We consider simplified dark matter models where a dark matter candidate couples to the standard model (SM) particles via an s-channel spin-2 mediator, and study constraints on the model parameter space from the current LHC data. Our focus lies on the complementarity among different searches, in particular monojet and multijet plus missing-energy searches and resonance searches. For universal couplings of the mediator to SM particles, missing-energy searches can give stronger constraints than WW, ZZ, dijet, dihiggs, tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document}, bb¯\documentclass[12pt]{minimal}
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\begin{document}$$b\bar{b}$$\end{document} resonance searches in the low-mass region and/or when the coupling of the mediator to dark matter is much larger than its couplings to SM particles. The strongest constraints, however, come from diphoton and dilepton resonance searches. Only if these modes are suppressed, missing-energy searches can be competitive in constraining dark matter models with a spin-2 mediator.
volume-issue-count12issue-article-count87issue-toc-levels0issue-typeRegularissue-online-date-year2017issue-online-date-month6issue-online-date-day8issue-pricelist-year2017issue-copyright-holderSIF and Springer-Verlag Berlin Heidelbergissue-copyright-year2017article-contains-esmNoarticle-numbering-styleContentOnlyarticle-toc-levels0article-registration-date-year2017article-registration-date-month4article-registration-date-day29article-grants-typeOpenChoicemetadata-grantOpenAccessabstract-grantOpenAccessbodypdf-grantOpenAccessbodyhtml-grantOpenAccessbibliography-grantOpenAccessesm-grantOpenAccessIntroduction
Convincing astrophysical and cosmological observations for the existence of dark matter (DM) provide us one of the strong motivations to consider physics beyond the standard model (SM). The search for DM is thus one of the main pillars of the LHC physics program.
As the nature of DM is known so little, a so-called simplified-model approach [1] has been widely adopted, and concrete simplified DM models have recently been proposed by the LHC DM working group to conduct the systematic DM searches at the LHC Run-II [2]. Following the proposal, the Run-I data as well as the early Run-II data have already been analysed to constrain simplified DM models with s-channel spin-1 and spin-0 mediators; see e.g. [3–12]. On the other hand, the model with a spin-2 mediator [13, 14] has not been fully explored for the LHC yet—it is one of the next-generation simplified DM models [15].
In this article, we consider simplified DM models where a DM candidate couples to the SM particles via an s-channel spin-2 mediator, and study constraints on the model parameter space from searches in final states with and without missing energy in the current LHC data. This work follows the DMsimp framework [16–18], which provides the DM model files for event generators such as MadGraph5_aMC@NLO [19] as well as for DM tools such as micrOMEGAs [20–22] and MadDM [23, 24]. The same framework was used previously to study the cases of s-channel spin-1 and spin-0 mediators.
We note that, to keep the analysis of the LHC constraints fully general, we do not impose any astrophysical constraints like relic density or (in)direct detection limits on the DM candidate, as these partly depend on astrophysical assumptions. Moreover, in a full model, the DM may couple to other new particles that are irrelevant for the collider phenomenology discussed here. We refer the reader to [13, 14] for the astrophysical constraints, and to [25] for a discussion of spectral features in the indirect detection.
The article is organised as follows. The simplified model is presented in Sect. 2, and the production and decays of the spin-2 mediator in Sect. 3. The re-interpretation of the LHC results is discussed in Sect. 4. Section 5 contains a summary and conclusions. Supplemental material for recasting is provided in the appendix.
Model
Gravity-mediated DM was proposed in [13, 14], where the dark sector communicates with the SM sector through a new spin-0 particle (radion) and spin-2 particles (Kaluza–Klein (KK) gravitons) in warped extra-dimension models as well as in the dual composite picture.
