Flavor universal resonances and warped gravity

Journal of High Energy Physics, Jan 2017

Warped higher-dimensional compactifications with “bulk” standard model, or their AdS/CFT dual as the purely 4D scenario of Higgs compositeness and partial compositeness, offer an elegant approach to resolving the electroweak hierarchy problem as well as the origins of flavor structure. However, low-energy electroweak/flavor/CP constraints and the absence of non-standard physics at LHC Run 1 suggest that a “little hierarchy problem” remains, and that the new physics underlying naturalness may lie out of LHC reach. Assuming this to be the case, we show that there is a simple and natural extension of the minimal warped model in the Randall-Sundrum framework, in which matter, gauge and gravitational fields propagate modestly different degrees into the IR of the warped dimension, resulting in rich and striking consequences for the LHC (and beyond). The LHC-accessible part of the new physics is AdS/CFT dual to the mechanism of “vectorlike confinement”, with TeV-scale Kaluza-Klein excitations of the gauge and gravitational fields dual to spin-0,1,2 composites. Unlike the minimal warped model, these low-lying excitations have predominantly flavor-blind and flavor/CP-safe interactions with the standard model. Remarkably, this scenario also predicts small deviations from flavor-blindness originating from virtual effects of Higgs/top compositeness at ∼ O(10) TeV, with subdominant resonance decays into Higgs/top-rich final states, giving the LHC an early “preview” of the nature of the resolution of the hierarchy problem. Discoveries of this type at LHC Run 2 would thereby anticipate (and set a target for) even more explicit explorations of Higgs compositeness at a 100 TeV collider, or for next-generation flavor tests.

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Flavor universal resonances and warped gravity

Received: September Flavor universal resonances and warped gravity Kaustubh Agashe 0 1 Peizhi Du 0 1 Sungwoo Hong 0 1 Raman Sundrum 0 1 Open Access 0 1 c The Authors. 0 1 0 University of Maryland , College Park, MD 20742 , U.S.A 1 Maryland Center for Fundamental Physics, Department of Physics their AdS/CFT dual as the purely 4D scenario of Higgs compositeness and partial compositeness, o er an elegant approach to resolving the electroweak hierarchy problem as well as the origins of avor structure. However, low-energy electroweak/ avor/CP constraints and the absence of non-standard physics at LHC Run 1 suggest that a \little hierarchy problem" remains, and that the new physics underlying naturalness may lie out of LHC reach. Assuming this to be the case, we show that there is a simple and natural extension of the minimal warped model in the Randall-Sundrum framework, in which matter, gauge and gravitational elds propagate modestly di erent degrees into the IR of the warped dimension, resulting in rich and striking consequences for the LHC (and beyond). The LHC-accessible part of the new physics is AdS/CFT dual to the mechanism of \vectorlike con nement", with TeV-scale Kaluza-Klein excitations of the gauge and gravitational elds dual to spin-0,1,2 composites. Unlike the minimal warped model, these low-lying excitations have predominantly warped; gravity; Phenomenology of Field Theories in Higher Dimensions - avor/CP-safe interactions with the standard model. Remarkably, this scenario also predicts small deviations from inating from virtual e ects of Higgs/top compositeness at O(10) TeV, with subdominant resonance decays into Higgs/top-rich nal states, giving the LHC an early \preview" of the nature of the resolution of the hierarchy problem. Discoveries of this type at LHC Run 2 would thereby anticipate (and set a target for) even more explicit explorations of Higgs compositeness at a 100 TeV collider, or for next-generation avor tests. Model with one intermediate brane 1 Introduction 2.1 2.2 2.3 Spin-1/Gauge KK Spin-2/Graviton KK Extended Bulk Gauge Symmetries/Dual to PNGBs of Vector-Like Con ne Current bounds in avor-universal limit Probing top/Higgs compositeness Model with two intermediate branes Conclusions/outlook A Details of choice of parameters Matching at the intermediate/Higgs brane A.2 Implications of above matching B Two dilaton system The scenario of Higgs compositeness [1] o ers a powerful resolution to the Hierarchy Problem. The Standard Model (SM) Higgs degrees of freedom remain much lighter than the Planck scale in the face of radiative corrections because they are only assembled at scale, as tightly bound composites of some new strongly interacting \preons". This is in close analogy to how the ordinary charged pion remains much lighter than the Planck scale in the face of QED radiative corrections, by being assembled as a quark-gluon composite GeV. But despite the simple plot, composite Higgs dynamics is notoriously di cult to model in detail because it requires understanding a new strongly-coupled dynamics, operating outside perturbative control. Remarkably, Higgs compositeness has an alternate \dual" formulation (for the duality of gravity and bulk gauge elds, see, for example, [2]; [3]; for duality of bulk fermions, [4]; [5, 6]) in the form of \warped" higher-dimensional theories of Randall-Sundrum type [7, 8], related to the purely 4D formulation via the famous AdS/CFT correspondence [9]. In the warped framework there can exist a regime of weakly-coupled higher-dimensional e ective eld theory, allowing more detailed phenomenological modeling as well as a prototype for UV completion, say within string theory [10, 11]. Figure 1 shows a schematic representation of particle physics in the simplest such setting, with a single microscopic extra-dimensional interval. The SM is now fundamentally 5-dimensional (for reviews of warped bulk SM, see, for example [12{16]), but its lightest modes appear as the familiar 4D SM particles, with phenomenological properties deriving from their extra-dimensional wavefunctions. In particular, the SM fermions naturally have disparate wavefunctions, which lead to an attractive mechanism for the origin of SM avor structure, AdS/CFT dual to the robust mechanism of Partial Compositeness ([17]; for AdS dual of this idea, see [4]). On top of the lightest modes are Kaluza-Klein (KK) excitations of the SM ( gure 2), which e ectively cut o quantum corrections to the Higgs mass and electroweak symmetry breaking (EWSB). Naturalness then implies that these KK states should have masses of the order TeV scale.1 This is the basis of ongoing LHC searches for KK-excited tops and bottoms (\top partners") and KK gauge bosons and spin-2 KK gravitons. Because of their strong extra-dimensional wavefunction-overlap with the top quark and Higgs, these KK resonances predominantly decay to t; h; WL; ZL [18]. From the viewpoint of 4D Higgs compositeness, the KK excitations are simply other composites of the same preons inside the Higgs (and the closely-related top quark). Lower-energy experiments are also sensitive to KK states via their virtual exchanges. Electroweak precision tests, now including the rapidly developing body of precision Higgs measurements, robustly constrain the KK spectrum, but are still consistent with KK discoverability at the LHC ([19{22]; for a more recent discussion, see, for example, [23]). However, as in the supersymmetric paradigm, the constraints from tests of CP violation are extremely stringent. Although the warped extra-dimensional framework (and partial compositeness) enjoys a powerful generalization of the SM GIM mechanism suppressing FCNCs [24{27], it is imperfect. Typically in parameter space avor and CP constraints imply MKK & O(10) TeV for the KK threshold [28{31]! What are we to make of this situation? While avor and CP tests have very high virtual reach for the warped/composite scenario, they do not appear as robust as electroweak constraints. It is indeed plausible that a more re ned mechanism for avor structure is occurring within Higgs compositeness so as to relax the bounds signi cantly, and admit KK states within LHC reach (for models using avor symmetries for relaxing the bounds, 1An elegant realization in warped extra dimension of the composite Higgs mechanism, i.e., where it is a PNGB like the pion, is via gauge-Higgs uni cation [5, 6]. It is in this case that the cuto of Higgs quantum corrections is the KK scale. However, this aspect plays little role in this paper. So, for brevity, we simply suppress this extra structure of the Higgs eld. Strong gravitational redshift, “Warping” 5D extended SM dimensional wavefunctions for various particles (zero modes/SM and a generic KK mode). see, for example, [32]; [33] in 5d; [34{36]; [37] in 4d). Because of this, it is imperative that LHC experiments continue to search for KK resonances along the lines of gure 1 and 2, in tandem with ongoing low-energy searches for new sources of avor and CP violation. But it is also possible that the hierarchy problem is imperfectly solved by Higgs compositeness at a scale & O(10) TeV, leaving a Little Hierarchy Problem between O(1)TeV. We simply do not understand fundamental physics and the principle of Naturalness underlying the SM hierarchy problem deeply enough to know if they should reliably predict the threshold of new physics to better than a decade in energy. Of course, such a possibility leads to the practical problem that MKK & O(10) TeV is outside LHC reach and yet frustratingly close! (It is noteworthy however that such new physics is might be within reach of proposed 100 TeV colliders). In this paper, we will pursue the scenario of Higgs compositeness at & O(10) TeV. This straightforwardly suppresses all virtual KK-mediated electroweak, avor and CP violating e ects enough to be robustly