Application of the Mandelstam Representation to Photoproduction of Pions from Nucleons

The Mandelstam representation is applied to the invariant amplitudes for photoproduction. By treating gauge invariance as a subsidiary condition, it is shown that the fixed-momentum-transfer dispersion relations of Chew, Goldberger, Low, and Nambu (CGLN) are probably valid without subtractions for the (-) amplitudes while a three-pion resonance would perhaps require a subtraction in the (+) amplitudes. The two-pion resonance will certainly require a subtraction for the (0) amplitudes, but to a good approximation the contribution of the two-pion intermediate state is found to produce a simple additive correction to the CGLN (0) formula. The strength of this new term is determined by a parameter $\Lambda$, which has been introduced elsewhere in treating the photon, three-pion problem. Otherwise, the form of the new term can be expressed in terms of nucleon electromagnetic form factors. Finally, the photoproduction amplitudes are calculated in the threshold region, and an estimate of the size of $\Lambda$ is made.