The waning of the WIMP? A review of models, searches, and constraints
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Abstract
Weakly Interacting Massive Particles (WIMPs) are among the bestmotivated dark matter candidates. No conclusive signal, despite an extensive search program that combines, often in a complementary way, direct, indirect, and collider probes, has been detected so far. This situation might change in near future due to the advent of one/multiTON Direct Detection experiments. We thus, find it timely to provide a review of the WIMP paradigm with focus on a few models which can be probed at best by these facilities. Collider and Indirect Detection, nevertheless, will not be neglected when they represent a complementary probe.
1 Introduction
 (i)
The relic abundance of DM needs to account for the observed CDM abundance;
 (ii)
the DM particle should be nonrelativistic at matterradiation equality to form structures in the early Universe in agreement with the observation. As a result, if the DM was produced as a thermal relic in the early Universe, its mass cannot be arbitrarily light. Specifically, cosmological simulations rule out DM masses below a few keV [4, 5, 6].
 (iii)
The DM should be electromagnetically neutral, as a result of null searches for stable charged particles [7, 8] as well as Direct Detection (DD) experiments, which we will review subsequently.
 (iv)
The DM particle must be cosmologically stable since its presence is ascertained today, implying that its lifetime is larger than the age of the Universe. Under certain assumptions, much stronger limits are applicable conservatively requiring a lifetime order of magnitude larger can be derived [9, 10, 11, 12, 13, 14, 15, 16].
 (v)
Cluster collisions, such as the Bullet Cluster [17], constrain the level of selfinteractions that DM particles can have (see however, Refs. [18, 19] for alternative scenarios).
In this work we will attempt to give an uptodate state of the art of ongoing WIMP searches and possible future prospects.
The extreme broadness of the topic makes, however, very difficult to satisfactory cover all the different DM search strategies. Consequently we will mainly focus on DM DD, motivated by the advent of highly sensitive one and multiTon detectors. We will then investigate, a selection of simple, but well motivated, WIMP models whether and to which extent the WIMP paradigm can be tested through this kind of search strategy.
Even within the WIMP framework, other DM search strategies like collider searches and Indirect Detection (ID) provide a complementary and essential contribution. This complementarity will be highlighted in some relevant cases of study.
Before discussing the main topic, we will anyway provide in the next sections a general and pedagogical introduction to the WIMP paradigm for the generation of the cosmological abundance of the DM relic density and to the three main categories of DM searches: DD, ID and collider searches.
2 The WIMP paradigm
The paradigm of thermal decoupling, based upon applications to cosmology of statistical mechanics and particle and nuclear physics, is enormously successful at making detailed predictions for observables in the early Universe, including the abundances of light elements and the CMB [21]. It is somewhat natural to invoke a similar paradigm to infer the abundance of DM as a thermal relic from the early Universe uniquely from the underlying DM particle properties.
The WIMP paradigm hence reduces, under the hypothesis of standard cosmological evolution of the Universe, the solution of the DM problem to the determination of a single particle physics input, i.e., the thermally averaged pair annihilation crosssection of the DM.
Since WIMPs freeze out in the nonrelativistic regime, and thus, \(v\ll c\) (where v is the relative velocity of the two annihilating WIMPs), a useful approximation consists of a velocity expansion (given in the Appendix) \(\langle \sigma v \rangle \simeq a + b v^2\). The velocity expansion is, however, not valid in some relevant cases, like for example annihilations through the resonant exchange of an schannel mediator [25]. For this reason, all the numerical results presented in this work will rely on the full numerical determination of \(\langle \sigma v \rangle \), as given in Eq. (14) and on the solution of the DM Boltzmann equation, as provided by the numerical package micrOMEGAs [26, 27, 28].
As a result, concrete realizations of WIMP models have been developed in different Beyond the Standard Model (BSM) frameworks, accessible to several different search strategies, as reviewed in the next sections.
3 Direct detection
Astrophysical and gravitational evidences indicates the existence of DM halos, surrounding all the visible structures like galaxies and the cluster of galaxies. The halos are formed by DM particles described by a time dependent velocity distribution. DD experiments aim at detecting DM, through scattering off nuclei, belonging to the halo surrounding our galaxy, flowing through the Earth. Several experiments have played an important role in this direction [29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59]. In this section we focus on DD experiments looking for WIMPs scattering, but there are important searches stemming from Neutrino telescopes by measuring the neutrino flux from the Sun [60, 61, 62, 63].
 1.
Energy threshold: drives the sensitivity to low WIMP masses, and consequently the sharpening of the DD limits on the scattering crosssection at low masses as shown in Fig. 3;
 2.
Control over the background and exposure: determine the overall sensitivity of the experiment pushing the limits to lower scattering crosssections assuming that they are statistically dominated;
 3.
Target: has an impact on the experimental sensitivity to low and heavy WIMP masses, as well as on the capability to probe SD scatterings.
It is important to highlight that from going to the measured scattering rate in Eq. (16) to the derivation of a limit on the WIMPnucleon scattering crosssection as a function of the WIMP mass, there are some assumptions that have to be made about the velocity distribution, nuclear form factor, type of WIMPnucleon scattering, and local DM density that suffers from large uncertainties [74, 75, 76]. In particular, the common assumptions are that there is a smooth halo of DM particles in our galaxy well described by a Maxwellian velocity distribution [77, 78, 79], that the nucleus can be treated as a hard sphere as indicated by the Helm form factor [70], and that the WIMPnucleon scattering is elastic. Our results rely on the same set of assumptions throughout this manuscript (see Refs. [80, 81, 82, 83, 84] for further discussions on these topics). Interestingly, if the uncertainties present in the astrophysical input are under control and precise measurements on the scattering crosssection can be realized, then one might even determine the nature of DM using DD experiments alone [85].

Current spinindependent limit:

Current spindependent limit:

Projected spinindependent limit:

Projected spindependent limit:
The limits (current and projected) which will be adopted throughout all this work are illustrated in Fig. 4. The two panels report, in a bidimensional plane of DM mass and scattering crosssection, the region which are/will be excluded, under the null result, by the concerned experiments. As evident that the experimental sensitivity appears to be very different between SI and SD interactions. This is due to the fact that the SI crosssection of the DM on a nucleus originates from a coherent sum of the contributions from the interactions of the DM with the single nucleons. Targets, like Xenon, with atomic number of the order of 100 feature then an enhanced sensitivity to SI interactions. The case of SD interactions is, instead very different, since the contribution to the different nucleons tend to interfere destructively, so that a sizable crosssection is obtained only for targets with an unpaired nucleon.
As customary, we have expressed the limits on SI interactions in terms of the scattering crosssection of the DM on protons. For SD interactions we have instead reported the scattering crosssection on neutron since Xenon targets, characterizing all the three considered experiments, are mostly sensitive to this process having Xenon isotopes with unpaired neutrons.
In most of the models considered in this work the DM candidate will have unsuppressed (i.e., independent from the DM velocity or the momentum transfer) SI interactions. Given the stronger sensitivity, we will consider just the limits from SI interactions in presenting our results while report the ones from SD interactions only when relevant.
4 Indirect detection
ID of the DM relies on the detection of the byproducts of WIMPs annihilations over the expected background at galactic or extragalactic scales, using Earth based telescopes such as HESS and CTA, or satellites such as AMS and FermiLAT [91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105]^{5}

The number density squared of particles, i.e., \(n_{\chi }^2=\rho ^2/m_{\chi }^2\);

The WIMP annihilation crosssection today, \(\sigma \);

The mean WIMP velocity v;

Volume of the sky observed within a solid angle \(\varOmega \);

Number of gammarays produced per annihilation at a given energy and for a given annihilation final state, also known as the energy spectrum (dN / dE).
From Eq. (18) it is clear that ID probes complementary properties of the DM particles. It is sensitive to how the DM is distributed, to the annihilation crosssection today, which might be different than the annihilation crosssection relevant for the relic density, and to the WIMP mass. Therefore, after measuring the flux of gammarays from a given source, we compare it with the background expectations. If no excess is observed, we can choose a DM density profile and select an annihilation final state needed for dN / dE, and then derive a limit on the ratio \(\sigma v/m_\chi ^2\) according to Eq. (18). This is the basic idea behind experimental limits. Although, more sophisticated statistical methods have been conducted such as likelihood analysis.
An interesting aspect of indirect DM detection, when it comes to probe WIMP models, is the fact that if the annihilation crosssection, \(\sigma v\), is not velocity dependent, bounds on \(\sigma v\) today are directly connected to the DM relic density. In particular, the observation of gammarays from dwarf spheroidal galaxies (dSphs) results in stringent limits on the plane of annihilation crosssection vs WIMP mass. If for a given channel the annihilation crosssection of \(10^{26}~\mathrm{cm}^3~\mathrm{s}^{1}\) is excluded for DM masses below 100 GeV, it also means that one cannot reproduce the right relic density for WIMP masses below 100 GeV.^{6} In other words, in this particular case, ID limits will trace the relic density curve. This effect will be clearly visible in many instances.
5 Collider searches
LHC proton–proton collisions might result in the production of WIMPs in association with one or more QCD jets, photons as well as other detectable SM debris. Since WIMPs are electrically neutral and cosmologically stable massive particles, they manifest at colliders as missing transverse momentum. For this reason searches for DM are based on the observation of the visible counterpart of the event such as charged leptons, jets or a photon, generally referred to as monoX searches. By selecting events with large missing transverse momentum/energy one can reduce the SM background and potentially disentangle a DM signal. However, as mentioned above, what colliders identify is missing energy, and therefore they cannot uniquely ascertain the presence of DM in a signal event. They can simply confirm the presence of a neutral and “stable” particle, that might have even decayed outside the detector.
Nevertheless, colliders offer an exciting and complementary search strategy to identify WIMPs. Indeed, assuming that the production of WIMPs at colliders is uniquely connected to the WIMPnucleon scatterings at underground laboratories, one can use the nonobservation signals with large missing transverse momentum to derive limits on the WIMPnucleon scattering crosssection [123, 124, 125, 126, 127, 128, 129, 130].
