Numerical calculation of 3-pion Coulomb scattering using scalar QED

TL;DR

This paper demonstrates that three-pion Coulomb scattering can be perturbatively calculated using scalar QED at high energies, revealing the accuracy of approximations and the need for higher-order terms at lower momenta.

Contribution

It provides the first perturbative calculation of three-pion Coulomb interactions in scalar QED and compares it to existing approximations across different momentum ranges.

Findings

At high momenta, the $O( ext{alpha}^2)$ calculation matches the Riverside approximation.

At lower momenta, higher-order terms are necessary for accurate description.

Two-pion correlations are well-described by lower-order calculations, unlike three-pion cases.

Abstract

The Coulomb scattering between three charged pions is shown to be perturbatively calculable using scalar QED for the relative momenta accessible to high-energy experiments of particle collisions. For the triplet relative momenta, $Q_3 \gtrsim 50$ MeV/$c$, the calculated three-pion correlation function at $O(\alpha^2)$ is very similar to the commonly-used triple-product of pair factors -- Riverside approximation. At lower $Q_3$, but above the lowest existing experimental measurement of $\sim15$ MeV/$c$, terms of $O(\alpha^3)$ or higher are needed to obtain a convincing description of three-pion Coulomb interactions. This is in contrast to two-pion correlations, for which the known exact solution is shown to be saturated by the lower orders at comparable pair relative momenta.