# Global analyses of Higgs portal singlet dark matter models using GAMBIT

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## Abstract

We present global analyses of effective Higgs portal dark matter models in the frequentist and Bayesian statistical frameworks. Complementing earlier studies of the scalar Higgs portal, we use GAMBIT to determine the preferred mass and coupling ranges for models with vector, Majorana and Dirac fermion dark matter. We also assess the relative plausibility of all four models using Bayesian model comparison. Our analysis includes up-to-date likelihood functions for the dark matter relic density, invisible Higgs decays, and direct and indirect searches for weakly-interacting dark matter including the latest XENON1T data. We also account for important uncertainties arising from the local density and velocity distribution of dark matter, nuclear matrix elements relevant to direct detection, and Standard Model masses and couplings. In all Higgs portal models, we find parameter regions that can explain all of dark matter and give a good fit to all data. The case of vector dark matter requires the most tuning and is therefore slightly disfavoured from a Bayesian point of view. In the case of fermionic dark matter, we find a strong preference for including a CP-violating phase that allows suppression of constraints from direct detection experiments, with odds in favour of CP violation of the order of 100:1. Finally, we present DDCalc 2.0.0, a tool for calculating direct detection observables and likelihoods for arbitrary non-relativistic effective operators.

## 1 Introduction

Cosmological and astrophysical experiments have provided firm evidence for the existence of dark matter (DM) [1, 2, 3, 4]. While the nature of the DM particles and their interactions remains an open question, it is clear that the viable candidates must lie in theories beyond the Standard Model (BSM). A particularly interesting class of candidates are weakly interacting massive particles (WIMPs) [5]. They appear naturally in many BSM theories, such as supersymmetry (SUSY) [6]. Due to their weak-scale interaction cross-section, they can accurately reproduce the observed DM abundance in the Universe today.

So far there is no evidence that DM interacts with ordinary matter in any way except via gravity. However, the generic possibility exists that Standard Model (SM) particles may connect to new particles via the lowest-dimension gauge-invariant operator of the SM, \(H^\dagger H\). It is therefore natural to assume that the standard Higgs boson (or another scalar that mixes with the Higgs) couples to massive DM particles via such a ‘Higgs portal’ [7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. The discovery of the Higgs boson in 2012 by ATLAS [28] and CMS [29] therefore opens an exciting potential window for probing DM.

Despite being simple extensions of the SM in terms of particle content and interactions, Higgs portal models have a rich phenomenology, and can serve as effective descriptions of more complicated theories [30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52]. They can produce distinct signals at present and future colliders, DM direct detection experiments or in cosmic ray experiments. In the recent literature, experimental limits on Higgs portal models were considered from Large Hadron Collider (LHC), Circular Electron Positron Collider and Linear Collider searches, LUX and PandaX, supernovae, charged cosmic and gamma rays, Big Bang Nucleosynthesis, and cosmology [36, 41, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76]. The lack of such signals to date places stringent constraints on Higgs portal models.

The first global study of the scalar Higgs portal DM model was performed in Ref. [77]. The most recent global fits [78, 79] included relic density constraints from *Planck*, leading direct detection constraints from LUX, XENON1T, PandaX and SuperCDMS, upper limits on the gamma-ray flux from DM annihilation in dwarf spheroidal galaxies with the *Fermi*-LAT, limits on solar DM annihilation from IceCube, and constraints on decays of SM-like Higgs bosons to scalar singlet particles. The most recent [79] also considered the \(\mathbb {Z}_3\) symmetric version of the model, and the impact of requiring vacuum stability and perturbativity up to high energy scales.

In this paper, we perform the first global fits of the effective vector, Majorana fermion and Dirac fermion Higgs portal DM models using the GAMBIT package [80]. By employing the latest data from the DM abundance, indirect and direct DM search limits, and the invisible Higgs width, we systematically explore the model parameter space and present both frequentist and Bayesian results. In our fits, we include the most important SM, nuclear physics, and DM halo model nuisance parameters. For the fermion DM models, we present a Bayesian model comparison between the CP-conserving and CP-violating versions of the theory. We also carry out a model comparison between scalar, vector and fermion DM models.

In Sect. 2, we introduce the effective vector and fermion Higgs portal DM models. We describe the constraints that we use in our global fits in Sect. 3, and the details of our parameter scans in Sect. 4. We present likelihood and Bayesian model comparison results respectively in Sects. 5 and 6, and conclude in Sect. 7. Appendix A documents new features included in the latest version of DDCalc. Appendix B contains all the relevant expressions for the DM annihilation rate into SM particles. All GAMBIT input files, samples and best-fit points for this study are publicly available online via Zenodo [81].

## 2 Models

We separately consider vector (\(V_\mu \)), Majorana fermion (\(\chi \)) and Dirac fermion (\(\psi \)) DM particles that are singlets under the SM gauge group. By imposing an unbroken global \(\mathbb {Z}_2\) symmetry, under which all SM fields transform trivially but \((V_\mu , \chi , \psi ) \rightarrow -(V_\mu , \chi , \psi )\), we ensure that our DM candidates are absolutely stable.

*H*is the SM Higgs doublet. The fermionic Lagrangians include both CP-odd and CP-even Higgs-portal operators, with \(\theta \) controlling their relative size. The choice \(\cos \theta = 1\) corresponds to a pure scalar, CP-conserving interaction between the fermionic DM and the SM Higgs field, whereas \(\cos \theta = 0\) corresponds to a pure pseudoscalar, maximally CP-violating interaction. We discuss a possible ultraviolet (UV) completion of such a model in Sect. 3.7 (see also Refs. [12, 23]).

Likelihoods and corresponding GAMBIT modules/backends employed in our global fit

Likelihoods | GAMBIT modules/backends | References |
---|---|---|

Relic density ( | DarkBit | [4] |

Higgs invisible width | DecayBit | [84] |

| gamLike 1.0.0 | [85] |

LUX 2016 (Run II) | DDCalc 2.0.0 | [86] |

PandaX 2016 | DDCalc 2.0.0 | [87] |

PandaX 2017 | DDCalc 2.0.0 | [88] |

XENON1T 2018 | DDCalc 2.0.0 | [89] |

CDMSlite | DDCalc 2.0.0 | [90] |

CRESST-II | DDCalc 2.0.0 | [91] |

PICO-60 2017 | DDCalc 2.0.0 | [92] |

DarkSide-50 2018 | DDCalc 2.0.0 | [93] |

IceCube 79-string | nulike 1.0.6 | [94] |

*h*is the physical SM Higgs field and \(v_0= (\sqrt{2} G_F)^{-1/2} = 246.22\) GeV is the Higgs VEV. Thus, the \(H^\dagger H\) terms in Eqs. (1–3) generate mass and interaction terms for the DM fields. The tree-level physical mass of the vector DM is

^{1}Using the details outlined in the appendix of Ref. [51], we arrive at the following post-EWSB fermion DM Lagrangianswhere \(\xi \equiv \theta + \alpha \),

^{2}In light of this, we compare the viability of a CP-conserving scenario to the most general case with arbitrary \(\xi \) in Sect. 6.

## 3 Constraints

The free parameters of the Lagrangians are subject to various observational and theoretical constraints. For the case of vector DM, the relevant parameters after EWSB are the vector DM mass \(m_V\) and the dimensionless coupling \(\lambda _{hV}\).^{3} The post-EWSB fermion Lagrangians contain three free parameters: the fermion DM mass \(m_{\chi ,\psi }\), the dimensionful coupling \(\lambda _{h\chi ,h\psi }/\varLambda _{\chi ,\psi }\) between DM and the Higgs, and the scalar-pseudoscalar mixing parameter \(\xi \).

In Table 1, we summarise the various likelihoods used to constrain the model parameters in our global fit. In the following sections, we will discuss both the physics as well as the implementation of each of these constraints.

### 3.1 Thermal relic density

*H*is the Hubble rate and \(\langle \sigma v_{\text {{rel}}}\rangle \) is the thermally averaged cross-section times relative (Møller) velocity, given by

*s*-channel; consequently, near the resonance region, where \(m_X \simeq m_h/2\), it is crucial to perform the actual thermal average as defined in Eq. (12) instead of expanding \(\sigma v_\text {rel}^\text {cms}\) into partial waves.

^{4}Moreover, we take into account the important contributions arising from the production of off-shell pairs of gauge bosons \(W W^*\) and \(Z Z^*\) [97]. To this end, for \(45 \;\text {GeV} \le \sqrt{s} \le 300 \; \text {GeV}\), we compute the annihilation cross-section into SM gauge bosons and fermions in the narrow-width approximation via

*P*(

*X*) is given by

*p*-wave suppressed.

As shown in Ref. [97], for \(\sqrt{s} \gtrsim 300\) GeV the Higgs 1-loop self-interaction begins to overestimate the tabulated Higgs boson width in Ref. [98]. Thus, for \(\sqrt{s} > 300\) GeV (where the off-shell production of gauge boson pairs is irrelevant anyway), we revert to the tree-level expressions for the annihilation processes given in “Appendix B”. Moreover, for \(m_X \ge m_h\), DM can annihilate into a pair of Higgs bosons, a process which is not included in Eq. (13). We supplement the cross-sections computed from the tabulated DecayBit values with this process for \(m_X \ge m_h\). The corresponding analytical expression for the annihilation cross-sections are given in “Appendix B”.

Finally, we obtain the relic density of *X* by numerically solving Eq. (11) at each parameter point, using the routines implemented in DarkSUSY [99, 100] via DarkBit.

In the spirit of the EFT framework employed in this work, we do *not* demand that the particle *X* constitutes all of the observed DM, i.e., we allow for the possibility of other DM species to contribute to the observed relic density. Concretely, we implement the relic density constraint using a likelihood that is flat for predicted values below the observed one, and based on a Gaussian likelihood following the *Planck* measured value \(\Omega _{\text {DM}}h^2 = 0.1188 \pm 0.0010\) [4] for predictions that exceed the observed central value. We include a \(5\%\) theoretical error on the computed values of the relic density, which we combine in quadrature with the observed error on the *Planck* measured value. More details on this prescription can be found in Refs. [80, 101].

In regions of the model parameter space where the relic abundance of *X* is less than the observed value, we rescale all predicted direct and indirect detection signals by \(f_{\text {rel}} \equiv \Omega _X/\Omega _{\text {DM}}\) and \(f_{\text {rel}}^2\), respectively. In doing so, we conservatively assume that the remaining DM population does not contribute to signals in these experiments.

### 3.2 Higgs invisible decays

### 3.3 Indirect detection using gamma rays

Arguably, the most immediate prediction of the thermal freeze-out scenario is that DM particles can annihilate today, most notably in regions of enhanced DM density. In particular, gamma-ray observations of dwarf spheroidal galaxies (dSphs) of the Milky Way are strong and robust probes of any model of thermal DM with unsuppressed annihilation into SM particles.^{5}

*i*from a target object labeled by

*k*can be written in the factorised form \(\varPhi _i \cdot J_k\), where \(\varPhi _i\) encodes all information about the particle physics properties of the DM annihilation process, while \(J_k\) depends on the spatial distribution of DM in the region of interest. For

*s*-wave annihilation, one obtains

*j*in the zero-velocity limit, and \(dN_{\gamma ,j}/dE\) is the corresponding differential gamma-ray spectrum. The

*J*-factor in Eq. (19) is defined via a line of sight (l.o.s.) integral over the square of the DM density \(\rho _X\) towards the target object

*k*, extended over a solid angle \(\varDelta \Omega _k\).