A large part of this work will be devoted to the study of the cases in which the interactions between DM and SM particles are due to a neutral spin0 or spin1 schannel mediator (see next sections). In this kind of scenario a compelling complementary collider probe is represented by the searches of its “visible” (i.e., into SM final states) decay channels.
We now review in some detail the specifics of the aforementioned search channels for WIMPs at colliders.
5.1 MonoX searches
MonoX searches stand for the search of WIMPs produced in association with one or more QCD jets or potentially other SM particles, such as \(\gamma \), h, Z etc. The idea is to search for events with a jet of high transverse momentum \(p_T\) within an event with large missing transverse momentum. In particular, the most recent studies performed at the LHC include up to four jets and require the leading jet to have \(p_T > 250\) GeV [131, 132], while others do not limit the number of jets while selecting events with at least one jet with \(p_T > 100\) GeV [133]. While being more inclusive, these recent searches have become more challenging due to the number of jets analyzed, requiring a substantial improvement on the background coming from \(Z+jet\) and \(W+jet\) channels.
There are important detector effects, such as fake jets, and QCD backgrounds that weaken the LHC sensitivity to WIMPs, and for these reasons monojet searches are subject to large systematics. Nevertheless, fortunately an enormous effort has been put forth in this direction with data driven background and optimized event selections, which combined with the increase in luminosity has led to an overall improvement on the LHC sensitivity to WIMPs.
That said, in the review, we will be using the latest results from CMS and ATLAS collaborations in the search for DM based on monoX searches [133, 134].
Now we discuss the WIMP production at colliders and also address another collider constraint relevant for the DM purposes which has to do with the invisible decay widths of the SMHiggs as well as Z bosons.
5.2 Invisible Higgs decays
If WIMPs are lighter than 62.5 GeV, the Higgs boson might invisibly decay into WIMP pairs. In this case, one can use bounds from LHC on the invisible branching ratio (Br) of the Higgs, \(\mathrm{Br (h\rightarrow inv)} \le 0.25\) at 95% C.L. [135, 136], to set constraints on WIMP models . Throughout the manuscript whenever applicable we compute the invisible decay rate of the Higgs into WIMPs and impose the aforesaid upper limits to obtain the limits displayed in the figures.
5.3 Invisible Z decays
The decay width of the Z boson has been precisely measured and therefore stringent limits can be derived on any extra possible decay mode of the Z boson. In some of the models as we will discuss further, the DM particle does couple to the Z boson, thus, when mass of the DM is smaller than half of the Z mass stringent limits are applicable. In particular, one can use only direct measurements of the invisible partial width using the single photon channel to obtain an average bound which is derived by computing the difference between the total and the observed partial widths assuming lepton universality. The current limit is \(\varGamma \mathrm{(Z\rightarrow inv)} \le 499 \pm 1.5\) MeV [137].
5.4 Searches of visible decays of the mediator
As already mentioned a relevant case of study of this review is represented by the case in which a new neutral field mediates interactions between SM fermion pairs and DM pairs. Provided that its mass is within the reach of a collider, it can be singly produced thanks to its coupling with the SM fermion pairs. The monoX searches discussed before essentially probe the decay channels of the mediator into DM pairs (the monoX is radiated by one of the initial state fermions). A potentially even more powerful probe is, however, provided by the visible decay channels of the BSM mediator. Its onshell production, and subsequent decay into SM states, may lead to spectacular signals represented by dijet and/or dilepton resonances^{7} peaked at the mass of the mediator field. Among the models which will be discussed here these kinds of signal are particularly prominent in the case of spin1 mediators, since their gaugelike couplings with the SM fermions allow a high production crosssection for these resonances. The considered models with spin0 mediators feature instead poorer prospects since a Yukawatype structure will be assumed for their couplings with the SM fermions. We will consequently specialize our discussion to the case of spin1 resonances.
Among dijet and dilepton resonance searches, the latter have typically the potential to exclude larger portions of the parameters space. This can be understood by noticing the different backgrounds, QCD jets and Drell–Yan production for dijet and dileptons respectively. Dilepton resonances typically represent a much clearer signal, hence provide the strongest constraints unless the couplings of the spin1 mediator with SM leptons are very suppressed, with respect to the ones with the SM quarks, or even null.
6 Model setup: dark portals
In order to maximally profit of the information from the different kind of experimental searches we need an efficient interface between the experimental outcome and theoretical models. The processes responsible for the DM relic density and its eventual detection can be described by simple extensions of the SM in which a DM candidate interacts with the SM states (typically the interactions are limited to the SM fermions) through a mediator state (dubbed portal). This idea is at the base of the socalled “Simplified Models” [140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154] which are customarily adopted especially in the context of collider studies, see e.g., Refs. [130, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166].
Keeping a similar spirit we will discuss a selection of DM setups, with increasing degree of refinement, which we will globally refer to as “Dark Portals”. These are characterized by spin0, spin1/2 and spin1 DM candidates (we will distinguish whenever relevant the cases of self and not selfconjugated DM) interacting with the SM fields, typically the electrically neutral bosons, and/or possibly a spin0 or spin1 mediator field.^{8}
In summary we will discuss a collection of models mostly built according to a bottomup approach and maintains a substantial degree of simplicity. Nevertheless, contrary to the aforementioned “simplified models”, they also aim at a more faithful description of theoretically motivated scenarios. For example, in most cases we will adopt specific choices of the parameters inspired by possible theoretical completions of the models under consideration. In the same fashion we will account, in our analysis, besides experimental constraints, the theoretical limitations of these frameworks (see e.g., Refs. [151, 152, 161, 162, 168] for more extensive discussions.)
Besides these considerations the choice of the models under study is dictated by the main purpose, already stated in the introduction, of assessing the capability of present and, more importantly, near future DD facilities of probing the WIMP paradigm. For each of the models under scrutiny we will determine the portion of the parameter space excluded by current DD limits, as set by the XENON1T experiment, and the parameter space which would be ruled out in case of absence of signals, after 2 years of exposure time at the XENON1T experiment itself, and at the multiTON detector LZ. Most of the considered models will then be characterized by efficient interactions of the DM with nucleons. At the same time we will discuss possible scenarios capable of surviving even eventual limits from high sensitivity future experiments like LZ. Limits from other kind of DM searches, i.e., ID and LHC, will not be neglected when competitive or complementary with the ones from DD.
 1.
SM portals: These represent a special case of schannel portals, i.e., Eqs. (21) and (22), in which \(S=h\) and \(Z^{'}=Z\) with h and Z being the SM Higgs and Z bosons, respectively.^{10} These are the most predictive models since the couplings of the mediators with the SM states are fixed. Moreover, \(\mu _\chi ^h=\mu _V^h=v_h\). These leave us with just two freeparameters, the DM mass and an adimensional coupling. On the other hand, these models are also the most strongly constrained ones so that they are already strongly disfavored by the present limits;
 2.
BSM schannel portals: These are the models fully described by Eqs. (21) and (22). Here the DM is coupled with the SM fermions either by a new spin0 (real) or by a spin1 electrically neutral state. These scenarios, besides the SM portals, are the most sensitive ones to DD of the DM;
 3.
Portals evading DD: Here we will instead consider the case of a pseudoscalar schannel mediator, in which constraints from DD are particularly weak so that the complementarity with other search strategies becomes crucial. We will also considered a more theoretically refined scenario in which the pseudoscalar mediator is part of a complex scalar field. The presence of an additional scalar component will reintroduce DD bounds. A broad region of the parameter space for thermal DM is nevertheless reopened by considering a very light pseudoscalar, which can be interpreted as a pseudoGoldostone boson of a global symmetry carried by the original complex scalar field;
 4.
tchannel portals: In this alternative version of the Dark Portals the mediator field has nontrivial quantum numbers with respect to the SM gauge group. In these kind of setups DM annihilation arise from tchannel interactions while schannel interactions are responsible for DD;
 5.
Portals to secluded sectors: We assume here that the mediator field cannot be directly coupled with the SM fermions. Even in these kind of constructions a portal can be originated by mass mixing with the SM Higgs, in the case of spin0 mediators, and by kinetic mixing with the Zboson in the case of a spin1 field, so that the DM actually interacts through a double schannel mediator, represented by a SM and a BSM state;
 6.
Portals with DM (partially) charged under \(SU(2) \times U(1)\): In these models the DM is charged under the EW symmetry group, being either the lightest electrically neutral component of a (or more) SU(2) multiplet (we will discuss in detail only the case of SU(2) doublets) or a mixed state between the latter and a SM singlet. Note that this category of models also contain renormalizable completions of the Higgs and Zportal models although the actual phenomenology features sensitive differences.
The SM portals are the simplest models since they are characterized by only two free parameters, the DM mass and couplings. As a consequence the comparison between DM relic density and DD limits/prospects (and eventual additional constraints) can be performed in full generality in a bidimensional plane of the two quantities. s and tchannel portals, as appear for the aforementioned model categories 2, 3 and 4, feature two mass scales, the DM mass and the mediator, and one or at most two relevant couplings (in some cases this actually implies additional assumptions, clarifications will be provided in the dedicated sections). For these scenarios we have adopted the two masses as freely varying parameters and assigned \(\mathscr {O}(1)\) values to the couplings unless this option is precluded by theoretical considerations.
There is very modest loss of generality in this choice. The DM annihilation and scattering crosssection feature a very similar dependence on the free couplings, lower values of the couplings correspond to weaker limit/prospect from DD but, at the same time, progressively disfavor the achievement of the correct relic density so that the overall picture is only marginally affected. For the last category of models the interplay between DD and relic density is less trivial. In this case we have considered, besides representations in bidimensional planes of two relevant parameters, also scans in which all the free parameters of the models are varied.
