LHC signals for Singlet Neutrinos from a Natural Warped Seesaw (I)

Recently, it was shown in arXiv:1512.06742 that a straightforward implementation of the type I seesaw mechanism in a warped extra dimensional framework is in reality a {\em natural} realization of"inverse"seesaw, i.e., the Standard Model (SM) neutrino mass is dominantly generated by exchange of pseudo-Dirac {\em TeV}-mass SM singlet neutrinos. By the AdS/CFT correspondence, this scenario is {\em dual} to these singlet particles being composites of some new strong dynamics, along with the SM Higgs boson, with the rest of the SM particles being mostly elementary. We study signals from production of these heavy neutrinos at the Large Hadron Collider (LHC). We focus on the scenario where the strong sector has a global $SU(2)_{\rm L} \times SU(2)_{\rm R} \times U(1)_{\rm X}$ symmetry; such a left-right (LR) structure being motivated by consistency with the electroweak (EW) precision tests. The singlet neutrinos are charged under $SU(2)_{\rm R} \times U(1)_{\rm X}$ symmetry, thus can be produced from $W^{ \pm }_R$ exchange, as in four-dimensional (4D) LR symmetric models. However, the direct coupling of light quarks to $W^{ \pm }_R$ is negligible, due to $W^{ \pm }_R$ also being composite; nonetheless, a sizable coupling can be induced by mixings among the various types of $W^{ \pm }$ bosons. Furthermore, $W^{ \pm }_R$ decays dominantly into the singlet and {\em composite} partner of charged lepton. This heavy charged lepton, in turn, decays into SM lepton, {\em plus} $Z$/Higgs, thus the latter can be used for extra identification of the signal. For a benchmark scenario with $W^{ \pm }_R$ of mass 2 TeV and singlet neutrino of mass 750 GeV, we find that, in both the di-lepton + di-jet + Higgs and tri-lepton + Higgs channels, significant evidence can be seen at the LHC14 for an integrated luminosity of 300/fb and that even discovery is possible with slightly more luminosity.


