# Global analysis of dark matter simplified models with leptophobic spin-one mediators using MasterCode

## Abstract

We report the results of a global analysis of dark matter simplified models (DMSMs) with leptophobic mediator particles of spin one, considering the cases of both vector and axial-vector interactions with dark matter (DM) particles and quarks. We require the DMSMs to provide all the cosmological DM density indicated by Planck and other observations, and we impose the upper limits on spin-independent and -dependent scattering from direct DM search experiments. We also impose all relevant LHC constraints from searches for monojet events and measurements of the dijet mass spectrum. We model the likelihood functions for all the constraints and combine them within the MasterCode framework, and probe the full DMSM parameter spaces by scanning over the mediator and DM masses and couplings, not fixing any of the model parameters. We find, in general, two allowed regions of the parameter spaces: one in which the mediator couplings to Standard Model (SM) and DM particles may be comparable to those in the SM and the cosmological DM density is reached via resonant annihilation, and one in which the mediator couplings to quarks are \(\lesssim \, 10^{-3}\) and DM annihilation is non-resonant. We find that the DM and mediator masses may well lie within the ranges accessible to LHC experiments. We also present predictions for spin-independent and -dependent DM scattering, and present specific results for ranges of the DM couplings that may be favoured in ultraviolet completions of the DMSMs.

## 1 Introduction

The nature of Dark Matter (DM) is one of the most pressing issues in contemporary physics. For over 80 years, astrophysical and cosmological observations have indicated its existence indirectly [1, 2, 3, 4, 5]. They have also suggested that DM is mainly cold, i.e., it has been non-relativistic since at least the epoch of cosmological recombination [6, 7]. Beyond this, we know very little, and candidates for DM range in mass from black holes heavier than the Sun, to various species of (meta)stable neutral particles, to very light particles with large occupation numbers best described by classical fields, see Ref. [8] for a review. If the DM is composed of weakly-interacting massive particles (WIMPs) that were in thermal equilibrium with Standard Model (SM) particles in the early Universe, freeze-out calculations suggest that the WIMP is likely to weigh \(\mathcal{O}\) (TeV), in which case it could be produced at accelerators, notably the LHC. Thus, the search for WIMP DM particles has been one of the principal research objectives of the LHC experiments, particularly ATLAS and CMS [9, 10, 11, 12, 13].

Several approaches have been taken to DM searches at the LHC, in both predictions for possible signals and interpretations of the search limits. Initially, many analyses were based on specific models that predict WIMPs capable of providing the cold DM, such as supersymmetry (SUSY) [14, 15] and some other models with new physics at the TeV scale [16, 17]. These searches were typically either for the production of heavier new particles followed by cascade decays to the dark matter particle, or direct production of dark matter particles in association with a single SM particle used for tagging purposes (the ’mono-X’ signature). DM models giving rise to long-lived signatures have also been studied recently [18, 19, 20].

We have used the MasterCode framework [21] previously to study several SUSY scenarios with various experimental signatures [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32], and other groups have made parallel analyses [33, 34, 35]. In particular, in Refs. [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] we provided detailed predictions for the preferred parameter spaces in various concrete SUSY models. However, there are many other scenarios for possible physics beyond the SM, and it is desirable to fashion search strategies that facilitate the interpretation of results with differing characteristics, minimizing theoretical biases.

Recently, growing attention is being paid to more general model frameworks and analyses motivated by more generic experimental signatures. One of the first such approaches was to construct an effective field theory (EFT) for the interactions between DM and SM particles [36, 37, 38, 39, 40], with a focus on the mono-X signatures. A shortcoming of this EFT approach is that, in order to obtain the appropriate cosmological DM density, the DM/SM interactions are likely to be mediated by particles in the TeV mass range, for which the approximation of a contact interaction may be inadequate. In particular, the LHC might also be capable of producing the mediator particle directly, providing an additional signature beyond the EFT framework [41, 42]. For this reason, interest has been growing in Dark Matter Simplified Models (DMSMs) [42, 43, 44, 45, 46, 47, 48, 49, 50], which postulate effective Lagrangians that include explicitly the mediator particle and its interactions with both DM and SM particles. Recommendations for the formulation of DMSMs for use by the LHC experiments have been made by the LHC Dark Matter Working Group (DMWG) [9, 10, 11, 12, 13].

In this paper we report the results of global likelihood analyses within the MasterCode framework of some of the DMSMs suggested by the DMWG and analyzed by ATLAS and CMS.

*s*-channel mediators of spin one. Avoiding the strong LHC constraints on mediators interacting with charged leptons [51, 52], we assume that their interactions are leptophobic. We assume that the DM particle is a neutral Dirac fermion \(\chi \). The Lagrangian for the spin-one mediator

*Y*interactions with the DM particle \(\chi \) is

*Y*-boson and quarks are generation-independent, and we also assume that they are the same for charge-2/3 and -1/3 quarks (\(g_q^{V,A} := g_{q_i}^{V,A} \;\forall \; i\)).

^{1}Being leptophobic, the

*Y*boson has no interactions with leptons. We consider two scenarios for the

*Y*couplings, one in which they are purely vectorial:

UV completions of the DMSMs we study could be achieved in a number of different ways. These could favour specific ranges of \(g_\mathrm{SM}\), \(g_\mathrm{DM}\) and their ratio, as we explore below. Since our spin-1 models involve massive vector bosons, they will likely involve a dark Higgs responsible for giving them masses, or else a Stückelberg mechanism [63]. However, our intention here is not to fit the parameters of a complete model, but rather to study correlations between the model parameters defined above, which are the minimal sets leading to an interplay between LHC and DM phenomenology that is of interest.