In this work, following the approach of simplified DM models, we consider DM particles which interact with the SM particles via an s-channel spin-2 mediator. The interaction Lagrangian of a spin-2 mediator (Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document}) with DM (X) is given by [13]LXY2=-1ΛgXTTμνXY2μν,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal{L}_{X}^{Y_2} = -\frac{1}{\varLambda } g^{T}_{X}\,T^X_{\mu \nu } Y_2^{\mu \nu }, \end{aligned}$$\end{document}where Λ\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda $$\end{document} is the scale parameter of the theory, gXT\documentclass[12pt]{minimal}
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\begin{document}$$g^T_X$$\end{document} is the coupling parameter, and TμνX\documentclass[12pt]{minimal}
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\begin{document}$$T_{\mu \nu }^X$$\end{document} is the energy-momentum tensor of a DM field. Here, we consider three types of DM independently; a real scalar (XR\documentclass[12pt]{minimal}
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\begin{document}$$X_R$$\end{document}), a Dirac fermion (XD\documentclass[12pt]{minimal}
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\begin{document}$$X_D$$\end{document}), and a vector (XV\documentclass[12pt]{minimal}
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\begin{document}$$X_V$$\end{document}). The interaction with SM particles is obtained byLSMY2=-1Λ∑igiTTμνiY2μν,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \mathcal{L}_\mathrm{SM}^{Y_2} = -\frac{1}{\varLambda } \sum _i g^{T}_{i}\, T^i_{\mu \nu } Y_2^{\mu \nu }, \end{aligned}$$\end{document}where i denotes each SM field, i.e. the Higgs doublet (H), quarks (q), leptons (ℓ\documentclass[12pt]{minimal}
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\begin{document}$$\ell $$\end{document}), and SU(3)C\documentclass[12pt]{minimal}
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\begin{document}$$SU(3)_C$$\end{document}, SU(2)L\documentclass[12pt]{minimal}
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\begin{document}$$SU(2)_L$$\end{document} and U(1)Y\documentclass[12pt]{minimal}
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\begin{document}$$U(1)_Y$$\end{document} gauge bosons (g, W, B). Following [26, 27] we introduce the phenomenological coupling parametersgiT={gHT,gqT,gℓT,ggT,gWT,gBT}\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g^T_i=\{g^T_H,\,g^T_q,\,g^T_\ell ,\,g^T_g,\,g^T_W,\,g^T_B\} \end{aligned}$$\end{document}without assuming any UV model.1 The energy-momentum tensors of the DM areTμνXR=-12gμν(∂ρXR∂ρXR-mX2XR2)+∂μXR∂νXR,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_{\mu \nu }^{X_R}= & {} -\frac{1}{2}g_{\mu \nu }( \partial _{\rho }X_R\partial ^{\rho }X_R - m^2_{X}X^2_R) \nonumber \\&+\,\partial _{\mu }X_R\partial _{\nu }X_R,\end{aligned}$$\end{document}TμνXD=-gμν(X¯Diγρ∂ρXD-mXX¯DXD)+12gμν∂ρ(X¯DiγρXD)+12X¯Di(γμ∂ν+γν∂μ)XD-14∂μ(X¯DiγνXD)-14∂ν(X¯DiγμXD),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_{\mu \nu }^{X_D}= & {} -g_{\mu \nu }( \overline{X}_Di\gamma _{\rho }\partial ^{\rho }X_D - m_{X}\overline{X}_DX_D) \nonumber \\&+\,\frac{1}{2}g_{\mu \nu }\partial _{\rho }(\overline{X}_Di\gamma ^{\rho }X_D) \nonumber \\&+\,\frac{1}{2}\overline{X}_D i(\gamma _{\mu }\partial _{\nu }+\gamma _{\nu }\partial _{\mu }) X_D \nonumber \\&-\,\frac{1}{4}\partial _{\mu }(\overline{X}_Di\gamma _{\nu }X_D) -\frac{1}{4}\partial _{\nu }(\overline{X}_Di\gamma _{\mu }X_D),\end{aligned}$$\end{document}TμνXV=-gμν-14FρσFρσ+mX22XVρXVρ+FμρFνρ+mX2XVμXVν,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} T_{\mu \nu }^{X_V}= & {} -g_{\mu \nu }\left( -\frac{1}{4}F_{\rho \sigma } F^{\rho \sigma } + \frac{m_X^2}{2} X^{}_{V\rho }X_{V}^{\rho }\right) \nonumber \\&+\,F_{\mu \rho }F^{\rho }_{\nu } +m^2_{X}X^{}_{V\mu }X^{}_{V\nu }, \end{aligned}$$\end{document}where Fμν\documentclass[12pt]{minimal}
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\begin{document}$$F_{\mu \nu }$$\end{document} is the field strength tensor. Those of the SM fields are similar; see e.g. [28] for the explicit formulae.
