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Published for SISSA by Springer Received: October 7, 2016

Revised: December 20, 2016 Accepted: December 30, 2016

Published: January 18, 2017

Daniele Barducci,a Aoife Bharucha,b Nishita Desai,c Michele Frigerio,dBenjamin Fuks,e,f Andreas Goudelis,e,f,g Suchita Kulkarni,g Giacomo Poleselloh and Dipan Senguptai

aLAPTh, Universit e Savoie Mont Blanc, CNRS, B.P. 110, F-74941 Annecy-le-Vieux, France

bCNRS, Aix Marseille U., U. de Toulon, CPT, Marseille, France

cInstitut fur Theoretische Physik, Universitat Heidelberg, Heidelberg, Germany

dLaboratoire Charles Coulomb (L2C), UMR 5221 CNRS-Universit e de Montpellier, F-34095 Montpellier, France

eSorbonne Universit es, UPMC Univ. Paris 06, UMR 7589, LPTHE, F-75005 Paris, France

f CNRS, UMR 7589, LPTHE, F-75005 Paris, France

gInstitute of High Energy Physics, Austrian Academy of Sciences,

Nikolsdorfergasse 18, 1050 Vienna, Austria

hINFN, Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

iLaboratoire de Physique Subatomique et de Cosmologie, Universit e Grenoble-Alpes, CNRS/IN2P3,53 Avenue des Martyrs, F-38026 Grenoble, France

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We consider minimal dark matter scenarios featuring momentum-dependent couplings of the dark sector to the Standard Model. We derive constraints from existing LHC searches in the monojet channel, estimate the future LHC sensitivity for an integrated luminosity of 300 fb1, and compare with models exhibiting conventional momentum-independent interactions with the dark sector. In addition to being well motivated by (composite) pseudo-Goldstone dark matter scenarios, momentum-dependent couplings are interesting as they weaken direct detection constraints. For a speci c dark matter mass, the LHC turns out to be sensitive to smaller signal cross-sections in the momentum-dependent case, by virtue of the harder jet transverse-momentum distribution.

Keywords: Beyond Standard Model, Cosmology of Theories beyond the SM, E ective eld theories

ArXiv ePrint: 1609.07490

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP01(2017)078

Web End =10.1007/JHEP01(2017)078

Monojet searches for momentum-dependent dark matter interactions

JHEP01(2017)078

Contents

1 Introduction 1

2 Theoretical framework and constraints 32.1 The minimal scenario: the Higgs portal 32.2 A pragmatic scenario with a scalar singlet mediator 42.3 Constraints on the parameters of the model 5

3 Numerical results 83.1 Analysis setup 83.2 Bounds derived from LHC monojet data 113.3 Complementarity of collider, cosmological and theoretical considerations 12

4 Conclusion 15

A Derivation of perturbative unitarity constraints 17

1 Introduction

Collider searches for nal states consisting of a hard jet and missing energy [1{5], dubbed monojet searches, provide a means to detect new invisible particles that are stable on detector or even cosmological scales. In the latter case, these particles could contribute to the dark matter (DM) energy density of the Universe and monojet searches could o er invaluable information about their existence. Furthermore it is well known that the jet transverse-momentum spectrum is one of the key observables that could unravel the nature of the dark matter couplings to the Standard Model from monojet probes [6, 7]. In this work, we study the e ect of derivative and non-derivative couplings between the Standard Model and the new physics sector on the monojet kinematics. Our preliminary results, including only 8 TeV LHC data, appeared in the proceedings of the \Les Houches 2015 | Physics at TeV colliders" workshop [8].

Models with derivative couplings are motivated by new physics setups featuring pseudo-Nambu-Goldstone bosons (pNGBs), i.e. light scalar elds connected to the spontaneous breaking of a global symmetry at an energy scale f. More concretely, this class of models includes composite Higgs scenarios where the set of pNGBs involves the Higgs boson and possibly extra dark scalar particles [9{14]. In this case, the pNGB shift symmetry indeed only allows for derivative (momentum-dependent) pNGB interactions suppressed by powers of the scale f. An explicit weak breaking of the shift symmetry, parameterized by a small coupling strength , is however necessary in order to induce pNGB masses, which

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JHEP01(2017)078

subsequently generates additional non-derivative momentum-independent couplings proportional to =f. In this work, we rely on a simpli ed e ective eld theory approach where the form of the Lagrangian is inspired by such pNGB setups, with all speci c and model-dependent assumptions for the new physics masses and couplings being, however, relaxed. As already pointed out in the literature, this e ective Lagrangian approach is appropriate for interpreting LHC missing energy signatures within frameworks featuring light dark matter particles interacting with the Standard Model via non-renormalizable derivative operators [13, 14]. Momentum-dependent interactions between the Standard Model and a dark sector may also have alternative motivations, as well as interesting phenomenological consequences alternative to monojet searches, see e.g. [15, 16].

Most ultraviolet-complete models of dark matter involve additional particles, potentially carrying Standard Model quantum numbers. Although dedicated LHC searches could detect such additional states, we consider a simple setup where the only new states that are accessible at the LHC are the dark matter particle and (sometimes) the mediator connecting the Standard Model and the dark sector. More speci cally, we rst consider an invisible sector solely comprised of a Standard Model-singlet real scalar eld which is taken to be odd under a Z2 discrete symmetry. The Standard Model elds are then chosen to be even

under the same Z2 symmetry, which forbids the decay of the particle into any ensemble

of Standard Model states and renders it a potential dark matter candidate. In a minimal scenario, the mediator is taken to be the Standard Model Higgs eld H that interacts with via both a renormalizable quartic coupling and a non-renormalizable derivative coupling. This framework, however, turns out to be strongly constrained by LHC measurements of the Higgs boson properties. As an alternative we therefore consider a slightly extended setup where we additionally introduce a real gauge-singlet Z2-even scalar mediator par

ticle s, a choice which allows us to avoid these constraints. While s couples to the dark sector both through a derivative (dimension- ve) and a non-derivative renormalizable operator, it is connected to the Standard Model only through a (potentially loop-induced) dimension- ve operator involving gluon eld strength bilinears. This simple model not only reproduces the observed dark matter abundance of the universe but also, assuming that momentum-dependent interactions dominate, can evade the direct detection constraints.

