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 $365\,\mbox{fb}^{-1}$ 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 $\sigma\left(e^+e^- \to h^\prime \chi_1 \chi_2\right)$, and the product of branching fractions $\mathcal{B}\left(\chi_2\to\chi_1 e^+ e^-\right)\times\mathcal{B}\left(h^\prime\to x^+x^-\right)$, where $x^+x^-$ indicates $\mu^+\mu^-, \pi^+\pi^-$, or $K^+K^-$, as functions of $h^{\prime}$ mass and lifetime at the level of $10^{-1}\,\mbox{fb}$. 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.
Comment: Submitted for publication with Physical Review Letters