Complying with the simplified-model idea, it is instructive to consider universal couplings between the spin-2 mediator and the SM particles:gSM≡gHT=gqT=gℓT=ggT=gWT=gBT.\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} g_\mathrm{SM}\equiv g^T_H=g^T_q=g^T_\ell =g^T_g=g^T_W=g^T_B. \end{aligned}$$\end{document}With this simplification, the model has only four independent parameters, two masses and two couplings:{mX,mY,gX/Λ,gSM/Λ},\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \{m_X,\,m_Y,\,g_X/\varLambda ,\,g_\mathrm{SM}/\varLambda \}, \end{aligned}$$\end{document}where we dropped the superscript T for simplicity. Such a universal coupling to SM particles is realised, e.g., in the original Randall–Sundrum (RS) model of localised gravity [29]. The parameters are related asmY/Λ=x1k/M¯Pl,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} m_Y/\varLambda =x_1\, k/\overline{M}_\mathrm{Pl}, \end{aligned}$$\end{document}where x1=3.83\documentclass[12pt]{minimal}
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\begin{document}$$x_1=3.83$$\end{document} is the first root of the Bessel function of the first kind, k is the curvature of the warped extra dimension, and M¯Pl=2.4×1018GeV\documentclass[12pt]{minimal}
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\begin{document}$$\overline{M}_\mathrm{Pl}=2.4\times 10^{18}\mathrm{\ GeV}$$\end{document} is the reduced four-dimensional Planck scale. On the other hand, in the so-called bulk RS model [30, 31], where the SM particles also propagate in the extra dimension, giT\documentclass[12pt]{minimal}
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\begin{document}$$g^T_i$$\end{document} can take different values depending on the setup.
In [28], the SM sector of the above model was implemented in FeynRules/NloCT [32, 33] (based on [34–36]), and the Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document} production and decay rates at next-to-leading order (NLO) QCD accuracy were presented. In this work, we include the three DM species (XR\documentclass[12pt]{minimal}
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\begin{document}$$X_V$$\end{document}) with the corresponding interactions, and add the model into the DMsimp framework [37] as the simplified DM model with a spin-2 mediator.
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\begin{document}$$\varGamma _Y/m_Y$$\end{document}, (upper panel) and mediator branching ratios (lower panel) as a function of the mediator mass mY\documentclass[12pt]{minimal}
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\begin{document}$$g_\mathrm{SM}=1$$\end{document}, where we assume a negligible branching ratio to the dark sector
Phenomenology at the LHCDecay of the spin-2 mediator
Regarding LHC phenomenology, let us begin by discussing the spin-2 mediator decays. The partial widths for the decays into a pair of spin-0 (S=XR,h\documentclass[12pt]{minimal}
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\begin{document}$$S=X_R,h$$\end{document}), spin-1/2 (F=XD,q,ℓ\documentclass[12pt]{minimal}
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\begin{document}$$V=X_V,g,\gamma ,Z,W$$\end{document}) DM or SM particles are given byΓS=gS2mY3960πΛ2βS5,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma _{S}= & {} \frac{g^2_S m^3_Y}{960\pi \varLambda ^2}\,\beta _S^{5}, \end{aligned}$$\end{document}ΓF=gF2NνNCFmY3160πΛ2βF31+83rF,\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma _{F}= & {} \frac{g^2_F N_\nu N^F_C m^3_Y}{160\pi \varLambda ^2}\,\beta _F^{3}\, \left( 1+\frac{8}{3}r_F\right) , \end{aligned}$$\end{document}ΓV=gV2NsNCVmY340πΛ2βVf(rV),\documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} \varGamma _{V}= & {} \frac{g^2_V N_s N^V_C m^3_Y}{40\pi \varLambda ^2}\,\beta _V\, f(r_V), \end{aligned}$$\end{document}where βi=1-4ri\documentclass[12pt]{minimal}
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\begin{document}$$r_i=m^2_i/m^2_Y$$\end{document}, gγ=gBcos2θW+gWsin2θW\documentclass[12pt]{minimal}
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\begin{document}$$g_\gamma =g_B\cos ^2\theta _W+g_W\sin ^2\theta _W$$\end{document} and gZ=gBsin2θW+gWcos2θW\documentclass[12pt]{minimal}
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\begin{document}$$f(r_V)= 1 + \frac{1}{12}\kappa ^2_H -r_V(3 -\frac{20}{3}\kappa _H -\kappa ^2_H) +r^2_V(6 -\frac{20}{3} \kappa _H +\frac{14}{3} \kappa _H^2)$$\end{document} with κH=gH/gV\documentclass[12pt]{minimal}
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\begin{document}$$\kappa _H=g_H/g_V$$\end{document}. For gluons and photons, κH=0\documentclass[12pt]{minimal}
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\begin{document}$$g_{Z\gamma }^2=[(g_W-g_B)\cos \theta _W\sin \theta _W]^2$$\end{document}. We see that, due to the different overall prefactors, the partial widths become larger in order of scalar, fermion, vector DM. Moreover, the different powers (5, 3, 1) of the velocity factor βi\documentclass[12pt]{minimal}
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\begin{document}$$\beta _i$$\end{document} indicate that the decay proceeds mainly via a D, P, and S wave for the scalar, fermion, and vector case, respectively.