Concerning, in particular, the DM mass, we have typically assumed a range of variation from 10 GeV to a few TeV. The choice of the lower bound is motivated by our focus on DD as DM search strategy. For masses below 10 GeV, experimental sensitivity is strongly reduced because of the approaching of the energy threshold of the experiments under consideration. We also remark that achieving the correct relic density becomes increasingly more difficult as the DM mass decreases, because of the reduced number of accessible final states (for this reason in the case of DM interactions mediated by BSM fields the lowest value of the DM mass shown in the figures is 100 GeV). Finally for several realizations, namely the cases in which the annihilation crosssection is velocity independent (swave), thermal DM below 10 GeV is already ruled out by ID as well as CMB measurements [1].
Concerning the maximal value of the DM mass, i.e., a few TeV, this should be regarded mostly as an assumption. We remark, nevertheless, that in bottomup setups it might result difficult to accommodate viable thermal DM without encountering issues of theoretical consistency. In other cases, like the ones discussed in Sect. 12, for a high value of the DM mass the correct relic density cannot be achieved because of a systematic suppression of the annihilation crosssection. Relevant viable WIMP scenarios, featuring DM heavier than a few TeV, like the socalled “minimal DM”, have been proposed in Refs. [169, 170, 171]. These kind of models will not be explicitly revised here; we will nevertheless discussed them in slightly more detail in Sect. 12.
7 SM portals
The first class of models which will be the object of study are the SM Dark Portals,^{11} i.e., models in which the DM interacts with the SM state through the Higgs or the Zboson. In the case when the DM is a pure SM singlet, gauge invariant renormalizable operator connecting the DM with the Z or the SMHiggs boson can be build only in the latter case and only for scalar and vectorial DM. In the other cases one should rely either on higher dimensional operators, or on the case that the coupling with the Higgs and/or the Z is originated by their mixing with new neutral mediators. The latter case can imply the presence of additional states relevant for the DM phenomenology and will be then discussed later on in the text. We will instead quote below some example of higher dimensional operator but we will not refer to any specific construction for our analysis. Alternatively one could assume that the DM has some small charge under \(SU(2)_L\) or \(U(1)_Y\), see e.g., Refs. [173, 174, 175, 176, 177, 178, 179]. We will not review these scenarios here.
7.1 Higgs portal
As already pointed out, in the case of a scalar and vectorial DM it is possible to rely on a dimension4 renormalizable operator; on the contrary a fermionic DM requires at least a dimension5 operator which depends on an unknown UltraViolet (UV) scale \(\varLambda \).
After EW symmetry breaking (EWSB), trilinear couplings between the Higgs field h and DM pairs are induced. In the case of fermionic DM it is possible to absorb the explicit \(\varLambda \) dependence by a redefinition of the associated coupling, i.e., \(\lambda _\psi ^H \frac{v_h}{\varLambda }\) as \(\lambda _\psi ^H \), so that it does not appear explicitly in computations.
In Fig. 5 we summarize our results for scalar, fermionic and vectorial DM, respectively. All the plots report basically three set of constraints.^{13} The first one (red contours) is represented by the achievement of the correct DM relic density.^{14} The DM annihilates into SM fermions and gauge bosons, through schannel exchange of the SMHiggs boson, and, for higher masses, also into Higgs pairs through both s and tchannel diagrams (in this last case a DM particle is exchanged). Since the coupling of the SMHiggs with SM fermions and gauge bosons depends on the masses of the particles themselves, the DM annihilation crosssection is suppressed, at the exception of the pole region \(m_\chi \sim m_h /2\), until the \(W^+W^\), ZZ and \(\overline{t} t\) final states are kinematically accessible. Even in this last case, the cosmologically allowed values for the couplings are in strong tension with the constraints from DD of the DM, which for all the considered spin assignation of the DM, arise from SI interactions of the DM with the SM quarks originated by tchannel exchange of the SMHiggs boson. As can be easily seen that the entire parameter space corresponding to thermal DM is already ruled out, at the exception, possibly, of the pole region, for DM masses at least below 1 TeV. Eventual surviving resonance regions will be ruledout in case of absence of signals at the forthcoming XENON1T, assuming a 2 years of exposure time. As expected, the most constrained scenario is the fermionic DM one because of the further suppression of the pwave suppression of its annihilation crosssection.
Notice that, scalar and vectorial DM, due to the swave annihilation crosssection, might also be probed through ID. The corresponding limits are nevertheless largely overpower by the ones from DD and hence, have been then omitted for simplicity. The limits from DD experiments are complemented at low DM masses, i.e., \(m_{\chi ,\psi ,V} < m_h/2\), by the one from invisible decay width of the Higgs. Indeed this constraint would exclude DM masses below the energy threshold of DD experiments. Our findings are in agreement with the other recent studies in the topic [143, 145, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205].
7.2 Zportal
In addition, after EWSB, an effective dimension4 interaction like \((g^2/16\cos ^2\theta _W)\lambda ^{ZZ}_{\chi \chi }{\chi }^2 Z^\mu Z_\mu \) can emerge from the dimension6 SM gauge invariant operator \(\lambda _{\chi \chi }\) \((D^\mu H)^\dagger \) \(D_\mu H {\chi }^2/\varLambda ^2\) such that \(\lambda ^{ZZ}_{\chi \chi }=\lambda _{\chi \chi }v^2_h/\varLambda ^2\). For simplicity we maintain a rescaling with powers of the \(SU(2)_L\) gauge coupling g.
As evident, in all but the Majorana Zportal case, thermal DM is already excluded, even for masses above the TeV scale, by current constraints from XENON1T. These constraints are even stronger with respect to the case of the SMHiggs portal. This is because, apart from the lighter mediator, the scattering crosssection on Xenon nuclei is enhanced by the isospin violating interactions of the Z with light quarks. Low DM masses, possibly out of the reach of DD experiments, are instead excluded by the limit on the invisible decay width of the Zboson. As already pointed out, the only exception to this picture is represented by the case of Majorana DM where the SI component of the DM scattering crosssection is largely suppressed due to the absence of a vectorial coupling of the DM with the Z. This scenario is nevertheless already (partially) within the reach of current searches for a SD component of the scattering crosssection. The increased sensitivity of XENON1T will allow to exclude DM masses below 300 GeV, except the “pole” region. The latter, however, will meet the same fate from a projected future LZ sensitivity.
8 BSM schannel portals
The results presented in the previous cases for the SMHiggs and Zboson portals will be generalized and discussed in more details in the case of generic, BSM spin0 and spin1 mediators interacting with a pair of scalar, fermion or vector DM fields. Contrary to the case of SM portals, interactions of the mediators with the gauge bosons are not mandatory. We will thus stick, in this section to the case, analogous to the socalled simplified models, in which the DM is coupled only to the SM fermions. The case of interactions with gauge bosons will be discussed separately later in the text, where unlike the aforementioned models, we will also assume interactions with both quarks and leptons.
8.1 Spin0 portals
8.1.1 Scalar dark matter
Current limits exclude then low values for both the mass of the DM and the one of the mediator. These limits will become, of course, progressively stronger, in case of absence of signals at XENON1T and/or LZ.
8.1.2 Fermionic dark matter
The main results of our analysis have been summarized in Fig. 10. We have once again considered the DM and scalar masses as free parameters and an analogous assignation of the couplings as in the previous subsection.
8.1.3 Vector dark matter
A similar construction is also possible from a gauge invariant \(D^\mu \mathbf{S} {(D_\mu \mathbf{S})}^*\) operator for a complex scalar field \(\mathbf{S}\) with \(D_\mu =\partial _\mu i \frac{1}{2} \eta ^S_V V_\mu \). However, in this scenario the third term of Eq. (42) would require new BSM charges for the SM fermions.
Table of couplings between the SM fermions and a \(Z'\) (see Eq. (46)) for the three different realizations of a \(Z'\) portal. Here \(s^2_W\equiv \sin ^2\theta _W\)
\(V_u^{Z'}\)  \(A_u^{Z'}\)  \(V_d^{Z'}\)  \(A_d^{Z'}\)  \(V_e^{Z'}\)  \(A_e^{Z'}\)  \(V_\nu ^{Z'}\)  \(A_\nu ^{Z'}\)  

SSM  \(\frac{1}{4}\frac{2}{3} s^2_W\)  \(\frac{1}{4}\)  \(\frac{1}{4}+\frac{1}{3} s^2_W\)  \(\frac{1}{4}\)  \(\frac{1}{4}+ s^2_W \)  \(\frac{1}{4}\)  \(\frac{1}{4}\)  \(\frac{1}{4}\) 
\(E_{6_\chi }\)  0  \(\frac{1}{2\sqrt{10}}\)  \(\frac{1}{\sqrt{10}}\)  \(\frac{1}{2\sqrt{10}}\)  \(\frac{1}{\sqrt{10}}\)  \(\frac{1}{2\sqrt{10}}\)  \(\frac{3}{\sqrt{10}}\)  \(\frac{1}{2\sqrt{10}}\) 
\(E_{6_\psi }\)  0  \(\frac{1}{2\sqrt{6}}\)  0  \(\frac{1}{2\sqrt{6}}\)  0  \(\frac{1}{2\sqrt{6}}\)  \(\frac{1}{4\sqrt{6}}\)  \(\frac{1}{2\sqrt{6}}\) 
8.2 Spin1 portals
In this subsection we will analyze, in analogous fashion as the previous subsection, the case of a schannel spin1 mediator. Unlike the scalar case, this kind of scenario offers a much richer collider phenomenology since one could assume gaugelike interactions (contrary to Yukawalike interactions for spin0 mediators) of the mediator with the SM light quarks and leptons, leading to visible signals, implying stronger constraints [223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241]. As a consequence we will consider a wider mass range, for both DM and the spin1 mediator, in our analysis.
In the case of spin1 DM we repropose the two constructions: the selfconjugate (Abelian) and not selfconjugate (nonAbelian) DM as already proposed in the SM Zportal setup.
