The Little Randall-Sundrum Model at the Large Hadron Collider

We present a predictive warped model of flavor that is cut off at an ultraviolet scale O(10^3) TeV. This"Little Randall-Sundrum (LRS)"model is a volume-truncation, by a factor $y \approx 6$, of the RS scenario and is holographically dual to dynamics with number of colors larger by $y$. The LRS couplings between Kaluza-Klein states and the Standard Model fields, including the proton constituents, are explicitly calculable without ad hoc assumptions. Assuming separate gauge and flavor dynamics, a number of unwanted contributions to precision electroweak, $Z b\bar b$ and flavor observables are suppressed in the LRS framework, compared with the corresponding RS case. An important consequence of the LRS truncation, independent of precise details, is a significant enhancement of the clean (golden) di-lepton LHC signals, by O(y^3), due to a larger"$\rho$-photon"mixing and a smaller inter-composite coupling.

Electroweak symmetry breaking (EWSB) in the Standard Model (SM) via the Higgs condensate v ≡ H ≃ 250 GeV is economical and consistent with data. However, quantum effects render v quadratically sensitive to an ultraviolet (UV) cutoff scale Λ UV . For Λ UV near the gravity scaleM P ∼ 10 18 GeV, a severe "hierarchy" O(10 −32 ) arises. One may question the urgency of this problem, as physics close toM P is unknown and inaccessible in the near future. Nonetheless, precision electroweak (EW) data require Λ UV > ∼ 10 TeV, near welltested scales, posing a challenge to a natural Higgs sector. This is often called the little hierarchy; for some recent proposals to address this problem see [1]. Precision flavor data demand Λ UV > ∼ 10 2 − 10 3 TeV, posing a much more severe "weak-flavor" hierarchy.
The Randall-Sundrum (RS) model [2] was originally proposed to solve the hierarchy problem and yielded distinct collider signatures [3]. However, with 4Dsequestered fermions [4,5], tension with precision data generates a little hierarchy [4,6], the resolution of which led to the inclusion of SM fermions [7] and gauge fields [4,5] in the 5D bulk. This also provided an attractive explanation of the SM flavor structure [7,8], but made the RS model less accessible to experiments [9,10,11,12,13,14]. In addition, the generic theory requires more structure to be consistent with oblique and non-oblique precision tests [15,16] and constraints from flavor changing neutral currents [17]. In what follows, "RS" denotes the original hierarchy model and all of its extensions.
While the RS construction has a compelling appeal, as it allows a simultaneous resolution of the SM hierarchy and flavor puzzles, it is premised on a strong assumption. That is, warping extends over many orders of magnitude, without any basic change in physics, from the weak scale to the Planck scale. Surely this assumption needs to be put to an experimental test and we will discuss below how this may indeed be possible, in a warped scenario with various attractive features.
In this work, we use a volume-truncated RS background only to address the hierarchy between the weak (IR) and flavor (UV) scales. SM couplings to new physics, and hence the LHC phenomenology, are explicitly set by the flavor structure without ad hoc assumptions. The 5D UV scale M 5 is taken to be O(10 3 ) TeV to suppress light-flavor operators in this "Little Randall-Sundrum (LRS)" model [19]. We note that all EW and flavor data are compatible with having Λ UV ∼ M 5 , where additional physics may arise.
Keeping Yukawa dynamics unchanged by our truncation, a number of unwanted contributions to precision EW and flavor data are suppressed within the LRS scenario, compared to the RS counterpart. In fact, we will show that any specific RS model is always more constrained than its corresponding LRS counterpart. An exciting consequence of the LRS truncation is a much improved prospect for discovery at the LHC, via clean di-lepton "golden" modes, since the couplings of the KK gauge bosons to light fermions are enhanced while their couplings to the heavy fields are suppressed. Here and below we assume, for simplicity, an IR-brane Higgs and tree level matching for the gauge couplings, as discussed later. Also, we will focus on the quark sector, however, leptons can be included straightforwardly.