consistent with all precision experiments to date. But we will ask what natural forms of new physics might lie within LHC reach if we go beyond the minimal structure of gure 1 and 2, without reintroducing con ict with precision tests. We can think of such non-minimal physics lying below the scale at which the hierarchy problem is solved as \vestiges of naturalness". If the LHC cannot reach the states central to solving the dominant part of the hierarchy problem (such as KK tops), the search for light vestiges, related to the central players but not among them, are the best hope for the LHC. In particular, we study literally a straightforward extension of gure 1 which exploits the fact that di erent types of elds can propagate di erent amounts into the IR of a warped extra dimension, as schematically depicted in gure 3. For simplicity, we focus on three categories of elds: (i) SM matter, including the Higgs, (ii) gauge gravity. Gravity is the dynamics of all spacetime and therefore must be present in the SM entire length of the extra dimension in the form of 5D General Relativity. Gauge elds and matter can however reside in a smaller region. elds can live in an even smaller region of the extra dimension than the gauge elds, but not the other way around because charged matter always radiate gauge elds. This explains the ordering shown in gure 3. The di erent regions are separated by \3-branes", (3 + 1) dimensional defects in the 5D spacetime. Figure 3 is a simple, robust and interesting generalization of the minimal structure of gure 1, 2. A quite di erent proposal using an intermediate brane in warped spacetime was made in [38] in the context of explaining 750 GeV diphoton excess at the LHC [39, 40]. Di erent matter elds propagating from the UV brane to di erent intermediate branes were studied in [41, 42]. Also, a set-up with (only) two branes, but a departure from pure AdS near the infrared brane, can result in the Higgs pro le being peaked a bit away from the IR brane [43]. The new physics to the IR of Higgs compositeness is (AdS/CFT dual to) that of \Vectorlike Con nement", proposed in references [44{46] as a phenomenologically rich structure that is remarkably safe from precision tests, and is a natural candidate for a light vestige of a more general dynamics that solves the hierarchy problem. In the framework of gure 3, vectorlike con nement incarnates as the extension of the IR of the extra dimension gure 1, resulting in di erent KK thresholds for matter, gauge elds and gravity as depicted schematically in gure 4. A simple but important result we will demonstrate is that the Goldberger-Wise (GW) mechanism [47] for brane/radion stabilization very naturally results in \little" hierarchies MKKmatter;Higgs From the purely 4D perspective of strong dynamics, the sequence of KK thresh MKKgrav , is dual to a sequence of strong con nement scales (for non-supersymmetric versions, see, for example, [48]; for supersymmetric cases, see, for example, [49]), glueball. Over the large hierarchy from the far UV (the Planck or uni cation scale) down to Higgs the strong dynamics is only slowly Higgs the strong dynamics con nes \preons" into composites, among which is the light SM-like Higgs. This is analogous to the emergence of pions and heavier hadrons as composites of quarks and gluons upon QCD con nement. But unlike QCD, the strong dynamics does not end at this point, but rather is reorganized into a new set of strongly interacting preons, now approximately decoupled from Higgs and avor physics. The IR preons do however carry SM gauge charges. meson there is a second stage of preon con nement, into \mesons" also carrying SM gauge charges. Without direct couplings to the Higgs and SM fermions, this second stage of con nement does not break the SM electroweak chiral symmetries, hence the name \vectorlike" con nement. Again, the strong dynamics need not end at this threshold, but can continue with a set of far-IR SM-neutral preons, which ultimately con ne into SMneutral \glueballs" at Since the new physics below Higgs couples to the SM states predominantly via avorblind gauge forces, it is naturally safe from the host of electroweak, avor and CP tests. Phenomenologically, production and decay of the new states below Higgs will be mediated by on- and o -shell SM gauge bosons. It is very important that experiments search broadly for this kind of physics. In this way, vectorlike con nement appears as set of \aftershocks" of Higgs compositeness, immune to earlier detection but plausibly lying within grasp of the LHC. We will study several aspects of this strongly motivated scenario in this paper. In references [44{46], vectorlike con nement was modeled on QCD-like dynamics as the simplest way of illustrating the rich possibilities, using real-world understanding of the strong interactions to stay in non-perturbative theoretical control. A feature of these models is that they typically contain several pseudo Nambu-Goldstone bosons (PNGBs) in the IR of the new physics related to the large chiral symmetry, which can dominate the phenomenology.2 However, the speci c phenomenological implications are model-dependent. Although QCD-like dynamics do not have a very useful AdS/CFT dual extra-dimensional 2For recent applications of vector-like con nement for explaining the 750 GeV diphoton excess at the LHC, see, for example, the early references [50{55]. Mass Gauge Fields Gravity in some number of A5 4D scalars dual to composite PNGB's. description, they are in the same \universality class" as extra-dimensional models of the type depicted in gure 5, where the 5D gauge group is extended beyond the SM. If UV and IR boundary conditions break some of the gauge symmetry generators, they result in physical extra-dimensional components of the gauge eld, \A5", which are 4D scalars, AdS/CFT dual to PNGB's [5].3 We will return to study this class of vectorlike con ning physics more closely in future work. Unlike in QCD-like constructions, in warped 5D effective eld theory we can suppress the existence of A5's by construction, allowing us to focus on other possibilities for the new phenomenology. 3Such states tend to be lighter than the typical KK scale and thus can be within LHC reach even in the minimal model of gure 1 with the IR brane at the avor/CP bound of O(10) TeV [56, 57]. One focus of this paper will be the possibility that lightest new states are the universal ones arising from 5D General Relativity, the scalar \radion" measuring the (dynamical) size of the nal IR segment of the extra-dimensional interval, and spin-2 KK gravitons. These are the hallmarks of warped extra-dimensional physics. Via the AdS/CFT correspondence these states are dual to special \glueballs" interpolated by the conserved energy-momentum tensor of the strong dynamics, the universal composite operator of any quantum theory. In particular, this symmetric tensor naturally interpolates spin-2 glueballs dual to KK gravitons, while its Lorentz-trace interpolates the \dilaton", a glueball dual to the radion. We will derive and discuss their phenomenological implications, pointing out (i) when they are likely to be the rst discovered new states beyond the SM, (ii) their special distinguishing features and the contrast with more QCD-like vectorlike con nement and other beyond-SM physics, (iii) how we can experimentally test whether the new physics is well-described by higher-dimensional dynamics. In table 1, we highlight a couple of signals from the gravity sector, namely radion in the model with one intermediate brane of gure 5 and KK graviton in the model with two intermediate branes of gure 3: further details will be provided in the relevant parts of the paper. For now, it is noteworthy that the decays in these cases dominantly occur to pairs of SM gauge bosons, cf. top/Higgs playing this role in the minimal model of gure 1. Also, we see that radion and KK graviton are allowed to be lighter than gauge KK modes.4 A second focus of the paper will be connecting the new physics the LHC can discover to the solution of the hierarchy problem beyond its reach. We will show that low-lying KK modes, though mostly decoupled from the Higgs and avor, will have subdominant decay channels into t; h; WL; ZL, the traditional signatures of Higgs compositeness. In this way, the LHC would have a valuable resonance-enhanced \preview" of the solution to the hierarchy problem by compositeness, only fully accessible to more energetic future colliders. In particular, we nd that spin-1 KK gauge bosons are well-suited for this task. Note that these are dual to composite vector \ " mesons, which arise as a robust feature in the framework of vector-like con nement also. A representative sample of the above novel probe of top/Higgs compositeness is shown in table 2: we will of course explain in later sections how we obtained these numbers (including assumptions made therein), but let us convey our main message using them for now. We focus on KK | excited (dual to composite) Z and gluon, where we x their mass and coupling to light quarks, hence production cross-section (as shown). However, decay branching ratios (BR's) to various nal states still vary for the same framework as Higgs: the left-most column corresponds to the standard composite Higgs model (i.e., single IR brane/scale, gure 1), whereas right extreme is the avor-blind limit, i.e., Higgs compositeness scale is decoupled, large Higgs). Remarkably, we see that decay BR's might be sensitive to 15 TeV Higgs compositeness scale [in the sense that such values of Higgs compositeness scale can result in O(1) deviations from both standard limits], which is the ball park of the generic lower limit on the Higgs compositeness 4It might be also possible to make KK graviton lighter than gauge KK using large brane-localized kinetic terms (BKT) for gravity [58]. For recent applications of this idea for explaining the 750 GeV diphoton excess at the LHC using KK graviton, see [59{64]. However, with too large BKT for gravity, the radion might become a ghost [58]. Radion / KK Graviton WW LHC13 (pp ! Radion / KK Graviton) s = 13 TeV LHC) and decay BR's of radion (left) and KK graviton (right) for a given choice of framework and parameters. For radion, model gravity(gauge) coupling of 1(3) [composite gravity (g?grav) and gluon (g?QCD) couplings, respectively, which we de ne in section 2.1]. For KK graviton, we instead considered model with two intermediate branes, in which KK graviton is naturally lighter than KK gauge boson. In this case, both inter-KK couplings are taken to be 3. ```L`HC`13`(p`p`!