Diphoton resonance from a warped extra dimension

Journal of High Energy Physics, Jul 2016

We argue that extensions of the Standard Model (SM) with a warped extra dimension, which successfully address the hierarchy and flavor problems of elementary particle physics, can provide an elegant explanation of the 750 GeV diphoton excess recently reported by ATLAS and CMS. A gauge-singlet bulk scalar with \( \mathcal{O} \)(1) couplings to fermions is identified as the new resonance S, and the vector-like Kaluza-Klein excitations of the SM quarks and leptons mediate its loop-induced couplings to photons and gluons. The electroweak gauge symmetry almost unambiguously dictates the bulk matter content and hence the hierarchies of the \( S\to\ \gamma \gamma, W\ W,ZZ,Z\gamma, t\overline{t} \) and dijet decay rates. We find that the S → Zγ decay mode is strongly suppressed, such that Br(S → Zγ)/Br(S → γγ) < 0.1. The hierarchy problem for the new scalar boson is solved in analogy with the Higgs boson by localizing it near the infrared brane. The infinite sums over the Kaluza-Klein towers of fermion states converge and can be calculated in closed form with a remarkably simple result. Reproducing the observed pp → S → γγ signal requires Kaluza-Klein masses in the multi-TeV range, consistent with bounds from flavor physics and electroweak precision observables. Useful side products of our analysis, which can be adapted to almost any model for the diphoton resonance, are the calculation of the gluon-fusion production cross section σ(pp → S) at NNLO in QCD, an exact expression for the inclusive S → gg decay rate at N3LO, a study of the \( S\to t\overline{t}h \) three-body decay and a phenomenological analysis of portal couplings connecting S with the Higgs field.

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Diphoton resonance from a warped extra dimension

Revised: May Diphoton resonance from a warped extra dimension Martin Bauer 0 1 2 5 6 Clara Horner 0 1 2 3 6 Matthias Neubert 0 1 2 3 4 6 0 Ithaca , NY 14853 , U.S.A 1 Johannes Gutenberg University , 55099 Mainz , Germany 2 Philosophenweg 16 , 69120 Heidelberg , Germany 3 PRISMA Cluster of Excellence & Mainz Institute for Theoretical Physics 4 Department of Physics & LEPP, Cornell University 5 Institut fur Theoretische Physik, Universitat Heidelberg 6 , W W , ZZ, Z , tt and dijet decay rates. We nd that We argue that extensions of the Standard Model (SM) with a warped extra dimension, which successfully address the hierarchy and particle physics, can provide an elegant explanation of the 750 GeV diphoton excess recently reported by ATLAS and CMS. A gauge-singlet bulk scalar with O(1) couplings to fermions is identi ed as the new resonance S, and the vector-like Kaluza-Klein excitations of the SM quarks and leptons mediate its loop-induced couplings to photons and gluons. The electroweak gauge symmetry almost unambiguously dictates the bulk matter content and hence the hierarchies of the S ! decay mode is strongly suppressed, such that Br(S ! Z )=Br(S ! Phenomenology of Field Theories in Higher Dimensions - the S ! Z ) < 0:1. The hierarchy problem for the new scalar boson is solved in analogy with the Higgs boson by localizing it near the infrared brane. The in nite sums over the Kaluza-Klein towers of fermion states converge and can be calculated in closed form with a remarkably simple result. Reproducing the observed pp ! S ! signal requires Kaluza-Klein masses in the multi-TeV range, consistent with bounds from avor physics and electroweak precision observables. Useful side products of our analysis, which can be adapted to almost any model for the diphoton resonance, are the calculation of the gluon-fusion production cross section (pp ! S) at NNLO in QCD, an exact expression for the inclusive S ! gg decay rate at N3LO, a study of the S ! tth three-body decay and a phenomenological analysis of portal couplings connecting S with the Higgs eld. 1 Introduction 2 3 4 6 7 RS model with a bulk scalar eld Diboson signals from warped space 3.1 3.2 4.1 4.2 Phenomenology of the diphoton resonance General discussion S-boson phenomenology in RS models Diboson couplings induced by KK fermion exchange Coupling to top quarks 5 Impact of Higgs portal couplings Three-body decay S ! tth Conclusions A RG evolution of the Wilson coe cients cgg and ctt B Inclusive S ! gg decay rate at N3LO in QCD new particles with a large multiplicity or sizable couplings to S have to enter the S ! loop for both gluon-fusion or bb-initiated production processes [5]. Producing the resonance from other quark-initiated states results in a tension with 8 TeV data, while photon-induced production would require non-perturbatively large couplings [ 6, 7 ]. Supersymmetric UV completions of the SM, which motivate such additional degrees of freedom, lack a neutral scalar candidate with appropriate couplings, and the full parameter space of the Minimal { 1 { Supersymmetric SM is excluded as a consequence [ 8 ]. One thus has to resort to models with a low supersymmetry-breaking scale, which allow for a sgoldstino explanation [9], or R-parity violating scenarios, in which the sneutrino can have large enough couplings to account for the excess [10, 11]. Composite Higgs models predict several composite resonances that can facilitate a large diphoton branching ratio [12{15]. Neutral composite scalars, which appear in non-minimal composite Higgs models with larger coset structure, as well as the dilaton/radion have been considered as possible candidates for S. While theoretically motivated, the latter implies a small radius of the extra dimension in order to enhance the couplings to diphotons [16], unless the Higgs-radion mixing is tuned to a particular value [17, 18]. In this regard, the sgoldstino and radion explanations have similar anomalous magnetic moment of the muon [8, 19{25]. In this paper we argue that Randall-Sundrum (RS) models featuring a warped extra dimension [26], with all SM elds (with the possible exception of the Higgs boson) propagating in the bulk, can explain the observed excess in a natural way. We introduce a bulk scalar singlet, whose only renormalizable interactions | with the exception of a possible Higgs portal | are couplings to bilinears of vector-like bulk fermions. Remarkably, for O(1) couplings of this new scalar the diphoton excess is explained for Kaluza-Klein (KK) masses in the multi-TeV range without any additional model building. This mass scale is su ciently large to avoid constraints from electroweak precision tests, avor physics and Higgs phenomenology. Our results are largely insensitive to the parameters of the RS model, such as the ve-dimensional (5D) masses of the fermions and their Yukawa couplings to the Higgs eld. To good approximation the loop-induced couplings of the new resonance S to diboson states just count the number of degrees of freedom propagating in the loop (times group-theory factors). We consider three implementations of the RS model with di erent fermion contents and present detailed predictions for the gluon-fusion production cross section (pp ! S) and the rates for the decays S ! , W W , ZZ, Z , gg, tt and tth, all of which are found within current experimental bounds. We note in passing that our scenario is particularly well motivated if one assumes the new scalar to take on a vacuum expectation value, which generates the fermion bulk mass terms, thus providing a mechanism for the avor-speci c localization of fermions along the extra