Precision calculation of threshold pi^- d scattering, pi N scattering lengths, and the GMO sum rule

TL;DR

This paper employs chiral perturbation theory to accurately compute the $\pi^- d$ scattering length, incorporating isospin-violating effects, and assesses the GMO sum rule's validity amid isospin breaking, leading to refined pion-nucleon interaction parameters.

Contribution

It provides a detailed calculation of the $\pi^- d$ scattering length with isospin violation effects and tests the GMO sum rule under these conditions, improving understanding of pion-nucleon interactions.

Findings

Calculated $\pi^- d$ scattering length with few percent accuracy.

Identified cancellation of leading virtual-photon effects, validating standard ChPT estimates.

Used sum rule to determine the charged-pion-nucleon coupling constant.

Abstract

We use chiral perturbation theory (ChPT) to calculate the $\pi^- d$ scattering length with an accuracy of a few percent, including isospin-violating corrections both in the two- and three-body sector. In particular, we provide the technical details of a recent letter, where we used data on pionic deuterium and pionic hydrogen atoms to extract the isoscalar and isovector pion-nucleon scattering lengths $a^+$ and $a^-$. We study isospin-breaking contributions to the three-body part of $a_{\pi^-d}$ due to mass differences, isospin violation in the $\pi N$ scattering lengths, and virtual photons. This last class of effects is ostensibly infrared enhanced due to the smallness of the deuteron binding energy. However, we show that the leading virtual-photon effects that might undergo such enhancement cancel, and hence the standard ChPT counting provides a reliable estimate of isospin violation in $a_{\pi^- d}$ due to virtual photons. Finally, we discuss the validity of the Goldberger-Miyazawa-Oehme sum rule in the presence of isospin violation, and use it to determine the charged-pion-nucleon coupling constant.