Renormalization in the New Tamm-Dancoff Theory of Meson-Nucleon Scattering

Abstract

The new Tamm-Dancoff equations for meson-nucleon scattering are set up in the lowest approximation and it is shown how explicit nonphysical singularities may be avoided in these equations. The particle self-energies appearing in the integral equation are renormalized, but the resulting modified propagator for the system then has a nonphysical singularity. For the states $T=j=\frac{1}{2}$, the vertex and self-energy expressions generated by the uncrossed graph are considered. The renormalized vertex may be constructed by the successive solution of two one-dimensional integral equations, the finite part of the self-energy then being obtained by quadratures. Vertex renormalization is uncertain to a constant factor in the $S_{\frac{1}{2}}$ state, and the $S_{\frac{1}{2}}$ theory therefore depends on two parameters. No numerical results are obtained, owing to a number of difficulties found in this theory—a comparison is made between these difficulties and those of the corresponding Bethe-Salpeter equation.