Simplified dark matter models with a spin-2 mediator at the LHC

TL;DR

This paper investigates simplified dark matter models with a spin-2 mediator at the LHC, analyzing how various search strategies constrain the model parameters and highlighting the most effective channels depending on coupling scenarios.

Contribution

It provides a comprehensive analysis of constraints on spin-2 mediated dark matter models from multiple LHC search channels, emphasizing the conditions under which missing-energy searches are most effective.

Findings

Diphoton and dilepton resonance searches provide the strongest constraints.

Missing-energy searches are more effective in low-mass regions or with large dark matter couplings.

The effectiveness of search channels depends on the mediator's coupling to SM particles and dark matter.

Abstract

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, $t\bar t$, $b\bar b$ 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.

Figures

Tables

Table 1. Table 1: Branching ratios of the spin-2 mediator for g SM = 1 subscript 𝑔 SM 1 g_{\rm SM}=1 and B ​ ( Y 2 → X ​ X ) = 0 𝐵 → subscript 𝑌 2 𝑋 𝑋 0 B(Y_{2}\to XX)=0 ; j ​ j 𝑗 𝑗 jj includes gluons and five flavours of quarks, and ν ​ ν 𝜈 𝜈 \nu\nu includes three flavours of neutrinos.

$m_Y$	branching ratios $[\%]$
$\left[\mathrm{GeV}\right]$	$jj$	$WW$	$tt$	$ZZ$	$\gamma \gamma$	$\nu \nu$	$ee$	$hh$
100	86.5	0	0	0	5.3	4.0	2.7	0
500	79.1	9.9	3.3	5.0	4.4	3.3	2.2	0.2
1000	78.5	9.4	5.7	4.7	4.3	3.2	2.1	0.3

Table 2. Table 2: Constraints from resonance searches used in this study. The observed 95% CL upper limits on resonant production cross section ( σ 𝜎 \sigma ) times branching ratio ( B 𝐵 B ) (times acceptance ( A 𝐴 A )) from each analysis are shown in Fig.  12 in Appendix  A.2 .

decay mode	reference	limit Tab./Fig.	limit on	$\sqrt{s}$(TeV)	$L$ (fb-1)
$jj$	ATLAS-CONF-2016-069 ATLAS-CONF-2016-069	Tab. 2 (Res)	$\sigma \left(\mathrm{Gaussian}\right)\times B\times A$	13	15.7
$jj\left(+j/\gamma \right)$	ATLAS-CONF-2016-070 ATLAS-CONF-2016-070	Tab. 4/3 (Res)	$\sigma \left(\mathrm{Gaussian}\right)\times B\times A$	13	15.5
$WW$	ATLAS-CONF-2016-062 ATLAS-CONF-2016-062	Fig. 6	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	13	13.2
$bb$	ATLAS-CONF-2016-060 ATLAS-CONF-2016-060	Fig. 7(b) (Res)	$\sigma \left(\mathrm{Gaussian}\right)\times B\times A\times {ϵ}_{2b}$	13	13.3
$tt$	CMS-PAS-B2G-15-002 CMS-PAS-B2G-15-002	Tab. 4 (1%)	$\sigma \left(Z^{\prime }\right)\times B$	13	2.6
$ZZ$	ATLAS-CONF-2016-082 ATLAS-CONF-2016-082	Fig. 10(d)	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	13	13.2
$\gamma \gamma$	CMS 1609.02507 Khachatryan:2016yec	Fig. 6(middle)	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	13+8	16.2+19.7
$\ell \ell$	ATLAS-CONF-2016-045 ATLAS-CONF-2016-045	Fig. 3(c)	$\sigma \left(Z^{\prime }\right)\times B$	13	13.3
$hh$	ATLAS-CONF-2016-049 ATLAS-CONF-2016-049	Fig. 11	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	13	13.3
$\gamma \gamma$	ATLAS 1407.6583 Aad:2014ioa	Fig. 4, HepData atlas:diphoton:hepdata	$\sigma \left(H\right)\times B\times A$	8	$20.3$
	CMS 1506.02301 Khachatryan:2015qba	Fig. 6	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	8	$19.7$
$WW$	ATLAS 1512.05099 Aad:2015ipg	Auxiliary Fig. 3	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	8	$20.3$
$ZZ$	ATLAS 1512.05099 Aad:2015ipg	Auxiliary Fig. 4	$\sigma \left(G_{\mathrm{RS}}\right)\times B$	8	$20.3$

