# Finite second-order Born term for Coulomb wavepacket scattering

## TL;DR

This paper demonstrates that using wavepacket states in Coulomb scattering calculations yields finite second-order Born terms, reproduces Rutherford scattering, and suggests a potential approach to handle divergences in quantum field theories.

## Contribution

It introduces a wavepacket-based method to obtain finite higher-order Born terms in Coulomb scattering, addressing divergence issues in traditional calculations.

## Key findings

- Second-order contribution is finite and negligible compared to first-order.
- First-order amplitude in the forward direction is finite and physically reasonable.
- Method may help address divergences in quantum field theories.

## Abstract

It has been known for some time that, for nonrelativistic Coulomb scattering, the terms in the Born series of second and higher order diverge when using the standard method of calculation. In this paper we take the matrix elements between square-integrable wavepacket state vectors. We reproduce the Rutherford cross section from the first-order contribution. We find that the second-order contribution is finite and negligible compared to the first-order contribution, away from the forward direction. At first order, the contribution to the amplitude in the forward direction is found to be finite and physically reasonable. We comment on how a similar procedure applied to the divergences of quantum field theories might render them finite.