*Fermi*-LAT data [85], which currently provides the strongest bounds on the annihilation cross-section of DM into final states containing gamma rays. We use the binned likelihoods implemented in DarkBit [101], which make use of the gamLike package. Besides the likelihood associated with the gamma-ray observations, given by

*J*-factors [85, 101]. We obtain the overall likelihood by profiling over the

*J*-factors of all 15 dwarf galaxies, as

### 3.4 Direct detection

*M*is the detector mass, \(T_{\text {exp}}\) is the exposure time and \(\phi \left( E\right) \) is the detector efficiency function, i.e., the fraction of recoil events with energy

*E*that are observable after applying all cuts from the corresponding analysis. The differential recoil rate

*dR*/

*dE*for scattering with a target isotope

*T*is given by

*q*(see e.g., Ref. [107]):

*A*is the mass number of the target isotope of interest, and \(F^2(E)\) is the standard form factor for spin-independent scattering [108]. As the typical momentum transfer in a scattering process is \(|q| \simeq (1- 100) \; \text {MeV} \ll m_X\), we note that direct detection constraints will be significantly suppressed for scenarios that are dominated by the pseudoscalar interaction, i.e., for \(\xi \simeq \pi /2\). For both the vector and fermion models, the spin-dependent (SD) cross-section is absent at leading order. Loop corrections are found not to give a relevant contribution to direct detection in the EFT approach, although they may lead to important effects in specific UV-completions [109, 110, 111].

For the evaluation of \(N_p\) in Eq. (22), we assume a Maxwell-Boltzmann velocity distribution in the Galactic rest frame, with a peak velocity \(v_{\mathrm{peak}}\) and truncated at the local escape velocity \(v_{\mathrm{esc}}\). We refer to Ref. [101] for the conversion to the velocity distribution \(f\left( \varvec{v},t\right) \) in the detector rest frame. We discuss the likelihoods associated with the uncertainties in the DM velocity distribution in Sect. 3.6.

^{6}to calculate the number of observed events

*o*in the signal regions for each experiment and to evaluate the standard Poisson likelihood

*s*and

*b*are the respective numbers of expected signal and background events. We model the detector efficiencies and acceptance rates by interpolating between the pre-computed tables in DDCalc. We include likelihoods from the new XENON1T 2018 analysis [89], LUX 2016 [86], PandaX 2016 [87] and 2017 [88] analyses, CDMSlite [90], CRESST-II [91], PICO-60 [92], and DarkSide-50 [93]. Details of these implementations, as well as an overview of the new features contained in DDCalc 2.0.0, can be found in “Appendix A”.

### 3.5 Capture and annihilation of DM in the Sun

Similar to the process underlying direct detection, DM particles from the local halo can also elastically scatter off nuclei in the Sun and become gravitationally bound. The resulting population of DM particles near the core of the Sun can then induce annihilations into high-energy SM particles that subsequently interact with the matter in the solar core. Of the resulting particles, only neutrinos are able to escape the dense Solar environment. Eventually, these can be detected in neutrino detectors on the Earth [112, 113, 114].

The capture rate of DM in the Sun is obtained by integrating the differential scattering cross-section \(d\sigma /dq^2\) over the range of recoil energies resulting in a gravitational capture, as well as over the Sun’s volume and the DM velocity distribution. To this end, we employ the newly-developed public code Capt’n General,^{7} which computes capture rates in the Sun for spin-independent and spin-dependent interactions with general momentum- and velocity-dependence, using the B16 Standard Solar Model [115] composition and density distribution. We refer to Refs. [116, 117] for details on the capture rate calculation. Notice that similar to direct detection, the capture rate is also subject to uncertainties related to the local density and velocity distribution of DM in the Milky Way. As mentioned earlier, these uncertainties are taken into account by separate nuisance likelihoods to be discussed in Sect. 3.6.

*C*(

*t*) is the capture rate of DM in the Sun, and \(A(t) \propto \langle \sigma v_\text {rel} \rangle N_X(t)^2\) is the annihilation rate of DM inside the Sun; this is calculated by DarkBit. We approximate the thermally averaged DM annihilation cross-section, which enters in the expression for the annihilation rate, by evaluating \(\sigma v\) at \(v=\sqrt{2T_\odot /m_X}\), where \(T_\odot = 1.35 \; \text {keV}\) is the core temperature of the Sun.

At sufficiently large *t*, the solution for \(N_X(t)\) reaches a steady state and depends only on the capture rate. However, the corresponding time scale \(\tau \) for reaching equilibrium depends also on \(\sigma v\), and thus changes from point to point in the parameter space. Hence, we use the full solution of Eq. (27) to determine \(N_X\) at present times, which in turn determines the normalization of the neutrino flux potentially detectable at Earth. We obtain the flavour and energy distribution of the latter using results from WimpSim [121] included in DarkSUSY [99, 100].

Finally, we employ the likelihoods derived from the 79-string IceCube search for high-energy neutrinos from DM annihilation in the Sun [94] using nulike [122] via DarkBit; this contains a full unbinned likelihood based on the event-level energy and angular information of the candidate events.

### 3.6 Nuisance likelihoods

Nuisance parameters that are varied simultaneously with the DM model parameters in our scans. All parameters have flat priors. For more details about the nuisance likelihoods, see Sect. 3.6

Parameter | Value (\(\pm \,\)range) | |
---|---|---|

Local DM density | \(\rho _0\) | \(0.2{-}0.8\) GeV cm\(^{-3}\) |

Most probable speed | \(v_{\mathrm{peak}}\) | \(240\,(24)\) km s\(^{-1}\) |

Galactic escape speed | \(v_{\text {esc}}\) | \(533\,(96)\) km s\(^{-1}\) |

Nuclear matrix element | \(\sigma _s\) | \(43\,(24)\) MeV |

Nuclear matrix element | \(\sigma _l\) | \(50\,(45)\) MeV |

Higgs pole mass | \(m_h\) | 124.1–127.3 GeV |

Strong coupling | \(\alpha _s^{\overline{MS}}(m_Z)\) | \(0.1181\,(33)\) |

The constraints discussed in the previous sections often depend on *nuisance parameters*, i.e. parameters not of direct interest but required as input for other calculations. Examples are nuclear matrix elements related to the DM direct detection process, the distribution of DM in the Milky Way, or SM parameters known only to finite accuracy. It is one of the great virtues of a global fit that such uncertainties can be taken into account in a fully consistent way, namely by introducing new free parameters into the fit and constraining them by new likelihood terms that characterise their uncertainty. We list the nuisance parameters included in our analysis in Table 2, and discuss each of them in more detail in the rest of this section.

Following the default treatment in DarkBit, we include a nuisance likelihood for the local DM density \(\rho _0\) given by a log-normal distribution with central value \(\rho _0 = 0.40\) GeV cm\(^{-3}\) and an error \(\sigma _{\rho _0}=0.15\) GeV cm\(^{-3}\). To reflect the log-normal distribution, we scan over an asymmetric range for \(\rho _0\). For more details, see Ref. [101].

For the parameters determining the Maxwell-Boltzmann distribution of the DM velocity in the Milky Way, namely \(v_\text {peak}\) and \(v_\text {esc}\), we employ simple Gaussian likelihoods. Since \(v_{\mathrm{peak}}\) is equal to the circular rotation speed \(v_{\mathrm{rot}}\) at the position of the Sun for an isothermal DM halo, we use the determination of \(v_{\mathrm{rot}}\) from Ref. [123] to obtain \(v_{\mathrm{peak}} = 240\pm 8\) km s\(^{-1}\).^{8} The escape velocity takes a central value of \(v_{\mathrm{esc}} = 533 \pm 31.9\) km s\(^{-1}\), where we convert the 90% C.L. interval obtained by the RAVE collaboration [126], assuming that the error is Gaussian.

*q*content of a nucleon

*N*. As described in more detail in Ref. [127], these are obtained from the following observable combinations

We also use a Gaussian likelihood for the Higgs mass, based on the PDG value of \(m_h = 125.09\pm 0.24\) GeV [130]. In line with our previous study of scalar singlet DM [78], we allow the Higgs mass to vary by more than \(4\sigma \) as the phenomenology of our models depends strongly on \(m_h\), most notably near the Higgs resonance region. Finally, we take into account the uncertainty on the strong coupling constant \(\alpha _s\), which enters the expression for the DM annihilation cross-section into SM quarks (see “Appendix B”), taking a central value \(\alpha _s^{\overline{MS}}(m_Z) = 0.1181 \pm 0.0011\) [130].

### 3.7 Perturbative unitarity and EFT validity

The parameter spaces in which we are interested are limited by the requirement of perturbative unitarity. First of all, this requirement imposes a bound on any dimensionless coupling in the theory. Furthermore, as neither the vector or fermion Higgs portal models are renormalisable, we must ensure that the effective description is valid for the parameter regions to be studied.

The dimensionless coupling \(\lambda _{hV}\) in the vector DM model is constrained by the requirement that annihilation processes such as \(VV\rightarrow hh\) do not violate perturbative unitarity. Determining the precise bound to be imposed on \(\lambda _{hV}\) is somewhat involved, so we adopt the rather generous requirement \(\lambda _{hV} < 10\) with the implicit understanding that perturbativity may become an issue already for somewhat smaller couplings.

*X*and the Higgs doublet as [12]

^{9}For this specific UV completion, we assume that the mixing between \(\varPhi \) and the Higgs field is negligible and can be ignored. The heavy scalar field can be integrated out to give a dimensionful coupling in the EFT approximation as

*q*of the interaction, i.e., \(m_\varPhi \gg q\) such that \(\varPhi \) can be integrated out. For DM annihilations, the momentum exchange is \(q \approx 2m_X\). Thus, the EFT approximation breaks down when \(m_\varPhi < 2m_X\) and our EFT assumption is violated when

For parameter points close to the EFT validity bound, the scale of new physics is expected to be close to or even below \(2m_\chi \). In this case, the annihilation cross-section \(\sigma v_\text {rel}\), used in predictions of both the relic density and indirect detection signals, may receive substantial corrections from interactions with \(\varPhi \), which are not captured in the EFT approach. The likelihoods computed for these points should hence be interpreted with care.

Note that this prescription is only the simplest and most conservative approach; additional constraints can be obtained by unitarising the theory (e.g. [132]).

## 4 Scan details

Parameter ranges and priors for the vector DM model

Parameter | Minimum | Maximum | Prior type |
---|---|---|---|

\(\lambda _{hV}\) | \(10^{-4}\) | 10 | Log |

\(m_V\) (low mass) | 45 GeV | 70 GeV | Flat |

\(m_V\) (high mass) | 45 GeV | 10 TeV | Log |

Parameter ranges and priors for the fermion DM models. Our choice for the range of \(\xi \) between 0 and \(\pi \) reflects the fact that only odd powers of \(\cos \xi \) appear in the observables that we consider, but never odd powers of \(\sin \xi \), which cancel exactly due to the complex conjugation. Thus, the underlying physics is symmetric under \(\xi \rightarrow -\xi \)

Parameter | Minimum | Maximum | Prior type |
---|---|---|---|

\(\lambda _{h\chi ,h\psi }/\varLambda _{\chi ,\psi }\) | \(10^{-6}\) GeV\(^{-1}\) | 1 GeV\(^{-1}\) | Log |

\(\xi \) | 0 | \(\pi \) | Flat |

\(m_{\chi ,\psi }\) (low mass) | 45 GeV | 70 GeV | Flat |

\(m_{\chi ,\psi }\) (high mass) | 45 GeV | 10 TeV | Log |

There are two main objectives for the Bayesian scans: firstly, producing marginal posteriors for the parameters of interest, where we integrate over all unplotted parameters, and secondly, computing the marginal likelihood (or Bayesian evidence). We discuss the marginal likelihood in Sect. 6. We use T-Walk, an ensemble Markov Chain Monte Carlo (MCMC) algorithm, for sampling from the posterior, and MultiNest [133, 134, 135], a nested sampling algorithm, for calculating the marginal likelihood. We use T-Walk for obtaining the marginal posterior due to the ellipsoidal bias commonly seen in posteriors computed with MultiNest [136].