Figure 1 shows the Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document} total width scaled by the mass, ΓY/mY\documentclass[12pt]{minimal}
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\begin{document}$$\varGamma _Y/m_Y$$\end{document}, and the decay branching ratios for the case that only decays into SM particles are allowed. MadWidth [38] provides the partial decay rates numerically for each parameter point. In Table 1 we provide the explicit values for a few representative mass points. We see that, for a universal coupling gSM\documentclass[12pt]{minimal}
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Production of the spin-2 mediator
Turning to the production modes, the potentially interesting channels are inclusive Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document} production cross sections at NLO QCD accuracy for pp collisions at 13 TeV are depicted in Fig. 3 as a function of the mediator mass.4 We employ MadGraph5_aMC@NLO [19] to calculate the cross sections and generate events with the LO/NLO NNPDF2.3 [40]. The factorisation and renormalisation scales are taken at the sum of the transverse masses of the final states as a dynamical scale choice. In our simplified model, the cross sections depend solely on gSM/Λ\documentclass[12pt]{minimal}
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\begin{document}$$Y_2+\mathrm{jet}$$\end{document} production. This is also the reason that the process has a huge K factor especially in the low-mass region [28].5
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Constraints from current LHC dataSearches with missing energy
The ATLAS and CMS experiments have been searching for new physics in a large variety of final states. As mentioned above, in the context of DM searches, the monojet signature is regarded as particularly interesting. In practice, at 13 TeV, the monojet analyses require one hard jet recoiling against , but allow for additional jets from QCD radiation. Therefore one can expect that multijet+ searches are also relevant [41, 42].
To work out the current constraints on the spin-2 mediator DM model from these searches, we consider the following early Run-II analyses:
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\begin{document}$$|\eta |<2.8$$\end{document} are allowed. Several inclusive and exclusive signal regions (SRs) are considered with increasing requirements from 250 to 700 GeV. The multijet+ analysis [43] is designed to search for squarks and gluinos in supersymmetric models, where neutralinos lead to missing energy. Several SRs are characterised by minimum jet multiplicity from two to six; is required for all SRs, while different thresholds are applied on jet momenta and on the azimuthal separation between jets and .
To reinterpret the above analyses in the context of our spin-2 mediator simplified DM model, we use CheckMATE2 [44], which is a public recasting tool providing confidence limits from simulated signal events and includes a number of 13 TeV analyses. We generate hadron-level signal samples by using the tree-level matrix-element plus parton-shower (ME+PS) merging procedure. In practice, we make use of the shower-kT\documentclass[12pt]{minimal}
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\begin{document}$$k_T$$\end{document} scheme [45], implemented in MadGraph5_aMC@NLO [19] with Pythia6 [46], and generate signal events with parton multiplicity from one to two partons. We impose and set Qcut=200\documentclass[12pt]{minimal}
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\begin{document}$$pp\rightarrow Y_2j$$\end{document} NLO cross sections shown in Fig. 3. (Note, however, that NLO corrections may also affect the shapes of the kinematic distributions, as shown for the spin-1 and spin-0 cases in [17]; a detailed study of this aspect will be reported elsewhere.)
