Clockwork/linear dilaton: structure and phenomenology

Journal of High Energy Physics, Jun 2018

Abstract The linear dilaton geometry in five dimensions, rediscovered recently in the continuum limit of the clockwork model, may offer a solution to the hierarchy problem which is qualitatively different from other extra-dimensional scenarios and leads to distinctive signatures at the LHC. We discuss the structure of the theory, in particular aspects of naturalness and UV completion, and then explore its phenomenology, suggesting novel strategies for experimental searches. In particular, we propose to analyze the diphoton and dilepton invariant mass spectra in Fourier space in order to identify an approximately periodic structure of resonant peaks. Among other signals, we highlight displaced decays from resonantly-produced long-lived states and high-multiplicity final states from cascade decays of excited gravitons.

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Clockwork/linear dilaton: structure and phenomenology

Accepted: May Clockwork/linear dilaton: structure and phenomenology Gian F. Giudice 0 1 2 4 Yevgeny Kats 0 1 2 3 4 Matthew McCullough 0 1 2 4 Riccardo Torre 0 1 2 4 Alfredo Urbano 0 1 2 4 Geneva 0 1 2 Switzerland 0 1 2 0 Via Bonomea 256 , 34136 Trieste , Italy 1 Via Dodecaneso 33 , 16146 Genova , Italy 2 Beer-Sheva 8410501 , Israel 3 Department of Physics, Ben-Gurion University 4 Theoretical Physics Department , CERN 5 Sezione di Trieste , SISSA The linear dilaton geometry in ve dimensions, rediscovered recently in the continuum limit of the clockwork model, may o er a solution to the hierarchy problem which is qualitatively di erent from other extra-dimensional scenarios and leads to distinctive signatures at the LHC. We discuss the structure of the theory, in particular aspects of naturalness and UV completion, and then explore its phenomenology, suggesting novel strategies for experimental searches. In particular, we propose to analyze the diphoton and dilepton invariant mass spectra in Fourier space in order to identify an approximately periodic structure of resonant peaks. Among other signals, we highlight displaced decays from resonantly-produced long-lived states and high-multiplicity nal states from cascade Phenomenology of Field Theories in Higher Dimensions 1 Introduction 2 Properties of the model Basic setup 2.3 Impact of cosmological constant terms KK mode mass spectrum and couplings nal states, branching fractions, lifetimes KK mode production cross sections at the LHC 3 LHC signatures 3.1 Standard signatures 3.2 Novel signatures 3.1.1 3.1.2 3.1.3 3.2.1 3.2.2 3.2.3 3.2.4 Continuum s-channel e ects at high invariant mass Continuum t-channel e ects in dijets at high invariant mass Distinct and e+e resonances Periodicity in the diphoton and dilepton spectra Turn-on of the spectrum at low invariant mass Cascades within the KK graviton and KK dilaton towers Resonant production of particles with displaced decays 4 Conclusions A Background solutions B Radius stabilization and comparison with RS C Graviton decays to SM particle pairs D Graviton decays to gravitons E F Dilaton/radion spectrum and couplings with SM elds Dilaton/radion decays to SM particle pairs G Trilinear graviton/dilaton/radion decays G.1 KK graviton decays into two scalars G.2 KK graviton decays into a KK graviton and a scalar G.3 KK graviton decays: branching ratios G.4 KK scalar decays into two scalars G.5 KK scalar decays into two KK gravitons { i { Representing the KK graviton tower by an e ective propagator An exact solution to Einstein's equations Introduction 59 60 60 61 HJEP06(218)9 It has been shown recently [1] that the clockwork mechanism, introduced in refs. [2{4] to reconcile super-Planckian eld excursions with renormalizable quantum eld theories, is a much broader tool with many possible applications and generalisations, some of which have been explored in refs. [1, 5{25]. A particularly interesting result is the observation that discrete clockworks have a non-trivial continuum limit that singles out a ve-dimensional theory with a special geometry. This geometry coincides with the one obtained in theories with a ve-dimensional dilaton which acquires a background pro le linearly varying with the extra-dimensional coordinate. This theory can address the Higgs naturalness problem in setups where the Standard Model (SM) lives on a brane embedded in a truncated version of the ve-dimensional linear-dilaton space. Earlier, the same setup was proposed in ref. [26] motivated by the seven-dimensional gravitational dual [27, 28] of Little String Theory [29, 30], which is a six-dimensional strongly-coupled non-local theory that arises on a stack of NS5 branes. Additionally, several recent studies have examined in more detail how the linear dilaton setup can be embedded in supergravity [31, 32]. The clockwork interpretation of the linear dilaton theory has helped in elucidating its relation with Large Extra Dimension (LED) [33{35] and Randall-Sundrum (RS) [36] theories, thus providing a coherent map of approaches to the hierarchy problem in extra dimensions. Because of the double interpretation of the same theory either as a linear dilaton setup emerging from an e ective description of non-critical string theory, including duals of Little String Theory, or as the continuum version of a clockwork model, we will refer to this theory as Clockwork / Linear Dilaton (CW/LD). So far, CW/LD has received very little experimental attention, in spite of its attractive and distinguishing features. Nonetheless, some phenomenological aspects of CW/LD have already been discussed in the literature. The distinctive KK graviton spectrum, with a mass gap followed by a narrowly spaced spectrum of modes, and some of its associated collider signatures have been pointed out in ref. [37]. A more detailed study of the KK graviton phenomenology, including in particular the case of a small mass gap, has been undertaken in ref. [38]. Finally, the KK dilaton / radion collider signatures have been studied in ref. [39]. In the current paper we aim at providing a comprehensive picture of the collider phenomenology of CW/LD, extending previous studies, deriving new constraints from present { 1 { LHC data, and suggesting new characteristic signatures, in the hope of motivating dedicated experimental searches. We start by reviewing in section 2 the structure of CW/LD and the consequences of its UV embedding in string theory for the low-energy parameters. We also discuss the e ect of adding cosmological constant terms to the e ective theory, arguing that the bulk theory should be supersymmetric to avoid destabilising the setup. Then, we explain the salient features of CW/LD for collider applications. The theory describes a tower of massive spin-two particles which, depending on the point of view, can be interpreted either as the Kaluza-Klein (KK) excitations of the ve-dimensional graviton or as the continuum version of the clockwork gears. Their mass spectrum and couplings are completely xed in terms of only two parameters: the fundamental gravity scale M5 and the mass k which characterizes the geometry of CW/LD. We encounter also a tower of spin-zero particles obtained from the combination of the single radion state with the KK excitations of the dilaton. In CW/LD the same scalar eld both induces the non-trivial geometry and stabilises the extra dimension (thus playing the role of the Goldberger-Wise eld [40] in RS). The mass spectrum and couplings of the scalar modes are not fully determined by M5 and k alone, but also depend on the brane-localised stabilising potential and on a possible Higgs-curvature coupling. However, many features of the scalar phenomenology are independent of these details. The rest of the paper is devoted to the collider phenomenology of CW/LD. In section 3.1 we study the e ect of s-channel and t-channel exchange of KK gravitons in diphoton, dilepton, and dijet distributions at high invariant mass. An interesting feature of CW/LD is that these processes can be reliably computed within the e ective theory, unlike LED where these processes are dominated by incalculable UV contributions. We also analyse the resonant production of new states decaying into diphoton and dilepton nal states. All signatures discussed in section 3.1 correspond to already existing searches performed by the LHC collaborations and results can be adapted to the case of CW/LD using studies of continuum distributions (previously done for LED) or resonant production (previously done for RS). In section 3.2 we explore new strategies that can be used at the LHC to discover or constrain CW/LD. The near-periodicity of invariant mass distributions with characteristic separations in the 1-5% range (at the edge of experimental resolution) has prompted us to suggest a data analysis based on a Fourier transform, similarly to what is routinely encountered in other elds, for example in analyses of CMB temperature uctuations. We show that such