Application of Dispersion Relations to the Photodisintegration of the Deuteron

The calculation of the matrix element of the process $\gamma +d\to n+p$ by dispersion techniques is considered. There are twelve invariant amplitudes; the covariant form of the transition amplitude is related to the noncovariant (Pauli matrix) form, and we further relate this to partial wave amplitudes, keeping, however, only the dipole amplitudes. The Born terms of the dipole amplitudes are derived, and the dispersion relations for the dipole amplitudes are written down and solved in a low-energy approximation in which the $n-p$ final-state rescattering is taken into account, but no other higher-order effects. In an Appendix these calculations are performed directly in the nonrelativistic limit to illustrate the essential simplicity of the technique. No light is shed on the well-known discrepancy between theory and experiment for the threshold $M1\mathrm{}$ amplitude; the nearest (anomalous) singularities, at least, will have to be included in order for the dispersion calculation to be sufficiently accurate. But we remark that the form of the amplitude implies a correlation between the threshold value of the amplitude and its energy dependence, a correlation that would be interesting to check experimentally.