In this paper, we rst provide details of our theoretical framework in section 2 and then examine the LHC constraints stemming from the monojet analysis performed by the ATLAS collaboration for proton-proton collisions at a center-of-mass energy of 13 TeV with 3.2 fb1 in section 3. We assess the e ects of momentum-dependent and momentum-independent dark matter couplings on monojet distributions and derive the corresponding bounds for both cases. Dijet searches for the mediator s at past and present colliders are also taken into account and discussed, and we nally entertain the possibility that the particle is responsible for the measured dark matter density in the Universe. In this spirit, we investigate the dependence of the relic abundance and the direct dark matter detection constraints on the model parameters. Our ndings are summarized in section 4, and technical details on the range of validity of our e ective description based on perturbative unitarity arguments are presented in appendix A.

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2 Theoretical framework and constraints

2.1 The minimal scenario: the Higgs portal

In order to study the impact of derivative and non-derivative couplings of dark matter to the Standard Model, we rst consider a minimal setup involving both momentum-dependent and momentum-independent couplings of the dark matter particle. We impose that the dark matter only couples to the Higgs eld, which plays the role of the mediator.

We supplement the Standard Model by a gauge-singlet real scalar eld that is chosen odd under a Z2 symmetry, where in contrast the Standard Model elds are taken to be

even. The particle then only interacts with the Standard Model through couplings to the Higgs doublet H, such that the model Lagrangian reads

L = LSM +

2@ @

2f2 (@ 2)@(HH) : (2.1)

This Lagrangian contains renormalizable operators compatible with the Z2 symmetry

( ! ) and a dimension-six operator involving derivatives. While several other non-

derivative dimension-six operators are allowed by the model symmetries, these are not expected to have a signi cant impact on the monojet analysis at large momentum transfer.1 As the e ect of these operators is negligible for our purposes, we have omitted these in our parameterization of eq. (2.1). In the context of composite Higgs models, the scalar eld may be a pNGB and f its decay constant. The theoretical motivations for this minimal model and the resulting dark matter phenomenology are described in ref. [9]. Further related studies are also available in the literature [10{12].

After electroweak symmetry breaking, the part of the Lagrangian containing the interactions of with the physical Higgs boson h is given by

JHEP01(2017)078

2 2 2

1 4 4

2 2HH +

14(v + h)2 [parenleftbigg]

2 + 1f2 @@ 2[parenrightbigg]

; (2.2)

and the mass m satis es

m2 = 2 + v2=2 : (2.3)

The trilinear scalar interaction in eq. (2.2) yields monojet events at the LHC via, for instance, the gluon fusion process gg ! gh( ) ! g , while the quartic interactions give rise

to mono-Higgs events gg ! h ! h . When 2m < mh, the Higgs boson is produced on-

shell and the strength of the derivative interaction vertex is proportional to p2h=f2 = m2h=f2, with ph being the nal-state Higgs boson four-momentum. The momentum-dependence reduces to a constant, so that momentum-dependent interactions become indistinguishable from their momentum-independent counterparts. In this regime, bounds from monojet searches are found to be weaker than the constraints stemming from the Higgs invisible width results [17{19],

(h ! ) =

v2 32mh

m2hf2

s1 4m2 m2h (m2h 4m2 ) [lessorsimilar] 0:15 SMh [similarequal] 0:7 MeV ; (2.4)

1Indeed, one can write several dimension-six operators involving two powers of and one or two derivatives. However, using equations of motions these can be rewritten in terms of non-derivative operators, except for that in eq. (2.1).

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at the 95% con dence level.We therefore focus on the complementary kinematic region where 2m > mh. The

monojet signal arises from o -shell Higgs-boson production, and the derivative interactions of the particle result in a strong momentum dependence at the di erential cross-section level. The subsequent di erences in the jet transverse momentum distribution could allow us to discriminate derivative from non-derivative dark matter couplings. This however comes at the price of a suppression of the monojet signal, since the relevant partonic cross-section ^

depends on the virtuality of the Higgs boson p2h as

(gg ! gh ! g ) /

(p2h 4m2 )(p2h m2h)2 + 2hm2h [parenleftbigg]

p2hf2

s1 4m2 p2h; (2.5)

where h denotes the Higgs total width. The denominator is clearly larger in the region where the Higgs is o -shell, or equivalently when p2h > 4m2 > m2h.

A preliminary monojet analysis within the considered theoretical framework has recently been performed [20], and the collider signatures of this o -shell Higgs portal model are discussed in ref. [21]. Our numerical analysis however indicates that in the light of current experimental data, the monojet signal is too weak to be observed at the LHC. The precise determination of the Higgs-boson mass and the important LHC constraints on its production cross-section and decay width indeed result in tight bounds on the free parameters of the model,

m [greaterorsimilar] mh=2; [lessorsimilar] 1; f [greaterorsimilar] 500 GeV 1 TeV ; (2.6)

where the latter bound applies to models in which the Higgs is a composite pNGB. As a consequence, the total monojet cross-section after including a selection on the jet transverse momentum of pjetT > 20 GeV is always smaller than 1 fb and 0.5 fb when only momentum-dependent and momentum-independent couplings are allowed, respectively, for a center-of-mass energy of ps = 13 TeV.

2.2 A pragmatic scenario with a scalar singlet mediator

Given the tight constraints discussed in the previous section, we extend our framework to analyse a scenario less severely constrained by data. In addition to the dark matter eld we introduce a second scalar mediator eld s, chosen to be even under the Z2 symmetry

and a singlet under the Standard Model gauge symmetries. We moreover impose that the scalar potential does not spontaneously break the Z2 symmetry, or equivalently that does

not acquire a non-vanishing vacuum expectation value (vev). We further demand, without any loss of generality, that the vev of the s eld vanishes as the latter could always be absorbed in a rede nition of the couplings.