Introduction
The seesaw mechanism [1] is a very attractive and hence perhaps the most popular one for explaining the extreme smallness of the Standard Model (SM) neutrino masses relative to those of the charged fermions. The basic idea is illustrated by the following schematic  [3,4], which is the subject of further study in this paper.
In the original seesaw, the typical choice is that the above Dirac mass term between the two neutrinos is of order the Higgs VEV, v (or somewhat smaller), and similarly, the Of course, the latter hierarchy can be technically natural (i.e., radiatively stable), but the point is that realizing all this might require additional dynamics. For example, if this scale corresponds to spontaneous breaking of a gauge symmetry [as in SU (2) L ×SU (2) R ×U (1) B−L or left-right (LR) symmetric models, i.e., N R is part of a doublet of SU (2) R ] by a scalar VEV, then we have to explain why this scalar mass term is much smaller than the Planck scale.
An alternative is to set the singlet mass scale to be close to the IR (low-scale seesaw), for example, weak scale: low/TeV-scale seesaw : M N ∼ M IR (∼ TeV) (no tuning) (4) but then the tuning is transferred to the Dirac mass term instead: Finally, the so-called "inverse" seesaw [2] seeks to have natural choices for both the Dirac mass term between doublet and singlet neutrinos (i.e., v) and the mass term for the singlet by itself (that too at the IR/weak scale). However, in the inverse seesaw, the singlet neutrino is Dirac fermion, requiring introduction of another left-handed (LH) singlet denoted by S: In addition the second singlet has a small Majorana mass term denoted by µ so that the SM neutrino mass formula ends up looking like: Of course, tuning is then shifted to the Majorana mass term for S: So, it seems that four-dimensional (4D) models of seesaw might not be entirely satisfactory as far as explaining fully the small observed SM neutrino mass. Recently, it was emphasized [4] that • a natural realization of seesaw mechanism occurs in the warped extra dimensional framework. 1 This framework is dual, following the AdS/CFT correspondence, to varying degree of compositeness of the SM particles. In a sense, this implementation actually features both highscale and inverse seesaw mentioned above. Namely, from a bottom-up viewpoint, the SM neutrino mass is generated by exchange of pseudo-Dirac singlet states as in inverse seesaw case. Remarkably, • the smallness of the required Majorana mass term (µ) for the inverse seesaw is itself due to a high-scale seesaw: schematically, we have (with M IR ∼ TeV as usual) Note that, even with the above nice feature, we still need (as allued to above) the other hierarchy for getting the observed SM neutrino mass, i.e., M UV M Pl : this seems to be a tuning at first sight, but we will see that this is also explained in warped/composite seesaw.
In detail, the dual CFT picture affords the most transparent understanding of this physics as follows (see more discussion in [4] and some using 5D model in Sec. 2 of this paper).
The SM Higgs boson arises as a composite of some new strong dynamics which confines at the ∼ TeV scale. Rest of the SM (i.e., all the gauge fields and fermions) start out as elementary degrees of freedom which are external to the strong dynamics, but they "mix" with appropriate composites of the latter. Thus, the actual SM particles are admixtures of the two sectors. Such "partial compositeness" of the SM fields allows them to couple to the SM Higgs, thus acquiring mass from its VEV. In particular, for the case of charged SM fermions, the picture is that external SU (2) L doublet and singlet fermions mix separately with respective composite ones, starting at the UV cut-off. Then, only in the far IR, i.e, at ∼ TeV scale, these two types of composites (and hence the corresponding external fermions as well) "connect" to each other via the Higgs VEV.
For the neutrino sector, the story starts out similarly, i.e., we add to the SM lepton sector, an external (chiral) SM singlet, denoted by N R , which mixes with an entire composite SM singlet tower from ∼ TeV upwards. However, from then on, there is a departure in the script (vs. that of charged fermions), again, kind of similarly to the usual seesaw models, but with some crucial difference as follows. Obviously, this concerns the "fate" of the external N R : namely, we assume that the strong dynamics in isolation preserves lepton number so that the composite singlets are purely Dirac to begin with. On the other hand, the external sector mass terms and interactions need not preserve lepton-number, for example, N R has a Majorana mass term, M N , which is close to the UV-cut-off, say, M Pl .
However, even though lepton-number is violated at the UV cut-off, we cannot write down a SM neutrino mass operator at this stage, since the SM Higgs boson VEV is not "born" yet. Instead, the relevant effect of Majorana N R is that its coupling to strong dynamics will inject lepton-number violation into the strong dynamics also; in particular, integrating out N R (again, close to the UV cut-off) generates Majorana mass terms for the composite singlet states: note that these Majorana mass terms are for the left chirality of composite, since that is the one with mass mixing term with external N R .
So, we start seeing the "ingredients" for a inverse seesaw model, with the seeds being sown in the UV; in particular, it is the two chiralities of the composite singlet who play the role of the N , S fields of the usual 4D model of this type! Thus, we naturally have warped/composite seesaw : M N S ∼ TeV/compositeness scale (10) Moreover, as already advertised above, we have an explanation for smallness of the Majorana mass term for S [i.e., µ in Eq. (7)]. Namely, for the TeV mass composites, this mass term will precisely be of the form of µ in Eq. (9) above, i.e., the "TeV" in the numerator there comes from the above-mentioned mass mixing term (between N R and LH composite) and M UV in denominator is just the (Majorana) mass term for N R with itself. We will argue in a bit that this "effective" UV scale can actually be naturally smaller than Planck scale. The final cog in this wheel is the Dirac mass term for the composite singlet with the SM SU (2) L doublet neutrino: similarly to the case of the charged fermions, this arises from coupling of composite singlet to Higgs VEV and composite doublet, latter mixing with the external SM neutrino. Of course, one difference from charged fermion case is "absence" of external leg on the singlet side (since N R decoupled); so schematically, we get i.e., with two external fermions, we would have gotten m τ vs. its "square root" here with only external doublet present 2 . In other words, • the composite singlets act as a "bridge" between EWSB in the IR and lepton-number violation in the UV, both of which are required in order to generate (Majorana) SM neutrino mass.
Note that plugging Eqs. (10), (9) and (11)  Of course, this is a feature in general of inverse seesaw models so that such signals have been studied before [5,6], but (as we will show here) the compositeness of the singlets make a difference!
In a series of papers (this being the first), we initiate the study of LHC signals for the ∼ TeV mass singlets in the natural realization of (inverse) seesaw in this warped/composite Higgs setting. We begin here by focussing on a specific, but well-motivated model within the above framework. Namely, • we assume that the strong dynamics has a global symmetry (in the EW sector) which being a combination of U (1) X and the U (1) contained in SU (2) R ] 5 . In the canonical case, we would identify X = (B − L) as in 4D LR models, but in general we could choose other representations under the extra U (1). The motivation for such an extension of the EW (global) symmetry in the present context is not the one for the 4D LR models, i.e., parity restoration at higher energy scales, but rather that it provides a custodial symmetry for suppressing the contributions of the strong dynamics to the EW precision tests, in particular, the T parameter. Thus, even with the choice of X = (B − L), there is then no need for an elementary (i.e., external to the strong sector) W ± R , i.e., charged gauge boson of SU (2) R group, in this model. Similarly, the combination of U (1) B−L and U (1) in SU (2) R which is orthogonal to U (1) Y -often denoted by Z -is not gauged, unlike in 4D LR models, i.e., the external sector does not respect the extended EW symmetry. 6 We will mostly use the elementary-composite sector picture (called two-site model [12], but augmented now by the composite singlet neutrinos) in our actual LHC signal analysis.
Even though we do not have elementary W ± R /Z in this model, given the above global symmetry of strong dynamics, we do have • composite W ± R and Z 7 , which do couple to singlet neutrinos (cf. composites of SM gauge bosons obviously do not); this simultaneous similarity (i.e., "existence" of W ± R and Z ) and difference (their compositeness vs. elementary nature) from 4D LR models will be crucial to the analysis of signals for the present model. 5 The warped 5D dual of this scenario is that the bulk EW gauge symmetry is extended as above and broken down to the SM subgroup on the UV brane. 6 As a bonus, with such a symmetry structure, we automatically realize the pure Diracness of composite singlets vs. large, possibly close to UV cut-off, Majorana mass term for the external singlet. 7 We will denote them simply by the same symbols, since there is no chance of confusion with elementary ones in this model. Also, strictly speaking, we have to assume degeneracy of spin-1 composites here in order to classify mass eigenstates in this way: we will consider the case of non-degeneracy in a follow-up paper, where we will give more details of this issue.
For the lepton sector, we indeed make the canonical choice of fermion representations, but now for the composites, since it is that sector which has the SU (2) R symmetry, i.e., • the composite (denoted by ψ e ) with which the external RH charged lepton mixes 8 is part of a doublet of the (global) SU (2) R of strong dynamics, whose other component is the composite RH neutrino (denoted by ψ N ), i.e., with which external N R mixes as mentioned above. 9 Both ψ's have Dirac mass ∼ TeV and are vector-like under the SM gauge and strong dynamics global symmetries.
We begin by considering the production of ψ N via decays of on-shell W ± R ; again such a signal has been studied extensively in the case of usual, 4D LR models [5], but the difference here is that W ± R is composite vs. quarks inside proton being mostly elementary. So, naively, this coupling seems to be negligible (i.e., ∝ tiny admixture of composite in SM light quarks or the corresponding Yukawa couplings). Nonetheless, we discuss how • a significant, albeit still mildly suppressed relative to SM, light quark-W ± R coupling is induced.
This arises by a combination of elementary-composite mixing for W ± 's corresponding to SU (2) L (denoted by W ± L ) 10 and composite W ± L − W ± R mixing induced by Higgs VEV, with the near degeneracy of these composites in a "minimal" model 11 amplifying the Higgs VEV effect (see reference [7,8] for the 5D version of this effect). (We will consider the case of non-degenerate spin-1 composites in a follow-up paper.) In fact, such a mild suppression of production of W ± R (as compared to usual 4D LR models) is perhaps "welcome" in the sense that the LHC early run 2 searches are already constraining 2 TeV W ± R in the usual case, but with compositeness, such low scale for W R would then (i.e., given smaller cross-section for the same mass) still be allowed. At the same time, as we will show, the coupling is sizable enough that discovery (for 2 TeV W ± R and ∼ 750 GeV ψ e,N 12 such that the above decay is allowed) by the end of run 2 (∼ 300 fb −1 ) would be possible. 8 called "electron" here for simplicity, even though we extend this to the second and third generations also 9 In detail, one might need two such SU (2)R doublet composites per generation -corresponding to two different 5D fields -in order to obtain the correct SM charged lepton vs. SM neutrino Dirac mass term, i.e., external charged lepton might actually mix with a different composite tower than the SU (2)R partners of composite SM singlets associated with the SM neutrino mass. However, this modification does not (qualitatively) affect the present discussion. 10 Recall that there is no elementary gauge boson mixing directly with composite W ± R . 11 This is dual to the 5D model with no IR brane-localized kinetic terms for bulk gauge fields. 12 We could contemplate even lighter singlet neutrino, but accomplishing such a hierarchy might require tuning, for example, too large brane-localized kinetic terms, given that gauge KK cannot be below ∼ 2 TeV due to constraints from EWPT.
Moving onto the relevant decays of W ± R 13 , first of all, the largest coupling of W ± R involves composite partner of SM e R and the composite singlet neutrino, i.e., ψ e and ψ N , cf. SM e R and singlet neutrino in the usual, 4D LR case. The singlet neutrino decays predominantly (as in 4D LR models) into SM doublet lepton and Higgs doublet (including physical Higgs and longitudinal W/Z) via the associated Yukawa coupling 14 : the channel we will focus on here (based on smaller background, thus more visibility) is e L + W . On the other side, ψ e will similarly decay: we will consider e L + Z long /h final state here. Thus, we see that there is • an "extra" Higgs/Z (vs. usual, 4D LR models) among the decay products of the W ± R , which, assuming it is tagged, can be used to reduce the SM background.
Moreover, it then allows us to possibly reconstruct the full decay of ψ e , thus determining its mass, which is same as that of ψ N [given the SU (2) R symmetry]. Including decays of W from ψ N , we then have • two search channels, i.e., dilepton+ dijet (hadronic decay of W ) and tri-lepton (leptonic decay of W), along with Higgs/Z boson. 15 We will study both of these and find them to be complementary, for example, rate is larger for the former (based simply on corresponding branching ratios of W ), but so is possibly SM background, given that leptons are typically "cleaner". (Of course, for the case of hadronic decay of the W from ψ N , that side is also fully visible and hence can furnish information on ψ e,N masses.) Finally, in addition to W ± R , we consider production of ψ e,N pairs from decays of on-shell Z . Once again, the "direct" coupling of quarks inside proton to Z is negligible; however, mixing does create a larger coupling (just like for the case of W ± R above). Note that in usual, 4D LR models, Z is typically heavier than W ± R , for example, assuming both (being elementary) get their mass from some scalar VEV, just like the case of SM W/Z. Hence, production cross-section of Z tends to be smaller than that of W ± R . However, in the seesaw model being studied here, • the W ± R and Z can be almost degenerate, since their masses arise from the compositeness scale so that Z signal can be comparable to W ± R . 13 Other decay channels for W ± R include various components of the Higgs doublet: these were studied in [8], but singlet neutrino was not included there.
14 Note that this coupling is indeed small, given that it involves degree of compositeness of SM (doublet) lepton, but there is not much of an "option" here in terms of decay channel, given that lepton-number is (approximately) preserved. 15 Note that even in the usual, 4D LR models, one can also get Higgs/Z boson from singlet neutrino decay, but then we lose lepton(s), i.e., final state with be lh+ MET, thereby increasing SM background (for example, SM W h production will then be relevant), as opposed to our case of Higgs along with di-or-tri-leptons.
Here is the outline of the rest of this paper. We begin in the next Sec. 2 with a brief review of the basic seesaw model in the warped extra dimensional framework and present details of the implementation in the context of the SU (2) R extension of the SM EW symmetry mentioned above. In Sec. 3, we outline the "simplified", i.e., two-site approach [12] to studying the 5D model that we will employ in our actual analysis of LHC signals. We then discuss our main results, starting with production cross-sections and decay branching ratios of various heavy particles in Sec. 4, followed by computations of SM background and thus the discovery potential for the new particles in Sec. 5. Here, we also mention/briefly discuss strategies (post-discovery) for distinguishing the composite/warped seesaw model from the usual, 4D LR one. We conclude and present some directions for future work in Sec. 6.

