Heavy baryon chiral perturbation theory (HBChPT) to leading order provides a kernel to solve the Bethe-Salpeter equation for the $P_{33}\hspace{0.5em}\left[\Delta \mathrm{}\right(1232)\mathrm{}$-channel] $\pi -N$ system, in the infinite nucleon mass limit. Crossed Born terms include, when iterated within the Bethe-Salpeter equation, both all one- and some two-pion intermediate states, hence preserving elastic unitarity below the two-pion production threshold. This suggests searching for a solution with the help of dispersion relations and suitable subtraction constants, when all in-elasticities are explicitly neglected. The solution allows for a successful description of the experimental phase shift from threshold up to $\sqrt{s}=1500\mathrm{}\hspace{0.5em}\mathrm{MeV}$ in terms of four subtraction constants. Next-to-leading order HBChPT calculations are also used to estimate the unknown subtraction constants which appear in the solution. Large discrepancies are encountered which can be traced to the slow convergence rate of HBChPT.

• Received 2 August 2000

DOI:https://doi.org/10.1103/PhysRevD.63.076001