Each of these DMSMs has four free parameters, the masses of the DM particle and the mediator, \(m_\chi \) and \(m_{Y}\), and the mediator couplings to the DM and SM particles, \(g_\mathrm{DM}\) and \(g_\mathrm{SM}\). Sampling these 4-dim parameter spaces is computationally tractable. However, experimental results are often interpreted for fixed values of the couplings, and thus previous investigations have generally fixed two out of the four parameters. A more general approach is desirable for combining the direct DM constraints with those from the LHC, particularly because obtaining the preferred cosmological value of \(\Omega _\chi h^2\) requires values of \(g_\mathrm{DM}\) and \(g_\mathrm{SM}\) that depend on \(m_\chi \) and \(m_{Y}\). Consequently, without fixing any parameter of the model, we implement the predictions of the models using MicrOMEGas 5.0.2 [64] and MG5_aMC 2.4.3 [65], which we run at LO for performance reasons, using DMSIMP model files [66, 67] to build a likelihood function for each of the astrophysical and accelerator constraints, and combine them using the MasterCode framework. We use MultiNest 2.18 [68, 69, 70] to identify the regions where the global likelihood is minimized. We do not include in our analysis indirect searches for astrophysical DM particles via their annihilations into SM particles, which were found in [55] to be unimportant in simple DM models when \(m_\chi > rsim 50 \,\, \mathrm {GeV}\).

The layout of this paper is as follows. In Sect. 2 we set out the constraints we use, and in Sect. 3 we describe our analysis framework. Our results are described in Sect. 4, Possible UV completions are discussed in Sect. 5, and our conclusions and some perspectives are set out in Sect. 6.

## 2 Astrophysical and LHC constraints

### 2.1 Dark matter constraints

#### 2.1.1 Dark matter density

*s*-channel process \(\chi \chi \rightarrow Y^{(*)} \rightarrow \) Standard Model particles, the leading expression for the cross-section takes the form [72, 73]

*v*, of the annihilating DM particles: \(c_Y \sim a \frac{m_q^2}{m_\chi ^2} + b v^2\). The total width of

*Y*is given by \(\Gamma _Y = \Gamma _Y^{\chi \chi } + N_c \sum _q \Gamma _Y^{qq}\), where \(N_c = 3\) is a colour factor and

*Y*becomes particularly important in the resonant region, where \(m_Y \simeq 2 m_\chi \) and \(\Gamma _Y^{\chi \chi } \ll \Gamma _Y\) because of phase-space suppression.

*s*-wave and the narrow-width approximation is applicable for \(g_\mathrm{SM}\ll 1\), and one finds that in the limit \(g_\mathrm{SM}\rightarrow 0\)

*Y*peak. However, we recall that \(\chi \chi \) annihilation takes place in a thermal bath, with collisions taking place at a range of centre-of-mass energies \(\ge 2 m_\chi \). For this reason, annihilations at an effective centre-of-mass energy within \({\mathcal {O}}(\Gamma _Y)\) of \(m_Y\) do not dominate when \(\Gamma _Y\) is very small. We find numerically that (11) captures the \(g_\mathrm{DM}\) dependence of \(\Omega _\chi h^2\), and that it depends only weakly on \(g_\mathrm{SM}\) over an intermediate range of \(g_\mathrm{SM}\), but that \(\Omega _\chi h^2\) increases at both large and small \(g_\mathrm{SM}\) :for \(g_\mathrm{SM}\gg g_\mathrm{DM}\) andfor \(g_\mathrm{SM}\ll g_\mathrm{DM}\). The dependence on \(g_\mathrm{DM}\) in this case depends on the relative sizes of the width and mass difference between \(m_Y\) and \(2 m_\chi \) [73]. If \(\Gamma _Y/m_Y > (1 - 4m_\chi ^2/m_Y^2)\) and \(m_Y > 2 m_\chi \) then \(\Omega _\chi h^2\) is proportional to \(g_\mathrm{DM}^2\). However, if the width is very small (for example if \(g_\mathrm{SM}\) is very small and \(m_Y < 2 m_\chi \)), then the denominator in (6) is dominated by the mass difference and \(\Omega _\chi h^2\) is inversely proportional to \(g_\mathrm{DM}^2\), as seen in Eq. (9) for the axial-vector case.

*t*-channel process \(\chi \chi \rightarrow Y Y\) for \(m_\chi > m_Y\). The leading contribution to the

*t*-channel cross-section is given by [72]

We use MicrOMEGAs 5.0.2 [64] to evaluate \(\Omega _\chi h^2\) numerically in each model parameter set, including a \({\pm 10}\)% range around Eq. (5). This enables us to obtain a substantial but unbiased sample of model points.

#### 2.1.2 Spin-independent DM scattering

*s*-channel region where \(m_Y \simeq 2 m_\chi \). The \(\sigma ^\mathrm{SI}_p\) constraint provides important limits on the vector mediator mass \(m_Y\) and the product \(g_\mathrm{DM}g_\mathrm{SM}\).

We allow for an overall uncertainty of 10% in the spin-independent cross section.

#### 2.1.3 Spin-dependent DM scattering

We also evaluate the constraint imposed by the upper limit on the spin-dependent DM-proton scattering cross-section \(\sigma ^\mathrm{SD}_p\) from the PICO-60 experiment [79], and that on the spin-dependent DM-neutron scattering cross-section \(\sigma ^\mathrm{SD}_n\) from the XENON1T experiment [80]. We evaluate theoretical predictions for \(\sigma ^\mathrm{SD}_p\) and \(\sigma ^\mathrm{SD}_n\) using MicrOMEGAs 5.0.2, which uses estimates of the spin-dependent scattering matrix elements consistent with [81].

We also allow for an overall uncertainty of 10% in the spin-dependent cross section.

#### 2.1.4 Halo model

The majority of direct searches for DM scattering assume a standard halo model (SHM) in which the local density of cold DM is 0.3 GeV/cm\(^3\), with negligible uncertainty. However, several recent analyses favour a larger local density, and here we follow [82] in assuming a central value for the local DM density of 0.55 GeV/cm\(^3\). For this reason, the direct DM scattering limits, estimated future sensitivity curves and neutrino ‘floors’ we use are rescaled relative to the published curves.

It has been suggested in a recent analysis of Gaia data that there may be a local debris flow that modifies the local velocity distribution from that given by the SHM [83]. However, this modification of the local velocity distribution was shown to have only a small effect on the interpretation of the XENON1T experiment for \(m_\chi \) in the range of interest in this paper, and a similar conclusion was reached in [82]. For another estimate of the local density of DM based on Gaia data, see [84].