Equations

$${\mathcal{L}}_X^{Y_2}=-\frac{1}{\Lambda }{g}_X^T{T}_{\mu \nu }^X{Y}_2^{\mu \nu },$$

$${\mathcal{L}}_{\mathrm{SM}}^{Y_2}=-\frac{1}{\Lambda }\sum _i{g}_i^T{T}_{\mu \nu }^i{Y}_2^{\mu \nu },$$

$$g_i^T=\{g_H^T,g_q^T,g_{\ell }^T,g_g^T,g_W^T,g_B^T\}$$

$$T_{\mu \nu }^{X_R}$$

$$+{\partial }_{\mu }{X}_R{\partial }_{\nu }{X}_R,$$

$$T_{\mu \nu }^{X_D}$$

$$+\frac{1}{2}{g}_{\mu \nu }{\partial }_{\rho }({\overline{X}}_{D}i{\gamma }^{\rho }{X}_D)$$

$$+\frac{1}{2}{\overline{X}}_{D}i({\gamma }_{\mu }{\partial }_{\nu }+{\gamma }_{\nu }{\partial }_{\mu })X_D$$

$$-\frac{1}{4}{\partial }_{\mu }({\overline{X}}_{D}i{\gamma }_{\nu }{X}_D)-\frac{1}{4}{\partial }_{\nu }({\overline{X}}_{D}i{\gamma }_{\mu }{X}_D),$$

$$T_{\mu \nu }^{X_V}$$

$$+F_{\mu \rho }{F}_{\nu }^{\rho }+m_X^2{X}_{V\mu }{X}_{V\nu },$$

$$g_{\mathrm{SM}}\equiv g_H^T=g_q^T=g_{\ell }^T=g_g^T=g_W^T=g_B^T.$$

$$\{m_X,m_Y,g_X/\Lambda ,g_{\mathrm{SM}}/\Lambda \},$$

$$m_Y/\Lambda =x_{1}k/{\overline{M}}_{\mathrm{Pl}},$$

$${\Gamma }_S$$

$${\Gamma }_F$$

$${\Gamma }_V$$

$$g_{\ell }^T\ll {\hat{g}}_{\mathrm{SM}}\equiv g_H^T=g_q^T=g_g^T=g_W^T=g_B^T.$$

Full text

Preprint: OCHA-PP-345

1 Introduction

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 Alves:2011wf 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 Abercrombie:2015wmb . 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. Khachatryan:2016mdm ; Aaboud:2016uro ; Aaboud:2016tnv ; Aaboud:2016qgg ; Aaboud:2016obm ; ATLAS-CONF-2016-086 ; Sirunyan:2016iap ; CMS-PAS-EXO-16-037 ; CMS-PAS-EXO-16-038 ; CMS-PAS-EXO-16-039 . On the other hand, the model with a spin-2 mediator Lee:2013bua ; Lee:2014caa has not been fully explored for the LHC yet—it is one of the next-generation simplified DM models Bauer:2016gys .

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 Mattelaer:2015haa ; Backovic:2015soa ; Neubert:2015fka , which provides the DM model files for event generators such as MadGraph5_aMC@NLO Alwall:2014hca as well as for DM tools such as micrOMEGAs Belanger:2006is ; Belanger:2008sj ; Barducci:2016pcb and MadDM Backovic:2013dpa ; Backovic:2015cra . 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 readers to Lee:2013bua ; Lee:2014caa for the astrophysical constraints, and to Garcia-Cely:2016pse for a discussion of spectral features in indirect detection.

The article is organised as follows. The simplified model is presented in Section 2, and the production and decays of the spin-2 mediator in Section 3. The re-interpretation of the LHC results is discussed in Section 4. Section 5 contains a summary and conclusions. Supplemental material for recasting is provided in the Appendix.