dimension [27, 28]. In this case, the bulk scalar would assume the role of the localizer eld rst introduced in the context of split fermion models [29]. We shall explore this intriguing possibility in future work. We are aware of only a few papers in which the possibility of an extra-dimensional origin of the diphoton signal has been explored. The authors of [ 30 ] considered a model with a at extra dimension. While such a framework does not address the hierarchy problem of the Higgs boson and the new scalar resonance, this work shares several technical similarities with our approach. However, the warped background of RS models makes our { 2 { This paper is organized as follows: in section 2 we brie y introduce the basic concalculations more demanding. In [ 31 ] it was assumed that the new resonance couples to the SM only via loops involving heavy vector-like leptons. In order to obtain the very large couplings required in this case [ 6, 7 ], the construction relies on more than one at extra dimension, and only SM lepton elds are placed in the bulk. This treatment gives up the attractive possibility of understanding the avor hierarchies from an extra-dimensional perspective. The authors found that the overlap integrals in their calculation required a cuto , which was introduced by hand and motivated based on stringy arguments. In [32{ 34] the new resonance was identi ed with the lowest spin-2 KK graviton in warped extradimension models. rate at N3LO, and the S ! struction of warped extra-dimensional models and derive expressions for the mass and the wave-function of the bulk scalar S. In section 3 we compute the e ective Wilson coe cients parameterizing the couplings of S to SM gauge bosons and top quarks by integrating out the heavy fermionic KK modes, considering both the minimal RS model as well as two di erent extensions with a custodial symmetry. Section 4 deals with the phenomenology of the resonance S. In the context of an e ective Lagrangian with local interactions of S with SM elds, we rst calculate the gluon-fusion production cross section (pp ! S) at nextto-next-to-leading order (NNLO) in QCD perturbation theory, the inclusive S ! gg decay , W W , ZZ, Z , tt decay rates at leading order. We then perform ts to the diphoton signal in the parameter space of the RS models, taking into account existing constraints from LHC Run 1 resonance searches. In section 5 we study the impact of possible Higgs portal interactions of S on the various branching fractions, including the S ! hh signal. The three-body decay mode S ! tth is studied in section 6, before we conclude in section 7. Some technical details are relegated to two appendices. 2 RS model with a bulk scalar eld We consider extensions of the SM with a warped extra dimension, described by an S1=Z2 orbifold parameterized by a coordinate orbifold xed-points: the UV brane at ; ], with two branes localized on the = 0, and the infrared (IR) brane at j j = . The curvature k and radius r of the extra dimension are assumed to be of Planck size, k 1=r MPl, and the metric reads [26] ds2 = e 2 ( ) dx dx r2d 2 = dx dx dt2 ; (2.1) t 2 2 1 MK2 K where ( ) = krj j is referred to as the warp factor. The quantity L = ( ) = kr measures the size of the extra dimension and is chosen so as to explain the hierarchy 34 between the Planck scale and the TeV scale [35]. With the help of the curvature and the warp factor evaluated on the IR brane, = e L 10 15, one de nes the KK scale MKK k . It sets the mass scale for the low-lying KK excitations of the model and controls the mass splitting between the KK modes. On the right-hand side of (2.1) we have introduced the dimensionless coordinate t de ned by t = e ( ) 2 [ ; 1], which will be { 3 { used throughout this work. It is related to the frequently used conformal coordinate z by the rescaling z = t=MKK. The hierarchy problem is solved by localizing the SM Higgs eld on or near the IR brane, e ectively cutting o UV-divergent contributions to the Higgs mass at the scale IR = MPl TeV [26]. If gauge bosons and fermions are promoted to 5D bulk elds, the avor problem can be addressed in a natural way by means of di erent localizations of the fermion zero modes along the extra dimension [27, 28]. The large hierarchies in the spectrum of fermion masses and mixing angles can then be reproduced by small variations of parameters in the underlying 5D Lagrangian [36]. The minimal RS model with bulk elds, which has the same gauge symmetry and matter content as the SM, is strongly constrained by electroweak precision observables [37, 38]. A recent tree-level analysis of the S and T parameters yields the lower bound [39] (minimal RS model) : (2.2) Since the masses of the low-lying KK excitations are typically several times heavier than MKK (for example, the lightest KK gluon and photon have a mass of 2:45 MKK [40]), this puts them out of the reach for discovery at the LHC. Bounds from electroweak precision observables are considerably relaxed if the electroweak sector respects the custodial symmetry present in the SM. This implies an enhanced bulk gauge group SU(3)c and its avor-changing counterparts [44] from receiving too large corrections. As a result, the bound on the KK scale is lowered to [39] (custodial RS model) : (2.3) Thorough discussions of this model containing many technical details can be found in [45, 46]. In the following, we will consider two di erent versions of the custodial RS model: one with a symmetric implementation of the quark and lepton sectors (custodial model I), and one in which the lepton sector is more minimal than the quark sector (custodial model II) [47]. Besides electroweak precision tests, RS models are constrained by avor observables [44, 45, 48{50] and Higgs phenomenology [46, 47, 51{53]. The most severe avor constraint comes from K K mixing [49]. In the minimal model the KK scale is so high that this bound can be satis ed with a modest 25% ne-tuning. For the lower values of the KK mass scale allowed in the custodial model, the avor constraints can either be solved by means of a 5 10 % ne-tuning or by enlarging the strong-interaction gauge group in the bulk [54]. Additional constraints arising from the phenomenology of the Higgs boson, such as its production cross section and decay rates into , ZZ and W W , are more model dependent and can readily be made consistent with present data by adjusting some model parameters. { 4 { We identify the diphoton resonance with the lightest excitation of a new bulk scalar eld S(x; ), which is a singlet of the full bulk gauge group. In order to allow for a coupling of this eld to the scalar density of the vector-like 5D fermion elds we need to implement S(x; ) as an odd eld on the S1=Z2 orbifold, such that S(x; ) = S(x; ). The relevant terms in the action read Z 2 2 S 2 X f sgn( ) f Mf f + S f Gf f ; where the sum extends over all 5D fermion multiplets f . Even in the minimal RS model there exists a 4-component vector-like 5D fermion eld for every Weyl fermion of the SM. The SM fermions correspond to the zero modes of these elds, which become massive after electroweak symmetry breaking. Consequently, for each SM fermion there exist two towers of KK excitations [55]. In extensions of the RS model with a custodial symmetry additional exotic matter elds are introduced, which have no zero modes but give rise to additional towers of KK excitations, thereby increasing the number of vector-like fermions of the model [41, 42]. The bulk masses Mf and couplings