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Top-philic Vector-Like Portal to Scalar Dark Matter

We investigate the phenomenology of scalar singlet dark matter candidates that couple dominantly to the Standard Model via a Yukawa interaction with the top quark and a colored vector-like fermion. We estimate the viability of this vector-like portal scenario with respect to the most recent bounds from dark matter direct and indirect detection, as well as to dark matter and vector-like mediator searches at colliders. Moreover, we take QCD radiative corrections into account in all our theoretical calculations. This work complements analyses related both to models featuring a scalar singlet coupled through a vector-like portal to light quarks, and to scenarios in which the dark matter is a Majorana singlet coupled to the Standard Model through scalar colored particles (akin to simplified models inspired by supersymmetry). Our study puts especially forward the complementarity of different search strategies from different contexts, and we show that current experiments allow for testing dark matter masses ranging up to 700 GeV and mediator masses ranging up to 6 TeV.


I. INTRODUCTION
The hypothesis that dark matter (DM) may consist in weakly interacting massive particles (WIMP) is currently being tested by various experiments including direct and indirect DM probes, as well as colliders searches. In this article, we study a minimal setup in which a real scalar DM particle S couples to the Standard Model (SM) through interactions with a vector-like fermion. Such a vector-like portal scenario has been the object of several previous studies, which have focused on couplings either to light leptons [1][2][3][4] or to light quarks [5]. A distinctive feature of this class of portals is that radiative corrections tend to play a major role in DM annnihilation phenomenology. In particular, virtual bremsstrahlung or annihilations into mono-energetic photons and gluons may be the dominant mechanism driving the DM relic abundance. By the same token, these may give rise to remarkable spectral features, like a gamma-ray line that consists of a smoking gun for many DM searches. For physical, but also technical reasons, previous studies have nevertheless been limited to couplings to light SM leptons or quarks. In this work we complement these studies by considering a scenario in which the DM particle solely couples, at tree-level, to the top quark through interactions with a vector-like quark T . We explore different approaches to investigate the phenomenological viability of the model from both collider and cosmology standpoints.
In our predictions, we take into account several higherorder corrections that include the QCD Sommerfeld effect and next-to-leading-order (NLO) QCD corrections to the DM annihilation cross section, both in the Early Universe and for what concerns indirect searches. For the latter purpose, we have in particular computed the O(α s ) corrections to the SS → tt annihilation process, which involves contributions from gluon emission by both the final-state top quarks as well as by the virtual intermediate t-channel vector-like mediator. Although the treatment of the associated infrared and collinear divergences is more involved for heavy quarks than when the DM candidate is coupled to light fermions, we only comment briefly on the associated difficulties and refer instead to Ref. [6] and Ref. [7] for details on the scalar DM and Majorana DM cases respectively. We complement these constraints stemming from the relic density of DM and its indirect detection null results by a study of the relevance of existing direct DM probes. Our calculations take into account the effective coupling of the dark scalar S to gluons through loops involving top quarks and T mediators [8].
On different lines, we estimate how collider searches for both the vector-like partner T and the DM particle S restrict the model. We extend a previous study relying on simplified model results from the Run 1 of the Large Hadron Collider (LHC) [9] by considering more recent LHC Run 2 supersymmetry searches that can be recasted to constrain any model featuring strongly-interacting quark partners decaying into a final-state comprised of missing energy and several SM objects. We moreover include NLO QCD corrections through the computation of the corresponding matrix elements and match the fixed-order predictions with parton showers [10], so that a state-of-the-art modeling of the LHC signals is used. We additionally investigate the reach of the dedicated DM searches at the LHC in the mono-X channels where the final-state signature consists in a pair of DM particles recoiling against a very hard SM object X.
The plan of this article is as follows. In section II we define the model and the associated parameter space. In section III, we discuss our calculation of the DM relic abundance and how the latest results constrain the parameter space. In section IV we further derive bounds stemming from DM direct and indirect detection searches, and we finally address the collider phenomenology of the model in section V. We emphasize the complementarity of the different approaches in section VI, in which we summarize the various cosmological and collider bounds that we have obtained.

II. THEORETICAL FRAMEWORK
We consider a simplified top-philic DM setup in which we extend the Standard Model with a real scalar DM candidate S with a mass m S and whose interactions with the Standard Model are mediated by exchanges of a heavy vector-like quark T of mass m T . The T quark is as usual considered as lying in the fundamental representation of the QCD gauge group SU (3) c , and we focus on a minimal option where it is a weak isospin singlet with an hypercharge quantum number set to 2/3. For the same minimality requirements, we further neglect the possible mixing of the T quark with the Standard Model up-type quark sector. In order for the S particle to be a stable DM candidate, we impose a Z 2 discrete symmetry under which all Standard Model fields are even and the new physics states are odd. This also prevents the S field from mixing with the Standard Model Higgs doublet Φ, provided the Z 2 symmetry is unbroken.