8.2.1 Scalar dark matter
Similar to the case of scalar mediator, our main parameters will be represented by the DM and \(Z'\) masses. For what regards the couplings of the \(Z'\) with the SM fermions we will consider some definite assignations, as dictated by the Sequential Standard Model (SSM), i.e., same couplings as the Zboson, and some GUTinspired realizations. According to this the coupling \(g'\) will be set to \(g \approx 0.65\) in the case of SSM and to \(g_\mathrm{GUT}=\sqrt{{5}/{3}}\,g \tan \theta _W \approx 0.46\) for the GUT realizations. Finally, unless differently stated, we will set \(\lambda _\chi ^{Z'}=1\). The different assignations of the \(V_f^{Z'}, A_f^{Z'}\) couplings (see Eq. (46)) considered in our analysis for the three cases of SSM, \(E_{6_\chi }\) and \(E_{6_\psi }\) realizations are exhibited in Table 1. For the same three realizations the effect of different constraints are summarized in Fig. 12.
Contrary to the previous case, these limits are in principle sensitive to modification of the decay branching fraction of the \(Z'\) as a consequence, for example, of couplings with the DM [139]. For the chosen assignation of the couplings, the decay branching fraction of the \(Z'\) into DM pairs is small so that the limits substantially coincide with the ones reported by experimental collaborations. Other limits stemming from flavour and muon \((g2)\) are weaker compared to the collider bounds [268, 269, 270, 271].
8.2.2 Fermionic dark matter
The combination of constraints is reported in Fig. 13 for a Dirac fermion DM, following the color coding of Fig. 12 (Fig. 14).
The SI crosssection, from tchannel exchange of a \(Z'\), in the case of Dirac fermion DM, exactly coincides with the similar one for a complex scalar DM. As a consequence the excluded regions in the \((m_\psi ,m_{Z'})\) plane are the same as shown in the previous subsection.
In the case of Majorana DM, DD principally relies on SD interactions, to which Xenon based detectors are also sensitive. We have then reported, together with the most recent constraints [53, 272], an estimation of the XENON1T and LZ sensitivities. As evident, even in the case of LZ, we have much weaker limits, not competitive with bounds from dilepton searches.
8.2.3 Vector dark matter
The impact of different theoretical and experimental limits on the two scenarios are shown, as customary, in the plane \((m_{Z'},m_V)\), in Fig. 15. In the case of nonAbelian DM the weaker limits from DD do not coincide with a larger viable region for thermal DM since the contemporary suppression of the DM annihilation crosssection into fermions (the annihilation into \(Z'Z'\) is strongly limited by unitarity) allows for the correct relic density only above the limit from LHC dilepton searches ad exception of a tiny region corresponding to the schannel resonance. Much worse is the situation in the case of a nonAbelian vectorial DM. Indeed, already current limits from DD overpower the ones from LHC and exclude thermal DM for both \(m_{Z'}\) and \(m_V\) below 5 TeV.
9 tchannel portals
We consider, in this section, the case in which the DM, instead of coupling in pairs, is coupled with one mediator state and a SM quark. Keeping the assumption that the DM is a SM singlet, the mediator field should carry at least nontrivial charges under the \(SU(3)_C\) and \(U(1)_Y\) of the SM.
To ensure the cosmological stability of the DM some suitable additional quantum numbers for both the DM and mediator should be assumed, so that couplings between any of these two states and only SM states are forbidden.^{22} It is also evident that the DM can be stable only if it is lighter than the tchannel mediator. In order to avoid possible occurrence of flavour violating effects, with consequent strong constraints on the couplings \(\lambda _{\varPsi _{u,d,Q}} (\lambda _{\Sigma _{u,d,Q}})\), we assume that the mediator carries also a flavour quantum number (a “flavored DM” [278, 279, 280] would be equally feasible), for example \(\Sigma _{u}\equiv \) \((\sigma _u,\sigma _c,\sigma _t)\) \((\varPsi _{u}\equiv (\psi _u,\psi _c,\psi _t))\), and that the interactions of Eq. (63) are flavour conserving. This is achieved by assuming the components of the mediator field to be degenerate in masses. We call these masses \(m_{\Sigma _{u,d,Q}}\) and \(m_{\varPsi _{u,d,Q}}\) and the couplings \(\lambda _{\varPsi _{u,d,Q}},\,\lambda _{\Sigma _{u,d,Q}}\) (being actually matrices) and considered them to be diagonal in the flavour space. For further simplification, we will assume all of these couplings to be equal and thus, drop the flavour indices.
The results of our analysis are presented, in the usual fashion, in Figs. 16 and 17 for scalar and fermionic (both Dirac and Majorana) DM, respectively.
Figures 16 and 17 clearly show that the cases of complex scalar and Dirac fermion DM, in which unsuppressed SI interactions are present, are already substantially ruled out.^{23}
More interesting is the case of a Majorana DM. A sizable portion of the parameter space corresponding to the correct DM relic density survives from the present constraints but is within the reach, for masses of both the DM and the mediator up to a few TeV, of the near future DD experiments. Possible complementarity with LHC searches is also worth of being explored.
Collider searches can probe the DM pair production through monoX events and pair production of the scalar or fermionic mediator with subsequent decay into a DM particle and a SM quark through dijet + missing energy events. The latter signature is also considered in supersymmetry (SUSY) searches since it would originate from squark pair production, with subsequent decay into a neutralino and a quark, under the assumption of decoupled gluino. Limits from SUSY searches cannot be straightforwardly applied to the scenario under consideration since the pair production crosssection of the scalar mediator field is enhanced by the presence of diagrams with t/u channel exchange of the DM (the corresponding diagrams in SUSY are suppressed since the concerned couplings of the neutralino with quarks and squarks cannot be of \(\mathscr {O}(1)\)). We have then applied the limits from the dedicated studies [140, 284] (see also [285, 286]), in which searches of both monojet and dijet + missing energy events have been considered, and reported them in the bottom panel of Fig. 17 (orange region with dashed orange border). Similar constraints have been also obtained in the case of Dirac and complex scalar DM. These are, however, overwhelmed by the constraints from DM DD and thus, omitted in the figures.
Our discussion can be extended further by considering mediators charged only under EW interactions while being colour singlet. Even if no interactions between the DM and the quarks are present at the treelevel, DD might still represent a more effective probe with respect to other DM searches.
The impact of the corresponding current/projected constraints is shown in the usual fashion in Fig. 18. Although sensitively weaker with respect to the case of charged scalar mediator, DD constraints are still capable of ruling out the region corresponding to the correct DM relic density, for order one assignation of the coupling. As further test of this scenario one could apply the limits from searches of direct production of sleptons in SUSY models [290]. These limits, however, are weaker with respect to the ones from DD and hence, for simplicity, have not been reported in Fig. 18.
10 Dark portals (partially) evading direct detection
10.1 Pseudoscalar portal
where \(v_E\) represents the DM speed in the Earth frame, \(\varDelta ^N_q\) denotes quark spin content of the nucleon, \(E_R\) is the nuclear recoil energy with recoil velocity q, and \(F^{NN'}_{\Sigma ^{''}}\) are (squared) form factors whose (approximate) analytical expressions are given in Ref. [296]. The crosssection of Eq. (75) corresponds neither to SI nor to SD (although the latter is a good approximation) interactions; for this reason we have expressed it directly in terms of a differential crosssection on a nucleus. As can be easily argued that the \(q^4\) dependence of Eq. (75) implies a very suppressed scattering rate so that a potentially detectable signal is produced only for \(m_a \sim \mathscr {O}(10\,\text{ MeV })\) [295]. These low values are, however, subject to bounds from low energy observables and rare flavor processes (see e.g., Ref. [297] for an extensive analysis). For these reason we will consider here (and also in this section) values of \(m_a\) above 1 GeV.^{25} We also remark that Xenon based detectors, like the ones considered in our study, would be in any case not suited for probing the interaction crosssection of Eq. (75) since it originates from a coupling of the DM mostly with unpaired protons, not present in Xenon, which instead has isotopes with an odd number of neutrons.
As evidenced in Fig. 19, this crosssection is very suppressed so that no constraints come from present experiments. On the contrary, for \(\mathscr {O}(1)\) values of both the \(\lambda _\psi ^a\) and \(c_a\) couplings, next generation detectors can partially probe the parameter space corresponding to the thermal DM.
More stringent constraints come from ID of the DM. Contrary to the scalar mediator case the DM annihilation crosssection into SM fermions is now swave dominated. These processes lead to potential signals in the gammaray continuum which can be probed by the FERMI satellite [105]. In addition, the DM can also pair annihilate into two photons, through an effective coupling between the pseudoscalar mediator and photons, generated by triangle loops of the SM fermions [105]. This process, responsible for the generation of gammaray lines, is strongly constrained by the negative results in present searches [102].
The summary of our analysis is presented, in the usual fashion, in Fig. 19. The figure features three panels corresponding to the assignations \((\lambda _\psi ^a,\,c_a)=(1,\,0.25)\), \((1,\,1)\), \((0.25,\,1)\).
As already anticipated that the most stringent constraints comes from indirect DM searches in dSphs.^{26} Indeed, the absence of signals excludes thermal DM for mass below approximately 50 GeV. The most disfavored scenario turns to be the one corresponding to the assignation \(\lambda _\psi ^a=c_a=1\) (middle plot of Fig. 19). Indeed a complementary constraint comes from searches of gammaray lines so that the excluded ranges of the DM masses reaches the order of 100 GeV. This specific assignation of the couplings has also been investigated at the LHC through searches of events with monojet and missing transverse momentum. Due to the absence of any signal, the regions of the parameter space corresponding to \(100 \lesssim m_a \lesssim 500\,\text{ GeV }\) as well as \( m_a > 2 m_\psi \) are currently excluded.
10.2 Scalar + light pseudoscalar portal
The main feature of this double schannel portal scenario is the lower amount of correlation between the direct DM detection and the relic density, due to the presence of a light mediator state (see also Ref. [301] for a similar idea).
As evident that the presence of annihilation channels, like aa and Sa, involving nonSM light states allows to achieve the correct relic density, compatibly with constraints from DM DD, for relatively low values of the DM mass without necessarily relying on schannel resonances. The presence of swave unsuppressed annihilation channels like \(\overline{f}f\) (contribution from schannel exchange of the pseudoscalar) and Sa^{28} makes mandatory to consider, besides DD, also limits from ID. However, for the chosen parameter assignation, these last constraints have no impact in the region corresponding to the correct DM relic density.
