Fixed-Angle Dispersion Relations for Nucleon Compton Scattering. I

The Mandelstam representation is used to derive fixed-angle dispersion relations for the twelve scalar amplitudes describing the process $\gamma +N\to \gamma +N$. The electromagnetic interaction is calculated to order $e^2$. The strong interactions are estimated by including one- and two-pion exchange on the left-hand cut. These depend on the ${\pi }^0\to 2\gamma$ decay lifetime and on the $T=0\mathrm{}$ $\pi \pi$ phase shift and the total cross section for photoproduction of pions on pions, respectively. The right-hand cut is estimated by allowing the exchange of a nucleon and a pion-nucleon pair, which depend on the amplitude for meson photoproduction on nucleons. The low-energy limit theorem provides an important tool for estimating the subtractions required in the dispersion relations.