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It turns out that, for an on-shell mediator of given mass, the selection efficiencies are independent of the mass and spin of the invisible decay products. Moreover, contributions from off-shell production are negligible for the scenarios considered here. The efficiencies can thus be evaluated as a function of the mediator mass only; see also Appendix A.1. In the following, we normalise the number of events with NLO cross sections, shown in Fig. 3, and the total branching ratio into invisible final states (DM and neutrino). We note that for a given mediator mass the leading jet for the spin-2 mediator case is harder and more forward than that for the spin-1 case. This is partly because the spin-2 mediator with a parton is produced not only through the qq¯\documentclass[12pt]{minimal}
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We note that we compared the CheckMATE results with those obtained by the equivalent analysis implementations in MadAnalysis 5 [47, 48] (recast codes [49, 50]) and Rivet 2.5 [51] for a couple of representative mass choices and found agreement at the level of 20% within all three tools.
The monophoton (as well as mono-Z / W) signature could also be interesting to explore the spin-2 model. However, as seen in Sect. 3.2, the production rate for a pair of DM with a photon is strongly suppressed. We checked that there is no constraint for the above benchmark points from the CMS 13 TeV monophoton analysis (12.9 fb-1\documentclass[12pt]{minimal}
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An interesting alternative to the universal coupling gSM\documentclass[12pt]{minimal}
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Resonance searches
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\begin{document}$$t\bar{t}$$\end{document}, hh) from ATLAS [52–55, 57, 59, 60] and CMS [9, 56, 58, 67–69], and give powerful constraints for mediator masses of a few hundred GeV up to several TeV. Lower masses are partly covered by Run-I results.7
Same as Fig. 5, but for the leptophobic scenario
Same as Fig. 6, but for the leptophobic scenario
Table 2 lists the current resonance search results which we use to constrain our spin-2 simplified model. The RS massive graviton is considered in the analyses for pairs of electroweak gauge or Higgs bosons [54, 57, 58, 60, 63, 64] as one of the new physics hypotheses. For the fermionic and jet final states in [52, 53, 55, 56, 59], on the other hand, Z′\documentclass[12pt]{minimal}
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\begin{document}$$Z'$$\end{document} and a model-independent Gaussian-shaped resonance have been studied. Except the dijet and di-b-jet analyses at 13 TeV and the low-mass diphoton analysis at 8 TeV from ATLAS, the limits are provided directly on the cross section in the given channel, and hence we obtain the model constraints by simply using the Y2\documentclass[12pt]{minimal}
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\begin{document}$$m_Y<2m_X$$\end{document}. Although the branching ratio is small, B(Y2→γγ)∼4\documentclass[12pt]{minimal}
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\begin{document}$$B(Y_2\rightarrow \gamma \gamma )\sim 4$$\end{document}% at high mass, the diphoton resonance searches give the most stringent limit for the whole mass range, resulting in Λ/gSM≳100TeV\documentclass[12pt]{minimal}
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\begin{document}$$m_Y\lesssim 1\mathrm{\ TeV}$$\end{document}. The dilepton channel, also having a branching ratio of about 4%, provides a similarly strong constraint for mediator masses above 200 GeV. The dijet and WW / ZZ resonance searches lead to a constraint of a few tens of TeV on Λ/gSM\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda /g_\mathrm{SM}$$\end{document} for around 1 TeV mediator mass. We note again that the limits are obtained based on the NLO production rates which are larger than the LO ones, especially for pp→(Y2→jj)γ\documentclass[12pt]{minimal}
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\begin{document}$$pp\rightarrow (Y_2\rightarrow jj)\gamma $$\end{document}; see Fig. 3. We also note that, as indicated by grey dotted lines in Fig. 9, the mediator width can be very large at high mass and low Λ/gSM\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda /g_\mathrm{SM}$$\end{document}; as the experimental analyses often assume a narrow width, this region has to be regarded with caution.