2 Model

Gravity-mediated DM was proposed in Lee:2013bua ; Lee:2014caa , 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 ($Y_{2}$) with DM ($X$) is given by Lee:2013bua

[TABLE]

where $\Lambda$ is the scale parameter of the theory, $g^{T}_{X}$ is the coupling parameter, and $T_{\mu\nu}^{X}$ is the energy–momentum tensor of a DM field. Here, we consider three types of DM independently; a real scalar ($X_{R}$), a Dirac fermion ($X_{D}$), and a vector ($X_{V}$). The interaction with SM particles is obtained by

[TABLE]

where $i$ denotes each SM field, i.e. the Higgs doublet ($H$), quarks ($q$), leptons ($\ell$), and $SU(3)_{C}$, $SU(2)_{L}$ and $U(1)_{Y}$ gauge bosons ($g,W,B$). Following Ellis:2012jv ; Englert:2012xt we introduce the phenomenological coupling parameters

[TABLE]

without assuming any UV model.111One may also assign independent coupling parameters for each flavour, especially for heavy flavours Das:2016pbk . The energy–momentum tensors of the DM are

[TABLE]

where $F_{\mu\nu}$ is the field strength tensor. Those of the SM fields are similar; see e.g. Das:2016pbk 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:

[TABLE]

With this simplification, the model has only four independent parameters, two masses and two couplings:

[TABLE]

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 Randall:1999ee . The parameters are related as

[TABLE]

where $x_{1}=3.83$ is the first root of the Bessel function of the first kind, $k$ is the curvature of the warped extra dimension, and $\overline{M}_{\rm Pl}=2.4\times 10^{18}{\rm\ GeV}$ is the reduced four-dimensional Planck scale. On the other hand, in the so-called bulk RS model Agashe:2007zd ; Fitzpatrick:2007qr , where the SM particles also propagate in the extra dimension, $g^{T}_{i}$ can take different values depending on the setup.

In Das:2016pbk , the SM sector of the above model was implemented in FeynRules/NloCT Alloul:2013bka ; Degrande:2014vpa (based on Hagiwara:2008jb ; deAquino:2011ix ; Artoisenet:2013puc ), and the $Y_{2}$ production and decay rates at next-to-leading order (NLO) QCD accuracy were presented. In this work, we include the three DM species ($X_{R}$, $X_{D}$, $X_{V}$) with the corresponding interactions, and add the model into the DMsimp framework FR-DMsimp:Online as the simplified DM model with a spin-2 mediator.

3 Phenomenology at the LHC

3.1 Decay 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 ($S=X_{R},h$), spin-1/2 ($F=X_{D},q,\ell$) and spin-1 ($V=X_{V},g,\gamma,Z,W$) DM or SM particles are given by

[TABLE]

where $\beta_{i}=\sqrt{1-4r_{i}}$ with $r_{i}=m^{2}_{i}/m^{2}_{Y}$, $g_{\gamma}=g_{B}\cos^{2}\theta_{W}+g_{W}\sin^{2}\theta_{W}$ and $g_{Z}=g_{B}\sin^{2}\theta_{W}+g_{W}\cos^{2}\theta_{W}$ with the weak-mixing angle $\theta_{W}$, and $f(r_{V})=1+\frac{1}{12}\kappa^{2}_{H}-r_{V}(3-\frac{20}{3}\kappa_{H}-\kappa^{2}_{H})+r^{2}_{V}(6-\frac{20}{3}\kappa_{H}+\frac{14}{3}\kappa_{H}^{2})$ with $\kappa_{H}=g_{H}/g_{V}$. For gluons and photons, $\kappa_{H}=0$ in $f(r_{V})$, while $\kappa_{H}=1$ for vector DM. The factors $N_{\nu}=1/2$ for neutrinos and $N_{s}=1/2$ for two identical particles, and are unity otherwise; $N_{C}^{F,V}$ is the number of colours. We note that $B(Y_{2}\to Z\gamma)=0$ for $g_{W}=g_{B}$ as the decay rate is proportional to $g_{Z\gamma}^{2}=[(g_{W}-g_{B})\cos\theta_{W}\sin\theta_{W}]^{2}$. 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 $\beta_{i}$ 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 $Y_{2}$ total width scaled by the mass, $\Gamma_{Y}/m_{Y}$, and the decay branching ratios for the case that only decays into SM particles are allowed. MadWidth Alwall:2014bza 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 $g_{\rm SM}$, decays into gluons and light quarks, leading to a dijet signature, are completely dominant ($\gtrsim 80\%$ depending on $m_{Y}$). The diphoton channel has 4–5% branching ratio; other diboson channels ($WW$ and $ZZ$) as well as $t\bar{t}$ are important as well when kinematically allowed. Finally, it is important to note that decays into neutrinos have 3–4% branching ratio, leading to missing energy signatures independent of decays to DM.222These decay branching ratios were already presented in Allanach:2002gn for the case of the RS graviton. We repeat them here for the sake of completeness. Our numbers agree with Allanach:2002gn apart from a factor 1/2 for decays into neutrinos. The width is proportional to $m_{Y}^{3}$, and from the upper panel in Fig. 1 we see that for $g_{\rm SM}/\Lambda\lesssim(3{\rm\ TeV})^{-1}$, the resonance is always very narrow ($\Gamma_{Y}/m_{Y}<1\%$) up to $m_{Y}\sim 1{\rm\ TeV}$. Note here, that $\Lambda$ is simply a scale parameter, not a physical cut-off of the theory.