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JHEP01(2017)078

The relevant Lagrangian is given by

L ,s = LSM +

2@ @

2m2 +

2@s@s

2m2sss

(2.7)

Here the dimensionless coupling cs governs the non-derivative interaction of the even scalar s with two odd scalars , in units of the scale f. The coupling c@s controls the strength of the leading, dimension- ve derivative interaction between s and . The other dimension- ve operators involving are non-derivative, and their e ect is irrelevant with respect to the cs operator. Finally, we also included an e ective coupling csg between s and the gluons, such that the mediator can be produced at the LHC via gluon fusion, and can give rise to a monojet signal via the process gg ! gs ! g . In ultraviolet-complete models,

this coupling can be generated by additional new physics. For example, a vector-like color-triplet fermion of mass M ms, that interacts via a Yukawa interaction y

+ cs f

c@s

csg

2 s +

f (@s)(@ ) +

s 16

f sGa Ga :

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s, will

induce a one-loop contribution of the form csg = (4=3)(y f=M ), according to the normal-isation of eq. (2.7). We implicitly assume that alternative s-production mechanisms are sub-leading with respect to gluon fusion. In particular, s could be produced via the Higgs portal, but in this case the cross section to produce at the LHC would be even smaller than in the minimal scenario of section 2.1. Indeed, the mixing between the Higgs and s is already constrained by Higgs coupling measurements, therefore we will assume it is small.

The Lagrangian given in eq. (2.7) only includes interactions that are relevant to our analysis, and the considered dimension- ve operator is the unique independent derivative dimension- ve operator inducing an interaction between s and . The model can hence be described in terms of six parameters,

ms; m ; f; cs ; c@s and csg: (2.8)

Strictly speaking, only ve of these parameters are independent as one can always choose c@s = 1 and determine the strength of the momentum-dependent interaction by varying f.

This choice is motivated by models where s, and the Higgs boson are pNGBs associated with the spontaneous breaking of a global symmetry at a scale f and where c@s is expected to be of order one. In this case, the f parameter is constrained by precision Higgs and electroweak data that roughly imposes f [greaterorsimilar] 500 GeV 1 TeV [22]. In our numerical analysis

of section 3, we consider four representative values for the s particle mass, ms = 50 GeV,

250 GeV, 500 GeV and 750 GeV, which allows us to cover a wide range of mediator masses.

2.3 Constraints on the parameters of the model

The model can be constrained in a number of ways. In particular, searches for dijet resonances could be promising since a singly-produced mediator via gluon fusion often decays back into a pair of jets (gg ! s( ) ! gg). For the case where is a viable dark matter

candidate, the model should in additional yield a relic density in agreement with Planck measurements and satisfy direct dark matter detection bounds. Before getting into a detailed investigation of these constraints, we rst study some properties of the model in order

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to understand the bounds that can be expected from collider, cosmological and theoretical considerations. A complete set of numerical results is then presented in section 3.

From the Lagrangian given in eq. (2.7), the partial decay widths of the s particle into gluon and pairs are calculated to be

(s ! gg) =

2sc2sgm3s

1283f2 ; (2.9)

(s ! ) =

f2 32ms

s1 4m2 m2s (m2s 4m2 ) ; (2.10)

in agreement with results obtained using the decay module of FeynRules [23, 24]. For the coupling values adopted in our analysis, the total width s is always small. This implies that the narrow width approximation can be used for any cross-section calculation involving a resonant s-contribution. The s-induced dijet cross section can hence be expressed as

(pp ! s ! gg) =

[integraldisplay]

c@s m2sf2 + cs

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0 dx1 [integraldisplay]

0 dx2 fg(x1; ms)fg(x2; ms)

2sc2sgm2s

1024f2 (^s m2s)BR[parenleftBig]

s ! gg

[parenrightBig]

;

(2.11)

where p^s denotes the partonic center-of-mass energy and fg(x; ) the universal gluon density that depends on the longitudinal momentum fraction x of the gluon in the proton and the factorization scale . For the considered values of ms, the most stringent dijet constraints arise from Sp pS [25] and Tevatron [26] data which provides upper limits on the new physics cross section for mediator masses of 140{300 GeV and 200{1400 GeV respectively. In comparison, the LHC Run I results further extend the range of covered mediator masses up to 4.5 TeV [27, 28]. For f = 1000 GeV, we nd that csg values up to about 100 (which corresponds to an e ective coupling of about 103) are allowed independently of the other parameters, and we adopt this upper limit henceforth.

Turning our attention to the dark matter phenomenology, we rst study the relic abundance h2[notdef] . This is numerically computed in section 3 with the MicrOmegas

package [29], in which we have implemented our model via FeynRules [24]. An approximate expression describing the relevant total thermally-averaged annihilation cross section

hv[angbracketright] can nonetheless be derived analytically. Restricting ourselves to the leading S-wave

contribution and ignoring all possible special kinematic con gurations featuring, e.g., intermediate resonances, the thermally-averaged cross section associated with annihilation into a pair of gluons is given by

hv[angbracketright]gg [similarequal]

2 163f4 m2s 4m2

2sc2sgm2 cs f2 + 4c@s m2

2 : (2.12)

In the case where m > ms, an additional 2 ! 2 annihilation channel contributes, ! ss,

whose leading S-wave contribution reads

hv[angbracketright]ss [similarequal]

r1 m2sm2 c@s m2s + cs f2

2 : (2.13)

16f4m2 m2s 2m2

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We impose the requirement that the relic density satis es the upper limit [30]

h2[notdef] h2[notdef]exp = 0:1188 [notdef] 0:0010 : (2.14)

Assuming a standard thermal freeze-out mechanism, and ignoring singular parameter space regions such as resonances, the dark matter relic density does not depend strongly on whether m > ms=2 or < ms=2. This condition is, however, crucial for the LHC: monojet searches can typically only reach couplings that correspond to thermal self-annihilation cross sections once the mediator can be produced and decay on-shell. Instead, in the o -shell regime, the LHC tends to probe parameter space regions where the dark matter abundance lies below h2[notdef]exp [31], but there are exceptions [32]. Finally, regardless of the

momentum-dependent or -independent nature of the dark matter interactions the dominant contribution to the dark matter annihilation comes from the S-wave.

