A Smooth Analytic Formula for the Yukawa SIDM Transfer Cross Section

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