The RS background is a slice of AdS 5 , bounded by two Minkowski 3-branes, with the metric [2] ds 2 = e −2σ η µν dx µ dx ν − r 2 c dφ 2 , where σ = kr c |φ|, k is the 5D curvature scale, r c is the radius of compactification, and φ ∈ [0, π]. The UV (Planck) brane is at φ = 0 and the IR (TeV) brane is at φ = π. Going from the UV brane to the IR brane, the 4D scale redshifts from k < ∼ M 5 to the weak scale κ ≡ ke −krcπ ∼ TeV. Solution to the hierarchy problem requires kr c ≃ 11, with the Higgs at or near the IR brane. A natural flavor structure is obtained, using bulk fermions with non-zero vector-like masses m i , i = u, d, . . . [7,8]. The resulting zero-mode fermions are exponentially localized in 5D, parameterized by c i ≡ m i /k. One may choose c i ∼ 1 so that light fermions are UV-localized and have small overlaps with the IR-localized Higgs. Due to the warping, the lightflavor cutoff scale is then much larger than the IR/weak scale. This suppresses dangerous light-flavor operators and yields the correct fermion mass hierarchy with O(1) parameters. However, not all precision data are accom-modated with bulk fermions.
Oblique Corrections: Here we would like to go over the important constraints on warped phenomenology from precision electroweak data. There are various contributions that can be parameterized in terms of the oblique Peskin-Takeuchi (S, T ) parameters [20] and we will discuss them in turn.
We begin by considering the case where the only gauged symmetries in the bulk are those of the 5D SM. First of all, there is a contribution that comes from the tree-level mixing of the gauge zero modes with the higher KK modes. In the RS model, these are given by [15], where encodes fermion localization; cos 2 θ W ≃ 0.77. For all realistic warped fermion profiles of interest in this work, ξ(c) ≪ 1. Without a "bulk" custodial symmetry, there is also a UV-sensitive loop contribution δT to the T parameter. This UV-sensitivity can be absorbed into a higher dimension operator. Assuming that this operator is generated by strong dynamics at the 4D cutoff scale Λ, it will have the form Current data favor |S| ∼ |T | ∼ 0.1 − 0.3 [21] summed over all contributions. We see that T tree from Eq. (2) is the dominant tree-level constraint, given its volume enhancement kr c π, which is roughly a factor of 35 in the RS model. To reduce the size of T tree in this setup then requires increasing κ (m KK ) to values that lead to a severe little hierarchy and null LHC signals. Alternatively, we see that reducing kr c π yields a significant suppression, keeping KK masses fixed. In our LRS construct, we will truncate the volume to kr c π = 6. Note that Even though the LRS truncation has suppressed the tree-level KK-tower mixing contribution, we must still address the loop and cutoff scale effects encoded in Eq. (4). These are the same in both the RS and the LRS scenarios, since the flavor sector is assumed to be independent of the gauge dynamics and hence unchanged by the LRS truncation. Barring unnatural cancellations, demanding that the cutoff contribution is less than O(0.1) pushes m KK to values of O(10) TeV.
Ref. [15] attributed T > ∼ 1, in the RS model, to the absence of bulk "custodial" protection and postulated a SU (2) L × SU (2) R × U (1) X 5D symmetry to eliminate tree-level contributions to T . It turns out that the loop contribution δT , governed by fermion KK modes, still remains, but is no longer UV-sensitive. Also, given the gauged 5D custodial symmetry, there is no cutoff contribution of the form in Eq. (4), at the IR-boundary. Ref. [15] concluded that including the SM effects, for m RS KK ∼ 3 − 4 TeV and a light Higgs, S and T can be accommodated at an acceptable level. Note that with a bulk custodial symmetry, the LRS construct will enjoy the same level of agreement with the oblique data as the models in Ref. [15].