Direct detection searches yield additional constraints on the phenomenologically viable regions of the model parameter space. These however do not constrain the strength of the momentum-dependent interactions, as the corresponding scattering cross section is proportional to the dark matter-nucleus momentum transfer which is very small compared to the mediator mass. On the other hand, the momentum-independent couplings in eq. (2.7) lead to an e ective interaction between particles and gluons,

L g = fG 2 G G with fG = scsgcs 32

1m2s : (2.15)

The spin-independent dark matter scattering cross section SI is then found to take the form [33, 34]

SI = 1

[parenleftbigg]

JHEP01(2017)078

[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

m mp

m + mp

9 s

mpm fGfT

2 ; (2.16)

where the term inside the brackets corresponds to the DM-nucleon reduced mass, and the squared matrix element depends on the gluon form factor fTG. The latter is derived from the quark form factors fTq [35],

fTG = 1

Xq=u,d,sfTq ; (2.17)

for which we take the values fTu = 0:0153, fTd = 0:0191 and fTs = 0:0447 [36]. The above expression for fTG would change if additional couplings between the mediator s and quarks were introduced. Our predictions for SI are compared, in the next section, to limits extracted from LUX data [37].2

Finally, additional restrictions can also be imposed on the model from perturbative unitarity requirements. For a given process, the e ective Lagrangian in eq. (2.7) is indeed expected to provide an accurate description of the underlying physics only as long as the typical momentum involved lies below a cuto scale which we have so far kept unspeci ed.

2While this work was being completed, the LUX collaboration has updated their results on the basis of 332 live days of exposure [38]. We do not include the latest limits in our analysis. Although more constraining, the new LUX results do not imply signi cant di erences in the allowed region of the parameter space.

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[vextendsingle][vextendsingle][vextendsingle][vextendsingle]

This scale can be deduced rigorously on a model-by-model basis, but its minimal acceptable value can be estimated without referring to any speci c ultraviolet completion. By enforcing the S matrix to be perturbatively unitary, we ensure that calculations performed on the basis of the Lagrangian of eq. (2.7) provide reliable predictions [39{41].

We provide the details of the calculation in appendix A, where we show that perturbative unitarity of the gg ! scattering amplitude imposes the constraints

MI <

64p22(1 m

1 4m 2 Q2

2(Q2 m2s) sQ4

1 4m 2 Q2

1/4 ; (2.19)

with MD = c@s csg. In typical hadron collider processes like those occuring at the LHC, the scale Q2 varies from one event to another. In order to simplify the discussion, we judiciously focus on large values of Q2 that are relevant for the high-energy tail of the di erential distributions where the e ective theory is expected to break down. Considering typical missing transverse-momentum distributions related to monojet events and the current LHC luminosity, the tail of the distribution extends to [notdef]Q[notdef] 2 TeV while most events relevant

for the extraction of LHC constraints feature a missing transverse energy in the [700, 1500] GeV range. In the next section, unitarity bounds are therefore computed for the conservative choice [notdef]Q[notdef] = 2 TeV.

3 Numerical results

We now estimate the constraining power of monojet searches both in the case of momentum-dependent and momentum-independent interactions. For simplicity, we consider scenarios featuring either momentum-independent (c@s = 0) or momentum-dependent (cs = 0)

couplings, and we set the composite scale f to 1 TeV. The mediator coupling to the gluon eld strength tensor is xed to csg = 10 and 100, as allowed by the dijet bounds discussed in section 2.2. We nally discuss the complementarity between theory, collider and cosmological constraints.

3.1 Analysis setup

In order to evaluate the LHC sensitivity to our model via monojet probes, we compare our theoretical predictions to o cial ATLAS results based on early 13 TeV data at an integrated luminosity of 3.2 fb1 [3]. This is achieved via an implementation of the analysis of ref. [3] in the MadAnalysis 5 framework [42, 43]. Details on our code and its validation are publicly available on Inspire [44] and within the MadAnalysis 5 Public Analysis Database [45].3 Our recasted analysis is in agreement with the ATLAS o cial results for well-de ned event

3https://madanalysis.irmp.ucl.ac.be/wiki/PublicAnalysisDatabase

Web End =https://madanalysis.irmp.ucl.ac.be/wiki/PublicAnalysisDatabase .

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Q2 )

1/4 ; (2.18)

JHEP01(2017)078

where MI = cs csg and

MD < 64p22f

samples at the 5% level, and we have also compared, for consistency, our results to those obtained when using LHC Run I data [8].

The analysis under consideration preselects events featuring one nal-state hard jet with a transverse-momentum pT larger than 250 GeV and a pseudorapidity satisfying

| [notdef] < 2:4, as well as at most four jets with pT > 30 GeV and [notdef] [notdef] < 2:8. Moreover, each jet is

required to be azimuthally separated from the missing momentum by an angle > 0:4, and events exhibiting muons or electrons with a transverse momentum greater than 10 GeV and 20 GeV respectively are vetoed. Preselected events are then categorized into seven inclusive and six exclusive signal regions. The seven inclusive regions are de ned by seven overlapping selections on the missing transverse energy, demanded to be larger than 250, 300, 350, 400, 500, 600 and 700 GeV. The same thresholds are further used to de ne six missing energy bins [250,300], [300,350], [350,400], [400,500], [500,600] and [600,700] GeV, which form the six exclusive signal regions.

In order to perform our study, we have implemented the model described in section 2.2 in the FeynRules [24] package, and generated a UFO library [46] that we have imported into MadGraph5 aMC@NLO [47]. Hard-scattering events describing the pp ! j

process (with an 80 GeV selection threshold on the jet pT ) have been generated for a center-of-mass energy of 13 TeV and matched to the parton showering and hadronization infrastructure of Pythia 6 [48]. The events are then processed by Delphes 3 [49] for a fast detector simulation using a tuned parameterization of the ATLAS detector and the Fastjet program [50] for jet reconstruction by means of the anti-kT algorithm [51] with an

R-parameter set to 0.4. Finally, the MadAnalysis 5 program is used to handle the event selection and to compute the associated upper limit at the 95% con dence level (CL) on the signal cross section according to the CLs technique [52, 53]. Although the considered analysis contains 13 signal regions, the upper bound on the cross section (interpreted at leading order) is determined only from the region that is expected to be the most sensitive. This region is determined using the background rate, its uncertainty and the observed number of events reported by the ATLAS collaboration.