5D natural warped seesaw model
In this section, we provide a brief review of seesaw model in 5D warped extra-dimensions.
After discussing general features of warped seesaw, we will focus on a model with the extended bulk gauge symmetry: SU (2) L × SU (2) R × U (1) X . Our studies of LHC signals are performed using the simplified two-site model of the full 5D warped model. Hence, our discussion about the full 5D model in this section will be brief, leaving details necessary for the phenomenology to Sec. 3 of the two-site model. More details about the 5D results, along with their 4D CFT dual description, can be found in [4].
We begin our discussion by taking usual Randall-Sundrum framework with all SM fermions and gauge bosons propagating the bulk of a slice of AdS 5 . For concreteness, we consider SM Higgs to be localized on the IR brane. The 5D SM gauge singlet field, N , which is the analog of the the right-handed neutrinos of the usual, 4D seesaw models, propagates the bulk.
Like all 5D fermion fields, N can be decomposed into both left (L( and right (R) chiralities (denoted by N L,R , respectively) from the 4D viewpoint. N R couples to SM SU (2) L lepton doublet, in particular left-handed neutrinos, and the Higgs on the IR brane with 5D Yukawa coupling y 5D . In addition, N R acquires large Majorana mass, which is taken to be localized on the UV brane. These can be summarized in the following 5D Lagrangian where since N is 5D fermion field, it is four component spinor, containing N L and N R 4D Weyl spinors. c N k is 5D mass parameter for N (in units of the AdS curvature scale, k) and The above model was studied in [3] using so-called KK-basis where KK decomposition was done without taking into account the large Majorana mass term from the beginning. The effects of the Majorana mass was added as a posteriori process and this leads to large Ma-jorana masses for zero-and KK-modes and large mixing among all modes. Hence, although analysis using KK-basis produces correct neutrino mass formula, using a basis that is vastly different from the mass basis obscures the physical picture. In particular, the results from KK-basis naively suggest (or give the misleading impression) that the above 5D warped seesaw model is indeed of Type I in the sense that the SM neutrino mass is generated by the dynamical exchange of a super-heavy singlet mode, i.e., at the (effective) seesaw scale (for more discussion of this point, see [4]).
However, as shown in [4], analysis based on the mass basis, including the Majorana mass TeV is allowed [9,10]. 17 There are also constraints from flavor/CP tests which generically 16 Since QCD gauge group int he bulk will not play any role in our study, we will simply drop it from hereon. 17 In fact, even with the extended bulk gauge group, KK scale is constrained generically to be O(3) TeV.
require O(10) KK scale, but here we assume addition flavor structure (for example, flavor symmetries) in order to ameliorate those bounds [11] .
On the UV brane, the gauge symmetry is broken down from SU (2) R × U (1) X to U (1) Y by choice of boundary conditions (BC). Specifically, the gauge fields associated with the broken generators (SU (2) R × U (1) X ) /U (1) Y will have Dirichlet BC, denoted henceforth as "−", whereas U (1) Y and SU (2) L has Neumann (+). All gauge fields are taken to be + on IR brane. In particular, only fields with (++) BC have zero-modes up on KK-decomposition, i.e., only gauge fields for SM gauge group in this case. We use W R and Z to denote the extra gauge fields, i.e., for charged SU Higgs field, which we choose to be localized on the IR brane, is a bi-doublet of SU (2) L × This representation results in a custodial symmetry, i.e., the Higgs VEV breaks S(2) L × SU (2) R down to SU (2) V , which suppresses contributions from the gauge sector to the Tparameter: note that U (1) X remains unbroken in this process. The Higgs VEV will also generate mixing between various modes of W R and W L , an effect which can be treated perturbatively and which will be very important for LHC signals for the singlet neutrinos.
We will make this point clearer in Sec. 3.
Moving onto representation of fermions under the extended gauge group, first note that (just like for gauge fields) SM fermions will arise as zero-modes of 5D fields with (++) BC. We begin with the leptons, where we choose the simplest possibility, i.e., X is same as (B − L) in this sector. Thus, we take L, i.e., the SM SU (2) L lepton doublet, to be a singlet of SU (2) R , while the right-handed charged lepton (denoted by l) is promoted to be a doublet of SU (2) R , denoted by L R [as in the canonical, 4D (gauged) left-right (LR) symmetric models]: where numbers in the parenthesis denote representation under SU (2) L and SU (2) R , while representation of U (1) X is shown as a subscript [we will explain momentarily why there are Remarkably, akin to usual, 4D LR symmetric models, we see that the SU (2) R partner of has the precisely the characteristics to play the role of the N field mentioned above, i.e., it is a (i) singlet under SM gauge group; (ii) it has a Yukawa coupling Thus, special regions of parameter space and/or additional contributions to these observables (perhaps from further model building) will be needed in order to have KK scale as low as O(1) TeV. Given that resonances with mass O(3) TeV or heavier is slightly beyond the LHC reach, having new colliders with higher energy reach are required and hence motivated for a better test.
with lepton doubleton the IR brane and (iii) a Majorana mass term for it can be written only only the UV brane, since that is the only location where SU (2) R × U (1) X (under which it is charged) is broken. As a by-product, such a choice gives rise to a way to produce N via decay of W R . In fact, this will be the production channel for our signal process.
In more detail, note that we will actually need two SU (2) R lepton doublets, namely: Here the SM lepton ( ) is obtained as the zero-mode from the 2nd multiplet above, i.e., with (++) BC; its SU (2) R partner (denoted byÑ ) is chosen to be − on the UV brane (thus having no zero-mode at all): this is consistent with the bulk gauge symmetry since e. different BC's for two components of doublet are allowed), while this symmetry is unbroken on IR brane (i.e. it should be same BC for both fields, which is + in this case). Note thatÑ then plays no role in the seesaw for the SM neutrino mass (hence will be dropped from now on). On the other hand, the BC's are "switched" in the 1st doublet, i.e.,˜ has no zero-mode, whereas the N here will be driving the SM neutrino mass seesaw (thus will be denoted as the singlet neutrino henceforth). Note that N has (++) BC to "begin with", but adding a UV brane localized Majorana mass term "repels" N profile away from UV brane, resulting in effective boundary condition of the form (− +) and hence removing the corresponding zero-mode.
The simple reason for having two SU (2) R doublets, instead of housing N and SM righthanded lepton in a single SU (2) R doublet, is the following. The N and SM right-handed lepton, i.e., , fields should have different 5D bulk mass parameters in order to produce correct masses for charged lepton and neutrino [3], i.e., we require c < −0.5 for the field giving charged lepton zero-mode so that this mode is localized near the UV brane 18 , thus giving the observed charged lepton mass, whereas we need c ∼ −0.3 for N (as mentioned above), i.e., that would-be zero-mode should be peaked near the IR brane instead. However, by SU (2) R -invariance, fields in a doublet should have a common 5D mass parameter. Thus, we need to "split" the SM charged lepton and singlet neutrino multiplets as shown above.
Following [10], i.e., in order to suppress corrections to the Zbb coupling, we choose the representations of the quarks to be somewhat non-minimal as follows.
Here, Q L denotes the SM left-handed quarks doublet and u R , d R are the SU (2) L singlets.
For the "extra"' fields in SU (2) R doublet or triplet representations above, we take Dirichlet-Neumann (− +) boundary condition in order to remove zero-mode (just like was done for 18 Such a profile also needs to be chosen for the L zero-mode. leptons above). As usual, t R zero-mode is taken to be localized near the IR brane, while (t, b) L , i.e., Q 3 L , has a (roughly) flat profile and rest of the quarks are peaked near the UV brane (just like the SM leptons).
We mentioned that the spectrum of N in 4D effective theory is a tower of pseudo-Dirac fermions. Since the Majorana splitting (O(MeV)) for these pseudo-Dirac pairs is very tiny comparing to its Dirac mass (O(TeV)), we are unlikely to be able to probe any effects from such Majorana splitting. Moreover, as far as investigating the discovery potential of the lightest pseudo-Dirac singlet mode is concerned, the existence of tiny Majorana splitting will not make any difference. For this reason and for simplicity, in our collider study, we ignore Majorana splitting and treat N as pure Dirac with (− +) boundary condition, which will have the same mass as its SU (2) R partner˜ .
The 5D fields discussed thus far are summarized in Fig. 1   to light quarks provided there is degeneracy between KK W ± R and KK W ± L ,similarly KK Z and KK Z . This coupling can then be used in production of KK W ± R and Z . Once produced, their decay is dominantly to modes localized near IR brane such as (light) KK fermions and/or top quark/Higgs boson, since those couplings are the largest.

