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 χ1 and χ2, with m(χ2)>m(χ1), is induced by a dark Higgs field and a corresponding dark Higgs boson h′. 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 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 σ(e+e-→h′χ1χ2), and the product of branching fractions B(χ2→χ1e+e-)×B(h′→x+x-), where x+x- indicates μ+μ-,π+π-, or K+K-, as functions of h′ mass and lifetime at the level of 10-1 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.