an analysis is competitive with other searches as a discovery mode, as well as being e ective for extracting model parameters. A distinguishing feature of CW/LD with respect to LED or RS is the mass gap of the KK tower, followed by a near-continuum of modes. We suggest that such a turn-on of the spectrum may be observable even when it occurs at low invariant mass, where the individual resonances are di cult to see due to the experimental resolution. Such searches would require the use of trigger-level analysis / data scouting or ISR-based triggers. An important new result presented in this paper is the calculation of the decay chains of graviton or scalar excited modes into lighter KK modes. We nd that such cascades { 2 { are the dominant decay mode for most of the scalar KK tower, and in certain parameter regions also a signi cant decay mode for part of the graviton KK tower. We study the properties of the high-multiplicity nal states that arise from such decays. We also explore the possibility of displaced vertices originating from resonant particle production, which is another signature characteristic of CW/LD. Finally we collect in the appendices a compendium of formul for the production cross sections and decay rates of the new particles in CW/LD. This should be a useful resource for those interested in experimental or phenomenological simulations of CW/LD in collider environments. Other appendices are dedicated to additional details and discussions concerning the structure of the theory. Properties of the model Basic setup We consider a 5D space in which the extra dimension is a circle, with the circumference parameterized by a coordinate y in the range R y R. The SM lives on a brane (TeV brane) at y = yT = 0, another brane (Planck brane) is at y = yP = R, and a Z2 orbifold symmetry identi es y $ y. The full action in the Einstein frame is where is a bulk potential and Sbulk = M53 SB = 2 X i=T;P S = Sbulk + SB + SGHY + SSM ; Z d x SGHY = 4M53 X d x dyp Ki (y SSM = 2 yT) ; g R dyp 3 V (S) = 4k2e 2S=3 T(S) = e S3 M53 4k + P(S) = e S3 M53 + 4k + 2 2 T (S P (S ST)2 ; SP) 2 (2.1) (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) parameterise the scalar potentials at the TeV and Planck branes near their local minima at S = ST;P. Here S is the dilaton eld, M5 is the ve-dimensional reduced Planck mass, { 3 { and LSM is the SM Lagrangian.1 in appendix A) are solved by k is a mass parameter, R is the 5D Ricci scalar, g is the determinant of the 5D metric, = g=g55 is the determinant of the induced metric at the boundaries, i are two masses that determine the strength of the dilaton boundary potentials, Ki are the extrinsic curvatures of the two boundaries, which determine the Gibbons-Hawking-York (GHY) term [41, 42], Einstein's equations and the dilaton equation of motion (which are presented in detail is assumed to be not much higher than the electroweak scale, while the exponentially greater scale MP is an illusion, created by the exponential factor in (2.10). To account for the hierarchy, one needs kR ' 1 ln MP r k ! M5 M5 potentials in eqs. (2.7){(2.8) naturally allow for this, one may argue that since the dilaton realises dilatations non-linearly one expects it would enter the action as an analytic function of eS. The boundary potentials in eqs. (2.7){(2.8) would correspond to non-analytic functions, since S log . However, it has been argued that such logarithmic terms can actually arise in string-theoretic setups [26] and we will assume this to be the case. To understand the origin of the hierarchy in di erent extra-dimensional setups it is useful to introduce the proper size of the extra dimension L5, given by L5 Z R R dy pg55 ; w R d5x p R d5x p g (y g (y R)=pg55 0)=pg55 1=4 ; 1=p8 G and the warp factor w, which can be de ned through the ratio of the total spacetime volumes on the two di erent branes where the extra pg55 comes from the de nition of the covariant -function. Both L5 and w are purely geometrical quantities that characterise the extra-dimensional compacti ed 1This is one possible choice for the dilaton coupling to the SM elds, however in general the coupling is model-dependent. For instance, one could have taken a coupling eSLSM in the Jordan frame, which would result in a di erent Einstein frame action. { 4 { space and their de nitions are explicitly invariant under coordinate reparametrization. The quantity L5 corresponds to the proper length of the compacti ed space (in the special case of a single extra dimension). The warp factor w encodes the information of how much each of the four spacetime dimensions is stretched between one end of the compacti ed space and the opposite end. In the case of a single at extra dimension of radius RLED, L5 = 2 RLED ; w = 1 ; MP2 = L5 M53 (LED): (2.14) We recognize in eq. (2.14) the familiar LED result that the largeness of MP2 =M52 comes entirely from the e ect of a large extra-dimensional volume (L5). HJEP06(218)9 For RS one nds L5 = 2 RRS ; w = ekRS RRS ; RRS is the location of the brane (in the coordinate system where g55 = 1). In RS the proper length L5 is a number of order one, in natural units, while the hierarchy MP =M5 comes entirely from the warp factor w. Notice also that RS reproduces LED when kRS ! 0 and RRS ! RLED. For CW/LD, we obtain L5 = 3 k e 32 k R 1 ; w = e 32 k R ; larger than its natural value 1=M5, as illustrated in gure 1, but not as extreme as in LED with one extra dimension. As a result, in CW/LD the hierarchy MP =M5 is explained by a combination of volume (as in LED) and warping (as in RS). While in RS the warp factor depends exponentially on the proper size, the warp factor of CW/LD is linearly proportional to L5, so that MP ' k(L5M5=3)3=2. However, MP is still exponentially sensitive to the parameters k and R which, as shown in section 2.5, determine the physical mass spectrum of the graviton excitations. The di erence between RS and CW/LD lies in the geometry of the corresponding compacti ed spaces. To appreciate this di erence it is useful to work in both cases in the coordinate basis where the line element is ds2 = A2(z)dx2 + dz2, such that the function A(z) measures the warping of four-dimensional spacetime as a function of the physical distance along the extra dimension. For RS one nds A(z) = ekRSz and, because of the steep exponential behavior, an order-one separation between the branes is su cient to obtain the large warp factor needed to explain the hierarchy. For CW/LD one nds A(z) = 2kz=3 and, because of the slower linear dependence on the coordinate z, an exponentially large separation between the branes is needed to obtain the required warping factor. Nonetheless, the stabilization of the compacti cation radius is naturally obtained with order-one parameters, and the KK masses and interaction scales are set by the typical size of k, R, and M5 and not by the size of L5, which corresponds to a much larger distance. Unlike the case of at geometry, where a single extra dimension would need to have a Solar System size and is therefore excluded, CW/LD makes the possibility of a single { 5 { D E L M 5 = 1 0 0 M 5 = T e V 1 T e V k [GeV] (blue) and 100 TeV (red). The horizontal dotted lines indicate the LED limit. The plot on the right zooms in on the phenomenologically most relevant range of k. relatively large extra dimension viable again. This is illustrated in gure 1, which shows the transition between the regime of roughly constant kR (and L5 / k 2=3) at xed MP to the regime at small k, in which the proper size of the extra dimension is frozen at 2 R and CW/LD turns into LED. This feature of CW/LD is important because the phenomenology of a single extra dimension di ers signi cantly from that of multiple extra dimensions, as we will discuss. (A similar possibility arises in the low-curvature RS model, as pointed out and analyzed in refs. [43, 44].) A remarkable feature of CW/LD is that the same eld S that determines the spacetime geometry also stabilizes the compacti cation radius by xing the factor k R in terms of the boundary conditions of S on the branes, set by the brane potentials for the dilaton eld appearing in SB [15, 39]. In this way the model automatically leads to radion stabilization. This is to be contrasted with the case of RS. The original RS model su ers from a stabilization problem, especially pressing because a massless radion with TeV-scale interactions is experimentally ruled out. Luckily, a simple solution in RS is readily found by adding a bulk scalar eld with a small mass term, which does not signi cantly perturb the AdS metric, but generates a stabilizing potential for the radion [40]. CW/LD has this feature already built in its structure. A more detailed discussion of the stabilisation is given in appendix B. 2.2 Stringy origins of the linear dilaton eld theory Since the linear dilaton action is rather peculiar from a eld theory perspective, it is instructive to consider the UV motivation for this setup. Let us brie y review the stringy setting for the string