EExtended Dark Matter EFT
Tommi Alanne ∗ and Florian Goertz † Max-Planck-Institut f¨ur Kernphysik, Saupfercheckweg 1, 69117 Heidelberg, Germany
Conventional approaches to describe dark matter phenomenology at collider and (in)direct de-tection experiments in the form of dark matter effective field theory or simplified models suffer ingeneral from drawbacks regarding validity at high energies and/or generality, limiting their appli-cability. In order to avoid these shortcomings, we propose a hybrid framework in the form of aneffective theory, including, however, both the dark matter states and a mediator connecting theformer to the Standard Model fields. Since the mediation can be realized through rather lightnew dynamical fields allowing for non-negligible collider signals in missing energy searches, theframework remains valid for the phenomenologically interesting parameter region, while retainingcorrelations dictated by gauge symmetry. Moreover, a richer new-physics sector can be consistentlyincluded via higher-dimensional operators. Interestingly, for fermionic and scalar dark matter witha (pseudo-)scalar mediator, the leading effects originate from dimension-five operators, allowing tocapture them with a rather small set of new couplings. We finally examine the correlations betweenconstraints from reproducing the correct relic density, direct-detection experiments, and mono-jetand Higgs + missing energy signatures at the LHC.
I. INTRODUCTION AND SETUP
The origin of the dark matter (DM) observed in theuniverse is one of the biggest mysteries in physics. Amultitude of experiments, which are probing very diverseenergies, are currently running or in preparation to ad-dress this question. Experiments aiming for a direct de-tection of DM particles via nuclear recoil typically probecollision energies in the keV range, while collider exper-iments, trying to produce DM particles, feature momen-tum transfers exceeding the TeV scale.Combining the results from all kinds of experimentsin a single, consistent, yet general framework is impor-tant in order to resolve the nature of DM. Bounds fromdirect detection experiments are usually interpreted inan effective field theory (EFT) approach, removing themediator that couples the DM particles to the StandardModel (SM) as an explicit dynamical degree of freedomat low energies. This is possible, if the mediator is as-sumed to be much heavier than the scale of the exper-iments, and its effects can thus be described by genericEFT operators, consisting only of the SM and DM fields.On the other hand, collider experiments such as theLHC run at much larger energies, and it turns out thatthe momentum transfers lie typically at (or above) inter-esting mass ranges for the mediator, to which the analy-ses are sensitive (unless the model is very strongly cou-pled [1]). Thus, the EFT description becomes invalidsince the mediator is missing in the spectrum [2, 3]. Inconsequence, collider searches are typically interpretedin terms of simplified models where the mediator is notremoved, and its interactions with the SM and the DMare simply parametrized by D ≤ ∗ [email protected] † fl[email protected] require gauge-invariance and are thus not well behavedat large energies. A further drawback of this approach isthat they are still specific models that do not allow for amaximally general description of the dark sector. In theend, one suffers either from a lack of generality or, evenworse, from a lack of validity.In this article, we want to propose a hybrid frame-work, which can alleviate the above problems: We willconsider the minimal amount of additional dynamical de-grees of freedom—a DM particle and a mediator—whichis able to generate the correct DM abundance and allowsfor testability at TeV scale energies, while retaining thegenerality and consistency of the EFT framework.We will, thus, consider the SM extended by a SM-singlet particle, D , that is stable on cosmological scales,and a mediator, M , that couples it to the SM, as wellas higher-dimensional EFT operators, consisting of thesefields. We will start by assuming the DM to be a fermion, D = χ , and the mediator a (pseudo)scalar, M = S ( ˜ S ),and then move on to consider also the case of scalar DM.In these setups, the leading EFT effects will be at thelevel of D = 5 operators, and their inclusion allows toparametrize physics of the dark sector beyond the singleDM particle and the mediator. In fact, while the inclusion of the latter particles makesthe theory valid at collider energies, the augmentationwith D = 5 operators accounts for the fact that thedark/new sector is likely to be non-minimal. Indeed,there is no stringent reason to believe that the sectorrelated with the DM consists only of very few particles,while the SM has a very rich structure. On the otherhand, it is conceivable that a few of the new particles (theDM particle and the mediator) are considerably lighter The generalization to (fermionic or scalar) DM with a vectormediator calls for the inclusion of D = 6 operators and a studyof the very rich phenomenology, and we will leave this for futurework. a r X i v : . [ h e p - ph ] M a r than the rest of the new physics (NP). For example, themediator could be a (pseudo-)Goldstone boson of a spon-taneously broken global symmetry. The goal of this ar-ticle is to povide the theoretical framework of this ex-tended DM EFT (e DMeft ) approach, demonstrating itsstrength in phenomenological analyses, as well as point-ing out emerging synergies and generic correlations be-tween observables, which are retained in the EFT ap-proach.