11 Portals to secluded sectors
In this section we will consider cases in which the mediator of the DM interactions has no direct coupling with the SM fermions. Dark portals can nevertheless be realized, at the renormalizable level. Indeed the SM features two Lorentz and gauge invariant bilinears, i.e., \(H^\dagger H\) and \(B^{\mu \nu },\) which together with the similar appropriate structures can connect the SM and BSM sectors. The first one, \(H^\dagger H\), can be coupled to another scalar bilinear. In case this second scalar field also has nonzero VEV, a mass mixing with the SMHiggs is generated so that the portal interactions with the SM fermions, as well as the gauge and the SMHiggs bosons itself are produced. The field strength \(B^{\mu \nu }\) can be coupled with the field strength of another U(1) gauge bosons. Such kinetic mixing term after the EWSB, is the origin of a mixing between the Z and the new \(Z'\) boson.
In both these two scenarios, the DM interacts with the SM fields (both fermions and bosons) through a double schannel mediator. The relevant phenomenological processes are substantially the same as already investigated in the cases of the SM and BSM schannel dark portals. Contrary to these scenarios, we cannot consider \(\mathscr {O}(1)\) couplings between the SM states and the BSM mediators, as well as between the DM and the SM ones, since the mixing between the SMHiggs and an additional scalar or the Z and the Z’ are required to be small by several experimental and theoretical constraints.
11.1 SMHiggs + spin0 portal
One should note that in \(V_{H,\mathrm SM}+V(H,\varPhi )\), the condition for getting a positive definite mass spectrum requires \(4\lambda _h \lambda _\varPhi > \lambda ^2_{hS}\) while \(\lambda _h,\,\lambda _\varPhi >0\) are necessary to get \(V_{H,\mathrm SM}+V(H,\varPhi )\) bounded from below. Combining these two conditions we see that \(\lambda _{hS}\) can take both the positive and negative values.
The mixing between \(\mathfrak {R}{H(0)}\) and \(\phi \) (see Eq. (82)) indicates that both the mass eigenstates \((h,\,S)\) will couple to the SM states as well as to the DM and thus, it represents a twoportal scenario.
The processes responsible for the DM relic density and DD have been already discussed in detail for the cases of the SMHiggs and scalar portal individually; we thus just illustrate the results of our analysis, as reported in Fig. 21. We just remind here that all the considered types of DM candidates feature SI interactions with nuclei.
The results are reported in the usual bidimensional planes, i.e., \((m_S,\,m_\chi )\), \((m_S,\,m_\psi )\) and \((m_S,\,m_V)\), respectively. The angle \(\theta \) has been conservatively set to \(\sin \theta =0.1\) in order to comply with the constraints from the SMHiggs signal strengths. Similarly, the coupling \(\lambda _{hS}\) has been set to a value of \(\,0.1\) in order to satisfy various constraints on the SMHiggs sector (see e.g., Ref. [304] for a detailed discussion). The couplings \(\lambda ^\chi _\varPhi \), \(\eta ^S_V\) have been set to 0.1 and 1, respectively while for the fermionic DM the associated Yukawa coupling \(y_\psi \) is not a free parameter in our construction.
It is evident that the outcome of our analysis presents some sensitive differences with respect to the case of the spin0 mediator discussed in the previous sections. In the case of a fermionic and a vectorial DM the limits from DD are rather effective, due to the presence of an additional light mediator. In the case of a vectorial DM the only viable region, for masses of the DM and the mediator below the TeV scale, corresponds to the case \(m_V > m_S\), thanks to the enhancement of the DM annihilation crosssection due to \(VV \rightarrow SS\) process. In the case of a fermionic DM the region surviving the current constraints corresponds to the “pole” \(m_\psi \sim m_S/2\). In both cases, the next generation of DD detectors will probe the WIMP paradigm for masses of the DM and the mediator up to a few TeV. More particular is the case of a scalar DM where the shape of the DD contours is rather different compared to the other spin0 mediators. This is due to the different choice of the energy scale, the VEV \(v_\varPhi \) rather than the mass of the mediator, which implies a larger crosssection at higher values of \(m_S\). On the contrary, as can be noticed from Eq. (90), for \(\lambda _H^\chi =0\) the DM couplings become smaller as \(m_h\) and \(m_S\) get close in their values.^{32} For the chosen parameter assignations, a large region of the viable thermal DM is present in the regime \(m_\chi > m_S\) for a rather light \(m_S\). A sizable part of this region will be excluded in the absence of signals at XENON1T and LZ.
11.2 Kinetic mixing
We will reconsider in this subsection the scenario in which the DM is coupled to the gauge boson of a new U(1) group. In this case, however, SM fermions won’t be charged under the new gauge group so that no direct couplings with the new gauge boson are induced. The “dark” and visible sectors can nevertheless be connected by a kinetic mixing operator \(B_{\mu \nu } X^{\mu \nu }\) which can already exist at the tree level, being both Lorentz and gauge invariant, or generated radiatively (for example, if the new gauge sector features new fermions having nontrivial quantum numbers also under the SM gauge group [315]).
By comparing the outcome of Fig. 22 with the scenarios where the \(Z'\) is directly coupled to the SM fermions, we notice that DD probes a more limited region of the parameter space. This is because the scattering crosssection depends on coupling suppressed by the smallness of the parameter \(\delta \). However, this kind of suppression affects also the DM annihilation processes, ad exception of the \(Z'Z'\) final state so that a strong tension with the experimental constraints, for example similar to the SSM (see Sect. 8.2), still persists. We, indeed notice that the thermal DM is excluded for masses below the TeV scale unless small values, \(\mathscr {O}\sim (0.01)\), of the kinetic mixing parameter are taken. Even in such a case, the correct relic density is achieved only at the pole \(m_{Z'}/2\) or for \(m_{\chi ,\psi ,V} > m_{Z'}\), where annihilation into \(Z'Z'\) is accounted without relying on the kinetic mixing parameter. These setups will nevertheless be excluded in the absence of signals in the next generation multiTON experiments.
As explicitly indicated in the plots of Fig. 22 that we have considered only the case \(m_{Z'}>m_Z\). The opposite regime (often dubbed as dark photon) would also be feasible, although the constraints on \(\delta \) would be even stronger because of Eq. (102) and additional constraints from muon \((g2)\) and parity violating effects in the atomic physics [247] as well as from lowenergy colliders [320]. We have checked that the low mass \(Z'\) regime is substantially excluded by the current DD limits, unless DM masses below the sensitivity reach of DD experiments are considered. Thus, we have not reported it explicitly.
12 DM (partially) charged under \(SU(2) \times U(1)\)
All the models considered until now have assumed the DM to be a SM singlet. In this section we will, instead, consider the case in which the DM particle has nontrivial quantum numbers under the EW component of the SM gauge group. A rather straightforward realization consists into considering the DM as the lightest neutral component of a SU(2) multiplet. Extensive discussions can be found in Ref. [321] and Refs. [169, 170] for scalar and fermionic DM, respectively. These models cannot strictly be classified as Dark portals since the DM relic density is mostly set by annihilations processes into Z and W boson which are determined by the gauge couplings between the latter and the components of the DM multiplet. These are, nevertheless, very interesting and predictive models, having the DM mass (for this reason they are dubbed as “Minimal DM”) as the only free parameter. For these models the DM relic density is typically strongly suppressed by efficient annihilation processes into \(W^{+}W^{}\) and ZZ final states unless the DM is sufficiently heavy, possibly above the TeV scale. For these high values of the DM masses the treatment presented in Sect. 2 should be refined since, besides coannihilation processes with the other states belonging to the DM multiplet, one should also account for Sommerfeld enhancement effects [170, 171, 321] and even effects from bound states formation [322]. These effects are typically associated with an enhancement, with respect to the conventional treatment, of the DM annihilation crosssection so that the correct DM relic density is reached, for some realizations of the Minimal DM, for masses even of the order of 10 TeV. Concerning detection strategies the best probe, at the moment, is represented by ID [323]. On the contrary, scattering crosssection on nucleons are typically rather suppressed [324, 325, 326] and hence, DD probes are not very efficient. Further, given the high values of the DM masses corresponding to the correct DM relic density, the collider searches could, similarly, hardly probe these scenarios in the near future [326].
As already mentioned that the minimal DM scenarios, despite being phenomenologically very interesting, do not strictly belong the category of Dark Portals. Moreover, they are not the best target for direct DM searches. For these reasons they will not be discussed here in further detail.
Models more similar to Dark portals, achieving the correct DM relic density at lower values of the DM mass, consist of the case in which the DM is the mixture of the neutral component of a SU(2) multiplet and a SM SU(2) singlet. In such a case the coupling of the DM with the gauge bosons is suppressed through the mixing between the SU(2) singlet and the multiplet(s). Similar to Dark portals, annihilation into the SM fermions, through schannel mediation of the SM Higgs (or the Z boson), can play a relevant role for the DM relic density. Viable relic density is potentially achieved also for the DM masses of the order of a few hundreds GeV. The most popular example of these kind of scenarios is represented by Supersymmetric theories where the DM candidate, the lightest neutralino, is indeed a mixed state comprised of the bino (SM singlet), the Wino (electrically neutral component of a SU(2) triplet) and two higgsinos (electrically neutral components of SU(2) doublets).
We also remark that these kind of constructions represent renormalizable completions of the SM Higgs and Z portal models discussed in the Sect. 7. As will be shown subsequently, the phenomenology of these more realistic constructions differs from the simple SM Higgs and Zportals in particular because of the presence of additional states belonging to the DM sector which can affect the DM relic density through coannihilations.