Our model is described by the effective Lagrangian where P R denotes the right-handed (RH) chirality projector and t the top quark field. The interaction strength between the mediator T , the DM and the SM sector (or equivalently the top quark) is denoted byỹ t . Like the DM, the vector-like mediator field T is odd under Z 2 but otherwise transforms as the RH top field under SU (3) × SU (2) × U (1) (and so has electric charge Q = +2/3). A similar effective Lagrangian has been considered in Ref. [5] in the case of a DM particle coupling to light quarks, and, more recently, in Refs. [11,12] for a coupling to the top quark. In contrast to these last two studies, our analysis significantly differs in the treatment of the radiative corrections that are relevant for the relic abundance, DM indirect and direct detection as well as for the modeling of the collider signals. For our phenomenological purpose, we assume that the last term in the Lagrangian of Eq. (2.1) and its associated λ coupling play a negligible role, so that we effectively set λ = 0 in the sequel. Such an assumption is justified in scenarios with a scalar DM candidate, which constrasts with models featuring its Majorana siblings. In the latter case, the analog of the T mediator is a scalar particle (i.e., a top squark) so that a Majorana DM candidate can consequently annihilate into a pair of Higgs bosons at the one-loop level [13]. This corresponds to a five-dimensional effective operator and the process is finite and calculable. The scalar DM analog yields to an ultraviolet-divergent correction to the quartic coupling λ that needs to be renormalized. We assume that λ = 0 so as to focus on the effects of the T mediator only (see also [11] for the case λ = 0). The relevant model parameter space is therefore defined by three parameters, namely the two new physics masses m T and m S , and the Yukawa couplingỹ t .

III.1. Radiative corrections
It has been recently shown that radiative corrections to the DM annihilation cross section play a significant role in the phenomenology of a real scalar DM candidate, either through internal bremsstrahlung or via new channels that open up (like for instance when DM annihilates into a pair of monochromatic gluons or photons) [1][2][3][4][5]. All these analyses have however been restricted to scenarios featuring a DM particle coupling to light SM quarks or leptons, so that the corresponding fermion masses could be neglected and the calculation performed in the so-called chiral limit. When nonvanishing SM fermion masses are accounted for, the computation of the radiative corrections to the annihilation cross section is plagued by infrared divergences that must be consistently handled, as it has been studied in details for Majorana DM [14]. The scalar DM case has been thoroughly analyzed by some of us [6], so that we summarize in this section the points that are the most relevant for our study.
The calculation of the annihilation cross section associated with the SS → tt process at O(α s ) involves contributions both from final state radiation (FSR) and from virtual internal bremsstrahlung (VIB) diagrams. The corresponding amplitudes exhibit a specific dependence on the kinematics, which reflects in distinguishable features in the spectrum of radiated photons or gluons. In particular, VIB tends to yield a final-state energy spectrum that peaks at high energies E γ,g m S . Whilst FSR contributions also lead to the emission of a hard gluon or photon [15], the related spectral feature is less remarkable than in the VIB case, unless VIB is relatively suppressed. For a fixed DM mass m S , the relative FSR and VIB weights are controlled by the mass of the vectorlike mediator m T and by the final state quark mass (i.e., the mass of the top quark m t ). FSR turns out to be less important as m t /m S decreases, since the contribution to the annihilation cross section is proportional to the leading order (LO) result which, in an s-wave configuration, is helicity suppressed. In the chiral limit, m t /m S → 0 and FSR can thus be neglected. On the other hand, the VIB spectral features are controlled by the m T /m S mass ratio, and the energy spectrum peaks toward E γ,g ∼ m S as T and S become more mass-degenerate.
For generic particle masses, both FSR and VIB features are present and must be accounted for. This consequently requires a consistent handling of the infrared and collinear divergences of the FSR amplitude, that are only cancelled out after including the virtual contributions as guaranteed by the Kinoshita-Lee-Nauenberg theorem. The associated computations are facilited when carried out in an effective approach (with a contact SStt interaction) that suits well for the annihilation of non-relativistic DM particles in the soft and collinear limit [6,14]. The hard part of the spectrum is then described by the SS → ttg (or ttγ) contribution as calculated from the full theory of Eq. (2.1), and the two results are matched by using a cutoff on the energy of the radiated gluon (photon). This approach allows us to get a regularized expression for the total SS annihilation cross section at the NLO accuracy that is valid for a broad range of parameters [6].
The procedure outlined above vindicates the fact that for a large part of the parameter space, one may rely on a simple approximation for the total annihilation cross section, σv tt | NLO ≈ σv tt m S < 300 GeV, σv ttg | mt=0 + σv tt m S > 300 GeV. (3.1) In this expression, σv tt is the s-wave contribution to the LO annihlation cross section, and σv ttg | mt=0 is the (s-wave) annihilation cross section as obtained in the chiral limit and when a single gluon radiation is included. Its explicit form can be found in Refs. [2,4]. The difference with the exact result is only large for m S m t and reaches at most 30% beyond (see Fig. 2 discussed in the framework of section III.2). When m S → m t , the treatment used for the derivation of σv ttg breaks down due to threshold corrections that affects the production of a top-antitop system nearly at rest [16]. For the (very narrow) range of parameters where such refinements are relevant, we make use of the LO result for the annihilation cross section. At the differential level, the NLO effects can potentially yield features for what concerns DM indirect detection. This is further elaborated in section IV.2.