For discrete choices of the mediator mass ms = 50, 250, 500 and 750 GeV, we scan over various ranges of the dark matter mass with 2m > ms. Since only one of the cs or c@s parameters are taken to be non-zero at a time, the computed cross section upper limits only depend on the kinematics of the events and not on the overall rate. The results are thus independent of the actual values of the cs , c@s and csg parameters, and we have consequently xed (cs , c@s ) to the nominal values (1, 0) and (0, 1). This choice enables an easy rescaling of the monojet cross section when di erent values of the input parameters are chosen and a straightforward derivation of limits on the momentum-dependent and momentum-independent interactions for a given set of masses and couplings.

In addition, we have also evaluated the LHC sensitivity to our model for a luminosity of 300 fb1, this time using Pythia 8 [54] for the parton showering and hadronization of the signal samples, along with e ciency factors and smearing functions aimed at reproducing the performance of the ATLAS detector during the rst run of the LHC [55]. Thanks to the higher statistics and the di erent shape of the missing energy distribution for signal and background, the optimal sensitivity to the signal is expected for tighter missing energy

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JHEP01(2017)078

= 50 GeV, m

= 100 GeV - MI

= 250 GeV, m

= 150 GeV - MI

= 100 GeV - MD

= 50 GeV, m

= 150 GeV - MD

= 250 GeV, m

Events ( Normlaized to one )

Events ( Normalized to one )

= 50 GeV, m

= 300 GeV - MI

= 250 GeV, m

= 400 GeV - MI

= 300 GeV - MD

= 50 GeV, m

= 400 GeV - MD

= 250 GeV, m

0 100 200 300 400 500 600 700 800 900 1000

( j

) [GeV]

( j

) [GeV]

Figure 1. Normalized distributions in the transverse momentum of the leading jet assuming a perfect detector. We consider a mediator mass of 50 GeV and 250 GeV in the left and right panels respectively, and a dark matter mass of 100 GeV and 300 GeV (left panel) or 150 GeV and 400 GeV (right panel). The solid lines re ect scenarios featuring momentum-independent (MI) interactions while the dashed lines correspond to scenarios featuring momentum-dependent (MD) interactions.

requirements than those adopted in ref. [3], thus motivating extending the number of signal regions. This, however, requires the extrapolation of the predictions for the expected Standard Model background and the associated uncertainties.

The 3.2 fb1 monojet publication of ATLAS provides the Standard Model expectation for the missing transverse-energy distribution [3], so that the latter can be used to extract the expected number of background events Nbg for 300 fb1. The estimation of the systematic uncertainties Nbg is however luminosity-dependent due to an extrapolation of the dominant Z+jets and W +jets backgrounds from the number of data events observed in appropriate control regions to the signal regions. We consequently parametrise Nbg as

N2bg =

JHEP01(2017)078

pNbg

k2 Nbg

2: (3.1)

The rst term on the right-hand side represents the statistical error on the number of events observed in the control regions and is controlled by the k1 parameter, while the second term consists of the systematic uncertainties connected to the extrapolation procedure from the control region to the signal regions and is driven by the k2 parameter. The ATLAS analysis nds the latter to be slowly varying with the missing transverse-energy selection and is of the order of a few percent [3]. We adopt the choice of k1 = 1:51 and k2 = 0:043, which parametrize the ATLAS results of ref. [3] at the percent level, and calculate 95% CL upper limits on the signal cross-section for overlapping signal regions de ned by minimum requirements on the missing transverse energy varying in steps of 100 GeV between 500 and 1400 GeV. The statistical procedure relies on a Poisson modelling with Gaussian constraints using the CLs prescription and the asymptotic calculator implemented in the RooStat package [56]. The lowest upper limit on the ducial production cross section (with a constraint on the jet transverse momentum of pT > 80 GeV) is then taken to be the nal result.

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3.2 Bounds derived from LHC monojet data

As a rst illustration of the di erences between scenarios featuring momentum-independent and momentum-dependent interactions, we show the leading jet pT distributions obtained with MadAnalysis 5 for the representative mass combinations (ms; m ) = (50; 100=300) GeV and (ms; m ) = (250; 150=400) GeV in the left and right panels of gure 1 respectively. Focusing on the shapes of the distributions that have been normalized to one, one observes that momentum-dependent interactions induce a harder jet pT spectrum. As a result, one expects that a larger fraction of events would pass a monojet selection when momentum-dependent interactions are present. For instance, choosing csg = 100, f = 1 TeV and either c@s = 2:5 in the momentum-dependent case or cs = 0:5 in the momentum-independent case we obtain, in both scenarios, a ducial cross section of 2.9 pb once an 80 GeV generator-level selection on the leading jet pT is enforced. The e ciency associated with a transverse-momentum selection of pT > 300 GeV on the leading jet is, however, relatively larger by about 50% in the momentum-dependent case. The di erence between the two scenarios is signi cantly reduced for larger dark matter masses.

As explained in section 3.1, for a given value of ms the constraints that can be derived from LHC dark matter searches only depend on m , since in the relevant subprocesses the mediator has to be o -shell. In gure 2, we present the upper limits on the monojet cross section at the LHC, UL(pp ! j), with a generator-level selection on the transverse mo

mentum of the leading jet of pT > 80 GeV. Existing constraints extracted from 3.2 fb1 of13 TeV LHC collisions are depicted by red lines for the momentum-independent (solid) and dependent (dashed) cases. As anticipated, the cross sections excluded at the 95% CL are signi cantly smaller in the momentum-dependent setup than in the moment-independent one, so that the former is more e ciently constrained than the latter. We additionally observe that the exclusion bounds become stronger with increasing m . As long as enough phase space is available, larger masses imply a larger amount of missing energy so that the signal regions of the monojet analysis are more populated and stronger limits can be derived, as shown in the gure.

Our results con rm the ndings of gure 1, the di erences between the momentum-independent and momentum-dependent cases being maximal for small values of m . Eventually, for dark matter masses of about 1 TeV, the limits become identical for both cases although the LHC loses its sensitivity for such heavy dark matter scenarios.

We also report in gure 2 projections for 300 fb1 of LHC collisions at a center-of-mass energy of 13 TeV. The blue solid and dashed lines respectively represent the momentum-independent and momentum-dependent cases. We observe a behavior that is similar to the lower luminosity one, although it is now driven by the additional higher missing-energy requirements. In the relevant bins, the signal acceptance is again found to be higher for the momentum-dependent dark matter coupling case, so that the corresponding exclusion bounds are stronger. Moreover, the two classes of dark matter operators can still only be distinguished up to a given dark matter mass, which is nonetheless larger than for lower luminosities.