When implementing the experimental constraints on \(\sigma ^\mathrm{SI}_p\) and \(\sigma ^\mathrm{SD}_p, \sigma ^\mathrm{SD}_n\), we assume an overall uncertainty of 30% [82, 85], in the local density.

### 2.2 LHC experimental constraints

#### 2.2.1 Monojet constraints

Spin-1 mediators decaying to missing energy are constrained by analyses targeting monojet final states by ATLAS [74] and CMS [75]. In this analysis we use the result obtained by CMS Collaboration [75] using 35.9/fb of data from collisions at 13 TeV. The ATLAS analysis of the axial coupling case has similar sensitivity. In particular, we use the 95 % CL upper bounds \(R_i^\mathrm{UL}(\mathbf{m})\), where *i* labels the vector and axial-vector mediators, imposed on \(\sigma (\mathbf{m}) / \sigma ^\mathrm{fix}(\mathbf{m})\) evaluated at each point in the \(\mathbf{m} = (m_\chi , m_Y)\) plane, where \(\sigma ^\mathrm{fix}\) is the cross-section evaluated with \((g_\mathrm{SM}, g_\mathrm{DM}) = (0.25, 1)\).

#### 2.2.2 Dijet constraints

^{2}We note that \(\sigma _{Y \rightarrow qq}\) and \(\sigma ({g_q^*})\) can be written as (using \(\Gamma _x \equiv \Gamma (Y \rightarrow xx)\))

*c*at \(m_Y = m_{Z'}\), and \(\Gamma _q = \big ( \frac{g_\mathrm{SM}}{g_q^*} \big )^2 \Gamma _q^*\). Using these relations, the equality in Eq. (17) occurs when \(g_\mathrm{SM}\) takes the value

^{3}The coloured lines correspond to the various experimental analyses, which can directly compared to the MasterCode \(\chi ^2\) evaluation indicated by colours in the plane. Note that we use the most sensitive upper limit for each mass value, and do not attempt to combine different experiments. Since the width of the band in Fig. 3 where the variation in the likelihood function is significant (as indicated by the colour transition) is typically smaller than the differences between the various experimental limits, this should be a good approximation. The excellent agreement between our 95% CL upper limit on \(g_\mathrm{SM}\) and that quoted by the experiments is a valuable cross-check of our implementation of the dijet invariant-mass constraints.

In addition to these invariant-mass searches, the CMS Collaboration has also published a limit on \(g_\mathrm{SM}\) as a function of \(m_{Z'}\) from an analysis of dijet angular distributions [94]. However, this limit is weaker than the limits from the dijet invariant mass distribution for the range of masses we study. Accordingly, we do not include it in our analysis. We also do not consider the \(Y \rightarrow t {\bar{t}}\) mode, since the sensitivity is weaker than in the dijet mode, due to the large \(t {\bar{t}}\) background.

Please note also that in evaluating the dijet and monojet constraints on model couplings we work only to leading order in QCD, in the absence of a complete set of the model-dependent NLO corrections. We expect this treatment to be conservative, in the sense that these corrections are generally positive, so that the upper limits on couplings that we obtain would be strengthened if NLO corrections were included.^{4}

## 3 Analysis framework

Global-fitting frameworks such as MasterCode [22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] have been used to study the constraints on different possible extensions of the SM, including SUSY models in particular, but can also be applied to other models with WIMP candidates for DM. Since DMSMs are not complete models, we do not consider it meaningful to compute *p*-values. Instead, here we use the MasterCode framework to correlate the impacts of constraints from different sectors, explore correlations and find regions of the model parameters that are still allowed.

We recall that MasterCode is a frequentist framework for combining constraints, written in Python/Cython and C++. As already mentioned, we use MG5_aMC 2.4.3 [65] to calculate mediator properties and the collider constraints on models implemented using DMSIMP [66, 67], and we use MicrOMEGAs 5.0.2 [64] for DM density and scattering calculations. We use the MultiNest 2.18 [68, 69, 70] algorithm for efficient sampling of the model parameter spaces. Since each of the DMSMs (3) and (4) that we study has a parameter space of only 4 dimensions, and since the constraint set is not large, sampling the model parameter spaces is not computationally onerous. In studies for our analysis we have made use of udocker software framework [97] to automatize the deployment of MasterCode inside Linux containers. This is a middleware suite developed in the context of the INDIGO datacloud project [98] to run docker containers in userspace, without requiring root privileges for installation or for execution.

*Z*could be subject to important constraints from precision electroweak data

^{5}and indirect searches for astrophysical DM annihilations should be considered,

^{6}but include all the masses for which LHC searches are sensitive. The couplings \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) were restricted to perturbative ranges \(< \sqrt{4 \pi }\).

The ranges of the DMSM parameters sampled, together with the numbers of segments into which they were divided during the sampling

Parameter | Range | No. of segments |
---|---|---|

\({m_Y}\) | \((0.1, 5) \,\, \mathrm {TeV}\) | 10 |

\(m_\chi \) | \((0, 2.5) \,\, \mathrm {TeV}\) | 8 |

\(g_\mathrm{SM}\) | \((0, \sqrt{4 \pi })\) | 2 |

\(g_\mathrm{DM}\) | \((0, \sqrt{4 \pi })\) | 2 |

Total # of segments | 320 |

In addition to the generic sampling ranges shown in Table 1, we have made dedicated scans of certain regions in order to clarify certain DMSM features. In particular, we gathered dedicated samples of the regions where \(2m_\chi /m_Y\) deviates from unity by \(< 10^{-3}\), so as to sample adequately annihilations near the *Y* peak when \(\Gamma _Y \ll m_Y\). We used similar sampling procedure for both the vector and axial-vector DMSMs, generating \(\sim 100\) million parameter sets in each case.

## 4 Results

### 4.1 Vector DM couplings

Green: annihilation via

*t*-channel \(\chi \) exchange into pairs of mediator particles*Y*that subsequently decay into SM particles, in the region where \(m_\chi \ge m_Y\);Yellow: rapid annihilation directly into SM particles via the

*s*-channel*Y*resonance, in the region where \(0.9<m_Y/(2m_\chi ) < 1.1\).