When decays into DM are allowed, their relative importance depends on $g_{X}$ and the type of DM (scalar, Dirac or vector) as illustrated in Fig. 2; see also Eqs. (10)–(12). Two mass scales are considered: $m_{Y}=100$ GeV and 1 TeV, with $m_{X}=10$ GeV and $g_{\rm SM}=1$.333As can be deduced from Fig. 1, above the $WW$ threshold up to high masses the picture does not change much apart from the $t\bar{t}$ and/or $hh$ channels being open or not. We see that decays into DM can be important and even dominant, but the resonance remains narrow for any choice of $\Lambda\gtrsim 3$ TeV for $m_{Y}\lesssim 1{\rm\ TeV}$. Another important observation is that for scalar DM ($X_{R}$), for $g_{X}\sim g_{\rm SM}$ the decay into $Y_{2}\to X_{R}X_{R}$ is practically irrelevant; one needs $g_{X}/g_{\rm SM}\approx 3$ for the decay into DM to exceed the one into neutrinos, and $g_{X}/g_{\rm SM}\approx 5$–$6$ to reach the 10% level. For Dirac ($X_{D}$) and vector ($X_{V}$) DM, the decays into DM and into neutrinos are of comparable magnitude at $g_{X}\sim g_{\rm SM}$, both contributing to missing-energy signatures. For $g_{X}/g_{\rm SM}=2$, the branching ratio of $Y_{2}\to X_{D}X_{D}$ $(X_{V}X_{V})$ attains about 10% (20%). These differences depending on the type of DM will be important later for the collider limits.

3.2 Production of the spin-2 mediator

Turning to the production modes, the potentially interesting channels are inclusive $Y_{2}$ production ($pp\to Y_{2}$), as well as the production with an extra hard tagging jet ($pp\to Y_{2}\,j$) or an electroweak boson (e.g. $pp\to Y_{2}\,\gamma$). With the $Y_{2}$ decaying into SM particles, the former gives resonant peak signatures (without missing energy). On the other hand, the latter two give the typical monojet or monophoton signatures when the mediator decays invisibly. Moreover, the latter two play a role in the low-mass resonance search in dijet events with initial-state radiation (ISR) as seen later.

The $Y_{2}$ 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.444See also Fig. 12 (bottom) for $\sigma(pp\to Y_{2})$ at $\sqrt{s}=8$ TeV. We employ MadGraph5_aMC@NLO Alwall:2014hca to calculate the cross sections and generate events with the LO/NLO NNPDF2.3 Ball:2012cx . 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 $g_{\rm SM}/\Lambda$ and scale with $(g_{\rm SM}/\Lambda)^{2}$. The dashed lines showing $g_{\rm SM}/\Lambda=(10{\rm\ TeV})^{-1}$ are therefore an order of magnitude below the corresponding solid lines for $g_{\rm SM}/\Lambda=(3{\rm\ TeV})^{-1}$. Also noteworthy is the fact that $pp\to Y_{2}$ is mostly gluon-initiated for the low-mass case Allanach:2002gn ; 97%, 83%, and 28% of the LO total rate for $m_{Y}=100{\rm\ GeV}$, $1{\rm\ TeV}$, and $5{\rm\ TeV}$, respectively, stem from $gg$ fusion. Since the radiation of an initial-state photon ($Z/W$) can only occur in the quark-initiated process, $Y_{2}+{\rm photon}\,(Z/W)$ production is very much suppressed as compared to $Y_{2}+{\rm jet}$ production. This is also the reason that the process has a huge $K$ factor especially in the low-mass region Das:2016pbk .555The $K$ factors in Fig. 3 are slightly different from the ones reported in Das:2016pbk due to different PDF choices and different kinematical cuts. See Das:2016pbk for details on theoretical uncertainties.

