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

Abstract

We use chiral perturbation theory (ChPT) to calculate the $\ddot{I}{€}^{-}d$ 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 $a_{\ddot{I}{€}^{-}d}$ due to mass differences, isospin violation in the $\ddot{I}€N$ 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 $a_{\ddot{I}{€}^{-}d}$ 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.

46 pages, 7 figures. Discussion of several points expanded, references added, version published in Nuclear Physics A