Reaction Amplitudes for Pseudoscalar Mesons on Spin-½ Fermions in Fourth-Order Perturbation Theory

The invariant reaction amplitudes for pseudoscalar mesons on spin-½ fermions are calculated in fourthorder perturbation theory. The masses of the incoming, outgoing, and internal lines of all the diagrams are arbitrary, subject to some tacit restrictions about anomalous thresholds. The amplitudes for the self-energy, vertex, and box graphs are calculated by dispersion-relation techniques. Formulas for absorptive parts and double spectral functions are explicitly given, and the self-energy and vertex graphs have been renormalized. The $A$ amplitude for the box diagram is expressed as a single-variable dispersion relation plus a once-subtracted double dispersion relation. Numerical as well as analytic methods were used to check the expressions obtained.