`K`K`Z`)``2`:5 fb for 3 TeV mass and inter-KK coupling of 3 Final state Higgs 3 TeV ( gure 1) 10 TeV 15 TeV one intermediate brane ( gure 5) two intermediate branes ( gure 3) inter-KK gravity coupling = 1 inter-KK gravity coupling = 3 inter-KK gauge coupling = 3 inter-KK gauge coupling = 3 MKKgauge = 3 TeV; m' = 1 TeV MKKgauge = 3 TeV; MKKgrav = 1 TeV di-leptons (e + ) di-jets (light quarks +b) ``LH`C`13 `(p`p!``K`K` `glu`o`n)``151 fb for 3 TeV mass and inter-KK coupling of 3 Final state Higgs 3 TeV ( gure 1) 10 TeV 15 TeV 1 top/Higgs compositeness scale ( Higgs), for xed spin-1 mass scale of 3 TeV and inter-KK Z/gluon s = 13 TeV LHC) of 2:5 fb (for KK Z) and 151 fb (for KK gluon). The paper is organized as follows. We begin in section 2 with laying out the structure of the model with gauge and gravity propagating in the same bulk, but matter/Higgs in a subspace, i.e., with the usual UV and IR branes along with a single intermediate brane demarking the matter/Higgs endpoint. In section 3, we then describe salient features of the LHC signals of this framework. In section 4, we discuss more general framework with two intermediate branes, in which gravity extends even beyond the gauge bulk. Section 5 provides our conclusions and outlook. Some technical details are relegated to the appendices. Model with one intermediate brane We consider gauge and gravity living in the same bulk starting at the UV brane, with scale UV . MPl and ending at the IR brane, with scale IR, which can be as low as of TeV: see gure 6. In the notation used in section 1, both which are also (roughly) the gauge and graviton KK scales in the 5D model. For now, we will assume the gauge symmetries to be only the SM throughout the bulk so that we do not have A5's; we will brie y discuss the latter possibility in section 2.4. The rest of the SM propagates from the UV brane to an intermediate brane (dubbed \Higgs" brane), taken to O(10) TeV consistently with (anarchic) avor bounds. We will discuss more details below, showing that even with contribution from composite states of strong dynamics below Higgs, our framework is indeed safe from EW and avor/CP violation precision tests. As usual, the lighter SM fermions are assumed to be peaked near the UV brane. We use the usual notation where M5 is the 5D Planck scale and k is the AdS curvature scale. The cubic self-coupling of graviton KK modes (or that of one graviton KK to any two modes localized near IR brane, for example, KK gauge) is then given (roughly) by Also, g?grav is dual to coupling of three composites, one of which being spin-2 (and for which we will use the same notation). Similarly, g5 is the (dimensionful ) 5D gauge coupling, with the coupling between (three) 4D modes (one of which is gauge KK) localized near IR brane (or three composites, with one being spin-1) given (roughly) by As usual, the sizes of both g?'s are constrained by perturbativity and tting observed/4D SM couplings (i.e., of zero modes). However, in the model at hand, there is a new ingredient, namely, the intermediate (Higgs/matter) brane which has tension, i.e., is gravitating, resulting in (i) k being di erent on the two sides of this brane and (ii) a new perturbativity constraint associated with branon (brane-bending) degree of We will discuss these issues in detail in appendix A; here we simply summarize. The following choices of couplings (in the far IR) su ce for having a nite regime of validity of 5D e ective eld theory (including the branon degree of freedom): while giving observable signals. We expect to have two radions (dual to dilatons in the CFT description), roughly corresponding to uctuations of Higgs brane relative to UV (heavier mode) and that of IR brane relative to Higgs brane. We now work out some of the details of this picture. We rst give a schematic review of the GW mechanism in the CFT language for the minimal model of gure 1 [3]. We start in the UV with where OGW is scalar operator with scaling dimension (4 > 0): we also use the convention where its naive/engineering dimension is the same so that the coupling constant above is dimensionless. We assume that OGW acquires a VEV in the IR, breaking the conformal symmetry spontaneously; this scale can be thought of as the VEV of the dilaton eld (denoted by of mass dimension +1). So, we get the dilaton potential UV where the second term on the r.h.s. is consistent with conformal symmetry and in the third term, d is an O(1) factor in the interpolation of terms, i.e. O scaling dimension of OGW remains (4 by OGW. Here, we assume that the ) even in the IR and we have dropped subleading Minimizing above potential in the IR, we see that the radius is stabilized, i.e., IR scale UV; assuming where a is an O(1) factor. Plugging this into the above potential, the lighter dilaton mass is then given by [65{68] is then (roughly) set (as above) to logarithm of hierarchy (the one relevant here is between Higgs and IR branes) and 0 is dual, in 5D, to the amount of detuning of IR brane tension. So, to summarize the various scales, we consider the case: above (and henceforth) indicates validity up to O(1) factors. In particular (and as is well-known), we see that O(0:1) (i.e., a mild tuning) together with ( d = 0) Once again, in the model at hand, we will have two copies of above module, roughly speaking corresponding to the two hierarchies, i.e., Higgs= UV (roughly the usual one) be done \sequentially", giving a heavy dilaton (mass dictated by Higgs) and lighter one IR): for the purpose here (i.e., LHC signals), we will focus on the latter, for Higgs can be simply taken to be a \ xed/UV" scale. The physical dilaton (denoted by ') corresponds to uctuations around the VEV, i.e., 5Note that we envisage the new, second hierarchy to be at most O(10), i.e., it is (much) smaller than the usual/ rst one, thus requiring an even more natural value of , i.e., 1= a few , cf. O(0:1) for the Once again, we treat the separation between UV and Higgs brane to be xed, thus reducing the (light) radion/dilaton analysis to the usual minimal case with only two branes. We then simply drop the label \IR" on dilaton and OGW. Coupling to SM gauge These can be deduced from the running of the SM gauge couplings as follows. We start with value gUV at UV and pass through various thresholds all the way to MZ [69]: + bstrong UV log + bstrong IR log btop; Higgs) log where bstrong UV (IR) are the contributions of UV and IR 4D strong dynamics (including, in the former case, the SM top quark and Higgs, which are composites), respectively, to the running of the SM gauge coupling and bSM is the usual SM contribution. where in second line, we have used the standard large-N relation that coupling of three composites, i.e., g?gauge (in this case, one being spin-1/gauge) is given by 4 =pNstrong. In fact, the 5D result is: OGWOt=H be discussed more below). Since OGW obtains a VEV at scale IR ( uctuations around which correspond to the dilaton), we can interpolate it in the IR as i.e., (as above) we can choose derivatives to not appear on ', which implies that we derivatives on top quark and Higgs elds): Ot=H 3 t 6 @ t t + c1ytttH + c2 @ Hy @ H + c3Hy H + bstrong = bstrong = where c's are independent /arbitrary coe cients. which (as expected) is a good match to the second line of eq. (2.12) above [using eq. (2.2)]. The dilaton can be considered to be uctuations around TeV scale, i.e., IR + ag?grav' [see eq. (2.8)]. We plug this into the gauge eld kinetic term in the form gauge eld, the dilaton coupling to SM gauge bosons [68, 70, 71]: IR Coupling to top quark/Higgs. For simplicity, we assume that the top quark/Higgs are strictly localized on the Higgs brane, which (as already mentioned) we are treating (e ectively) as \UV" brane for the purpose of obtaining couplings of the light radion. In the 5D model, we can couple the Higgs and top quarks to the 5D GW eld (used for stabilization) evaluated at the Higgs brane, thereby generating a coupling of radion to the top quark/Higgs. We will work out the size of this induced coupling in the compositeness picture, the above coupling in the 5D model being dual to: that from the Higgs kinetic term. Let us consider dilaton decay from each term in turn. A quick, explicit computation simple argument based on angular momentum conservation for scalar decay into a fermionantifermion pair shows that it must be so. So, the rst two terms actually contribute similarly to the third term, i.e., \mass" term (where we have included yt, i.e., SM top Yukawa, as avor spurion in the power counting). On the other hand, for ' ! HyH, i.e., decay into scalars, there is no such constraint from angular momentum conservation: indeed, we explicitly nd that kinetic term for H gives amplitude / pH 1:pH 2 m2'=2 (in the limit of mH H term (for on-shell H) is / m2H , i.e., actual mass term, which is have assumed that the SM Higgs complex doublet H is a PNGB so that its \mass squared" is SM loop factors smaller than O(10) TeV and IR a few TeV, we see that this contribution is | roughly and numerically | comparable to m'). Note that contribution mttt + (@ H)y @ H which gives a (much) smaller decay width for dilaton into top/Higgs as compared to SM gauge bosons in We conclude from the above analyses that the production of the radion/dilaton is dominated by gluon fusion; dilaton decays mostly to two SM gauge bosons, all via eq. (2.14). Spin-1/Gauge KK eigenstate: see below). We focus here on the lightest spin-1 composite, denoted by ~ (reserving for the mass Flavor universal coupling. The avor universal part of coupling of (to matter/Higgs elds) is given by a generalization of the well-known phenomenon of from QCD (for simpli ed discussion | using elementary-composite sectors | see, for example,[72]) (see also gure 7), which we brie y review here. We start with the kinetic and mass terms 2 IR ~ ~ 6Quotes are used here since these are actually multiplied by '. 