{ 11 {

JHEP01(2017)078

Monojet xs UL [pTj>80 GeV] ms= 50 GeV LHC13 3.2/fb fb

5000

Monojet xs UL [pTj>80 GeV] ms= 250 GeV LHC13 3.2/fb fb

104

5000

1000

500

UL [ fb ]

1000

500

100

100 Solid

Dashed

Dashed=MD

JHEP01(2017)078

50 100 150 200

100 200 300 400 500 600 700 800

m [GeV]

1000

Monojet xs UL [pj >80 GeV] m = 500 GeV fb

Monojet xs UL [pTj>80 GeV] ms= 750 GeV

LHC13 3.2/fb fb

1000

500

UL [ fb ]

100

Solid

Dashed=MD

Solid

Dashed=MD

300 400 500 600 700 800

400 500 600 700 800 900 1000

m [GeV]

Figure 2. 95% CL upper limits (UL) on the monojet production ducial cross section (that includes a generator-level selection of pT > 80 GeV on the leading jet). We consider proton-proton collisions at a center-of-mass energy of 13 TeV with an integrated luminosity of 3.2 fb1

(recasting, red lines) and 300 fb1 (projections, blue lines) for ms = 50 GeV (top left), 250 GeV (top right), 500 GeV (bottom left) and 750 GeV (bottom right) as a function of m . The solid lines correspond to the momentum-independent case, whereas the dashed lines correspond to the momentum-dependent case.

3.3 Complementarity of collider, cosmological and theoretical considerations

In order to estimate the regions of the model parameter space that are viable with respect to current data, we investigate the interplay between the LHC monojet bounds presented in the previous section and the dark matter and theoretical considerations discussed in section 2.2. Assuming momentum-independent dark matter interactions, the LUX results exclude the spin-independent direct detection cross section predicted by eq. (2.16) in the entire parameter space region accessible with the 13 TeV LHC monojet results. More precisely, for a dark matter mass of 50 GeV that is close to the LUX sensitivity peak and for the minimal csg = 10 choice, the maximum cs allowed values are of the order of 1:2 [notdef] 103, 0.03, 0.13 and 0.28 for ms = 50, 250, 500 and 750 GeV respectively. Increasing

the dark matter mass to a slightly higher value of 200 GeV that is still within the LHC

{ 12 {

100

orc

c s

0.5

0.1 50 100 150 200

JHEP01(2017)078

200 300 400 500 600 700 800

[GeV]

500

100

orc

c s

Dashed=MD csg=10

ms= 750 GeV

0.5 300 400 500 600 700 800

0.5 400 500 600 700 800 900 1000

[GeV]

Figure 3. Constraints on the ci couplings of eq. (2.7) driven by monojet searches. The red lines depict constraints from existing 3.2 fb1 of data whereas blue ones correspond to predictions for an integrated luminosity of 300 fb1. We x f = 1 TeV, ms = 50 GeV (top left), 250 GeV (top right), 500 GeV (bottom left) and 750 GeV (bottom right) and the results are represented as functions of m for csg = 10. The shaded regions correspond to momentum-dependent coupling values for which the universe is overclosed, while above the black lines the perturbative unitarity of the e ective theory is lost. The solid and dashed lines correspond to the momentum-independent (MI) and dependent (MD) cases respectively.

reach, these numbers increase to 0.008, 0.2, 0.5 and 0.9.4 Therefore, in the momentum-independent case, an observable monojet signal could be explained only by missing energy unrelated to dark matter. Thus, in the following, we show the constraint from the dark matter relic density only for the momentum-dependent case.

In gures 3 and 4 we superimpose constraints arising from the 13 TeV LHC monojet search results and the corresponding projections (red and blue lines respectively) on those obtained by imposing the relic density bound of eq. (2.14) (the latter for the momentum-dependent case only), assuming a standard thermal freeze out dark matter scenario. The bounds on the cs coupling are stronger for larger f values while those on the c@s parameter are weaker. In the shaded regions, annihilation is not e cient enough, so that the

4These numbers assume that the local dark matter density is 0 = 0.3 GeV cm3.

{ 13 {

0.50

orc

0.50

c s

0.10

0.05

0.01 50 100 150 200

JHEP01(2017)078

200 300 400 500 600 700 800

[GeV]

100

orc

c s

Dashed=MD

csg=100 ms= 750 GeV

0.50

csg=100 ms= 500 GeV

0.1

0.10

0.05 300 400 500 600 700 800

400 500 600 700 800 900 1000

[GeV]

Figure 4. Same as in gure 3 but for csg = 100.

Universe is overclosed. Along the borders of these regions, the relic density limit is exactly reproduced, the shape of these borders being fairly well described by the approximate results of eq. (2.12) and eq. (2.13). In the un-shaded region, the predicted abundance is smaller than the observed Planck value.

Since in our parameter space scan, no resonant con guration can occur, the c@s values satisfying the dark matter abundance bounds vary relatively mildly with the dark matter and mediator masses. The minor apparent features (especially in the ms = 250 GeV and 500 GeV scenarios) that can be observed are related to the opening of the additional dark matter annihilation channel into s pairs. For our choices of parameters, however, this channel only contributes subleadingly to the relic density, its maximal impact being found to be of the order of 15%. The annihilation cross section hence approximately scales as (c@s [notdef] csg)2, so that a smaller value of csg implies almost proportionally larger allowed

values for c@s . Note, however, that further decreasing csg (and, correspondingly, increasing c@s ) would lead to a qualitatively di erent picture. In this case, it is the t-channel process ! ss that would constitute the leading dark matter annihilation channel. The corre

sponding cross section, approximated by eq. (2.13), is independent of csg, implying that by simply tuning c@s to an appropriate value the observed dark matter abundance in the Uni-

{ 14 {

verse can be reproduced regardless of how feebly dark matter couples to gluons. In such a scenario, the model becomes essentially invisible to collider searches and, hence, irrelevant for our work. For the sake of completeness we note that in this case interesting constraints, the study of which falls well beyond our purposes, can be obtained by considering, e.g., the impact of late decays of the mediator particles on primordial nucleosynthesis.

