We present a smooth, single-line analytic formula for the velocity-dependent transfer crosssection of self-interacting dark matter (SIDM) with a Yukawa mediator. The formula replacesthe widely used but discontinuous three-piece Tulin-Yu-Zurek (2013) parametrization with acontinuously differentiable expression that achieves 0.48% mean absolute error across 4.7 decadesin cross section. Both asymptotic limits, the constant scattering-length plateau at low velocityand the Born power law σT ∝ v −4 at high velocity, are locked by construction. The double-softplus functional form directly encodes the quantum mechanical structure of the scattering:two distinct partial-wave transitions corresponding to the l=0 → l=1 activation and the onset ofthe Born regime. Ground truth is computed from first principles via the Variable Phase Methodapplied to the radial Schrödinger equation for 498 velocity values
Daniel James Arthur Clark (2026) studied this question.