Spectrum of KK modes
Mass of KK gauge boson is dictated by boundary condition of corresponding 5D gauge field.
The mass of first KK mode of gauge fields with (+ +) boundary condition is typically O(1)× warped-down k and we denote it as m gauge . On the other hand, first KK mode of gauge fields with (− +) boundary condition has slighter smaller mass than m gauge .
KK fermion masses are determined by boundary condition and 5D mass m 5 , or c = m 5 k . For fermion fields with c chosen such that the corresponding (would-be) zero-mode is localized near the UV brane, we find that the KK mass is larger than KK gauge mass m gauge , regardless of its boundary condition [assuming brane localized kinetic terms (BKT) are negligible]. This is the case for all leptonic fields, except forL R . In order to produce SM neutrino mass,L R has c ∼ −0.3 and resulting KK mass is naturally smaller than m gauge . However, in the minimal setup, its mass is still bigger than 1 2 m gauge , preventing the decay of W R into N (1) and˜ (1) . As is well-known, turning on BKT's could lower the mass of corresponding KK modes. We can show O(1) BKT on the IR brane forL R can result in mass of N (1) and (1) smaller than 1 2 m gauge . Another interesting fact about BKT is that, both N L , which are our signal channels. A similar analysis can be applied to the quark sector: we find that, in the absence of BKT's, the only KK fermions which are a bit lighter than the W (1) R (but still heavier than 1/2 m gauge ) are the SU (2) R partners of Q 3 L (like the case ofL R above). We assume that BKT's for these states are not turned on (unlike forL R ) so that KK W R cannot decay into pairs of these extra fermions. The decay channel for W L and the above extra fermions is kinematically open; however, given the (roughly) flat profile of Q 3 L , this coupling is nonetheless suppressed compared to the coupling to N (1) and˜ (1) so that this decay mode can be neglected.
In this paper, we study the on-shell production of KK gauge bosons W (1) R and its decays to N (1) -˜ (1) pair. Particles heavier than W (1) R are dropped for simplicity of study. Below, we summarize the spectrum of particles of interest: (18) where KK gauge bosons included in our phenomenological study are W (1) 3 Two site approach to natural warped seesaw Full 5D warped model contains all the degrees of freedom with perturbative couplings. In this sense, it is fully calculable 5D effective theory and any relevant questions can be answered by explicit computation. However, for a specific phenomenological search, only a finite subset of degrees of freedom and related couplings are involved and a simplified model consisted of only relevant particles and couplings will be much more efficient in practice. Two site model of [12] provides one way to obtain a simplified 4D effective theory by a consistent truncation of a full 5D warped model to the first KK modes. This approach not only simplifies phenomenological studies, but also can encompass phenomenology of broader class of 5D warped models, or its 4D composite models, thereby allowing more inclusive/systematic searches. scale. Such a feature is known as Partial Compositeness in 4D strong dynamics, a robust mechanism that solves flavour hierarchy problem of the SM. 5D dual of partial compositeness is the localization of the zero-mode profile along the extra-dimension, localization near the IR (UV) brane being dual to more composite (elementary).
Two site model of the natural warped seesaw that we reviewed in Sec. 2 can be described as follows. We begin by discussing the singlet neutrino N R . In the elementary sector, there is elementary field N R that has large Majorana mass term m N . In the composite sector, as already mentioned in the introduction, there is a composite singlet Dirac fermion (χ L , χ R ) with O(TeV) Dirac mass. Finally, there is mass mixing between N R and χ L , i.e. they have the same quantum number, with the size of the mixing being characterized by the relevant scale, i.e. of the order of O(TeV). These can be summarized by the following Lagrangian (dropping kinetic terms for simplicity): where Notice that integrating out N R generates the Majorana mass for left-handed χ L of the composite singlet fermion, that is, it transmits lepton-number violation into the composite sector.
it is clear that the composite fermion (χ L , χ R ) becomes pseudo-Dirac and the exchange of this pseudo-Dirac singlet fermion between the two left-handed SM neutrinos then is the dynamical origin of the SM neutrino mass. Namely, it is the inverse seesaw for SM neutrino mass generation. Notice, however, that the way the small Majorana splitting is generated is by the "exchange" of super-heavy N R , which can be viewed as Type I seesaw.
As mentioned in Sec. 2, since the Majorana splitting is much smaller than Dirac mass, we simply drop it and treat (χ L , χ R ) as a pure Dirac fermion for our collider analysis presented in Sec. 5. For the rest of the study, we simply use (N (1) R ) to denote (χ L , χ R ) and put them and their SU (2) R partner, denoted as (˜ what was discussed for singlet neutrino above). In this study, just for simplicity, we treat all SM fermions to be purely elementary, except (b L , t L ) and t R . As we discussed in Sec. 2, the mass for KK modes of all SM fermions are higher than gauge KK; however, the KK modes from theL R multiplet in Eq. (16) are taken to be lighter. This is mapped into the two site model by the fact that all "excited" composite modes of the SM fermions 19 are heavier than composite vector mesons, thus for simplicity, we neglect them in what follows. However, the composite SU (2) R doublet containing the singlet neutrino (discussed above) is light. Higgs is chosen to be pure composite state as a standard choice. The diagonalized Lagrangian before EWSB (see next section for this effect) containing all these degrees of freedom is given by where we provide each part below one by one. First of all, L gauge is given by where ρ µ = (W Lµ , B µ ) (using the 5D notation, i.e., KK of SM gauge fields),ρ µ = (W All covariant derivatives in the Lagrangians are with respect to the unbroken SM gauge group, namely Dµ = ∂ µ − igA µ . Gauge couplings are SM gauge couplings g = (g W , g Y ) and The elementary-composite mixing angle φ = (φ W , φ Y ) is defined as sin φ = g g . Finally, m denotes the composite spin-1 mass before mixing with elementary states, hence this is also the mass forρ's (i.e., W ± R and Z ) who do not have such mixing. Whereas, for composite partners of SM gauge bosons, i.e., ρ's, the mass is modified by this mixing as indicated above.
Note that we are providing phenomenologically most relevant terms only, dropping terms with 3 or more ρ's orρ's. This is valid approximation since we are working to leading order and, at the leading order, only two body decays of the heavy particles, e.g. ρ's orρ's, are relevant.
Moving onto the fermion sector, L fermion (for the fields relevant for our collider study) is given by i.e. sin φ ψ light = 1 (corresponding to the zero-modes being localized near the UV brane in the 5D model), and Q 3 L is slightly composite (roughly flat profile) and t R is fully composite (localized near the IR brane), i.e. sin φ t R = 0. Finally, L Higgs has the form