theory.2 eld theory we are considering, beginning with the worldsheet CFT in We will only consider the spacetime metric and dilaton in the Polyakov 2For more details see e.g. refs. [45{49] and the recent papers [31, 32]. { 6 { the superpartner elds that are also present in the massless spectrum.3 The worldsheet action is S = 1 Z + 4 0 S(X)R(2) + : : : : (2.17) Here XM are the target space coordinates of an as-yet un xed spacetime dimension D, 0 is the string tension, and h is the worldsheet metric. To determine if the action is Weyl-invariant at the quantum level we may consider the -functions that describe the renormalization of the massless elds, which are in superstring theory In matching to the e ective theory, these -functions arise as the equations of motion for massless elds in the target spacetime. Thus the relevant action is given by [50{52]4 S = MDD 2 Z 2 dDxp g eS D 10 This is analogous to the 5D linear dilaton action in the Jordan frame S = M53 Z 2 d x 5 p g eS from which, with a Weyl transformation g ! e 2S=3g, one obtains the Einstein frame bulk action (2.2). From this matching we see that k2 is in fact related to the string tension. The action (2.20) is also often called the string frame action. It describes the low energy e ective theory for the dilaton and graviton below the scale of string excitations. It is known that the string e ective action is classically scale-invariant.5 To see this the dilaton eld as = e 2S=(D 2). The e ective action becomes symmetry realised linearly we may make a Weyl transformation g ! e 2S=(D 2)g and write S = MDD 2 Z 2 dDxp g D 10 + : : : + O( 0) : (2.22) rescaled S ! S In this basis we see that under the transformation gMN ! 2gMN , ! = 2, the action is D 2 . Thus, since the overall coe cient factors out of the classical equations of motion, they are invariant under this transformation. Alternatively, in the standard Jordan-frame action eq. (2.21) the transformation is realised non-linearly as S ! S + , where is a constant, which again only rescales the total action. 3We do not consider the bosonic string theory due to its inherent problems with tachyons. 4Note that this e ective action may be written in numerous forms, all related by eld rede nitions. We 5Discussions of this point are found in refs. [54, 55], however we will essentially follow the discussion of will use the convention of ref. [53]. ref. [46] section 13.2, albeit in a di erent basis. { 7 { (2.18) (2.19) (2.20) (2.21) This scale invariance is only classical, and will not be respected quantum mechanically as the normalisation of the action is physical. Nonetheless, if the action only contains the dilaton and metric there are selection rules on how the quantum corrections enter. They may be determined by considering ~ to transform as ~ ! ~e (essentially, one can think of e S as being ~). As fractional powers of ~ will not arise in the perturbative expansion, we see that the only additional scalar potential terms in the bulk action will be proportional to the dimensionless quantity V / (~e S)n, where n is an integer. Essentially the perturbative series is an expansion in e S, since this plays the role of a coupling constant [55]. This argument shows that the classical scale invariance can persist perturbatively if only the dilaton and metric are present. However, non-perturbative e ects, or the presence of additional elds with couplings that do not follow the classical scale invariance, may spoil any selection rules. An additional structural aspect motivated by the UV picture is that, since we are considering a superstring origin for this 5D action, the full e ective action should also be supersymmetric. Due to the supersymmetry of the e ective action, the classical scale invariance of the action in fact survives quantum corrections. This means that additional terms, such as a cosmological constant, are not expected to arise perturbatively in the bulk action. We note that in superstring theory for D 6= 10 neither the dilaton Weyl anomaly nor the dilaton potential vanish. However, one may consider the linear dilaton background D 10 for which the Weyl anomalies vanish, at the expense of a curved metric in the Einstein frame. Since this background allows for a vanishing -function, it describes a worldsheet CFT. Such theories are known as non-critical string theories, in that the critical number of dimensions has not been chosen, however they still describe string worldsheet CFTs [53, 56]. This describes the basic features of the non-critical string UV motivation for the linear dilaton eld theory. In addition to the non-critical string motivation, the linear dilaton theory in 7D has been shown to arise in the Little String Theory (LST) limit of critical superstring theory with a stack of NS5 branes. This limit corresponds to vanishing string coupling [27]. It has been argued that LST-like theories are dual to backgrounds which asymptote to string theory in the linear dilaton background (see section 3 of ref. [57]). However, in this case one has two extra dimensions. The linear dilaton form of the e ective action persists to 5D when two of the additional dimensions are compacti ed [26]. In this framework, M5 and k are determined by the string scale Ms, the number of the NS5 branes N , and the volume of the six compacti ed dimensions V6 as M5 we have M5=k Ms2(N V6)1=3, hence the ratio depends on the UV parameters. Ms3V61=3=N 1=6 and k p Ms= N [37]. Thus In this work we will not consider the phenomenology of the additional RR two-form eld, nor the additional states required by supersymmetry. All of these states would likely have the usual LD spectrum, with a mass gap and densely packed states. Furthermore, the superpartners, such as the dilatino and gravitino, may be charged under remnants of the { 8 { R-symmetries that may make the lightest states stable. These neutral fermions will only be pair-produced in colliders and thus their greatest e ect will likely be to contribute to the decay channels of the KK gravitons. Furthermore, we do not include the e ects of genuine string excitations which, if entering below the cuto M5, could lead to additional signatures. Having considered the stringy origins of the action, including the bulk supersymmetry, we can also take a viewpoint that is agnostic of the UV completion, and consider the action from a purely non-supersymmetric eld theory perspective. There are two terms that enter Einstein's equations in the form of a cosmological constant (CC): one from the bulk and one from the branes. These terms may be written in the Jordan frame action as SCC = Z dDxp g D e D2 2 S ; SBCC = Z d D 1 p x B D 1 e 2(DD 21) S ; (2.24) where is the determinant of the induced metric. These terms violate classical scale invariance, which is why they did not arise in the string e ective action at tree level. However, from a purely eld theory perspective classical scale invariance is impotent unless the UV theory and non-perturbative corrections also respect it. If we are being agnostic as to the UV, including considering non-supersymmetric theories, then we cannot rely on such arguments to forbid these cosmological constant terms. Continuing in this vein, let us determine how large these terms can be before they signi cantly modify the solution. Since the inclusion of the cosmological constant terms generally leads to Einstein's equations that cannot in general be solved analytically,6 we will instead perform a perturbative analysis valid for small cosmological constants, which we will parameterise as " (bulk) and "B (brane). We consider a bulk potential which, in Einstein frame, is given by Sbulk = Z d x 5 p g M53 " 2 R(g) 1 # (2.25) (2.26) 3 "B 4 4 (y R) ; and a boundary potential given by7 SB = Z d4x dyp 1 3 " 40 ( 4kM53 2 (1 + "B)e S=3 + 3) + " B 3 e S=3 + 3 " 32 3 " 20 3 "B 4 1 (y) = e 2k R=3. The speci c form of the terms at the boundaries has been chosen to give a vanishing 4D cosmological constant, thus that particular ne-tuning, present in 6For its novelty, in appendix J we include an isolated exact solution that satis es a di erent set of boundary conditions, and is thus not of interest here. 7The junction conditions, which follow immediately from integrating the equations of motion following eq. (2.25) at the branes give, for a given solution to the bulk equations of motion, conditions on the absolute value and gradient of the brane potentials i(S) and d i(S)=dS. Thus there are an in nite class of brane potentials that can satisfy these constraints. We present only those that correspond to the usual linear dilaton brane potential and a cosmological constant on the branes. The quadratic brane terms proportional to T;P are irrelevant for this discussion. { 9 { the boundary conditions S(0) = 0 ; (0) = 0 ; and the junction conditions are all satis ed for the following dilaton and metric pro les all extra-dimensional models, has already been performed. This means that any further tuning of " or "B is now in addition to the tuning for a vanishing 4D CC. Since is exponentially small in our scenario, the limit " B ! 