A. Fermionic DM with a scalar or pseudoscalarmediator
The effective Lagrangian of the model described above,with a fermion singlet, χ , and a (CP even) scalar, S , in-cluding operators up to D = 5 (following normalisationsof Ref. [4]) reads L S χ eff = L SM + 12 ∂ µ S ∂ µ S − µ S S + ¯ χi / ∂χ − m χ ¯ χχ − λ (cid:48) S v S − λ (cid:48) S √ v S − λ S S − λ (cid:48) HS v | H | S − λ HS | H | S − y S S ¯ χ L χ R + h . c . − S Λ (cid:2) c λS S + c HS | H | S + c λH | H | (cid:3) (1) − S Λ (cid:104) ( y Sd ) ij ¯ Q i L Hd j R + ( y Su ) ij ¯ Q i L ˜ Hu j R + ( y S(cid:96) ) ij ¯ L i L H(cid:96) j R + h . c . (cid:105) − y (2) S S + y (2) H | H | Λ ¯ χ L χ R + h . c . − S Λ 116 π (cid:2) g (cid:48) c SB B µν B µν + g c SW W Iµν W Iµν + g s c SG G aµν G aµν (cid:3) . Here Q i L and L i L are the i -th generation left-handedSU(2) L quark and lepton doublets, resp., d j R , u j R , and (cid:96) j R are the right-handed singlets for generation j , and H is the Higgs doublet. The Higgs doublet develops a vac-uum expectation value (vev), |(cid:104) H (cid:105)| ≡ v/ √ (cid:39)
174 GeV,triggering electroweak symmetry breaking (EWSB). Inunitary gauge, the Higgs field is expanded around thevev as H (cid:39) / √ , v + h ) T . Here, h is the physicalHiggs boson, with mass m h ≈
125 GeV. We assume thatthe mediator does not develop a vev and have, thus, in-cluded a linear term in S .Besides the SM couplings, there are several new inter-actions, both at the renormalizable ( D = 4) level and The EFT of the SM plus just a scalar singlet S has been exploredin Refs [4–6]. in the form of effective D = 5 operators. In the scalarsector, the cubic and quartic terms in the singlet poten-tial are parametrized by the couplings λ (cid:48) S and λ S , resp.,while the Higgs-portal couplings involving one or two sin-glets are denoted by λ (cid:48) HS and λ HS , resp. Note that, afterEWSB, the latter coupling provides a contribution to themediator mass, which is given by m S = (cid:113) µ S + λ HS v . (2)In addition, there is a Yukawa coupling between thescalar mediator, S , and the DM fermions, χ , denoted by y S . At the D = 5 level, all interactions are suppressedby one power of the scale of heavy NP, Λ, which medi-ates contact interactions between the various fields. Inthe pure scalar sector, gauge invariant terms feature four( c λH ), two ( c HS ), or zero ( c λS ) Higgs fields. In order tocouple the mediator, S , to SM fermions, the presence of aHiggs doublet is required, allowing for D = 5 Yukawa-likecouplings, ∼ ( y Sd ) ij , ( y Su ) ij . Scalar couplings to the DMfermions at the D = 5 level, on the other hand, involveeither two scalar singlets or two doublets, due to gaugeinvariance, parametrized by y (2) S and y (2) H , respectively.Finally, there are effective couplings of S to the U(1) Y ,SU(2) L , and SU(3) c field strengths squared, denoted by c SB , c SW , and c SG . In the following, we assume the interac-tions with S to conserve CP, and thus all coefficients inthe Lagrangian of Eq. (1) are real.We conclude this discussion noting that, if the newsector residing at the scale Λ is governed by a coupling g ∗ ,the effective coefficients above can be assigned a certainscaling in this coupling (see e.g. [7–10]), which in our case(without assuming the Higgs to be a pseudo-Goldstonestate) reads c λS ∼ c HS ∼ c λH ∼ g ∗ , y Sf ∼ y f g ∗ , y (2) S,H ∼ g ∗ , c SV ∼ g ∗ . This allows to order the operators accordingto their expected importance in a certain coupling regimeand can for example be used to reduce the number of D =6 operators to be considered to leading approximation inthe case of a vector mediator. The e
DMeft
Lagrangian, Eq.(1), allows to describephenomena relevant for collider searches for DM as wellas for direct (and indirect) detection experiments, as wewill now explore in more detail. For example, for a non-negligible ( y Sq ) ij / Λ, the operators S ¯ Q iL Hq jR can mediateinteractions of DM with a nucleus, coupling the mediatorto the DM via the S ¯ χχ interaction, see the upper panel ofFig. 1, where the scalar mediator is depicted by double-dashed lines, while the fermionic DM is represented byfaint double-lines. The same combination of operatorsinduces, on the other hand, DM signals at the LHCin two different, but correlated, incarnations. Mono-jetand Higgs+missing transverse energy ( / E T ) signatures are As shown recently [11], such a counting could also allow to liftthe ambiguity in determining masses of new states in an EFTapproach. ( y Sq ) ij v Λ y S q iL q jR χ L χ R v ( y Sq ) ij y S Λ m S q iL q jR χ L χ R ( y Sq ) ij v Λ y S q iL q jR χ L χ R g ( y Sq ) ij y S q iL q jR χ L χ R h FIG. 1. Relevant diagrams contributing to nuclear inter-action with fermionic DM (first row) and corresponding DMobservables at hadron colliders (monojet and Higgs+ / E T , sec-ond row), turning on the interactions ∼ ( y Sq ) ij and ∼ y S . Thediagrams are similar for scalar and pseudo-scalar mediators,where for the latter case, the operator coefficients in Eq. (1)are to be replaced by the corresponding (tilded) coefficientsin Eq. (3). See text for details. generated, radiating off a gluon from the (initial state)quarks and considering the physical Higgs within H inthe S ¯ Q iL Hq jR operator, resp., while the S → ¯ χχ tran-sition is responsible for the / E T , as shown in the lowerpanel of Fig. 1. Note that, since the Yukawa-like couplings ∼ ( y Sq ) ij can feature flavour-changing neutral currents (FCNCs),a flavour-protection mechanism like minimal flavour vi-olation could be thought of, which would lead to asuppressed coupling to the light valence quarks andthus small effects at colliders at the tree level (see alsoRef. [16]). In an agnostic approach, however, all cou-plings could be treated as free, and some tuning withingthe Yukawa structure could be allowed, considering only‘direct’ experimental constraints allowing, in principle,for considerable effects for valence quarks. Finally notethat, in case light-quark contributions were suppressed,the operators could still induce a coupling to gluons atthe one-loop level, via heavy-quark triangle diagrams.If other D = 5 operators (as well as the portal cou-pling, λ (cid:48) HS ) are set to zero for the moment, all threeprocesses above scale in terms of effective coefficients as y S ( y Sq ) ij / Λ, and we can explore the complementarity andcombined information of both types of experiments in oneframework. We will examine this in more detail in the At low (nuclear) energies, we can also integrate out the mediatorto arrive at a four-fermion interaction, as considered in the usualDM EFT [12–15]. Now, however, the effective coefficient is fixedby Eq. (1). c SG / Λ y S gg χ L χ R c SG y S Λ m S gg χ L χ R c SG / Λ y S gg χ L χ R g FIG. 2. Relevant diagrams contributing to nuclear interac-tion with fermionic DM (first row) and corresponding DM ob-servables at hadron colliders (monojet, second row), turningon the interactions ∼ c SG and ∼ y S . The diagrams are simi-lar