III.2. Relic abundance
In order to determine the relic abundance of the dark S particle, we consider the freeze-out mechanism for DM production in the Early Universe and make use of the MicrOMEGAs code [17], which we have modified in order to accommodate some of the particularities of our model. These include dark matter annihilations into a tW b three-body final state once one lies below the top threshold, the radiative corrections mentioned in section III.1 and Sommerfeld effects. The latter especially affect vector-like fermion annihilation and dark matter co-annihilation with a mediator, those corrections contributing to the relic abundance by at most 15% (see appendix A). In addition, DM annihilations into a tW b system play a non negligible role for DM masses lying in the [(m t + m W )/2, m t ] mass window, and we have included these contributions by evaluating them numerically with CalcHEP [18]. Finally, we have added the loop-induced SS → gg and SS → γγ processes in the computation of the DM annihilation cross section [3,4]. The annihilation into gluons is in particular significant for DM masses below the top threshold.
We present the results in Fig. 1, under the form of two two-dimensional projections of our three-dimensional parameter space. In the left panel of the figure, we show the region of the (m S ,ỹ t ) plane for which there exists a mediator mass value yielding to a relic density Ωh 2 = 0.12 compatible with the Planck results [19]. The gradient of colors in Fig. 1 is associated to relative mass difference between the DM and the mediator given by r − 1 with Similarly, we present in the right panel of the Fig. 1 the region of the (m S , r − 1) plane for which there exists aỹ t coupling value, shown through a color code, leading to the observed relic abundance. In order to enforce the chemical equilibrium of the S and T particles in the Early Universe, we have imposed thatỹ t > 10 −4 , which at the same time guarantees a correct treatment of the coannihilations by MicrOMEGAs. For smaller coupling values, conversion-driven freeze-out could give rize to the correct DM abundance [13,20], and we leave this possibility open for future work. For comparison purposes, we also superimpose to our results in Fig. 1 a black dotted contour which corresponds to the limit of the viable parameter space of a model similar to the one considered in this work but where the DM is up-philic and thus couples to the right-handed up quark u R . We refer to Ref. [5] for more details. The viable part of the parameter space can be divided into three distinct regions according to the DM mass m S . • m S > 5 TeV. For very heavy DM, the mass of the top quark only plays a subleading role. This is clearly visible in the right panel of Fig. 1, where the viable region of the parameter space of the top-philic scenario matches the one expected in the up-philic case. In this regime, m S m t and the chiral limit approximation for the DM annihilation cross section is valid. Moreover, VIB corrections are large, as illustrated in Fig. 2 where we show, for all benchmark points giving rise to the right DM abundance in Fig. 1, the ratio of the exact NLO result [6] to the LO predictions σv tt . The importance of the NLO corrections will be further discussed in the context of DM indirect detection bounds in section IV.2.
• m t < m S < 5 TeV. In this regime where the DM mass is moderate, the tree-level s-wave SS → tt contribution to the annihilation cross section dominates, as additionally illustrated in Fig. 2 where the NLO to LO ratio is close to 1. Notice that the feature observed for m S ∼ m t in Fig. 2 is spurious as correct predictions must include threshold effects that we have ignored. The LO annihilation into a pair of quarks is, in contrast, completely negligible in the light quark case for which the relic density is driven by loop-induced annihilations into gluons [5]. The phenomenologically viable region of the parameter space in the top-philic scenario consequently strongly deviates from the corresponding one in the up-philic model, as shown in the right panel of Fig. 1. Given that finite quark mass effects are significant, larger r parameters are found acceptable for a given DM mass in the top-philic case.
