Classical particle scattering for power-law two-body potentials

TL;DR

This paper rigorously analyzes classical particle scattering for power-law potentials, deriving deflection angles in different regimes, highlighting differences between attractive and repulsive interactions, and examining effects of potential regularization.

Contribution

It provides a comprehensive derivation of scattering deflections for power-law potentials, including asymptotic behaviors and effects of regularization, with exact solutions for validation.

Findings

Deflection angle series for large impact factors

Asymptotic expressions for small impact factors

Impact of potential regularization on scattering behavior

Abstract

We present a rigorous study of the classical scattering for anytwo-body inter-particle potential of the form $v(r)=g/r^\gamma$, with$\gamma\textgreater{}0$, for repulsive ($g\textgreater{}0$) and attractive ($g\textless{}0$)interactions. We give a derivation of the complete power series of thedeflection angle in terms of the impact factor for the weak scatteringregime (large impact factors) as well as the asymptotic expressionsfor the hard scattering regime (small impact factors). We see a verydifferent qualitative and quantitative behavior depending whether theinteraction is repulsive or attractive. In the latter case, thefamilies of trajectories depend also strongly on the value of$\gamma$. We also study carefully the modifications of the resultswhen a regularization is introduced in the potential at small scales.We check and illustrate all the results with the exact integration ofthe equations of motion.