Higgs induced gauge mixing
When Higgs gets a VEV and the electroweak symmetry is spontaneously broken, it generates mixing among gauge bosons and fermions. In order to obtain mass spectrum, then, mass matrices should be diagonalized. In this section, we shall discuss the diagonalization of mass matrices and show, in particular, that the mass eigenstates of massive vector bosons consist L and W R . That is, EWSB induces a significant mixing between W (1) L and W (1) R , and this will be the main production channel for W (1) R (as mentioned earlier). We choose g W and g R to be the same for our benchmark points.
The mass matrix for charged vector bosons is given by Note that we assume the same purely composite mass m for all composite gauge fields, i.e., before mixing with elementary states; this mixing does perturb the mass for excited SM gauge bosons as seen above. We will return to the more general case of non-degenerate composites in a follow-up paper.
Performing explicit diagonalization of the above matrix analytically can be quite challenging.
However, we can use the following method to get an approximate result. Our procedure will be valid as the following relations hold: We demand that the mass matrix can be fully diagonalized by the following transformation by U . where with Here s, S, s represent the sines of θ 12 , θ 13 , and θ 23 , whereas c, C, c represent cosines of associated angles. Making use of the approximations of Eq. (27), one readily finds that The last formula corresponds to the mixing between two composite vector bosons, W L and W (1) R . Since 1 4 g 2 v 2 m 2 , we would naively think that this mixing is small. However, using and, as far as g g , this mixing angle is O(1) ! That is, in most of the parameter space of interest, we get a significant mixing between W and using sin 2 φ W ≈ tan 2 φ W ≈ g 2 W g 2 W 1, Eq. (32) can be rewritten as follows.
The relation with the mass basis denoted by W , W L and W R is given by Since 1 4 g 2 v 2 m 2 , θ 12 ≈ θ 13 1, we can approximate s = S. After dropping all terms with two or more s or S, we can get The typical size of these mixing angles are Given the above large mixing between W L, R induced by the Higgs VEV, it is clear that light quarks will now couple similarly (and significantly) to both the mass eigenstates, cf. in the basis prior to EWSB, the coupling to one of the states, i.e., W R , was negligible.
The masses of the physical states will also be perturbed due to the EWSB effects. Here, for simplicity, we kept only two massive states W (1) L and W (1) R to obtain the mass splitting, assuming that the small fraction of W (0) in the mass eigenstate does not make any difference.
With such mass splitting, the physical mass for W L and W R are given by where the +(−) sign is for m W L (m W R ).
Similar analysis can be done for neutral gauge bosons. The mass matrix is given by where Here c = 1 − tan 2 θ W and θ W is Weinberg angle in the composite sector. We assume that the composite sector has the same Weinberg angle as the SM. Mass eigenstates are denoted by Z, Z 1 and Z , and are related to gauge basis fields by The typical size of mixing angles are And the spectrum of the mass eigenstate is

Lepton Mixing
Apart from mixing in the gauge sector, EWSB also induces mixing in the fermion sector.
As discussed earlier, composite "excited" modes for SM particles are neglected because they are heavier than composite vector bosons and composite states of singlet neutrino. For this reason, we focus on the mixing among SM lepton doublet L and the composite SU (2) R doubletL R . The relevant parts of the Lagrangian containing Yukawa coupling of L andL R are as follows: where y is the Yukawa coupling and i denotes the generation index of leptons, i = {e, µ, τ }.
When the Higgs field gets a VEV, the Lagrangian Eq. (44) generates neutrino Mixing as can be seen from From this, we can obtain physical mass eigenstates denoted as N L , N R , and ν L : where the mixing is given by The same Lagrangian also introduces electron mixing after EWSB (we can safely neglect the SM electron Yukawa coupling or mass term here as compared to the others): Again, from this, we can obtain physical mass eigenstates denoted asẽ L ,ẽ R , and e L : L .
In principle, there is a similar effect from mixing of SM SU (2) L singlet charged lepton (after EWSB) with composite SU (2) L doublets; however, since we assumed that such composites are heavy, we can neglect it. Moreover, electrons and neutrinos have the same mixing V N . This is because (1) N and˜ are in the same SU (2) R doublet with the same mass m D , together ν L and l L being in the same SU (2) L doublet and (2) these two mixings originate from the same Yukawa coupling.

Overview of LHC signals
In this section, based on our discussion in previous section, we first summarize couplings relevant to our collider study in Sec. 5. Then, we specify the choice of parameters used for actual analysis, together with related bounds. We then discuss production and dominant decay channels of heavy gauge bosons, i.e. W L and W R . In particular, we show that W R → N˜ is indeed the dominant decay channel for most of the parameter space of interest, providing abundance production of N and˜ . We end our discuss by providing formulae for decay widths of N and˜ .

Relevant Couplings
There are three types of couplings that we need to consider: (1) couplings between W L /W R and SM fermions (2)  (1) The first type of coupling can be obtained by using Eq. (23) and EWSB induced mixing Eq. (36): These couplings are responsible for the production of W L and W R via light quarks fusion inside proton. Notice that they suppressed by the factor g W g W and mixing angle compared to 4D LR models. However, as we will show in Sec. 5, these couplings, even with such suppressions, still render large enough signal production to be discoverable in near future.
(2) The second type of coupling can be understood from Eq. (23) and mixing induced by EWSB Eq. (36): These couplings lead to the decays of W L and W R to N and˜ .
These couplings lead to the decays of N and˜ to H/W/Z and /ν.

Parameter Choice
The composite sector generally contains many parameters, such as g 's andg 's. In our study, as our benchmark points, we assume all φ's are the same, i.e. the ratio g/g are the same for all SM gauge groups. This choice is mainly for the sake of simplicity, and other choices with small variations will not lead to much difference in the final results. Besides, we fix g W = g R , or equivalently we assume there exists Z 2 symmetry connecting SU (2) L and SU (2) R . This is well motivated by the consistency with EW precision tests, e.g. to suppress the corrections to the coupling Z → bb. With these choices, we are left with basically only one free gauge coupling in composite sector g W . The composite gauge coupling g W has a lower bound ∼ 3, which comes from the requirement that the Landau pole does not appear below the GUT scale. We choose g W = 3 as a benchmark points.
The mass parameter m for heavy gauge bosons is constrained by EW precision tests.
With extended symmetry group SU (2) L × SU (2) R × U (1) X , the bound is given by 3 TeV in most parts of parameter space. Partly motivated by the discoverability at the LHC, we choose m = 2 TeV for our study. Such a low mass might be achieved in some corners of the parameter space or by invoking additional effects in EW precision tests (see for example [14]). Also, we choose cos φ Q 3 L = 0.21, which may be on the edge of constraints from the EW precision test. This, again, can potentially be allowed by introducing additional structure in the model.
Next, |V N | 2 is constrained by various experiments and the results are summarized in [15] . Considering consistency with these experimental bounds, we choose the |V N | 2 = 0.001 for all three generations.
In order for W L and W R to be able to decay to the pair N -˜ , mL R needs to be smaller than half of m . In principle, this mass is also constrained correlated with constraints of |V N | 2 . With the choice we make |V N | 2 = 0.001, however, there is no effective bound on mL R . Nevertheless, given that heavy gauge bosons, N , and˜ all "live" in the same composite sector, too big hierarchy between mL R and m will lead to unwanted tuning. Taking into account all these considerations, we choose mL R = 750 GeV in our study.