For the frequentist analysis, we are interested in mapping out the highest likelihood regions of our parameter space. For this analysis we largely use Diver, a differential evolution sampler, efficient for finding and exploring the maxima of a multi-dimensional function. Details of T-Walk and Diver can be found in Ref. [136].

Due to the resonant enhancement of the DM annihilation rate by *s*-channel Higgs exchange at \(m_X \approx m_h/2\), there is a large range of allowed DM-Higgs couplings that do not overproduce the observed DM abundance. When scanning over the full mass range, it is difficult to sample this resonance region well, especially with a large number of nuisance parameters. For this reason, we perform separate, specific scans in the low-mass region around the resonance, using both T-Walk and Diver. When plotting the profile likelihoods, we combine the samples from the low- and high-mass scans.

Conversion criteria used for various scanning algorithms in both the full and low mass regimes. The Open image in new window chosen for T-Walk varies from scan to scan; we use the default T-Walk behaviour of Open image in new window = \(N_{\text {MPI}}\) + \(N_{\text {params}}\) + 1 on 1360 MPI processes. For more details, see Ref. [136]

Scanner | Parameter | Value |
---|---|---|

T-Walk | 1370 (1) | |

\(<0.01\) | ||

1380 | ||

MultiNest | 20,000 | |

\(10^{-2}\) | ||

Diver | 50,000 | |

\(10^{-5}\) |

The convergence criteria that we employ for the different samplers are outlined in Table 5. We carried out all Diver scans on 340 Intel Xeon Phi 7250 (Knights Landing) cores. As in our recent study of scalar singlet DM [79], we ran T-Walk scans on 1360 cores for 23 h, providing us with reliable sampling. The MultiNest scans are based on runs using 240 Intel Broadwell cores, with a relatively high tolerance value, which is nevertheless sufficient to compute the marginal likelihood to the accuracy required for model comparison. We use the importance sampling log-evidence from MultiNest to compute Bayes factors.

For profile likelihood plots, we combine the samples from all Diver and T-Walk scans, for each model. The plots are based on \(1.46\times 10^7\), \(1.70\times 10^7\) and \(1.73\times 10^7\) samples for the vector, Majorana and Dirac models, respectively. We do all marginalisation, profiling and plotting with pippi [137].

## 5 Results

### 5.1 Profile likelihoods

In this section, we present profile likelihoods from the combination of all Diver and T-Walk scans for the vector, Majorana and Dirac models. Profile likelihoods in the vector model parameters are shown in Fig. 1, with key observables rescaled to the predicted DM relic abundance in Fig. 2. Majorana model parameter profile likelihoods are shown in Figs. 3 and 4, with observables in Fig. 5. For the Dirac model, we simply show the mass-coupling plane in Fig. 6, as the relevant physics and results are virtually identical to the Majorana case.

#### 5.1.1 Vector model

Figure 1 shows that the resonance region is tightly constrained by the Higgs invisible width from the upper-left when \(m_V < m_h/2\), by the relic density constraint from below, and by direct and indirect detection from the right. Nevertheless, the resonant enhancement of the DM annihilation at around \(m_h/2\), combined with the fact that we allow for scenarios where \(V_\mu \) is only a fraction of the observed DM, permits a wide range of portal couplings. Interestingly, the perturbative unitarity constraint (shown as dark grey) in Eq. (30) significantly shortens the degenerate ‘neck’ region that appears exactly at \(m_h/2\). Most notably, this is in contrast with the scalar Higgs portal model [78, 79] where no such constraint exists.

The high-mass region contains a set of solutions at \(m_V \simeq 10\,\text {TeV}\) and \(\lambda _{hV} \gtrsim 1\), which are constrained by the relic density from below and direct detection from the left. This second island is prominent in both our previous studies of the scalar Higgs portal model [78, 79] as well as other studies of the vector Higgs portal [51]. The precise extent of this region depends on the upper bound imposed on \(\lambda _{hV}\) to reflect the breakdown of perturbativity. While the constraint that we apply ensures that perturbative unitarity is not violated [131], higher-order corrections may nevertheless become important in this region. The perturbative unitarity constraint from Eq. (30) excludes solutions that would otherwise exist in a separate triangular region at \(m_\chi \simeq m_h\), \(\lambda _{hV}\simeq 1\).

*Planck*relic abundance, the best-fit point remains almost unchanged, at \(\lambda _{hV} = 4.5\times 10^{-4}\) and \(m_V = 62.46\) GeV. We give details of these best-fit points, along with the equivalent for fermion models, in Table 7.

^{10}[139] will also be able to probe large amounts of the high-mass region; however it does not have the exclusion power that direct detection does for Higgs portal models. Again, the best-fit point remains out of reach.

#### 5.1.2 Majorana fermion model

We show profile likelihoods in the \((m_{\chi },\lambda _{h\chi }/\varLambda _\chi )\) plane in Fig. 3, with the low-mass region in the left panel and the full mass region in the right panel. Here, there are no longer two distinct solutions: the resonance and high mass regions are connected. From the left panel in Fig. 4, where we plot the profile likelihood in the \((m_{\chi },\xi )\) plane, we can see that these regions are connected by the case where the portal interaction is purely pseudoscalar, \(\xi = \pi /2\), leading to an almost complete suppression of constraints from the direct detection experiments, as given in Eq. (25).

The high mass region prefers \(\xi \sim \pi / 2\), with a wider deviation from \(\pi /2\) permitted as \(m_\chi \) is increased, due to direct detection constraints, which become less constraining at higher WIMP masses. There is an enhancement in the permitted range of mixing angles at \(m_\chi \gtrsim m_h\), due to the contact term \((\propto \overline{\chi }\chi hh)\), where DM annihilation to on-shell Higgses reduces the relic density, providing another mechanism for suppressing direct detection signals, thus lifting the need to tune \(\xi \).

*less*

*p*-wave suppressed (Eq. 14), and indirect detection comes to dominate the constraint at the edge of the allowed parameter space just above the resonance.

In the low-mass resonance region, virtually all values of the mixing angle are permitted, seen clearly in the left panel of Fig. 4, as even purely scalar couplings are not sufficient for direct detection to probe the remaining parameter space. The right panel also shows this in the lower ‘bulb’: couplings between \(10^{-3}\) and \(10^{-5}\,\mathrm{GeV}^{-1}\) are only permitted in the resonance region, without any constraint on the mixing angle.

Contributions to the delta log-likelihood \((\varDelta \ln \mathcal {L})\) at the best-fit point for the vector, Majorana and Dirac DM, compared to an ‘ideal’ case, both with and without the requirement of saturating the observed relic density (RD). Here ‘ideal’ is defined as the central observed value for detections, and the background-only likelihood for exclusions. Note that many likelihoods are dimensionful, so their absolute values are less meaningful than any offset with respect to another point (for more details, see Sect. 8.3 of Ref. [80])

Log-likelihood contribution | Ideal | \(\varDelta \ln \mathcal {L}\) | |||||
---|---|---|---|---|---|---|---|

\(V_\mu \) | \(V_\mu \) + RD | \(\chi \) | \(\chi \) + RD | \(\psi \) | \(\psi \) + RD | ||

Relic density | 5.989 | 0.000 | 0.106 | 0.000 | 0.107 | 0.000 | 0.242 |

Higgs invisible width | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.000 | 0.000 |

\(\gamma \) rays ( | \(-\) 33.244 | 0.105 | 0.105 | 0.102 | 0.120 | 0.129 | 0.134 |

LUX 2016 (Run II) | \(-\) 1.467 | 0.003 | 0.003 | 0.020 | 0.000 | 0.028 | 0.028 |

PandaX 2016 | \(-\) 1.886 | 0.002 | 0.002 | 0.013 | 0.000 | 0.018 | 0.017 |

PandaX 2017 | \(-\) 1.550 | 0.004 | 0.004 | 0.028 | 0.000 | 0.039 | 0.039 |

XENON1T 2018 | \(-\) 3.440 | 0.208 | 0.208 | 0.143 | 0.211 | 0.087 | 0.087 |

CDMSlite | \(-\) 16.678 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

CRESST-II | \(-\) 27.224 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

PICO-60 2017 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

DarkSide-50 2018 | \(-\) 0.090 | 0.000 | 0.000 | 0.002 | 0.000 | 0.005 | 0.005 |

IceCube 79-string | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.001 | 0.001 |

Hadronic elements \(\sigma _{s}\), \(\sigma _l\) | \(-\) 6.625 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Local DM density \(\rho _0\) | 1.142 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Most probable DM speed \(v_\text {peak}\) | \(-\) 2.998 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Galactic escape speed \(v_\text {esc}\) | \(-\) 4.382 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

\(\alpha _{\text {s}}\) | 5.894 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Higgs mass | 0.508 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 | 0.000 |

Total | 86.051 | 0.322 | 0.428 | 0.308 | 0.439 | 0.307 | 0.553 |

Details of the best-fit parameter points for vector, Majorana and Dirac DM Higgs portal models, both with and without the requirement that the predicted relic density is within \(1\sigma \) of the *Planck* observed value. Here, \(X \in \{V, \chi , \psi \}\) and the dimensionful nature of the coupling is implied for the fermion cases. We do not include the values of nuisance parameters, as they do not differ significantly from the central values of their likelihoods

Model | Relic density condition | \(\lambda _{hX}\) | \(m_X\) (GeV) | \(\xi \) (rad) | \(\Omega _Xh^2\) | \(\varDelta \ln \mathcal {L}\) |
---|---|---|---|---|---|---|

Vector | \(\Omega _V h^2 \lesssim \Omega _{DM}h^2\) | \(4.9\times 10^{-4}\) | 62.46 | – | \(9.343\times 10^{-2}\) | 0.322 |

\(\Omega _V h^2 \sim \Omega _{DM}h^2\) | \(4.5\times 10^{-4}\) | 62.46 | – | \(1.128\times 10^{-1}\) | 0.428 | |

Majorana | \(\Omega _\chi h^2 \lesssim \Omega _{DM}h^2\) | \(4.5\times 10^{-2}\,\text {GeV}^{-1}\) | 138.4 | 1.96 | \(6.588\times 10^{-8}\) | 0.308 |

\(\Omega _\chi h^2 \sim \Omega _{DM}h^2\) | \(6.3\times 10^{-6}\,\text {GeV}^{-1}\) | 61.03 | 1.41 | \(1.128\times 10^{-1}\) | 0.439 | |

Dirac | \(\Omega _\psi h^2 \lesssim \Omega _{DM}h^2\) | \(6.3\times 10^{-4}\,\text {GeV}^{-1}\) | \(9.950\times 10^3\) | 2.06 | \(3.813\times 10^{-2}\) | 0.307 |

\(\Omega _\psi h^2 \sim \Omega _{DM}h^2\) | \(3.6\times 10^{-4}\,\text {GeV}^{-1}\) | \(9.895\times 10^3\) | 2.07 | \(1.155\times 10^{-1}\) | 0.553 |

In Fig. 5, we show the relic density (top) and scaled cross-sections for direct (centre) and indirect detection (bottom). For plotting purposes, we compute \(\sigma _{\mathrm{SI}}\) at a reference momentum exchange of \(q=50\) MeV, typical of direct detection experiments. Substantial fractions of allowed parameter space lie close to current limits, but unsurprisingly, large portions of the parameter space will not be probed by future direct detection experiments, due to the momentum suppression. This is also true for indirect detection, where cross-sections are velocity suppressed. However, given that the two suppressions have opposite dependences on the mixing parameter, the two probes will be able to compensate for each others’ weaknesses to a certain extent.

