Soft pattern of Rutherford scattering from heavy target mass expansion

TL;DR

This paper analyzes the soft behavior of Rutherford scattering involving heavy targets and light projectiles, revealing universal patterns and spin-dependent effects through $1/M$ and velocity expansions, using effective field theory methods.

Contribution

It provides a detailed analysis of the universal and spin-dependent features of Rutherford scattering in the soft limit, including higher-order corrections and effective field theory approaches.

Findings

Unpolarized cross sections exhibit universal forms at leading orders in $1/M$ expansion.

Universality partially breaks down at higher orders in velocity and mass expansion.

Effective field theory reproduces the soft behavior for composite Dirac fermion targets.

Abstract

We investigate the soft behavior of the tree-level Rutherford scattering process. We consider two types of Rutherford scattering, a low-energy massless point-like projectile (say, a spin-${1\over 2}$ or spin-$0$ electron) to hit a static massive composite target particle carrying various spins (up to spin-$2$), and a slowly-moving light projectile hits a heavy static composite target. For the first type, the unpolarized cross sections in the laboratory frame are found to exhibit universal forms in the first two orders of $1/M$ expansion, yet differ at the next-to-next-to-leading order (though some terms at this order still remain to be universal or depend on the target spin in a definite manner). For the second type, at the lowest order in electron velocity expansion, through all orders in $1/M$, the unpolarized cross section is universal (also not sensitive to the projectile spin). The universality partially breaks down at relative order-$v^2/M^2$, though some terms at this order are still universal or depend on the target spin in a specific manner. We also employ the effective field theory approach to reproduce the soft behavior of the differential cross sections for the target particle being a composite Dirac fermion.