In the following subsection we will review some representative cases of the general scenarios discussed above. We will first review the Inert Doublet Model (IDM) [327, 328, 329, 330, 331, 332, 333, 334, 335] in which the DM belongs to a scalar SU(2) doublet coupled with the SM Higgs sector. For what concerns the case of mixing between SU(2) singlets and multiplet we will consider the simplest possibility, the so called SingletDoublet models [173], specializing to the cases of Dirac [177] and Majorana [176, 336, 337, 338] fermionic DM. For the discussion of cases of mixing with higher multiplets we refer, for example, to Refs. [339, 340, 341, 342].
12.1 Inert doublet model
As will be shown subsequently that the most relevant DM observables depend on the combination of couplings \(\lambda _L=\frac{1}{2}(\lambda _3+\lambda _4+\lambda _5)\). We will thus adopt, as free parameters for our analysis, \(\lambda _L\) as well as the masses of the new Higgs states. The constraints from vacuum stability and perturbativity are then translated into bounds on \(m_{H^0},m_{A^0},m_{H^{\pm }}\) through Eq. (112). The latter are also constrained by the EWPTs. Finally, a lower bound of \(79.3\,\text{ GeV }\) on the mass of the charged Higgs is applied from LEP [137].
Concerning the DM relic density, there are similarities with the SM Higgs portal only at the low DM masses, namely below the mass \(m_W\) of the SM W boson. Here the DM relic density is mostly determined by annihilation into SM fermions mediated by the h boson. As the mass of the DM approaches towards \(m_W\), the annihilation into WW finals states enhances, with respect to the SM Higgs portal model discussed in Sect. 7, by tchannel diagrams involving the exchange of \(H^\pm \) and contact four field interactions between a \(H^0\) and a \(W^{\pm }\) pairs. The contributions of these diagrams to the crosssection depend on the \(SU(2)_L\) gauge coupling g rather than the quartic couplings of the scalar potential. A strong enhancement of the crosssection is already produced for \(m_{H^{0}} \lesssim m_{W}\) by the process \(H^0 H^0 \rightarrow W W^{*} \rightarrow Wf \overline{f}^{'}\) (this process is illustrated in detail, including an analytical expression of its amplitude, in Ref. [328]).
An overview of the DM phenomenology of the IDM in the low DM mass regime is presented in Fig. 23. The two panels of the figure, corresponding to the cases of \(\lambda _L>0\) and \(\lambda _L<0\) report, similar to the case of SM Higgs portal model, the combined constraints from the DM relic density, DD including the projected sensitivities of XENON1T and LZ and, invisible branching fraction of Higgs boson, in the bidimensional plane \(m_{H^0},\lambda _L\). Moreover, we assumed a sizable mass splitting between the DM and the extra scalar states in order to forbid coannihilation effects. As already anticipated, for \(m_{H^0} \lesssim m_h/2\) the outcome is substantially analogous to the SM Higgs portal model; the viable DM region is excluded by the combination of DD and Higgs invisible branching fraction constraints, except for the pole \(m_{H^0} \sim m_h/2\) which will be fully probed by the next generation DD experiments. For \(m_{H^0}>m_h/2\), an additional viable region is present corresponding to a scenario where the DM relic density is mostly determined by the annihilation process \(H^0 H^0 \rightarrow Wf \overline{f}^{'}\). Given the dependence of the corresponding rate on the gauge coupling, the correct DM relic density is achieved for very low values of \(\lambda _L\), passing the current DD constraints.
The model points featuring the correct DM relic density have been reported in Fig. 24 in the bidimensional plane \((m_{H^0},\lambda _L)\) together with the existing limits and projected sensitivities from direct DM detection and Higgs invisible branching fraction. As evident that the presence of coannihilations does not allow to evade the strong constraints in the low DM mass regime. On the contrary, a new viable DM region is present for \(m_{H^0}\gtrsim 500\,\text{ GeV }\) where although the present and expected future limits are effective but still there will be a sizable amount of model points that would survive even in case of absence of signals at LZ.
We emphasize that in this high DM mass regime, coannihilations are responsible for producing the correct DM relic density. The DM annihilations into gauge bosons are rather strong to get the correct DM relic density. That said, today these coannihilations are absent since the coannihilating particles first got decoupled from the thermal bath and then decayed back into DM. Therefore, we are left with the DM self annihilations into gauge bosons which are of the order of \(10^{25}  5\times 10^{26}\) \(\mathrm{cm}^3\, \mathrm{s}^{1}\). This falls perfectly within the sensitivity reach of Cherenkov Telescope Array (CTA). Indeed, it has been shown that such telescope array provides the most effective probe for this model, being possibly able to rule out the model till DM masses of 2.8 TeV [344, 345] (see also [346, 347, 348] for possible indirect signals in the Inert Doublet Model).
12.2 Singletdoublet DM
We will discuss in the following texts two scenarios of fermionic DM originating from the mixing between a SM singlet and the electrically neutral component of a \(SU(2)_L\) doublet. As already pointed out that this kind of setup allows for renormalizable couplings between the DM and Higgs and gauge bosons, in particular the Z, and thus, represents a theoretical framework where the SM portal models discussed in Sect. 7 can be embedded. We also remark that the new fermions which couple with the Higgs should belong to a real representation (i.e., Majorana fermions) or form vectorlike pairs, since chiral fermions, with masses originating from the EWSB, are experimentally strongly disfavored [349].
12.2.1 Dirac DM
The model features four free parameters, the two masses \(M_{N},M_{L}\) and the couplings \(y_1,y_2\). As can be easily realized that the couplings of the DM with the SM fields essentially depends on the angles \(\theta _L\) and \(\theta _R\) which determine the amount of its component charged under \(SU(2)_L \times U(1)_Y\).
The DM relic density is determined by a large variety of annihilation processes, including fermion pair final states (from schannel exchange of the Z and Higgs boson), \(W^+ W^\) final state (from schannel exchange of the Z / h boson and t channel exchange of \(\psi ^{\pm }\)), ZZ final state (from schannel exchange of Higgs boson and tchannel exchange of \(\psi _{1,2}\)) and finally Zh final state (from schannel exchange of the Z boson and tchannel exchange of \(\psi _{1,2}\)). To these, one should possibly add coannihilation processes in analogous combination of the final states in case the additional neutral and charged fermions have masses close enough to the same of the DM.
Similar to all the models discussed in this work, we will provide an overview of the interplay between the requirement of the correct DM relic density and the experimental constraints, mostly from DD, by considering a bidimensional plane in the mass parameters \((M_L,M_N)\) in this case and, a fixed assignation of the couplings. This kind of study is presented in Fig. 25. Contrary to most of the previously studied cases we have not chosen \(\mathscr {O}(1)\) values of the couplings here but a sensitively lower value, i.e., \(y_1=y_2=0.1\). This is because the possibility of vectorlike fermions having \(\mathscr {O}(1)\) couplings with Higgs is theoretically rather contrived since it would result potentially dangerous effects on the stability of the scalar potential [179, 350, 351].
The behavior of the contour of the correct DM relic density in the bidimensional plane \((M_L,M_N)\) is explained as follows. In order to match the DM annihilation crosssection with the thermally favored value, a sizable mixing between the singlet and the doublet components is needed, implying \(M_N \sim M_L\) (in this regime coannihilations are also important). The correct DM relic density can be obtained for a maximal value of the DM mass of approximately 733 GeV [177]. At this value the correct relic density can be achieved when the DM mostly coincides with the neutral component of a SU(2) doublet (hence coannihilating with its almost degenerate charged partner) and, is guided by gauge interactions of the DM with the Z and the W bosons. Note that a similar effect concerning the issue of mass degeneracy of the DM multiplet, as the one discussed in the previous subsection, occurs also in this case. Here, the presence of a state nearly mass degenerate with the DM is associated with an increase of the DM relic density, rather than a decrease as happens in the conventional coannihilation paradigm.
In order to evade DD constraints one should instead approach the pure singlet limit, i.e., \(\left( \sin ^2 \theta _L+\sin ^2 \theta _R\right) \ll 1\). This would, however, imply a very suppressed pair annihilation crosssection for the DM. The correct DM relic density can nevertheless be achieved through the pair annihilations of the \(\psi _2\) and \(\psi ^{\pm }\) states, provided that they are almost degenerate in masses with the DM. This would require \(M_N \sim M_L\) and to avoid an enhancement of \(\left( \sin ^2 \theta _L+\sin ^2 \theta _R\right) \), \(y_{1,2} \ll 1\).
As can be seen from the figure that values of the DM scattering crosssection span over a wide range of values. The highest ones, which can be even above \(10^{42}\,{\text{ GeV }}^2\), correspond to the case of a highly mixed DM. In such cases the interactions of the DM with the Z boson are of similar size as expected when they are coupled via ordinary gauge couplings and are, by far, excluded by DD. At the same time, model points with the correct DM relic density are obtained also for the case of a almost pure singletlike DM with \(y_1,y_2 \sim 10^{6}\), implying negligible Zmediated scattering rate, even surviving upon the absence of signals at LZ. For these points the DM relic density is entirely determined by the (co)annihilation processes of the quasi mass degenerate doublet partner. Notice, that the viable points are present only for \(m_{\psi _1} \lesssim 500\,\text{ GeV }\). Above this value, all the points with the correct DM relic density are characterized by \(M_L < M_N\), corresponding to a doubletlike DM. These points cluster at very high values of the DM scattering crosssection, beyond the range reported in the yaxis, and thus, are largely ruledout by DD.
The presence of extra fermions besides the DM would in principle offer collider tests for the scenario under consideration. The heavier neutral fermion \(\psi _2\) and \(\psi ^{\pm }\) could indeed be produced through EW processes at colliders, followed by subsequent decay into the DM and an on or offshell gauge boson. However, the strong bounds form direct DM detection can be evaded only through a very compressed spectrum for the new fermions. Furthermore, the decay rate of the heavier fermions into the DM is suppressed when the DM is mostly singletlike. This, as already discussed, is a way to evade DD constrains. The only possible detection prospects, compatible with constraints from the DM phenomenology, would be represented by the detection of the charged fermion \(\psi ^{\pm }\), stable at the collider scales, giving large missing transverse momentum with charge tracks.
