Phenomenology of the inert doublet model with a global U(1) symmetry

The inert doublet model is a minimal dark matter model with strong theoretical motivations, where the stability of dark matter is usually achieved by imposing a $Z_2$ parity. We promote the $Z_2$ parity into a global $U(1)$ symmetry and study its phenomenological implications. There are two characteristic features of the model: both the CP-even and CP-odd neutral inert scalars, $h_1$ and $h_2$, become DM candidates; the number of model parameters is one less than that with $Z_2$ parity. We first analyze the constraints from LEP experiments, electroweak precision tests, theoretical stability, Higgs precision data, dark matter relic density, and direct detection experiments. It is found that if the model is required to explain at least 10% of the observed relic density, the theory is extremely limited such that the dark matter mass is about $70\;{\rm GeV}$ and the charged Higgs boson is not very heavy. Focusing on this narrow parameter space, we calculate the production cross sections of almost all the possible mono-$X$ and mono-$XX'$ processes at the LHC. The mono-$W\gamma$ process is shown to have high discovery potential with the help of the decay of the intermediate-mass charged Higgs boson into $W^\pm h_{1,2}$. A search strategy is designed to increase the potential discovery of the model for the mono-$W\gamma$ signal at both the HL-LHC and the FCC-hh. The optimal cut on $E_T^{\rm miss}/\sqrt{H_T}$ is suggested to maximize the signal significance, being about $0.76$ at the HL-LHC and about $7.5$ at the FCC-hh.