Direct detection of dark matter with resonant annihilation

In the scenario where the dark matter (DM) particles $\chi\bar\chi$ pair annihilate through a resonance particle $R$, the constraint from DM relic density makes the corresponding cross section for DM-nuclei elastic scattering extremely small, and can be below the neutrino background induced by the coherent neutrino-nuclei scattering, which makes the DM particle beyond the reach of the conventional DM direct detection experiments. We present an improved analytical calculation of the DM relic density in the case of resonant DM annihilation for $s$- and $p$-wave cases and invesitgate the condition for the DM-nuclei scattering cross section to be above the neutrino background. We show that in Higgs-portal type models, for DM particles with $s$-wave annihilation, the spin-independent DM-nucleus scattering cross section is proportional to $\Gamma_{R}/m_{R}$, the ratio of the decay width and the mass of $R$. For a typical DM particle mass $\sim50$ GeV, the condition leads to $\Gamma_{R}/m_{R} \gtrsim \mathcal{O}(10^{-4})$. In $p$-wave annihilation case, the spin-independent scattering cross section is insensitive to $\Gamma_{R}/m_{R}$, and is always above the neutrino background, as long as the DM particle is lighter than the top quark. The real singlet DM model is discussed as a concrete example.