*Y*funnel dominates, and another with \(m_\chi > m_Y\) where annihilation into pairs of mediator particles

*Y*dominates. The boundaries of both allowed regions are very sharp, reflecting the steepness of the resonance peak in the

*s*-channel case and the phase-space limit in the

*t*-channel case.

^{7}

*t*-channel region is located at smaller \(m_Y\) than the

*s*-channel region for any value of \(m_\chi \), subject to the kinematic constraint \(m_Y < m_\chi \). In this region the annihilation cross-section of this channel, \(\chi \chi \rightarrow Y Y\), is proportional to \(g_\mathrm{DM}^4\) and independent of \(g_\mathrm{SM}\). Therefore, the relic abundance can be brought to the observed value by tuning \(g_\mathrm{DM}\), unless \(m_\chi \lesssim {100}\) GeV, whilst the experimental constraints from the LHC and the direct DM detection can be avoided by taking \(g_\mathrm{SM}\) small enough.

As can be seen in Fig. 4, the lower limit \(m_Y > 100 \,\, \mathrm {GeV}\) in our sample enforces a corresponding lower limit \(m_\chi > rsim 100 \,\, \mathrm {GeV}\) in the *t*-channel region. This is because the annihilation process \(\chi \chi \rightarrow YY\) is kinematically blocked for \(m_\chi < m_Y\), so the cross-section for \(\chi \chi \) annihilation becomes very small, resulting in DM overdensity. On the other hand, in the *s*-channel region the relic density constraint requires \(g_\mathrm{DM}\) to be very small for \(m_Y \sim 100 \,\, \mathrm {GeV}\), in which case \(\Gamma _Y/m_Y \ll 1\) and \(\chi \chi \) annihilations are rapid enough only if \(m_\chi \simeq m_Y/2 > rsim 50 \,\, \mathrm {GeV}\), the inequality being due to our scanning limit on \(m_Y\). The uncoloured vertical band at small mass in this and subsequent figures is a reflection of this restriction on the range of \(m_Y\) sampled.

*s*-channel region is \(m_Y \sim {\mathcal {O}}(700) \,\, \mathrm {TeV}\) for \(g_\mathrm{DM}\lesssim \sqrt{4 \pi }\).

*t*-channel region in Fig. 4, whereas the larger values of \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) outside this region appear in the

*s*-channel funnel region.

^{8}This region is bounded at large values of the product \(g_\mathrm{SM}\) \(g_\mathrm{DM}\) by the upper limit on \(\sigma ^\mathrm{SI}_p\) and the limited scanning range \(m_Y < 5 \,\, \mathrm {TeV}\), as indicated [see Eq. (15)]: the upper right portion of the figure with larger values of \(g_\mathrm{SM}\) \(g_\mathrm{DM}\) would be allowed for larger \(m_Y\). In the region \(0.07> g_\mathrm{DM}> 3 \times 10^{-4}\), \(\Omega h^2 \propto m_Y^2 / g_\mathrm{DM}^2\) and can be sufficiently small if \(m_\chi \simeq m_Y/2\) and \(m_Y < 5 \,\, \mathrm {TeV}\). In this case, Eqs. (15) and (21) imply

We have indicated by a diagonal dotted line where \(g_\mathrm{DM}= g_\mathrm{SM}\). As discussed later, we might expect \(g_\mathrm{DM}\) and \(g_\mathrm{SM}\) to be similar in magnitude in many UV completions of DMSMs, and we study later the restriction to the band between the dashed lines in Fig. 5 that is shaded darker yellow, where \(1/3< g_\mathrm{DM}/g_\mathrm{SM}< 3\), see Sect. 5.

We do not show the one-dimensional likelihood function for \(g_\mathrm{SM}\), which is quite featureless apart from a sharp rise for \(g_\mathrm{SM} > rsim 0.3\), nor that for \(g_\mathrm{DM}\), which is also featureless apart from a steep rise for \(g_\mathrm{DM}\lesssim 3 \times 10^{-4}\) that is an artefact of the limit on our scanning range for \(m_Y\).

*t*-channel region there is a strong upper limit on \(g_\mathrm{SM}\lesssim 10^{-3}\), which is enforced by the combination of the upper limit on \(\sigma ^\mathrm{SI}_p\), which constrains the product \(g_\mathrm{DM}\, g_\mathrm{SM}\), and the lower limit on \(g_\mathrm{DM}\) visible in the right panel, which is another result of our limited scan to \(m_Y > 100 \,\, \mathrm {GeV}\). Since \(\sigma ^\mathrm{SI}_p\) scales as \(g_\mathrm{DM}^2 g_\mathrm{SM}^2/m_Y^4\), the limit is stronger for smaller \(m_Y\), and \(g_\mathrm{SM}\) must be smaller than \(10^{-4}\) for \(m_Y \lesssim 100\) GeV. The upper limit on \(m_Y\) in the

*t*-channel region visible for small \(g_\mathrm{SM}\) is an artefact of the limited scanning range \(m_\chi < 2.5 \,\, \mathrm {TeV}\). The upper limit on \(g_\mathrm{SM}\) in the

*s*-channel region comes mainly from the LHC dijet constraint, with the \(\sigma ^\mathrm{SI}_p\) constraint also playing a role when \(m_Y \lesssim 500 \,\, \mathrm {GeV}\).

We see in the left panel of Fig. 6 that \(g_\mathrm{SM}\lesssim 10^{-2}\) for \(m_Y \sim 100 \,\, \mathrm {GeV}\) and \(\lesssim 0.1\) for \(m_Y > 1 \,\, \mathrm {TeV}\). Comparing with Fig. 11 of [55], where leptophobic models with universal quark couplings were analyzed,^{9} and identifying \(g_\mathrm{SM}\) with the combination \(g Y^\prime _q\) of the parameters defined in those models, we see that the precision electroweak data do not constrain our model sample.^{10} The two-branch structure of the *t*-channel region is a consequence of likelihood profiling. In fact, parameter space regions characterized by both *s*- and *t*-channel mechanisms overlap when projected in that specific region of the \(m_Y\)-\(g_{SM}\) plane and, since they have very similar \(\chi ^2\), sometimes one is selected over the other due to a statistical fluctuation.

