The propagation of a massive spin-$\frac{3}{2}$ field, with application to $\pi N$ scattering

In the present paper, we investigate the propagation of a massive spin-$\frac{3}{2}$ field, aiming at a direct application in hadronic models of the pion-nucleon ($\pi N$) interaction. Suitable expressions for the contributions to the standard invariant amplitudes $A$ and $B$ are derived, applicable in the general case of isospin decomposition. We first deal with the details of the lengthy calculation involving the Rarita-Schwinger propagator and confirm the validity of the expressions which had appeared in the literature in the early 1970s. We subsequently derive the corresponding contributions when following two other approaches, one featuring the Williams propagator, the other being known as Pascalutsa's method.