Table 6 shows a breakdown of the contributions to the likelihood at the best-fit point, which lies in the high mass region at \(m_\chi = 138.4\) GeV, \(\lambda _{h\chi }/\varLambda _\chi =4.5\times 10^{-2}\) GeV\(^{-1}\) and \(\xi = 1.96\) rad (Table 7). When we demand that \(\chi \) saturates the observed DM relic abundance, the best-fit point shifts to the lower end of the resonance region at \(m_\chi = 61.03\) GeV, \(\lambda _{h\chi }/\varLambda _\chi =6.3\times 10^{-6}\) GeV\(^{-1}\) and \(\xi = 1.41\) rad.

#### 5.1.3 Dirac fermion model

The results from our low- and high-mass scans of the Dirac fermion model are very similar to those for the Majorana model. We therefore only show results in the \((m_\psi , \lambda _{h\psi }/\varLambda _\psi )\) plane in Fig. 6.

In Table 6, we show a breakdown of the contributions to the likelihood at the best-fit point. This point lies towards the upper end of the high mass region, where \(\lambda _{h\psi }/\varLambda _\psi =6.3\times 10^{-4}\) GeV\(^{-1}\), \(m_\psi = 9.95\) TeV and \(\xi = 2.06\) rad. If \(\psi \) makes up all of the DM, the best-fit point shifts slightly to the bottom of the high mass triangle at \(\lambda _{h\psi }/\varLambda _\psi =3.6\times 10^{-4}\) GeV\(^{-1}\), \(m_\psi = 9.9\) TeV and \(\xi = 2.07\) rad. We compare the locations of these best-fit points to those from the vector and Majorana models in Table 7.

#### 5.1.4 Goodness of fit

In Table 6, we show the contribution to the log-likelihood for the best-fit points of the vector, Majorana and Dirac DM models. By equating \(\varDelta \ln \mathcal {L}\) to half the “likelihood \(\chi ^2\)” of Baker and Cousins [142], we can compute an approximate *p*-value for each best-fit point against a null hypothesis. We take this null to be the ‘ideal’ case, which we define as the background-only contribution in the case of exclusions, and the observed value in the case of detections.

*p*-value between roughly 0.4 and 0.7. Requiring the relic density of \(V_\mu \) to be within \(1\sigma \) of the

*Planck*value, the

*p*-value becomes \(p\approx 0.35\)–0.65. For both the Majorana and Dirac fermion models, we also find \(p \approx 0.4\)–0.7, falling to 0.35–0.65 with the relic density requirement. All of these are completely acceptable

*p*-values.

### 5.2 Marginal posteriors

The marginal posterior automatically penalises fine-tuning, as upon integration of the posterior, regions with a limited ‘volume of support’ over the parameters that were integrated over are suppressed.^{11} As usual, the marginal posteriors depend upon the choice of priors for the free model parameters, which are summarised in Tables 3 and 4. We choose flat priors where parameters are strongly restricted to a particular scale, such as the mixing parameter and the DM mass in scans restricted to the low-mass region. For other parameters, in order to avoid favouring a particular scale we employ logarithmic priors. Note that in this treatment for the fermionic DM models we have not chosen priors that favour the CP-conserving case. We instead present posteriors for this well motivated case separately, and later in Sect. 6 we perform a Bayesian model comparison between a CP-conserving fermionic DM model and the full model considered here.

#### 5.2.1 Vector model

To obtain the marginal posterior distributions, we perform separate T-Walk scans for the low and high mass regimes, shown in Fig. 7. Within each region we plot the relative posterior probability across the parameter ranges of interest.

In the left panel of Fig. 7, the scan of the resonance region shows that the neck region is disfavoured after marginalising over the nuisance parameters, particularly \(m_h\), which sets the width of the neck. This dilutes the allowed region due to volume effects.

In the full-mass-range scan, the fine-tuned nature of the resonance region is clearly evident. Although the best-fit point in the profile likelihood lies in the resonance region, the posterior mass is so small in the entire resonance region that it drops out of the global \(2\sigma \) credible interval.

#### 5.2.2 Majorana fermion model

Nevertheless, direct detection does have a significant impact on the high-mass region, in spite of the mixing parameter \(\xi \). While the \(2 \sigma \) contour is roughly triangular, the points with highest posterior probability (i.e. within the \(1 \sigma \) contours) are split into two smaller triangles. The approximately rectangular region that separates these two triangular regions is disfavoured by the combination of volume effects and direct detection, which requires \(\xi \) to be tuned relatively close to \(\pi /2\).

To better understand the role of tuning in \(\xi \) in the process of marginalisation, we show the marginalised posterior in the \((m_\chi , \xi )\) and \((\xi ,\lambda _{h\chi }/\varLambda _\chi )\) planes in Figs. 9 and 10, respectively. Figure 9 provides a clear understanding of the differences between the marginalised posteriors in Fig. 8 and the profile likelihood in Fig. 3. In the resonance region (left panel), the neck region is less prominent in the marginalised posterior because direct detection limits become very constraining as soon as \(m_\chi > m_h/2\) and the mixing parameter is forced to be very close to \(\pi /2\). In the full-range scan (right panel) we see the annihilation channel \(\overline{\chi } \chi \rightarrow h h\) open up, thus allowing a greater range of values for \(\xi \), leading to an enhancement in the marginalised posterior probability. This clearly corresponds to the \(1 \sigma \) triangular region in the mass-coupling plane at \(m_\chi \approx m_h\), in the right hand panel of Fig. 8.

In the left panel of Fig. 10, which focuses on the resonance region, we see two separate solutions for the mixing angle and coupling: the larger island at lower coupling corresponds to the triangular region at \(m_\chi < m_h/2\), permitting all values of \(\xi \), and the thinner solution at larger couplings reflects the solution at \(m_\chi > m_h/2\), where the scalar coupling between the Higgs and the Majorana DM needs to be sufficiently small (i.e. \(\xi \sim \pi /2\)) to evade direct detection limits. The two regions appear disconnected because the intermediate parameter points require so much tuning that they fall outside of the \(2\sigma \) credible regions upon marginalisation. Considering the full mass range (see the right panel in Fig. 10), we find that the lower ‘bulb’ seen in the profile likelihood in Fig. 4 is hardly visible in the marginalised posterior when integrating over the nuisance parameters, due to a lower posterior volume in the resonance region.

For comparison, we consider the CP-conserving case with fixed \(\xi = 0\) in Fig. 12. As expected from the discussion above, we find that the permitted parameter space shrinks vastly with respect to the case where the mixing parameter is allowed to vary (see Fig. 8). In the resonance region (left panel), we see that direct detection, the invisible Higgs width and relic density impose strong constraints from the left, upper-left and below, respectively. No neck region exists because the direct detection constraints are too strong, overlapping with constraints on the invisible width of the Higgs boson. In the full-range scan (right panel), we find that the only surviving parameter space is split into the resonance region, and two small islands, at \(m_\chi \sim m_h\) and \(m_\chi \sim 5\) TeV. These islands are constrained by direct detection and the EFT validity requirement. Both will be ruled out by the next generation of direct detection experiments, if no DM signal is observed.

Our analysis of the Dirac fermion model parameter space is identical to the Majorana fermion one, whether \(\xi \) is fixed or left as a free parameter, so to avoid repetition we omit those results.

It should be clear from the comparison between Figs. 8 and 12 that the CP-conserving case \((\xi = 0)\) is strongly disfavoured relative to the case where \(\xi \) is allowed to vary. We will make this qualitative observation more precise in the following section.

## 6 Bayesian model comparison

### 6.1 Background

*not*apply any further prior in favour of CP conservation. The volume integrals involved in the Bayes factor automatically implement the concept of naturalness via Occam’s razor, penalising models with more free parameters if they do not fit the observed data any better than models with less parameters.

From Eq. (35), we can see that the evidence of a model depends on the prior choices for its parameters. This prior on the model parameters (along with the priors on the models themselves) makes the results of Bayesian model comparison inherently prior-dependent. However, the influence of common parameters treated with identical priors in both models approximately cancels when taking the ratio of evidences, as in Eq. (34). The overall prior dependence of the Bayes factor can thus be minimised by minimising the number of non-shared parameters between the models being compared. The best case is where one model is nested inside the other, and corresponds simply to a specific choice for one of the degrees of freedom in the larger model. In this case, the leading prior dependence is the one coming from the chosen prior on the non-shared degree of freedom. Thus, we first investigate the question of CP violation in the Higgs portal, which we can address in this manner, before going on to the more prior-dependent comparison of the broader models.

### 6.2 CP violation in the Higgs portal

- 1.
Assuming the parametrisation that we have discussed thus far for the Majorana model, taking a uniform prior for \(\xi \) and a logarithmic prior for \(\lambda _{h\chi }/\varLambda _\chi \). This corresponds to the assumption that some single mechanism uniquely determines the magnitude and phase of both couplings.

- 2.Assuming that the scalar and pseudoscalar couplings originate from distinct physical mechanisms at unrelated scales, such that they can be described by independent logarithmic priors. The post-EWSB Lagrangian in this parametrisation contains the termsIn this case, the parameters \(\xi \) and \(\lambda _{h\chi }/\varLambda _\chi \) from the first parametrisation are replaced by \(g_\text {s}/\varLambda _\text {s}\) and \(g_\text {p}/\varLambda _\text {p}\). In this parametrisation, the Bayes factor may be sensitive to the range of the prior for the couplings, as the normalisation factor does not cancel when computing the Bayes factor for the CP-conserving scenario. We choose \(-6 \le \log _{10} ({g/\varLambda }) \le 0\) for the couplings when computing the Bayes factors in this parametrisation, in line with the prior that we adopt for \(\lambda _{h\chi }/\varLambda _\chi \) in parametrisation 1.$$\begin{aligned} \mathcal {L}_\chi \supset - \frac{1}{2}\left( \frac{g_\text {s}}{\varLambda _\text {s}} \overline{\chi } \chi + \frac{g_\text {p}}{\varLambda _\text {p}} \overline{\chi } i\gamma _5 \chi \right) \left( v_0h + \frac{1}{2} h^2 \right) \, . \end{aligned}$$(38)

*a priori*is very unlikely to be zero.

In Table 8, we give the odds ratios against the CP-conserving case in each of these parametrisations. The value given in the final column of this table is the ratio of the evidence for the CP-violating model to the CP-conserving case. Depending on the choice of parametrisation, we see that there is between 140:1 and 70:1 odds against the CP-conserving version of the Majorana Higgs portal model. The similarity in order of magnitude^{12} between these two results is expected, as it reflects the relatively mild prior-dependence of the Bayes factor when performing an analysis of nested models that differ by only a single parameter. Given the similarity of the likelihood functions in the Majorana and Dirac fermion models, the odds against the pure CP-conserving version of the Dirac fermion Higgs portal model can also be expected to be very similar.

