Inelastic dark matter models that have two dark matter particles and a massive dark photon can reproduce the observed relic dark matter density without violating cosmological limits. The mass splitting between the two dark matter particles ${\chi }_1$ and ${\chi }_2$, with $m({\chi }_2)>m({\chi }_1)$, is induced by a dark Higgs field and a corresponding dark Higgs boson $h^{\prime }$. We present a search for dark matter in events with two vertices, at least one of which must be displaced from the interaction region, and missing energy. Using a $365{\mathrm{fb}}^{-1}$ data sample collected at Belle II, which operates at the SuperKEKB $e^{+}{e}^{-}$ collider, we observe no evidence for a signal. We set upper limits on the product of the production cross section $\sigma (e^{+}{e}^{-}\to h^{\prime }{\chi }_1{\chi }_2)$, and the product of branching fractions $\mathcal{B}({\chi }_2\to {\chi }_1{e}^{+}{e}^{-})\times \mathcal{B}(h^{\prime }\to x^{+}{x}^{-})$, where $x^{+}{x}^{-}$ indicates ${\mu }^{+}{\mu }^{-},{\pi }^{+}{\pi }^{-}$, or $K^{+}{K}^{-}$, as functions of $h^{\prime }$ mass and lifetime at the level of ${10}^{-1}\mathrm{fb}$. We set model-dependent upper limits on the dark Higgs mixing angle at the level of ${10}^{-5}$ and on the dark photon kinetic mixing parameter at the level of ${10}^{-3}$. This is the first search for dark Higgs bosons in association with inelastic dark matter.

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• Dark matter
• Extensions of Higgs sector
• Extensions of gauge sector
• Particle dark matter
• Lepton colliders
• Particle data analysis