations thereof still lead to stabilization (with backreaction taken into account) with the only difference being that the solutions can be expressed only in terms of complicated integrals.
The situation with the corresponding R y is more intriguing. With c now being infinitesimally small, it is tempting to consider the possibility of a rolling R y , especially since a slowly varying R y would have very interesting and attractive cosmological consequences. On the other hand, R y can indeed be stabilized, by introducing a second scalar on one of the two 4-branes, viz. at x 5 = π or at x 5 = 0. The first alternative naturally leads to R y being stabilized to a value of the order of r z . The second alternative, on the other hand, leads to a situation whereby a scalar field of apparently TeV-range mass (on account of the warping) leads to R y being stabilized at a scale somewhat higher than the electroweak one. If the SM fields were considered to be five-dimensional ones, defined on the entire brane at x 5 = 0, this immediately leads to a UED-like scenario with the TeV-scale protected naturally. The orbifolding inherent to the system would not only eliminate unwanted modes, but also introduce for a KK-parity that, in turn, provides for a Dark Matter candidate on the one hand and eliminates many contributions to rare decays and precision variables on the other, thereby improving agreement with observed phenomenology. It might be argued, though, that with c being different from zero, the KK-parity is not exact. This is indeed so, but with the extent of Z 2 -breaking being determined by the 10 Note that a non-zero value for the five-dimensional cosmological constant does not preclude a vanishing four-dimensional cosmological constant (witness the original RS model), and, indeed, we do obtain the latter even in the general case. Furthermore, a vanishing value of the former is not a requirement for our analysis, and serves only to simplify the algebra.
(vanishingly small) induced cosmological constantΩ, the lifetime of such DM-candidates would be exceedingly long.
Before ending, we revisit the question of fine tuning in such models. Both the original formulation [17] as well as the extended version (Sec.2) seemed to be dependent on the presence of particular values of brane tensions.
Exactly analogous to the original RS model, this could be interpreted as a fine-tuning endemic to this class of models. Naively, the choice of values for b 2 and the constant of separation Ω represent additional fine-tunings.
We have shown here, though, this is not so. The exact value of Ω (equivalently, b 2 ) has relatively little bearing on the phenomenology. Indeed, for any two values of b 2 , in the range 0 ≤ b 2 < ∞, the difference between the resultant warp factors differs by at most 50%, and that too, only for small k. can be easily worked out by interpolating between the results for b 2 = 0 and b 2 → ∞ respectively. We have, consequently, chosen to demonstrate the results in these two limits as they admit simple analytical solutions whereas the general b 2 would need numerical methods to be employed.
Having argued that a specific value of Ω (or, equivalently, b 2 ) does not imply any fine-tuning over and above that endemic to RS models, we now turn to the latter, or more specifically, to the analogue thereof. As we have argued earlier, within the original RS model (sans modulus stabilization), the brane tension had to be just so, for the bulk solution and the orbifolding to be valid simultaneously. Furthermore, these were not related to the modulus. Here too, a similar situation seems to hold (see eq.8), with the recognition that the ratio k/r z is determined entirely by fundamental scale M 6 and the bulk cosmological constant Λ 6 . Indeed, it has parallels with the RS model wherein the branes were allowed to have a nonvanishing cosmological constant. On introducing the stabilization mechanism the brane tensions V 3,4 were identified with the stabilized values of the brane-localized potentials U 1,2 (φ) of the bulk scalar φ.
A deeper understanding is afforded if one considers possible quantum corrections to the bulk Einstein-Hilbert action. While no such actual calculation is available, these, presumably, would appear as diffeomorphisminvariant higher derivative terms. Assuming that these could be parametrized as a polynomial in R, Ref. [32] considers, for example, a 5-dimensional bulk theory defined in the Jordan frame by with the constants a i ∼ O(10 −1 ) and M the cut-off scale. In the Einstein frame (obtained from the Jordan frame through a conformal transformation), the extra degree of freedom associated with the higher-derivatives can be recast in terms of a scalar field with a very nontrivial potential. Most interestingly, this degree of freedom can play the role of the Goldberger-Wise scalar, thereby allowing for a "geometric stabilization" of the modulus.
A similar stabilization can occur in six-dimensions too with the field φ 1 parameterizing such higher derivative terms appearing in the bulk action. Additional possibilities arise in the shape of brane-localized f (R)-terms (since the branes are characterized by matter fields, the quantum corrections to the Einstein-Hilbert action would, in general, be different on them). These extra terms would play the role of the brane-localized fields (masquerading as our φ 2 ) with their own potentials. Being very steep [32], these would enforce the system being in the vacuum state, thereby according a quantum origin to V 3,4 . Of course, once again, this entire paradigm depends upon the exact form of f (R), including the coefficients, for both the bulk and the branes.
However, the conjecture that the entire stabilization process is but a consequence of an effective geometric action born of quantum corrections, is, undoubtedly, a very interesting one, especially in the quest to understand the fine-tuning problem (such as that associated with choosing Ω = 0).
It is also worthwhile to consider extending the formalism developed herein to still higher dimensions. For example, it has been shown [33,34,35] that a six-dimensional UED model not only suppresses proton decay through a higher dimensional operator, but also gives a topological origin for the number of chiral fermion generations. The extension of the formalism presented here to seven dimensional nested warping [17] would accord a dynamical origin to the scale of the model. These and other issues are currently under investigation.