0 corresponds to the solution whenever the bulk CC is non-vanishing, which is one possibility of interest here, and the limit " ! 0 corresponds to only having a CC on the 0-brane, up to tiny corrections on the R-brane, which is the second possibility of interest. We present both solutions together for convenience. Working in the conformally at metric ds2 = e2 (y)( dx dx + dy2), we nd that the bulk Einstein's equations and S equation of motion HJEP06(218)9 S = 2ky + 2 3 = ky + 3" 32 " 80 1 2ky 6e4ky=3 e 2ky + B 5 2ky 5e 2ky ; 5(1 + ky) e 2ky + B 1 ky e 2ky : " 6 The physical impact of these modi cations can be expressed in numerous ways. One way is to derive from the expression of the 4D Planck mass the value of kR, which characterizes the physical mass spectrum of the KK modes. We nd that, at leading order in " and "B and for the same input values of M5 and k, the usual CW/LD value of R given by eq. (2.11) is modi ed as follows R R";"B = 1 + " B 3 " 32 18 25 kR e4 kR=3 Let us consider the e ect of ". For the linear dilaton solution in the conformally at coordinates we have kR 10, thus e4 kR=3 1018. This means that unless j"j . 10 16, " j j the value of kR and, consequently, the mass spectrum will be signi cantly di erent from the usual CW/LD prediction. Similarly, the metric will have deviated signi cantly from the linear dilaton form in moving from the IR brane to the UV brane. Of course, nothing radical will happen, with the metric owing smoothly from the linear dilaton form at 10 16, to a dS or AdS-like solution. We see this re ected in the reduced (comoving) radius required to achieve the desired hierarchy of Planck scale. In the AdS case the mass spectrum and wavefunctions would correspond to kRS kp"=12, at larger ". Thus one needs to require an extremely small value of " to retain the linear dilaton solution. From a eld theory perspective this is an enormous tuning, however by considering the superstring motivation the bulk theory is supersymmetric and this protects against the generation 1 3 3 2 " 4 (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) of D if the action started out with classical scale invariance, which is the case for the string e ective action. The requirement is actually a little stronger than requiring solely supersymmetry, since the vacuum expectation value of the superpotential must also vanish to avoid supersymmetric AdS solutions. It has recently been shown that the CW/LD background is indeed consistent with a bulk supersymmetry [31, 32]. Supersymmetry must be broken on the SM brane, however locality still protects against the generation of supersymmetry breaking in the bulk action (see e.g. ref. [58] for related discussions). Thus, given that the motivation for the linear dilaton setup is coming from string theory, the consequent bulk supersymmetry naturally protects against the generation of ". B Now consider " . The story for this parameter is very di erent. The reason is that supersymmetry breaking must exist at least at the TeV scale on the SM brane, and thus there is no symmetry, including supersymmetry, that can protect against the generation of "B M5=k. Studying the perturbation to the metric we see that the metric retains B the linear dilaton form for " B not too large. Similarly, the comoving radius required is only signi cantly modi ed for " 1. Thus, although we have no control over the size of the " B brane cosmological constant term on the TeV brane, the setup is robust to its presence as long as "B is not too large, which in turn does not require signi cant ne-tuning unless k M5. To summarise, bulk supersymmetry protects against the presence of a bulk cosmological constant that would otherwise radically alter the setup.8 On the other hand, the lack of supersymmetry on the SM brane does not imperil the solution as the brane cosmological constant does not feed into a bulk cosmological constant, due to locality, and it does not generate a signi cant deviation from the linear dilaton solution unless "B is O(1). Thus, if the UV motivations are taken seriously, in particular the bulk supersymmetry, then the theory does not exhibit signi cant tuning beyond that required for a vanishing 4D cosmological constant. On the other hand, if one were agnostic as to the UV structure one could only conclude that the necessity for j"j additional tuning. 10 16 corresponds to an extreme At this stage it is worth commenting on the deconstruction of the continuum model, which gives the discrete clockwork graviton model presented in section 2.5 of ref. [1]. While this is a perfectly valid, di eomorphism invariant, multi-graviton model, since it follows from a deconstruction of a solution of Einstein's equations accompanied by the dilaton, the question of the cosmological constant for the continuum model is equally relevant for the discrete. There is nothing to forbid a cosmological constant at each site. Translation invariance enforces that all cosmological constants must be equal, just as " is positionindependent in the continuum, however no symmetry protects the overall magnitude of each term and the cosmological constant could imperil the clockwork wavefunctions. As with the continuum model, the only plausible solution to this problem is supersymmetry. Thus, if one wants the discrete setup to be safe from large quadratic corrections, one really requires an entire 4D copy of Supergravity, with the associated gravitino at each site. Thus the discrete model of ref. [1] alone is not su cient to address the hierarchy problem. 8This discussion has focussed on the presence of cosmological constant terms, however in the absence of supersymmetry the dilaton potential itself is also not protected from dangerous radiative corrections, the presence of which may lead to similarly signi cant corrections to the form of the bulk geometry. 2.4 In CW/LD, M5 and k are two independent input parameters. As the collider phenomenology changes signi cantly by varying the ratio k=M5, we think it is interesting to explore a wide range of this ratio and to study the corresponding experimental features, some of which are novel and characteristic. However, we rst want to discuss in this section which values of k=M5 are reasonable. The CW/LD action in eq. (2.25) has an enhanced symmetry (a shift in S) for k ! 0. This suggests that small k=M5 is technically natural. On the other hand, in the string theory setup, both M5 and k might be expected to arise from the single dimensionful string tension parameter 0, although their relation is not uniquely determined. In the little string theory limit a hierarchy of O(N 1=3) is possible, where N is the number of NS5-branes in the theory. While the form of the bulk action is motivated by string theory in the UV, the boundary action has no such motivation. In particular, we have no guidance as to the parameters ST, SP, which determine the form of the dilaton solution to the equations of motion. We may thus write y=0 = 0 and Sy=0 = S0, in which case 2k R = jST SPj and S0 = ST. On the equations of motion the overall factors can be absorbed into a rede nition of the 5D Planck scale M5;e = M5e 0 , ke = ke 0 S0=3, thus the only e ect of taking general boundary conditions on these constants is to modify the relationship between the observables M5;e and ke . As a result, since we do not have guidance on the scale of these parameters from the UV, in this paper we are instead open minded as to considering relatively small k=M5, especially because this possibility gives rise to a variety of interesting collider phenomena. It should also be kept in mind, however, that to have the linear dilaton solution with k M5 does require tuning of the brane CC at that order. In this work, instead of using lower bounds on k based on theoretical considerations, we will be guided by the phenomenological limits from beam dump experiments, supernova emission, and nucleosynthesis. For M5 in the domain of interest to present and future collider experiments, the lower limits on k are in the 10 MeV / 1 GeV range [38], taking into account that astrophysical and cosmological bounds are subject to large numerical uncertainties. 