In the context of DM searches, the monojet signature is expected to give important constraints on the model. The fiducial cross sections for $pp\to Y_{2}j$ with $p_{T}^{j}>200$ GeV and $|\eta^{j}|<5$ are shown in Fig. 3, where one can estimate the monojet cross section by taking into account the $Y_{2}$ branching ratio into DM particles (and/or neutrinos) when $m_{Y}>2m_{X}$. In Fig. 4 we also plot the fiducial cross sections for $pp\to j+\not{E}_{T}$ as a function of the DM mass, separating the contributions from neutrinos (black lines) and DM (red lines) produced through the spin-2 mediator. For definiteness, we take $m_{Y}=500{\rm\ GeV}$, $\Lambda=3{\rm\ TeV}$, $g_{\rm SM}=1$ and compare $g_{X}=1$, 2 and 10 for Dirac DM. As already seen in Fig. 2, their relative importance depends on $g_{X}$. For $m_{Y}<2m_{X}$, a pair of DM is produced via the off-shell mediator and the cross section is strongly suppressed. Therefore, the neutrino contribution always dominates the monojet signature for the $m_{Y}<2m_{X}$ region even if $g_{X}/g_{\rm SM}=10$. For the other DM types, scalar and vector, the picture is similar, but the relative importance to the neutrino channel is different; see Fig. 2. This is one of the characteristic features of the spin-2 mediator DM model with universal couplings, as compared to the $s$-channel spin-1 and spin-0 models, whose mediators do not couple to charged leptons and neutrinos in the minimal setup Abercrombie:2015wmb .

4 Constraints from current LHC data

4.1 Searches 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 $\not{E}_{T}$, but allow for additional jets from QCD radiation. Therefore one can expect that multijet+$\not{E}_{T}$ searches are also relevant Haisch:2013ata ; Buchmueller:2015eea .

To work out the current constraints on the spin-2 mediator DM model from these searches, we consider the following early Run-II analyses:

• •

ATLAS monojet with $3.2$ fb*-1* Aaboud:2016tnv ,

• •

ATLAS 2–6 jets + $\not{E}_{T}$ with $3.2$ fb*-1* Aaboud:2016zdn .

In the monojet analysis Aaboud:2016tnv , a simplified DM model with an $s$-channel spin-1 mediator is considered. Events are required to have at least one hard jet with $p_{T}>250{\rm\ GeV}$ and $|\eta|<2.4$, and a maximum of four jets with $p_{T}>30{\rm\ GeV}$ and $|\eta|<2.8$ are allowed. Several inclusive and exclusive signal regions (SRs) are considered with increasing $\not{E}_{T}$ requirements from 250 GeV to 700 GeV. The multijet+$\not{E}_{T}$ analysis Aaboud:2016zdn 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; $\not{E}_{T}>200{\rm\ GeV}$ is required for all SRs, while different thresholds are applied on jet momenta and on the azimuthal separation between jets and $\not{E}_{T}$.

To reinterpret the above analyses in the context of our spin-2 mediator simplified DM model, we use CheckMATE2 Dercks:2016npn , 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-$k_{T}$ scheme Alwall:2008qv , implemented in MadGraph5_aMC@NLO Alwall:2014hca with Pythia6 Sjostrand:2006za , and generate signal events with parton multiplicity from one to two partons. We impose $\not{E}_{T}>200{\rm\ GeV}$ and set $Q_{\rm cut}=200$ GeV for the merging separation parameter at the parton level; these values are chosen for an efficient event generation without affecting the final results. The event rate is normalised to the $pp\to Y_{2}j$ 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 Backovic:2015soa ; a detailed study of this aspect will be reported elsewhere.)