W L /W R production and decay
As mentioned already, W L and W R are produced via couplings in Eq. (50). Decay width for dominant decay channels are shown below, which are computed using couplings Eq. (51).
Since the analytic expression for decay widths of mass eigenstates W L and W R are quite complicated, we instead provide expressions for gauge fields in gauge basis, namely W R . This will be sufficient for the purpose of our discussion. All the decay widths present in this paper are given with the assumption m > 2mL R mass of SM particles, thus masses of SM particles are reasonably neglected. Decay widths for W (1) L are given by where ψ and ψ denote SM fermions, and N c shows the degree of freedom of corresponding fermion ψ: 3 for quarks and 1 for leptons. Next, decay widths for W (1) R are given by where subscript i is generation index.
From Eq. (53) and Eq. (54), we see that W From there, we see that W R indeed decays dominantly to N -˜ pair, providing production mechanism for them. This can be contrasted to the case of 4D LR models, where the dominant decay is into jets. For our collider study in Sec. 5, however, we used full model including Higgs induced mixing and mass splitting.  in decay widths:

N and˜ production and decay
In principle, there will be three body decays via virtual W R . However, we have checked that, for the choice of parameters we made, such three body decays are suppressed compared to 2 body decays.
So far, we have focused on production and decay of charged gauge bosons, W L and W R , and resulting production of singlet neutrino N . In addition to these, however, the model also contains neutral gauge bosons Z 1 and Z (see Sec. 3). The relevant couplings for these neutral gauge bosons can be obtained in a similar way as those for charged ones. In particular, just like Eq. (50) for charged gauge bosons, Z 1 (Z ) couplings to light quarks is basically g 2 Z g Z times a factor for EWSB induced mixing, and it is via this couplings that neutral gauge bosons are produced at the LHC. In our framework (i.e. 5D/composite LR model), since Z 1 and Z arise as composite vector mesons of the strong dynamics in the same way as the charged ones do, they have the same/comparable mass as W L (W R ). This, then, naturally leads to the comparable production rates for Z 1 and Z , i.e. they are not suppressed compared to W L and W R . This feature can be contrasted to the case of 4D LR models, where production of Z is suppressed compared to W R due to the fact that Z , as an elementary particle, is heavier than W R . Moving onto the decays of the neutral gauge bosons, for the same reason for the charged gauge bosons, Z 1 and Z also have significant branching ratio to a pair of N . In this way, we see that, production and decay of these neutral gauge bosons provide another way to abundantly produce a pair of singlet neutrinos N . This signal channel, however, has almost the same process topology as 4D LR. Instead, we are planning to study the production of the singlet neutrino via on-shell decay of neutral gauge boson in our follow-up paper, but in a slightly different set up with interesting features/differences that only 5D framework can furnish.

Discovery Potential
In this section, we present our results for phenomenological studies of the LHC signals for the model described in Sec. 3. In particular, we study the pair production of the singlet neutrino (N ) and its SU (2) R partner (˜ ) via the one-shell decay of W R and W L , and their subsequent decays to SM particles. We consider two benchmark points depending on how N and˜ cascade decay to SM particles: Di-lepton-and Tri-lepton-channels.
For Di-lepton channel, the production and the cascade decays of N and˜ are as follows: Hence, the final states of the Di-lepton channel consist of jjbb, where, for the lepton pair, only opposite sign combination can arise since we are ignoring small Majorana splitting for N . That is, this process is lepton-number conserving. In particular, this channel contains two leptons, and hence the name for the channel.
For Tri-lepton channel, on the other hand, we take the leptonic decay for SM W boson from N . In detail, we get: Hence, the final states of the Tri-lepton channel consist of νbb. This time, it contains three leptons, explaining the name of the channel.
Notice that in both channels, we add contributions from both H and Z decaying into bb.
This is because resolutions of LHC detectors may not be good enough to distinguish those two cases, and at the same time, we will achieve a slight increase in the signal rate.
The Feynman diagrams for both signal processes are shown in Fig. 4. The topology of our signal processes are characterized by several resonance peaks in various invariant mass variables. In particular, invariant masses of W R and N/˜ , which we take to be M W R = 2 TeV and M N = 750 GeV in our study, will draw sharp distinctions between signal and SM backgrounds. For Tri-lepton channel, however, due to the presence of neutrino and the multiplicity of leptons (i.e. combinatorics issue), naively, one would think that resonance peaks are less pronounced. However, as we show below, by reconstructing the longitudinal component of the neutrino's momentum and by figuring out the identification of each lepton, i.e. which lepton is to be paired with bb, neutrino, and ν, respectively, we are able to construct all invariant mass peaks.
Event simulations are performed by employing a sequence of simulation tools. We first created our two-site simplified model files using FeynRules [16] based on Heavy Vector Triplets models [17]. Then we used them as inputs model in a Monte Carlo event generator MG5@aMC [18] to generate parton level events. In this procedure, parton distribution functions parameterized by NN23LO1 [19] is used. All the simulations are done at the leading order with a √ s = 14 TeV pp collider. The generated parton level events are then streamlined to Pythia 6.4 [20] to take care of showering and hadronization/fragmentation.
Since all our channels contain only regular jets, i.e. no boosted gauge bosons leading to fat jets, we directly pass on the output from Pythia 6.4 to Delphes 3 [21]. Delphes 3, interfaced with FastJet [22,23], provides a way to incorporate the detector effects and jet formation. The jets are constructed with the anti-k t algorithm [23] with a radius parameter In Sec. 5.1, we present our results for Di-lepton channel. Results for Tri-lepton channel follow in Sec. 5.2. We also briefly discuss phenomenological distinctions between our 5D left-right symmetry model and that of 4D. In particular, we will point out several salient features of our case by which two frameworks can be distinguished once discovery is made.