for pseudo-scalar mediators, employing the correspondingtilded coefficients, as discussed before. next section. As an alternative option, we will also con-sider the coupling of the mediator to the proton/nucleusvia the S G aµν G aµν operator, trading ( y Sq ) ij for c SG , whichallows for the production of DM in gluon fusion. The cor-responding diagrams for the processes discussed aboveare shown in Fig. 2, where similar correlations can beexplored (with the Higgs-associated production now be-ing absent at leading order). In the same context, an-other interesting opportunity is to produce the mediatorin weak-boson fusion (WBF), by turning on c SW or c SB , asdepicted by the diagrams in the last row of Fig. 15 andcommented on further below.On the other hand, the Higgs boson might play a cru-cial role in coupling the DM to quarks and gluons. First,it can provide a portal to the mediator, via the couplings λ HS , λ (cid:48) HS , connecting the SM to the dark sector. Inparticular if the second operator is present, the medi-ator can be produced in gluon-fusion Higgs productionvia mixing with the Higgs field connecting then to theDM via y S . The corresponding diagrams are given inFig. 3, where again unavoidably the Higgs+ / E T channelis present, fixed by gauge invariance. Finally, the oper-ators in Eq. (1) also allow for interactions of DM withhadrons mediated directly by Higgs exchange, if the coef-ficient y (2) H is non-vanishing. Turning on this single cou-pling provides an instantaneous link between the Higgsfield and the DM via a contact interaction, inducing allprocesses discussed before, with the diagrams given inFig. 4. Once more, the Higgs+ / E T channel is induced bygauge invariance.Let us conclude this discussion by emphasizing againthat the Lagrangian of Eq. (1) allows to consistently com- FIG. 3. Relevant diagrams contributing to nuclear inter-action with fermionic DM (first row) and corresponding DMobservables at hadron colliders (monojet and Higgs+ / E T , sec-ond row), turning on the interactions ∼ λ (cid:48) HS and ∼ y S . Notethat the corresponding diagrams are not present for pseudo-scalar mediators, which only appear in pairs. bine various processes and to include information fromdifferent kinds of sources. For example, DM might beproduced by a combination of different mechanisms, e.g.via the mediator S (triggered by ( y Sq ) ij , c SG , or a portal),but also via direct Higgs exchange—due to the effective y (2) H , even without H − S mixing—where each contri-bution leads to characteristic correlations between LHCphysics and direct (as well as indirect) detection experi-ments. We note that the direct Higgs couplings also enterin invisible Higgs decays constraining their size, while res-onance searches are directly sensitive to the properties ofthe mediator. The e DMeft allows to describe and com-bine all these different phenomena in a general (inclusive)and consistent way including resonance searches for themediator particle, which would not be possible in a sim-ple DM EFT or a simplified-model approach. Yet, it issimple enough to keep predictivity and to be straight-forwardly implemented into tools for automated eventgeneration and implementation of constraints. Finally,matching UV complete models to the e
DMeft will alsoallow to interpret experimental results obtained in thelatter framework in terms of such explicit models, with-out the need to repeat the analysis for numerous differentsetups.Before exploring in more detail the LHC and direct-detection phenomenology in the e
DMeft approach, wewill now turn to the remaining scenarios of DM coupledto the SM via a potentially light mediator, which canbe analyzed in a similar way. First, we will consider thecase of a CP-odd scalar, ˜ S , which has several interestingfeatures. On the one hand, direct detection bounds witha CP-odd scalar mediator are much weaker due to mo- FIG. 4. Relevant diagrams contributing to nuclear inter-action with fermionic DM (first row) and corresponding DMobservables at hadron colliders (monojet and Higgs+ / E T , sec-ond row), turning on the interaction ∼ y (2) H . mentum suppression of the cross section [17, 18]. More-over, besides the D = 5 pure scalar interactions, the por-tal with a single mediator vanishes, which automaticallyavoids the mixing between the scalars. The Lagrangiannow becomes L ˜ S χ eff = L SM + 12 ∂ µ ˜ S ∂ µ ˜ S − µ S ˜ S + ¯ χi / ∂χ − m χ ¯ χχ − λ ˜ S S − λ H ˜ S | H | ˜ S − y ˜ S ˜ S i ¯ χ L χ R + h . c . − ˜ S Λ (cid:104) ( y ˜ Sd ) ij i ¯ Q iL Hd jR + ( y ˜ Su ) ij i ¯ Q iL ˜ Hu jR + ( y ˜ S(cid:96) ) ij i ¯ L iL H(cid:96) jR + h . c . (cid:105) (3) − y (2)˜ S ˜ S + y (2) H H † H Λ ¯ χ L χ R + h . c . − ˜ S Λ 116 π (cid:104) g (cid:48) c ˜ SB B µν ˜ B µν + g c ˜ SW W Iµν ˜ W Iµν + g s c ˜ SG G aµν ˜ G aµν (cid:105) . Here, the contact interactions with gauge bosons fea-ture dual field strength tensors, while all terms can beinterpreted analogously as in the CP-even scalar case. Inparticular, the very same discussion as before on genericcorrelations between different observables can be per-formed. Note, however, that due to the vanishing single-mediator portal, no production via the Higgs is possi-ble at the level of D ≤ ∼ y ˜ Sq , c G,B,W , y (2) H even more in-teresting. We will again assume no new sources of CPviolation, and the coefficients in Lagrangian in Eq. (3)are real. B. Scalar DM with a scalar mediator
We finally move to the case of (singlet) scalar DM,denoted by D = χ s still considering a scalar mediator, M = S . The Lagrangian for this setup, at the D = 5level, reads L S χ s eff = L SM + 12 ∂ µ S ∂ µ S + 12 ∂ µ χ s ∂ µ χ s − λ (cid:48) S v S− µ S S − m χ s χ s − λ (cid:48) S √ v S − λ S S − λ χ s χ s − λ (cid:48) HS v | H | S − λ HS | H | S − λ (cid:48) Sχ s √ v S χ s − λ Sχ s S χ s − λ Hχ s | H | χ s − S Λ (cid:2) c λS S + c HS | H | S + c λH | H | + c Sχ s S χ s + c λχ s χ s + c Hχ s | H | χ s (cid:3) − S Λ (cid:104) ( y Sd ) ij ¯ Q iL Hd jR + ( y Su ) ij ¯ Q iL ˜ Hu jR + ( y S(cid:96) ) ij ¯ L iL H(cid:96) jR + h . c . (cid:105) − S Λ 116 π (cid:2) g (cid:48) c SB B µν B µν + g c SW W Iµν W Iµν + g s c SG G aµν G aµν (cid:3) . (4)We assume again that the mediator, S , does not developa vev, while a Z symmetry assures stability of χ s .For pseudo-scalar, M = ˜ S , the terms in the lasttwo square brackets are replaced similarly as before(Eq. (1) → Eq. (3)). However, all further contributionswith an odd number of S are absent, and further dynam-ics would be required for s -channel mediation betweenthe DM and SM fields, and therefore, we do not considerthis scenario here.Crucial (new) terms in Eq. (4) are the portal-like in-teractions connecting the DM to the mediator, λ Sχ s , λ (cid:48) Sχ s (and the corresponding D = 5 operators, containingodd powers of S and even powers of χ s , i.e., the terms ∼ c Sχ s , c λχ s , c Hχ s ), replacing y S /y ˜ S of the fermionic DMcase. Furthermore, there is a new direct D = 4 portal tothe DM via the Higgs field, given by λ Hχ s , instead of thecorresponding term ∼ y (2) H for fermionic DM, as well asa quartic DM self-interaction ∼ λ χ s . We will discuss thecorresponding changes in the processes described abovefor fermionic dark matter, including the respective dia-grams, in the next section. II. COMPLEMENTARY CONSTRAINTSWITHIN ONE FRAMEWORK: eDM