Gf are hermitian matrices in generation space. By means of eld rede nitions one can arrange that Mf are real, diagonal matrices. From now on we will always work in this so-called bulk mass basis. The values of the bulk masses determine the pro les of the SM fermions along the extra dimension, which generically turn out to be localized near one of the two branes [27, 28]. Note that there is the intriguing possibility that the bulk masses could be generated dynamically in models where the scalar eld S acquires a vacuum expectation value w, such that Mf = wGf . While we leave the detailed construction of such models to future work, we shall assume that the structure of the couplings Gf follows the structure of Mf . In (2.4) we have not considered the possibility of a portal coupling S j j2 connecting the eld S with the Higgs doublet. We will investigate the phenomenological impact of such a coupling on the various decay rates of the resonance S in section 4, nding rather strong constraints. An extra-dimensional setup, in which the Higgs sector is localized on the IR brane, where the Z2-odd scalar eld S vanishes, might provide a dynamical explanation for the suppression of the portal interaction. We emphasize, however, that even with such sequestering a Shh coupling is inevitably induced at one-loop order, since the 5D bulk fermions can mediate between the IR brane, where the Higgs eld is localized, and the bulk, where the eld S lives. In our phenomenological analysis in section 4 we will therefore allow for the presence of a loop-suppressed portal interaction. The solution of the eld equations satis ed by the KK modes of the scalar eld S is obtained in complete analogy to the case of a bulk scalar eld studied in [39, 56, 57]. Imposing the KK decomposition (with t = e ( )) (2.4) the pro le functions nS(t) are obtained from the equation of motion e ( ) p r X n S(x; ) = Sn(x) nS(t) ; 2 + t2x2n nS(t) t = 0 ; { 5 { (2.5) (2.6) where Nn is a normalization constant. In order to obtain a relatively light mass mS m1S 750 GeV for the lightest scalar resonance, we impose the mixed boundary condition nS(1) = nS0(1) on the IR brane, which can be engineered by adding brane-localized terms to the action. In the limits Dirichlet boundary condition ! 0 and ! 1 one recovers the special cases of the nS(1) = 0 and the Neumann boundary condition nS0(1) = 0, respectively. In the general case, we obtain and due to the smallness of rn / approximation. It follows that the mass of the lightest resonance is given by 2 the right-hand side can be set to zero to excellent (2.7) 1(xn) ; o (2.9) (2.10) (2.11) = O(1) (2.12) x 2 1 750 GeV can be achieved with a moderate tuning of parameters. For example, with MKK = 2 TeV we need 0:69 for = 0:5, 0:51 for = 1, 0:17 for = 5 and 0:09 for = 10. The properly normalized pro le function of the lightest resonance is given by (dropping irrelevant terms vanishing for 1S(t) = r L(1 + ) t1+ 1 x 2 1 4 t 2 1 + 1 2 + ! 0) + O(x14) : where xn = mnS=MKK and 2 = 4 + 2=k2. To obtain canonically normalized kinetic terms for the KK modes, we must impose the normalization condition L t 2 Z 1 dt Sm(t) nS(t) = mn : Requiring the Dirichlet boundary condition general solution nS( ) = 0 on the UV brane, one nds the nS(t) = Nn t [J (xnt) rn J (xnt)] ; rn = J ( xn) J ( xn) (1 (1 + ) ) xn 2 2 ; (2.8) controls the localization of the bulk scalar, and in analogy to the case of a bulk Higgs boson we will assume that > 0 (i.e., 2 > 4k2) [58]. For values the scalar has a wide pro le along the extra dimension, while for 1 it is localized near the IR brane; in fact, we have 1S(t) !=1 r L(1 + ) 1 2 + (t 1) : While there is no particular reason why the bulk scalar should be localized near the IR brane, we will nd that our results take a particularly simple form in this limit. { 6 { In the models we consider, the masses of the KK excitations of gauge bosons and fermions are bound by constraints from electroweak precision and avor observables to lie in the multi-TeV range. The 750 GeV resonance is considerably lighter, and it is thus justi ed to integrate out the tower of fermion KK modes in computing the decays of S to diboson or fermionic nal states. Below the KK mass scale we de ne the e ective Lagrangian Le = cgg 4 s S G a G ;a + cW W 4 s2 S W a W w ;a + cBB 4 c2 S B w B S QLY^u ~ uR + S QLY^d dR + S LLY^e eR + h.c. ; (3.1) is the scalar Higgs doublet, and sw = sin w and cw = cos w are functions of the weak mixing angle. Since the mass of the new resonance is much larger than the electroweak scale, it is appropriate to write the e ective Lagrangian in the electroweak symmetric phase. Upon electroweak symmetry breaking the second and third operator in the rst line generate the couplings of S to pairs of electroweak gauge bosons. In particular, the resulting diphoton coupling is Le 3 c 4 S F F ; with c = cW W + cBB : The terms in the second line in (3.1) describe the couplings of S to fermion pairs (with or without a Higgs boson). In our model these couplings have a hierarchical structure, and the dominant e ect by far is the coupling to the top quark. Rewriting Re[(Y^u)33] = ctt yt (after transformation to the mass basis), where yt = 2mt=v is the top-quark Yukawa p coupling, we can express the corresponding term as Le 3 ctt mt 1 + S tt + : : : : h v The Wilson coe cients in the e ective Lagrangian are suppressed by the mass scale of the heavy KK particles, cii / 1=MKK. In the remainder of this section we will calculate these coe cients at the matching scale KK = few MKK corresponding to the masses of the low-lying KK modes, which give the dominant contributions. It is well known that the two-gluon operator has a non-trivial QCD evolution [59, 60] and mixes with the operator in (3.3) under renormalization [61]. These e ects are discussed in detail in appendix A. When the strong coupling s and the Yukawa coupling yt are factored out from the de nitions of cgg and ctt as we have done above, evolution e ects from the high matching scale KK to the scale = mS only arise at NLO in renormalization-group (RG) improved perturbation theory. At this order they give rise to the simple relations cgg( ) = 1 + 1 matching scale, it is inevitably induced through RG evolution; however, this is a very small e ect. For and cgg( ) KK = 5 TeV and = 750 GeV we nd ctt( ) ctt( KK) + 0:0028 cgg( KK) 1:0045 cgg( KK). Higher-order QCD corrections to the Wilson coe cients at the high matching scale are likely to have a more important impact. For instance, they enhance the top-quark contribution to the Higgs-boson production cross section in the SM by about 20% [62, 63]. To be conservative we will not include such enhancement factors in our analysis. 3.1 Diboson couplings induced by KK fermion exchange Since the scalar eld S is a gauge singlet, its couplings to gauge bosons are induced by fermion loop diagrams, such as those shown in gure 1. The relevant couplings in (2.4) are parameterized by the matrices Gf , while the pro les of the fermions along the extra dimension depend on the (diagonal) bulk mass matrices Mf . It is conventional to de ne dimensionless bulk mass parameters by cf = Mf =k, where the plus (minus) sign holds for fermion elds whose left-handed (right-handed) components have even pro le functions under the Z2 symmetry. In the minimal RS model the SU(2)L fermion doublets have even left-handed components, while the SU(2)L fermion singlets have even right-handed components. In extensions of the RS model elds transforming as SU(2)L triplets also have even right-handed components. Using the same sign conventions, we de ne dimensionless couplings gf of S to fermions via gf = pk(1 + ) 2 + Gf : (3.5) This de nition is analogous to the de nition of the dimensionless Yukawa couplings in RS models with a bulk Higgs eld studied in [39, 52, 64]. The -dependent terms ensure that the dimensionless couplings remain well-behaved in the limit ! 