2.5 KK mode mass spectrum and couplings The KK gravitons have masses m0 = 0 ; and couple to the SM stress-energy tensor T L 1 (n) h~(n) T G (0)2 = MP2 ; G (n)2 = M53 R G 1 + = M53 R 1 k 2 m2n 1 : n 2 R2 ; as ; k2R2 n2 (2.34) (2.35) (2.36) 50 150 200 for M5 = 10 TeV, k = 200 GeV, however the result does not strongly depend on these parameters. The blue band indicates the typical ATLAS and CMS resolutions in the diphoton and dielectron channels ( 1%). The zero mode is the usual massless graviton, while the rest of the KK modes appear after a mass gap of order k and their couplings to the SM are not suppressed by MP . A unique property of this scenario is that the KK modes form a narrowly-spaced spectrum above the mass map. At the beginning of the spectrum, the relative mass splitting is m2 m1 m1 3 ' 2 (kR)2 1:5% ; i.e., comparable to the diphoton and dielectron invariant mass resolutions in ATLAS and CMS, which are typically around 1% [59{64]. The splittings then increase, as shown in gure 2, reaching a maximum value Eventually they start decreasing, becoming asymptotically mn+1 mn mn max 1 ' 2kR 5% for n kR : mn+1 mn mn 1 ' n for n kR ; (2.37) (2.38) (2.39) thus dropping below the experimental resolution at n The physics of the KK dilatons (including the radion degree of freedom) depends on the details of the brane potentials that stabilize the extra dimension [39, 65]. We rst focus on the limit of rigid boundary conditions for the dilaton eld, obtained when the mass parameters T;P in the brane-localized potential in eq. (2.3) are in nitely large. The general case is presented in detail in appendix E. The dominant features of the phenomenology of the model as a whole are largely independent of the details of the brane-localized potential. For rigid boundary conditions, the KK dilatons have masses m0 = r 8 9 k ; n 2 R2 ; (2.41) 1 : (2.42) (2.43) = 0 corresponds to rigid boundary conditions (see appendix E for details). The horizontal dotted gray lines correspond to the spectrum m2n = k2 + n2=R2. The dashed black lines correspond to the analytical approximations discussed in appendix E, eqs. (E.31){(E.32). and couple to SM particles via the trace of T as9 L interaction scale of dilatons with respect to gravitons, when compared at equal masses. If the boundary conditions are relaxed by lowering the mass parameters T;P in the brane potential, the n = 0 and n = 1 modes can become signi cantly lighter, and both are massless in the T;P ! 0 limit (unstabilized limit). At the same time, all the higher KK dilaton modes shift down by one mode (as in \Hilbert's Hotel") so that the massive mode spectrum in the unstabilized limit ( T;P = 0) is again described by the expression for m2n in eq. (2.40) with n starting from 1. This is shown in gure 3. The coupling strengths to T get modi ed as well. In the unstabilized limit, modes with mn k have (n)2 3 ' 4 M53 R m4n 2 k4 ; 9These expressions follow from eqs. (3), (4) and (39) of ref. [65], and we have con rmed them. However, while the expression for (n)2 agrees with the result given in eq. (41) of ref. [65] in the limit n taken there, our expression for (0)2 is larger than theirs by a factor of 2. Our expressions for both and (n)2 (in the same limit) agree with those in eqs. (4.4){(4.5) of ref. [39] after taking into account that kR (0)2 the four- and ve-dimensional Planck masses, MP l and M , of ref. [39] are related to our MP and M5 as MP2 l = MP2 =2, M 3 = M53=2. We thank the authors of ref. [39] for clarifying this to us. gg 34% Pi qiqi 38% W +W hh 3.2% where the dimensionless parameter , which vanishes in the rigid limit ( T;P ! de ned in appendix E. By comparing this with eq. (2.42), one can see that the interactions with T get signi cantly suppressed as one goes away from the rigid limit. In addition, there appear couplings to LSM, the SM Lagrangian, where in the unstabilized limit for mn k 1 ' L (n) nLSM ; (n) 2 ' ' 3M53 R : nal states, branching fractions, lifetimes Since the KK gravitons couple to the SM via T , the relative branching fractions into the various SM particles are the same as in any 5D model with the SM on a brane. These are shown in table 1 for KK gravitons that are much heavier than the SM particles. The detailed expressions, including phase space e ects, which must be taken into account for lighter KK gravitons, are given in appendix C. The total decay rate of a mode-n KK graviton into SM particles in this limit is In the absence of other types of decays, the resulting lifetime is c n 6:6 m2n 1 : Gn!SM = M5 We see that the KK graviton decays can be prompt, but it is also possible, especially if M5 & 10 TeV, that KK modes below a certain mass will be displaced, or even stable on detector scale. The last factor in eq. (2.47) gives a further enhancement to the lifetimes of the lightest modes besides the mn 3 factor. For instance, for the rst mode we nd (1 k2=m21) 1 k2R2 100. As a result, it is possible that, within the same theory at a given M5 and k, some KK gravitons decay promptly, while others lead to displaced vertices. Importantly, we nd that decays of KK gravitons to SM particles are not the full story. Graviton self-interactions allow a heavy KK graviton to decay to a pair of lighter KK gravitons.10 The method of calculation and the detailed expressions for the rates of these decays are presented in appendix D. Their total rate, in the limit n kR 1, is given by Gn!P G`Gm ' 3 595 m7n=2 214 2 k1=2RM53 ; 10Numerical results for graviton-to-graviton decays were also considered for RS models in [66]. (2.44) (2.45) (2.46) (2.47) (2.48) G→ GG0.010 → G Γ G Γ to the asymptotic expression given in eq. (2.48). Here we have taken M5 = 10 TeV, k = 10 GeV, however the dependence on these parameters is weak. while for smaller values of n it is reduced relative to this expression in the fashion shown in gure 4. This rate can be quite sizable, and even dominate over decays to SM particles for large n: Gn!P G`Gm Gn!SM 4:1 r mn : k This has signi cant e ects on the phenomenology. First, decays to SM particles are diluted, as shown for the example of the diphoton channel in gure 5 (left). Second, signatures due to decays to lighter KK gravitons can be important because the branching fraction for such decays can be large, as shown in gure 5 (right). Even though the contribution to the total width ( n) from decays to lighter KK gravitons can be signi cant, note from (2.48) that all modes within the range of validity of the theory (mn . M5) have n mn. Even the stricter condition n < mn mn 1, which allows us to treat the KK excitations as individual resonances, is ful lled as long as This is satis ed by all modes with mn . M5 if mn . 6:8 k M5 1=7 M5 : k & 1:5 Note also that the extra contribution to the decay rate does not preclude displaced decays, since only high-n modes are a ected (see gure 4). A KK graviton can also decay to a KK scalar and a KK graviton, or to a pair of KK scalars.11 The branching fractions for such decays are usually small, as shown in gure 6. 11Cascade decays of gravitons to KK scalars were also considered for RS models in [67]. k = 0.1 GeV 10 GeV 1 TeV k = 0.1 GeV 10 GeV 1 TeV G G m [GeV] m [GeV] for k = 0:1, 10 and 1000 GeV, as a function of the KK graviton mass. In the left plot, the thick black curve shows the result that would be obtained without accounting for decays to lighter KK k = 0.1 GeV 10 GeV 1 TeV k = 0.1 GeV 10 GeV 1 TeV R( 1 B G( 1 0.1 0.1 1 10 m [GeV] or a pair of KK scalars (right) for k = 0:1, 10 and 1000 GeV, as a function of the KK graviton mass, assuming rigid boundary conditions for the dilaton eld. We show separately contributions involving two scalar zero modes (dotted), one scalar zero mode (dashed) and no zero modes (solid). Let us now consider the decays of KK scalars. As we have already mentioned, their couplings to the SM are model dependent. The decay rates have been computed in ref. [39] and we present the full expressions in appendix F. For rigid boundary conditions (which implies couplings to T only), assuming the Higgs-curvature interaction (which we will discuss below) to be absent, and neglecting the SM particle masses relative to the KK scalar mass, the total partial width to SM particles is dominated by W +W , ZZ and hh, where the last expression applies for n 6= 0. The corresponding lifetime is and is given by c n 0 1 8mk2n2 A ; 9m2n 1 GeV k 3 1 TeV mn kR 10 0 1 8k2 1 9m2n k2 A : m2n For large n, the last factor in eq. (2.53) is equal to one, while for n = 1 it is equal to 11, showing a smaller enhancement than in the graviton case. However, there is a higher overall tendency for the decays to be displaced than in the KK graviton case, at least for the low modes, where KK tower cascades are irrelevant. The decay rate into SM particles is even smaller below the W +W threshold, where gg becomes the dominant SM decay channel, with the rate approximately given by eq. (2.52) times 49 ( s=2 )2. In that regime c n 66 cm M5 10 TeV 3 3 1 GeV k 9m2n k2 A : m2n (2.52) (2.53) (2.54) Relaxed boundary conditions, which allow KK scalars to decay via a direct coupling to LSM, eq. (2.44), rather than only through T , generate sizable branching ratios to gg and relative to the total rate to pairs of SM particles, of O(30%) and O(3%), respectively. The partial widths to W +W , ZZ, hh are modi ed by O(1) factors, so the total SM rates of the heavy modes stay in the same ballpark. The light modes, on the other hand, can become signi cantly shorter lived. Because the couplings of the scalar KK modes to the SM are somewhat suppressed, cascade decays within the KK tower play an important role. We compute these decays in appendix G. We nd that the KK scalar decay rates, with the exception of the rst few modes, are dominated by the cascades, as shown in gure 7. the KK mode spectrum for di