7We have checked that other possible contributions to the radion couplings to top/Higgs are comparable to or smaller than the above. Aelem∗ gg⋆gealeumge = cos Aelem + sin ~ = cos ~ where Aelem denotes gauge eld external to the 4D strong dynamics (thus often called \elementary"): all SM matter (fermions and Higgs boson, denoted generically by q above) couple to it with strength gelem. Similarly, all composite fermions of strongly coupled sector are denoted by and composite vector meson ~ couples to them with strength g?gauge. Note that the second term in the second line of eq. (2.19), ~ Aelem, is obtained by starting from AelemJstrong IR and then using the usual interpolation for (the lightest) spin-1 composites ( mesons): As we will see, even though the above mass terms break elementary gauge symmetry, there is a residual gauge invariance (corresponding to a massless eld) which we identify with the nal SM gauge symmetry [72]. We diagonalize the mass terms by de ning the physical states (admixtures of ~ and Aelem): which however is not relevant for collider signals. Note that using mixing in rst line of eq. (2.27), one naively obtains couplings of ' to A A ; however, after properly adding contributions from the other two terms in the second line of eq. (2.19), we can see that these terms vanish. In addition, after radius stabilization/explicit breaking of conformal symmetry, we get a mixed coupling of dilaton, i.e., to and SM gauge eld as follows. In the IR, we can interpolate the GW operator as Plugging above in eq. (2.4), RG-running down to IR and then promoting IR ! Finally, plugging the mass eigenstates from eq. (2.22) into above gives:9 8More precisely, dependence of gSM on IR originates from dependence of gelem on IR via the relation gS2M9The same procedure also results in couplings of the form ' couplings of dilaton/radion to pairs of SM gauge elds from radius stabilization [68, 71] and 's; however, these are sub-dominant to the pre-existing ones, hence we will not discuss them further. F , i.e., corrections to the where the last term is the (universal) coupling of SM fermions to . Also, as anticipated above, A is massless (thus corresponds to the SM gauge eld), with being the SM gauge coupling. Henceforth, we will assume gelem and coupling of SM fermions to Couplings to radion/dilaton. As discussed above, couplings of dilaton/radion can be obtained by using it as a \compensator" for IR, giving eq. (2.14) from dependence of gSM8 IR (via RG evolution of the gauge coupling) and a coupling to two ~'s (which gets converted mostly into two 's): gSM = gS2M=g?gauge. (g⋆gaUuVge)2 From eq. (2.7), here we have 1= log ( Higgs= IR) 1= a few, since the relevant hier Higgs= IR as indicated (again, it is not the large one: UV= IR), and from this we also see that ( IR= Higgs) is an O(1) factor. Thus, the -dilaton-SM gauge boson coupling in eq. (2.30) can be (roughly) comparable to the last term in eq. (2.24), i.e., coupling (assuming g?grIaRv conservation arguments. low) is not allowed by a combination of Bose-Einstein statistics and angular momentum 1). Note that decay of to two ' (cf. spin-2 beFlavor non -universal couplings to top/Higgs. On the other hand, the avor nonuniversal part of the couplings (relevant only for top quark/Higgs: negligible for light fermions, at least for LHC signals) arises from where this coupling of top/Higgs to IR strong dynamics is generated by integrating out physics of top/Higgs compositeness at scale Higgs, with a coupling characteristic of gauge sector of the UV strong dynamics (see appendix B for further explanation of the UV and IR CFT's with stabilization mechanism). This runs down to the IR: where we have used the interpolation relation of eq. (2.20). Clearly, the production of at the LHC proceeds via light quark coupling in last term in eq. (2.24), while decays occur via same coupling and that in eq. (2.32) and (2.30), assuming ' is lighter than . Electroweak and avor/CP violation precision tests. The physics of top/Higgs compositeness with characteristic mass scale Higgs (where the UV strong dynamics connes) contributes to EW and avor/CP violation precision tests. However, as we already indicated at the beginning of section 2, these contributions are safe from experimental constraints for the choice of O(10) TeV. Notice that the (small) avor non-universal parts of the couplings of spin-1 resonances of the IR strong dynamics [see eq. (2.32)] | which are suppressed by Higgs | also give contributions (via their virtual exchange) to EW and avor/CP violation precision tests. However, as we will show now, such e ects are comparable to the direct (albeit still virtual) e ects of Higgs scale physics hence are safe/on the edge (just like the latter). We begin our discussion by considering contributions of IR strong dynamics to precision tests observables using the above non-universal coupling only once, for example, the operator corresponding to the S-parameter: C W 3 B W3 and B being the neutral SU(2) and hypercharge gauge elds and g (g0) are the respective gauge couplings. Integrating out physics at and above the scale the IR e ective theory the above operator with coe cient CUV Higgs generates in gg0= 2 usual, naive dimensional analysis). The contribution from the IR strong dynamics can be obtained by computing the diagram shown in gure 8. Such a diagram can be generated by sewing together eq. (2.31) (non-universal coupling) and the (universal) coupling A Jstrong IR [mentioned below eq. (2.19)], via the common Jstrong IR. gelem g?gaUuVge 2 AelemhJstrong IRJstrong IRi 2 J t=H The current-current correlator hJ (p)J ( p)i contains the piece which contributes to the S-parameter operator. nd a log-divergence in the S-parameter in the theory below Finally, matching to the S-parameter operator and using eq. (2.12) for overall size of correlator, we get Cstrong IR IR strong dynamics to S-parameter is then comparable to that from physics at However, there is an important feature we want to emphasize. Namely, the contribution of IR strong dynamics to S-parameter shows a mild logarithmic enhancement! This enhancement, however, is not harmful because, with custodial symmetry protection, the constraint from EW precision test on the Higgs compositeness scale in the minimal model of gure 1 can be as low as 3 TeV [19{23] so that, even with the above enhancement in the extension in gure 6, the overall size is small enough with Next, we consider cases where two non-universal couplings are involved, giving (for example) a 4-top quark operator, which after rotation to mass basis for quarks will give avor-violating e ects even for light fermions such as K K mixing [24{31]. Clearly, the above). For the IR strong dynamics contribution, we combine eq. (2.31) with itself in this 10We have also checked explicitly that the contribution to the S-parameter from the sum over tree-level exchanges of composite resonances (in the 4D picture with strong dynamics) or gauge KK modes (from the 5D model) gives a log-divergence. from here; instead the bound on be discussed in section 3.2.1. Spin-2/Graviton KK IR is dominated by the direct LHC searches which will case. Here, the current-current correlator can instead give a quadratic divergence, which reduces the initial the entire IR strong dynamics to such avor/CP violating processes are comparable to that of the physics of the UV strong dynamics, hence safe. We stress that, for a Higgs, the contribution to precision tests from IR strong dynamics is (roughly) independent of IR so that there is no relevant constraint on We denote the composite spin-2 by H . In general, H couples to not only T composites, but also other possible Lorentz structures built out of the latter Here, for simplicity and because it dominates in warped 5D e ective eld theory, we will as a representative structure (others will anyway give roughly similar size for coupling/amplitude). If experiments show spin structures other than T , it would point to strong dynamics without a good 5D dual. Coupling to SM gauge bosons. The coupling of H to SM gauge bosons is obtained gure 9) by rst coupling it to ~'s with strength g?grIaRv (i.e., a 3-composite vertex), followed by mixing of ~'s with external gauge eld (as outlined above), i.e., gg⋆gealeumge Coupling to radion/dilaton. In addition, we have the coupling to two dilatons/radions: IR IR gauge bosons) elds. Of course, this is relevant for decay of composite spin-2/KK graviton only if m' . IR=2 and in this case, dominates over other decays: see, for example, [64]. Flavor non -universal coupling (to top/Higgs). Finally, coupling to top quark/Higgs follows from a procedure similar to spin-1 above, i.e., we have (t=H)T (strong IR) of IR strong dynamics) and this coupling of top/Higgs to IR strong dynamics is generated by integrating