All orders factorization and the Coulomb problem

TL;DR

This paper demonstrates a comprehensive factorization approach for Coulomb corrections in nuclear beta decay at large Z or low velocities, providing all-order calculations and clarifying theoretical relationships.

Contribution

It introduces a constructive proof of all-orders factorization for Coulomb corrections and computes the relevant hard and soft functions, linking effective field theory with traditional Fermi function treatments.

Findings

All-orders factorization formula derived for Coulomb corrections.

Explicit computation of hard and soft functions in the factorization.

Clarification of the connection between EFT amplitudes and Fermi function.

Abstract

In the limit of large nuclear charge, $Z\gg 1$, or small lepton velocity, $\beta \ll 1$, Coulomb corrections to nuclear beta decay and related processes are enhanced as $Z\alpha/\beta$ and become large or even non-perturbative (with $\alpha$ the QED fine structure constant). We provide a constructive demonstration of factorization to all orders in perturbation theory for these processes and compute the all-orders hard and soft functions appearing in the factorization formula. We clarify the relationship between effective field theory amplitudes and historical treatments of beta decay in terms of a Fermi function.