*t*-channel region at low \(m_Y\) is due to the DM density constraint. In the

*s*-channel region the relic constraint imposes a weaker bound on \(g_\mathrm{DM}\), which is given roughly by

Figure 7 displays the \((m_\chi , g_\mathrm{SM})\) plane in the left panel and the \((m_\chi , g_\mathrm{DM})\) plane in the right panel. The upper bounds on \(g_\mathrm{SM}\) and the lower bounds on \(g_\mathrm{DM}\) are the same as in Fig. 6, all increasing with \(m_\chi \).

*t*- and

*s*-channel regions. The upper limits on \(\sigma ^\mathrm{SI}_p\) at various confidence levels are determined by the combined experimental likelihood for the LUX [76], PANDAX-II [77] and XENON1T [78] experiments, which we have rescaled to account for the different local DM density that we assume. In the

*s*-channel region, small values of \(\sigma ^\mathrm{SI}_p\) are allowed when \(m_\chi \sim m_Y/2\) and small values of \(g_\mathrm{DM}\) and/or \(g_\mathrm{SM}\) are favoured by the relic density constraint, and small values of \(\sigma ^\mathrm{SI}_p\) are allowed in the

*t*-channel region because small values of \(g_\mathrm{SM}\) are allowed, as discussed previously. On the other hand, we see that \(\sigma ^\mathrm{SI}_p\) may well lie within the range to be probed by upcoming experiments such as LUX-ZEPLIN (LZ) [100] and XENONnT [101], though \(\sigma ^\mathrm{SI}_p\) may also be much smaller than the current experimental sensitivity, even below the neutrino ‘floor’ indicated by the dashed orange line in Fig. 8, which is based on [82], updating [102, 103].

### 4.2 Axial-vector DM couplings

We now turn to the case of DM with axial-vector couplings. Figure 9 displays the \((m_Y, m_\chi )\) plane in this case: it is also colour-coded according to the dominant DM annihilation mechanism using the same shading scheme as for the vector case, and the 1- and 2-\(\sigma \) contours are indicated by red and blue lines, respectively. We see an *s*-channel funnel feature that is rather broader than in the case of DM with vector couplings shown in Fig. 4, and in this case we shade yellow the region where \(0.6< m_Y/(2 m_\chi ) < 2\).

*p*-wave or \(m_q\) suppressed (see the discussion around Eq. (7) of [48]). This opens up more parameter space in the \((m_{Y}, m_{\chi })\) plane, with the deviation from \(m_Y \sim 2 m_{\chi }\) bounded only by the dijet constraint. We note that this constraint is weaker when \(m_Y > 2 m_\chi \) because the decay channel \(Y \rightarrow \chi \chi \) is open. We also note that, as in the vector case, at low masses the funnel region merges with the

*t*-channel annihilation region where \(m_\chi > m_Y\), so that the preferred parameter space is simply connected also in the axial-vector case.

Also as in the vector-like case, we see again in the axial case in Fig. 9 the lower bound \(m_\chi > rsim 50 \,\, \mathrm {GeV}\) due to the interplay of the sampling limit \(m_Y > 100 \,\, \mathrm {GeV}\) and the relic density constraint. Above this value of \(m_\chi \), the one-dimensional \(\Delta \chi ^2\) likelihood function for \(m_\chi \) is featureless, as is that for \(m_Y > 100 \,\, \mathrm {GeV}\).

*s*-channel region extends to larger values of \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) than in the vector case, because of the suppression of axial-vector-mediated

*s*-channel annihilation discussed above. Values of \(g_\mathrm{DM}\) as large as the sampling limit \(\sqrt{4 \pi }\) are allowed. We see a separation between the regions of this plane where the

*s*- and

*t*-channel mechanisms are dominant when \(g_\mathrm{SM}\sim 10^{-2}\) and \(g_\mathrm{DM}\sim 10^{-1}\). The upper bound on \(g_\mathrm{SM}\) comes from the dijet constraint. As in the vector case, the dotted line is where \(g_\mathrm{SM}= g_\mathrm{DM}\), and the deeper shading indicates where \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\), favouring the large-\(g_\mathrm{SM}\) part of the

*s*-channel annihilation region, see the discussion in Sect. 5. As in the vector case, there is a region at low \(g_\mathrm{DM}\), rising at small \(g_\mathrm{SM}\), whose exclusion by the relic density constraint is an artefact of the sampling restriction \(m_Y > 100 \,\, \mathrm {GeV}\).

We display in Fig. 11 the \((m_Y, g_\mathrm{SM})\) and \((m_Y, g_\mathrm{DM})\) planes (left and right panels, respectively) in the scenario with axial-vector couplings. We see in the left panel that the *t*-channel DM mechanism is important for \(m_Y \lesssim 2.6 \,\, \mathrm {TeV}\). The limit visible in Fig. 9, which is due to the sampling limit \(m_\chi < 2.5 \,\, \mathrm {TeV}\). Annihilation via the *s*-channel becomes more important as \(m_Y\) increases, as also seen in Fig. 9, and is the only mechanism for \(m_Y > rsim 2.6 \,\, \mathrm {TeV}\) up to the sampling limit of \(5 \,\, \mathrm {TeV}\). The upper limit on \(g_\mathrm{SM}\) for \(m_Y < 5 \,\, \mathrm {TeV}\) is due to the dijet constraint. Unlike the vector case seen in the left panel of Fig. 6, the *s*-channel region is excluded for small \(g_\mathrm{SM}\). This is due to the *p*-wave nature of the *s*-channel annihilation via the axial mediator, which leads to the scaling behaviour Open image in new window, as discussed around Eq. (9).

*t*-channel mechanism is relevant for \(m_Y \lesssim 3 \,\, \mathrm {TeV}\), whereas the

*s*-channel mechanism is dominant for larger \(m_Y\). The lower limit on \(g_\mathrm{DM}\) is given by the relic density constraint since. even exactly at \(m_Y = 2 m_\chi \), Open image in new window and DM is overproduced for sufficiently small \(g_\mathrm{DM}\).