out physics at the scale Higgs, with a coupling characteristic of gravity sector of the UV strong dynamics. After IR theory hadronizes, eq. (2.37) becomes using the interpolation T (t=H) Production of composite spin-2/KK graviton occurs via coupling to gluons in eq. (2.35). Decays of composite spin-2/KK graviton is dominated by the same couplings, i.e., into all SM gauge bosons and to pair of dilatons via eq. (2.36), assuming m' < IR=2. We give a summary of relevant couplings in table 3. Given the above avor non-universal couplings of KK graviton of the IR strong dynamics (cf. those of gauge KK discussed earlier), it is clear that contributions from KK graviton exchange to precision tests are suppressed compared to those of gauge KK by there is no additional constraint here from the KK graviton sector. Extended Bulk Gauge Symmetries/Dual to PNGBs of Vector-Like ConRelation to vector-like con nement. We can enlarge the bulk gauge symmetries beyond the SM. We then consider breaking them down to the smaller groups (while preserving the SM subgroup of course) on the various branes by simply imposing Dirichlet boundary condition, i.e., SM where each stage of gauge symmetry breaking delivers (scalar) A5's, localized at the corresponding brane (including possibly the SM Higgs boson in the rst step). Such a framework is shown in gures 5 and 10. These A5's are dual to PNGBs arising from spontaneous breakdown of global symmetries of the strong dynamics corresponding to the gauged ones shown in eq. (2.40) [5]. In particular, the 4D physics dual to the last stage of breaking (rightmost gures 5 and 10), i.e., SM symmetries being unbroken, is known in the literature as vector-like con nement [44{46]. While from the 5D viewpoint, presence of A5's seems rather \non-minimal", it is quite natural to have PNGB's in 4D strong dynamics as illustrated by ordinary QCD. In fact, QCD-like strong dynamics was rst used to realize the general idea of vector-like con nement. Note that A5's are massless at tree-level (in the presence of only the above boundary condition breaking), acquiring a potential via loops, with mass scale being set by corresponding . Thus they are naturally light, as expected from them being dual to PNGB's. Gauge and graviton KK modes (and even possibly the radion) can then decay into pairs of A5's, drastically modifying the LHC signals of the gauge and graviton KK (or radion) based only on the couplings shown earlier. In this paper, we take the minimal 5D perspective in assuming that A5's are absent, cf. the expectation based on QCD-like 4D strong dynamics. Hence, gauge KK will decay dominantly into pairs of SM fermions, while SM gauge bosons will be the search channel for KK graviton and radion, as mentioned earlier. exibility a orded by 5D leads to broader class of models, with more diverse phenomenology than contemplating just 4D QCD-like strong dynamics. Coupling to two SM gauge bosons. There is an interesting comparison with dilaton/radion that we would like to draw by considering the simplest mechanism for production and decay of (single) A5 (dual to PNGB). Namely, PNGB famously has a coupling to two weakly-coupled gauge bosons via the (gauged) Wess-Zumino-Witten term, for example, we have 0F leading to the decay in real-world QCD. This interaction is dual to the one originating for the A5 from the Chern-Simons term in the 5D model (see also discussions in [50, 74]): where a, b, c are gauge adjoint indices. HIR ⊃ SM Crucially, we see that, ir respective of considerations of parity as a fundamental symmetry, the coupling of A5 to two SM gauge bosons via Chern-Simons term has \CP-odd" structure, i.e., involves F ~ . This feature is in contrast to the \CP-even" coupling, , of dilaton/radion as we see in eq. (2.14). Let us compare to vector-like con nement, in particular, QCD-like dynamics: this theory respects parity even in the IR and PNGB's are parity-odd (as per the Vafa-Witten theorem [75, 76]), which enforces a coupling to pairs of SM gauge bosons to be to the combination F ~ . However, we see that there is a more general (than parity) argument for such a structure from Chern-Simons term in 5D. Moreover, the 5D Chern-Simons term is dual to anomalies in global currents of the 4D strong dynamics, i.e., K of eq. (2.41) | appropriately made dimensionless | is related to the coe cient of the chiral anomaly in 4D. In this sense, we see that there is actually a similarity in the couplings of A5 (PNGB) and dilaton to two SM gauge bosons, i.e., both are driven by anomalies: chiral for former vs. scale anomaly for dilaton [as seen clearly in rst line of eq. (2.14), i.e., the \bstrong"-form]. Phenomenology General features. We rst discuss some overall points, before studying each particle in detail. Assuming IR, the couplings of the (lightest) KK/composite spin-1 gauge bosons to the SM matter (fermions and Higgs) are signi cant (albeit mildly suppressed relative to the SM values) and (approximately) avor-blind : see last term in eq. (2.24) and eq. (2.32). On the other hand, radion and KK/composite graviton couple predominantly to pairs of SM gauge bosons and negligibly to SM matter: see eqs. (2.35), (2.38), (2.14) and (2.18). This feature is in sharp contrast to standard minimal model of gure 1, where concerned. So, dilepton, diphoton and dijet nal states are usually | and correctly | m' =1 (2) TeV are tot part of table 1 in introduction. neglected, but now they acquire signi cance or even the dominant role. At the same time, the (small) avor non-universality arising in these couplings (i.e., Higgs/top compositeness scale) can be probed by precision studies of these avor-universal resonances (of mass IR), thereby distinguishing it from (purely) vector-like con nement (which corresponds to decoupling of top/Higgs compositeness scale), rather experimentally one can see the latter as a vestige of a full solution to the Planck-weak hierarchy. Finally, in the case of a uni ed bulk gauge symmetry, i.e., entire SM gauge group is subgroup of simple IR bulk gauge group (HIR of eq. (2.40)), we should of course also nd that resonances come in complete degenerate uni ed multiplets. This is dual to the IR strong dynamics having a simple global symmetry partially gauged by SM. Dilaton production. Note that dilaton can be somewhat lighter than higher spin composites [see eq. (2.9)], thus possibly the rst particle to be discovered. Rough estimates 4:4) fb.11 One of these sample points was mentioned as Moving onto decays of dilaton, these are dominantly to two SM gauge bosons (based on the couplings discussed earlier, assuming IR). It is noteworthy that in the uni ed case, i.e., SM gauges a subgroup of a simple global symmetry group of 4D strong dynamics, considering SU(5) uni cation as an example here, we obtain the following relation [see eq. (2.14)]: C'gg = where C'V V denotes the coupling of the dilaton to two corresponding SM gauge bosons and gV 's are corresponding SM gauge couplings, both being renormalized at a relevant energy scale (roughly at m'). This striking feature can be checked by measuring dilaton BR's. Numerically, BR's to , ZZ, W W and gg (in this uni ed case) are 1%, 3% and 95%, respectively. However, note that the above universality (among the SM gauge groups) feature applies for any HIR-singlet composite scalar. In this sense dilaton is not unique. The current bounds on cross-section BR to di-photons from resonant di-photon searches at the LHC [77, 78] are 0.5 (0.2) fb for 1 (2) TeV mass. Similarly, di-jet searches [79] give a bound of 200 fb (1 pb) for 2 (1) TeV mass. Both of these are satis ed for the above illustrative choice of parameters, although the 1 TeV case is on the edge of the di-photon bound. Note that values of g?grav larger than 1 would then be ruled out (keeping other parameters the same). However, for the model with two intermediate branes to be discussed in section 4, we will show that such values of g?grav can indeed satisfy 11All cross-section numbers are for LHC13 and have been obtained using implementations of above models The CP-even structure of the couplings to SM gauge bosons for dilaton vs. CP-odd for A5/PNGB's (discussed above: see eqs. (2.41) and (2.14)) is an important issue. It can be discriminated by (for example) decays to ZZ ! four leptons, using the additional observables therein, i.e., corresponding to polarization of Z (as compared to using just angular distribution of spin-summed SM gauge boson taken as \ nal" state, which is the same for both cases) [80]. Spin-1 composite Here, we have more than one type, each with several competing decay channels. So, we need more detailed analysis for obtaining bounds/signals. We give some general arguments rst. In the uni ed case, based on same mass and composite coupling as in eq. (2.24), we should nd for SU(5) uni cation as an example (similarly to the radion above) qq! g = where qq! V denotes the production cross section of the composite spin-1 resonance which mixes with external gauge boson V and gV 's are corresponding SM gauge couplings renormalized at a relevant energy scale (roughly at MKKgauge). In the non-uni ed case, while the above relations do not apply, the following correlation between radion decays and spin-1 production cross-section can nonetheless be tested: as seen from eqs. (2.14) and (2.24), we expect coupling of dilaton to SM gauge boson (gauge coupling)2 (corresponding) composite spin-1 cross-section same for all SM gauge groups Remarkably, in spite of apparent lack of uni cation (i.e., bstrong is di erent for di erent gauge groups), we nd that the above ratio is universal! Moreover, it applies only for the case of composite scalar being dilaton, i.e., the above relation is not valid for a generic scalar composite. In contrast, in the uni