A Higgs in the Warped Bulk and LHC signals

Warped models with the Higgs in the bulk can generate light Kaluza-Klein (KK) Higgs modes consistent with the electroweak precision analysis. The first KK mode of the Higgs (h_{1}) could lie in the 1-2 TeV range in the models with a bulk custodial symmetry. We find that the h_{1} is gaugephobic and decays dominantly into a t\bar{t} pair. We also discuss the search strategy for h_{1} decaying to t\bar{t} at the Large Hadron Collider. We used substructure tools to suppress the large QCD background associated with this channel. We find that h_{1} can be probed at the LHC run-2 with an integrated luminosity of 300 fb^{-1}.


Introduction
The Randall-Sundrum model (RS model) [1], as originally proposed, is a five-dimensional model with a warped metric with the fifth dimension y compactified on an S 1 /Z 2 orbifold of radius, R. Two branes are located at y = 0 and y = πR ≡ L and are called the UV and the IR branes respectively. Starting with a bulk gravity action one can show that the solutions to the Einstein equation imply for the warp factor A(y) A(y) = ±k|y| , (1.2) where k 2 ≡ −Λ/12M 3 with M being the Planck scale. A value of kL ∼ 30 is sufficient, through the warp factor, to generate a factor of v/M ∼ 10 −16 (where v is the vacuum expectation value of the SM Higgs field) thereby stabilising the gauge hierarchy. This suppression factor is, however, material for all fields localised on the IR brane and, indeed, in the original RS model this was the case for all SM fields with only gravity localised in the bulk. With SM fields localised on the brane, mass scales which suppress dangerous higher-dimensional operators responsible for proton decay or neutrino masses also become small and this spells a disaster for the RS model. Wisdom gleaned from AdS/CFT correspondence also gives an understanding of the need to go beyond the original RS model. The fields localised on the IR brane turn out, through the correspondence, to be composites of operators in the fourdimensional field theory that is dual to the RS model. The latter then turns out to be dual to a theory where all the SM fields are composite, which is not viable. However, a theory of partial compositeness is viable and can survive experimental constraints. This corresponds to a RS model where the SM fields are localised in the bulk.
This was, in fact, the motivation to move the SM fields into the bulk and construct what are called the Bulk RS models. For reviews, see Refs. [2,3]. In such models, often, the Higgs is still kept localised on the IR brane so that the gaugehierarchy solution discussed above continues to hold. The big gain that accrues in the Bulk RS models is that the differential localisation of SM fermions in the bulk gives rise in a natural way to the Yukawa-coupling hierarchy [4][5][6]. The other features of Bulk RS models are that they give rise to small mixing angles in the Cabibbo-Kobayashi-Maskawa (CKM) matrix, provide a natural way of obtaining gauge-coupling universality and allow for the suppression of flavour-changing neutral currents [7][8][9][10][11].
As shown in later work on Bulk RS models 1 , even the Higgs need not be sharply localised on the IR brane but only somewhere close to it in order to address gaugehierarchy. This freedom allows for more interesting model-building possibilities. It is this latter class of models which will be the focus of the present paper.
The serious issue to contend with in Bulk RS models is that of electroweak precision. In models with only gauge bosons propagating in the bulk, the constraints on the masses of the Kaluza-Klein (KK) gauge bosons are very strong (of the order of 25 TeV) though this is somewhat ameliorated by also allowing SM fermions in the bulk, especially with fermions of the first and second generation localised close to the UV brane. Even in this case, there are unacceptably large couplings of the KK gauge bosons to the Higgs resulting in severe T -parameter constraints. One way of addressing this problem is called the Custodial symmetry model. In this model we have an enlarged gauge symmetry [13,14] in the bulk i.e an SU (3) c × SU (2) L × SU (2) R × U (1) y that acts like the custodial symmetry of the SM in protecting the ρ parameter and this extended group is then broken on the IR brane to recover the SM gauge group. This extended symmetry takes care of the T -parameter but non-oblique Z → bb corrections, coming from the fact that the fermions are not all localised at the same point in the bulk, persist which are then addressed by a suitable choice of fermion transformations under the custodial symmetry group. The bound on the lightest KK gauge boson mode comes down to about 3 TeV [15,16]. 2 The upshot of the above discussion is that, it is possible to get the masses of the KK modes of SM particles within the reach of collider searches. Indeed, there is already a significant amount of literature suggesting search strategies for KK gauge 1 For a review, see [12]. 2 There are other approaches in dealing with the electroweak precision constraints such as the deformed metric model [17,18] or a model using brane kinetic term [19], but we will not consider these approaches here.
