# Inert Doublet Model¶

Jon Butterworth, Louie Corpe, Tania Robens, Gustavs Zilgalvis

The following is a study of the Inert Doublet Model [91, 116, 137], a simple two-Higgs doublet model with the scalar potential

$\begin{split}\begin{array}{c} V=-\frac{1}{2}\left[m_{11}^{2}\left(\phi_{S}^{\dagger} \phi_{S}\right)+m_{22}^{2}\left(\phi_{D}^{\dagger} \phi_{D}\right)\right]+\frac{\lambda_{1}}{2}\left(\phi_{S}^{\dagger} \phi_{S}\right)^{2}+\frac{\lambda_{2}}{2}\left(\phi_{D}^{\dagger} \phi_{D}\right)^{2}+\lambda_{3}\left(\phi_{S}^{\dagger} \phi_{S}\right)\left(\phi_{D}^{\dagger} \phi_{D}\right) \\ +\lambda_{4}\left(\phi_{S}^{\dagger} \phi_{D}\right)\left(\phi_{D}^{\dagger} \phi_{S}\right)+\frac{\lambda_{5}}{2}\left[\left(\phi_{S}^{\dagger} \phi_{D}\right)^{2}+\left(\phi_{D}^{\dagger} \phi_{S}\right)^{2}\right] \end{array}\end{split}$

that obeys a discrete $$Z_2$$ symmetry and provides a dark matter (DM) candidate [144, 147]. The UFO files come from the Feynrules model library https://feynrules.irmp.ucl.ac.be/wiki/InertDoublet and can also be found here: https://gitlab.com/hepcedar/contur/-/tree/master/data/Models/2HDM/InertDoublet

Events are generated with Herwig7 [94], including all 2-to-2 processes involving a beyond-the-SM (BSM) particle in the matrix-element or an outgoing leg.

In this model, $$H^{0}$$ is considered to be the DM candidate, and only regions where it is the lightest new particle are considered.

Taking into account that larger $$\lambda$$ values and larger mass splittings between $$A^{0}$$ and the charged Higgs are excluded by precision measurements of electroweak observables and theory considerations, we use parameter values $$\lambda_{2} = 2$$, $$\lambda_{L} = \frac{1}{2}(\lambda_{3}+\lambda_{4}+\lambda_{5}) = 0.005$$, $$m_{H^{\pm}}=m_{A^0}+20$$ GeV, and scan $$m_{H^{0}}$$ and $$m_{A^{0}}$$ over the range [50, 500] GeV in which we have the most sensitivity. A 10x10 grid is used, with 30,000 simulated events for each grid point. The regions where $$H^{0}$$ is not the lightest particle are excluded.

The figures above show some of the dominant cross sections at each point according to the Herwig event generator. The lower half of the plane, coloured in orange, is not considered. White sections of the plots indicate regions where the cross-section was not sampled because it was negligible compared to other processes. The largest cross-section is for low-mass $$m_{H^{0}}$$ pair-production, shown in the upper-left figure. In this case the DM candidate is produced with no other SM objects, and would therefore be a challenging missing-transverse-energy ($$E_T^{\rm miss}$$) -only signature. In other parts of the plane, $$H^{0}$$ may also be produced with other new particles such as $$A^{0}$$, or $$H^{\pm}$$, or pair-production of these other BSM particles may occur. These BSM particles then decay back to a $$H^{0}$$ in association with a SM vector boson, which follow their normal decays to quarks or leptons. For instance, pair-produced $$A^{0}$$ particles may decay to pairs of $$H^{0}$$ and two SM Z bosons.

The figures below show the results of propagating these production and decays to final states, which could be observed at the LHC. The final states with the highest cross-sections are shown. Although the $$E_T^{\rm miss}$$-only signature would have the highest cross-sections, it is experimentally very challenging to analyse this final state at the LHC. It is clear that the signatures more easily observable at the LHC would come from $$H^{0}$$ production associated with jets, leading to $$E_T^{\rm miss}$$ +jets final states. Only few such measurements (as opposed to searches) are available at time of writing. Signatures with one or more leptons with $$E_T^{\rm miss}$$ and jets may also play a role.

The heatmaps below show the results of applying the CONTUR method to this model, using for all currently-available LHC data preserved in Rivet (7, 8 and 13 TeV runs in Rivet as of 07/07/2020)].

As expected from the cross section plots, there is very little sensitivity, with only a small amount of exclusion at about the 60% level showing at low $$m_{H^0}$$. As can be seen in the left-hand plot, the $$E_T^{\rm miss}$$ +jets analyses are the most sensitive across the plane. However, other signatures play a role in particular at low $$m_{H^{0}}$$, where the overall sensitivity is greatest. These additional signatures include the EEJET and WW pools, which collect lepton+jet and lepton+ $$E_T^{\rm miss}$$ +jets signatures, and the four-lepton invariant mass measurement, which is sensitive in the rare cases when DM candidates are produced with four leptons, such as $${A^{0}}{A^{0}} \rightarrow {H^{0}}{H^{0}} ZZ \rightarrow {H^{0}}{H^{0}} \ell \ell \ell \ell$$.

The $$E_T^{\rm miss}$$ +jets measurements and searches are available in Contur only for 3.2/fb, so the 139/fb full-run-2 updates are expected to boost sensitivity.

The CLs exclusion is low in all regions of the heatmap, showing that the Inert Doublet Model is very challenging to exclude, as it appears to evade many of the measurements made at the LHC thus far. It is therefore a good candidate for dedicated searches with higher integrated luminosity.