Scattering of non-relativistic finite-size particles and puffy dark matter direct detection

TL;DR

This work addresses non-relativistic scattering between finite-size particles and its implications for direct detection of puffy dark matter. It develops a first-principles framework by constructing a modified Yukawa potential through volume integration of spherical density profiles and solving the Schrödinger equation with a partial-wave expansion to obtain elastic and momentum-transfer cross sections. Key findings include the regularization of the short-range potential, size-driven extensions of the Born regime, and the emergence of resonant and classical scattering regimes when particle sizes differ, as well as bounds on nugget-type dark matter from bound-state stability. The results refine predictions for DM–nucleus interactions in direct-detection experiments and illuminate how target size and DM substructure influence observable cross sections, with practical impact on exclusion limits and model viability for both standard and nugget DM scenarios.

Abstract

In this work we consider the scattering between non-relativistic particles with different finite sizes. We first calculate their interaction potential and apply the partial wave method to obtain their scattering cross section. Our findings show that the particle size can significantly affect the scattering between non-relativistic particles. Then we apply such a study to direct detection of puffy dark matter. We find that the finite size of the target nucleus may introduce non-perturbative effects that differ from the scenario of point-like dark matter. For large-size dark matter particles, this non-perturbative regime in the dark matter nucleus scattering cross section effectively disappears; while for small values of the size-to-range ratio in the scattering process, a significant non-perturbative regime can maintain. Finally, for the direct detection of nugget-type puffy dark matter with a small number of constituent particles, we find that the stability conditions for the formation of bound-state dark matter can provide constraints on the dark matter nucleus scattering cross section.

Figures (8)

• Figure 1: The interaction potential between two particles: the red line represents the Yukawa potential between two point particles, the blue lines (for $r<R_{\rm N}$) and the cyan-green lines (for $r>R_{\rm N}$) represent the modified Yukawa potential between a point-like dark matter particle and a finite-size target nucleus with a radius $R_N$.
• Figure 2: Interaction potential between particles: the red line represents the Yukawa potential between two point particles, the blue lines (for $r<2 R_{\chi}$) and the cyan-green lines (for $r>2R_{\chi}$) represent the modified Yukawa potential between a puffy dark matter particle and a finite-size target nucleus for different $R_{\chi}$ values.
• Figure 3: The parameter space of $y$ versus $b$ for different values of $bf(y)$, showing the Born and nonperturbative regimes separated by the black dashed line for point-like particles or the solid red curve for the finite-size particles.
• Figure 4: Similar as Fig.\ref{['fig3']}, but for a finite-size dark matter particle scattering off a finite-size nucleon with different $R_{\chi}/R_N=n$ values.
• Figure 5: The parameter space of $a$ versus $b$ for point dark matter particle scattering off a proton with different $R_{\rm N}m_{\phi}$ values, showing the values of $\sigma_{\rm point-N}k^2/(4\pi)$.