Dilepton + dijet + H/Z channel
We begin by considering the production of N −˜ pair and their decays at the LHC. In our current study, we consider (N,˜ ) as a SU (2) R doublet and as a consequence the production of this doublet pair should be proceeded via decay of W (34)), and when the mass splitting, M 2 , is small enough we acquire significant mixing, leading to enhanced production for signal. This can be realized when the masses of W R , as elaborated in Sec. 4.3, it will dominantly decay into (N,˜ ) pair. Therefore, making use of all these features, we can secure enough statistics for signal production at 14 TeV LHC. In Di-lepton channel, N decays to W ± ∓ and SM W boson, in turn, decays hadronically producing two jets. On the other hand,˜ ± decays to ± H/Z, which is then followed by decay of H/Z to bb. As is evident from these cascade decays of N and˜ ± , (i) signal process does not contain any neutrinos and hence no missing There are several SM backgrounds we need to consider and we describe them one by one now.
(1) ttjj: The relevant process is pp > ttjj > − + ννbbjj, where t > b (W + > + ν), and similarly fort, is considered. Being a purely QCD process, this is the background with largest cross section. Background reduction will be achieved by means of a combination of various invariant mass cuts. Particularly useful ones will be M All and M bb /M jj cuts. In principle, missing transverse momentum / E T , the opposite of the vectorial p T sum of reconstructed objects in the event, can provide useful reduction, although we found other cuts are more efficient.
(2) ttH/Z: The relevant process is pp > ttH/Z > − + ννbbbb, where t > b (W + > + ν), and similarly fort, and H/Z > bb are considered. If two b's in the signal process are b-tagged as a part of selection criteria, then in order for this background to pass the selection criteria, two of four b's must be un-tagged as regular two jets, leading to a large reduction of the background. Moreover, M All , M bb /M jj and M jj cuts will be useful.  Defining N , N b and N j as the number of isolated leptons, b-tagged jets and non-b-tagged jets, respectively, we select events using the following selection criteria: In addition, we impose a set of basic cuts p T j /p T b > 20 GeV and p T > 10 GeV at parton level event simulation, partly to avoid possible IR-divergence issues for background simulations.
We reimpose such cuts on objects (hardest two jets, two b-jets, and two leptons) that pass selection criteria of Eq. (58). We use p T to evaluate hardness of the reconstructed objects and take the hardest two. In Fig. 5  be paired with b-pair, and similarly for j-pair. We found, for example, that naively plotting the invariant mass of b-pair with both leptons (similarly j-pair with both leptons) does not reveal sharp peak at M N and resulting distribution is broadly extended with large overlap with background distributions. In order to achieve sharper distinction, we make use of the fact that the masses of N R and˜ R are equal due to SU (2) R invariance. Namely, we identify the lepton that goes with b-pair ( b ) and the one that goes with j-pair ( j ) by minimizing As can be seen from for two major backgrounds: ttjj and ttH/Z. Therefore, these variables will supplement above described variables to attain additional suppression of background events. Finally, the missing transverse momentum variable also helps a bit. This is expected based on the insight that the backgrounds ttjj and ttH/Z have larger / E T than signal. We provide the cut flows for signal and the major SM backgrounds in Table 1. We find that the Di-lepton channel may provide a sensitivity to uncover warped seesaw nature by ∼ 3.5σ with an integrated luminosity of L = 300 fb −1 and even by ∼ 11σ with L = 3000 fb −1 .

Tri-lepton + H/Z channel
In this section, we present the results for Tri-lepton channel. Similarly to the Di-lepton channel discussed in previous section, N −˜ pair is produced via the decay of W L . In Tri-lepton channel, N decays to W ± ∓ and SM W boson, in turn, decays leptonically producing ν. Like in Di-lepton channel,˜ ± decays to ± H/Z, with subsequent decay of H/Z to bb. As is evident from these cascade decays of N and˜ ± , (i) signal process now does contain neutrino, leading to missing energy and (ii) there are three leptons in final states. The existence of neutrino (or missing particle in general) and the large multiplicity of leptons can be a potential obstacle in reconstruction of resonance peaks. However, we will show below that such difficulty can be, at least partly,   These variables do not correspond to any of resonance peaks appeared in the signal process.
However, they will still provide very strong distinctions between the signal and backgrounds.
There are several SM backgrounds we need to consider and we describe them one by one now.
(3) − + Wjj: The relevant process is pp > − + W ± jj, W ± > l ± ν(ν). Since we will select events with two b's are tagged, only very small fraction of events with two regular jets mistagged as b-tagged jets will contribute to the backgrounds. Mistage rate is typically 1% [24] and uds-jet mistag rate can even be as small as 0.3% [25]. The cross section of the process is σ ∼ 180 fb and the surviving events with two mistagging is ∼ O(0.01) fb. This corresponds to roughly ∼ O(3) events at an integrated luminosity of L = 300 fb −1 . It will be very unlikely that any of these events will in the signal region given the number of invariant mass cuts that it should pass. Hence we will not explicitly consider this background for our analysis.
Defining N and N b as the number of isolated leptons and b-tagged jets, respectively, we select events using the following selection criteria: In addition, we impose a set of basic cuts p T b > 20 GeV and p T > 10 GeV at parton level event simulation, partly to avoid possible IR-divergence issues for background simulations.
We reimpose such cuts on objects (hardest two b-jets, and three leptons) that pass selection criteria of Eq. (60). We use p T to evaluate hardness of the reconstructed objects.
Next, we discuss the way we reconstruct longitudinal component of neutrino's four momentum. Together, we also discuss how we figure out lepton identifications. Namely, we want to know, out of three leptons selected as described above, which one is produced together with bb from the decay of˜ ± (we call it b ) and which one is produced directly from the decay of N (we call it W ), and finally which one is the decay product of SM W (we call it ν ). 20 First of all, for a given choice of lepton (a candidate for ν ), the z-component of the neutrino's momentum can be obtained by requiring where M W is the mass of the SM W boson. For p ν µ , we use the fact that neutrino is massless, (p ν µ ) 2 = 0. Then, the above equation is a quadratic equation for the z-component of p ν µ , and if solutions exist, there are two solutions, unless determinant vanishes by numerical coincidence.
In this case, we pick up p ν z that minimizes the sum of z-component of all particles' momenta, i.e. sum of p z of two b's, three 's and ν. This is based on the insight that W R is mostly produced at rest. In case when the determinant of the quadratic equation is less than 0, so that no solution exists, we set This choice again is motivated by the intuition that W R is mostly produced at rest. Once z-component of neutrino's momentum (or equivalently full p ν µ ) is reconstructed this way for a given choice of (again a candidate for ν ), we then determine b and W by minimizing (63) 20 The subscript is designed to indicate a set of particles that the lepton accompanies.  This criteria is motivated as before by SU (2) R invariance and resulting mass degeneracy.
In this way, for each choice of ν , we determine full p ν µ and identify, for the remaining two leptons, which lepton is b and which lepton is W . We repeat this procedure for all three possible choices of ν . Final decision is made for the combination { ν , W , b } that renders minimum value for Eq. (63). In Fig. 6, we show distributions of various invariant mass variables for signal and background events that pass selection criteria and basic cuts.  Fig. 6 shows M W recon R distribution and it is indeed peaked at/near 2 TeV, a input value for W (1) R . This is to be compared to the M All distribution shown in the right panel of the top row in Fig. 6.
Again, M All is the invariant mass for all reconstructed visible particles, i.e. two b's and three 's, but without neutrino. Although M All distribution also develops a peak with good separation from background distributions (dotted red (ttw) and dotted green (irred)), the position of the peak is shifted toward the smaller value, reflecting the existence of neutrino.
We found that both M W recon R and M All , separately, provide very efficient cuts. Overall, we see that above described prescription for reconstructing { ν , W , b }-identification and full p ν µ is very effective and successful. We also note that M distribution for irred is sharply peaked at M Z showing that two leptons come from on-shell decay of Z boson. Finally, Mdistribution for backgrounds are clustered for smaller values and well-separated from that of signal. We provide the cut flows for signal and the major SM backgrounds in Table 2. We find that the Tri-lepton channel may provide a sensitivity to discover N ,˜ and W R by ∼ 4σ with an integrated luminosity of L = 300 fb −1 and even by ∼ 13σ with L = 3000 fb −1 .
We close our discussion by pointing out several phenomenological features that can draw distinction between 4D LR and 5D/composite LR models. 21 First of all, the production of W ± R in 4D LR models is via the unsuppressed coupling to 21 For distinguishing between various 4D seesaw models, see, for example, [26] Cuts 13.00 -- Table 2: Cut flows for signal and major background events in terms their cross sections. The cross sections are in fb. The numbers in the first row ("No cuts") are cross sections obtained with basic cuts at the generation level to avoid divergence (for both signal and backgrounds). In the second row, the same basic cuts are reimposed to both signal and background events along with multiplicity requirements for b-jet and leptons. Once the cross section decreases such that the net number of events at L = 3000 fb −1 is less than 1, we report it as "0". quarks, whereas in the case of 5D LR, it is via suppressed/smaller couplings, leading to smaller production rate.
For 5D/composite LR, the production of N via the decay of W ± R accompanies its SU (2) R partner˜ . This, in turn, renders additional Higgs/Z. Therefore, in 5D/composite LR models, there are two extra resonance bumps, those of˜ and Higgs/Z. Both structures were crucial in reducing background. Perhaps more importantly, once discovery is made, these extra resonance peaks will be critical in discriminating 4D vs. 5D LR nature.
The distribution of the di-lepton invariant mass will have (i) different shape and (ii) different dependence of endpoints on M W ± R and M N . To be more specific, for usual 4D LR, the signal process is two-step cascade decay, leading to smooth distribution, except perhaps at endpoint, where, depending on spin correlations, there could be a sharp/"vertical" drop [27]. For 5D/composite LR, on the other hand, having heavỹ , in addition to N , in the decay of M W ± R , the shape of the distribution will be that of antler with a cusp, i.e., a derivative discontinuity, in roughly middle of distribution [28]. The end point for 4D LR is located at ∼ M 2 W R − M 2 N , that of 5D LR being different from this.