EFT
We will now explore phenomenological aspects of allthe models described above, making use of the correla-tions predicted in our e
DMeft approach, as anticipated before. In particular, we will address the question ofhow many events in DM searches at the LHC can be ex-pected, given limits from direct detection experiments,without restricting to an explicit NP model. The pres-ence of the mediator in our EFT will ensure the validityof the analysis for collider searches. We will finally dis-cuss the inclusion of further processes like loop-inducedDM production via D = 5 operators, invisible Higgs de-cays, Higgs pair production, as well as resonance searchesin the same framework.As main observables, we consider the direct-detectioncross section for the scattering of DM off a nucleus, N , σ N ≡ σ ( N D → N D ) , (5)and the cross section for a monojet signal at the 13 TeVLHC σ j ≡ σ ( pp → j + / E T ) . (6)In addition, in the cases where the mediator is producedvia the new D = 5 operators coupling it to quarks, weexpect also the cross section for producing the Higgs inassociation with missing transverse energy to be compa-rable to the monojet cross section and consider also σ h + / E T ≡ σ ( pp → h + / E T ) . (7)We can now calculate how many events in LHC DMsearches (monojet and Higgs+ / E T ) are possible, given thelimits from direct detection, in the framework of differ-ent e DMeft s discussed in the last section, each repre-senting a large class of NP models. The LHC resultscould thus potentially rule out certain DM incarnationsvery generally or support/constrain them in synergy withdirect-detection experiments. In that context, note thatthe currently most stringent direct detection constraintcorresponds to N = Xe and is given by XENON1T [19].Going through the different scenarios, we will addressthis question, always turning on a set of (one or two)NP couplings that allow for a different production mech-anism of the DM or different DM–nucleus interaction,considering quark- and gluon-induced production, pro-duction through the Higgs-mediator portal, and produc-tion trough direct Higgs–DM interactions.The requirement to generate the correct relic abun-dance implies further constraints on the same opera-tors that are entering DM–SM scattering and thus di-rect detection. On the other hand, the limits from di-rect detection for a subdomimant DM component areweaker, since the direct detection experiments assume aone-component DM with the observed relic abundance.Therefore, in order to correctly compare with the limits,the produced DM abundance must always be estimated.Another source of constraints arises from DM annihi-lations potentially producing an excess of e.g. gammarays over the galactic background. The most stringentcurrent limits come from the Fermi-LAT satellite exper-imemt [20] constraining the canonical thermal cross sec-tion up to ∼
100 GeV DM masses. We focus here onscenarios with heavier DM candidate and leave furtheranalysis of indirect observables to future work.
A. Quark-induced production
We start by considering the production of the medi-ator, M = S , in q ¯ q annihilation and subsequent decayto DM (as well as the crossed process leading to D − N scattering), see Figs 1 and 5.
1. Fermionic DM
For fermionic DM, D = χ , the relevant couplings for aCP-even mediator, S , are ( y Sq ) ij and y S , see Eq. (1). Theformer allows the production of S via a gauge-invariantcoupling to SM quarks and the latter its decays to DM.The corresponding Feynman diagram is given in the up-per left corner of Fig. 1. At low energies relevant fordirect-detection experiments, the mediator can be inte-grated out leading to the diagram in the upper right cor-ner, which governs D− N scattering at low momenta. Forsimplicity, we only consider ( y Su ) , although the analysiscan easily be extended to include all quark flavours.For a CP-odd mediator, ˜ S , the couplings above are re-placed by the corresponding tilded coefficients in Eq. (3).However, in this case, the tree-level interactions with nu-clei are momentum suppressed, and this scenario is outof the reach of current direct detection experiments (andthus LHC cross sections are basically unconstrained).Nevertheless, as discussed in Ref. [21], the future exper-iments will start probing this scenario as well. Scenarioswith CP-odd mediator have also been considered recentlyin e.g. Refs [22–24]. Here, we concentrate on a CP-evenmediator, and leave the phenomenology of CP-odd me-diators, utilizing the full strength of the e DMeft ap-proach, for future work.The cross section for the DM scattering off nuclei, interms of e
DMeft couplings, reads [25] σ N = y S [( y Su ) )] ( f uN ) m N µ N v π Λ m S m u , (8)where f uN is the form factor defined by (cid:104) N | m q ¯ qq | N (cid:105) ≡ m N f qN , µ N ≡ m χ m N m χ + m N (9)is the reduced mass of the DM-nucleon system, and m N = ( m p + m n ) / (cid:104) N | y Sq ¯ qq | N (cid:105) from the EW scale (where we define our cou-plings) down to direct-detection energies and the corre-sponding threshold effects from integrating out the heavyquark flavours following Ref. [26] (see also[27, 28]). Forthe case with only ( y Su ) non-zero, this effect is trivial. ( y Sq ) ij v Λ λ ′ Sχ s v q iL q jR χ s χ s v ( y Sq ) ij λ ′ Sχs Λ m S q iL q jR χ s χ s ( y Sq ) ij v Λ λ ′ Sχ s v q iL q jR χ s χ s g ( y Sq ) ij λ ′ Sχ s v q iL q jR χ s χ s h FIG. 5. Relevant diagrams contributing to nuclear inter-action with scalar DM (first row) and corresponding DM ob-servables at hadron colliders (monojet and Higgs+ / E T , secondrow), turning on the interactions ∼ ( y Sq ) ij and ∼ λ (cid:48) Sχ s . We fix y S such that we obtain the fraction f rel ≡ Ω χ / Ω DM of the total dark matter abundance. Inthe direct-detection limits, a single component DM with f rel = 1 is assumed, and thus we require f rel σ N ≤ σ X1T when we compare with the limits for XENON1T.If m χ < m S , the DM annihilation cross section is dom-inated by the s -channel process to SM quarks via the D = 5 operator, and the cross section is thus propor-tional to y S [( y Su ) ] . This is the same combination ap-pearing in the direct-detection cross section and valuesavoiding overabundance are disfavoured by experiments(which does not necessarily hold if the DM couples toheavy quarks). Therefore, we concentrate here on themass range m χ > m S , where the dominant annihila-tion channel is χχ → SS . The annihilation cross sectioncan thus be estimated (neglecting subleading m S contri-butions, which are considered in the numerical results)by [21] (cid:104) σ v (cid:105) ( χχ → SS ) ≈ . × − cm s − y S (cid:18) m χ (cid:19) , (10)where v ∼ .