Odds ratios for CP violation for the singlet Majorana fermion Higgs portal model. Here the odds ratios are those against a pure CP-even Higgs portal coupling, as compared to two different parametrisations (and thus priors) of the model in which the CP nature of the Higgs portal can vary freely

Model | Comparison model and priors | Odds |
---|---|---|

\(\xi =0\) | \(m_\chi \): log \(\lambda _{h\chi }/\varLambda _\chi \): log \(\xi \): flat | 70:1 |

\(g_\text {p}/\varLambda _\text {p} = 0\) | \(m_\chi \): log \(g_\text {s}/\varLambda _\text {s}\): log \(g_\text {p}/\varLambda _\text {p}\): log | 140:1 |

### 6.3 Scalar, vector, Majorana or Dirac?

We also carry out model comparison between the different Higgs portal models: scalar, vector, Majorana and Dirac. As these models are not nested, they each have unique parameters. This means that there is no *a priori* relationship between their respective parameters that would allow the definition of equivalent priors on, e.g., masses or couplings in two different models. The prior dependence of the Bayes factor is therefore unsuppressed by any approximate cancellations when taking the ratio of evidences in Eq. (34). We caution that the resulting conclusions are consequently less robust than for the nested Majorana models. For this exercise, we update the fit to the scalar model from Ref. [78] to incorporate the likelihood function and nuisances that we use in the current paper.

We find that the scalar Higgs portal model has the largest evidence value in our scans, but is comparable to the fermion DM models. In Table 9, we give the odds ratios against each of the Higgs portal models, relative to the scalar model. The data have no preference between scalar and either form of fermionic Higgs portal model, with odds ratios of 1:1. The vector DM model is disfavoured with a ratio of 6:1 compared to the scalar and fermion models; this constitutes ‘positive’ evidence against the vector DM model according to the Jeffreys scale, though the preference is only rather mild. Overall, there is no strong preference for Higgs portal DM to transform as a scalar, vector or fermion under the Lorentz group.

Odds ratios against each singlet Higgs portal DM model with \(\mathbb {Z}_2\) symmetry, relative to the scalar model

Model | Parameters and priors | Odds |
---|---|---|

| \(m_S\): log \(\lambda _{hS}\): log | 1:1 |

\(V_{\mu }\) | \(m_V\): log \(\lambda _{hV}\): log | 6:1 |

\(\chi \) | \(m_\chi \): log \(\lambda _{h\chi }/\varLambda _\chi \): log \(\xi \): flat | 1:1 |

\(\psi \) | \(m_\psi \): log \(\lambda _{h\psi }/\varLambda _\psi \): log \(\xi \): flat | 1:1 |

## 7 Conclusions

In this study we have considered and compared simple extensions of the SM with fermionic and vector DM particles stabilised by a \(\mathbb {Z}_2\) symmetry. These models are non-renormalisable, and the effective Higgs-portal coupling is the lowest-dimension operator connecting DM to SM particles. Scenarios of this type are constrained by the DM relic density predicted by the thermal freeze-out mechanism, invisible Higgs decays, and direct and indirect DM searches. Perturbative unitarity and validity of the corresponding EFT must also be considered.

We find that the vector, Majorana and Dirac models are all phenomenologically acceptable, regardless of whether or not the DM candidate saturates the observed DM abundance. In particular, the resonance region (where the DM particle mass is approximately half the SM Higgs mass) is consistent with all experimental constraints and challenging to probe even with projected future experiments. On the other hand, larger DM masses are typically tightly constrained by a combination of direct detection constraints, the relic density requirement and theoretical considerations such as perturbative unitarity. Our results show that with the next generation of direct detection experiments (e.g., LZ [138]), it will be possible to fully probe the high-mass region for both the vector and CP-conserving fermion DM model. Future indirect experiments such as CTA [139] will be sensitive to parts of viable parameter space at large DM masses, but will have difficulty in probing the resonance region.

An interesting alternative is fermionic DM with a CP-violating Higgs portal coupling, for which the scattering rates in direct detection experiments are momentum-suppressed. By performing a Bayesian model comparison, we find that data strongly prefers the model with CP violation over the CP-conserving one, with odds of order 100:1 (over several priors). This illustrates how increasingly tight experimental constraints on weakly-interacting DM models are forcing us to abandon the simplest and most theoretically appealing models, in favour of more complex models.

We have also used Bayesian model comparison to determine the viability of the scalar Higgs portal model relative to the fermionic and vector DM models. We find a mild preference for scalar DM over vector DM, but no particular preference between the scalar and the fermionic model. This conclusion may however quickly change with more data. Stronger constraints on the Higgs invisible width will further constrain the resonance region and the combination of these constraints with future direct detection experiments may soon rule out the vector model.

Our study clearly demonstrates that, in the absence of positive signals, models of weakly-interacting DM particles will only remain viable if direct detection constraints can be systematically suppressed. This makes it increasingly interesting to study DM models with momentum-dependent scattering cross-sections. A systematic study of such theories will be left for future work. Conversely, Higgs portal models provide a natural framework for interpreting signals in the next generation of direct and indirect detection experiments. An advanced framework for such a reinterpretation using Fisher information will be implemented in future versions of GAMBIT.

## Footnotes

- 1.
Note that for the Majorana case, the 4-component spinor can be written in terms of one two-component Weyl spinor. This transformation simply corresponds to a phase transformation of this two-component spinor.

- 2.
This is not the case for the maximally CP-violating choice \((\cos \theta = 0)\) as EWSB induces a scalar interaction term with \(\cos \xi \propto v_0^2\) [82].

- 3.
The quartic self-coupling \(\lambda _V\) does not play any role in the DM phenomenology that we consider, and can be ignored. However, it is vital if constraints from electroweak vacuum stability and model perturbativity are imposed [83]. For a global fit including vacuum stability of scalar DM, see e.g., Ref. [79].

- 4.
We assume DM to be in a local thermal equilibrium (LTE) during freeze-out. As pointed out in Ref. [96], this assumption can break down very close to the resonance, thereby requiring a full numerical solution of the Boltzmann equation in phase space. As this part of the parameter space is in any case very difficult to test experimentally (see Sect. 5), we stick to the standard approximation of LTE.

- 5.
We do not include constraints from cosmic-ray antiprotons; although they are potentially competitive with or even stronger than those from gamma-ray observations of dSphs, there is still no consensus on the systematic uncertainty of the upper bound on a DM-induced component in the antiproton spectrum [68, 103, 104, 105].

- 6.
- 7.
- 8.
Reference [124] argues that the peculiar velocity of the Sun is somewhat larger than the canonical value \(\mathbf{{v}}_{\odot ,\text {pec}} = (11, 12, 7)\) km s\(^{-1}\) [125], leading to \(v_{\mathrm{rot}} = 218\pm 6\) km s\(^{-1}\). In the present study we do not consider uncertainties in \(\mathbf{{v}}_{\odot ,\text {pec}}\) and therefore adopt the measurement of \(v_{\mathrm{rot}}\) from Ref. [123].

- 9.
Note that the \(\gamma _5\) term can be generated by a complex mass term \(\widetilde{m}_X\) in the original fermion Lagrangian and performing a chiral rotation. Thus, full CP conservation \((\cos \theta = 1)\) is equivalent to having a real mass term.

- 10.
- 11.
By ‘volume of support’, we mean the regions of the parameter space that have a non-negligible likelihood times prior density.

- 12.
Odds ratios are best conceived of in a logarithmic sense, so a factor of 2 difference is of negligible importance.

- 13.
Note that DDCalc 2.0.0 no longer maintains a command line interface, so that the example files are in fact the only executables that are generated when compiling DDCalc.

- 14.
The normalisation of the non-relativistic operators corresponds to a DM particle that is not self-conjugate. Hence, for a self-conjugate particle all operator coefficients have to be multiplied by a factor of two.

- 15.
The expected number of background events is quoted as \(1.8\pm 0.5\). Assuming the uncertainty in this estimate to be Gaussian, the likelihood is maximized for a background expectation of \(1.8 - 0.5^2 = 1.55\) events.

## Notes

### Acknowledgements

We thank Shyam Balaji, Archil Kobakhidze and Ross Young for helpful discussions, and Lucien Boland, Sean Crosby and Goncalo Borges from CoEPP Research Computing for providing computing assistance and allocating resources. We acknowledge PRACE for awarding us access to Marconi at CINECA, Italy. We are grateful to the UK Materials and Molecular Modelling Hub for computational resources, which is partially funded by EPSRC (EP/P020194/1). This work was supported by STFC (1810964, ST/K00414X/1, ST/N000838/1, ST/P000762/1), the Swedish Research Council (contract 621-2014-5772), the Norwegian Research Council (FRIPRO project 230546/F20), NOTUR (Norway; NN9284K), DFG (Emmy Noether Grant No. KA 4662/1-1), NSERC, the Research Excellence Fund (CFREF), H2020 (ERC Starting Grant ‘NewAve’ 638528, Marie Skłodowska-Curie Individual Fellowship ‘DarkGAMBIT’ 752162), ARC (CE110001104 CoEPP, Future Fellowships FT140100244, FT160100274), the Australian Postgraduate Award and the Centre for the Subatomic Structure of Matter (CSSM).

