A Smooth Analytic Formula for the Yukawa SIDM Transfer Cross Section

We present a smooth, single-line analytic formula for the velocity-dependent transfer cross
section of self-interacting dark matter (SIDM) with a Yukawa mediator. The formula replaces
the widely used but discontinuous three-piece Tulin-Yu-Zurek (2013) parametrization with a
continuously differentiable expression that achieves 0.48% mean absolute error across 4.7 decades
in cross section. Both asymptotic limits, the constant scattering-length plateau at low velocity
and 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 of
the Born regime. Ground truth is computed from first principles via the Variable Phase Method
applied to the radial Schrödinger equation for 498 velocity values