Off-shell Mandelstam representation for the second Born term of a Yukawa potential

The well-known result for the on-shell Mandelstam representation of the nonrelativistic second Born term of a Yukawa potential is extended to the general off-shell case for real-valued momentum and positive energies. The resulting representation is shown to differ from the on-shell case in two aspects. The first is a simple extension of the spectral function and corresponding Mandelstam boundary to the off-shell case which includes two additional parameters. These parameters simply reflect the extension to the case of arbitrary magnitudes for the initial- and final-state momentum off-shell. The second aspect is a change of sign of the principal-value (real) part of the representation off-shell which is not reflected by the imaginary part alone. This sign remains the same as that of the on-shell amplitude for all values of off-shell momentum only at sufficiently low energies. The critical scattering energy, above which the off-shell results differ in sign in a low-momentum region from those of the on-shell case, is determined by the mass of the exchanged particle. For the case of pion-exchange between nucleons, this energy corresponds to about 40 MeV in the laboratory. Potential implications for the calculation of two-pion--exchange contributions to nuclear potentials are discussed