*t*-channel mechanism dominates and at larger \(g_\mathrm{SM}\) where the

*s*-channel mechanism dominates. There is a region at low \(m_\chi \) where the

*s*-channel mechanism dominates. The lower limit on \(g_\mathrm{DM}\) comes from the relic density constraint as can be seen in Eqs. (12), (13) and (14).

The one-dimensional \(\Delta \chi ^2\) function for \(g_\mathrm{DM}\) in the axial case rises sharply below \(\sim 4 \times 10^{-3}\), and that for \(g_\mathrm{SM}\) rises above \(\sim 0.3\). The \(\Delta \chi ^2\) functions for \(m_\chi \) and \(m_Y\) are featureless above 50 and 100 GeV, respectively.

*s*- and the

*t*-channel regions. However, we note that, whereas the short-term advantage may lie with searches for \(\sigma ^\mathrm{SD}_n\), since that currently provides the stronger constraint, the longer-term advantage may lie with searches for \(\sigma ^\mathrm{SD}_p\), since the expected ‘floor’ is lower in that case.

## 5 Possible ultraviolet completions

The vector- and axial-like leptophobic DMSMs analyzed here were chosen following the recommendations of the LHC Dark Matter Working Group [9, 10, 11, 12, 13], on the basis of their phenomenological simplicity and without regard for their possible ultraviolet (UV) completions. In any such UV completion, the spin-one boson could be expected to have comparable couplings to SM and DM particles, *modulo* possible group-theoretical factors and mixing angles.

Looking beyond the requirements of specific grand unified or string scenarios, one should consider the important consistency conditions on gauge couplings that are imposed by the cancellation of anomalous triangle anomalies required for renormalizability. These entail, characteristically, that the gauge couplings to different particle species are related by rational algebraic factors that are \(\mathcal{O}(1)\) before mixing. Supersymmetry is an important example of a framework where such mixing factors are important, but the couplings of supersymmetric WIMPs to the SU(2)\(\times \)U(1) gauge bosons of the SM are typically not much smaller than those of SM particles.

The construction of DMSMs with anomaly-free U(1)\('\) gauge bosons has been studied in [54], where it was found that in order to be leptophobic, such DMSMs would necessarily contain non-trivial dark sectors containing other particles besides the DM particle. Explicit examples have been given of leptophobic DMSMs with purely vector- or axial-like couplings to quarks, and DMSMs with purely vector- or axial-like couplings to the DM particle would be subject to further constraints. We do not discuss the construction of such models here, but expect on the basis of the argument in the previous paragraph that they would in general have \(g_\mathrm{DM}/g_\mathrm{SM}= \mathcal{O}(1)\). Additionally, one might expect that in a UV completion featuring unification in a non-Abelian gauge group the spin-1 mediator couplings would be \( > rsim \mathcal{O}(0.1)\), favouring parts of the funnel regions away from the regions where *t*-channel annihilation is dominant.

We have included in Figs. 5 and 10 diagonal dotted lines where \(g_\mathrm{DM}= g_\mathrm{SM}\) and shaded the strips where \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\), which are bounded by dashed lines. We see that in the vector case it traverses the funnel region where the DM particles annihilate via the *s*-channel, and does not approach the regions where DM particles annihilate mainly via *t*-channel exchanges into pairs of mediator particles, which are dominant when \(g_\mathrm{DM}\gg g_\mathrm{SM}\). On the other hand, in the axial-vector case both *t*- and *s*-channel annihilations are possible in the strip with \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\), as we discuss later. We now discuss in more detail the implications of the constraint \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\) for the DMSM parameter spaces.

*s*-channel annihilation to be fast enough. The upper bounds on \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) are, on the other hand, largely due to the spin-independent DM-nucleus scattering constraint, with the dijet constraint also playing a role in constraining \(g_\mathrm{SM}\) for \(m_Y > rsim 1.8\) TeV (\(m_\chi > rsim 0.9\) TeV). These features are also visible in the region of Fig. 5 with the darker yellow band.

Figure 15 shows the corresponding parameter planes for the axial-coupling case with the selection \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\). In the upper panels, we see that the *t*-channel region is confined to a small region where \(m_Y \lesssim 500\) GeV and \(g_\mathrm{SM} > rsim 0.1\). We see from the right panel of Fig. 11 that \(g_\mathrm{DM}\) must be larger than \(\sim 0.3\) (0.9) at \(m_Y = 500\) GeV (2 TeV) to account for the DM density. The \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\) condition then implies that \(g_\mathrm{SM} > rsim 0.1\) (0.3) for \(m_Y = 500\) GeV (2 TeV). However, such a large value of \(g_\mathrm{SM}\) is not compatible with the dijet constraint in this range of \(m_Y\). Therefore, no region with \(m_Y > rsim 500\) GeV is allowed in the *t*-channel annihilation region. As discussed above, *s*-channel annihilation undergoes a *p*-wave or \(m_q\) suppression in the axial-vector mediator case, and \(g_\mathrm{SM} > rsim 10^{-3}\) and \(g_\mathrm{DM} > rsim 3 \times 10^{-3}\) (see the darker yellow band in Fig. 10), are needed to compensate this suppression. However, there is a narrow band of \(g_\mathrm{SM}\) values that is also compatible with the dijet constraint for all the sampled range of \(m_Y\). We see that the *s*-channel region is allowed for all the sampled range of \(m_Y\), for the most part also if \(g_\mathrm{SM}> 0.1\).

*t*-channel region appears in the coupling range \(0.03 \lesssim g_\mathrm{SM}\lesssim 0.3\), and in the right plot its range is \(0.05 \lesssim g_\mathrm{DM}\lesssim 0.5\), sandwiched by the DM density condition from below and the LHC dijet constraint from above. As seen in this projection, the

*t*-channel region is less restricted in the \(m_\chi \) direction, and any value of \(m_\chi > rsim 300 \,\, \mathrm {GeV}\) is in principle compatible with this mechanism. In the

*s*-channel region all values of \(m_\chi \) between 50 GeV and 2.5 TeV are allowed. We note also that \(g_\mathrm{DM}, g_\mathrm{SM}> 0.1\) are possible for all the range of \(m_\chi \).

