Sommerfeld factor for arbitrary partial wave processes

TL;DR

This work addresses non-perturbative Sommerfeld enhancements for arbitrary partial waves in non-relativistic two-body scattering. It derives a general expression for the Sommerfeld factor $S_l$ by mapping the Bethe-Salpeter equation to an effective Schrödinger equation with a Yukawa-like potential, thereby relating non-perturbative and perturbative cross sections. An approximate analytic expression for the Sommerfeld factor with Yukawa interactions across arbitrary angular momentum $l$ is presented, exact in the Coulomb limit and validated against numerical results for $l=0$ and $l=1$, demonstrating accuracy at the 5–10% level in many regimes. The findings show that higher partial waves can dominate in certain parameter regions and have potential implications for dark matter relic abundance and indirect detection signals, while acknowledging limitations near resonances where numerical methods remain essential.

Abstract

The Sommerfeld factor for arbitrary partial wave processes is derived in the non-relativistic limit. The s-wave and p-wave numerical results are presented for the case of Yukawa interactions. An approximate analytic expression is also found for the Sommerfeld factor of Yukawa interactions with arbitrary partial waves, which is exact in the Coulomb limit. It is demonstrated that this result is accurate to within 10% for some common scenarios. The non s-wave Sommerfeld effect is determined to be significant, and can allow higher partial waves to dominate cross sections.

Figures (4)

• Figure 1: Bethe-Salpeter equation in diagrammatic form.
• Figure 2: Diagram with non-perturbative scattering before annihilation
• Figure 3: Sommerfeld Factor for a Yukawa interaction of a s-wave state where the contours not labelled vary by a factor of 10. The solid lines represent $S_0^{}$ with the wavefunction determined from numerical simulations, and the red dashed lines are the approximate analytic result, $\tilde{S_0^{}}$. These contours are almost coincident except around the resonances where there is a noticeable shift. For clarity the contours of $\tilde{S_0^{}} \geq 10^3$ in (a) are not shown.
• Figure 4: Sommerfeld Factor for a Yukawa interaction of a p-wave state where the contours not labelled vary by a factor of 10. The solid lines represent $S_1^{}$ with the wavefunction determined from numerical simulations, and the red dashed lines show the approximate analytic result, $\tilde{S_1^{}}$. The contours are almost coincident in the region $y/x \gtrsim 1$ but have a mismatch in the region $y/x \lesssim 1$. For clarity the contours of $\tilde{S_1^{}} \geq 10^4$ in (a) are not shown.