ed case, the above correlation is not independent of the two separate relations discussed earlier, i.e., eqs. (3.1) and (3.2). Note that the universal constant on r.h.s. of eq. (3.3) involves g?grav [apart from other known factors: see eq. (2.14) and last term of eq. (2.24)]. Thus, independent determination of g?grav, for example, from KK graviton measurements could provide an interesting test of this framework using eq. (3.3). This would apply to both uni ed and non-uni ed cases Current bounds in avor-universal limit Based on the suppressed (as compared to the SM, but still non-negligible) and universal coupling in the last term of eq. (2.24), we nd that spin-1 masses of a few TeV are still consistent with the LHC searches performed so far in multiple channels . We now move onto more details, discussing bounds on KK Z rst, followed by KK gluon. (i) Di-lepton: note that composite/KK Z in this case is (approximately) like sequential SM Z0, but with coupling to light quarks inside proton (the dominant production mechanism) being reduced by gEW=g?EW. We nd that predicted cross-section of sequential SM Z0 exceeds the bound [81, 82] by 70 (25) for MZ0 Translating this bound to our case, we get (setting gEW Of course, only the smaller values of g?gauge( 3) are compatible with a controlled 5D description, but the somewhat larger values ( 5) are still reasonable from the viewpoint of (purely) 4D strong dynamics, for example, coupling in real-world QCD is roughly of this size. We can of course interpolate for other composite spin-1 masses. To be more precise, we will have to add bound from composite photon (above was just composite Z) but as an estimate what we did should su ce. Similarly, we can obtain a bound on KK W in our model based on the searches for W 0's (via their leptonic decays) at the LHC [83, 84]: we nd that it is (roughly) comparable to that on the KK Z and KK gluon (as we discuss below). for heavy vector triplet (HVT) model [85], which is (roughly) similar to standard warped/composite case of gure 1 (i.e., couplings to Higgs/top dominate): the current the above HVT model decay to dibosons with a BR of 100% , since couplings to dibosons are (much) larger than to the SM fermions, latter being assumed to be avor-universal. On the other hand, in the (fully) avor-universal limit that we are considering here, we can readily estimate that the BR to dibosons is reduced to (roughly) 4%, in which case, bound is weaker than 2 TeV (rescaling from the experimental plots). So, we conclude that di-lepton bound for our KK Z case is a bit stronger than di-boson. Just for completeness' sake, we mention that there is also a Z0 bound of 2{2.5 TeV from the di-jet search [79]. However, this assumes coupling to light quarks inside proton is same as SM Z, vs. smaller here. Similarly, Z0 bound from di-top is 2:5 TeV [88, 89], but that is for a model with enhanced (even with respect to the SM) coupling to rst and third generations [90]; hence for our case, bound should be weaker. Overall, then di-jet and d-top bounds for KK Z are sub-dominant to that from di-lepton discussed earlier. 89]: the predicted cross-section [all for g?QCD 5, as assumed in [91], which is quoted in [88, 89]] is larger than bound by above bounds are assuming BR to top quarks 1 (as in the standard scenario) so that for our case (i.e., with BR to top quarks of 1=6 instead), we get As usual, we can interpolate for other composite spin-1 masses. (ii) Di-jet: here, we can re-scale from axigluon bounds [79], i.e., coupling to our composite gQCD= g?QCD gluon is smaller by a factor of 2 , since coupling of axigluon [see discussion in [92] referred to by [79]] is larger than QCD by p2. The cross-section is constrained to be smaller than the prediction for axigluon by 50 (30) for axigluon mass of 2 (2.5) TeV. So, using the above couplings, we get for our case: Similarly, we can nd the bound for other values of IR. So, di-top and di-jet bound are (roughly) comparable in the case of KK gluon. Probing top/Higgs compositeness Next, we discuss the possibility of being able to see some remnants of top/Higgs compositeness in the properties of composite resonances at IR. As seen from eqs. (2.32), (2.38) and (2.18), spin-1 couplings (cf. dilaton and spin-2) at the LHC are most sensitive to avor non-universal corrections. In particular, for spin-1 composite, the net coupling [combining eqs. (2.24) and (2.32)] to SM fermions is then given (schematically) by: Here, h is an O(1) factor which depends on details of the model (whether a 4D composite theory or 5D dual). Note that the 5D model gives opposite sign for the avor non-universal coupling (to top/Higgs) of spin-1 vs. avor universal one, i.e., h > 0, whereas from purely 4D CFT viewpoint, h < 0 cannot be ruled out. Eq. (3.10) shows that the non-universal contributions (second term above) start becoming relevant (i.e., comparable to the universal rst term) for: 3 TeV; a universal g?gaUuVge equality occurs (roughly) for 0:6 and gQCD 1, we see that above 10 (15) TeV for KK gluon (Z) which is (roughly) the avor bound, i.e., (in general) we do expect sensitivity to top/Higgs compositeness! Again, note that in the standard scenario, i.e., IR, the nonuniversal contribution actually dominates: see eq. (3.10). KK gluon vs. KK Z. In particular, KK gluon might be especially promising in this regard, since for the avor-universal case, di-jet bounds on KK gluon seem comparable to di-top as indicated above, which suggests that there should be signi cant sensitivity to above perturbations, for example, non-universal coupling to top being comparable to universal might then show up even at discovery stage! Whereas, in avor-universal limit, it seems bounds from di-boson/di-top are somewhat weaker than from di-lepton nal state for KK Z, thus suggesting that probe of top/Higgs compositeness (again, for the case when avor non-universal couplings are comparable to avor universal ones) might have to wait for post -discovery precision-level studies. On the other hand, as discussed above, for the same top/Higgs compositeness scale, avor non-universal e ects are actually a bit larger for KK Z than for KK gluon. So, overall, the two modes might be complementary in Details of analysis. Estimates of various BR's illustrating the above ideas are given in table 2: these were already mentioned in the introduction, including the tables. We now present more details. First, as a reminder, in this table 2, we x KK Z/gluon mass to be 3 TeV and the composite gauge coupling (g?gaUuVge) to be 3. Hence, the production cross-section is the same throughout the tables, but we vary Higgs compositeness scale. These numbers are obtained simply using the net coupling given in eq. (3.10). Just for the sake of concreteness, we choose a \central" value for the O(1) coe cient h in eq. (3.10) gluon and KK Z, respectively. Then, for each Higgs, we vary h between a factor of 2 Mostly for simplicity, we assume only tR (and Higgs) is (fully) composite, i.e., (t; b)L's compositeness is smaller. Also, we will assume of h > 0 (based on 5D model, as mentioned above). We then see that for values of Higgs= IR around eq. (3.11), there is actually a possibility of \cancellation" between the two terms in eq. (3.10); this feature is re ected in these tables in BR's to top/dibosons becoming smaller than avor-universal limit as we start lowering the Higgs compositeness scale from a high value. Note that, as re ected by our O(1) variation of h factor, we are not really contemplating a ne-tuning here, rather only pointing out that a mild suppression is possible in this way. Eventually, i.e., for even Higgs, of course the non-universal part of couplings to top/Higgs dominates over universal one so that BR's to top/Higgs become larger, as they asymptote to the values of the minimal model of gure 1. Finally, we have to consider the decay of (composite) spin-1 to a dilaton and a SM gauge boson. Based on eqs. (2.14) and (2.30), it is straightforward to show that there exists choices of the relevant parameters such that this decay is (much) smaller than to the SM fermions. For simplicity, here we assume that is the case in tables shown above. Having said this, a dilaton and a SM gauge boson is an interesting nal state (followed by dilation ! two SM gauge bosons), which (to the best of our knowledge) has not been studied before. In fact, in ongoing work, we are determining (other) regions of parameter space where this new decay channel actually dominates over the SM fermion pair mode and analyzing the corresponding LHC signals. Also, in this case, the BR to SM fermion pairs is suppressed, thereby relaxing the bound on gauge KK particles that were discussed earlier. As anticipated earlier (but now seen more explicitly in the tables), as we lower Higgs compositeness scale from decoupling limit, at O(10) TeV, we start seeing avor-blindness (middle vs. rightmost columns), that too \earlier" for KK Z than for KK gluon. At the same time, these BR's signi cantly di erent than standard Higgs compositeness case (leftmost column). So, the moral here is that composite Z/gluon can provide \glimpse" into Higgs/top compositeness, provided that this scale is not too far from the lower limit from avor/CP violation, i.e., Other values of KK masses. For the sake of completeness, we mention that the (total) s = 13 TeV 28, 0.3 fb (Z) and 1834, 17 fb (gluon), respectively (of course, the 2 TeV case might be ruled out as per above discussion, unless we invoke extra decay modes, for example to light A5's). From eq. (3.11), it is clear that as we vary composite spin-1 masses in this way, one could then be sensitive to lower/higher top/Higgs compositeness scale. Comparison to other probes of top/Higgs compositeness. Let us summarize by comparing the