bosons [20][21][22][23][24][25][26] and KK fermions [27,28] at the Large Hadron Collider (LHC). In contrast, KK modes of bulk Higgs have not received their due attention. The zero mode of the bulk Higgs has been studied in [29,30] and in [31] the CP-odd excitation of the bulk Higgs in the deformed metric model has been studied. It is to the search for the first KK excitation of the Higgs in the context of the custodial symmetric model at the LHC that we devote the rest of this paper.

Bulk Higgs Models
The action with the Higgs propagating in the bulk [12] is given by λ 0 , λ 1 represent the scalar potential on the UV and IR brane respectively. M 0 , M 1 are boundary mass terms on the UV and IR brane respectively. A quartic term is added on the IR brane to ensure electroweak symmetry breaking. a represents the dimensionless bulk mass parameter defined in the units of curvature, k.
H(x, y) is a scalar field that can be expanded in terms of its KK tower as f h n h n / √ πR where h n (x) is the nth KK field with mass m n and f h n (y) is the profile. The equation The electroweak symmetry breaking occurs on the TeV brane and the zero mode(h 0 with m 0 = 0) gets its mass from the boundary potential on the TeV brane. Thus, the vev and the zero mode follow the same bulk profile and one can say that the 5D vev of the Higgs field is entirely carried by the zero mode. The potential on the UV brane is chosen such that the profile of the zero mode and the vev localises on the TeV brane and the boundary condition on the TeV brane fixes the mass of the Higgs with the identification M 1 = bk. b represents the dimensionless brane mass parameter in units of k. Thus, we have Similarly, the bulk equation of motion of h 1 gives us the profile having mass given by From figure 1 we see that, depending on the value of b, the mass of h 1 can be as low as the third of the first gauge boson KK mode mass. This implies that the h 1 mass can be as low as a TeV in the custodial symmetry mode. When b 2, the mass of the h 1 is heavier and can not be directly probed at the LHC. In our analysis, we have considered a h 1 with mass of 1 TeV and beyond. It is also important to note that the best-fit point from the electroweak analysis presented in Ref. [16] gives a value of b = 2. This value of b is consistent with a h 1 mass of 1 TeV and, in other words, such a value of h 1 mass passes the acid test of electroweak constraints. It may be noted that the normalisation of the profile for the zero mode fixes the coupling of the SM Higgs with all the other SM particles. Thus, we do not expect any deviation from the observed signal strength measurement of the SM Higgs at the LHC [32][33][34].
The SM Higgs mixes with the radion, which is the field parametrising the fluctuation between the two branes. In the limit of negligible back reaction, the kinetic term involving the radion and the Higgs induces the mixing [35]. As the vev of the bulk Higgs is carried out by the zero mode, the orthogonality condition prevents the mixing of the first KK mode with the radion.
One can calculate the following tree-level interaction of the KK modes with the SM particles from the action (A.2), • h 1 → V 0 V 0 : The term that governs the coupling is The zero mode of the KK gauge bosons(i.e the W µ , Z µ ) have a flat profile and hence, the tree level coupling vanishes following the orthogonality condition of the KK profiles.
• h 1 → h 0 h 0 : Unlike the radion-h 1 mixing term, this interaction comes from the quartic scalar potential added on the TeV brane (A.2) where the trilinear coupling of the SM Higgs is The decay of the h 1 to SM Higgs is given by • h 1 → t 0 t 0 : Relative to the Yukawa coupling term of the SM Higgs to tops, where Considering the reduced normalised profiles we can write the Yukawa coupling of h 1 to fermions with respect to the SM Yukawa coupling as follows:- For a flat 5D metric, the reduced normalised profiles for zero-mode fermions is given by Using the c L =0.4 and c R =0 and b = 2 we obtain y h1t L t R = 1.0755y SM . (2.12) The partial decay widths of the KK higgs to the pair of gluons, photons, tops and SM Higgs are given by, In the above equations, I q = 3[2λ q +λ q (4λ q −1)f (λ q )] and I l = 2λ l +λ l (4λ l −1)f (λ l ), , are the form factors.