1 of an IR brane-localized scalar eld. These matrices are hermitian but, in general, not diagonal in generation space. Since with the exception of the top quark all SM fermions have masses much below the electroweak scale, the values of most of the bulk mass parameters cf cluster near or below the critical value 1=2, below which the zero-mode fermion pro le is localized near the UV brane. For example, a typical set of bulk mass parameters adopted in [55] ranges from 0:74 for cu1 to 0:47 for cQ3 . The only exception is the parameter c t cu3 +0:34 of the right-handed top quark, which is positive so as to realize a localization near the IR brane. In our phenomenological analysis below we will for simplicity assume that the diagonal elements of the matrices gf all have the same sign and magnitude, with the possible exception of gt (gu)33. In close analogy with the case of the induced hgg and h couplings of the Higgs boson in models where all SM eld propagate in the bulk, we nd that the sums over the in nite towers of KK fermion states in gure 1 converge and can be calculated in closed form using 5D fermion propagators [65{67]. In the unbroken phase of the electroweak gauge symmetry (i.e. for v = 0), there is no mixing between fermion states belonging to { 8 { HJEP07(216)94 S fn fn MK2 K ! m2S S di erent multiplets of the gauge group and the fermion propagators are diagonal matrices in generation space. The Wilson coe cients are then given by sums over the contributions from the di erent fermion multiplets. Mixing e ects induced by electroweak symmetry breaking yield corrections of order (mf =mS)2 relative to the leading terms we will compute. Even for the top quark these corrections are at most a few percent and can safely be neglected. The fermion representations of the custodial RS models have been discussed in detail in [43, 45{47]. We begin with a brief description of the quark sector. As a consequence of the discrete PLR symmetry, the left-handed bottom quark needs to be embedded in an SU(2)L SU(2)R bi-doublet with isospin quantum numbers TL3 = TR3 = 1=2. This assignment xes the quantum numbers of the remaining quark elds uniquely. In particular, the righthanded down-type quarks have to be embedded in an SU(2)R triplet in order to obtain a U(1)X -invariant Yukawa coupling. We choose the same SU(2)L SU(2)R quantum numbers for all three quark generations, which is necessary to consistently incorporate quark mixing in the anarchic approach to avor in warped extra dimensions. Altogether, there are fteen di erent quark states in the up sector and nine in the down sector (for three generations). The boundary conditions give rise to three light modes in each sector, which are identi ed with the SM quarks. These are accompanied by KK towers consisting of groups of fteen and nine modes of similar masses in the up and down sectors, respectively. In addition, there is a KK tower of exotic fermion states with electric charge Q = 5=3, which exhibits nine excitations in each KK level. In order to compute the Wilson coe cients cgg, cW W and cBB in (3.1) it is most convenient to decompose these multiplets into multiplets under U(1)Y . There are two SU(2)L doublets and one triplet cQ : 0 u(+) 1 L L L 0 ( ) A ; 7 6 c 1 : R BB U 0 ( ) CCC ; B R DR0( ) A2 as well as four singlets cu : uc (+) R 2 3 ; cd : DR(+) 1 3 ; U ( ) R 2 3 ; We only show the chiral components with even Z2 parity; the other chiral components are odd under the Z2 symmetry. The subscript denotes the hypercharge of each multiplet. { 9 { 3 ( ) R 5 3 : (3.6) (3.7) The superscripts on the elds specify the type of boundary conditions they obey on the UV brane. Fields with superscript (+) obey the usual mixed boundary conditions allowing for a light zero mode, meaning that we impose a Dirichlet boundary condition on the pro le functions of the corresponding Z2-odd elds. These zero modes correspond to the SM quarks. Fields with superscripts ( ) correspond to heavy, exotic fermions with no counterparts in the SM. For these states, the Dirichlet boundary condition is imposed on the Z2-even elds so as to avoid the presence of a zero mode. The UV boundary conditions for the elds of opposite Z2 parity are of mixed type and follow from the eld equations. Above we have indicated the bulk mass parameters associated with the various multiplets.1 The three parameters contained in the matrix c 1 can be related to the other ones by extending the PLR symmetry to the part of the quark sector that mixes with the left-handed down-type zero modes [46]. It then follows that c 1 = cd. Whether or not this equation holds turns out to be irrelevant to our discussion. In the custodial model I the lepton sector is constructed in analogy with the quark sector [45]. It consists of two SU(2)L doublets and one triplet (3.8) (3.9) B BB N R0( ) CC ; C ER0( ) A 0 as well as four singlets c : c (+) R 0 ; ce : ER(+) 1 ; NR ( ) 0 ; ( ) R 1 : Again we only show the chiral components with even Z2 parity. There are fteen di erent lepton states in the neutrino sector and nine in the charged-lepton sector. The boundary conditions give rise to three light modes in each sector, which are identi ed with the SM neutrinos and charged leptons. These are accompanied by KK towers consisting of groups of fteen and nine modes in the two sectors, respectively. In addition, there is a KK tower of exotic lepton states with electric charge Q = +1, which exhibits nine excitations in each KK level. The three parameters contained in the matrix c 3 can be related to the other ones by requiring an extended PLR symmetry, in which case c 3 = ce. In the custodial model II the lepton sector is more minimal [47]. It consists of one SU(2)L doublet and two singlets cL : L (+) ! e(+) L 1 2 ; ce : ec (+) R 1 ; N R0( ) 0 : (3.10) The choice of the boundary conditions is such that the zero modes correspond to the light leptons of the SM, without a right-handed neutrino. Note that the minimal RS model is 1Fields belonging to the same SU(2)R multiplet have equal bulk mass parameters. The two doublets associated with cQ form a bi-doublet under SU(2)L SU(2)R, while the three singlets associated with cd form a triplet under SU(2)R. obtained by simply omitting all multiplets containing elds carrying a superscript \( )" from the above list. The calculation of the one-loop diagrams in gure 1 proceeds in complete analogy with the corresponding calculation for a bulk Higgs eld performed in [39]. One evaluates the amplitude in terms of an integral over 5D propagator functions, employs the KK representation of these functions in terms of in nite sums, simpli es the resulting expression and recasts it in the form of an integral over a single 5D fermion propagator. Adapting these steps to the present case, we obtain the expressions (for v = 0) cgg = which only di er in group-theory factors. The sum in the rst line runs over quark states only. Here df is the dimension of the SU(2)L multiplet, Tf is the Dynkin index of SU(2) (Tf = 1=2 for doublets, Tf = 2 for triplets, and Tf = 0 for singlets), Yf is the hypercharge of the multiplet, and