23. We fix y S as a function of f rel bycomparing this to the standard thermal cross section (cid:104) σ v (cid:105) = 3 · − cm s − and using Ω h ∝ (cid:104) σ v (cid:105) − toscale this with f rel . The latest limit [19] then leads tothe bound (see Fig. 6) | ( y Su ) | Λ (cid:46) . × − f − / (cid:16) m S (cid:17) TeV − . (11)Attaching on the other hand a gluon to the initial statequarks or emitting a Higgs boson—possible directly viathe contact interaction ∼ ( y Sq ) ij —leads to final statesconsidered in LHC searches for DM, i.e. monojet andHiggs+ / E T signatures. The corresponding diagrams aregiven in the lower panel of Fig. 1.We calculate the maximal cross sections for these pro-cesses at the LHC at 13 TeV center-of-mass energy us-ing MadGraph [29] and employing the direct-detectionlimit, Eq. (11), for two benchmark scenarios: ( m S =400 GeV , m χ s = 500 GeV) and ( m S = 500 GeV , m χ s =1 TeV). We arrive at (fixing for simplicity f rel = 1) σ j | m χ =500 GeV (cid:46) . · − fb ,σ j | m χ =1 TeV (cid:46) . · − fb , (12)and σ h + / E T | m χ =500 GeV (cid:46) . · − fb ,σ h + / E T | m χ =1 TeV (cid:46) . · − fb . (13)These cross sections are tiny, but serve here as referencevalues for this utterly simplified scenario with only twoaddtional couplings turned on. For example, already al-lowing for heavy quark flavors to couple to the mediator,interpolating between the light quark case and the gluoncase discussed in the next section, is expected to increasethe cross section significantly.
2. Scalar DM
For scalar DM, D = χ s , the corresponding couplingsare still ( y Sq ) ij for the q ¯ q production of the mediator,while the decay to DM is now induced by the portal term λ (cid:48) Sχ s . The Feynman diagrams for the processes discussedabove are analogously presented in Fig. 5, where χ s isrepresented by faint dashed double-lines. Note that fora CP-odd mediator the above portal linear in ˜ S is notpresent.The scattering cross section off nuclei now becomes [25] σ N = ( λ (cid:48) Sχ s ) ( y S ) ( f uN ) m N µ N v π Λ m S m u m χ s . (14)Again, to estimate the relic density, we concentrateon m χ s > m S , whence the annihilation cross sectionyields [21] (cid:104) σ v (cid:105) ( χ s χ s → SS ) ≈ . × − cm s − · (cid:18) λ (cid:48) Sχ s v √ m S (cid:19) (cid:18) m χ s (cid:19) . (15)Trading λ (cid:48) Sχ s for f rel similarly as above, we show thebounds from direct detection again in Fig. 6. For thebenchmark scenarios ( m S = 400 GeV , m χ s = 500 GeV)and ( m S = 500 GeV , m χ s = 1 TeV), we obtain limits | ( y Su ) | Λ | m χs =500 GeV (cid:46) . × − f − / TeV − , | ( y Su ) | Λ | m χs =1 TeV (cid:46) . × − f − / TeV − . (16) g SDD = λ S χ s g SDD = y S - - - m DM [ GeV ] y u S Λ [ T e V - ] FIG. 6. Limits from the direct-detection experiments for thecoupling ( y Su ) / Λ as a function of the DM mass. Blue (yel-low) lines correspond to fermionic (scalar) DM with CP-evenmediator, and g Sχχ denotes the corresponding mediator–DMcoupling. In the plot, we have fixed m S = 500 GeV, and thesolid (dashed) lines correspond to f rel = 1 ( f rel = 0 . Using these and fixing f rel = 1, the maximal values ofthe LHC cross sections become σ j | m χs =500 GeV (cid:46) . · − fb ,σ j | m χs =1 TeV (cid:46) . · − fb , (17)and σ h + / E T | m χs =500 GeV (cid:46) . · − fb ,σ h + / E T | m χs =1 TeV (cid:46) . · − fb . (18)which again are tiny as expected for this simplified sce-nario. B. Gluon-fusion production
We now turn to the case of coupling the mediator toa gg state, allowing its production in gluon-gluon fusion,which is complementary to the case of external q ¯ q states.Since the coupling to the DM is still assumed to be in-duced by y S and λ (cid:48) Sχ s , for the cases of fermionic DM χ with a CP-even mediator S and scalar DM χ s , respec-tively, we only replace ( y Su ) by c SG to couple the media-tor to gluons. The relevant diagrams are given in Figs 2and 7. Note that, due to the absence of the Higgs con-tact interaction in the NP sector, the Higgs+ / E T channelis significantly suppressed with respect to the monojetsignature, requiring a GGh interaction, induced in theSM via a top loop, and we do not consider that here.
1. Fermionic DM
Starting with the S GG interaction at the EW scalewill now induce the S ¯ qq couplings at the nuclear energy c SG / Λ λ ′ Sχ s v gg χ s χ s vc SG λ ′ Sχs Λ m S gg χ s χ s c SG / Λ λ ′ Sχ s v gg χ s χ s g FIG. 7. Relevant diagrams contributing to nuclear interac-tion with scalar DM (first row) and corresponding DM ob-servables at hadron colliders (monojet, second row), turningon the interactions ∼ c SG and ∼ λ (cid:48) Sχ s . scale [26]. Therefore, the cross section for scattering offnuclei now becomes σ N = y S ( c SG ) m N µ N π Λ m S (cid:88) q = u,d,s ( c qG f qN ) + 29 c gG f gN , (19)where c qG and c gG account for the running and thresholdeffects down to the nuclear-energy scale, the gluonic formfactor is defined as (cid:104) N | − g s π G µν G µν | N (cid:105) ≡ m N f gN ,and we again employ the results of Ref. [26].Comparing now with XENON1T results, leads to thebound c SG Λ (cid:46) . × f − / (cid:16) m S (cid:17) TeV − , (20)shown in Fig. 8.The monojet cross section is thus constrained for thebenchmark scenarios ( m S = 400 GeV , m χ = 500 GeV)and ( m S = 500 GeV , m χ = 1 TeV) and f rel = 1 to σ j | m χ =500 GeV (cid:46) . · fb ,σ j | m χ =1 TeV (cid:46)
250 fb . (21)These values are significantly higher than in the quark-induced production, connected to the relative growth ofthe gluon parton distribution functions with respect tothe up-quark one from nuclear to collider energies andthe negative interference between the quark and gluonform factors at nuclear scales. Therefore this scenarioprovides an interesting possibility to partially evade thestrong direct-detection limits. g SDD = λ S χ s g SDD = y S m DM [ GeV ] c G S Λ [ T e V - ] FIG. 8. Limits from the direct-detection experiments for thecoupling c SG / Λ as a function of the DM mass. Blue (yellow)lines correspond to fermionic (scalar) DM with CP-even medi-ator, and g Sχχ denotes the corresponding mediator–DM cou-pling. In the plot, we have fixed m S = 500 GeV, and the solid(dashed) lines correspond to f rel = 1 ( f rel = 0 .