Having listed the couplings above for completeness, we would like to point out that the branching ratio of h 1 decaying to tops is overwhelmingly large as shown in figure 3. Thus, we focus on the tt decay mode of h 1 in this analysis.
Since, the couplings of the KK Higgs to the massive gauge bosons vanish at the tree level, the production of the KK Higgs via vector boson fusion is heavily suppressed. As the Yukawa coupling of the tops with the KK Higgs is of O(1), the KK Higgs can be produced in association with tops or via gluon-gluon fusion with tops running in the loop. The associated production of the KK Higgs with tops is suppressed by two orders of magnitude in this mass range. Thus, the only dominant production mode of the KK Higgs is via gluon-gluon fusion. Even before we launch into our analysis, we should check what constraints existing collider data from tt production places on a 1 TeV resonance decay. Recently, the ATLAS [36] and the CMS [37] collaborations at the LHC have presented their measurements of 3 h 1 at the LHC As discussed earlier, the h 1 is produced via gluon-gluon fusion with tops propagating in the loop and it further decays to tt at the LHC. Thus, our signal is characterised by two tops. Model files have been obtained with FEYNRULES [38], and the signal events are generated by interfacing it with MADGRAPH [39] with the parton distribution function NNLO1 [40]. Since, we are considering the scalar having mass beyond TeV, the tops coming from the scalar are in the boosted regime with most of the tops having transverse momentum in the range of 200 -500 GeV as can be seen from the figure 4 and the decay products of the top will mostly lie in a single hemisphere as can be seen in figure 5. To optimize the signal, we have considered the hadronic decay of tops that can be tagged using the HEPTopTagger [41,42] algorithm. The backgrounds for our signal can be categorised as • Reducible background: The dominant reducible background in this topology is the dijet background. Once we demand top tagging, this background reduces drastically. It can be controlled further using a high-transverse momentum (p T ) cut on the tagged top.
• Irreducible background: The irreducible background arises from the pair pro-duction of tops via QCD processes. As expected, the tagging efficiencies of the two tops are similar to the signal and hence, we need to use the decay kinematics to isolate the signal.
The SM tt and jet events are generated using PYTHIA 8 [43]. The showering and the hadronisation of the signal event as well as the background events have been carried out using PYTHIA 8. To generate background events with larger statistics we have divided our analyses into different phase space regions 3 depending upon the mass of the KK Higgs that we are probing. In figure 4, we have plotted the distribution of transverse momentum at the parton-level for leading (sub-leading) tops from h 1 having mass of 1.5 TeV, SM tt background. As discussed earlier, the transverse momentum of tops coming from h 1 are mostly peaked near half of the h 1 mass whereas the SM backgrounds largely peak at the lower transverse momentum region. Also, the decay product of the tops coming from the signal can be encompassed within a fat jet of radius ∼ 1.5 ( figure 5). Keeping this in mind, we split our analysis into two regions. In the first region, we have reconstructed the jets using the Cambridge Aachen (C-A) algorithm [44,45] with jet radius (R = 1.5), p T > 250 GeV and |η| < 2.4. In the second region we have used a slightly higher value of transverse momentum to reconstruct the fat jet i.e p T > 350 GeV. The first part is optimised for the search of the h 1 in the range of 1 TeV whereas the second region is proposed when its mass is around 1.5 TeV and beyond. These two fat jets are then considered as an input for the HEPTopTagger. The algorithm of the HEPTopTagger is briefly described here, • Inside the fat jet one looks for hard substructure using a loose mass-drop criterion. For a splitting of the fat jet J → j 1 j 2 , one demands that for m j2 < m j1 , m j2 > 0.2m J . The splitting continues till m j1 < 30 GeV. The fat jets having at least 3 subjets are allowed.
• Once we get 3 subjets, the subjets are filtered with R f ilt = 0.3 and 5 filtered subjets are retained. Only those fat jets are considered which give total jet mass close to the top mass. These filtered subjets with correct top mass reconstruction are then reclustered into three subjets.
• These three subjets are then made to satisfy top decay kinematics. One can construct three pairs of invariant mass with these three subjets out of which two of them are independent. In the two dimensional space determined by the pair of invariant mass, top-like jets represent a thin triangular annulus (as one of them always reconstructs a W). On the other hand, the background is concentrated in the region of small pair-wise invariant mass.