Conclusions and Outlook
Searches have been done (and are ongoing) at the LHC for TeV-mass SM singlet neutrinos involved in the generation of super-small SM neutrino mass via various 4D models of seesaw.
However, we have tried to present a case here that many these require a small parameter in order to obtain the right size of the SM neutrino mass, thus in some cases reducing the original attraction of the seesaw. In fact, we feel that there might not be any strong motivation for singlets in these models to be at ∼ TeV other than getting a signal at the LHC from them. In earlier work, some of us had demonstrated that a completely natural realization of TeV-scale seesaw occurs instead in a warped extra-dimensional framework, which is dual (as per the AdS/CFT correspondence) to the SM Higgs being a composite particle arising from some new strong dynamics.
In this paper (and a follow-up), we initiated the study of the LHC phenomenology of this framework of a natural TeV-scale seesaw. In particular, here, we showed that signals similar to the 4D models arise in this warped/composite framework as well. At the same time, the details of the phenomenology are different in an interesting manner. Hence, one can suitably adapt existing searches for singlet neutrinos in 4D models to the natural 5D one.
The easiest way to see how these features arise is using a (effective) two-site picture of this framework. Namely, we have two sectors of the theory: elementary and composite. The SM Higgs is contained in the composite sector, whose characteristic mass scale is ∼ TeV so as to address the Planck-weak hierarchy problem; whereas, the rest of SM particles are admixtures of those in the two sectors, i.e., partially composite/elementary. Specifically, the degree of compositeness of the non-Higgs SM particles reflects the size of their mass, i.e., the top quark is significantly composite, while the light quarks are negligibly so. Moreover, lepton-number is preserved by the composite sector, but broken at the UV cut-off in the elementary sector. So, if we include an elementary SM singlet RH neutrino (N R ), then it will naturally have a super-large, even Planck-scale, Majorana mass. However, by itself, this lepton-number violation is not quite sufficient to induce Majorana mass for SM neutrino, since we also require EWSB/Higgs VEV for this purpose. Thus, this information about lepton-number violation has to be transmitted from the elementary to the composite sector, where the SM Higgs resides. In this way, one can "sew" together the two necessary ingredients in order to generate the SM neutrino mass.
A simple and natural way for sharing lepton-number violation between the two sectors is for the above elementary N R to also mix with composite sector TeV-mass singlets. These singlet states are purely Dirac to begin with, but as a result of the above coupling to elementary N R , they acquire a tiny Majorana mass component. It can be shown that it is the exchange of these (now pseudo-Dirac) singlet states generates -without any tuning -the right size of the SM neutrino mass. Thus, the TeV-mass singlets play a crucial role in this entire process: their observation at the LHC would provide a vital test of this mechanism of the SM neutrino mass generation. Just to emphasize, the TeV-mass for these composite singlets is natural, being directly related to the electroweak scale (cf. usual 4D models, where some extra assumptions are typically needed in order to get such a mass for the singlet neutrinos).
The obvious next question is how to produce these TeV-mass composite neutrinos N R at the LHC, given that they are SM singlets. The analogous 4D models provide a recipe: typically this is achieved in these models in the context of extending the SM EW symmetry to the left-right (LR) structure, i.e., SU (2) L × SU (2) R × U (1) B−L , with SU (2) R × U (1) B−L broken down to SM hypercharge at the TeV scale. The point is that N R -while being SM singlet -is a doublet of SU (2) R , thus can be produced via decay of charged W R . W R is, in turn, produced via qq annihilation with the associated W R couplings of SM EW strength.
Indeed, a similar LR symmetric pattern is motivated in the warped/composite Higgs framework, albeit for a different reason (i.e., than parity restoration in usual 4D models).
The purpose of the extra symmetry is to protect ρ parameter from receiving large corrections.
So, we assume this extension only in the composite sector as simply a global symmetry.
There is then no elementary charged W R gauge boson (unlike for the SM W L ), but we do have composite charged W R 's. However, in this way, it seems naively that we do not have a way to produce W R , since the SM quarks inside proton are mostly elementary, leading to a negligible direct coupling to composite-sector W R .
Remarkably, we found that elementary-composite W L mixing, followed by composite W L -W R mixing via Higgs VEV, induces the required coupling of composite charged W R 's to quarks. It is the degeneracy among spin-1 composites which ensures that the second mixing effect is rather large for a few TeV composite W 's. The end result is that coupling of light quarks to W R in these models is suppressed compared to the typical SM EW coupling, but it still sizable. Consequently, although production rates for W R are smaller than in 4D LR case, as we showed here, it is still enough for discovery. We would like to emphasize here that this subtle effect has been discussed earlier in the context of LHC signals for these spin-1 states in general, i.e., independent of neutrino mass considerations. However, this feature was not really exploited before, in the sense that decay modes of W R studied in that context (for example, W/Z/Higgs) were also accessible via W L , i.e., production of W R was not really "needed" (cf. here N R only couples to W R ).
Note that, in the W R decay, the composite N R is accompanied by composite charged lepton, since the associated coupling is, for example, larger than coupling to one composite and one elementary states (cf. in 4D models, it would be simply the SM charged lepton).
Composite charged lepton decays into SM charged lepton, plus Higgs/longitudinal Z, while N R decays (just like in 4D models) into SM charged lepton and W , latter decaying either leptonically or hadronically. Thus, the final state is either (di-lepton + W -jet + Higgs/Z) or (tri-lepton + MET + Higgs/Z). Note that the dileptons in first channel are of opposite sign, given the pseudo-Dirac nature of these singlets (cf. same-sign dileptons from Majorana singlets in some 4D LR models).
We performed a detailed analyses of both these channels for singlet neutrino production via decay of composite W R , finding that, for both channels, significant evidence can be observed for ∼ 2 TeV W R and composite N R /composite charged lepton of mass 750 GeV, with an integrated luminosity of 300 fb −1 , and even discovery with slightly more integrated luminosity. It is clear that the extra boson in final state permits distinguishing this framework from 4D LR models. In addition, this feature is crucial for reducing the SM background, especially given smaller rate than in 4D LR models and the absence of the "smoking-gun", i.e., same-sign dileptons; indeed, it is noteworthy that in spite of these seeming challenges, we are able to extract a reasonable signal.
Finally, we would like to provide a "preview" of part II, where we will consider signals of singlet neutrinos from production and decay of particles absent in 4D LR models. In particular, one idea is to relax the degeneracy of spin-1 composites that was assumed here.
In the light of the above discussion, this direction actually results in suppressing the charged W R signal, but we will show that a "new" type of signal appears from a neutral heavy boson, i.e., which is not accompanied by a charged channel (unlike in the 4D LR case, where charged spin-1 channel is actually dominant, W R being lighter than the corresponding extra neutral gauge boson). We will also study production of composite SU (2) L doublet leptons inherent to this framework (cf. absent in the 4D LR models); singlet neutrinos can be produced in their decays via a Yukawa coupling, i.e., independent of the couplings of N R to W R , thus of