2. Scalar DM
For scalar DM (with a CP-even mediator), we obtain σ N = ( λ (cid:48) Sχ s ) ( c SG ) m N µ N v π Λ m S m u · (cid:88) q = u,d,s ( c qG f qN ) + 29 c gG f gN , (22)and the resulting bounds are shown in Fig. 8. For thebenchmark scenarios ( m S = 400 GeV , m χ s = 500 GeV)and ( m S = 500 GeV , m χ s = 1 TeV), the limits read c SG Λ | m χs =500 GeV (cid:46) f − / TeV − ,c G Λ | m χs =1 TeV (cid:46) f − / TeV − . (23)In consequence, the monojet cross section is boundedby σ j | m χs =500 GeV (cid:46)
670 fb ,σ j | m χs =1 TeV (cid:46)
140 fb . (24)Again the cross sections are comparable to the fermionic-DM case, and the conclusion that the gluon-induced pro-duction provides an interesting scenario also for collidersearches. C. Higgs–mediator portal
Another interesting option is to couple the DM to theSM via the portal term involving the mediator and the
FIG. 9. Relevant diagrams contributing to nuclear inter-action with scalar DM (first row) and corresponding DM ob-servables at hadron colliders (monojet and Higgs+ / E T , secondrow), turning on the interactions ∼ λ (cid:48) HS and ∼ λ (cid:48) Sχ s . Higgs field, ∼ λ (cid:48) HS , turning on this coupling in additionto y S ( λ (cid:48) Sχ s ) for fermionic (scalar) DM. This allows for aSM-like production of the scalar mediator via its mixingwith the Higgs field, while its coupling to the DM remainsas in the cases discussed above. The relevant diagramsare given in Figs 3 and 9. Note that in the figures for sim-plicity, we employ a mass-insertion approximation, wherethe mixing of the Higgs with the new scalar is treated asan interaction, marked by a black cross. In the numericalanalysis below, we instead diagonalize the H − S system.Moreover, the Higgs+ / E T channel receives the contri-bution from the second Higgs field present in the portal,as depicted by the lower-right diagrams in Figs 3 and 9,respectively.We diagonalise the Higgs– S system via a rotation h = h cos α + S sin α,H = − h sin α + S cos α, (25)with the angle, α , given bytan 2 α = 2 λ (cid:48) HS v λ H v − µ S . (26)Further, we trade λ H (the coefficient of the quartic Higgsoperator) and µ S for the masses of the eigenstates, m h and m H , the lighter of which we identify with the 125-GeV Higgs boson. m H =
400 GeV m H =
500 GeV
500 1000 5000 10 m DM [ GeV ] λ H S FIG. 10. Limits from the direct-detection experiments for thecoupling λ (cid:48) HS as a function of the DM mass for the Higgs–mediator portal scenario with fermionic DM. Blue (yellow)lines correspond to a mass of the heavy scalar eigenstate of400 GeV (500 GeV). Solid (dashed) lines correspond to f rel =1 ( f rel = 0 .
1. Fermionic DM
For fermionic DM, we finally obtain σ N = y S f N m N µ N πv sin α cos α (cid:18) m h − m H (cid:19) = y S f N m N µ N ( λ (cid:48) HS ) v πm h m H . (27)For the relic density calculation, we now add the ¯ tt chan-nel mediated by the Higgs doublet. We focus on m χ > m t and use the estimate [21] (cid:104) σ v (cid:105) ( ¯ χχ → ¯ tt ) ≈ . × − cm s − · y S sin α cos α (cid:18) m χ (cid:19) . (28)For the scalar channels, we use Eq. (10) after scaling thecouplings with the appropriate mixing coefficients givenin Eq. (25).Comparing now with XENON1T results, leads to thebounds shown in Fig. 10, and the monojet cross sectionis bounded due to the direct detection limit as σ j | m χ =500 GeV (cid:46) . · − fb ,σ j | m χ =1 TeV (cid:46) . · − fb . (29)0 m H =
400 GeV m H =
500 GeV400 600 800 1000 1200 14000.20.40.60.8 m DM [ GeV ] λ H S FIG. 11. Limits from the direct-detection experiments for thecoupling λ (cid:48) HS as a function of the DM mass for the Higgs–mediator portal scenario with scalar DM. Blue (yellow) linescorrespond to a mass of the heavy scalar eigenstate of 400 GeV(500 GeV). Solid (dashed) lines correspond to f rel = 1 ( f rel =0 .
2. Scalar DM
For scalar DM, we arrive at σ N = ( λ (cid:48) Sχ ) f N m N µ N πm χ s sin α cos α (cid:18) m h − m H (cid:19) = ( λ (cid:48) Sχ s ) f N m N µ N ( λ (cid:48) HS ) v πm χ s m h m H . (30)The annihilation cross section to top quarks for m χ s > m t now becomes [21] (cid:104) σ v (cid:105) ( χ s χ s → ¯ tt ) ≈ . × − cm s − · λ (cid:48) Sχ s sin α cos α (cid:18) m χ s (cid:19) , (31)and for the scalar channels we use Eq, (15) scaling thecouplings with the mixing coeffients, see Eq. (25). Thedirect detection bounds are shown in Fig. 11, and thecorresponding LHC cross sections are now predicted tobe below σ j | m χs =500 GeV (cid:46) . · − fb ,σ j | m χs =1 TeV (cid:46) . · − fb . (32)Although this process is not observable in this restrictedscenario, the portal contribution could become relevantin a combined analysis with more operators turned on. D. Higgs–DM portal
Finally, the DM could also be coupled directly to theSM via the Higgs–DM portal, promoting the SM-like
FIG. 12. Relevant diagrams contributing to nuclear inter-action with scalar DM (first row) and corresponding DM ob-servables at hadron colliders (monojet and Higgs+ / E T , secondrow), turning on the interaction ∼ λ Hχ s . Higgs boson itself to a mediator to the DM sector, simplyby turning on the D = 5 operator ∼ y (2) H or D = 4 portal ∼ λ Hχ s , for fermionic or scalar DM, respectively. Thiscorresponds to the traditional SM+singlet DM modelstudied extensively in the literature; see e.g. Refs [25, 30–33].While the invisible width of the Higgs boson signifi-cantly constrains these operators in case the DM is light, m χ ( s ) < m h / E T channelis induced at leading order via the second Higgs field re-quired by gauge invariance, see the lower-right diagrams.