We consider two such 'top-tagged' jets for our further analysis. At this stage, we have very few (almost negligible) events coming from the dijet background. The h 1 is produced mostly at rest: as a result the top pairs are back to back. We have plotted the distribution of the absolute value of difference in rapidity (|∆η|) of the 'toptagged' pair coming from the signal as well as from the SM background in figure 6. For the tt background, the distribution peaks near |∆η| ∼ 0 whereas the tops coming from the signal have a larger spread. We found that a minimum cut on ∆η helps us to isolate signal from background. When the mass of the KK Higgs is around 1 TeV, we have selected events with transverse momentum of the 'top-tagged' pairs (p l T and p sl T ) greater than 350 GeV. The combination of minimum cut on the transverse momentum and the minimum cut on pseudorapidity helps us to suppress the dijet background further. The efficiency of the minimum cut on ∆η increases as the mass of the KK Higgs increases. Thus, for the KK Higgs having a mass of 1.5 TeV and beyond, a minimum cut on pseudorapidity is sufficient to reduce QCD as well as tt.
After the angular cut, we made sure that the tops coming from the signal reconstruct the h 1 mass. We enhance the signal efficiency by demanding that the invariant mass lies within a window about the h 1 mass. The distribution of the invariant mass of the pair of top-tagged jets (m tt ) for the signal and tt background is plotted in figure 7. Due to the effect of final state radiation (FSR), the peak of the invariant mass gets smeared mostly in the lower region of m tt , as can be seen in figure 7. The cut flow Since the number of background events are comparable to the number of signal events, we calculated the significance 4 using [46], where S is the number of signal events and B is the number of background events. The discovery reach for the h 1 of 1.0 TeV is about 650 fb −1 luminosity for √ s = 14 TeV. As the mass of the KK Higgs increases, the dijet as well as tt backgrounds fall rapidly and one can probe it with even lower luminosity. In figure 8, we plotted the luminosity required to discover the KK Higgs with 5σ discovery. In the range of 1 TeV, due to large SM backgrounds, we applied stronger cuts which reduces the signal. Thus, we need more than 600 fb −1 of integrated luminosity to discover it. Once the mass increases, the SM background falls and it is possible to observe the KK Higgs having a mass around 1.2 TeV with about 200 fb −1 of integrated luminosity at √ s = 13 TeV. Beyond 1.8 TeV, the production cross section decreases severely due to s-channel suppression and thus, we need about 1000 fb −1 of integrated luminosity.

Conclusion
In order to address the gauge hierarchy problem, it is sufficient to have Higgs close to the IR brane (b ≥ 2) and not necessarily brane localised. We find that with b = 2, which is the best fit value consistent with the electroweak analysis, one can have h 1 much lighter than the first KK mode of gauge bosons. The orthogonality relations among the KK profiles prevent the coupling of the h 1 to the massive gauge bosons at the tree level. We observe that the branching ratio of h 1 decaying to a pair of SM Higgs is about 1%. Thus, the h 1 decays dominantly to a pair of tt. We have focussued on the tt decay mode of h 1 where both the tops are decaying hadronically. Such a h 1 is produced at the LHC via gluon-gluon fusion. The reducible background for this topology is the SM dijet background and the irreducible background is the SM tt background. We find that using the substructure of the boosted top, especially tagging the fat jets using HEPTopTagger, QCD background reduces drastically. We find that on applying cuts on the kinematic variable such as transverse momentum (p T ) and absolute value of the rapidity difference (∆η) of the tagged-top jets, we could suppress the irreducible background as well. In fact, one can discover a h 1 having a mass lying in the range of 1.1 -1.6 TeV at 13 TeV center of mass energy with an integrated luminosity of about 300 fb −1 . The high luminosity LHC on the other hand will be able to probe the full range between 1 and 2 TeV.
To conclude, we have shown that it is possible to explore the first KK mode of Higgs hitherto considered beyond the reach of LHC.

Acknowledgements
We would like to thank Abhishek Iyer for discussion and for collaboration at the early stages of this work. KS was visiting CRAL, IPNL Lyon and Theory Division, CERN while this work was carried out and would like to gratefully acknowledge their hospitality.