• m S < m t . In this regime, the DM abundance is driven either by annihilations into a tW b system via a virtual top quark (for m S m t ), through loop-induced annihilations into gluons, or through co-annihilations with the mediator. Any other potential contribution, like DM annihilations into pairs of SM particles through the Higgs portal (as it occurs in the scalar singlet DM scenario [21,22]) is here irrelevant since we have set the λ quartic coupling in Eq. (2.1) to zero. Co-annihilations particularly play an important role near m T + m S m t , as the ST → t → tg channel is resonantly enhanced. This corresponds to the light-yellow region in the left panel of Fig. 1 for m S ∼ 70 − 80 GeV, and to the blue peak in the right panel of the figure for the same m S values. Annihilations into monochromatic gluons are only important when the mass of the mediator is large enough to close all co-annihilation channels, and annihilations into a tW b three-body system are only relevant close to threshold,

IV.1. Direct detection constraints
In the limit in which the quartic coupling of S to the Higgs boson vanishes, the DM nucleon scattering crosssection can be computed from the evaluation of the oneloop diagrams shown in Fig. 3. This allows one to derive an effective Lagrangian for the DM coupling to gluons, where the Wilson coefficient C g S includes both short and long-distance contributions (relatively to the momentum scale involved in the loop) [8,23]. The resulting effective spin-independent coupling f N of the scalar DM particle S to a nucleon N of mass m N is then given by [24] where the quark mass fractions f (N ) Tq and the analytical expression for C g S can be found in Ref. [25]. We compute the total spin-independent cross section σ A for DM scattering off a nucleus with charge Z and a mass number A by taking the coherent sum of the proton and the neutron contributions, where f p and f n denote the respective DM couplings to a proton and a neutron derived from Eq. (4.2) with N = p or n, respectively, and m A is the nucleus mass. In Fig. 4, we present the dependence of the DM scattering cross section on protons σ SI calculated as depicted above, for all DM scenarios of Fig 1. For m S m t , the models featuring the largest σ SI values are those with the largestỹ t value and for which the relic density is driven by annihilations into a pair of gluons. As in the left panel of Fig. 1, the yellow region around m S ∼ 80 GeV corresponds to scenarios for which resonant co-annihilations of the S and T particles into a top quark play a leading role. Above the top mass threshold, the Yukawa coupling required to match a correct relic abundance drops, and so does the elastic scattering cross section. The figure   FIG. 4. DM-proton spin-independent scattering cross section as a function of the DM mass mS. For each scenario, the coupling to the top quark and the value of the r − 1 parameter (shown trough the color code) are fixed to reproduce the observed relic density. The continuous red line represents the 90% confidence level exclusion of the Xenon 1T experiment [26], the orange dashed line the Xenon 1T reach [27] and the red dashed line the neutrino floor [28].
finally exhibits a bump above m S 2.5 TeV, which corresponds to setups in which m S + m t ∼ m T . The C g S coefficient is then consequently enhanced, which directly impacts the elastic cross section [25].
For most DM models σ SI lies however below the neutrino floor, except for some scenarios with a DM candidate lighter than the top quark. The constraints originating from the results of the Xenon 1T experiment after 34 days of exposure [26] are also indicated, together with predictions under the assumption of 2.1 years of data acquisition [27]. Although a large part of the parameter space region lying above the neutrino floor is within the range of Xenon 1T, a significant fraction of it will stay unconstrained in the near future by DM direct detection searches. The corresponding excluded region projected in the (m S , r − 1) plane is presented in the summary of Fig. 11, after accounting for the latest bounds from the Xenon 1T experiment (red region), together with the region that could be tested up to the neutrino floor (red dashed contour).

IV.2. Indirect detection constraints
In Figs. 5 and 6, we present, for all scenarios satisfying the relic density constraints of section III, the value of the DM annihilation cross section at zero velocity into varied final states and using different approximations. In the upper panel of  .1), as well as the SS →ttg annihilation cross section in the chiral limit [2] (lower panel). We superimpose to our results the indirect detection limits obtained from the cosmic ray (CR) analysis of Ref. [29] (green continuous line), as well as the bounds that could be expected after 15 years of Fermi-LAT running when dwarf spheroidal galaxy data in the bb channel is analyzed [30] (dot-dashed orange line). An estimation of the upper limits expected from gamma-ray line H.E.S.S. data [31] is also presented (see the text for details) in the lower panel (gray). The color code represents the value of the r −1 parameter and we have considered DM models satisfying the relic density constraints of section III.
FIG. 6. Predictions for the SS → gg annihilation cross section, to which we superimpose the upper limits extracted from the cosmic ray analysis of dwarf spheroidal galaxy data in the bb channel from Fermi-LAT [32] using current results (dark green continuous line) and projected results assuming 15 years of data acquisition (orange dot-dashed line) [30]. We also indicate the upper limits obtained from the gamma-ray line analysis of Fermi-LAT [33] and H.E.S.S. [31] by gray dotted and double-dot-dashed lines, respectively (see the text for details). The color code represents the value of the r − 1 parameter and we have considered DM models satisfying the relic density constraints of section III.
only show the gluon emission contributions, SS → ttg, computed in the chiral limit for m S > 300 TeV. The (loop-induced) contributions of the SS → gg channel to the annihilation cross section are evaluated in Fig. 6.
Comparing the upper and central panels of Fig. 5, we observe that QCD emissions play a significant role for m S > 2 TeV, as already visible in Fig. 2 (in which the exact NLO results from Ref. [6] have been employed). In contrast, Fig. 6 shows that annihilations into pairs of gluons are only relevant for m S < m t (see also section III). Moreover, σv gg exhibits a minimum around m S ∼ 280 GeV independently of the value of r. This minimum is connected to a change of sign at the level of the loop-amplitude that always happens for m S ∈ [270, 290] GeV (see appendix B for an analytic expression of σv gg ). As in Fig. 1, the yellow region around m S ∼ 60 − 70 GeV in Fig. 6 corresponds to models with a DM abundance dominated by the resonant coannihilatation of a T S system into a top quark.
