We study self-interacting dark matter (SIDM) scenarios, where the s-wave self-scattering cross section almost saturates the Unitarity bound. Such self-scattering cross sections are featured by strong velocity dependence in a wide range of velocities. They may be indicated by observations of dark matter halos in a wide range of masses, from Milky Way's dwarf spheroidal galaxies to galaxy clusters. We pin down the model parameters that saturate the Unitarity bound in a composite asymmetric dark matter scenario. We discuss implications and predictions of such model parameters for dark-matter direct-detection experiments and particle phenomenology like the beam-dump experiments. If we have a time, I may also comment on another motivative SIDM model, a gauged Lμ - Lτ model, and its implications. This talk is based on 2007.15522.
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