Precision calculation of threshold image scattering, πN scattering lengths, and the GMO sum rule

We use chiral perturbation theory (ChPT) to calculate the image scattering length with an accuracy of a few percent, including isospin-violating corrections in both the two- and three-body sectors. In particular, we provide the technical details of a recent letter (Baru et al., 2011) , where we used data on pionic deuterium and pionic hydrogen atoms to extract the isoscalar and isovector pion–nucleon scattering lengths image and image. We study isospin-breaking contributions to the three-body part of image due to mass differences, isospin violation in the πN 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 image 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.