We superimpose to our predictions limits extracted from varied observations. DM annihilations into topantitop systems can be constrained with antiproton cosmic ray data [29] (continuous green lines in Fig. 5). We also show Fermi-LAT gamma-ray constraints from dwarf spheroidal analysis for annihilations into a bb final state [32] (continuous dark green line in Fig. 6) and the corresponding prospects from 15 years of Fermi-LAT running [30] (dot-dashed orange lines). Whilst the Fermi- LAT collaboration has not published any specific limits for what concerns the gamma-ray spectrum issued from DM annihilations into the tt and gg final states, both spectra are expected to show a similar behavior as for annihilations into a bb system, as illustrated in Fig. 7 for m S below (upper panel) and above (lower panel) the top mass. An estimate of the limits for tt and gg final states can be obtained following the methodology advocated in Ref. [35], using exclusion limits from DM annihilations into bb pairs that are rescaled using 4) where N X γ is the number of photon expected from an X final state. We have nevertheless verified that N bb γ < N gg,tt by determining N X γ using the hadronization model of Pythia 8 [34], so that the obtained bounds can be seen as conservative. Gamma-ray spectrumm S = 2 TeV, r = 1.1 tt + γ + g + gg + γγ tt + γ + g gg γγ Gamma-ray spectrumm S = 10 TeV, r = 1.1 tt + γ + g + gg + γγ tt + γ + g gg γγ The shape of the gamma-ray spectrum could also potentially be used to get hints on DM, as radiative corrections may give rise to specific gamma-ray spectral imprint such as line-like features. However, these are most of the time overwhelmed by the continuum originating from the hadronization of the annihilation products. There are however two regimes in which they may be potentially important, namely in the low mass range (m S < m t ) where annihilations into a photon pair could be relevant, and in the multi-TeV regime where radiative emission is crucial (as shown on the different panels of Fig. 5). The typical gamma-ray spectral signature of the annihilation of a pair of very heavy S particles into tt, γγ and gg systems is presented in Fig. 8, our predictions being derived as sketched in appendix C.
• m S 5 TeV. This regime is the one for which VIB emissions play a significant role and for which the approximation of Eq. (3.1) holds. DM annihilations into a top-antitop system produced in association with a photon can then be simply deduced, 5) where N c = 3 denotes the number of colors. Moreover, α and α s stand for the electromagnetic and strong coupling constants and we use Z-pole values as references, α = 1/128 and α s = 0.112. Although results from the H.E.S.S. collaboration can potentially constrain the model, there is no official VIB dedicated analysis and one must thus refer to the independent analysis of Ref. [36] and the recent constraints that can be extracted from the gamma ray spectrum issued from the galactic center [31]. This suggests that the annihilation cross section can be of at most σv ttγ ∼ 10 −27 cm 3 /s for DM masses of about 10 TeV, which can be translated as σv ttg ∼ 10 −25 cm 3 /s. This is illustrated in the lower panel of Fig. 5 where we show the H.E.S.S. constraints derived in Ref. [31], after including both the rescaling factor of Eq. (4.5) and a factor of 2 accounting for the photon multiplicity.
• m S < m t . In this regime, σv gg can be as large as about 2 · 10 −26 cm 3 /s (see Fig. 6), and there is a well defined prediction for annihilations into a pair of photons [37], where C F = 4/3. The strongest constraints on the production of gamma-ray lines at energies around and below m t originate from the Fermi-LAT collaboration [33] and we indicate them in Fig. 6 after including the rescaling factor of Eq. (4.6) (gray dotted line). H.E.S.S. bounds at larger DM masses are also indicated, following Ref. [31] (double-dot-dashed line). In both cases, we use the limits associated with an Einasto DM density profile.
To conclude this section, we project the DM indirect detection constraints from the cosmic ray analysis (green region at large mass) and existing (dark green region at low mass) and future (orange region with a dotdashed contour) Fermi-LAT constraints from the gammaray continuum from dwarf Spheroidal galaxies in the summary of Fig. 11. The color code is the same as in Figs. 5 and 6. A substantial part of the parameter space, for m S < 1 TeV region, turns out to be constrained by probes of the gamma-ray continuum and antiproton cosmic rays. Moreover, for moderately heavy DM candidates, these constraints are complementary to those originating from direct DM searches studied in Sec. IV.1. As for the relic density, annihilations into pairs of gluons are relevant for light DM (m S < m t ) whilst annihilations into top-antitop systems help to test heavier candidates with masses ranging up to m S ∼ 400 GeV and 450 GeV when observations based on gamma rays and antiprotons are respectively used. The major difference with the relic density considerations is that close to the top-antitop threshold, the non-zero DM velocity at the freeze-out time allows for DM annihilations into a tt pair, which is kinematically forbidden today. A threebody tW b final state must therefore be considered instead, which does not yield further constraints. Finally, the predicted annihilation cross sections σv gg and σv ttg appear to be too small to allow us to constrain the models using searches of specific features in the gamma-ray spectrum (considering an Einasto DM density profile).

V. COLLIDER CONSTRAINTS
Searches for new physics have played an important role in past, current and future physics programs at colliders. In the context of the class of scenarios investigated in this work, in which the Standard Model is extended by a bosonic DM candidate and a fermionic vector-like mediator, the results of many collider analyses can be reinterpreted to constrain the model.