1. Fermionic DM
For fermionic DM, we obtain the direct detection crosssection σ N = ( y (2) H ) f N m N µ N π Λ m h . (33)We trade y (2) H / Λ for f rel from the thermal ¯ χχ → ¯ tt, W W, ZZ, hh cross section for fixed m χ s . The dom-inant contribution corresponds to the t -channel annihi-lation to hh , which can be obtained using Eq. (10) withsubstitution y S → y (2) H v/ Λ. We show the direct detectionlimits in the ( m DM , f rel ) plane in Fig. 13. The monojetcross section is in turn limited to σ j | m χ =500 GeV (cid:46)
49 fb ,σ j | m χ =1 TeV (cid:46) . . (34)1 λ H χ s y H ( )
200 300 400 500 600 700 800 90010 - - m DM [ GeV ] f r e l FIG. 13. The blue (yellow) curve shows the maximum allowedfraction of the DM abundance, f rel , as a function of the DMmass fixed by limits from direct detection for the Higgs–DMportal scenario with fermionic (scalar) DM. We only showthe mass range m DM (cid:46)
900 GeV where the relic abundanceis reproduced for λ Hχ s < π . This scenario could provide another interesting probe forthe collider experiments, but is constrained to deliveronly a tiny fraction of the DM relic abundance.
2. Scalar DM
For the thermal cross section, including χ s χ s → ¯ tt, W W, ZZ, hh , we use the result from [34]: (cid:104) σ v (cid:105) = λ Hχ s πm χ s (4 − r h ) (cid:104) r t (1 − r t ) / + (cid:88) V = W,Z δ V r V (2 + (1 − /r V ) ) √ − r V + 2 (cid:18) λ Hχ s λ H (1 − r h / r h r h − r h (cid:19) √ − r h (cid:35) , (35)where r i = m i /m χ s , and δ W = 1 , δ Z = 1 / σ N = λ Hχ s µ N m N f N πm χ s m h , (36)so we plot the direct detection limits in the ( m DM , f rel )plane in Fig. 13. The LHC cross section in that final caseis thus constrained to σ j | m χs =500 GeV (cid:46) .
38 fb . (37)We note that the cross section is somewhat smaller thanthat of the fermionic-DM case, making the latter scenariomore promising for the collider experiments. ( y Sq ) ii Λ y S gg χ L χ R h vλ H ˜ S y (2)˜ S Λ gg χ L χ R FIG. 14. Potentially important loop diagrams contributingto DM interactions in the e
DMeft , turning on the operators ∼ ( y Sq ) ij and ∼ y S , or ∼ λ H ˜ S and ∼ y (2)˜ S , respectively. Theright diagram involves the quartic portal term ∼ λ H ˜ S , whichis basically the only relevant term that allows to producescalar DM via a (pair-produced) pseudo-scalar mediator atthe D ≤ y (2)˜ S / Λ → λ ˜ Sχ s ).See text for details. E. Further Processes
While dedicated analyses are left for future work, herewe already comment on further potentially interestingapplications of the e
DMeft framework ranging from theinclusion of loop processes in the EFT, over new produc-tion mechanisms, up to analyses of invisible Higgs decays,Higgs pair production and collider searches for the medi-ator.
1. Loop mediated processes in e
DMeft
While the only loop diagrams we encountered so farcontained the SM-like
GGh triangle, loops involving D = 5 vertices allow for interesting new means to cou-ple the DM to hadrons. First of all, the Yukawa-likeoperator ∼ ( y Sq ) ij can now be inserted coming with thetop quark ( q = u ; ij = 33) and coupling the mediatorto a gluon pair via a (top-)quark loop, see the left dia-gram in Fig. 14. This operator is expected to be sizable,featuring no (minimal-flavour-violation-like) flavour sup-pression. The loop suppression is thus lifted by the ex-pected enhancement with m t /m q , since the operator isbasically unconstrained for the top. A detailed study offlavour constraints on the coefficients, combining themwith complementary bounds from limits on the invisibleHiggs width and other searches, such as to derive con-clusive limits on DM production in the e DMeft , will bepresented elsewhere.Finally, at the two-loop level, DM production via aportal to a pseudo-scalar mediator becomes possible, seethe right diagram in Fig. 14. Applied to the case ofscalar DM (i.e. replacing y (2)˜ S / Λ → λ ˜ Sχ s ), this opensthe possibility to obtain potentially viable models with aCP-odd mediator.2 c SW,B / Λ y S q iL q jR χ L χ R c SW,B y S m q Λ m S q iL q jR χ L χ R c SW,B / Λ y S q iL q jR χ L χ R g c SW,B / Λ y S q iL q jR χ L χ R h c SW,B / Λ y S q iL,R q jL,R χ L χ R g c SW,B / Λ y S q iL,R q jL,R χ L χ R h FIG. 15. Diagrams contributing to nuclear interaction withfermionic DM (first row) and DM observables at hadron col-liders, turning on the interactions ∼ c SW and ∼ y S . The dia-grams are similar for pseudo-scalar mediators, employing thecorresponding tilded coefficients, as discussed before. See textfor details.
2. Weak-boson fusion
In the same context, a production of the mediator inweak-boson fusion, as depicted by the last two diagramsin Figs 15 and 16, is interesting regarding ‘monojet’ andHiggs+ / E T signals. In fact, if the corresponding D = 5operators feature a sizable coupling c SW,B (or S → ˜ S , fora CP-odd mediator), DM signals at the LHC can be sig-nificant, while limits from direct-detection experimentsare met, since the corresponding processes with externalquark bi-linears are suppressed by a quark-mass insertion(and a loop factor), see the first (and second) row of Figs15 and 16.