## References

- 1.F. Zwicky, Die Rotverschiebung von extragalaktischen Nebeln. Helvetica Phys. Acta
**6**, 110–127 (1933)ADSzbMATHGoogle Scholar - 2.V.C. Rubin, W.K. Ford Jr., Rotation of the Andromeda Nebula from a spectroscopic survey of emission regions. ApJ
**159**, 379 (1970)ADSCrossRefGoogle Scholar - 3.D. Clowe, M. Bradač, A direct empirical proof of the existence of Dark Matter. ApJ
**648**, L109–L113 (2006). arXiv:astro-ph/0608407 ADSCrossRefGoogle Scholar - 4.Planck Collaboration: P.A.R. Ade et al.: Planck 2015 results. XIII. Cosmological parameters. Astropart. Phys.
**594**, A13 (2016). arXiv:1502.01589 - 5.G. Bertone, D. Hooper, J. Silk, Particle Dark Matter: evidence, candidates and constraints. Phys. Rep.
**405**, 279–390 (2005). arXiv:hep-ph/0404175 ADSCrossRefGoogle Scholar - 6.G. Jungman, M. Kamionkowski, K. Griest, Supersymmetric Dark Matter. Phys. Rep.
**267**, 195–373 (1996). arXiv:hep-ph/9506380 ADSCrossRefGoogle Scholar - 7.V. Silveira, A. Zee, Scalar phantoms. Phys. Lett. B
**161**, 136–140 (1985)ADSCrossRefGoogle Scholar - 8.J. McDonald, Gauge singlet scalars as cold Dark Matter. Phys. Rev. D
**50**, 3637–3649 (1994). arXiv:hep-ph/0702143 ADSCrossRefGoogle Scholar - 9.C.P. Burgess, M. Pospelov, T. ter Veldhuis, The minimal model of nonbaryonic dark matter: a singlet scalar. Nucl. Phys. B
**619**, 709–728 (2001). arXiv:hep-ph/0011335 ADSCrossRefGoogle Scholar - 10.H. Davoudiasl, R. Kitano, T. Li, H. Murayama, The new minimal standard model. Phys. Lett. B
**609**, 117–123 (2005). arXiv:hep-ph/0405097 ADSCrossRefGoogle Scholar - 11.B. Patt, F. Wilczek, Higgs-field portal into hidden sectors. arXiv:hep-ph/0605188
- 12.Y.G. Kim, K.Y. Lee, S. Shin, Singlet fermionic Dark Matter. JHEP
**05**, 100 (2008). arXiv:0803.2932 ADSGoogle Scholar - 13.S. Andreas, C. Arina, T. Hambye, F.-S. Ling, M.H.G. Tytgat, A light scalar WIMP through the Higgs portal and CoGeNT. Phys. Rev. D
**82**, 043522 (2010). arXiv:1003.2595 ADSCrossRefGoogle Scholar - 14.M. Aoki, S. Kanemura, O. Seto, Higgs decay in Higgs portal Dark Matter models. J. Phys. Conf. Ser.
**315**, 012024 (2011)CrossRefGoogle Scholar - 15.S. Kanemura, S. Matsumoto, T. Nabeshima, H. Taniguchi, Testing Higgs portal Dark Matter via \(Z\) fusion at a linear collider. Phys. Lett. B
**701**, 591–596 (2011). arXiv:1102.5147 ADSCrossRefGoogle Scholar - 16.M. Raidal, A. Strumia, Hints for a non-standard Higgs boson from the LHC. Phys. Rev. D
**84**, 077701 (2011). arXiv:1108.4903 ADSCrossRefGoogle Scholar - 17.Y. Mambrini, Higgs searches and singlet scalar dark matter: combined constraints from XENON 100 and the LHC. Phys. Rev. D
**84**, 115017 (2011). arXiv:1108.0671 ADSCrossRefGoogle Scholar - 18.X.-G. He, J. Tandean, Hidden Higgs Boson at the LHC and light Dark Matter searches. Phys. Rev. D
**84**, 075018 (2011). arXiv:1109.1277 ADSCrossRefGoogle Scholar - 19.A. Drozd, B. Grzadkowski, J. Wudka, Multi-scalar-singlet extension of the standard model—the case for Dark Matter and an invisible Higgs Boson. JHEP
**04**, 006 (2012). arXiv:1112.2582 [**Erratum: JHEP 11, 130 (2014)**] - 20.A. Djouadi, O. Lebedev, Y. Mambrini, J. Quevillon, Implications of LHC searches for Higgs-portal dark matter. Phys. Lett. B
**709**, 65–69 (2012). arXiv:1112.3299 ADSCrossRefGoogle Scholar - 21.T. Nabeshima, Higgs portal dark matter at a linear collider. in
*International Workshop on Future Linear Colliders (LCWS11) Granada, Spain, September 26-30, 2011*(2012). arXiv:1202.2673 - 22.H. Okada, T. Toma, Can a Higgs portal Dark Matter be compatible with the anti-proton cosmic-ray? Phys. Lett. B
**713**, 264–269 (2012). arXiv:1203.3116 ADSCrossRefGoogle Scholar - 23.L. Lopez-Honorez, T. Schwetz, J. Zupan, Higgs portal, fermionic Dark Matter, and a standard model like Higgs at 125 GeV. Phys. Lett. B
**716**, 179–185 (2012). arXiv:1203.2064 ADSCrossRefGoogle Scholar - 24.A. Djouadi, A. Falkowski, Y. Mambrini, J. Quevillon, Direct detection of Higgs-portal Dark Matter at the LHC. Eur. Phys. J. C
**73**, 2455 (2013). arXiv:1205.3169 ADSCrossRefGoogle Scholar - 25.D.G.E. Walker, Unitarity constraints on Higgs portals. arXiv:1310.1083
- 26.N. Okada, O. Seto, Gamma ray emission in Fermi bubbles and Higgs portal Dark Matter. Phys. Rev. D
**89**, 043525 (2014). arXiv:1310.5991 ADSCrossRefGoogle Scholar - 27.Z. Chacko, Y. Cui, S. Hong, Exploring a dark sector through the Higgs Portal at a Lepton collider. Phys. Lett. B
**732**, 75–80 (2014). arXiv:1311.3306 ADSCrossRefGoogle Scholar - 28.ATLAS Collaboration: G. Aad et al., Observation of a new particle in the search for the standard model Higgs boson with the ATLAS detector at the LHC. Phys. Lett. B
**716**, 1–29 (2012). arXiv:1207.7214 - 29.CMS Collaboration: S. Chatrchyan et al., Observation of a new boson at a mass of 125 GeV with the CMS experiment at the LHC. Phys. Lett. B
**716**, 30–61 (2012). arXiv:1207.7235 - 30.S. Baek, P. Ko, W.-I. Park, Invisible Higgs decay width vs. Dark Matter direct detection cross section in Higgs Portal Dark Matter models. Phys. Rev. D
**90**, 055014 (2014). arXiv:1405.3530 ADSCrossRefGoogle Scholar - 31.N. Craig, H.K. Lou, M. McCullough, A. Thalapillil, The Higgs Portal above threshold. JHEP
**02**, 127 (2016). arXiv:1412.0258 ADSCrossRefGoogle Scholar - 32.F. Bishara, J. Brod, P. Uttayarat, J. Zupan, Nonstandard Yukawa couplings and Higgs portal Dark Matter. JHEP
**01**, 010 (2016). arXiv:1504.04022 ADSCrossRefGoogle Scholar - 33.M.A. Fedderke, T. Lin, L.-T. Wang, Probing the fermionic Higgs portal at lepton colliders. JHEP
**04**, 160 (2016). arXiv:1506.05465 ADSGoogle Scholar - 34.A. Freitas, S. Westhoff, J. Zupan, Integrating in the Higgs portal to Fermion Dark Matter. JHEP
**09**, 015 (2015). arXiv:1506.04149 CrossRefGoogle Scholar - 35.M. Duch, B. Grzadkowski, M. McGarrie, A stable Higgs portal with vector dark matter. JHEP
**09**, 162 (2015). arXiv:1506.08805 ADSCrossRefGoogle Scholar - 36.C.-H. Chen, T. Nomura, Searching for vector Dark Matter via Higgs portal at the LHC. Phys. Rev. D
**93**, 074019 (2016). arXiv:1507.00886 ADSCrossRefGoogle Scholar - 37.A. DiFranzo, P.J. Fox, T.M.P. Tait, Vector Dark Matter through a radiative Higgs portal. JHEP
**04**, 135 (2016). arXiv:1512.06853 ADSGoogle Scholar - 38.A. Aravind, M. Xiao, J.-H. Yu, Higgs portal to inflation and Fermionic Dark Matter. Phys. Rev. D
**93**, 123513 (2016). arXiv:1512.09126 [**Erratum: Phys. Rev. D 96(6), 069901 (2017)**] - 39.P. Ko, H. Yokoya, Search for Higgs portal DM at the ILC. JHEP
**08**, 109 (2016). arXiv:1603.04737 ADSCrossRefGoogle Scholar - 40.A. Cuoco, B. Eiteneuer, J. Heisig, M. Krämer, A global fit of the \(\gamma \)-ray galactic center excess within the scalar singlet Higgs portal model. JCAP
**6**, 050 (2016). arXiv:1603.08228 ADSCrossRefGoogle Scholar - 41.G. Dupuis, Collider constraints and prospects of a scalar singlet extension to Higgs portal Dark Matter. JHEP
**07**, 008 (2016). arXiv:1604.04552 ADSCrossRefGoogle Scholar - 42.A. Das, N. Okada, O. Seto, Galactic center excess by Higgs portal dark matter. J. Phys. Conf. Ser.
**718**, 042054 (2016)CrossRefGoogle Scholar - 43.P. Di Bari, P.O. Ludl, S. Palomares-Ruiz, Unifying leptogenesis, Dark Matter and high-energy neutrinos with right-handed neutrino mixing via Higgs portal. JCAP
**1611**, 044 (2016). arXiv:1606.06238 CrossRefGoogle Scholar - 44.G. Arcadi, C. Gross, O. Lebedev, S. Pokorski, T. Toma, Evading direct Dark Matter detection in Higgs portal models. Phys. Lett. B
**769**, 129–133 (2017). arXiv:1611.09675 ADSCrossRefGoogle Scholar - 45.S. Banerjee, N. Chakrabarty, A revisit to scalar Dark Matter with radiative corrections. arXiv:1612.01973
- 46.J.A. Casas, D.G. Cerdeño, J.M. Moreno, J. Quilis, Reopening the Higgs portal for single scalar dark matter. JHEP
**05**, 036 (2017). arXiv:1701.08134 ADSzbMATHCrossRefGoogle Scholar - 47.M. Heikinheimo, T. Tenkanen, K. Tuominen, WIMP miracle of the second kind. Phys. Rev. D
**96**, 023001 (2017). arXiv:1704.05359 ADSCrossRefGoogle Scholar - 48.C.-F. Chang, X.-G. He, J. Tandean, Exploring spin-3/2 Dark Matter with effective Higgs couplings. Phys. Rev. D
**96**, 075026 (2017). arXiv:1704.01904 ADSCrossRefGoogle Scholar - 49.E.W. Kolb, A.J. Long, Superheavy Dark Matter through Higgs portal operators. Phys. Rev. D
**96**, 103540 (2017). arXiv:1708.04293 ADSCrossRefGoogle Scholar - 50.S. Baum, M. Carena, N.R. Shah, C.E.M. Wagner, Higgs portals for thermal Dark Matter. EFT perspectives and the NMSSM. JHEP
**04**, 069 (2018). arXiv:1712.09873 ADSCrossRefGoogle Scholar - 51.A. Beniwal, F. Rajec, Combined analysis of effective Higgs portal dark matter models. Phys. Rev. D
**93**, 115016 (2016). arXiv:1512.06458 ADSCrossRefGoogle Scholar - 52.S. Bhattacharya, P. Poulose, P. Ghosh, Multipartite interacting scalar Dark Matter in the light of updated LUX data. JCAP
**1704**, 043 (2017). arXiv:1607.08461 ADSCrossRefGoogle Scholar - 53.C. Balázs, T. Li, Simplified Dark Matter models confront the gamma ray excess. Phys. Rev. D
**90**, 055026 (2014). arXiv:1407.0174 ADSCrossRefGoogle Scholar - 54.H.M. Lee, C.B. Park, M. Park, Supersymmetric Higgs-portal and X-ray lines. Phys. Lett. B
**744**, 218–224 (2015). arXiv:1501.05479 ADSCrossRefGoogle Scholar - 55.C. Balázs, T. Li, C. Savage, M. White, Interpreting the Fermi-LAT gamma ray excess in the simplified framework. Phys. Rev. D
**92**, 123520 (2015). arXiv:1505.06758 ADSCrossRefGoogle Scholar - 56.Z. Chacko, Y. Cui, S. Hong, T. Okui, Hidden Dark Matter sector, dark radiation, and the CMB. Phys. Rev. D
**92**, 055033 (2015). arXiv:1505.04192 ADSCrossRefGoogle Scholar - 57.S. Baek, P. Ko, M. Park, W.