12.2.2 Majorana DM
We conclude by commenting briefly on the possible collider tests of the model under consideration. As already pointed out in the previous subsection, because of the sizable couplings with the SM gauge bosons, the charged fermion \(\psi ^{\pm }\) and the neutral fermions \(\psi _{2,3}\) can be produced through DrellYann processes, followed by successive decay into the DM and a (either onshell or offshell) \(W^\pm /Z\) boson, respectively, leading to events with 2  3 leptons and missing energy. Given the similarity with SUSY setups, one could recast the results from the corresponding searches of chargino/neutralino production [290, 356] in this framework. The present relevant collider constraints are, however, not competitive with the ones from other DM searches [176, 337] and thus, for simplicity, have not been reported in Fig. 27.
13 Summary and discussion
We have discussed impact of the current and possible future DD limits, possibly complemented by the ones from ID and/or collider searches, in several simplified realizations of the WIMP DM.
The first and simplest classes of models considered are the ones in which the interactions of a pair of SM singlet scalar, fermionic and vectorial DMs and a pair of SM fermions, are mediated by electrically neutral schannel (portal) mediators. In the minimal most case the particle spectrum of the SM should be complemented by just a new state, i.e., the DM candidate, since portal interactions can be mediated either by the SMHiggs or by the Zboson, although in the last case a theoretically consistent construction is more contrived. In the case of SMHiggs portal, for all the DM spin assignations, SI interactions with nucleons are induced. The consequent very strong limits, due to the light mediator, are incompatible with the thermal relic density ad exception of DM masses above the TeV scale or the “pole”, i.e., \(m_\mathrm{DM} \simeq m_h/2\), region. This last scenario would nevertheless be ruled out in the absence of signals at XENON1T (assuming a 2 years of exposure time) and LZ. In the Zportal scenario current limits on the SI crosssection already exclude the pole region. These strong limits can nevertheless be partially overcome in two setups: (i) a fermionic DM with only axial couplings with the Z, as naturally realized in the case of a Majorana fermion DM and, (ii) a vectorial DM coupled through Chern–Simons term. In these two cases the DM features SD interactions with nuclei, whose constraints are sensitively weaker. In particular, in the case of a Majorana fermion DM, the thermal DM with mass of a few hundreds GeV would remain viable even in the absence of signals at the next generation detectors.
The SMHiggs and Zportal setups are easily extended to the cases of BSM spin0 and spin1 mediators, respectively. In the case of scalar mediators we have imposed, in order to preserve \(SU(2)_L\) invariance, a Yukawa structure for the couplings of the mediator with the SM fermions. This, on one side, implies a suppression of the DM annihilation crosssection for masses below the one of the topquark (unless the SS final state is kinematically accessible). At the same time, possible collider signals are also strongly suppressed so that the corresponding limits are not competitive with respect to the ones from DD and have been neglected for simplicity. Despite of the different velocity dependencies of the annihilation crosssections, the regions of the correct DM relic density are then mostly determined by Yukawa structure of the couplings between the mediator and the SM fermions. The correct DM relic density is indeed obtained, far from the resonance regions, only when the \(\overline{t} t\) and/or SS annihilation channels are kinematically open. Regarding DD, the limits are associated to SI component of the DM scattering crosssection for all the different assignations of the DM spin. The shape of the DD isocontours are, however, different for the various DM scenarios. This is due to the different assignations of the couplings with the dimension of mass for the scalar and vectorial DM. Theoretical considerations suggest, indeed, to parametrized these couplings in terms of a fundamental mass scale, the mass of the mediator and the DM mass in the cases of scalar and vectorial DM, respectively, and an unknown dimensionless coupling. The current limits still allow masses of a few hundreds GeV for both the DM and the mediator while XENON1T, in the absence of signals after 2 years of exposure, will exclude mediator masses up to approximately 1 TeV and DM masses up to a few TeV. Given the several free parameters, for clarity of the picture, we have focused our investigation on the masses of the new particle states and fixed the couplings to be close to \(\mathscr {O}\,(1)\) (see for alternative, e.g., Ref. [357]). We notice on the other hand that lowering the couplings would simultaneously suppress both the DD rate and the DM annihilation crosssection, in particular the SS channel becomes negligible as soon as the DM couplings \(\lambda ^S_\chi \), \(g_\psi \), \(\eta ^S_V\) deviate sensitively from \(\mathscr {O}\,(1)\) values. As a consequence, in this setup, the thermal DM is achieved only in the pole region which requires particular fine tuning because of the typical small decay width of the scalar mediator.
The scenario of spin1 BSM schannel mediator is even more constrained than the spin0 case. Indeed, the constraints from SI crosssection are typically much stronger, because of an effective enhancement of the crosssection due to the isospin violating interactions of the \(Z'\) with nucleons, so that masses of the DM and the mediator approximately below 5 TeV are already excluded. In the case of no signals at the next generation DD experiments, the exclusion regions will extend up to masses of \({\mathscr {O}}\sim 10~{\mathrm{TeV}}\), beyond the reach of LHC. In addition, the (reasonable) assumption of a \(Z'\) coupled with both the SM quarks and leptons implies a strong complementarity with the LHC searches of dilepton resonances. The corresponding limits, exclude, for the models considered here, masses of the \(Z'\) between 2 and 3 TeV (the exclusion can be even above 4 TeV in other realizations [358]), even in setups in which the SI component of the DD crosssection is suppressed or absent. We remark again that although in our analysis we have limited to some fixed assignations of the couplings, our results, nevertheless, have general validity because of the strong correlation between the DM relic density and its scattering rate of nucleons. For example, reducing the sizes of the couplings would actually reduce the viable parameter regions since the correct DM relic density would then be achieved only in correspondence of the schannel resonances.
Despite of the fact that our work is focused on scenarios already probed by the current and will be tested by the near future DD experiments, we have nevertheless also discussed a setup in which DD is, in general, evaded: the pseudoscalar portal. Under the assumption of CP conservation only a fermionic DM is considered in this case. Most of the parameter space is substantially insensitive to DD (we remind here that we have, conservatively, considered values of the pseudoscalar mass above 1 GeV in order to avoid flavour constraints) since tree level interactions with nucleons are momentum suppressed and, furthermore, are not subject to coherent enhancement. A rather limited region of the parameter space might still be probed by 1 and multiTON detectors because of an oneloop induced SI crosssection. The thermal DM is nevertheless sensitively constrained from ID. In addition, there is again a strong complementarity from the collider constraints, dominating for this scenario from monojet searches. A light pseudoscalar mediator can be interpreted as the pseudoGoldstone boson of a spontaneously broken global U(1) symmetry. We have then considered the case of a complex scalar mediator which can be decomposed into a scalar and a light pseudoscalar components. Although in this case sizable DD limits are reintroduced, the thermal DM still remains viable in the large portions of the parameter space due to the presence of efficient annihilation processes in the aa and Sa final states.
Relaxing the hypothesis of a SM singlet BSM mediator and assigning it nontrivial quantum numbers under the SM gauge groups, one can construct a simple and predictive class of model which we have labeled as tchannel portal. In this case a single DM state is coupled with the mediator and a SM fermion, according to the gauge charge assignation of the mediator (for simplicity we have restricted most of our analysis to couplings with the righthanded uptype quarks). Contrary to the other scenarios considered in this work, here the DM pair annihilation occurs through tchannel exchange of the mediator while DD scattering is induced by its schannel exchange. Focusing, for simplicity, mostly to the case in which the mediator field has the same quantum numbers with respect to the SM gauge group as the righthanded uptype quarks, the scenarios, i.e., a complex scalar and a Dirac fermion DM, in which SI interactions are present are excluded for \(\mathscr {O} (1)\) values of the couplings and for mediator masses up to \(\mathscr {O}\sim 10\) TeV. On the contrary, thermal Majorana fermion DM is still viable for masses below a TeV and will be extensively probed by the next generation of DD experiments.
We have then performed some steps towards more theoretically motivated realizations of dark portals. As well known, the bilinears \(H^{\dagger } H\) and \(B^{\mu \nu }\) (together with a new BSM field strength) are Lorentz and gauge invariant, so naturally lead to portal interactions with a dark sector, even if this is completely secluded. We have thus, considered the cases of (i) a scalar mediator coupled both to the DM and the SMHiggs boson and mixes with the latter because of a nonzero VEV and, (ii) a \(Z'\) coupled to the Z boson through a kinetic mixing term, also responsible for a mixing between the two spin1 states. These two mixing in turn allow the BSM states to interact with the other SM states. Also concerning the coupling of the DM with the mediators we have considered less generic assignations with respect to the cases considered in the previous frameworks. In the case of SMHiggs \(+\) spin0 portal we have explicitly considered a dynamical origin for the DM mass. Indeed, a fermionic DM has been assumed to have Yukawa interaction so that its mass is originated by the VEV of the new scalar field. Similarly, a vectorial DM has been assumed to be the vector boson of a spontaneously broken dark U(1) gauge symmetry and its mass is again related to the VEV of the new scalar field, also charged with respect to the same U(1) group.
As the last case of study we have considered models in which the DM features SM gauge interactions, either being the lightest neutral component of a \(SU(2)_L\) multiplet (in our studied examples we have focused on the case of \(SU(2)_L\) doublets) or a mixing between the latter and a SM singlet. These models are characterized by the fact that the DM sector is composed of multiple states. Particularly relevant is the case in which some of these states are very close in mass to the DM. In this setup the DM annihilation crosssection features a twofold enhancement with respect to the other Dark portals. First of all the annihilation into \(W^{+}W^{}\) is enhanced by the tchannel exchange of electroweakly charged states belonging to the DM sector (this, at the same time, requires that the DM has a nonnegligible SU(2) component). A further enhancement comes from coannihilation effects. Despite of a limited number of free parameters, these models are capable of encompassing many theoretically motivated scenarios like for example SUSY models. Interestingly, the SingletDoublet models considered at the end of Sect. 12 allows renormalizable couplings between the DM with the SM Higgs and the Z bosons. From the phenomenological point of view, the SingletDoublet model with a Dirac fermion DM features the greatest similarities with the SM Zportal model, sharing with the latter the extremely strong constraints from DD. This model appears viable only in a very fine tuned coannihilation configuration. On the contrary, only the response of the future experimental facilities can fully probe the case of a Majorana fermion DM. More particular is the case of the Inert Doublet Model. Its main limitation is represented by the very efficient annihilation crosssection into gauge bosons. The viable DM relic density can be obtained when this annihilation channel is kinematically allowed and only very specific assignations of the masses of the new particle sector are considered.