In our model, the extra scalar particle is rendered stable (and thus a viable candidate for DM) by assuming a Z 2 symmetry under which all new states are odd and all Standard Model states are even. As a consequence, the collider signatures of the model always involve final states containing an even number of odd particles that each decay into Standard Model particles and a DM state. This guarantees the presence of a large amount of missing transverse energy as a generic model signature.
For top-philic models, the relevant signatures can be classified into two classes, the model-independent mono-X searches that target the production of a pair of DM particles in association with a single energetic visible object X, and the production of a pair of top-antitop quarks in association with missing energy.
Before going through the most recent constraints originating from LHC searches for DM, we will account for LEP results. In electron-positron collisions, top partners can be produced electroweakly, and yield a signature made of a pair of top-antitop quarks and missing transverse energy / E T . Reinterpreting the results of the LEP searches for the supersymmetric partner of the top quark, vector-like (top) partners are essentially excluded if their mass satisfies m T 100 GeV [38]. This excludes the lower left corner of the viable parameter space of the summary of Fig. 11 (magenta region) corresponding to DM masses of typically m S < 78 GeV.
At the LHC, pairs of mediators can be copiously produced by virtue of the strong interaction. Top-antitop production in association with missing energy consists of the corresponding signature, as each mediator then decays, with a 100% branching fraction, into a system comprised of a top quark and a DM particle, pp → TT → tStS . Contributions to this process are illustrated by the first two Feynman diagrams of Fig. 9. Such a top-antitop plus missing energy signature has been widely studied by both the ATLAS and CMS collaborations, in particular in Run 2 searches for the superpartners of the top quark (assuming a decay into a top quarks and missing energy carried by a neutralino) [39][40][41][42][43][44][45][46][47] and in dedicated DM searches [48]. Additionally, the model can also be probed through classical DM searches using mono-X probes. Amongst all mono-X searches, we focus on the monojet one given the relative magnitude of the strong coupling with respect to the strength of the electroweak interactions. In this case, the considered signature exhibits the presence of a hard QCD jet recoiling against a large quantity of missing energy carried away by a pair of DM particles. Such a process, is loop-induced in our model, as illustrated by the last Feynman diagram of Fig. 9. Although early monojet analyses were vetoing events featuring any extra hadronic activity through additional hard jets, it has been demonstrated that the latter could consist in useful handles to get a better sensitivity to the signal [49]. For this reason, recent ATLAS and CMS monojet analyses now include several signal regions in which more than one hard jet is allowed [50-53].

V.1. Simulation details
In order to reinterpret relevant results of the LHC in the context of the considered top-philic DM scenario and to determine their impact, we have implemented the Lagrangian of Eq. (2.1) into the FeynRules program [54]. With the help of a joint usage of the NLOCT [55] and FeynArts [56] packages, we have analytically evaluated the ultraviolet and so-called R 2 counterterms required for numerical one-loop computations in four dimensions. The information has been exported under the form of an NLO UFO model [57] containing, in addition to the treelevel model information, the R 2 and NLO counterterms.
We rely on the MadGraph 5 aMC@NLO [58] platform for the generation of hard-scattering events, at the NLO accuracy in QCD for the vector-like quark pair production process of Eq. (5.2) and at the LO accuracy for the loop-induced monojet process of Eq. (5.3). In our simulation chain, we respectively convolute the LO and NLO matrix elements with the LO and NLO sets of NNPDF 3.0 parton distribution functions [59], that we access through the LHAPDF 6 library [60]. Moreover, the unphysical scales are always set to half the sum of the transverse mass of all final-state particles.
The decay of the heavy T quark into DM and a top quark, is factorized from the production processes and is handled with the MadSpin [61] and MadWidth [62] programs, together with those of all Standard Model heavy particles. For each considered new physics setup, we have consequently checked that the narrow-width approximation could be safely and constitently used, which is guaranteed by the fact that the mediator decay width satisfies Γ T /m T < 0.2. The resulting partonic events are matched with parton showers by relying on the Pythia 8 code [34] and the MC@NLO prescription [63]. Whilst hadronization is also taken care of by Pythia, we simulate the response of the ATLAS and CMS detectors by means of the Delphes 3 program [64] that internally relies on the anti-k T jet algorithm [65] as implemented in the FastJet software [66] for object reconstruction. For each of the analyses that we have recasted, the Delphes configuration has been tuned to match the detector setup described in the experimental documentation. We have used the MadAnalysis 5 framework [67][68][69] to calculate the signal efficiencies for the different considered search strategies and to derive 95% confidence level (CL) exclusions with the CL s method [70].

V.2. Reinterpreted LHC analyses
In order to assess the reach of LHC searches for DM in top-antitop quark production in association with missing energy (pp → tt + / E T ), we reinterpret a CMS analysis of collision events featuring a pair of leptons of opposite electric charge [47]. While other final states in the single lepton and fully hadronic decay mode of the top-antitop pair are relevant as well, all these LHC searches are so far found to yield similar bounds. For this reason, we have chosen to focus to a single of those channels, namely the cleaner dileptonic decay mode of the top-antitop pair.