In the vector-like case, the main changes in the one-dimensional \(\Delta \chi ^2\) functions for \(g_\mathrm{DM}\) and \(g_\mathrm{SM}\) due to the selection \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\) are reductions in their upper limits to \(\sim 0.3\), and there are negligible changes in the one-dimensional \(\Delta \chi ^2\) functions for \(m_Y\) and \(m_\chi \). In the axial case the main changes when the selection is made are that \(g_\mathrm{DM}\lesssim 0.5\) and \(g_\mathrm{SM} > rsim 10^{-3}\), while all values of \(m_\chi > rsim 50 \,\, \mathrm {GeV}\) and \(m_Y > rsim 100 \,\, \mathrm {GeV}\) still have very low \(\Delta \chi ^2\).

Finally, we show in Fig. 16 the preferred ranges of \((m_\chi , \sigma ^\mathrm{SI}_p)\) in the vector-like model (left panel) and \((m_\chi , \sigma ^\mathrm{SD}_n)\) in the axial model (right panel) after the selection \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\). We see that there are good prospects for detecting spin-independent DM scattering in the vector-like case, since \(\sigma ^\mathrm{SI}_p\) lies partially above the neutrino ‘floor’ within the range of \(m_\chi \) sampled, though smaller values of \(\sigma ^\mathrm{SI}_p\) could also occur for any value of \(m_\chi > 50 \,\, \mathrm {GeV}\). In the case of spin-dependent scattering after the selection \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\), we see in the right panel of Fig. 16 that in the *t*-channel exchange region the predicted values of \(\sigma ^\mathrm{SD}_n\) are relatively close to the XENON1T limit [80], and hence may offer prospects for detection with the planned LZ experiment [100] within the sampled range of DM masses. However, values of \(\sigma ^\mathrm{SD}_n\) may be considerably lower in the *s*-channel exchange region.^{11} The disfavored region for \(10^{-45}~\mathrm {cm}^2 \lesssim \) \(\sigma ^\mathrm{SD}_n\) \(\lesssim 10^{-42}~\mathrm {cm}^2\) and \(m_\chi > rsim 300\) GeV is caused by the constraint coming from dijet searches.

*t*-channel region corresponds to small values of \(m_Y\) with larger values of \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) than in the

*s*-channel region, and in the lower panels these projections show that the

*t*-channel region may have larger values of \(g_\mathrm{SM}\) and \(g_\mathrm{DM}\) than in the

*s*-channel region for a large range of values of \(m_\chi \). The intermediate ‘hole’ is enforced by the LHC dijet constraint, in particular.

It is interesting that, with the selection \(1/3< g_\mathrm{SM}/g_\mathrm{DM}< 3\) motivated by the prospect of UV completion, searches for \(\sigma ^\mathrm{SI}_p\) and \(\sigma ^\mathrm{SD}_n\) are able to probe both the *s*- and *t*-channel regions of the DMSM parameter spaces

## 6 Conclusions and perspectives

In this paper we have used MasterCode to make a global analysis of the parameter spaces of dark matter simplified models with leptophobic spin-one mediator particles *Y* with either vectorial or axial couplings to SM particles and to the dark matter particle \(\chi \). Each of these models is characterized by four free parameters: \(m_Y\) and \(m_\chi \), the coupling \(g_\mathrm{DM}\) of the mediator to the dark matter particle, and the coupling \(g_\mathrm{SM}\) of the mediator particle to quarks, which we assume to be independent of flavour.

We have implemented constraints on the model parameter spaces due to LHC searches for monojet events and measurements of the dijet invariant mass spectrum, as well as the cosmological constraint on the dark matter density and direct upper limits on spin-independent and -dependent scattering on nuclei. We have scanned mediator masses \(m_Y \le 5 \,\, \mathrm {TeV}\) and dark matter particle masses \(m_\chi \le 2.5 \,\, \mathrm {TeV}\), delineating the regions of the model parameters with \(\Delta \chi ^2 < 2.30 (5.99)\), which are favoured at the 68% (95%) CL and regarded as proxies for 1- (2-)\(\sigma \) regions, respectively. Within these regions we have identified two main mechanisms for bringing the cosmological dark matter density into the range allowed by cosmology, namely annihilation via *t*-channel \(\chi \) exchange and annihilation via the *Y* boson in the *s* channel. With an eye to possible ultraviolet completions of the simplified models studied here, we have also explored the portions of the favoured parameter space where \(1/3< g_\mathrm{DM}/g_\mathrm{SM}< 3\), as discussed below.

In the vector-like case, we find a relatively clear separation between the regions where the *t*- and *s*-channel mechanisms dominate, with the former being more important at smaller mediator masses, small values of \(g_\mathrm{SM}\) and relatively large values of \(g_\mathrm{DM}\). The one-dimensional likelihood functions for both \(m_\chi \) and \(m_Y\) are quite small and flat above thresholds \(\sim 50 \,\, \mathrm {GeV}\) and \(\sim 100 \,\, \mathrm {GeV}\). Thus the LHC still has interesting prospects for discovering DM and mediator particles in these simplified models. Any value of \(g_\mathrm{SM}\) between \(\sim 10^{-6}\) and the dijet limit of \(\sim 0.3\) is possible without any \(\chi ^2\) penalty, as is the case for \(g_\mathrm{DM} > rsim 10^{-3}\), where the lower limit is due to the upper limit on the sampling range for \(m_Y\). The spin-independent dark matter scattering cross section \(\sigma ^\mathrm{SI}_p\) may be very close to the present experimental upper limit, within the range accessible to the upcoming LZ and XENONnT experiments. However, in both the *s*- and the *t*-channel cases \(\sigma ^\mathrm{SI}_p\) may also be much below the neutrino ‘floor’.