The suppression of T tree in the LRS scenario, without bulk custodial symmetry, can be understood as follows. After EWSB, the entire KK tower of states mixes and the zero mode (SM) wavefunctions get deformed away from a constant. This generates a large tree-level oblique correction to T , controlled by the KK-Higgs (i.e. KK-IR brane) coupling. The deformation of the SM W/Z wavefunction is then proportional to kr c π. Hence, the LRS contributions get suppressed by a factor Note that S from Eq. (1) is basically the same in the RS and LRS models. This is because the dominant contributions to S come from a universal shift in the light fermion-gauge field couplings [15]. This shift depends on the product of the mixing between the zero modes and KK gauge states (after EWSB) and the universal couplings of KK gauge fields to the light fermions. While the former decreases with shrinking volume the latter is increased so that the product is unchanged.
In brief, the LRS truncation does suppress the contribution from KK-tower mixing in the gauge sector, compared to the RS case, quite efficiently. However, a bulk custodial symmetry is still required in both the RS and LRS setups to control loop and cutoff scale contributions to T and to bring the scale of KK masses down to ∼ 3 TeV [15,22]. As we will show next, a more dramatic improvement can be achieved regarding the non-oblique and precision flavor observables.
Non-oblique Correction & Flavor Physics: Refs. [23,24] have shown that RS models flow to next-to-minimalflavor violation, where flavor changing effects are primarily from mixing with the third generation. The extra flavor breaking sources are quasi-aligned with their SM counterparts and the misalignment is at most of order the CKM matrix, but new sources of CP violation are present. Non-oblique Zbb and FCNC constraints are more involved since they depend on the amount of flavor non-universality, determined by the fermion zero-mode IR-brane profile values, f Q,u,d [23].
As shown in Ref. [23,25], once the overall scale of 5D Yukawa coupling λ 5 and t R -localization c-parameter are fixed, the localization of all other fermions is set by the measured masses and CKM mixing angles, assuming anarchical 5D Yukawa matrices via the relation where m x denotes 4D masses and x = u, d correspond to the up and down quarks respectively; f x are eigenvalues of F x . Here, we will keep λ 5 unscaled by LRS-truncation and fixed at its "RS value" [17,23] (see discussion below). Then, the amount of non-universality is unchanged, but the strength of the KK-mediated effects get decreased like the truncated LRS volume, parameterized by kr c π. This is the reason the non-universal precision observables are significantly suppressed in our model. For concreteness, in Table I, we give a set of c x and f x values that reproduce the SM quark masses and mixing angles; λ 5 /k = 2 in accordance with Ref. [23].  Bulk-RS constraints from the Zbb coupling require non standard fermion representations under the custodial symmetry, as well as a Z 2 symmetry [16] , in order to have m RS KK ∼ 3 TeV; otherwise, m RS KK > ∼ 5 TeV [15]. Without the custodial symmetry, there are various contributions to Zbb. The first originate from the gauge zeromode-KK-tower mixing due to EWSB and the enhanced coupling of the gauge KK modes to IR-localized b L . These corrections are proportional to (kr c π/m 2 KK )f 2 Q 3 [15], where f Q 3 is assumed to have the RS value. Thus, the LRS bound from these contributions is With √ y ≈ 2.4 we get m LRS KK > ∼ 2 TeV. The second type of correction to Zbb is due to O(1) mixing between b L and the exotic SU (2) R partner of t R [28]. This latter contribution will be absent for a choice of representation in which t R is a L-R isosinglet [16]. Note that without a bulk custodial symmetry, there is no exotic t R partner. However, there is a third type of correction to Zbb from the mixing of the KK modes of b R and the b L zero mode. This contribution is not truncated in the LRS model and is of or- To keep deviations in the Zbb coupling below 0.3%, we then need m KK (b R ) > ∼ 4 TeV. Interestingly, m KK > ∼ 3 TeV for gauge fields already implies the former bound for the KK modes of b R , in the LRS framework presented here. We hence conclude that all of the above constraints from Zbb can be satisfied for gauge sector m KK > ∼ 3 TeV, without any protective symmetries (however, as discussed above, this would be inconsistent with the bound from the T parameter).