We also include in the gures the perturbative unitarity limit of validity of our e ective parameterization (black dashed line) choosing [notdef]Q[notdef] = 2 TeV in eq. (2.19). For csg = 10 the perturbative unitarity bound reads c@s [lessorsimilar] 190 in the momentum-dependent case, and cs [lessorsimilar] 760 in the momentum-independent case. This bound depends weakly on ms and m , however it is proportional to 1=csg, as observed on comparing gure 3 with gure 4. We observe that although the unitarity limits do exhibit some overlap with the current and projected LHC reach, our e ective description is consistent over most of the parameter space.

Our ndings show that existing monojet constraints are not yet strong enough to probe regions of parameter space where can account for the entire dark matter energy density of the Universe. We therefore recover the fairly well-known result that in the \o -shell" regime of dark matter models, the LHC tends to be sensitive to dark matter candidates for which the relic density is underabundant [57, 58]. On the other hand, collider searches probe large values of the s coupling while the Planck results instead constrain small values, where the Universe tends to be overclosed. In this sense, there is an interesting complementarity between collider and cosmological measurements. Besides, we observe that for an integrated luminosity of 300 fb1, the LHC will be able to access a part of the low-mass region of our model where the observed dark matter abundance can be exactly reproduced, for dark matter masses up to about 140 GeV. Whether or not it will be able to actually distinguish between the two scenarios will be the subject of forthcoming work. Another remark is related to the fact that our results are valid regardless of the stability of the states at cosmological timescales. In other words, letting aside model-building considerations, our analysis holds for metastable particles as well, as long as they do not decay within the LHC detectors.

4 Conclusion

Momentum-dependent couplings between dark matter and the Standard Model are well-motivated both from a theoretical and a phenomenological perspective. Indeed, broad classes of ultraviolet-complete dark matter models predict e ective derivative operators at low energy, in particular whenever the dark matter particle is an approximate Goldstone boson of the underlying theory. This is quite natural in the context of composite Higgs models for the electroweak symmetry breaking. On the phenomenological side, scenarios involving momentum-dependent couplings can reproduce the observed dark matter abundance in the Universe, while evading the stringent bounds from direct detection experiments.

Monojet searches at the LHC are important probes of dark matter. In the context of a simpli ed model where a pair of dark matter particles interacts with the Standard Model via a scalar mediator s, we considered two types of dark matter-mediator couplings, momentum-dependent or momentum-independent, corresponding to two operators with di erent Lorentz structures. The high-energy tail of the monojet di erential

{ 15 {

JHEP01(2017)078

distribution being harder in the momentum-dependent case, the associated cross-section is expected to be more e ciently constrained by current LHC data. We demonstrated this by studying the monojet cross section upper limits in the two scenarios, employing early13 TeV LHC data. We showed that, indeed, one can probe smaller cross sections in the momentum-dependent case. The di erence in sensitivity appears when the mediator is produced o -shell, m > ms=2, and provided enough phase space is available, m [lessorsimilar] 1 TeV.

This di erence ranges from a factor of order one, for m ms 500 GeV, up to one order

of magnitude for lower masses, m ms 50 GeV. We moreover estimated the reach of

the LHC assuming an integrated luminosity of 300 fb1 in proton-proton collisions at a center-of-mass energy of 13 TeV.

In the momentum-dependent case, that is free from direct detection constraints, we compared the monojet upper bounds on the dark matter couplings with the requirement of not exceeding the observed dark matter abundance in the Universe. While the present bounds correspond to under-abundant relic densities, the projected 300 fb1 bounds become sensitive to the observed dark matter relic density for su ciently light masses, m ; ms [lessorsimilar] 100 GeV. We also carefully checked that, in the relevant parameter space of the model, our description in terms of e ective operators is consistent with perturbative unitarity.

Our study indicates that, in the near future, the LHC can cover the most signi cant portion of the parameter space in the case of a light, o -shell mediator to the dark sector. Indeed, one can progressively close the gap between the collider upper limits on the dark matter couplings, and their values preferred by cosmological observations, assuming a standard thermal history. Were a monojet signal observed, the di erences in the monojet pT distribution between the momentum-dependent and the momentum-independent couplings could provide handles on the nature of the dark matter interactions with the Standard Model. One should keep in mind that, contrary to our simplifying assumption, both types of coupling can be present, but it is likely that one provides the dominant contribution. A discrimination among the two scenarios appears feasible, once the statistics become su cient to analyse the shape of the distributions.

Acknowledgments

The authors are grateful to F. Maltoni for discussions about the unitarity constraints and to S. Lacroix for his fruitful participation to the early stage of this project. We would also like to thank the organizers of the 2015 Les Houches | Physics at TeV colliders workshop where this work was initiated.

MF has been partly supported by the OCEVU Labex (ANR-11-LABX-0060) and the A*MIDEX project (ANR-11-IDEX-0001-02) and by the European Unions Horizon 2020 research and innovation programme under the Marie Skodowska-Curie grant agreements No 690575 and No 674896. BF has been supported by the Theorie LHC France initiative of the CNRS, SK by the New Frontiers program of the Austrian Academy of Sciences, DS by the French ANR project DMAstroLHC (ANR-12-BS05-0006).

{ 16 {

JHEP01(2017)078

A Derivation of perturbative unitarity constraints

We follow the analysis presented in ref. [40], using conventions essentially coinciding with those used by Jacob and Wick in their seminal paper on the computation of scattering amplitudes in terms of helicity eigenstates [59, 60]. In order to compute the range of validity of our e ective eld theory framework, we rely on the optical theorem,

Mi!f Mf!i = iXX

[integraldisplay]

d XLIPS (2)4 4(pi pX) Mi!XMX!f ; (A.1)

where X represents a complete set of intermediate states in the amplitudes M and d XLIPS the associated Lorentz-invariant phase space measure. This relation is exact and would hold if we could compute the amplitudes non-perturbatively, and should also hold order-by-order in perturbation theory. The case of interest to us is the one where f i, which gives

2Im [Mi!i] = iXX

[integraldisplay]

d XLIPS (2)4 4(pi pX) [notdef]Mi!X[notdef]2 : (A.2)

For 2 ! 2 reactions and adopting the center-of-momentum reference frame, all kinematic

variables can be integrated over, except the angle between the collision axis and one of the nal-state particle momenta,

2Im [Mi!i] =Xf f

[integraldisplay]

JHEP01(2017)078

16 [notdef]Mi!f[notdef]2 ; (A.3)

where f reads

f =

d cos

p[s (m1 + m2)2] [s (m1 m2)2]s ; (A.4)

with ps being the center-of-mass energy and m1 and m2 the masses of the outgoing particles.

