Searching for elusive dark Higgs boson in spin-1/2 inelastic dark matter models at Belle II

Spin-1/2 inelastic dark matter (DM) models are popular among sub-GeV to GeV thermal DM scenarios due to the dominant role of co-annihilation in determining the DM relic abundance. In these models, the dark Higgs boson plays a crucial role in generating the mass of the new gauge boson, the dark photon ($A^{'}$), and in establishing the mass splitting between the excited ($\chi_2$) and ground ($\chi_1$) states of DM. In particular, the Compton scattering $\chi_1 A' \rightarrow \chi_2^* \rightarrow \chi_1 A'$ and its $t$-channel crossed process, $\chi_1 \chi_1 \rightarrow A' A'$, remain unitary for high energy longitudunal dark photon, only if the contribution of the dark Higgs boson is included. However, experimental searches for the dark Higgs boson have received relatively little attention. In particular, when the dark Higgs boson mass exceeds twice that of the DM excited state, its decay signatures become semi-visible or invisible, making detection challenging with current light scalar search strategies. In this work, we explore the prospects for detecting the elusive dark Higgs boson in inelastic DM models at Belle II via dark Higgs-strahlung and rare $B$ meson decay processes. Our analysis indicates that both the inclusive signature of two displaced dilepton vertices and the additional missing energy from dark Higgs boson decays serve as robust indicators of its presence. Furthermore, we assess the future potential for detecting the dark Higgs boson with the proposed far detector related to Belle II, GAZELLE.