The Gell-Mann-Goldberger formula for long-range potential scattering

A derivation of the Gell-Mann-Goldberger (GG) formula and cut-off versions of this formula for the T-matrix involving long-range potentials is given. The derivation is based on the time-dependent and recently developed stationary formalisms for scattering via long-range potentials. A stationary S-operator expression for two-body Coulomb-like scattering is derived. Using the well-known expression for the off-energy-shell “T-matrix” for a pure Coulomb potential the energy-shell limit of this stationary expression is shown to yield the pure Coulomb scattering amplitude. A proof of the convergence of the perturbation series corresponding to the Gell-Mann-Goldberger formula for the two-body Coulomb-like T-matrix is given.