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 $q\bar{q}$ and $qg$ initial states but also dominantly through the $gg$ initial state, and partly because the spin-2 mediator is also emitted from a gluon as well as from the $gggY_{2}$ and $q\bar{q}gY_{2}$ four-point vertices.

Figure 5 shows the ratio of signal events over the number of events excluded at 95% confidence level (CL), $S/S^{95}$, as a function of the mediator mass, for the three types of DM (taking $g_{X}=1$ or 2 with $\Lambda=3{\rm\ TeV}$, $g_{\rm SM}=1$ and $m_{X}=10$ GeV as a benchmark case). As expected from the discussion in the previous section, the scalar DM case is the least constrained, with the $\not{E}_{T}$ coming dominantly from the neutrino channel; for $g_{X}=1$ (2), we find the limit $m_{Y}\gtrsim 600$ (750) GeV from the monojet analysis and $m_{Y}\gtrsim 750$ (850) GeV from the multijet+$\not{E}_{T}$ analysis.666While 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+$\not{E}_{T}$ analysis tends to give the stronger limit. For Dirac DM the limit increases to $m_{Y}\gtrsim 950$ (1300) GeV owing to the contribution from $Y_{2}\to X_{D}X_{D}$. Finally, for vector DM we have $m_{Y}\gtrsim 1100$ (1550) GeV. For the monojet analysis, the inclusive SR with the $\not{E}_{T}$ cut of 500, 600, and 700 GeV (denoted as IM5, IM6, and IM7 in Aaboud:2016tnv ) gives the limit for the low ($100\sim 300{\rm\ GeV}$), middle ($300\sim 450{\rm\ GeV}$), and high ($\gtrsim 450{\rm\ GeV}$) mass region, respectively. For the multijet+$\not{E}_{T}$ analysis, the 2-jet loose (2jl) SR gives the limit for the mass range of $100\sim 300{\rm\ GeV}$, while the 2-jet medium (2jm) SR does for $\gtrsim 300{\rm\ GeV}$. See Aaboud:2016zdn for the detailed selection criteria.

As the production rate scales as $1/\Lambda^{2}$, the upper limit of $\Lambda$ can be estimated from the plots. For instance, for vector DM with $m_{Y}=100{\rm\ GeV}$, $\Lambda$ should be larger than around 10 TeV for $g_{\rm SM}=g_{X}=1$. It should be noted that, due to the $K$ factors of $1.7-1.2$ for $m_{Y}=100-2000{\rm\ GeV}$ (see Fig. 3), these limits are slightly stronger than what would be obtained with LO production rates.

The 95% CL exclusion in the $m_{X}$ vs. $m_{Y}$ plane is shown in Fig. 6. Due to the different threshold behaviours, as seen in Eqs. (10)–(12), the excluded region near $m_{Y}=2m_{X}$ strongly depends on the type of DM.

We note that we compared the CheckMATE results with those obtained by the equivalent analysis implementations in MadAnalysis 5 Conte:2014zja ; Dumont:2014tja (recast codes ma5:monojet ; ma5:multijet ) and Rivet 2.5 Buckley:2010ar 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 Sec. 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*) CMS-PAS-EXO-16-039 .

An interesting alternative to the universal coupling $g_{\rm SM}$ is a leptophobic scenario with

[TABLE]

In this case, the $\not{E}_{T}$ signatures come exclusively from decays into DM, because $Y_{2}$ decays into neutrinos are switched off. Moreover, constraints from dilepton resonance searches, which as we will see in the next subsection are quite severe, are evaded. The results for the leptophobic scenario are presented in Figs. 7 and 8 in analogy to Figs. 5 and 6. As expected, the $m_{Y}<2m_{X}$ region is no longer constrained. Also, for $g_{X}=1$, the exclusion becomes considerably weaker for all the DM types; in particular there is no more constraint for scalar DM. For $g_{X}=2$, except scalar DM, the mediator decays into DM dominates the neutrino decay mode even for the universal coupling scenario (see Fig. 2), and hence the $m_{Y}$ limits are very similar.