the color factor Ncf equals 3 for quarks and 1 for leptons. The variables x and y (with y 1 y) are Feynman parameters. The quantity Tf ( p2) denotes an integral over the product of mixed-chirality components of the 5D fermion propagator with momentum p2 and 5D coordinates t = t0 with the pro le of the scalar resonance S. Explicitly, we nd in the Euclidean region2 p 2E = p2 > 0 Tf (p2E) = r 2 + L p 1 + Z 1 dt 1S(t) Tr ( gf ) 3 matrices in generation space. The KK representation of the propagator functions reads fAB(t; t0; p2E) = X n mn p2E + m2n F A(n)(t) FB (n)y(t0) ; where the normalization of the fermion pro les F A(n)(t) with A = L; R is such that [69] Z 1 dt F A(m)y(t) F A(n)(t) = mn : (3.13) (3.14) The zero modes are massless in the limit where v = 0 and hence give no contribution to the result at leading order. Note that the sum over KK modes in (3.13) is logarithmically divergent by naive power counting, since the masses of the KK modes have approximately equal spacing. Nevertheless an explicit calculation of the in nite sum leads to a nite and well-behaved answer, 2This relation holds under the assumptions that Tf (p2E) vanishes for pE ! 1, and that pE dTf =dpE vanishes for pE = 0 and pE ! 1. We have checked that these conditions are satis ed in our models. hinting at a non-trivial interplay of the pro le functions for the various KK fermions. The calculation of the propagator functions has been discussed in detail in the literature [39, 66{ 68]. It requires solving a second-order di erential equation subject to appropriate boundary conditions. We obtain fRL(t; t; p2E) = 2 1 2MKK d( )(cf ; pE; t) ; (3.15) where the overall sign is the same as that in (3.5), and the superscript \( )" refers to the boundary conditions (normal or twisted) obeyed by the fermion multiplet f . The functions d( ) are diagonal matrices, whose entries depend on the bulk mass parameters. Explicitly they are given by (omitting the matrix notation for simplicity) D1(cf ; a; t; p^E) = I cf 12 (ap^E) Icf 12 (p^Et) Icf + 12 (ap^E) I cf + 12 (p^Et) ; D2(cf ; a; t; p^E) = I cf 12 (ap^E) Icf + 12 (p^Et) Icf + 12 (ap^E) I cf 12 (p^Et) ; with p^E pE=MKK, are given in terms of modi ed Bessel functions. In our case these functions are evaluated (by analytic continuation to the time-like region) at momenta of order p2 m2S MK2 K, so that it is possible to expand these complicated expressions in a power series. This yields d( )(cf ; pE; t) = k0( )(cf ; t) + p^2E k2( )(cf ; t) + O(p^4E) ; k0(+)(cf ; t) = 1 + k2(+)(cf ; t) = 1 + 2cf 2F 2(cf ) t1+2cf 2t2 1 t 1 2cf 1 ; k0( )(cf ; t) = 1 ; + 2(1 2) F 4(cf ) (1 t1+2cf 1 4cf2 )(3 + 2cf ) 1 1 4cf2 4cf2 is the well-known zero-mode pro le [27, 28], which is exponentially small for all fermions with the exception of the right-handed top quark. We note the exact boundary values 2(1 + cf ) t3+2cf (1+2cf )2(3+2cf ) 1 + 2(t1+2cf (1 2cf )(3+2cf ) 1 t 1 2cf : F 2(cf ) = 1 1 + 2cf 1+2cf where (dropping irrelevant terms in ) (+)(cf ; ) = Analogous expressions hold for the Wilson coe cients cW W and cBB, as is evident from (3.11). In gure 2 we show the exact numerical results for ( )(cf ; ) as functions of the bulk mass parameter cf for various values of . Even for MKK as low as 2 TeV we nd that the corrections of O(m2S=MK2 K) are very small and can safely be neglected. Moreover, for all fermions other than the right-handed top quark it is an excellent approximation to neglect the exponentially small quantity F 2(cf ), while for the right-handed top quark we can replace F 2(ct) 1 + 2ct. Note that in the limit a scalar resonance localized on the IR brane, we obtain the exact result ! 1, corresponding to ( )(cf ; ) ! 0, and hence the Wilson coe cients in this limit are simply given in terms of sums over the diagonal elements of the matrices gf , meaning that they essentially count the number of 5D fermionic degrees of freedom. For simplicity, we will adopt this approximation in displaying the following results. In our numerical work we will use the correct expressions, which are obtained by replacing gt (gu)33 ! 3+ +22cctt gt. 1+ We now collect our results for the Wilson coe cients in the three versions of the RS model, adopting these approximations. For the custodial model I we nd Tr 2gQ + 1 3 2 gd + 3 2 g 1 Tr 3gQ + 6g 1 + gL + 2g 3 16ge 3MKK gt 6MKK ; 12ge ; MKK Tr 25 3 gQ + 3 gu + 10gd + 4g 1 + gL + 2ge 236ge 9MKK 4gt ; 9MKK cgg = For the custodial model II we nd instead 1, from which it follows that kn( )(cf ; 1) = ! 0 where possible) m2S 4MK2 K t 2 Using these results, it follows that (taking cgg = dt (2 + ) t1+ 1 1 1 3MKK f=q 2 7m2S 120MK2 K k2( )(cf ; t) + : : : X df Tr gf 1 + ( )(cf ; ) ; 4 4 Tr 3gQ + 6g 1 + Tr 25 3 1 2 gL 19ge ; while cgg is unchanged. In the minimal RS model, the corresponding expressions read cgg = 1 2 gd Tr gQ + Tr Tr 3 2 1 6 gQ + gQ + 1 2 4 gL 3 gu + 2ge ; MKK 1 3 gd + 1 2 gL + ge 11ge 6MKK gt 6MKK ; 26ge 9MKK 4gt 9MKK : (3.25) In the last step in each line we have assumed, for simplicity, that all diagonal entries of the matrices gf are equal to a universal value ge . While there is no particular reason why this should be true, the near equality of all cf parameters other than ct suggests that such an approximation might be reasonable. Note that any reference to the parameter has disappeared, except for the small correction term multiplying gt. In our phenomenological discussion of diboson decays in section 4 the ratio of the will play an important role. Neglecting the small 2:19 + 0:04 gt=ge in the custodial model I, Wilson coe cients cBB and cW W correction terms, we nd cBB=cW W in the minimal RS model. 3.2 Coupling to top quarks cBB=cW W 2:60+0:05 gt=ge in the custodial model II, and cBB=cW W 1:44+0:22 gt=ge The resonance S has tree-level couplings to the SM fermions, which are induced after electroweak symmetry breaking. These interactions are very similar to the couplings of a bulk Higgs to fermions studied in [39]. The largest e ects arise in the up-quark sector. We will discuss them for the case of the minimal RS model, but the nal result is the same in the custodial models. Using the zero-mode pro le functions derived in [55], we nd that the corresponding terms in the e ective Lagrangian read (neglecting terms of order m2h=MK2 K) Lferm = X S(x) u(Lm)(x) u(Rn)(x) (2 + ) m;n xn a^(mU)y F (cQ) tcQ gQ F (cQ) Z 1 0 t1+cQ + xm a^(mu)y F (cu) t1+cu 1+2cu t cu 1 + 2cu dt t1+ 1 + 2cQ 1+2cQ t cQ a^(U) n gu F (cu) tcu a^(u) + h.c. ; n (3.26) HJEP07(216)94 where xn = mn=MKK, and n = 1; 2; 3 label the three lowest-lying states u, c and t. The integrand involves a product of a Z2-even fermion pro le with a Z2-odd one, and for the SM fermions the latter one arises from the mixing of the zero modes with their KK excitations induced by electroweak symmetry breaking. As a consequence, the overlap integrals scale with the masses of the fermions involved. The 3-dimensional vectors ^a(nU) and a^(nu) describe the mixings in avor space and are normalized to unity. Their entries are strongly hierarchical, with the largest entry at position n. The most important interaction involves the coupling of S to a pair of top quarks. For the Wilson coe cient ctt in (3.3), we nd to a good approximation ctt 1 1 MKK MKK gQ 1 (gQ)33 + F 2(cQ) 3 + + 2cQ 2 + 3 + + 2ct + gu 1 gt : F 2(cu) 3 + + 2cu 33 (3.27) 4 Phenomenology of the diphoton resonance In this section we express the production cross section and the rates for the decays of the resonance S into SM particles in terms of the Wilson coe cients in the e ective Lagrangians (3.1) and (3.3). These results are general and can be applied to any model in which the couplings of S to SM particles are induced by the exchange of some heavy new particles. We will then apply these general results to the case of the RS models studied in At Born level, the cross section for the production of the resonance S in gluon fusion at the previous section. 