In the axial case the *t*- and *s*-channel regions are more connected. The one-dimensional likelihood functions for \(m_\chi \) and \(m_Y\) are again featureless above thresholds \(\sim 50 \,\, \mathrm {GeV}\) and \(100 \,\, \mathrm {GeV}\), respectively, and that for \(g_\mathrm{SM}\) is quite featureless, as is that for \(g_\mathrm{DM} > rsim 10^{-2}\). We find that the spin-dependent dark matter scattering cross sections \(\sigma ^\mathrm{SD}_p\) and \(\sigma ^\mathrm{SD}_n\) may also be very close to the present experimental upper limit, within the range accessible to the upcoming PICO-500 and LZ experiments, though much lower values below the corresponding ‘floors’ are also possible.

Finally, we have also explored the possibility that \(1/3< g_\mathrm{DM}/g_\mathrm{SM}< 3\), as might be suggested in some ultraviolet completions of the dark matter simplified models considered here. In this case, values of \(g_\mathrm{SM}, g_\mathrm{DM}\sim 0.1\) are possible, as might also be suggested in some scenarios with unified gauge interactions, and wide ranges of DM and mediator masses are again accessible to the LHC experiments. In the case of vector-like couplings, this reduced range of \(g_\mathrm{DM}/g_\mathrm{SM}\) disfavours the *t*-channel mechanism. However, in the axial case the *t*-channel mechanism is still possible, yields values of \(\sigma ^\mathrm{SD}_n\) that are relatively close to the upper limit set by the PICO-60 experiment, and may well be within reach of the upcoming PICO-500 and LZ experiments, whereas lower values of \(\sigma ^\mathrm{SD}_n\) are possible if the *t*-channel mechanism dominates.

In this paper, we have performed a first global study - including all the relevant constraints from Run 2 of the LHC and direct scattering searches and not fixing any of the underlying parameters - of some DMSMs proposed by the LHC Dark Matter Working Group [9, 10, 11, 12, 13] using the MasterCode framework. Our analysis also provides a more detailed study of the vector and axial leptophobic models considered in [55]. These models are undoubtedly over-simplified, and it would be interesting and useful to extend this type of analysis to other models that may be more realistic, which generally contain more parameters. For example, one should study models with spin-one mediators that are not leptophobic, and also models with spin-zero mediators that may be either scalar or pseudoscalar. One could also study models that are not flavour-universal in the quark sector, which may be motivated by the anomalies reported in *B* meson decays. Consistent ultraviolet completions of DMSMs should include a mechanism for anomaly cancellation, which typically include more ‘dark’ particles whose possible phenomenological signatures could also be considered. The MasterCode tool is a very suitable tool for such analyses, and we plan to use MasterCode for such analyses in the future.

## Footnotes

- 1.
- 2.
We note that \(\sigma (g_q^*)\) is the cross-section limit assuming the finite width given by \(g_q^*\). Due to the presence of the \(Y \rightarrow \chi \chi \) mode, \(\Gamma _Y\) and \(\Gamma _{Z'}\) may be different. However, when this limit is important, the \(Y \rightarrow qq\) branching ratio must be large, which justifies our approximation that \(\Gamma _Y \sim \Gamma _{Z'}\) in this regime.

- 3.
The discontinuity in the limit at \(m_Y = 2 \,\, \mathrm {TeV}\) is due to the improved sensitivity of the recent ATLAS limit using 139/fb of data at 13 TeV [93].

- 4.
- 5.
See [55] for a treatment of these constraints, which we discuss in more detail below.

- 6.
Searches for \(\gamma \)-rays from hadronic DM annihilations are generally insensitive to the cross-section required to obtain the correct cosmological DM density for \(m_\chi > rsim 50 \,\, \mathrm {GeV}\) [55], see also [99] and references therein. Indirect constraints from searches for energetic solar neutrinos are not competitive with direct searches for spin-dependent DM scattering, as discussed below.

- 7.
In other projections, the

*s*- and*t*-channel regions may overlap, and have quite similar \(\chi ^2\), so that points with different colours appear interspersed. - 8.
Close to the boundary between these regions there is an area left unshaded because of incomplete sampling.

- 9.
We recall that in such models the

*Z*and*Y*have only kinetic mixing, which is loop-induced and hence suppressed. - 10.
Even in the leptophobic models we consider here, couplings to leptons will be generated at the one-loop level but, because of the lack of interference, would contribute to cross sections and rates only at the two-loop level, and hence can be neglected.

- 11.
Note that we have again not included in the right panel the upper limits and prospective detection ‘floors’ presented by the Super-Kamiokande [104] and IceCube [105] Collaborations, which are based on specific assumptions about the DM annihilation channels that are not applicable in the leptophobic DMSMs studied here.

## Notes

### Acknowledgements

We would like to thank Y. Mambrini and M. Voloshin for helpful discussions. G.W. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC 2121 “Quantum Universe” – 390833306. The work of K.S. has been partially supported by the National Science Centre, Poland, under research grants DEC-2014/15/B/ST2/02157, DEC-2015/18/M/ST2/00054 and DEC- 2015/19/D/ST2/03136. The work of M.B. and D.M.S. has been supported by the European Research Council via Grant BSMFLEET 639068. The work of J.C.C. is supported by CNPq (Brazil). The work of M.J.D. is supported in part by the Australia Research Council. The work of J.E. is supported in part by STFC (UK) via the research grant ST/L000258/1 and in part via the Estonian Research Council via a Mobilitas Pluss grant, and the work of H.F. is also supported in part by STFC (UK) via grant ST/N000250/1. The work of S.H. is supported in part by the MEINCOP Spain under contract FPA2016-78022-P, in part by the Spanish Agencia Estatal de Investigación (AEI) and the EU Fondo Europeo de Desarrollo Regional (FEDER) through the project FPA2016-78645-P, in part by the AEI through the grant IFT Centro de Excelencia Severo Ochoa SEV-2016-0597, and by the “Spanish Red Consolider Multidark” FPA2017-90566-REDC. The work of M.L. is supported by XuntaGal. The work of K.A.O. is supported in part by DOE grant desc0011842 at the University of Minnesota. During part of this work, to deploy MasterCode on clusters we used the middle-ware suite udocker [97], which was developed by the EC H2020 project INDIGO-Datacloud (RIA 653549). We also thank Imperial College London and the University of Bristol for making available to us cluster computing resources that have been used intensively to carry out this work.

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