3. Higgs pair + / E T Exploring the production of Higgs pairs in associationwith missing energy could be an additional interestingprobe of dark sectors. Indeed, several operators in thee
DMeft allow the production of Higgs pairs along withDM and can be tested in this process. Fig. 17 shows sam-ple diagrams that become potentially important in casethe bi-quadratic portal ∼ λ HS is weak. Beyond the usualcase of the mediator decaying to the DM, they also con- c SW,B / Λ λ ′ Sχ s v q iL q jR χ s χ s c SW,B λ ′ Sχs vm q Λ m S q iL q jR χ s χ s c SW,B / Λ λ ′ Sχ s v q iL q jR χ s χ s g c SW,B / Λ λ ′ Sχ s v q iL q jR χ s χ s h c SW,B / Λ λ ′ Sχ s v q iL,R q jL,R χ s χ s g c SW,B / Λ λ ′ Sχ s v q iL,R q jL,R χ s χ s h FIG. 16. Diagrams contributing to nuclear interaction withscalar DM (first row) and DM observables at hadron colliders,turning on the interactions ∼ c SW and ∼ λ (cid:48) Sχ s . See text fordetails. tain potentially resonant decays of M to a Higgs pair,after emission of a DM pair, which could allow to seea peak in the hh invariant mass spectrum and boostedHiggs bosons [35] . Although the cross sections are notexpected to be large, non-negligible NP couplings couldstill feature interesting effects in hh + / E T production.A dedicated analysis is needed to examine the actualprospects of this process in the light of the expected lim-ited number of events.
4. Invisible Higgs decays
Note that the e
DMeft can also significantly effect sin-gle Higgs physics. For example, for light DM ( m D ≤ m h )the e DMeft operators involving the Higgs field (includ-ing D = 4 Higgs portals) are constrained more and moreseverely from limits on invisible Higgs decays. Corre-sponding diagrams, involving the portal coupling λ (cid:48) HS together with y S or λ (cid:48) Sχ s for the decay to DM are givenin the right-hand side of Fig. 18, while those with direct Note that, for the given couplings turned on, there are addi-tional (potentially) similar important contributions, attachingthe Higgs or mediator lines differently in the diagrams above(including Higgs emissions from SM lines). χ L χ Rv λ ′ HS y (2) S Λ vλ ′ HS gg hh hh vc λH Λ y S gg χ L χ R χ s χ s vc Hχs Λ vλ ′ HS gg hh hh v λ ′ HS c
Hχs Λ gg χ s χ s FIG. 17. Selected diagrams contributing to Higgs pair pro-duction in association with / E T , turning on the interactions ∼ λ (cid:48) HS , y (2) S or ∼ c λH , y S in the case of fermionic DM (firstrow) and ∼ λ (cid:48) HS , c Hχ s in the case of scalar DM (second row).See text for details. h − D − D interactions y (2) H or λ Hχ s are depicted in theleft-hand side of the same figure. A production of themediator via D = 5 e DMeft operators avoiding theseconstraints becomes particularly interesting.Moreover, the additional scalar particle S can have aninteresting impact on the nature of the electroweak phasetransition, which in the SM is not first order, such as toallow for electroweak baryogenesis [36, 37]. With the helpof a light S , electroweak baryogenesis can become viable(see, e.g. [38–41]). In turn, the DM phenomenology willbe affected. Including such a scenario in our frameworkis also left for the future. S Resonance search
Finally, one can also search directly for the mediatorby looking for a resonant enhancement in the di-jet, t ¯ t ,di-lepton, or di-boson spectrum. These processes can alsobe described consistently in the e DMeft , since the inclu-sion of the mediator as a dynamical degree of freedom is adefining feature of the setup, and their study can deliverimportant insight on the nature of DM. Sample diagramsfor the first two processes, involving the couplings λ (cid:48) HS and ( y Sq ) ij are shown in Fig. 19, and the other resonantprocesses are generated via similar diagrams (includingthe production of the resonance in gluon fusion and thedecay via Higgs mixing). Again, the black cross denotesa mass insertion, while in the (diagonal) mass basis justthe heavy mediator is exchanged in the s-channel, withits coupling to the t ¯ t state governed by the Higgs admix-ture. Resonance searches are in fact a powerful comple-mentary tool to understand dark sectors and to probethe coupling structure of the mediator even in the case v y (2) H Λ h χ L χ R v λ ′ HS y S h χ L χ R vλ Hχ s h χ s χ s v λ ′ HS vλ ′ Sχ s h χ s χ s FIG. 18. Contributions to the invisible width of the Higgsboson from turning on the interactions ∼ y (2) H , ∼ λ (cid:48) HS , y S , ∼ λ Hχ s , ∼ λ (cid:48) HS , λ (cid:48) Sχ s . v λ ′ HS ( y Sq ) ij v Λ gg q iL q jR v λ ′ HS ( y Sq ) ij v Λ q iL q jR t L t R FIG. 19. Resonant contributions of the mediator to di-jetand t ¯ t final states, turning on the interactions ∼ λ (cid:48) HS and ∼ ( y Sq ) ij . Replacing the latter by c SW,B,G or ( y S(cid:96) ) ij leads tofurther interesting contributions to di-boson, di-jet, and di-lepton final states. where its production is dominated by a single operator.Being able to combine this information with that of theother DM observables in a single consistent, yet general,framework is a particular strength of the e DMeft . III. CONCLUSIONS
We have presented a new framework to describe DMphenomenology, which can be used to consistently con-front limits from direct-detection experiments and therelic abundance with possible collider signatures at highenergies, while maintaining a high degree of model-independence. Both the DM and mediator fields re-main as propagating degrees of freedom, whereas addi-tional new physics is described in the form of higher-dimensional operators. As an application of the frame-work, focusing on the level of D = 5 operators, we de-rived possible cross sections in monojet and Higgs+ E miss signatures at the LHC. Considering the limits from di-4rect detection and reproducing (a certain fraction of) theDM relic density, for scalar mediator, we found that themost promising case for the collider searches is gluon in-duced production, which could lead to sizable LHC crosssections both for fermionic and scalar DM. Cases withpseudoscalar mediators are still rather unconstrained bydirect-detection experiments, and e DMeft will revealits full strength with the future generation of direct-detection experiments. These results are valid in a moregeneral context than those derived in conventional (sim-plified) models. A plethora of possible further applica-tions of the framework is left for future work.
ACKNOWLEDGMENTS
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