Soft pattern of Rutherford scattering from heavy target mass expansion

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.