14 Conclusions
We have reviewed the theoretical foundations of the WIMP paradigm and discussed the limits, prospect and challenges of direct DM detection in a multitude of models encompassing scalar, vectorial and fermionic DM setups. In the light of extensive programme of direct DM searches and including a broad variety of complementary probes from indirect and collider searches, we assessed the status of the WIMP paradigm in the context of simplified models, accounting for the current and projected limits.
In particular, we have reviewed well known portals such as the SMHiggs portal and the Zportal. We have also addressed the popular dark \(Z^{\prime }\) portal and many others models that possess in their spectrum more than one mediator. Moreover, we have also investigated new models dictated by the kinetic mixing, often used in dark photon models.
We concluded that the simplest constructions, i.e., the SM dark portals will be substantially ruled out, ad exception of the case of a fermionic DM with only axial couplings with the Zboson (e.g., a Majorana fermion DM), in the absence of signals in the next generation of DD experiments.
The most straightforward extension of the SM dark portals, represented by the introduction of BSM schannel mediators, are, similarly, strongly constrained in the presence of SI interactions of the DM with the nuclei. In particular, the case of spin1 mediator is strongly disfavored because of the presence of complementary constraints from searches of resonances at the LHC, pushing the DM mass towards the multiTeV scale.
The tension with DD constraints can be relaxed somehow in the nexttominimal scenarios, featuring multiple mediators or new states lighter than the DM (we have reviewed the example of a light pseudoscalar).
In summary, we combined a plethora of experimental data set and theoretical models, computed the DM relic density, direct, indirect and collider observables to have a clear picture of where the WIMP paradigm stands and the future prospects. It is clear that most of the WIMP models will be scrutinized in the next decades, highlighting the paramount role of the next generation of experiments.
Moreover, our work shows that some DM constructions will survive the null results from the collider, direct and indirect experiments,suggesting that a further step in sensitivity reach is needed to falsify the WIMP paradigm.
Footnotes
 1.
See Ref. [22] for an exception (“relentless” DM) for modified expansion histories.
 2.
We make also use of: \(\frac{ds}{dt}=\left[ \frac{3s}{T}\left( 1+\frac{T}{3h_\mathrm{eff}}\frac{dh_\mathrm{eff}}{dT}\right) \right] \frac{dT}{dt}\). (8)
 3.
In scenarios where the DM is not the only new particle state, other processes like coannihilations, might also be relevant for the DM relic density. A more general definition of \(\langle \sigma v \rangle \), including such processes, can be found e.g., in Ref. [24].
 4.
The velocity distribution is understood as the probability of finding a WIMP with velocity v at a time t.
 5.
 6.
There are still some exceptions to this direct relation between nonvelocity dependent annihilation crosssection and relic density as discussed in detail in Ref. [25].
 7.
In the model considered in this work the mediator fields are mostly coupled with the SM fermions. The discussion can be straightforwardly extended to the case of sizable couplings with the gauge and Higgs bosons.
 8.
 9.
We will also consider the case of a pseudoscalar mediator. As will be clarified later, this is a more specific scenario since, under the main assumptions considered in this work, it can occur only for a fermionic DM. We refer to the dedicated section later for the corresponding Lagrangian.
 10.
As can be easily guessed that the DM annihilations also yield WW, ZZ , Zh and hh as possible final states.
 11.
An analogous study has been performed in Ref. [172]. Our results are in substantial agreement with the ones reported in this reference.
 12.
We limit, for simplicity, to the lowest dimensional operators. Higher dimensional operators are discussed, for example, in Ref. [180].
 13.
We will report in the main text just the results of the analysis. Analytical expressions of the relevant rates are extensively reported in the Appendix.
 14.
In the mentioned figure and throughout all the paper we will, as convention, use the label “PLANCK”, as the corresponding experiment, for isocontours corresponding to the correct DM relic density.
 15.
Similar to the case of the Higgs portal we just quote, as an example, the lowest dimensional operator. This is however, not the only possible option.
 16.
Similar to the SMHiggs portal case we will report in the main text only the main results while discussing the computation in more detail in the Appendix.
 17.
This statement is strictly valid in the case, considered here, of \(\lambda _\chi ^S=1\). In the case \(\lambda _\chi ^S \ll \lambda _S\) the contribution of the trilinear couplings \(\lambda _S\) to the annihilation crosssection into SS might be sizable and event dominant. However, we expect in such a case, since the annihilation crosssection would scale at least as \((\lambda _\chi ^{S})^2\), the DM to be in general overabundant.
 18.
 19.
The difference arises from the fact that in the case of spin0 mediator the quantities \(f_p\) and \(f_n\) are originated by matrix element \(\langle N q q N\rangle \) which is related to the mass of the nucleon. In the case of spin1 mediator one instead evaluates the matrix element \(\langle Nq \gamma ^\mu q N\rangle \) which is, instead, related to the electric charge of the nucleon.
 20.
Given the nontrivial gauge charges, the tchannel mediators can also couple with the gluons, the photon and the Zboson. We do not explicitly write such interactions for simplicity. In the case of the scalar mediator a renormalizable 4field coupling with the SMHiggs doublet, i.e., \(H^\dagger H \Sigma ^\dagger _q \Sigma _q\), can arise at the treelevel. We will assume that the corresponding coupling is negligible.
 21.
In Eq. (63) the labels u and d refer globally to up and downtype quarks. The couplings and the masses of the mediator fields carry also a generation index which is not explicitly reported (see main text for further clarification).
 22.
 23.
One could weaken the constraints by taking smaller values of the associated couplings. However, the relic density would then be achieved only in the very finetuned coannihilation regime.
 24.
For this reason it is also dubbed as coy DM [294].
 25.
Values of \(m_a\) below 10 GeV are still partially in tension with low energy constraints for \(c_a=1\). We will ignore this issue since it has marginal impact on our discussion.
 26.
For the mass range assumed for the DM, the ID signal probed by dSph relies on the \(\overline{b} b\) final state for \(m_\psi < m_t\) and on the \(\overline{t} t\) final state for \(m_\psi > m_t\). For simplicity we have adopted everywhere the limit from the \(\overline{b}b\) final state. This is a good enough approximation for the purposes of our study.
 27.
 28.
For simplicity, we have considered in our analysis only limits from searches in dSphs. As pointed out in Refs. [299, 302, 303] the annihilation into Sa can lead to box shaped gammaray signals which could be probed in the near future by CTA. However, our choice for \(m_a\), implies a too suppressed decay branching ratio for \(a\rightarrow \gamma \gamma \) process. For this reason we have not considered explicitly this possible signal in our analysis.
 29.
Stability of a scalar DM is protected either by a U(1) (complex scalar) or by a \({\mathbb {Z}}_2\) symmetry (real scalar) and thus, the associated Lagrangian contains \({\chi }^2\) or \(\chi ^2\) term, respectively. In this analysis we have used the notation \({\chi }^2\) for generality although the DM is considered to be a real scalar.
 30.
 31.
We have adopted the same definition of the covariant derivative as of Ref. [312].
 32.
In the case of a scalar DM it would even be possible to generate a “blind spot” in the scattering crosssection for a rather specific choice of the couplings \(\lambda ^\chi _{H,\varPhi }\) [221, 313] which would induce a destructive interference between the amplitudes associated to the h and S exchange. A different solution for relaxing DD constraints in the presence of multiple scalar singlets coupled to the SMHiggs has recently been proposed in Ref. [314].
 33.
Note that in the definition of \(g_X\) we have also encoded the value of U(1) charge of the DM.
 34.
The case of an Abelian vectorial DM in the kinetic mixing scenario will be addressed in a dedicated future publication.
 35.
In the case of extreme mass degeneracy between the \(H^0\) and \(A^0\) states, one should consider an additional SI interaction, namely \(H^0 p \rightarrow A^0 p\), mediated by the Z boson [327]. We will neglect this possibility implicitly by assuming a mass splitting of at least 1 GeV between these two states. This, in turn, implies that the coupling \(\lambda _5\) is always not negligible.
 36.
There are elegant ways to overcome this underabundant DM regime in the two Higgs Doublet Model using axions [343].
 37.
SI interactions are also induced by interactions of the DM with the SM Higgs boson. Their contributions are subdominant and for simplicity we will omit them in our discussion. It has nevertheless been included in our numerical analysis.
 38.
 39.
Figure 27 reports only limits from SI interactions. For the chosen assignation of the parameters, limits from SD interactions are always subdominant. For simplicity, thus, these have not been reported in the figure.
Notes
Acknowledgements
The authors thank Werner Rodejohann, Miguel Campos, Alexandre Alves, Carlos Yaguna and Chris Kelso for discussions. The authors also thank Francesco D’Eramo for his valuable comments. P. G. acknowledges the support from P2IO Excellence Laboratory (LABEX). M. D. acknowledges the support from the Brazilian PhD program “Ciência sem Fronteiras”CNPQ. S. P.’s work was partly supported by the U.S. Department of Energy grant number DESC0010107. This work is also supported by the Spanish MICINN’s ConsoliderIngenio 2010 Programme under grant MultiDark CSD2009  00064, the contract FPA2010  17747, the FranceUS PICS no. 06482 and the LIATCAP of CNRS. Y. M. acknowledges partial support the ERC advanced grants Higgs@LHC and MassTeV. This research was also supported in part by the Research Executive Agency (REA) of the European Union under the Grant Agreement PITNGA2012316704 (“HiggsTools”).
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