The S-Wave Pion-Nucleon Scattering Lengths from Pionic Atoms using Effective Field Theory

The pion–deuteron scattering length is computed to next-to-next-to-leading order in baryon chiral perturbation theory. A modified power-counting is then formulated which properly accounts for infrared enhancements engendered by the large size of the deuteron, as compared to the pion Compton wavelength. We use the precise experimental value of the real part of the pion–deuteron scattering length determined from the decay of pionic deuterium, together with constraints on pion–nucleon scattering lengths from the decay of pionic hydrogen, to extract the isovector and isoscalar S-wave pion–nucleon scattering lengths, a− and a+, respectively. We find a−=[0.0936(±0.0011)]Mπ−1 and a+=[−0.0029(±0.0009)]Mπ−1.