4.1 General discussion the LHC is given by where Z 1 dx y x (pp ! S) = 64 s 2( ) m2S cg2g( ) ffgg(m2S=s; ) ; s ffgg(y; ) = fg=p(x; ) fg=p(y=x; ) (4.1) (4.2) is the gluon-gluon luminosity function. The factorization and renormalization scales should be chosen of order mS. It is well known from the analogous Higgs production cross p s 8 TeV 13 TeV MSTW2008 [73] NNPDF30 [74] PDF4LHC15 [75] (cgg=TeV)2, for di erent sets of parton distribution functions. The quoted errors are estimated from scale variations. section that higher-order QCD corrections have an enormous impact on the cross section. We have calculated these corrections up to NNLO and including resummation e ects using an adaption of the public code CuTe [70] developed in [71, 72]. Table 1 shows our results for the ratio (pp ! S)=cg2g( ) for the default scale choice = mS and di erent sets of parton distribution functions (PDFs). Taking the MSTW2008 PDFs as a reference, we obtain N8TNeLVO(pp ! S) = (44:9 +12::64 +12::87) fb N13NTLeOV(pp ! S) = (203 +160+56) fb cgg(mS) 2 TeV cgg(mS) 2 TeV ; ; where the errors refer to scale variations and the variations of the PDFs. The program CuTe also predicts the pT distribution of the produced S bosons, and we nd that this distribution peaks around 22 GeV. The higher-order corrections enhance the cross section by more than a factor 2 compared with the Born cross section in (4.1). The Wilson coe cients in (3.1) also contribute to possible decays of the new resonance into the electroweak diboson nal states , W W , ZZ and Z , and into hadronic nal states such as gg and tt. The partial decay rates for the former channels are (4.3) (4.4) (S ! ) = 624m33S (cW W + cBB)2 ; where xW;Z = m2W;Z =m2S. While the Wilson coe cients are evaluated at the scale the gauge couplings and Weinberg angle are evaluated at the scale appropriate for the nal-state bosons. We use (mZ ) = 1=127:94 for Z and W bosons, = 1=137:04 for the photon, and s2 W = 0:2313 for the weak mixing angle. Apart from known quantities, the two Wilson coe cients cW W and cBB determine these four rates entirely. It follows that any ratio of two rates is a function of the ratio cBB=cW W , which in turn is characteristic for the model under investigation. This is illustrated in gure 3. Note that the S ! Z decay rate in particular strongly depends on the value of cBB=cW W . In the RS models considered in the previous section this decay mode turns out to be strongly suppressed. the S ! rate as functions of cBB=cW W . Assuming 2:11 < cBB=cW W < 2:23 in custodial model I, 2:50 < cBB=cW W < 2:65 in custodial model II, and 1:00 < cBB=cW W < 1:66 in the minimal RS model. The partial rates for decays into hadronic nal states are where all running quantities should be evaluated at mS. In the second case mt( ) is the running top-quark mass (we use mt(mS) = 146:8 GeV), whereas the mass ratio xt = mt2=m2S entering the phase-space factor involves the top-quark pole mass mt = 173:34 GeV. In many scenarios the dijet decay mode S ! gg is the dominant decay channel and hence enters in the calculation of the branching fractions for all other decay modes. It is therefore important to calculate this partial rate as accurately as possible. Using existing calculations of the Higgs-boson decay rate (h ! gg) up to O( s5) in the heavy top-quark limit [78, 79] it is possible to derive an exact expression for the S ! gg decay rate to the same accuracy. This is discussed in detail in appendix B. We nd that the impact of radiative corrections is signi cantly smaller than in the Higgs case, and that the perturbative series at exhibits very good convergence. We obtain KgNg3LO(mS) 1:348. = mS 4.2 S-boson phenomenology in RS models We are now ready to explore the phenomenological consequences of our calculations. The challenge is to reproduce the observed diphoton rate in (1.1), while at the same time respecting existing bounds on dijet, diboson and tt resonance searches3 from Run 1 of the 3Note that the reported bound for the S ! tt channel might in fact be considerably weaker due to interference e ects not considered in the experimental analyses [80]. The potential impact of these e ects jj < 2:5 pb [85] diboson and tt resonance searches performed in Run 1 of the LHC ( s = 8 TeV). p HJEP07(216)94 reproduced at 1 (light blue) and 2 (dark blue). The black dashed line corresponds to the central value shown in (1.1). The two upper panels refer to the custodial models I and II, while the lower left panel refers to the minimal RS model. Regions excluded by bounds from resonance searches in Run 1 data (at 95% CL) are shaded gray with boundaries drawn in red (dijets), purple (tt), blue (W W ), orange (ZZ) and green (Z ). We use = 1 and ct = 0:4. The lower right panel shows the variation of the central t values with the localization parameter of the scalar pro le, for = 1 ing bounds at p LHC, which are collected in table 2. Assuming that the new resonance S is predominantly produced via gluon fusion, as is the case in the RS models we study, the corresponds = 13 TeV are obtained by multiplying these numbers with the boost factor 4.52 corresponding to the ratio of the production cross sections in (4.3). In gure 4, we present plots in the MKK=ge gt=ge plane showing the 1 and 2 t regions to the diphoton excess in light and dark blue. The central t values are shown by the dashed black line. Regions excluded by bounds from resonance searches in data collected during the 8 TeV run of the LHC are shaded gray with a boundary drawn in red (dijet searches), purple (tt searches), blue (W W searches), orange (ZZ searches) and green (Z searches). Throughout we use = 1 and ct = 0:4 for the parameters entering the contribution from the SU(2)L-singlet top quark. The lower right plot shows the variation of the central t result for di erent values of the localization parameter , namely = 1 (red). It is apparent that there is only a minor dependence on this parameter. Changing ct does not substantially alter the t either. We observe that in the two versions of the RS model with a custodial symmetry the diphoton signal can be reproduced over a wide range of parameters without any ne tuning and without violating any of the bounds from other searches. Depending on the choice of gt=ge one obtains values for MKK=ge in the range between 2 and 8 TeV. If the KK scale MKK is close to the lower bound (2.3) allowed by electroweak precision tests, this requires couplings ge in the range 0.25 to 1, which are well inside the perturbative region. In this scenario some of the low-lying KK excitations could have masses around 4 TeV, in which case they might be discovered in Run 2 of the LHC. If the KK mass scale is signi cantly higher a direct discovery of KK excitations will not be possible at the LHC. Nevertheless, even for MKK 5 TeV (implying KK resonance masses near 10 TeV) the diphoton signal can be explained with a modest coupling ge 1. In the minimal RS model the parameter space in which the diphoton signal can be explained is more constrained. We nd values in the range MKK=ge 0:4 1, which for a KK scale as high