# Simplified Vector Dark Matter Model (1st gen)¶

This is the model discussed in the ‘white paper’ [26]. It is a simplified model with a dark matter Majorana fermion, $$\psi$$, which interacts with the SM through a new vector particle, $$Z^\prime$$. The couplings of the mediator $$Z^\prime$$ to the dark matter $$\psi$$ and to the SM are specified as

\begin{aligned} \label{eq:vector_mediator} {\cal L} \supset {{g_{\rm DM}}}\,\overline{\psi} \gamma_{\mu}\gamma_5 \psi\,Z'^{\mu} + {{g_{q}}}\sum_{q} \bar q \gamma_{\mu} q \,Z'^{\mu} \,,\end{aligned}

where the sum in the second term includes only the first generation SM quarks, $$q \in \{u,d\}$$. The model has only four free parameters - two couplings and two masses: $$g_{\rm DM}$$, $$g_{q}$$, $$M_\psi \equiv {{M_{\rm DM}}}$$, and $$M_{Z^{\prime}}$$. The width of the mediator, $$\Gamma_{Z'}$$, is determined by these four parameters.

Following Ref. [48] we have chosen to couple the mediator to dark matter and to the SM quarks through an axial-vector and vector current, respectively. An axial-vector coupling of the mediator to dark matter leads to spin-dependent dark matter-nucleon interactions and thus weaker bounds from direct dark matter searches. Such a coupling structure naturally arises for Majorana fermion dark matter.

To investigate the exclusion power of the particle-level measurements considered, we scan a range in plausible mediator masses ($$M_{Z^{\prime}}$$) and dark matter masses ($$M_{\rm DM}$$) within this model for three choices of the coupling of the mediator to the SM ($$g_{q}$$). These coupling choices correspond to

• an ‘optimistic’ scenario $${{g_{q}}}= 0.5, {{g_{\rm DM}}}= 1$$: strong signals, close to the edge of exclusion or excluded already,
• a ‘challenging’ scenario $${{g_{q}}}= 0.25, {{g_{\rm DM}}}= 1$$: low couplings, hard to exclude, and
• an ‘intermediate’ scenario $${{g_{q}}}= 0.375, {{g_{\rm DM}}}= 1$$, between the two. We also consider
• a scenario where the coupling of dark matter to the mediator is suppressed, $${{g_{q}}}= 0.375, {{g_{\rm DM}}}= 0.25$$.

For all these scenarios, the calculated width of the mediator is less than 10% of $$M_{Z^{\prime}}$$, as shown in the table.

Table of maximal $$\Gamma_{Z'}/{{M_{Z^{\prime}}}}$$ occuring over the mass ranges for the four pairs of coupling values.

$$g_{q}$$ $$g_{\rm DM}$$ $$M_{Z^{\prime}}$$[GeV] $$M_{\rm DM}$$[GeV] $$\Gamma_{Z'}/{{M_{Z^{\prime}}}}$$
0.25 1 3000 100 0.0626
0.375 1 3000 100 0.0751
0.5 1 3000 100 0.0925
0.375 0.25 3000 100 0.0257

## Heatmaps for First-generation Vector DM model¶

The sensitivities derived from multiple distributions such as those discussed in the previous section are combined into heatmaps’ which delineate exclusion regions and contours in the parameter space of $$M_{\rm DM}$$and $$M_{Z^{\prime}}$$.

Heatmaps displaying 2D parameter space scans in mass corresponding to a fixed $$g_{\rm DM}=1$$ and variable $$g_{q}$$. The confidence level of exclusion represented corresponds to testing the full signal strength hypothesis against the null background-only hypothesis. The combination of measurements entering into the confidence level presented here is the maximally sensitive allowed grouping, considering all available measurements.

Legend:

Heatmaps for $$g_{q}=0.5$$ , $$g_{DM} = 1$$

Heatmap and contour for all available data (measurements from 7, 8 and 13 TeV runs in Rivet as of 21/9/2017)

To see where the sensitivity comes from, here’s what we have from the inclusive, dijet and multijet measurements:

this is the exclusion from the W+jet and Z+jet measurements:

and here’s what the various diboson measurements do.

The photon measurements (inclusive, photon+jet, diphoton) don’t add anything significant for this model.