[47] focuses on the analysis of 35.9 fb −1 of LHC collisions featuring the presence of a system of two isolated leptons of opposite electric charges which is compatible neither with a lowmass hadronic resonance nor with a Z boson. The presence of at least two hard jets is required, at least one of them being b-tagged, as well as a large amount of missing transverse energy. The latter is required to possess a large significance and to be well separated from the two leading jets. After this preselection, the analysis defines three aggregated signal regions depending of the value of the missing energy and of the stransverse mass m T 2 [71,72] reconstructed from the two leptons and the missing momentum.
In addition, we include in our investigations the CMS-SUS-16-052 analysis which is dedicated to probing the more compressed regions of the parameter space with 35.9 fb −1 of LHC collisions [73]. In this analysis, it is assumed that the top partner cannot decay on-shell into a top quark plus missing energy system, so that the search strategy is optimized for top partners decaying into systems made of three mostly soft fermions (including bquarks) and missing energy via an off-shell top quark. Event selection requires the presence of one hard initialstate-radiation jet and of at most a second jet well separated from the first one. Moreover, one asks for a single identified lepton, a large amount of missing energy and an important hadronic activity. The threshold values that are imposed and the detailed properties of the lepton, the missing energy and the hadronic activity allow to define two classes of three signal regions targeting varied new physics configurations.
We have also confronted the process of Eq. (5.2) to the LHC Run 1 results, and in particular to the null results of the 8 TeV search labeled CMS-B2G-14-004 [74][75][76]. This search focuses on singly-leptonic final states containing at least three jets (including at least one b-tagged jet) and a large amount of missing energy well separated from the jets. The event selection moreover constrains the transverse mass of the system comprised of the lepton and of the missing transverse momentum, as well as the m W T 2 transverse variable [77]. For the reinterpretation of the LHC search results for mono-X DM signals, we have considered two ATLAS analyses targeting a monojet-like topology, i.e. at least one very hard jet recoiling against some missing momentum and a subleading jet activity. Although those analyses [50,78] focus on a small integrated luminosity of LHC collisions (3.2 fb −1 ), they are already limited by the systematics so that the constraints derived from early Run 2 data are not expected to get more severe in the future [79]. In the ATLAS-EXOT-2015-03 analysis [50,80], the target consists in a monojet-like topology where the subleading jet activity is rather limited, the event selection being allowed to contain only up to three additional jets. Seven inclusive and seven exclusive signal regions are defined, the differences between them being related to various requirements on the missing energy. In contrast, the ATLAS-SUSY-2015-06 analysis [78,81] allows both for a small and larger subleading jet activity, the event selection being dedicated to final states containing two to six jets. Seven signal regions are defined, depending on the number and on the kinematic properties of the jets and on the missing momentum.
All the above analyses are implemented and validated in the MadAnalysis 5 framework, and have thus been

V.3. Collider constraints
In Fig. 10, we report our findings in the (m T , m S ) mass plane. As the vector-like-quark production process of Eq. (5.2) dominates regardless of the actual value of thẽ y t coupling, the latter is irrelevant for what concern constraints stemming from the LHC. All results can thus be represented in the considered two-dimensional plane. In the figure, we superimpose to the cosmology considerations discussed in the previous sections (namely the relic density and direct detection bounds, the indirect bounds being not reproduced, so as to avoid cluttering of the figure) the constraints that can be obtained by reinterpreting the results of the LHC searches for new physics discussed in section V.2.
For each new physics configuration and each signal region of each considered analysis, we evaluate the number of signal events s surviving the selection with MadAnalysis 5 and extract a CL s exclusion from the observed number of events n data populating the region and the expected number of background eventsb ± ∆b. To this aim, we undertake 100.000 toy experiments in which we generate the actual number of background events b by assuming that the corresponding distribution is a Gaussian of meanb and width ∆b. We then consider two Poisson distributions of parameters b and b + s to evaluate the p-values of the signal-plus-background and backgroundonly hypotheses, knowing that n data events have been observed. From these p-values, we derive the associated CL s value.
The colored regions shown in Fig. 10 are excluded at the 95% CL by at least one signal region of the considered analyses. The dark blue region corresponds to what we obtain with the reinterpretation of the results of the two CMS searches for DM in the top-antitop plus missing energy channel, namely CMS-SUS-17-001 and CMS-SUS-16-052. Whilst our results only focus on Run 2 data, we have verified that the obtained limits are compatible with the less stringent Run 1 constraints derived from the results of the CMS-B2G-14-004 analysis. The light blue area depicted on the figure corresponds to bounds that can be extracted from the reinterpretation of the results of the ATLAS-EXOT-2015-03 and ATLAS-SUSY-2015-06 searches for new physics in the multijet plus missing energy channel.
We have found that mediator masses ranging up to 1 TeV are excluded, provided that the DM mass is light enough for having enough phase space to guarantee the decay of the mediator into a DM particle and a top quark in a far-from-threshold regime. Whilst generic multijet plus missing energy searches are quite sensitive when the DM mass is small, they quickly lose any sensitivity for larger m S values. This stems from the monojet-like selection of the considered analyses, that can only be satisfied if enough phase space is available for the T decay process.