The scattering amplitudes Mi!f can be expanded in partial waves as

Mi!f (s; cos ) = 8

Xj=0(2j + 1)T ji!f(s) dj f i( ) ; (A.5)

where j is the total angular momentum of the nal state (2-body) system, i and f are the initial and nal-state (2-body system) helicities, T ji!f(s) are the amplitudes describing

the transition between the (de nite helicity) states i and f for a given value of j and d are the Wigner d-functions. Multiplying both sides of the equation by dj[prime]

f i( ), integrating

over cos from 1 to 1 and using the identity

[integraldisplay]

22j + 1 j[prime]j ; (A.6)

the j-th partial wave amplitude between the de nite helicity states i and f is given by

T ji!f(s) =

1 16

1 d cos dj f i( )dj

[prime]

f i( ) =

[integraldisplay]

1 d cos Mi!f(s; cos )dj f i( ) : (A.7)

{ 17 {

One therefore obtains,

Im(T ji!i) =

Xf[negationslash]=i f[notdef]T ji!f[notdef]2 ; (A.8)

which yields the following restrictions for the transition amplitudes T ji!f(s),

iRe

hT ji!i(s)[bracketrightBig] 1 ; iIm hT ji!i(s)[bracketrightBig] 2 ;

Xf[negationslash]=i i f

[vextendsingle][vextendsingle][vextendsingle]

T ji!f(s)[vextendsingle][vextendsingle][vextendsingle]

Xf f[notdef]T ji!f[notdef]2 = i[notdef]T ji!i[notdef]2 +

2 1 : (A.9)

In order to compute the helicity amplitudes, we need explicit forms for the wave-functions of the external particles. We work in the Dirac representation throughout our calculation. Spinors of de nite helicity = [notdef]1=2, propagating in the direction ( ; ) and

describing particles with mass m and energy E are represented as

u(E; ; ; ) = pE + m ~ ( ; )

2 pE m ~ ( ; ) [parenrightBigg]

and v(E; ; ; ) = pE m ~ ( ; )

2 pE + m ~ ( ; ) [parenrightBigg]

JHEP01(2017)078

;

(A.10)

where the Weyl spinors ~ are given by

~1/2( ; ) = cos 2

ei sin 2

[parenrightBigg]

and ~1/2( ; ) =

ei sin 2 cos 2

[parenrightBigg]

: (A.11)

The conjugate spinors can be computed as usual with

u = u 0 and similarly for

v, with

0 being taken in the Dirac representation. Polarisation vectors of massless vector elds are represented as

[notdef] =

1p2e[notdef]i (0; cos cos + i sin ; cos sin i cos ; [notdef] sin ) ; (A.12)

and four-momenta are nally written as

p = (p; p sin cos ; p sin sin ; p cos ) : (A.13)

The initial and nal state helicities i and f appearing in eq. (A.7) are de ned as i = 1 2 and f = 3 4, as we consider a 2 ! 2 collision where the colliding particles

are labelled as 1, 2, 3 and 4. By convention, the particles 1 and 3 are chosen to propagate in the ( ; ) = (0; 0) and ( f; 0) direction respectively, the choice = 0 not a ecting the results since all distributions of nal-state particles are azimuthally symmetric. Consequently, the particles 2 and 4 propagate in the ( ; +) = (; ) and ( f; ) direction respectively.

For the new physics model considered in this paper, we treat the momentum-dependent and momentum-independent operators separately. Extracting the Feynman rules from the momentum-independent part of the Lagrangian in eq. (2.1), the transition amplitude for the gg ! process reads

MMI = scs csg4 1 (p1p 2 g (p1 [notdef] p2)) 2

1k2 m2s

: (A.14)

{ 18 {

The only non-zero partial amplitudes are associated with the transition (+; +) ! (0; 0),T 0(+,+)!(0,0) =

csgcs ss

642(m2s s)

s )

1/4 ; (A.16)

where an extra factor of 1/2 has been added to account for the identical nal-state particles. For the momentum-dependent part of the Lagrangian, the transition amplitude reads

MMD = scs csg4 1 (p1p 2 g (p1 [notdef] p2)) 2

csgc@s ss2

642f2(m2s s)

2(s m2s) ss2

1 4m 2 s

1/4 : (A.19)

Focusing on the process gg ! , and for given values of masses and couplings, these

relations can be used to extract the maximal allowed value for s for which our e ective description makes sense perturbatively. Conversely, for a given value of s it can be used in order to bound the parameters of our model, see section 3.3.

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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Abstract

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Abstract

We consider minimal dark matter scenarios featuring momentum-dependent couplings of the dark sector to the Standard Model. We derive constraints from existing LHC searches in the monojet channel, estimate the future LHC sensitivity for an integrated luminosity of 300 fb ^-1, and compare with models exhibiting conventional momentum-independent interactions with the dark sector. In addition to being well motivated by (composite) pseudo-Goldstone dark matter scenarios, momentum-dependent couplings are interesting as they weaken direct detection constraints. For a specific dark matter mass, the LHC turns out to be sensitive to smaller signal cross-sections in the momentum-dependent case, by virtue of the harder jet transverse-momentum distribution.

Details

Title

Monojet searches for momentum-dependent dark matter interactions

Author

Barducci, Daniele; Bharucha, Aoife; Desai, Nishita; Frigerio, Michele; Fuks, Benjamin; Goudelis, Andreas; Kulkarni, Suchita; Polesello, Giacomo; Sengupta, Dipan

Pages

1-24

Publication year

2017

Publication date

Jan 2017

Publisher

Springer Nature B.V.

e-ISSN

10298479

Source type

Scholarly Journal

Language of publication

English

DOI

https://doi.org/10.1007/JHEP01(2017)078

ProQuest document ID

1865252232

Monojet searches for momentum-dependent dark matter interactions

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