erent values of M5 and k. Interestingly, even for a single choice of the parameters M5 and k, the model typically contains particles with a very wide range of lifetimes. For xed M5 and k, the variation of the KK mode lifetimes with mn is more evident for gravitons than for scalars. Since no symmetry prevents its appearance, it is natural to introduce on the SM brane a renormalizable curvature-Higgs interaction L = RHyH ; (2.55) where is a dimensionless coupling. After electroweak symmetry breaking, this interaction induces a kinetic mixing, proportional to v=M5, between the Higgs and the radion component of the KK scalars. Through a eld rede nition (see ref. [39] for the explicit calculation) the kinetic mixing can be turned into a mass mixing. It is not di cult to where in the last term we integrated by parts. Thus, the scalar uctuations satisfying the constraint equation 3 2 (G0 + 2 0G) = do not mix with the vector and are therefore the physical scalar uctuations that are not eaten by the massive graviton modes. One may be concerned that we may have omitted an important term in performing integration by parts, however the vector eld carries odd parity under the orbifold symmetry, thus the boundary term vanishes. If the previous constraint is employed we may continue without consideration of the massive graviton degrees of freedom as we have isolated the scalar uctuations that are not kinetically mixed with them. To proceed we will consider the following general metric ds2 = e2 (y) (1 + 2F (x; y)) dx dx + (1 + 2G(x; y)) dy2 ; S(x; y) = S0(y) + '(x; y) ; T~MN TMN with (y) = 2ky=3. We consider the linearized (i.e. at the rst order in the scalar uctuations) Einstein's equations, which we recast in the form RMN = T~MN =M53 + BMN , where 13 gMN TP QgP Q and BMN accounts for boundary contributions at the two branes. As customary, some equations are dynamical, some equations provide constraints. In particular, the o -diagonal components enforce the condition 2F (x; y) = G(x; y) ; as found in eq. (E.4), while the 5 component can be straightforwardly integrated, and we nd eq. (E.6). As for the rest of the Einstein's equation, we nd (E.6) (E.7) (E.8) (E.9) (E.10) (E.11) (E.12) (E.13) (E.15) R = R55 = 2 T~ =M53 = T~55=M53 = Bi=T;P = B5i=5T;P = e 2 3 e 4e 23 S00'0 + e 2 3 9M53 (6F F + F 00 + 9 0F 0 + 6F 3( 0)2 + 00 ; F 2F 00 6 0F 0 ; 2F V (S0) + 'V 0(S0) ; 3M53 G i=T;P(S0) + ' 0i=T;P(S0) (y yi=T;P) : 2GV (S0) + 'V 0(S0) ; 3G) i=T;P(S0) + 3' 0i=T;P(S0) (y yi=T;P) ; (E.14) In addition, we need the equation of motion for the uctuations of the dilaton eld. We nd ' + '00 + 3 0'0 + 6S00F 0 + 4F 3 0S00 + S000 2 3 V 00(S0)'e2 e 2 X 3 = i=T;P e M53 2F 0i(S0) + ' 0i0(S0) (y yi) : (E.16) Finally, we need to impose junction conditions at the boundaries. There is only one relevant junction condition, corresponding to the boundary term of eq. (E.16). We nd '0 i=T;P + 2F S00 i=T;P = 3' 0i0=T;P(S0(yi))e M53 ; where, for a generic rst derivative A0(y), the jump [A0] at the generic point y is de ned as ). Using this de nition, we can rewrite eq. (E.17) in the form We can now extract a dynamical equation for F (x; y). Following ref. [117], we consider in the bulk the combination R55 4 R = 1 M53 ~ T55 4 ~ T ; and we nd we nd (E.17) (E.18) (E.19) (E.20) (E.22) (E.23) 3F 00 3 0F 0 +6F 3( 0)2 + 00 = 6 ( 0)2 + 00 Using the constraint in eq. (E.6), and the explicit expressions for the background solution, 1 3 (S00)2 : (E.21) 23 S00'0 2F F + F 00 + 2kF 0 = 0 : This equation can be conveniently rewritten in the form + d 2 dy2 k 2 ekyF (x; y) = 0 : Because of the relation 2F (x; y) = G(x; y), the 5D eld G(x; y) satis es the same equation of motion. Similarly, starting from the eq. (E.16) in the bulk, and using the constraint in eq. (E.6), it is possible to show that also the scalar eld uctuation '(x; y) respects the same equation ' + '00 + 2k'0 = 0. However, only a certain combination of these elds has a canonical kinetic term in the bulk. By direct diagonalization of the quadratic action, as illustrated in ref. [65], one nds it to be v(x; y) it also satis es the 5D equation of motion v + v00 + 2kv0 = 0. This is the equation we have to solve together with the appropriate boundary conditions. To this end, we need an explicit expression for the brane potential. The latter can be obtained by means of the solution-generating method championed in ref. [113], and we nd the explicit expressions reported in eqs. (2.7){(2.8). The mass parameters T=P do not a ect the background solution (since they do not contribute to the background junction conditions) but they enter in the junction conditions for the perturbations in eq. (E.17). Notice the presence of a massless dilaton in the limit T=P ! 0 since in this limit a shift in S (in the Jordan p6ekyM53=2[F (x; y) '(x; y)=3], and 1 X n=0 frame) corresponds to an overall rescaling of the action. We now have all the ingredients needed to solve the scalar equation of motion. Using 2F = G, F = , G = , we adopt the notation of ref. [39]. We introduce the KK decompositions (x; y) = n(y) n(x) ; '(x; y) = X 'n(y) n(x) ; v(x; y) = X vn(y) n(x) : (E.24) 1 n=0 1 n=0 Notice that the 4D part of the three KK decompositions is equal. This is a direct consequence of the de nition of v(x; y) and the constraint in eq. (E.6). We de ne n to be the solution of the equation of motion of a free 4D scalar eld of mass mn, namely m2n n(x) = 0. At this level of the analysis we are interested in the mass spectrum mn. We can therefore solve the equation of motion focusing on a speci c mode | equivalently, n(y), 'n(y) or vn(y) | and imposing the corresponding boundary conditions. We consider the mode n, for which the equation of motion is d 2 dy2 + m2n k 2 eky n = 0 : We can write a generic solution in the form n(y) = Nne ky [sin( ny) + wn cos( ny)] ; with n2 = m2n k2, and Nn, wn constants. eqs. (E.18){(E.19). Using the constraint in eq. (E.6), we nd The boundary conditions are given by 9 9 4k 0n0 + 3 0n 4k 0n0 + 3 0n 2k n = 2k n = 4k + 9 T 4k + 9 P 2 2 n + n + 3 3 4k n 4k n 0 0 We start considering the rigid limit i ! 1. The boundary condition at the TeV brane xes wn = 3 n=k. The boundary condition at the Planck brane can be solved analytically. We nd with n = 1; 2; 3; : : : . The lowest state in the KK tower corresponds to the radion. Notice that the mass of the radion vanishes in the absence of warping, k = 0. The identi cation of the radion with the lowest state in the KK tower can be understood by means of the following argument. In the rigid limit i ! 1, if k = 0 we have from eq. (E.29) a massless state and an ordinary KK tower with spectrum m2n = n2=R2, n = 1; 2; 3; : : : . Note that there is no zero mode in the KK tower. The point is that in this setup it is straightforward to identify the states m2n with the KK decomposition of the scalar S. Indeed, in this situation we just have a free scalar eld in the bulk of a at extra dimension (since k = 0) with an in nite mass term on the two branes. After KK reduction, the equation of motion for the y-dependent variable reduces to an ordinary Schrodinger equation for a free particle in a one-dimensional well 0 6 y 6 R with in nite potential (E.25) (E.26) (E.27) (E.28) (E.29) on the two walls. As well known from Quantum Mechanics, the quantization condition precisely implies m2n = n2=R2, n = 1; 2; 3; : : : , thus making possible the identi cation of these states with the k = 0 limit of m2n in eq. (E.29). Consequently, the remaining scalar degree of freedom can be identi ed with the radion eld. The second state in the KK tower corresponds to the dilaton, with mass m2dil = k2 + 1=R2. For a generic value of T;P, the boundary condition at the TeV brane gives wn = 3 n T 2(k2 + n2) + k T : (E.30) HJEP06(218)9 The boundary condition at the Planck brane can be solved numerically. We show our result in gure 3, with T;P 2k= T;P, and taking for simplicity T = P At the rst order in T | and for generic value of n | we nd . 