Finally, for reference, the heatmap and contour for all available 7 TeV data (similar dataset to the 2016 Contur paper)

And lower couplings: Heatmaps for $$g_{q}=0.25$$ , $$g_{DM} = 1$$

Heatmap and contour for all available data (measurements from 7, 8 and 13 TeV runs in Rivet as of 21/9/2017)

## Comparison to Data¶

Some examples for this model:

Explanation of Contur plot format

## Discussion¶

As expected, the exclusion is much weaker in the ‘challenging’ case and quite strong in the ‘optimistic’ scenario. For the first three scenarios, at $${{M_{Z^{\prime}}}}> 2{{M_{\rm DM}}}$$ the decay of the mediator to dark matter dominates over the decay to jets. This leads to the diagonal structure across the plots, with the sensitivity above the diagonal, in the left portion of the map, coming mainly from the jet measurements. In the fourth scenario, even when the decay to DM is kinematically allowed, the jet signatures continue to contribute, and so the diagonal structure is less visible.

At low values of $$M_{Z^{\prime}}$$the sensitivity comes mainly from the $$V+$$jets signatures. In the challenging and intermediate scenarios, a dip in sensitivity around $${{M_{Z^{\prime}}}}\approx 700$$ GeV is visible, where the sensitivity from inclusive jets and $$V+$$jets do not quite overlap. In the optimistic scenario, they overlap, and the whole upper left region of the map is excluded. In addition, the cross section $$\times$$ branching ratio for quarks $$\rightarrow Z^\prime \rightarrow$$ quarks remains large enough that the diagonal cutoff in sensitivity of the jet channels at $${{M_{Z^{\prime}}}}\approx 2{{M_{\rm DM}}}$$ is blurred.

To the bottom right region of the diagonal the decay of the mediator to dark matter is kinematically allowed, and for $${{g_{\rm DM}}}=1$$ it will dominate over the decay to quarks. Hence the sensitivity in the inclusive jet (and $$V+$$jet) signatures drops in all scenarios except the fourth. This is the region where a measurement of $$E_T^{\rm miss}$$+jets would be useful (and indeed it is where the searches performed using such signatures contribute, see, for example, [48][45]). Current sensitivity in the intermediate and challenging scenarios comes from the $$l^+l^- + {{E_T^{\rm miss}}}$$ measurement, and dies away at $${{M_{Z^{\prime}}}}\approx 750$$ GeV. In the fourth scenario, the decays to dark matter are relatively suppressed and so the $$l^+l^- + {{E_T^{\rm miss}}}$$ signature makes little contribution. However, as already discussed, the exclusion from the jet measurements remains strong.

Note that as expected, the sensitivity from the 7 TeV dijet measurements used here is qualitatively similar, but inferior, to the exclusions obtained combining the searches in 8 TeV and 13 TeV jet data — see, for example, [41]. This should change once measurements are available from these later running periods (indeed, the CMS measurement is already made [51], but is not yet available in or HepData). The other channels extend the sensitivity, and this will also improve as more measurements are incorporated.

As mentioned in section [sec:model], the parameters of the simplified model are constrained by perturbative unitarity. In the region $${{M_{\rm DM}}}\gtrsim \sqrt{\pi/2}\, {{M_{Z^{\prime}}}}/g_{\rm DM}$$, indicated by the blue shaded area in Fig. [fig:cont], the dark matter relic density cannot be calculated reliably [48]. Since we only consider couplings $$g_{\rm DM}$$and $$g_{q}$$well within the perturbative regime, perturbative unitarity is respected in the production of mediators at the LHC and does not provide any further restrictions on the parameter space of our model [40]. The physics of dark matter is, of course, constrained by astrophysical and cosmological observations, including in particular the dark matter relic density, and direct and indirect searches for dark matter, see, for example, Refs. [48][45][46] for combined analyses of collider and astrophysical constraints of simplified dark matter models with vector mediators. However, all those constraints are based on additional assumptions on the thermal history of the Universe and astrophysical properties of dark matter, and they do not affect BSM searches at the LHC. Since we have adopted the simplified dark matter model to illustrate the power of the approach for BSM searches at the LHC in general, rather than providing a detailed cosmological and astrophysical analysis of dark matter, we do not show the corresponding constraints in the figures.