-I. Park, C. Yu, Beyond the Dark matter effective field theory and a simplified model approach at colliders. Phys. Lett. B
**756**, 289–294 (2016). arXiv:1506.06556 ADSzbMATHCrossRefGoogle Scholar - 58.T. Mondal, T. Basak, Galactic center gamma-ray excess and Higgs-portal Dark Matter. Springer Proc. Phys.
**174**, 493–497 (2016). arXiv:1507.01793 CrossRefGoogle Scholar - 59.K. Cheung, P. Ko, J.S. Lee, P.-Y. Tseng, Bounds on Higgs-Portal models from the LHC Higgs data. JHEP
**10**, 057 (2015). arXiv:1507.06158 ADSCrossRefGoogle Scholar - 60.M. Duerr, P. Fileviez Pérez, J. Smirnov, Scalar Dark Matter: direct vs. indirect detection. JHEP
**06**, 152 (2016). arXiv:1509.04282 - 61.H. Han, S. Zheng, Higgs-portal scalar Dark Matter: scattering cross section and observable limits. Nucl. Phys. B
**914**, 248–256 (2017). arXiv:1510.06165 - 62.G. Krnjaic, Probing light thermal Dark-Matter with a Higgs portal mediator. Phys. Rev. D
**94**, 073009 (2016). arXiv:1512.04119 ADSCrossRefGoogle Scholar - 63.H. Han, J.M. Yang, Y. Zhang, S. Zheng, Collider signatures of Higgs-portal scalar Dark Matter. Phys. Lett. B
**756**, 109–112 (2016). arXiv:1601.06232 - 64.F.S. Sage, R. Dick, Gamma ray signals of the annihilation of Higgs-portal singlet dark matter. arXiv:1604.04589
- 65.K. Assamagan et al., The Higgs portal and cosmology (2016). arXiv:1604.05324
- 66.X.-G. He, J. Tandean, New LUX and PandaX-II results illuminating the simplest Higgs-portal Dark Matter models. JHEP
**12**, 074 (2016). arXiv:1609.03551 ADSCrossRefGoogle Scholar - 67.M. Escudero, A. Berlin, D. Hooper, M.-X. Lin, Toward (finally!) ruling out Z and Higgs mediated Dark Matter models. JCAP
**1612**, 029 (2016). arXiv:1609.09079 ADSCrossRefGoogle Scholar - 68.A. Cuoco, J. Heisig, M. Korsmeier, M. Krämer, Probing dark matter annihilation in the Galaxy with antiprotons and gamma rays. JCAP
**1710**, 053 (2017). arXiv:1704.08258 ADSCrossRefGoogle Scholar - 69.J. Liu, X.-P. Wang, F. Yu, A tale of two portals: testing light, hidden new physics at future \(e^+ e^-\) colliders. JHEP
**06**, 077 (2017). arXiv:1704.00730 ADSCrossRefGoogle Scholar - 70.C. Cai, Z.-H. Yu, H.-H. Zhang, CEPC precision of electroweak oblique parameters and weakly interacting Dark Matter: the scalar case. Nucl. Phys. B
**924**, 128–152 (2017). arXiv:1705.07921 ADSzbMATHCrossRefGoogle Scholar - 71.T. Kamon, P. Ko, J. Li, Characterizing Higgs portal dark matter models at the ILC. Eur. Phys. J. C
**77**, 652 (2017). arXiv:1705.02149 ADSCrossRefGoogle Scholar - 72.H. Tu, K.-W. Ng, Supernovae and Weinberg’s Higgs portal dark radiation and Dark Matter. JHEP
**07**, 108 (2017). arXiv:1706.08340 ADSzbMATHCrossRefGoogle Scholar - 73.A. Fradette, M. Pospelov, BBN for the LHC: constraints on lifetimes of the Higgs portal scalars. Phys. Rev. D
**96**, 075033 (2017). arXiv:1706.01920 ADSCrossRefGoogle Scholar - 74.M. Hoferichter, P. Klos, J. Menéndez, A. Schwenk, Improved limits for Higgs-portal dark matter from LHC searches. Phys. Rev. Lett.
**119**, 181803 (2017). arXiv:1708.02245 ADSCrossRefGoogle Scholar - 75.J. Ellis, A. Fowlie, L. Marzola, M. Raidal, Statistical analyses of Higgs- and Z-portal Dark Matter models. Phys. Rev. D
**97**, 115014 (2018). arXiv:1711.09912 ADSCrossRefGoogle Scholar - 76.B. Dutta, T. Kamon, P. Ko, J. Li, Prospects for discovery and spin discrimination of dark matter in Higgs portal DM models and their extensions at 100 TeV \(pp\) collider. Eur. Phys. J. C
**78**, 595 (2018). arXiv:1712.05123 ADSCrossRefGoogle Scholar - 77.K. Cheung, Y.-L.S. Tsai, P.-Y. Tseng, T.-C. Yuan, A. Zee, Global study of the simplest scalar phantom Dark Matter model. JCAP
**1210**, 042 (2012). arXiv:1207.4930 ADSCrossRefGoogle Scholar - 78.GAMBIT Collaboration: P. Athron, C. Balázs et al., Status of the scalar singlet dark matter model. Eur. Phys. J. C
**77**, 568 (2017). arXiv:1705.07931 - 79.P. Athron, J.M. Cornell, Impact of vacuum stability, perturbativity and XENON1T on global fits of \(\mathbb{Z}_2\) and \(\mathbb{Z}_3\) scalar singlet dark matter. Eur. Phys. J. C
**78**, 830 (2018). arXiv:1806.11281 ADSCrossRefGoogle Scholar - 80.GAMBIT Collaboration: P. Athron, C. Balázs et al., GAMBIT: the global and modular beyond-the-standard-model inference tool. Eur. Phys. J. C
**77**, 784 (2017). arXiv:1705.07908 - 81.The GAMBIT Collaboration, Supplementary data: global analyses of Higgs portal singlet dark matter models using GAMBIT (2018). https://doi.org/10.5281/zenodo.1400653
- 82.M.A. Fedderke, J.-Y. Chen, E.W. Kolb, L.-T. Wang, The fermionic Dark Matter Higgs portal: an effective field theory approach. JHEP
**08**, 122 (2014). arXiv:1404.2283 ADSCrossRefGoogle Scholar - 83.F. Kahlhoefer, J. McDonald, WIMP Dark Matter and unitarity-conserving inflation via a gauge singlet scalar. JCAP
**1511**, 015 (2015). arXiv:1507.03600 ADSCrossRefGoogle Scholar - 84.GAMBIT Models Workgroup: P. Athron, C. Balázs et al., SpecBit, DecayBit and PrecisionBit: GAMBIT modules for computing mass spectra, particle decay rates and precision observables. Eur. Phys. J. C
**78**, 22 (2018). arXiv:1705.07936 - 85.Fermi-LAT Collaboration: M. Ackermann et al., Searching for Dark Matter annihilation from Milky Way Dwarf spheroidal galaxies with six years of fermi large area telescope data. Phys. Rev. Lett.
**115**, 231301 (2015). arXiv:1503.02641 - 86.LUX Collaboration: D.S. Akerib et al., Results from a search for dark matter in the complete LUX exposure. Phys. Rev. Lett.
**118**, 021303 (2017). arXiv:1608.07648 - 87.PandaX-II Collaboration: A. Tan et al., Dark Matter results from first 98.7 days of data from the PandaX-II experiment. Phys. Rev. Lett.
**117**, 121303 (2016). arXiv:1607.07400 - 88.PandaX-II Collaboration: X. Cui et al., Dark Matter results from 54-ton-day exposure of PandaX-II experiment. Phys. Rev. Lett.
**119**, 181302 (2017). arXiv:1708.06917 - 89.XENON Collaboration: E. Aprile et al., Dark Matter search results from a one tonne \(\times \) year exposure of XENON1T. arXiv:1805.12562
- 90.SuperCDMS Collaboration: R. Agnese et al., New results from the search for low-mass weakly interacting massive particles with the CDMS low ionization threshold experiment. Phys. Rev. Lett.
**116**, 071301 (2016). arXiv:1509.02448 - 91.CRESST Collaboration: G. Angloher et al., Results on light dark matter particles with a low-threshold CRESST-II detector. Eur. Phys. J. C
**76**, 25 (2016). arXiv:1509.01515 - 92.PICO Collaboration: C. Amole et al., Dark Matter search results from the PICO-60 C\(_3\)F\(_8\) bubble chamber. Phys. Rev. Lett.
**118**, 251301 (2017). arXiv:1702.07666 - 93.DarkSide Collaboration: P. Agnes et al., DarkSide-50 532-day Dark Matter search with low-radioactivity Argon. arXiv:1802.07198
- 94.IceCube Collaboration: M.G. Aartsen et al., Search for Dark Matter annihilations in the Sun with the 79-string IceCube detector. Phys. Rev. Lett.
**110**, 131302 (2013). arXiv:1212.4097 - 95.P. Gondolo, G. Gelmini, Cosmic abundances of stable particles: improved analysis. Nucl. Phys. A
**360**, 145–179 (1991)CrossRefGoogle Scholar - 96.T. Binder, T. Bringmann, M. Gustafsson, A. Hryczuk, Early kinetic decoupling of Dark Matter: when the standard way of calculating the thermal relic density fails. Phys. Rev. D
**96**, 115010 (2017). arXiv:1706.07433 ADSCrossRefGoogle Scholar - 97.J.M. Cline, K. Kainulainen, Electroweak baryogenesis and Dark Matter from a singlet Higgs. JCAP
**1301**, 012 (2013). arXiv:1210.4196 ADSCrossRefGoogle Scholar - 98.LHC Higgs Cross Section Working Group: S. Dittmaier et al., Handbook of LHC Higgs cross sections: 1. Inclusive observables. arXiv:1101.0593
- 99.T. Bringmann, J. Edsjö, P. Gondolo, P. Ullio, L. Bergström, DarkSUSY 6: an advanced tool to compute Dark Matter properties numerically. JCAP
**1807**, 033 (2018). arXiv:1802.03399 ADSCrossRefGoogle Scholar - 100.P. Gondolo, J. Edsjo, DarkSUSY: computing supersymmetric dark matter properties numerically. JCAP
**0407**, 008 (2004). arXiv:astro-ph/0406204 ADSCrossRefGoogle Scholar - 101.GAMBIT Dark Matter Workgroup: T. Bringmann, J. Conrad et al., DarkBit: a GAMBIT module for computing Dark Matter observables and likelihoods. Eur. Phys. J. C
**77**, 831 (2017). arXiv:1705.07920 - 102.G. Belanger, B. Dumont, U. Ellwanger, J.F. Gunion, S. Kraml, Global fit to Higgs signal strengths and couplings and implications for extended Higgs sectors. Phys. Rev. D
**88**, 075008 (2013). arXiv:1306.2941 ADSCrossRefGoogle Scholar - 103.A. Urbano, W. Xue, Constraining the Higgs portal with antiprotons. JHEP
**03**, 133 (2015). arXiv:1412.3798 CrossRefGoogle Scholar - 104.A. Cuoco, J. Heisig, M. Korsmeier, M. Kramer, Constraining heavy Dark Matter with cosmic-ray antiprotons. JCAP
**1804**, 004 (2018). arXiv:1711.05274 ADSCrossRefGoogle Scholar - 105.A. Reinert, M.W. Winkler, A precision search for WIMPs with charged cosmic rays. JCAP
**1801**, 055 (2018). arXiv:1712.00002 ADSCrossRefGoogle Scholar - 106.M.W. Goodman, E. Witten, Detectability of certain Dark Matter candidates. Phys. Rev. D
**31**, 3059 (1985)ADSCrossRefGoogle Scholar - 107.K.R. Dienes, J. Kumar, B. Thomas, D. Yaylali, Overcoming velocity suppression in Dark-Matter direct-detection experiments. Phys. Rev. D
**90**, 015012 (2014). arXiv:1312.7772 ADSCrossRefGoogle Scholar - 108.J. Lewin, P. Smith, Review of mathematics, numerical factors, and corrections for dark matter experiments based on elastic nuclear recoil. Astropart. Phys.
**6**, 87–112 (1996)ADSCrossRefGoogle Scholar - 109.G. Arcadi, M. Lindner, F.S. Queiroz, W. Rodejohann, S. Vogl, Pseudoscalar mediators: a WIMP model at the neutrino floor. JCAP
**1803**, 042 (2018). arXiv:1711.02110 ADSCrossRefGoogle Scholar - 110.N.F. Bell, G. Busoni, I.W. Sanderson, Loop effects in direct detection. arXiv:1803.01574
- 111.T. Abe, M. Fujiwara, J. Hisano, Loop corrections to dark matter direct detection in a pseudoscalar mediator dark matter model. arXiv:1810.01039