4.2 Resonance searches

Direct resonance searches can also be used to explore $s$-channel mediator DM models, see e.g. Chala:2015ama ; Arina:2016cqj for the spin-1 and spin-0 mediator models, respectively. Results from Run-II data are already available for a large variety of final states (dijet, dilepton, diphoton, $WW$, $ZZ$, $b\bar{b}$, $t\bar{t}$, $hh$) from ATLAS ATLAS-CONF-2016-045 ; ATLAS-CONF-2016-062 ; ATLAS-CONF-2016-069 ; ATLAS-CONF-2016-070 ; ATLAS-CONF-2016-082 ; ATLAS-CONF-2016-060 ; ATLAS-CONF-2016-049 and CMS Khachatryan:2016yec ; Sirunyan:2016iap ; CMS-PAS-EXO-15-002 ; CMS-PAS-EXO-16-031 ; Khachatryan:2016qkc ; CMS-PAS-B2G-15-002 , 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.777 We thank the referee for pointing us to the ATLAS analysis Aad:2014ioa , which looked for narrow scalar resonances in the diphoton invariant mass spectrum down to 65 GeV.

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 Khachatryan:2016yec ; ATLAS-CONF-2016-062 ; ATLAS-CONF-2016-082 ; ATLAS-CONF-2016-049 ; Khachatryan:2015qba ; Aad:2015ipg as one of the new physics hypotheses. For the fermionic and jet final states in ATLAS-CONF-2016-045 ; ATLAS-CONF-2016-060 ; CMS-PAS-B2G-15-002 ; ATLAS-CONF-2016-069 ; ATLAS-CONF-2016-070 , on the other hand, $Z^{\prime}$ 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 $Y_{2}$ production cross section and the branching ratio discussed in Section 3. For the analyses with different hypotheses from the spin-2 resonance, we assume that the acceptance and efficiency are similar. When limits are given on the fiducial cross section, $\sigma\times B\times A$, we generate LO events normalised by the NLO cross section and apply the fiducial cuts at the parton level by using MadAnalysis5 Conte:2012fm .

We recall that, for a given mediator mass, the $Y_{2}$ production cross section depends solely on $g_{\rm SM}/\Lambda$, while the branching ratio depends also on the parameters related to DM, i.e. $g_{X}$ and $m_{X}$, as well as on the type of DM. In the decoupling limit of the dark sector, the constraints on $\Lambda/g_{\rm SM}$ are the most stringent. When decays to DM are relevant, the branching ratios to SM particles become smaller and hence the constraints are weakened.

Figure 9 shows the constraints on $\Lambda/g_{\rm SM}$ from the observed 95% CL upper limits of the resonance searches listed in Table 2 as a function of the mediator mass, where we assume a negligible branching ratio to DM particles, i.e. $g_{X}\ll 1$ and/or $m_{Y}<2m_{X}$. Although the branching ratio is small, $B(Y_{2}\to\gamma\gamma)\sim 4$% at high mass, the diphoton resonance searches give the most stringent limit for the whole mass range, resulting in $\Lambda/g_{\rm SM}\gtrsim 100{\rm\ TeV}$ for $m_{Y}\lesssim 1{\rm\ TeV}$. 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 $\Lambda/g_{\rm SM}$ 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 $pp\to(Y_{2}\to jj)\gamma$; 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 $\Lambda/g_{\rm SM}$; as the experimental analyses often assume a narrow width, this region has to be regarded with caution.

The weakening of the constraints when $Y_{2}$ 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 $g_{X}=10$ and $m_{X}=10{\rm\ GeV}$. For instance, at $m_{Y}=1{\rm\ TeV}$, the dilepton (electron and muon) branching ratio becomes 0.8%, i.e. the dilepton production rate becomes smaller by a factor of five, reducing the limit on $\Lambda/g_{\rm SM}$ by $1/\sqrt{5}$. As seen in Fig. 2, the above assumption gives the largest DM branching ratio within the scenarios we consider.888In Fig. 9 there is hardly any difference between the $g_{X}\ll 1$ and $g_{X}=1$ cases. Therefore, the diphoton resonance searches, and for $m_{Y}>200{\rm\ GeV}$ also the dilepton resonance searches, provide stronger constraints on the universal coupling scenario than the searches with missing energy.

