Many particle physics models for dark matter self-interactions - motivated to address long-standing challenges to the collisionless cold dark matter paradigm - fall within the semi-classical regime, with interaction potentials that are long-range compared to the de Broglie wavelength for dark matter particles. In this work, we present a quantum mechanical derivation and new analytic formulas for the semi-classical momentum transfer and viscosity cross sections for self-interactions mediated by a Yukawa potential. Our results include the leading quantum corrections beyond the classical limit and allow for both distinguishable and identical dark matter particles. Our formulas supersede the well-known formulas for the momentum transfer cross section obtained from the classical scattering problem, which are often used in phenomenological studies of self-interacting dark matter. Together with previous approximation formulas for the cross section in the quantum regime, our new results allow for nearly complete analytic coverage of the parameter space for self-interactions with a Yukawa potential. We also discuss the phenomenological implications of our results and provide a new velocity-averaging procedure for constraining velocity-dependent self-interactions. Our results have been implemented in the newly released code CLASSICS.
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