- 112.W.H. Press, D.N. Spergel, Capture by the sun of a galactic population of weakly interacting massive particles. ApJ
**296**, 679–684 (1985)ADSCrossRefGoogle Scholar - 113.J. Silk, K.A. Olive, M. Srednicki, The photino, the sun and high-energy neutrinos. Phys. Rev. Lett.
**55**, 257–259 (1985)ADSCrossRefGoogle Scholar - 114.A. Gould, Resonant enhancements in WIMP capture by the earth. ApJ
**321**, 571 (1987)ADSCrossRefGoogle Scholar - 115.N. Vinyoles, A.M. Serenelli, A new generation of standard solar models. ApJ
**835**, 202 (2017). arXiv:1611.09867 ADSCrossRefGoogle Scholar - 116.A.C. Vincent, A. Serenelli, P. Scott, Generalised form factor Dark Matter in the sun. JCAP
**1508**, 040 (2015). arXiv:1504.04378 ADSCrossRefGoogle Scholar - 117.A.C. Vincent, P. Scott, A. Serenelli, Updated constraints on velocity and momentum-dependent asymmetric Dark Matter. JCAP
**1611**, 007 (2016). arXiv:1605.06502 ADSCrossRefGoogle Scholar - 118.A. Gould, WIMP distribution in and evaporation from the sun. ApJ
**321**, 560 (1987)ADSCrossRefGoogle Scholar - 119.G. Busoni, A. De Simone, W.-C. Huang, On the minimum dark matter mass testable by neutrinos from the sun. JCAP
**1307**, 010 (2013). arXiv:1305.1817 ADSCrossRefGoogle Scholar - 120.G. Busoni, A. De Simone, P. Scott, A.C. Vincent, Evaporation and scattering of momentum- and velocity-dependent dark matter in the Sun. JCAP
**1710**, 037 (2017). arXiv:1703.07784 ADSCrossRefGoogle Scholar - 121.M. Blennow, J. Edsjo, T. Ohlsson, Neutrinos from WIMP annihilations using a full three-flavor Monte Carlo. JCAP
**0801**, 021 (2008). arXiv:0709.3898 ADSCrossRefGoogle Scholar - 122.IceCube Collaboration: M.G. Aartsen et al., Improved limits on dark matter annihilation in the Sun with the 79-string IceCube detector and implications for supersymmetry. JCAP
**1604**, 022 (2016). arXiv:1601.00653 - 123.M.J. Reid, Trigonometric parallaxes of high mass star forming regions: the structure and kinematics of the Milky Way. ApJ
**783**, 130 (2014). arXiv:1401.5377 ADSCrossRefGoogle Scholar - 124.J. Bovy, The Milky Way’s circular velocity curve between 4 and 14 kpc from APOGEE data. ApJ
**759**, 131 (2012). arXiv:1209.0759 ADSCrossRefGoogle Scholar - 125.R. Schoenrich, J. Binney, W. Dehnen, Local kinematics and the local standard of rest. MNRAS
**403**, 1829 (2010). arXiv:0912.3693 ADSCrossRefGoogle Scholar - 126.T. Piffl, The RAVE survey: the Galactic escape speed and the mass of the Milky Way. Astron. Astrophys.
**562**, A91 (2014). arXiv:1309.4293 CrossRefGoogle Scholar - 127.J.M. Cline, K. Kainulainen, P. Scott, C. Weniger, Update on scalar singlet dark matter. Phys. Rev. D
**88**, 055025 (2013). arXiv:1306.4710 ADSCrossRefGoogle Scholar - 128.H.-W. Lin, Lattice QCD for precision nucleon matrix elements. arXiv:1112.2435
- 129.F. Bishara, J. Brod, B. Grinstein, J. Zupan, From quarks to nucleons in Dark Matter direct detection. JHEP
**11**, 059 (2017). arXiv:1707.06998 ADSzbMATHCrossRefGoogle Scholar - 130.Particle Data Group: C. Patrignani et al., Review of particle physics. Chin. Phys. C
**40**, 100001 (2016)Google Scholar - 131.O. Lebedev, H.M. Lee, Y. Mambrini, Vector Higgs-portal Dark Matter and the invisible Higgs. Phys. Lett. B
**707**, 570–576 (2012). arXiv:1111.4482 ADSCrossRefGoogle Scholar - 132.N.F. Bell, G. Busoni, A. Kobakhidze, D.M. Long, M.A. Schmidt, Unitarisation of EFT amplitudes for dark matter searches at the LHC. JHEP
**8**, 125 (2016). arXiv:1606.02722 ADSCrossRefGoogle Scholar - 133.F. Feroz, M. Hobson, Multimodal nested sampling: an efficient and robust alternative to MCMC methods for astronomical data analysis. MNRAS
**384**, 449 (2008). arXiv:0704.3704 ADSCrossRefGoogle Scholar - 134.F. Feroz, M. Hobson, M. Bridges, MultiNest: an efficient and robust Bayesian inference tool for cosmology and particle physics. MNRAS
**398**, 1601–1614 (2009). arXiv:0809.3437 ADSCrossRefGoogle Scholar - 135.F. Feroz, M.P. Hobson, E. Cameron, A.N. Pettitt, Importance nested sampling and the MultiNest algorithm. arXiv:1306.2144
- 136.GAMBIT Scanner Workgroup: G.D. Martinez, J. McKay et al., Comparison of statistical sampling methods with ScannerBit, the GAMBIT scanning module. Eur. Phys. J. C
**77**, 761 (2017). arXiv:1705.07959 - 137.P. Scott, Pippi—painless parsing, post-processing and plotting of posterior and likelihood samples. Eur. Phys. J. Plus
**127**, 138 (2012). arXiv:1206.2245 CrossRefGoogle Scholar - 138.LZ Collaboration: D.S. Akerib, C.W. Akerlof et al., Projected WIMP sensitivity of the LUX-ZEPLIN (LZ) Dark Matter experiment. arXiv:1802.06039
- 139.Cherenkov Telescope Array Consortium: B.S. Acharya et al., Science with the Cherenkov Telescope Array. arXiv:1709.07997
- 140.H. Silverwood, C. Weniger, P. Scott, G. Bertone, A realistic assessment of the CTA sensitivity to dark matter annihilation. JCAP
**1503**, 055 (2015). arXiv:1408.4131 ADSCrossRefGoogle Scholar - 141.M. Pierre, J.M. Siegal-Gaskins, P. Scott, Sensitivity of CTA to Dark Matter signals from the galactic center. JCAP
**6**, 24 (2014). arXiv:1401.7330 ADSCrossRefGoogle Scholar - 142.S. Baker, R.D. Cousins, Clarification of the use of Chi square and likelihood functions in fits to histograms. Nucl. Instrum. Meth.
**221**, 437–442 (1984)CrossRefGoogle Scholar - 143.H. Jeffreys,
*The Theory of Probability. Oxford Classic Texts in the Physical Sciences*(Oxford University Press, Oxford, 1939)Google Scholar - 144.R.E. Kass, A.E. Raftery, Bayes factors. J. Am. Stat. Assoc.
**90**, 773–795 (1995)MathSciNetzbMATHCrossRefGoogle Scholar - 145.J.O. Berger, L.R. Pericchi, Objective Bayesian methods for model selection: introduction and comparison. Lect. Notes Monogr. Ser.
**38**, 135–207 (2001)MathSciNetCrossRefGoogle Scholar - 146.A.L. Fitzpatrick, W. Haxton, E. Katz, N. Lubbers, Y. Xu, The effective field theory of Dark Matter direct detection. JCAP
**1302**, 004 (2013). arXiv:1203.3542 ADSCrossRefGoogle Scholar - 147.N. Anand, A.L. Fitzpatrick, W.C. Haxton, Weakly interacting massive particle-nucleus elastic scattering response. Phys. Rev. C
**89**, 065501 (2014). arXiv:1308.6288 ADSCrossRefGoogle Scholar - 148.J.B. Dent, L.M. Krauss, J.L. Newstead, S. Sabharwal, General analysis of direct dark matter detection: From microphysics to observational signatures. Phys. Rev. D
**92**, 063515 (2015). arXiv:1505.03117 ADSCrossRefGoogle Scholar - 149.P. Klos, J. Menéndez, D. Gazit, A. Schwenk, Large-scale nuclear structure calculations for spin-dependent WIMP scattering with chiral effective field theory currents. Phys. Rev. D
**88**, 083516 (2013). arXiv:1304.7684 ADSCrossRefGoogle Scholar - 150.CRESST Collaboration: G. Angloher et al., Description of CRESST-II data. arXiv:1701.08157
- 151.F. Kahlhoefer, S. Kulkarni, S. Wild, Exploring light mediators with low-threshold direct detection experiments. JCAP
**1711**, 016 (2017). arXiv:1707.08571 ADSCrossRefGoogle Scholar - 152.SuperCDMS Collaboration: R. Agnese et al., Low-mass dark matter search with CDMSlite. Phys. Rev. D
**97**, 022002 (2018). arXiv:1707.01632 - 153.LZ Collaboration: D.S. Akerib et al., LUX-ZEPLIN (LZ) Conceptual design report. arXiv:1509.02910
- 154.G. Bertone, N. Bozorgnia, Identifying WIMP dark matter from particle and astroparticle data. JCAP
**1803**, 026 (2018). arXiv:1712.04793 ADSCrossRefGoogle Scholar - 155.S. Fallows, Toward a next-generation dark matter search with the PICO-40L bubble chamber. In
*15th International Conference on Topics in Astroparticle and Underground Physics, TAUP2017*(2017). https://indico.cern.ch/event/606690/contributions/2623446/attachments/1497228/2330240/Fallows_2017_07_24__TAUP__PICO-40L_v1.2.pdf (**24–28 July**) - 156.PICO Collaboration: C. Amole et al., Dark Matter search results from the PICO-2L C\(_3\)F\(_8\) bubble chamber. Phys. Rev. Lett.
**114**, 231302 (2015). arXiv:1503.00008 - 157.DARWIN Collaboration: J. Aalbers et al., DARWIN: towards the ultimate dark matter detector. JCAP
**1611**, 017 (2016). arXiv:1606.07001 - 158.M. Schumann, L. Baudis, L. Bütikofer, A. Kish, M. Selvi, Dark matter sensitivity of multi-ton liquid Xenon detectors. JCAP
**1510**, 016 (2015). arXiv:1506.08309 ADSCrossRefGoogle Scholar - 159.C.E. Aalseth, DarkSide-20k: A 20 tonne two-phase LAr TPC for direct dark matter detection at LNGS. Eur. Phys. J. Plus
**133**, 131 (2018). arXiv:1707.08145 CrossRefGoogle Scholar - 160.J. Billard, L. Strigari, E. Figueroa-Feliciano, Implication of neutrino backgrounds on the reach of next generation dark matter direct detection experiments. Phys. Rev. D
**89**, 023524 (2014). arXiv:1307.5458 ADSCrossRefGoogle Scholar - 161.T.D.P. Edwards, C. Weniger, A fresh approach to forecasting in astroparticle physics and Dark Matter searches. JCAP
**1802**, 021 (2018). arXiv:1704.05458 ADSCrossRefGoogle Scholar - 162.M. Drees, K.-I. Hikasa, Note on Qcd corrections to Hadronic Higgs decay. Phys. Lett. B
**240**, 455 (1990) [**Erratum: Phys. Lett. B 262, 497 (1991)**]Google Scholar

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