The weakening of the constraints when Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document} decays into DM are allowed is demonstrated for the dilepton channel in Fig. 9, depicted by a dotted line, where we assume vector DM and take gX=10\documentclass[12pt]{minimal}
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\begin{document}$$m_Y=1\mathrm{\ TeV}$$\end{document}, the dilepton (electron and muon) branching ratio becomes 0.8%, i.e. the dilepton production rate becomes smaller by a factor of 5, reducing the limit on Λ/gSM\documentclass[12pt]{minimal}
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\begin{document}$$1/\sqrt{5}$$\end{document}. As seen in Fig. 2, the above assumption gives the largest DM branching ratio within the scenarios we consider.8 Therefore, the diphoton resonance searches, and for mY>200GeV\documentclass[12pt]{minimal}
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\begin{document}$$m_Y>200\mathrm{\ GeV}$$\end{document} also the dilepton resonance searches, provide stronger constraints on the universal coupling scenario than the searches with missing energy.
Summary of the constraints on Λ/gSM\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda /g_\mathrm{SM}$$\end{document} from searches with and without missing energy at the 13 TeV LHC as a function of the spin-2 mediator mass, for mX=10GeV\documentclass[12pt]{minimal}
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\begin{document}$$g_X/g_\mathrm{SM}=1$$\end{document} (left) and 10 (right). The labelling of the constraints from resonance searches is the same as in Fig. 9. For gX/gSM=1\documentclass[12pt]{minimal}
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\begin{document}$$g_X/g_\mathrm{SM}=1$$\end{document} the differences among the different types of DM for the limits from the resonance searches are not visible. For gX/gSM=10\documentclass[12pt]{minimal}
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\begin{document}$$g_X/g_\mathrm{SM}=10$$\end{document}, however, they are quite relevant so only the vector DM case is shown. The figure assumes a universal gSM\documentclass[12pt]{minimal}
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\begin{document}$$g_\mathrm{SM}$$\end{document} but is also valid for the leptophobic case when ignoring the ℓℓ\documentclass[12pt]{minimal}
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To avoid such severe constraints from resonance searches, it is interesting to consider scenarios beyond the universal coupling case. The dilepton constraints could be avoided, for example, in the leptophobic scenario, gℓT=0\documentclass[12pt]{minimal}
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\begin{document}$$g^T_\ell =0$$\end{document}, as already discussed in the previous subsection. To avoid the diphoton constraints is somewhat more complicated. One possibility would be the gravity-mediated DM model [13, 14], where the KK graviton mainly couples to massive particles—DM, Higgs, massive gauge bosons and top quarks—while the couplings to photons, gluons and light quarks are highly suppressed. In such scenarios, the branching ratios and the production cross sections of the spin-2 resonance strongly depend on the setup and can be very different from those in the universal coupling case. In fact associated production of the mediator with a W or Z boson, or mediator production in vector boson fusion may be more relevant than s-channel production in qq¯\documentclass[12pt]{minimal}
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\begin{document}$$q\bar{q}$$\end{document} or gg fusion. While such setups can in principle be studied easily in the simplified-model framework by appropriately choosing the free parameters gXT\documentclass[12pt]{minimal}
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\begin{document}$$g_i^T$$\end{document} in Eq. (3), such an analysis is beyond the scope of this paper. A final caveat is that non-universal couplings to gluons and quarks, ggT≠gqT\documentclass[12pt]{minimal}
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\begin{document}$$g^T_g\ne g^T_q$$\end{document}, give rise to a unitarity violating behaviour at higher order in QCD [36]. We therefore only consider phenomenological scenarios with ggT=gqT\documentclass[12pt]{minimal}
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Summary
We considered a simplified DM model where the DM candidate couples to the SM particles via an s-channel spin-2 mediator, Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document}, and studied the constraints from the current LHC data. In particular, we compared the constraints from searches with and without missing energy.
