Off-shell contributions to the energy shift in the π-d atom

The validity of the usual zero-energy approximation relating to the energy shift and the scattering length in pionic atoms is examined for the case of the pion-deuteron system. A form of three-body perturbation theory is used to derive an expansion for the energy shift directly in terms of the off-shell pion-nucleon amplitudes. With the help of separable models to determine the off-shell behaviour of the pion-nucleon amplitudes in the S11 and S13 channels, results of numerical calculations of the energy shift are compared with the zero-energy approximation. The errors caused by firstly neglecting the momentum spread in the atomic wavefunction of the pion, and secondly by the off-shell variation of the pion-nucleon t-matrices are found to be significant. (Auth.)