as the bound (2.2) enforced by electroweak precision tests requires large couplings ge 5 12, close to the perturbativity limit. One also needs to require that the ratio gt=ge is negative so as to avoid the strong constraint from tt resonance searches (see, however, footnote 3). We nd it useful to de ne a benchmark point for each model and study the individual branching fractions for the various S decay modes for these points. Speci cally, we choose the points indicated by the orange stars in gure 4, for which (with = 1 and ct = 0:4) Minimal model : Custodial model I : Custodial model II : MKK=ge = 0:7 TeV; MKK=ge = 4:0 TeV; MKK=ge = 3:0 TeV; gt=ge = gt=ge = gt=ge = 0 : 1:5 ; 0:5 ; (4.6) In the upper portion of table 3 we collect the branching ratios into the various nal states for these benchmark models. Note that the S ! tt decay rate is only calculated at lowest order in QCD and hence a icted with some uncertainty. The S ! tt branching ratio has rst been pointed out in [81] and has recently been reemphasized in [82, 83]. Minimal Custodial I Custodial II Minimal Custodial I Custodial II Minimal gg and for the benchmark parameter points de ned in (4.6). In the center and lower portions of the table we show the branching ratios in the presence of a small portal coupling 1 = 0:02 and 0.04, respectively, see section 5. The small contributions to the S ! hh and S ! tth branching ratios resulting from the portal coupling 2 in (5.2) and (6.2) have been set to 0. is rather sensitive to the choice of gt=ge , while the remaining branching fractions only mildly depend on this parameter. The three-body decay mode S ! tth will be discussed in section 6. In the last column we show the total decay width of S, which is very small in our models. Given the existing Run 1 dijet bound shown in table 2, it is impossible to obtain a total width exceeding a few GeV in any model in which the decay S ! gg has a signi cant branching ratio. This is below the experimental resolution of approximately 10 GeV on m . In our framework we can therefore not accommodate the best t value tot 45 GeV reported by ATLAS [1]. Rather, the numbers shown in the table correspond to values tot=mS (1:1 4:3) 10 4. We recall, however, that the large width tot 0:06 mS is only slightly preferred by the ATLAS analysis, leading to an improvement of the t by 0:3 over a narrow-width scenario. An independent analysis in [84] concludes that the large-width scenario is disfavored by a combination of the ATLAS and CMS analyses of the 13 TeV data, and only slightly preferred taking into account the 8 TeV data, because it is easier to absorb the signal of a broad resonance in the background model (the local signi cance changes at most by 0:5 between these options).4 We observe that there are rather striking di erences between the three RS models considered here, even though any of the three benchmark points reproduces the diphoton signal and is consistent with all other bounds. In particular, the S ! W W , ZZ and tt branching ratios vary signi cantly from one model to another, indicating that future measurements of these modes will provide very interesting clues about the underlying model. Note also that in all cases we nd that the S ! Z branching fraction is very 4After the submission of our paper, CMS has reported a combined analysis of the 8 TeV and 13 TeV data using three templates with tot=mS = 1:4 10 4, 1:2 10 2 and 5:6 10 2 [CMS Collaboration, CMSPAS-EXO-16-018]. The best t is obtained for the narrow-width assumption with tot=mS = 1:4 10 4. A value of this order is indeed predicted in our models. S S h h t t¯ S S h S h W +, Z W −, Z S h h h small, so that it will be challenging to observe this mode in our scenarios. On the other hand, not seeing the S ! Z signal would be as important a nding as seeing it. 5 Impact of Higgs portal couplings The most general renormalizable Lagrangian includes besides the operators in (3.1) potential Higgs portal interactions (see e.g. [ 5, 8, 91, 92 ]) Le = 1 mS S j j 2 2 2 S2 j j2 3 2 1 mS S (v + h)2 4 2 S2 (v + h)2 : (5.1) In RS models the couplings 1 and 2 can be suppressed at tree level by localizing the Higgs sector on or near the IR brane, where the Z2-odd bulk eld for the resonance S vanishes. However, starting at one-loop order the portal couplings will be induced through diagrams analogous to that shown in gure 1, but with the external gauge elds replaced by Higgs bosons. Note that this diagram exists even if only the \right-chirality" couplings of the Higgs bosons are included. Also, below the electroweak scale the e ective Lagrangian (3.1) gives rise to a contribution to the portal coupling 1 proportional to ctt from top-quark loop graphs. RS models thus provide a rationale for why the portal interactions should be suppressed (by small overlap integrals or a loop factor), but it would be unjusti ed to omit them altogether. 1 After electroweak symmetry breaking, the rst portal interaction gives rise to three interesting (and potentially dangerous) e ects. First, the presence of a tadpole for the eld S requires that we de ne the physical eld by the shift S ! S this shift in the Lagrangian (3.1) generates corrections to the SM Yukawa couplings and wave-function corrections to the gauge elds. Otherwise these corrections do not have observable e ects. Second, there is a tree-level decay S ! hh, generated by the upper two diagrams shown in gure 5, whose decay rate (at leading order in the portal coupling 1) ( 1v2)=(2mS). Performing is given by (S ! hh) = 32 mS 21 1 + 3m2h m2S Third, there is a mass mixing between the scalar resonance S and the Higgs boson. At leading order in the portal coupling 1 this gives rise to tree-level contributions to the (S ! W W ) = 0:5 for the custodial model I and gt=ge = 0 for the custodial model II as in (4.6), = 1 and ct = 0:4. The constraints from Run 1 resonance searches refer to W W (blue), ZZ (orange), Z (green), tt (purple), dijets (red) and hh (brown). See text for further explanations. S ! tt, W W , ZZ decay amplitudes induced by the lower two diagrams in gure 5. The corrected expression for the S ! tt decay width is then given by xh 2 : The corrected expression for the S ! W W decay rate reads mS p 16 1 + 1 1 xh 4xW 2 m2S c2W W 2 s 2 W 2 (1 (1 4xW +12x2W ) 6mS cW W 4xW + 6x2W ) s 2W 1 1 xh xW (1 2xW ) ; (5.3) (5.4) and similarly for S ! ZZ. The mixing between the elds S and h also has an impact on the properties of the Higgs boson. The physical Higgs boson is given by the combination sin S), where sin 2 = 2 1mSv=(m2S m2h) 0:67 1, with mS and mh refering to the physical masses after the eld rede nitions. Existing measurements of the Higgs branching fractions constrain cos to be larger than 0.86 at 95% CL [93, 94], which implies the rather weak constraint j 1j < 1:3 in our model. The S ! hh decay channel and the admixture of a tree-level coupling to W W and ZZ induced by the mixing with the Higgs boson can have a signi cant impact on our phenomenological analysis. In gure 6, we present the 1 and 2 t regions to the diphoton excess in the custodial RS models in the presence of the portal coupling 1 . These plots are analogous to those shown in gure 4, except that we have xed the values of gt=ge to those of the benchmark points in (4.6). The meaning of the colors of the various curves S h h t h t¯ t t¯ S S h t h t¯ t h t¯ h h S t t¯ t t¯ xh 1 xh : is the same as before. In all cases the LHC Run 1 bound on the S ! ZZ rate provides the strongest constraint, excluding portal