Sommerfeld enhancement: general results from field theory diagrams

TL;DR

This paper derives a general field theory framework for Sommerfeld enhancement in particle annihilation, showing that the total amplitude can be expressed as a convolution involving the Schrödinger equation solution and an arbitrary bare amplitude, with explicit results for massless vector potentials.

Contribution

It introduces a general method to compute Sommerfeld enhancement from field theory diagrams, accommodating arbitrary bare amplitudes and providing explicit formulas for massless vector potentials.

Findings

The annihilation amplitude equals a convolution of Schrödinger solution and bare amplitude.

Explicit Sommerfeld enhancement formulas for massless vector potentials.

Enhancement factors depend on the partial wave and velocity, e.g., P wave enhancement is 2pi(alpha/v)^3.

Abstract

Assuming that two incoming annihilating particles interact by a generally massive attractive vector potential,we find, by taking the non-relativistic limit of the field theory ladder diagrams, that the complete annihilation amplitude A is equal to: the convolution of a solution of the Schroedinger equation (including the vector potential) with the Fourier transform of the bare (i.e. ignoring the attraction) annihilation amplitude A0. The main novelty is that A0 is completely arbitrary. In particular for a massless vector potential we find for the l-partial-wave cross-section the Sommerfeld enhancement 2pi/(l!)^2 (alpha/ v)^{2l+1} (v relative velocity), e.g. for the P wave the enhancement 2pi(alpha/ v)^3.