above signals of top/Higgs compositeness scale of O(10) TeV to other approaches. One of the standard probes would be existing/upcoming low-energy avor experiments, which will be sensitive to O(10) almost by construction, since O(10) TeV was chosen to barely satisfy the current avor/CP violation bounds. Of course, this would provide the most indirect view, for example, even if we see a signal, we cannot be sure about which underlying new physics it corresponds to, i.e., whether it is the warped/composite Higgs framework or some thing else. On the other hand, the most direct signal is possible at a future 100 TeV hadron collider, where the associated, i.e., O(10) TeV, physics of compositeness can be produced without any suppression. In fact, this could serve as a motivation to build such a machine. Here, we showed how extending the usual, minimal framework to include a intermediate brane ( gure 3) results in novel probe of the general framework. Namely, it creates a new threshold, i.e., a few TeV resonances intermediate in mass between O(10) scale and the SM/weak scale itself, whose leading couplings are avor-universal, rendering such a mass scale safe from avor bounds. This angle actually combines some of the virtues of both the above approaches, for example, we can directly produce the relevant particles at the ongoing LHC. Of course, simply discovering these few-TeV particles in avor-blind channels | even if very exciting! | would not quite constitute a smoking-gun of top/Higgs compositeness which lies at the core of this framework. Remarkably, we have seen above that the non-universal contributions to the couplings of these few TeV particles | stemming from top/Higgs compositeness | are not far behind. Hence, precision studies of these new states can indeed unravel these e ects. Clearly, this sensitivity to O(10) TeV compositeness scale is intermediate between explicit production of compositeness physics by a 100 TeV collider and indirect low energy avor tests. Finally, we mention (other) virtual e ects of this Higgs physics at the LHC such as on precision Higgs or top couplings measurements or analysis of continuum top/Higgs production. However, given O(10) TeV, even the high-luminosity LHC will not be sensitive to the e ects in these searches. The point is that such probes lack the resonanceenhancement12 that the above lighter spin-1 studies a ord: again, both these e ects do have a (common) (few TeV= Higgs)2 suppression. Spin-2 composite The (total) cross-sections (again, from gluon fusion, at p s = 13 TeV LHC) are above, decays are dominated by two SM gauge bosons, unless m' < 12 MKKgrav, in which case, decay to the dilatons dominates (due to stronger coupling). Furthermore, in the uni ed case, we get coupling of spin-2 to two SM gauge bosons / corresponding (SM gauge coupling)2. Thus, (neglecting decays to dilaton, for example, assuming m' > 12 MKKgrav) , ZZ, W W , and gg are 0.7 %, 1%, 3% and 95%, respectively (like for radion). It is also clear that current bounds on cross-section from resonant di-photon search are satis ed for above choice of parameters, since there is not much di erence between spin-0 and spin-2 here in so far as experimental bounds are concerned. Signi cance of spin 2. Even though the nal state for composite/KK graviton might be similar to dilaton (i.e., two SM gauge bosons), obviously, spin-2 vs. spin-0 can be distinguished using angular distributions. In fact, as already mentioned earlier, a random spin-2 has three di erent angular amplitudes [73] vs. KK graviton having only one (i.e., coupling to T only), hence providing disambiguation between generic strong dynamics and extra-dimensional frameworks (i.e., dual to a special structure of strong dynamics). Finally, it is interesting that mere discovery of spin-2 implies that there is an in nite tower of heavier states (whether composite or KK) because the theory of (massive) spin-2 is nonrenormalizable (vs. spin-0 or 1), thus guaranteeing more and rich discoveries in the future! Model with two intermediate branes Our work opens up other possibilities also: most signi cantly, we can have the gauge brane meson) from gravity ( glueball) as in gure 3. In this case, KK graviton/radion will be the lightest; in particular, radion can be lighter than KK graviton, as seen from eq. (2.9).13 So, we have (parametrically speaking) m' . stabilization of the inter-brane separations (in this case, we have three of them) can be done via a generalization of what was done for the model with one intermediate brane above. In more detail, the couplings of KK graviton and radion to SM gauge bosons will be 12In fact, these states are quite narrow. For example, with the assumptions made above and for Higgs 15 TeV, we estimate that =M for KK Z is O(0:1%). 13In fact, (very) recently [93] studied a 4D model (with new | pure glue | strong dynamics) which is sort of dual of the above gauge-gravity split case (with the lightest scalar glueball being roughly the with two intermediate branes. top/Higgs in the model of gure 5 studied here. As discussed in sections 2.3.1 and 2.3.2, these couplings result from exchange of (heavy) physics at meson. Essentially, we perform the replacements T (t=H) eq. (2.18), along with ! T (gauge) in eq. (2.38) and Higgs kinetic term ! F meson in both equations. On the other hand, couplings of dilaton/spin-2 to top/Higgs and those of spin-1 to all SM matter remain the same. Here, we simply summarize all these couplings in table 4 (cf. table 3). Note that for xed mass of the spin-1 composites ( meson), the couplings of the lightest states in this model (i.e., KK graviton/radion) relevant for their production (i.e., to gluons) become weaker as we lower glueball, i.e., their mass. On the other hand, PDF's relevant for production are enhanced in this process, providing some compensation. Remarkably, it turns out that within the range of interest the former e ect (i.e., weaker couplings) tends to dominate so that the cross-sections actually reduce (i.e., bounds and visibility get weaker) as we lower the KK graviton/radion mass. A sample point is as follows: g?grIaRv = 3 gives (total) cross-section of meson = 3 TeV, m' = glueball = 1 TeV, g?QCD = 3 and 1 fb, respectively, for KK graviton and dilaton (former being larger mostly due to multiple polarizations). The decay BR's are similar to the model with one intermediate brane case. Note that gauge KK/spin-1 composite cross-section at this point are comparable to/larger than that of graviton/dilaton; in fact, the gauge KK would be strongly constrained (if not ruled out), assuming decays directly to SM particles (as discussed above). However, the spin-1 states can decay directly into non-SM particles such that they are e ectively \hidden" from SM pair-resonance searches such as dileptons or dijets. For example, light A5's (dual to PNGB's) can provide such channels.14 In this way, KK graviton/dilaton can actually be the most visible channel. Table 1 in the introduction had already displayed this interesting possibility. Based on the discussion in section 2.3.3 of KK graviton contributions to precision tests, it is clear that the only relevant constraint on the KK graviton mass scale, i.e., in this model comes from direct LHC searches; in particular, using the cross-sections given above and bounds given earlier, we see that glueball is then allowed to be as low as (or even smaller). 14For a recent application of this idea in the context of the 750 GeV diphoton excess at the LHC, see [59]. The LHC Run 1 complemented by electroweak/ avor/CP precision tests have thus far seen no deviations from the SM. In light of this we must conclude that the principle of Naturalness, that predicts new physics below the TeV scale, is either (i) at the cusp of discovery at the LHC, (ii) playing itself out in some exceptional dynamics (such as Twin Higgs theory [94]) that evades our standard experimental probes, or (iii) that the principle is compromised in some way. Our e orts must be directed at all these options. Higgs compositeness (AdS/CFT dual to warped extra-dimensions) within the LHC reach remains a strongly motivated possibility for (i), but requires some new re nement of the warped GIM mechanism. This paper is directed instead to the option (iii) in the same, broad framework. Indeed, it is noteworthy that the minimal incarnation of this paradigm gure 1) can readily and elegantly t the experimental facts if we take the related new physics to live at O(10) TeV, solving the \big hierarchy problem" between the electroweak and Planck scales, but leaving unexplained a \little hierarchy problem". It is not the modest associated ne-tuning that disturbs us here but the fact that the solution to the hierarchy problem would then lie out of LHC reach! We have shown that a simple extension of the above model can also readily t all the experimental facts if the physics of naturalness is deferred until O(10) TeV. Namely, when di erent elds propagate di erent amounts into the IR of the extra dimension (see gure 3), there can naturally be lighter TeV-scale \vestiges" of the heavy naturalness physics within LHC reach, in the form of new spin-0,1,2 resonances, identi ed as KK excitations of the extra dimension or composites in the dual mechanism of vectorlike con nement. Although they would constitute a rich new physics close at hand, they escape the strong constraints avor/CP tests by virtue of their largely avor-blind, gauge-mediated couplings to the standard model. We have described several striking features of their phenomenology in the 5D Randall-Sundrum framework and its AdS/CFT dual. In particular, search channels at the LHC such as dileptons, dijets and diphotons for the TeV-mass resonances acquire signi cance in this framework, cf. decays