For universal couplings of the mediator to SM particles, we found that diphoton resonance searches provide the strongest constraints, Λ/gSM≳100\documentclass[12pt]{minimal}
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\begin{document}$$m_Y$$\end{document}. The dilepton channel provides a similarly strong constraint for mediator masses above 200 GeV. Monojet and multijet+ searches are competitive only if the mediator decays into photons and leptons are heavily suppressed; in this case they could provide complementary constraints to the other resonance searches in particular in the low-mass region below 0.5–1 TeV, depending on gX/gSM\documentclass[12pt]{minimal}
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\begin{document}$$g_X$$\end{document} and the type of dark matter. The dependence on the DM mass is less pronounced unless one approaches the threshold region. For mX=10\documentclass[12pt]{minimal}
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\begin{document}$$g_X/\varLambda =g_\mathrm{SM}/\varLambda =(3\mathrm{\ TeV})^{-1}$$\end{document}, we found mY≳750\documentclass[12pt]{minimal}
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\begin{document}$$m_Y\gtrsim 750$$\end{document}, 950, and 1100 GeV for scalar, Dirac, and vector DM, respectively. This increases to 850, 1300, and 1550 GeV when doubling gX\documentclass[12pt]{minimal}
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\begin{document}$$g_X$$\end{document}. We note that the obtained limits are based on the NLO-QCD predictions, which give a larger production rate than at the LO. The K factor depends on the mediator mass and the production channel.
The complementarity among the different searches is illustrated in Fig. 10, where we have rescaled the reach of the jets + searches from 3.2 to 15 fb-1\documentclass[12pt]{minimal}
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\begin{document}$$^{-1}$$\end{document} in order to make a fair comparison. We see that, for the same amount of data, in the case of gX≃gSM\documentclass[12pt]{minimal}
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\begin{document}$$g_X\simeq g_\mathrm{SM}$$\end{document} the missing-energy searches are roughly competitive with the dijet and heavy diboson (WW, ZZ) searches, pushing Λ/gSM\documentclass[12pt]{minimal}
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\begin{document}$$\varLambda /g_\mathrm{SM}$$\end{document} beyond 20 TeV. (As mentioned, when the dilepton and diphoton constraints hold, they give even stronger limits.)
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\begin{document}$$g_X/g_\mathrm{SM}=10$$\end{document} (or gX/g^SM=10\documentclass[12pt]{minimal}
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\begin{document}$$g_X/\hat{g}_\mathrm{SM}=10$$\end{document}), also the resonance constraints strongly depend on the type of DM. Therefore, in the right plot in Fig. 10 only the vector DM case is shown. We see that the jets+ searches give stronger constraints than the dijet and heavy diboson searches up to mediator masses of about 1.2 TeV. The dilepton and diphoton constraints are weakened by about a factor of 2 but still give the strongest constraints.
We hope our work will be useful to find reasonable benchmark scenarios for spin-2 mediated DM searches at the LHC as well as to construct viable UV-completed models which can give predictions for those parameters. We also note that our study on resonance searches in Sect. 4.2 can be applied not only for spin-2 mediated DM models but also for usual RS-type graviton searches; see also, e.g. [71]. As a final remark we like to point out that in a full model the presence of KK excitations might alter the LHC phenomenology as compared to the simplified-model scenarios discussed here. Examples are limits on gauge KK modes providing additional constraints on light gravitons, or KK excitations of the DM fields contributing to signatures. While this goes well beyond the simplified-model picture, it is certainly an interesting topic for future studies.
Signal acceptance times efficiency, A×ϵ\documentclass[12pt]{minimal}
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\begin{document}$$A\times \epsilon $$\end{document}, as a function of the mediator mass for the most relevant SRs, i.e. IM5, IM6 and IM7 of the ATLAS monojet search [5] and 2jl and 2jm of the ATLAS 2–6 jets + search [43], evaluated with CheckMATE2 [44]
Acknowledgements
We would like to thank G. Das, C. Degrande, V. Hirschi and H.-S. Shao for help with the NLO calculations, and M.-H. Genest, F. Maltoni, V. Sanz and M. Zaro for valuable discussions. We are also thankful to C. Doglioni and K. Krizka for discussions on ATLAS-CONF-2016-070. This work was supported in part by the French ANR, Project DMAstro-LHC ANR-12-BS05-0006. U. L. is supported by the Investissements d’avenir, Labex ENIGMASS. K. M. is supported by the Theory-LHC-France Initiative of the CNRS (INP/IN2P3). K. Y. acknowledges support for a long-term stay at LPSC Grenoble from the Program for Leading Graduate Schools of Ochanomizu University; she also thanks the LPSC Grenoble for hospitality while this work was completed.