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, $g^{T}_{\ell}=0$, 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 Lee:2013bua ; Lee:2014caa , 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 $q\bar{q}$ or $gg$ fusion. While such setups can in principle be studied easily in the simplified model framework by appropriately choosing the free parameters $g_{X}^{T}$ and $g_{i}^{T}$ 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, $g^{T}_{g}\neq g^{T}_{q}$, give rise to a unitarity violating behaviour at higher order in QCD Artoisenet:2013puc . We therefore only consider phenomenological scenarios with $g^{T}_{g}=g^{T}_{q}$.

5 Summary

We considered a simplified DM model where the DM candidate couples to the SM particles via an $s$-channel spin-2 mediator, $Y_{2}$, 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, $\Lambda/g_{\rm SM}\gtrsim 100$ TeV for $Y_{2}$ masses up to $\sim 1$ TeV. For $\Lambda/g_{\rm SM}=10$ (3) TeV, the exclusion extends up to 4 (beyond 5) TeV in $m_{Y}$. The dilepton channel provides a similarly strong constraint for mediator masses above 200 GeV. Monojet and multijet+$\not{E}_{T}$ 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 $g_{X}/g_{\rm SM}$.

For $m_{Y}<2m_{X}$, $\not{E}_{T}$ signatures arise solely from $Y_{2}$ decays into neutrinos, leading to $m_{Y}\gtrsim 700$ GeV for $g_{X}/\Lambda=g_{\rm SM}/\Lambda=(3{\rm\ TeV})^{-1}$, based on 3.2 fb*-1* of data at $\sqrt{s}=13$ TeV. For $m_{Y}>2m_{X}$, the limit crucially depends on $g_{X}$ and the type of dark matter. The dependence on the DM mass is less pronounced unless one approches the threshold region. For $m_{X}=10$ GeV and $g_{X}/\Lambda=g_{\rm SM}/\Lambda=(3{\rm\ TeV})^{-1}$, we found $m_{Y}\gtrsim 750$, 950, and 1100 GeV for scalar, Dirac, and vector DM, respectively. This increases to 850, 1300, and 1550 GeV when doubling $g_{X}$. 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 + $\not{E}_{T}$  searches from $3.2$ to $15$ fb*-1* in order to make a fair comparison. We see that, for the same amount of data, in case of $g_{X}\simeq g_{\rm SM}$ the missing energy searches are roughly competitive with the dijet and heavy diboson ($WW$, $ZZ$) searches, pushing $\Lambda/g_{\rm SM}$ beyond 20 TeV. (As mentioned, when the dilepton and diphoton constraints hold, they give even stronger limits.)

For $g_{X}/g_{\rm SM}=10$ (or $g_{X}/\hat{g}_{\rm SM}=10$), 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+$\not{E}_{T}$ 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 two 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 Sec. 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. Alvarez:2016ljl . 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 $\not{E}_{T}$ signatures. While this goes well beyond the simplified model picture, it is certainly an interesting topic for future studies.

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.

Appendix A Supplemental material for recasting

A.1 Searches with missing energy

As mentioned in the main part of the paper, in case of the monojet and the 2–6 jets + $\not{E}_{T}$ searches, the signal comes solely from on-shell mediator production with the $Y_{2}$ decaying into neutrinos and/or DM. The signal selection efficiency (more precisely acceptance times efficiency, $A\times\epsilon$) depends only on the properties of the mediator, but not on those of the invisible decay products. Figure 11 shows $A\times\epsilon$ for those SRs which, depending on $m_{Y}$, 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 $A\times\epsilon$ tables for all SRs are available in numerical form at recasting:lpsc .

A.2 Resonance searches

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 $\sigma\times B$, while those by dashed lines provide the limit on $\sigma\times B\times A\,(\times\epsilon\ {\rm for}\ b\bar{b})$; see Table 2 for more detailed information.

As indicated in Table 2, the dijet (+ ISR jet/photon) and $t\bar{t}$ 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 wepplotdigitizer , a public software. To avoid that other people have to redo this exercise, our digitised data files are available at recasting:lpsc and on the new PhenoData database phenodata . 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 ATLAS-CONF-2016-070 . We found that final state radiation (FSR) may be also 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.