Determination ofπNscattering lengths from pionic hydrogen and pionic deuterium data

The $\pi N$ s-wave scattering lengths have been inferred from a joint analysis of the pionic hydrogen and the pionic deuterium x-ray data using a nonrelativistic approach in which the $\pi N$ interaction is simulated by a short-ranged potential. This potential is assumed to be isospin invariant and its range, the same for isospin $I=3/2\mathrm{}$ and $I=1/2,$ is regarded as a free parameter. The proposed model admits an exact solution of the pionic hydrogen bound state problem, i.e., the $\pi N$ scattering lengths can be expressed analytically in terms of the range parameter and the shift $\left(\epsilon \right)$ and width $\left(\Gamma \right)$ of the $1s$ level of the pionic hydrogen. We demonstrate that for small shifts and short ranges from the exact expression, one retrieves the standard range independent Deser-Trueman formula. The $\pi d$ scattering length has been calculated exactly by solving the Faddeev equations and also by using a static approximation. It has been shown that the same very accurate static formula for $\pi d$ scattering length can be derived (i) from a set of boundary conditions; (ii) by a reduction of Faddeev equations; and (iii) through a summation of Feynman diagrams. By imposing the requirement that the $\pi d$ scattering length, resulting from the Faddeev-type calculation, be in agreement with pionic deuterium data, we obtain bounds on the $\pi N$ scattering lengths. The dominant source of uncertainty in the deduced values of the $\pi N$ scattering lengths are the experimental errors in the pionic hydrogen data.