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As mentioned in the main part of the paper, in the case of the monojet and the 2–6 jets + searches, the signal comes solely from on-shell mediator production with the Y2\documentclass[12pt]{minimal}
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\begin{document}$$Y_2$$\end{document} decaying into neutrinos and/or DM. The signal selection efficiency (more precisely acceptance times efficiency, A×ϵ\documentclass[12pt]{minimal}
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\begin{document}$$A\times \epsilon $$\end{document}) depends only on the properties of the mediator, but not on those of the invisible decay products. Figure 11 shows A×ϵ\documentclass[12pt]{minimal}
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\begin{document}$$A\times \epsilon $$\end{document} for those SRs which, depending on mY\documentclass[12pt]{minimal}
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\begin{document}$$m_Y$$\end{document}, can be the most sensitive ones in each of the two ATLAS analyses considered in this paper. As a service to the reader and potential user of our work, the complete A×ϵ\documentclass[12pt]{minimal}
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\begin{document}$$A\times \epsilon $$\end{document} tables for all SRs are available in numerical form at [72].
A.2 Resonance searches
Observed 95% CL upper limits on resonant production cross section times branching ratio (times acceptance) as a function of the resonance mass from each experimental paper; see Table 2 for more detailed information. Dashed lines denote limits including cut acceptance. For reference, NLO production cross sections of the spin-2 mediator are shown by dotted lines for different values of gSM/Λ\documentclass[12pt]{minimal}
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\begin{document}$$g_\mathrm{SM}/\varLambda $$\end{document}
In Fig. 12 we show observed 95% CL upper limits on resonant production cross section times branching ratio (times acceptance) as a function of the resonance mass from each experimental paper. The analyses denoted by solid lines present the limit on σ×B\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \times B$$\end{document}, while those by dashed lines provide the limit on σ×B×A(×ϵforbb¯)\documentclass[12pt]{minimal}
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\begin{document}$$\sigma \times B\times A\,(\times \epsilon \ \mathrm{for}\ b\bar{b})$$\end{document}; see Table 2 for more detailed information.
As indicated in Table 2, the dijet (+ ISR jet/photon) and tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} analyses at 13 TeV as well as the ATLAS 8 TeV diphoton analysis provide tables with the numbers corresponding to the lines in the exclusion plots, which is very convenient for our purpose. The other analyses do not provide explicit values, and hence we have to extract these data from the exclusion plots ‘by hand’, e.g. using WebPlotDigitizer [73], a public software. To avoid that other people have to redo this exercise, our digitised data files are available at [72] and on the new PhenoData database [74]. We encourage the experimental collaborations to provide digitised data together with their plots, in order to make it easier to use their results.
Finally, we notice a caveat regarding the re-interpretation of the low-mass resonance search in dijet plus ISR final states [53]. We found that final-state radiation (FSR) may also be important and give rise to a non-trivial structure in the dijet invariant mass spectrum. Technically, simulated event shapes can differ by including FSR or not in the matrix elements, which may affect the parameter fitting procedure for a bump search.
One may also assign independent coupling parameters for each flavour, especially for heavy flavours [28].
These decay branching ratios were already presented in [39] for the case of the RS graviton. We repeat them here for the sake of completeness. Our numbers agree with [39] apart from a factor 1/2 for decays into neutrinos.
As can be deduced from Fig. 1, above the WW threshold up to high masses the picture does not change much apart from the tt¯\documentclass[12pt]{minimal}
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\begin{document}$$t\bar{t}$$\end{document} and/or hh channels being open or not.
See also Fig. 12 (bottom) for σ(pp→Y2)\documentclass[12pt]{minimal}
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The K factors in Fig. 3 are slightly different from the ones reported in [28] due to different PDF choices and different kinematical cuts. See [28] for details on theoretical uncertainties.
While both analyses have very similar sensitivity, i.e. their expected limits are basically the same, the monojet results have over- and under-fluctuations in some SRs. Therefore the expected and observed limits slightly differ from each other for the monojet analysis. Overall, the multijet+ analysis tends to give the stronger limit.
We thank the referee for pointing us to the ATLAS analysis [61], which looked for narrow scalar resonances in the diphoton invariant mass spectrum down to 65 GeV.
In Fig. 9 there is hardly any difference between the gX≪1\documentclass[12pt]{minimal}
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