being dominated by top/Higgs in the minimal model of gure 1. In addition (and in contrast to the minimal model of gure 1), the radion and KK graviton (i.e., the gravitational sector) can be readily lighter than other states and, in part of the parameter space, can even lead to rst discovery. But avor-blindness, however rich the physics, also suggests blindness to the solution to the hierarchy problem. Fortunately, we saw there are small deviations from blindness in resonance decays into top/Higgs rich nal states. These processes thereby give a resonance-enhanced \preview" of Higgs compositeness at the LHC, even though the Higgs compositeness scale and its ultimate resolution of the hierarchy problem is out of LHC reach! This provides a pathway in which LHC discoveries might set the stage for even higher energy explorations. A roughly comparable analogy within the supersymmetric paradigm is (mini-) Split SUSY [95, 96], in which the stops most directly relevant to the hierarchy problem lie above LHC reach (helping to explain the larger-than-expected Higgs boson mass) while spin-1/2 super-partners are signi cantly lighter. Seeing the lighter super-partners at the LHC with their SUSY-speci c quantum numbers would also give a \preview" of the supersymmetric solution to the hierarchy problem, which could be fully con rmed by going to higher energy colliders. In future work, it will be interesting to study in more detail the LHC signals for the TeV mass resonances which were outlined here, including what we can learn about the physics of top/Higgs compositeness at O(10) TeV from their precision analysis. In addition, we plan to initiate investigation of more direct signals of the latter physics which might be possible at a future 100 TeV hadron collider. We are now at the beginning of LHC Run 2, and it is essential that theory lays out the most plausible and powerful mechanisms within reach. In the language of 4D strong dynamics we have shown that vectorlike con nement arising in the IR of Higgs compositeness is such a plausible form of new physics, already exciting at the LHC and able to pave the way for an even more ambitious program of discovery at future higherWe would like to thank Zackaria Chacko for discussions. This work was supported in part by NSF Grant No. PHY-1315155 and the Maryland Center for Fundamental Physics. SH was also supported in part by a fellowship from The Kwanjeong Educational Foundation. Details of choice of parameters Matching at the intermediate/Higgs brane We assume the same 5D Planck scale (M5) throughout the bulk. However, in short, the bulk cosmological constant (CC) | and hence AdS curvature scale (k) | will be di erent in the matter/Higgs and gauge/gravity (only) bulks due to presence of (tension on) the intermediate/Higgs brane. In more detail, we de ne THiggs = 12M53 (kIR where \UV" and \IR" denote the bulks on the two sides of the Higgs brane and RAdS is the AdS curvature radius. Solving Einstein's equations across the the Higgs brane (with tension, THiggs) gives [97]: Since we require THiggs > 0 in order to avoid a branon (brane-bending degrees of freedom, denoted by Y ) ghost [98], we see that i.e., curvature scale increases in the IR. Let us consider in the following how this new feature modi es the usual choice of parameters. Implications of above matching Consider the gravity sector of the model rst. Clearly, we then have two di erent g?grav's on the two sides of the Higgs brane: As usual, we have bulk gravity becoming strongly coupled at [99] Suppose we would like to have at least NKmKin number of weakly-coupled KK modes (i.e., that much gap between 5D cut-o and curvature scale as our control parameter). Then we gg?g?grrUaIaRvVv = from the condition that bulks, i.e., for both g?grIaRv; UV mental/5D parameters (for example, between k and M5), we would also impose that g?grav grav strong & NKmKink. Note that this is required in each of the two . Of course, in order to avoid large hierarchies amongst funda Moving onto gauge sector, we similarly have The strong coupling scale is given by : g?gaUuVge = where NSM denotes size of the SM gauge group (take it here to be 3 for color) and factor of 3 in denominator above (i.e., enhancement of loop expansion parameter) comes from counting helicities of spin-1 eld. So, the associated request (i.e, imposing sgtaruogneg & NKmKink) is for each of the two bulks. On the other hand, tting to the observed/SM gauge coupling gives lower limits on g?gauge as follows (note that there is no analog of Landau pole for gravity, hence no lower limit on g?grav on this count). Consider the running of the SM gauge couplings from the UV cut-o to the IR shown in eq. (2.11). Plugging in the low-energy values of gSM and bSM into eq. (2.11), we nd (assuming 10 TeV and IR from the requirement that 1=gU2V (much) smaller hierarchy. 1015 GeV. However, ggauge mostly unconstrained, since it contributes over a > 0, i.e., Landau poles for SM gauge couplings are Finally, there is another requirement that the strong coupling scale of the Y selfinteractions be (at least modestly) above the curvature scale, i.e., So we need [as usual, imposing sbtrraonnogn & NKmKink and using eq. (A.2)] (including giving observable LHC signals): for a minimal request of g?grUavV < g?grIaRv . 3 and g?gaUuVge (and corresponding to kIR=kUV Note that g?gaIuRge and g?gaUuVge are \forced" to be close to each other, due to a combination of perturbativity (upper bound on g?gaUuVge) and Landau pole (lower bound) constraints. One possibility to relieve this tension is to reduce the UV-IR hierarchy, for example, lower the UV scale to the avor scale of 105 TeV [100], while keeping IR scale eq. (2.11), we see that g?gaUuVge & 2 might then be allowed (keeping both g?gauge's at/below 3 for perturbativity). Two dilaton system Here we discuss the CFT dual of stabilization of the model with one intermediate brane studied in the main text. In short, as usual, we start with a CFT at a UV cut-o This CFT con nes, i.e., scale invariance is broken, at int, which is to be identi ed with Higgs, i.e., scale of the Higgs brane in the speci c model, but here we would like to keep the notation more general. As already mentioned, this scale can be parametrized by the VEV of dilaton/radion eld [denoted by int of mass dimension +1, uctuations around which correspond to the physical dilaton ('int)], i.e., We can check that the following choices of couplings barely satisfy all the above needs The departure from the standard (i.e., minimal model of gure 1) script involves the resulting (daughter) theory (i.e., below int) owing to a new xed point. This \IR" CFT then con nes at an even lower scale IR, corresponding to the VEV of another eld, IR (associated with a second dilaton, 'IR). In more detail, in order to stabilize the two inter-brane separations (dual to determining the various mass scale hierarchies), we perturb the (UV) CFT by adding a single scalar operator (dual to the GW eld) in the UV: UV UUVV OGW where scaling and naive/engineering dimension of OGUVW is (4 above is dimensionless). As usual, we assume that there is only one scalar operator with scaling dimension close to 4, rest of them being irrelevant (hence being dropped from the Lagrangian). We ow to int (as usual, promoting appropriately 's to 's throughout): where d1; 2 are O(1) factors. Let us elaborate on the various terms above. The rst three terms above (in rst line) are as discussed earlier (i.e., for the usual minimal model). Whereas, the rst new term (in second line above) comes from using the interpolation: in the RG evolved explicit conformal symmetry breaking term in eq. (B.2). Here, (with obvious choice of notation) OGIRW is an operator of the IR CFT of scaling dimension (4 again, we assume that there exists only one such operator. On the other hand, the second term in second line of eq. (B.3) arises from spontaneous conformal symmetry breaking at int, i.e., even if OGUVW were \absent". Given above assumption about scaling dimensions of scalar operators of the IR CFT, it is clear that both terms in second line above must involve the same operator (as the leading term), i.e., coupling of int to other scalar operators of the IR CFT will be irrelevant. Finally, i.e., RG owing to the far IR scale of IR and adding (for a second time) the usual term consistent with the (IR) conformal symmetry, we obtain the complete potential for the two scalar elds ( 's): UV We have to minimize the above potential in order to determine the scales int and IR in terms of UV and the scaling dimensions [we can assume that the various 's are O(1)]. As usual, we assume UV; IR are modestly smaller than 1. In this case, we can proceed with the minimization in steps as follows. At \leading-order" (LO), it is reasonable to assume eq. (B.5) (i.e., potential for int by itself) to be: int is mostly determined (as in the minimal two brane case) by rst line of potential for IR, will give (again, as usual): 1= UV 1= IR As a (partial) consistency check of the above procedure (for obtaining the values of VEV's), we can consider the mixing (if you will, the NLO) term involving both the dilatons arising from the last two terms of second line of eq. (B.5), where int can be thought of as uctuations around int: We see that this results in a mixing angle between two dilatons of small enough. As a further check, we can show that the rst derivatives of the full potential in eq. (B.5) at above values of VEV's vanish, up to terms suppressed by (powers of) wise" minimization of the potential. 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Kaustubh Agashe, Peizhi Du, Sungwoo Hong, Raman Sundrum. Flavor universal resonances and warped gravity, Journal of High Energy Physics, 2017, 16, DOI: 10.1007/JHEP01(2017)016