We finally review the strongest constraints on generic bulk RS models, from ∆F = 2 processes due to tree level exchange of KK gluons. Ref. [23] showed that, with an IR localized Higgs, the ratio of RS and SM (V −A)×(V −A) contributions h RS ∝ (F 2 Q ) 2 ij (in the down quark mass basis). One can write At present, h RS < ∼ 0.3 [24,26,27]. However, the dominant contribution δ(ǫ K ) to ǫ K from (V − A) × (V + A) operators [26] is given by δ(ǫ K ) ∝ kr c π(F 2 Q ) 12 (F 2 d ) 12 [17] which is O(20) times smaller. This is not enough due to a matrix element chiral enhancement of O(11) and a O (7) factor from the running between the KK and weak scales, requiring m RS KK > ∼ 8 TeV. In the LRS case, both contributions are suppressed by y and thus m LRS KK > ∼ 3 TeV, no stronger than the oblique constraints. We note that the RS CP electric dipole moment problem [23] persists in our setup, as it is governed by 5D Yukawa interactions which are unchanged (for possible solutions to this problem see [17,29]).
Phenomenology: Gauge KK modes couple to UVlocalized light fermions (important initial states at colliders), with strength g KK ∼ g 4 / √ kr c π; g 4 is a typical 4D SM gauge coupling. We get g RS KK ∼ g 4 /6 and g LRS KK ∼ g 4 /2.5.  This leads to improvements in the LHC sensitivity, at fixed m KK , for the following reasons: (i) Typically broad states [10] become narrower by a factor y ∼ (0.2/0.08) 2 , (ii) branching ratio (BR) into light states (e.g. e + e − ) increases by a factor y 2 , (iii) from (i) and (ii) one can show that the signal S goes up by y 3 ∼ 250, while the background B drops as 1/y, over the resonance width. Hence, S/B in the LRS model is expected to go up by a factor y 4 ∼ 1500, a remarkable enhancement! These features lead to a larger LRS discovery reach and a way to test the validity of this setup and the underlying assumptions. As the LHC reach for KK gluons in bulk-flavor RS models is 3-4 TeV [10], the corresponding LRS reach can be as big as ∼ 5 TeV. The enhanced g KK in the LRS model could also allow access to the elusive EW gauge KK modes [14]. For example, the Z ′ → ℓ + ℓ − , ℓ = e, µ, golden decay modes which were close to hopeless within the RS case [14] could lead to discovery in the LRS setup. Using the same cuts as in Ref. [14], we find roughly 2000 (3) events with S/B and S/ √ B ≫ 1 [in fact O(100)] for M Z ′ = 2 (5) TeV and 100 fb −1 [30]. Note that given the significance of the signal, a discovery would be unambiguous over this range. Enhancement in production rate is expected for the SM KK fermions whose LHC discovery, in the RS model, would be quite challenging [31]. Furthermore, any bulk couplings mediated via λ 5 are relatively stronger due to LRS scaling. This would, in principle, be a direct test of our setup. Also, BR's of the neutral modes into composite states such as W L W L , Z L h and tt compared to those into light fermions will provide a robust test of the LRS construct.
TeV-scale spin-2 "graviton" resonances are distinct RS signatures [3,9]. Since M 5 ∼ 10 3 TeV, we eliminate the zero mode graviton, using UV-brane Dirichlet boundary conditions; we find x Gluon-fusion production of KK gravitons is dominant at hadron colliders, with a cross section proportional to [(k/M 5 )/kr c π] 2 (k/M 5 )(x G /m G ) 2 [9]. In a generic LRS model, both k/M 5 and kr c π shrink by a factor of y, decreasing this cross section by a factor of O(y). Thus, observation of the LRS spin-2 resonances is unlikely at the LHC (probably more unlikely than in the RS case [9]).