As soon as the T → tS decay channel is closed, the T quark becomes long-lived enough to hadronize before decaying and it could potentially travel on macroscopic distances in the detector. Whilst the unknown modeling of vector-like quark hadronization would introduce uncontrolled uncertainties on the predictions, none of the currently available computer tools allows for a proper handling of long-lived colored particles. Moreover, all considered LHC analyses have been designed for being sensitive to promptly-decaying new-physics states, and are thus expected to lose sensitivity when new physics particles are long-lived. For this reason, we restrict ourselves to provide LHC constraints in the region of the parameter space where the T quark can promptly decay into a top quark and a DM particle.

VI. SUMMARY
The WIMP paradigm is being tested by various experiments, both in astrophysics, cosmology and at colliders. At the same time, there is a significant interest on top-philic new physics, as the top quark is widely considered, due to its large mass, as a perfect laboratory for the study of the electroweak symmetry breaking mechanism. In this work, we have extensively investigated a simple DM scenario that brings naturally these topics together. It is based on a real scalar particle coupled to the top quark through a Yukawa coupling with a heavy vector-like quark. As in the SM sector, the top quark has the largest coupling to the Higgs boson, it is at least   [28]. Indirect DM searches: the dark green (at low mass) and light green (at large mass) regions are excluded by Fermi-LAT gamma-rays constraints [32] and by the CR analysis of Ref. [29]. The orange region delimited by a dot-dashed line is the projected sensitivity of Fermi-LAT after 15 years of exposure [30]. Collider searches: constraints on top partner production at LEP [38]  conceivable that it also features the largest coupling to a new dark sector. The model rests only on very few parameters (one coupling strength and two masses), so that it provides a good starting point to compare the impact of different experimental results from varied origins.
In the present case, we focus on DM direct and indirect detection searches, as well as on collider probes. We have studied the constraints on the DM model, paying special attention to the potential impact of the QCD radiative corrections on all the considered bounds (i.e., the DM relic abundance, the DM direct and indirect searches and the collider searches). In this way, our study complements and extents similar earlier works based on Majorana DM candidates [13,14,82].
Our analysis reveals that, although there is a complementarity between the different searches, only a small fraction of the viable parameter space of this very simple DM scenario is tested by the current experiments. This is illustrated in Fig. 11 which summarizes our results. On the long term, the most fruitful strategy to further test such a DM scenario would be to increase the energy reach at colliders. channels. If the vector-like fermionic mediator is not too heavy relatively to the DM particle, it may still be abundant at the time of freeze-out and hence either annihilates or co-annihilates [83], TT → gg or qq , with the latter processes involving t-channel DM exchanges. The TT , T T andTT channels are impacted by either attractive or repulsive Sommerfeld effects through gluonic exchanges [84]. In order to account for these effects, we have followed the procedure depicted in Ref. [5] with the only difference that we have taken explicitly into account the top-quark mass effects on the annihilation cross sections. In addition, we have verified, through various approaches that our treatment is correct in the s-wave approximation. Our treatment agrees with the results of Ref. [85] in the s-wave approximation. Going beyond this approximation is however known to lead to corrections to the relic density of less than 1%, as shown e.g., in supersymmetry [86].
On general grounds, we should also take into account the possible formation of TT , T T andTT bound states, which would imply a modification of the Sommerfeld corrections. It has however been concluded, using a setup similar to ours, that bound states have only a moderate impact on the DM relic density [87]. As the Sommerfeld corrections affect the relic abundance by less than at most 15%, we ignore bound state formation from the present calculations.

(B7)
Appendix C: On QCD corrections to the SS → ttg(γ) In this section, we comment on our derivation of the differential cross section for the SS → ttg(γ) process at the partonic level and on the methods that have been used to cope with the hadronization of the colored final state particles.
At O(α s ), the analytical expression for the SS → ttg amplitude includes contributions of gluon emission by the final state quarks and the intermediate particle T . When relevant, final-state radiation typically gives rise to double Sudakov logarithms associated with soft and collinear divergences, which must be consistently taken into account, in particular in regimes where they are large.
In our model, VIB emission is finite and moreover only relevant for the radiation of highly-energetic gluons. When VIB contributions are negligible (i.e., for sufficiently large m T /m S ratios), we have simply discarded them and used Pythia 8 [34] to handle both final-state radiation and hadronization. This effectively resums the large logarithms via the use of an appropriate Sukadov form factor. When VIB contributions are relevant, the corresponding three-body hard-scattering process has been explicitly evaluated with CalcHEP [18], and we have made use of Pythia 8 to simulate the subsequent hadronization. Finally, for low energies, we have restricted our computation to the two-body SS → tt process and relied on Pythia 8 for the simulation of final-state radiation and hadronization. The matching of the separate contributions to the gamma-ray spectrum has been achieved by implementing an explicit cutoff on the gluon energy at the partonic level. More details can be found in Ref. [6], which is similar in spirit but differs in details from the prescription proposed in Ref. [14].