couplings j 1j & 0:06 (0.07) in the custodial RS model I (II). The impact of a small portal coupling 1 = 0:02 or 0.04 on the various branching ratios is shown in the central and lower portions of table 3. Even for such a small coupling the S ! W W , ZZ branching ratios can be signi cantly enhanced, and a sizable S ! hh branching fraction can open up. 6 Three-body decay S ! tth The e ective Lagrangian (3.3) contains a tree-level coupling of the new resonance S to a tt pair and a Higgs boson. It is interesting to ask if this coupling might explain the enhanced tth production rates reported by ATLAS and CMS [95, 96]. At tree level, the three-body decay S ! tth is mediated by the diagrams shown in gure 7. Note that both portal couplings introduced in (5.1) contribute here. Introducing the dimensionless variables z = mt2t=m2S and w = mt2h=m2S, we obtain the Dalitz distribution d2 (S ! tth) where B = 2xt 1 1 xh 1 1 mS ctt z w z + xt + xh w 1 + 1 3xh + xh 1 2xt xt v 2 m2S 1 1 + 1 w z + xt + xh w xt mS ctt 1 (1 pxh)2, where The phase space for the variables w and z is wmin(z) wmax(z) and 4xt z wmax=min(z) = (1 xh)2 4z 1 4z 4xt) p(1 xh)2 4zxh 2 : pz(z 1 (6.1) (6.2) (6.3) expect that the rate for pp ! S ! tth at p Our results for the S ! tth branching ratio obtained by integrating over the Dalitz plot are shown in the penultimate column in table 3. This branching ratio is typically two orders of magnitude smaller than the S ! tt branching fraction. This will be true in any model in which the decays of the new resonance can be described in terms of local operators. Given the existing upper bound from tt resonance searches in Run 1 shown in table 2, we would s = 8 TeV cannot be larger than about 12 fb. This falls short by far to explain the enhanced Higgs production rate in association with tt. In our speci c models, the predicted tth production rates do not exceed 8 fb. 7 The recent hint of a 750 GeV diphoton excess, observed in the data from Run 2 of the LHC by both ATLAS and CMS, could be the rst direct manifestation of physics beyond the SM. If veri ed by future analyses, this excess will most likely have been created by the decay of a new, scalar boson with a mass of 750 GeV produced either in gluon fusion or in bb-initiated scattering. Many possible interpretations of such a boson have been proposed. Remarkably, the simple addition of a single new scalar to the SM as well as several well motivated UV completions | including the Minimal Supersymmetric SM | have already been excluded. In this paper, we have shown that the diphoton signal can be reproduced from a very straightforward extension of the popular warped extra-dimension models introduced by Randall and Sundrum. In these models, our spacetime is enlarged by a warped extra dimension in order to give natural explanations of the gauge and avor hierarchy problems of the SM. We have identi ed the diphoton resonance with the lightest excitation of an additional bulk scalar eld. Such a scalar might serve as the localizer eld providing a dynamical generation of the bulk mass parameters of the 5D fermions and is thus a natural ingredient of RS models with bulk matter elds. In addition to a model with a minimal particle content, we have discussed two implementations of RS models providing a custodial protection mechanism for electroweak precision observables. These models feature a larger number of fundamental 5D fermions compared with the SM. As a result, the gluon-fusion production process and the decay into two photons are enhanced by the large multiplicity of vector-like KK fermion states propagating in the loop. By summing up the contribution from the in nite towers of KK fermions into 5D propagator functions we were able to derive remarkably simple analytic expressions for the e ective couplings to gluons or photons, which to very good approximation simply count the number of fermionic degrees of freedom, weighted by group-theory factors. For the custodial RS models, we have found that with O(1) couplings of the resonance S to fermions and KK masses in the multi-TeV range one can explain the diphoton signal without violating any of the Run 1 bounds from resonance searches in various diboson and dijet channels. Useful side products of our analysis, which can be adapted to any model for the diphoton resonance that at the scale = mS can be mapped onto an e ective Lagrangian with local interactions, are the calculation of the gluon-fusion production cross section (pp ! S) at NNLO in QCD, an expression for the inclusive S ! gg decay rate at N3LO, a study of the S ! tth three-body decay mode, and a phenomenological analysis of portal couplings connecting S with the Higgs eld. We conclude that our simple extension of the established RS models can deliver one of the most elegant and minimal explanations for the observed diphoton excess. Assuming that the resonance will survive future veri cations, it could hint at the existence of a warped extra dimension and open the door to detailed studies of the parameter space of RS models. Acknowledgments HJEP07(216)94 (A.1) (A.2) (A.3) We are grateful to Joachim Kopp for useful discussions which have initiated this project, to Daniel Wilhelm for adapting the CuTe program to the calculation of the pp ! S cross section, and to Ulrich Haisch for drawing our attention to the possible impact of interference e ects in S ! tt resonance searches. M.B. acknowledges the support of the Alexander von Humboldt Foundation. The work of C.H. and M.N. was supported by the Advanced Grant EFT4LHC of the European Research Council (ERC), the DFG Cluster of Excellence Precision Physics, Fundamental Interactions and Structure of Matter (PRISMA EXC 1098), grant 05H12UME of the German Federal Ministry for Education and Research (BMBF) and the DFG Graduate School Symmetry Breaking in Fundamental Interactions (GRK 1581). A RG evolution of the Wilson coe cients cgg and ctt The mixing among the Wilson coe cients ctt and cgg in the e ective Lagrangians (3.1) and (3.3) is described by the RG equations d d cgg( ) = gg( s) cgg( ) ; ctt( ) = tg( s) cgg( ) ; d d where s = s( ), and gg is given by the exact expression [59, 60] gg( s) = s2 dd s ( s) 2 s in terms of the -function ( s) = d s( )=d . The leading contribution to tg( s) has been obtained rst in [61]. The exact solutions to the evolution equations are cgg( ) = ctt( ) = ctt( 0) ( s( ))= s2( ) ( s( 0))= s2( 0) cgg( 0) ; 2 ( 0) s ( s( 0)) cgg( 0) Z s( ) s( 0) d tg( ) 2 : s 2 The perturbative expansions of the anomalous dimension and -function read ( s) = 2 s 1 2 where 1 = 24CF [61]. 0 + 1 4 s + : : : ; tg( s) = 1 4 + : : : ; (A.4) ! gg decay rate at N3LO in QCD An analytic expression for the inclusive S ! gg decay rate at N3LO in QCD perturbation theory can be derived using existing calculations of the Higgs-boson decay rate (h ! gg) up to O( s5) obtained in [78, 79]. Taking the heavy top-quark limit and dividing the result by the three-loop expression for the matching coe cient Ct(mt; ) obtained in [62, 63], we nd with and (for Nc = 3 colors and nf = 6 light avors) Lh ; 49 2 16 2821025 where Lh = ln( 2=m2S). Numerically, we turbative expansion KgNg3LO = 1 + 0:32768 + 0:03325 excellent convergence. nd for the K-factor at = mS the per0:01290 1:348, which exhibits Open Access. 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Martin Bauer, Clara Hörner, Matthias Neubert. Diphoton resonance from a warped extra dimension, Journal of High Energy Physics, 2016, 94, DOI: 10.1007/JHEP07(2016)094