We also note that there can be LRS collider signatures in terms of deviations from the SM top-quark couplings. Within the RS scenario, there are two types of contributions to t → cZ of similar size [32]. One is through mixing between Z zero and KK modes which will be suppressed by y and probably unobservable in the LRS framework. However, the second one proceeds through mixing between t R and the KK modes of t L . This mixing is controlled by the 5D Yukawa which is left unchanged in our LRS construct and, therefore, t → cZ should be within the reach of the LHC. A similar effect yields an O(20%) shift in the Zt RtR coupling which is probably beyond the LHC sensitivity but may be observed at a future linear collider [33].  II: Summarized comparison of constraints and predictions in the RS and the LRS scenarios. For simplicity and definiteness, the Higgs is assumed to be on the IR-brane. The constraints correspond to lower bounds on gauge KK masses, in TeV. Here, we assume a custodial symmetry for the T parameter; a left-right Z2 symmetry is imposed to protect the Zbb coupling, unless denoted by * . The predictions in the last row correspond to a Z ′ of mass {2, 5} TeV, respectively. Holography: We now present a qualitative discussion of the LRS, using the AdS/CFT correspondence [34], following previous interpretations of geometric RS results [35,36] in the dual context of a strongly coupled large N 4D gauge theory [37]. We begin by studying the effects of LRS-truncation on the classical geometric relation between g 4 and the 5D gauge coupling g 5 [38]: 1/g 2 4 = τ UV + τ IR + log(k/κ)/(kg 2 5 ).
Here, τ UV and τ IR will be treated as small UV and IR quantum threshold corrections, respectively. For a generic comparison of couplings, we neglect τ UV,IR and keep g 4 fixed to its measured value. Thus, reducing the volume suppression kr c π (the log) requires lowering the value of kg 2 5 . In the dual CFT, this is interpreted as the contribution of CFT "quarks" to the running of external gauge couplings from the fundamental scale, M 5 , down to the TeV scale (just like the contribution of SM quarks to α QED running) [35,36,39]. Thus the relation kg 2 5 ∼ 4π/ √ N should hold between the dual theories. Explicit calculations [4,5] confirm that couplings among gauge KK modes, i.e. IR localized fields, are enhanced, compared to the corresponding zero mode gauge coupling, by √ kr c π ∼ kg 2 5 . This, in the dual CFT picture, corresponds to the coupling of three composites given by 4π/ √ N at large N [37,40]. Consequently, the truncated LRS volume is dual to N LRS ∼ yN RS > N RS , making the inter-composite interactions weaker. The weakened CFT interactions with the Higgs, a composite state, account for the decrease in T from Eq. (2).
In our LRS construct, we held the 5D Yukawa coupling λ 5 unscaled, lowering the 4D IR-brane cutoff to about 10 TeV, as in the RS case. In the dual language, this corresponds to separate dynamics for this sector, characterized by a "flavor" CFT with N F ∼ N RS < N LRS (N F ∼ 3 − 4 [17,23]). This independent CFT is linked to dynamical breaking of the 5D flavor group which is not completely broken by the bulk masses (unlike H which breaks the "chiral" symmetry) [23]. If the Higgs is dynamically realized as a pseudo-Goldstone boson (PGB) a similar different scaling for its potential should be applied, i.e. the dynamics which generates the PGB potential is characterized by N = N F . Otherwise, increasing N would induce a more severe fine tuning for the PGB potential [41]. However, in all the known models including the most realistic ones [41] the dominant contributions to this potential come from the top sector, corresponding to the flavor CFT, and not from the weakly gauged one, consistent with the above assumptions [42] 1 . This also explains why keeping S ∼ (v/f π ) 2 N LRS at the RS value does not lead to extra fine-tuning of v, since the "decay constant" f π here is from the weakly gauged dynamics which does not govern the Higgs potential. The constancy of the S parameter under truncation can be understood as follows. The main contribution to S is from the universal vertex corrections [15]. This is controlled by gauge zero-KK mode mixing, which scales as 1/ √ N , and the universal KK couplings to light fermions, which scales as √ N (see below). Therefore, S remains unchanged.