1 8k2 ' 9 2 T 9 1 2 : 2 i T ; (E.31) (E.32) (E.33) (E.34) (E.35) (E.36) We now move to discuss the couplings of the scalar perturbations with SM elds. To that end, we need to determine the proper normalization factor in eq. (E.26). By expanding the 5D action up to the quadratic order in the scalar uctuations, and using the KK decompositions in eq. (E.24), we nd the following normalization condition in 4D Z Sn4D kin = Cn d4x n(x) m2n n(x) ; Cn = 27M53 Z R 16k2 0 dye2ky 0 2 + n 8k2 3 2n + 3 8k 0n n The coupling of the radion and the scalar KK modes with the SM Lagrangian at the TeV brane are described by the action Sint = Z d x 4 p Te '=3LSM : We can now extract the value of Nn by plugging in Cn the solution found in eq. (E.26). T 4k3 2Ti sin2 R n 3 h4k2 k2 + n2 2 + 4k k2 + n2 2 T + k4 + 2k2 n2 + 9 n4 (2 R n sin 2 R n)+3 n k4 + n4 + 2k2 n2 + 2T (2 R n + sin 2 R n) : 2 = p g g S S = Z d4x n(x)Tr[T ] d x 4 p gT g ; 2 ; 'n(0) 3 2 1 Silnint = De ning the coupling constants T we nd | at the rst order in T=P, and for generic n | the following couplings with the trace of the energy-momentum tensor 1 6 k 1 + coth k 3R M53 1 + 9 ' 6 1 s k M53 1 + 9 ; 2knR (n) = p3 M53R (n2 + k2R2) (9n2 + k2R2) (1 T) : Expanding at the rst order in the scalar uctuation, and using the de nition of the energy-momentum tensor (here for a generic metric g) (E.37) (E.38) (E.39) (E.40) (E.41) (E.42) (E.43) while for the direct coupling with the SM Lagrangian we have 1 ' 1 ' (0) = (n) = n T p k 3 R3=2 k 1 + coth k 3R M53 s n2 + k2R2 M53(9n2 + k2R2) : 2 T ' 27 s k M53 ; In the rigid limit the direct coupling with the SM Lagrangian vanishes. This is a consequence of the junction conditions in eqs. (E.18){(E.19) in which a large value of 0T0 ;P forces the scalar eld uctuation ' on the two branes to vanish. We note that the analytical approximations in eqs. (E.40){(E.43) do not feature an explicit P-dependence. However, at higher orders, an explicit P-dependence appears since it unavoidably enters via the scalar masses. For this reason, in gure 24 we show the full numerical results for the couplings in eq. (E.39) under the assumption T = P already adopted in gure 3. In light of the discussion we put forward in section 2.5 about the KK mode production cross sections, it is also interesting to present a useful analytical limit for the couplings in eq. (E.39) for a generic value of T and large n. By approximating the KK scalar spectrum with m2n n2=R2, we nd, at the leading order in a kR=n expansion 1 ' Equipped with these results, we can now move to compute the dilaton/radion decays to SM particle pairs. ) ( Φ HJEP06(218)9 SM Lagrangian (right panel). The dashed black lines (for simplicity, only for radion and dilaton) represent the analytical approximations in eqs. (E.40){(E.43). We only show the rst few modes of the KK tower. F Dilaton/radion decays to SM particle pairs Using the results of ref. [39] (accounting for the di erence in conventions described in footnote 9), the partial decay widths of a KK scalar of mass mn are ZZ = 1 xZ + fh(mn) n 1 + fh(mn) + W +W 1 xW + 4 xW 3 2 fh(mn) n 1+ fh(mn) + ) ( φ xZ 2 xW 2 1 8 2 x 2 h 4 m2 )2 h 2 + bQCD + n 2 2 0 ; bQED + n 0 ; 34 2n (1 34 2n (1 xZ )1=2 0 ; xW )1=2 0 ; 4 2 (F.1) (F.2) 36 m2h 18 m2h m2n m2n m2 h m2 h 72 m4h m2n(m2n 14 2n (1 m2 ) h m2 h xh)1=2 0 ; 4 (F.3) (F.4) (F.5) (F.6) where where 0 8 4mm2ni2 ; fh(mn) = 1 m2n 6 m2n ; m2 (n) is given in eqs. (2.41){(2.42), n (n)= ('n), is the Higgs-curvature coupling as de ned in eq. (2.55), Nf = 3 for each quark, 1 for each charged lepton and 1=2 for each (Majorana) neutrino, bQCD = 11 2nQ=3, where nQ is the number of quarks lighter than mn, and bQED = (4=3) P f Nf Qf2 , where the summation is over fermions lighter than mn and Qf is the electric charge. In the region of the W and top masses, threshold e ects are accounted for by taking HJEP06(218)9 with b2 = 19=6 and bY = 41=6, and (relevant also for production) bQED = b2 + bY fh(mn)(2 + 3xW + 3xW (2 xW )f (xW )) 8 + 3 fh(mn)xt(1 + (1 xt)f (xt)) bQCD = 7 + fh(mn) xt (1 + (1 xt)f (xt)) ; f (x) = < 8 > > > > : > >>> sin 1 1 4 ln 1 p 1 + p 1 p 1 1 2 x i 2 (F.7) (F.8) (F.9) (F.10) (G.1) G Trilinear graviton/dilaton/radion decays In this section we discuss trilinear decays involving scalars and gravitons. The interactions responsible for these processes are encoded in the action obtained by expanding the metric including both tensor and scalar perturbations ds2 = e2 (y) h(1 + 2F (x; y)) ( + 2h (x; y)=M53=2)dx dx + (1 4F (x; y)) dy2i ; S(x; y) = S0(y) + '(x; y) : Although technically involved, the computation does not pose particular complications. After expanding the 5D action up to the cubic order in the uctuations, one just needs to apply the KK reduction | already introduced in appendix D and appendix E for, respectively, tensor and scalar perturbations | in order to extract the 4D Lagrangian interactions. In this process a non-trivial check is the exact cancellation of the GibbonsHawking-York terms against the total derivatives coming from the Ricci scalar. In addition to the trilinear graviton vertex already studied in appendix D, the inclusion of scalar perturbations gives rise to three additional structures that we shall now discuss in detail. Let us start from the trilinear 5D action involving one graviton and two scalar elds (labelled `Tensor-Scalar-Scalar', TSS in the following). We nd d4xdye3 h (G.2) 2 3 X + e5 X i=T;P Z This trilinear action is responsible for KK graviton decay into two scalars, Gl ! and KK scalar decay into a graviton and a scalar, l ! Gm + n. Next, we consider the trilinear 5D action involving two graviton and one scalar elds (labelled TTS in the following). We nd STTS = d4xdy6e3 h h 12F 0 2 + S00(@y')=9 + 4S002F=3 + e5 'V 0(S0)=6 + F V (S0)=3 Z d x 4 e 4 M53 F 4 h h ' 0i[S0(yi)] + [S0(yi)] : (G.3) This trilinear action is responsible for KK graviton decay into one scalar and one graviton, Gl ! Gm + n, and KK scalar decay into two gravitons, l ! Gm + Gn. Finally, we have the trilinear 5D action involving three scalar elds (labelled SSS in the following). We nd SSSS = d4xdyM53 e 3 1536F 3 0 2 12F 2S00'0 2F '0 2 + 240(@yF )F 2 0 + 12F 2'S00 0 + 4F 2'S000 F '2V 00(S0) '3V 000(S0)=6 6 (G.4) This trilinear action is responsible for KK scalar decay into two scalars, l ! m + n. In the following, we shall compute and discuss each one of these processes. We describe the computations in full generality | that is for an arbitrary choice of | but we specialize all our numerical results as well as analytical approximations in the rigid limit = 0. G.1 KK graviton decays into two scalars The 4D Lagrangian density for a xed triad (n; m; l) of KK states is TSS L(n;m;l) = 1 M53=2p R C(n;m;l)(k; R) h~(n) (@ TSS l) ; (G.5) TSS where the dimensionless function C(n;m;l)(k; R) encodes the integral over the extra dimendye3 k n(y) 3 ~ m(y) ~ l(y) + '~m(y)'~l(y) : (G.6) 2 3 Notice that, compared with eq. (E.26), we de ned here the dimensionless scalar wavefunction ~ n pM53=k n. Furthermore, from the constraint in eq. (E.6) we get 3 2k '~n(y) = 2k ~ n(y) + 32 ~ 0n(y) : (G.7) Gl! m n (left panel) and Gl!Gm n (right panel) nal state masses. The example shown is M5 = 10 TeV, k = 10 GeV, for mode l = 1000. The corresponding mass of the decaying particle is m1000 ' 1 TeV. From eq. (G.5) we nd the decay width for the process Gl ! m + n Gl! m n (k; R) = jC(l;m;n)(k; R)j2 TSS 480 2(1 + mn)ml7RM53 5=2(ml2; m2m; m2n) ; (G.8) where (x; y; z) x2 + y2 + z2 the density plot of the decay width 2xy 2xz 2yz. In the left panel of gure 25 we show Gl! m n as a function of the nal state masses. From this plot it is evident that the decay width is dominated by states close to the phase space boundary. If compared with the trilinear graviton decay computed in appendix D, it is easy to see that the KK graviton decay into two scalars is largely sub-dominant. The main reason is that in the process Gl ! m + n conservation of angular momentum requires a nal state in d-wave, L = 2. By direct computation, it is possible to show that the decay width the two Gl! m n is indeed proportional to the fourth power of the relative velocity between nal state particles, de ned as vrel = jp~m=Em p~n=Enj, in agreement with the usual correspondence between total orbital angular momentum L and velocity vrel, which implies the scaling vr2eLl in the non-relativistic limit. Consequently, close to the phase space boundary the decay width receives a further suppression due to the fact that nal state particles are produced almost at rest with very small relative velocity. Finally, we provide an analytical approximation for the sum over the kinematically allowed nal state particles, in parallel with the discussion presented for the graviton decay into gravitons. We nd Gl!P m n = Gl!P GmGn 6k2 595ml2 : (G.9) where Gl!P GmGn is the inclusive width computed in eq. (D.13). This result clearly shows the relative suppression between the two channels. KK graviton decays into a KK graviton and a scalar The 4D Lagrangian density for a xed triad (n; m; l) of KK states is TTS where C(n;m;l)(k; R) encodes the integration over the extra dimension, and we nd h TTS TTS C(n;m;l)(k; R) + T(n;m;l)(k; R) P(n;m;l)(k; R)i TTS nh~(m)h~(l) (G.10) dye3 pk h48k2 ~ '~n m l + HJEP06(218)9 8k2 3 4k 3 TTS L(n;m;l) = 1 M53=2( R) TTS C(n;m;l)(k; R) = Z R 0 TTS TTS The two functions T(n;m;l)(k; R) and P(n;m;l)(k; R) arise, respectively, from interactions localized on the TeV and Planck brane. We nd T(Tn;TmS;l)(k; R) = 4k3=2 h'~n(0) m(0) l(0) + 2 ~ n(0) m(0) l(0)i ; P(n;m;l)(k; R) = 4k3=2e2k R h'~n( R) m( R) l( R) + 2 ~ n( R) m( R) l( R)i : (G.13) TTS We can now compute the decay width for the process Gl ! Gm + n. We nd Gl!Gm n = jS(TnT;mS;l) + S(n;l;m)j TTS 2 2880M53ml7m4m 3R2 1=2(ml2; m2m; m2n) F (ml2; m2m; m2n) ; (G.14) where F (x; y; z) TTS P(n;m;l)(k; R). x4 + x3(26y 4z) + 2x2(63y2 28yz + 3z2) + 2x(13y 2z)(y z)2 + (y z)4, and we introduced the short-hand notation S(n;m;l) TTS TTS TTS C(n;m;l)(k; R) + T(n;m;l)(k; R) In the right panel of gure 25 we show the density plot of the decay width Gl!Gm n nal state masses. From this plot we see that the decay width is dominated by heavy gravitons and light scalars in the nal state. We provide an analytical approximation for the sum over the kinematically allowed nal state particles, in parallel with the discussion presented for the graviton decay into gravitons. We nd (G.11) (G.12) Gl!P Gm n = Gl!P GmGn (G.15) 1524k3=2 595ml3=2 : G.3 KK graviton decays: branching ratios We can now summarize our results. In gure 26 we show the branching ratios of the graviton KK modes. For clarity, in the nal states involving KK scalars we plotted separately the case with a radion in the nal state. All the decay widths are inclusive, meaning that we consider the sum over all kinematically allowed nal state particles. For the inclusive processes, in the limit l kR 1 we nd the simple scaling relations k l M53 m2k2 l M53 ; HJEP06(218)9 = = ϕ = = ϕ ϕ = ϕ ϕ [ m3k l M53 : = ] = ϕ ϕ = ϕ Branching ratios of the graviton KK modes for k = 10 GeV and rigid boundary conditions. We take M5 = 10 TeV. that should be compared with the decay width into SM states Gl!SM (G.17) As shown in gure 26, and already discussed in the main text, graviton decays into gravitons competes with the SM decay channels at small k since the ratio of their decay widths parametrically grows as (ml=k)1=2. On the contrary, KK graviton decays into, respectively, a KK graviton and a scalar and two scalars feature the suppressions and (k=ml)2 if compared to the graviton into SM decay. Let us consider these cases in more detail. It is useful to remember that in the graviton case the wavefunction integrals prefer small mass splittings, which is physically understood from the view of extra-dimensional momentum conservation in the k ! 0 limit. For the decay of graviton into a graviton and a scalar there is no wavefunction suppression. This is because in the sti limit the boundary conditions for the dilaton are Dirichlet-Dirichlet and, when integrated over the wavefunction for two gravitons, one no longer requires that ml mm + mn as the integrand is y-parity odd in the limit k ! 0. Naively one would expect the total width should then scale as m4=M53. l However, in the limit k ! 0 the dilaton acquires an additional Z2 symmetry that would completely forbid such decays. Thus one expects the amplitude should still scale proportional to k due to the vertex factor itself. Thus, when squared, this explains the form of the decay width in eq. (G.16), and the extra suppression relative to the Gl ! P GmGn decays. The suppression for Gl ! P m n decays can be understood from momentum conservation. The wavefunction integral suppression is still in action as in the k ! 0 limit the integrand is y-parity even. In addition to this the decay of a spin-2 graviton to two scalars will be d-wave suppressed, hence the matrix element is itself proportional to the nal state momenta, which are proportional to the mass splitting. When squared this leads to an additional factor of (k=ml)2 relative to the Gl ! KK scalar decays into two scalars The 4D Lagrangian density for a xed triad (n; m; l) of KK states is LSSS (n;m;l) = 1 M53=2 hC(SnS;Sm(;1l))(k; R) n(@ SSS l) + B(n;m;l)(k; R) n m l ; i where the coe cient of the derivative interaction comes from the extra-dimensional integral Z R 0 C(SnS;Sm(;1l))(k; R) = 6 dye3 k3=2 ~ n(y) ~ m(y) ~ l(y) ; C(SnS;Sm(;2l))(k; R) + T(n;m;l)(k; R) + P(n;m;l)(k; R). The rst term arises from the following extra SSS SSS dimensional integral C(SnS;Sm(;2l))(k; R) = Z R 0 dye3 k3=2 3 ~ n ~ 0m ~ 0l + 64k2 ~ n ~ m ~ l 6k ~ n ~ m'~0l + ~ n'~0m'~0l 20k ~ 0n ~ m ~ l + 4k2 ~ n'~m'~l 81 16 k2'~n'~m'~l ; SSS SSS while T(n;m;l)(k; R) and P(n;m;l)(k; R) describe interactions localized, respectively, on the TeV and Planck branes. We (G.18) (G.19) (G.20) SSS T(n;m;l)(k; R) SSS P(n;m;l)(k; R) nd 1 6 4k 9 4k 4k '~n(0)'~m(0) ~ l(0) '~n(0) ~ m(0) ~ l(0) ; (G.21) [(27 P + 4k)'~n( R)'~m( R)'~l( R) (G.22) +(72k + 162 P)'~n( R)'~m( R) ~ l( R) +216k'~n( R) ~ m( R) ~ l( R)i : Notice that both TeV and Planck brane contributions vanish in the rigid limit. Equipped with this result, we can compute the decay width for the process l ! m + n. We nd l! m n (k; R) = 1=2(ml2; m2m; m2n) 16 (1 + mn)ml3M53 ml2 + m2m + m2n C(SnS;Sm(;1l))(k; R) (G.23) + X h SSS 2 C(n;m;l)(k; R) + T(n;m;l)(k; R) + P(n;m;l)(k; R)i 2 SSS SSS : perm: In the left panel of gure 27 we show the density plot of the decay width l! m n as a function of the nal state masses. From this plot we see that the decay width is dominated by the region close to the phase space boundary. l!GmGn = jS(Tl;TnS;m) + S(l;m;n)j TTS 2 2880(1 + mn)M53ml3m4nm4m 3R2 1=2(ml2; m2m; m2n) F (M m2; Mn2; Ml2) : In the central panel of gure 27 we show the density plot of the decay width function of the nal state masses. From this plot we see that the decay width is dominated by the region close to the phase space boundary with an additional sizable contribution coming from the decay into two light gravitons. G.6 KK scalar decays into a KK graviton and a scalar The decay process l ! nd the width m + Gn is controlled by the Lagrangian density in eq. (G.5). We l! mGn(k; R) = jC(n;m;l)(k; R)j2 TSS 96M53ml3m4n 2R 5=2(ml2; m2m; m2n) : In the right panel of gure 27 we show the density plot of the decay width function of the nal state masses. From this plot we see that the decay width prefers the presence of a light scalar in the nal state. G.7 KK scalar decays: branching ratios We can now summarize our results. We already discussed in gure 7 the branching ratios of the scalar KK modes for k = 10 GeV, rigid boundary conditions and = 0. For the inclusive processes, in the limit l kR 1 we nd the simple scaling relations l!GmGn (central panel) and l! mGn (right panel) as a function of the nal state masses. The example shown is M5 = 10 TeV, k = 10 GeV, for mode l = 1000. The corresponding mass of the decaying particle is m1000 ' 1 TeV. G.5 KK scalar decays into two KK gravitons The decay process l ! Gm + Gn is controlled by the Lagrangian density in eq. (G.10). We nd the width m2k2 l M53 ; l!P m n l!P mGn l!P GmGn m2k2 l M53 ; (G.24) l!GmGn as a (G.25) l! mGn as a mlk3 M53 ; (G.26) l!SM 3 mlk M53 : As already noticed and discussed in the main text, the suppression of the SM decay modes makes all decay channels in eq. (G.26) important for the phenomenology. H Dijet angular distributions Analyses of the dijet angular distributions (such as refs. [74{76]) use the variable , de ned HJEP06(218)9 in terms of the rapidities of the two jets as which can be expressed in terms of Mandelstam variables as (G.27) (H.1) (H.2) (H.3) : (H.4) (I.1) Therefore, the angular distributions of the various dijet processes can be obtained from the expressions for d dt (s; t; u) available in appendix A of ref. [43] as d d (s; ) = ( + 1)2 d dt + 1 s 1= + 1 1= + 1 s + 1 The e ective graviton propagators in our model are di erent from those of ref. [43] and are derived in appendix I. I Representing the KK graviton tower by an e ective propagator For computing matrix elements involving KK gravitons, it is sometimes useful to sum the propagators (times couplings) of the whole KK graviton tower and work in terms of the e ective propagator exp(jy1 y2j) ; 8 t = < s + 1 for 0 < t < for t < s Se (s) ' = 1 X n=1 G 1 (n)2 s 1 M53 k 1 dm 1 m2n + i 1 r 1 s + k2 + k k2 m2 s m2 + i the sum of squared propagators, Se (s) ! for s for jsj k 2 k 2 1 1 X n=1 G 1 (n)4 (s 2M56R k 1 1 ' 4M56s m2n)2 + s 2n(s) 2 R (s) k 2 3=2 for s k2 ; 1 m2)2 + s 2(s) Another useful quantity, relevant for on-shell KK graviton production (at p s > k), is (I.2) (I.3) (I.4) where (s) is the width of KK gravitons with mass near ps. For s k2, there is an extra suppression factor of approximately 1 k 2 3=2 as can be seen by using m2 ' s in the k-dependent prefactor in the integrand. Note the important di erence between the result of eq. (I.3) and the nave square of eq. (I.2), which has been also discussed in refs. [44, 83] in the context of the low-curvature RS model. When using the formulas of ref. [43] as envisioned in appendix H, one should use jS(s)j2e For S(s) with p whenever jS(s)j2 with p s > k appears, but Se (s) when S(s) is not squared. s < k, S(t), or S(u), one should use Se (s), Se (t), or Se (u), respectively. J An exact solution to Einstein's equations Working in the comoving coordinate ds2 = e2 (z) dx dx + dz2 we nd that a solution of Einstein's equation and the dilaton equation of motion from the action in eq. (2.25) is S(z) = log + 1 ; (z) = kz + log + 1 ; (J.1) 3 2 4kz 1 4kz p3" where we set the integration constants such that (0) = 0 and S(0) = 0. Note that there is no smooth limit to either AdS or CW/LD geometry, suggesting this is a branch of the general solution to the equations of motion that is rather unique. 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Gian F. Giudice, Yevgeny Kats, Matthew McCullough, Riccardo Torre, Alfredo Urbano. Clockwork/linear dilaton: structure and phenomenology, Journal of High Energy Physics, 2018, 9, DOI: 10.1007/JHEP06(2018)009