The non-oblique and FCNC contributions depend on the amount of non-universality in the couplings of the KK states to different generations. On the CFT side, this corresponds to the amount of partial compositeness for a given N F . The amount of compositeness follows from the observed masses and mixing angles [23], once λ 5 is set and the location of t R is decided. By fixing these to the RS value, the LRS amount of partial compositeness is then unchanged, and hence the non-universal observables are suppressed by truncation. This, generically, yields a better agreement with the data. It implies that interactions proportional to λ 5 (such as between the Higgs and two KK fermions) are stronger than the corresponding KK gauge interactions.
An important consequence of volume truncation is enhanced ρ − photon mixing, proportional to √ N [35,39], leading to larger couplings of light SM fermions to gauge composite/KK modes. The composite (KK) partial widths into elementary fermions scales as N , while the total width drops as 1/N . Hence, S ∼ N 3 and B ∼ 1/N , over the resonance width. Both effects yield stronger LRS signals at the LHC than for the RS case, since y = N LRS /N RS ≫ 1, as discussed before. This is analogous to how e + e − → ρ → µ + µ − is modified when N c is increased.
Finally, we emphasize that unless mentioned explicitly (as for λ 5 ) we have rescaled all the couplings in the theory according to the LRS value of N . This is why we did not get an enhancement in the KK graviton production. Also, we have neglected brane-kinetic terms to allow a transparent comparison of our model with generic RS models where such terms have sub-dominant effects on the above observables. Lastly, we note that M 5 ∼ 10 3 TeV does not suppress baryon and lepton number violation sufficiently. Such issues lie beyond the scope of this work, but can be addressed with discrete symmetries or in a UV-completion of the LRS model. However, dimension-9 operators suppressed by the LRS M 5 lead to n −n oscillations at acceptable levels [43] and may be accessible in near future experiments.
In summary, we presented the "Little Randall-Sundrum (LRS)" model of hierarchy between the fla-vor and weak scales which is much less constrained than M P -weak warped scenarios. Here, the 5D cutoff scale M 5 ∼ 10 3 TeV is chosen to suppress unwanted lightquark operators sufficiently and the weak scale is obtained from O(M 5 ) scales by warping; the flavor puzzle is addressed by fermion localization, as in the RS model. Even without a bulk custodial symmetry, the "tree-level" lower bound on LRS gauge KK masses is at ∼ 5 TeV; the RS bound is m RS KK > ∼ 12 TeV. Loop and higher dimension contributions to T raise m KK > ∼ O(10) TeV, without a custodial symmetry, in both RS and LRS models. Custodial symmetry can be imposed if desired, leading to an oblique lower bound at ∼ 3 TeV. As we kept the overall Yukawa scale unchanged, the most severe RS-type constraints are much better behaved here: non-oblique constraints from Zbb, without a protective Z 2 symmetry, are absent for m KK > ∼ 3 TeV and the FCNC bounds are largely relaxed. We have summarized the comparison between the RS and LRS frameworks regarding various precision constraints in Table II. As can be seen from this table, the LRS framework is never more constrained than its RS counterpart and in many cases it is much more compatible with data. Typical LRS graviton KK modes are more elusive than those ofM P -weak warped models. However, the light-fermion LHC-production rate and BR's for LRS gauge KK modes are much bigger than the corresponding RS values and yield a signal ∼ y 3 times larger; y ∼ 6 is the LRS truncation factor. We hence conclude that the LRS model is a good candidate for new physics and may soon be uncovered at the LHC, or perhaps probed at a future linear collider.
We thank Csaba Csaki, Shri Gopalakrishna, Ami Katz, and Rob Pisarski for useful discussions. We especially thank Kaustubh Agashe for many helpful comments. G.P. thanks the Aspen Center for Physics and the HEP groups at Boston and Harvard universities for hospitality while working on this paper. The work of A.S. and H.D. is supported in part by the United States Department of Energy under Grant Contracts DE-AC02-98CH10886 and the work of G.P. is supported by the NSF under grant 06353354.