Determination of Pion-NucleonS-Wave Scattering Lengths by theNDMethod

The pion-nucleon $S$-wave scattering lengths are calculated using the method originally developed by Balázs, wherein an effective-range two-pole approximation is made for the numerator function. The residues of the effective-range poles are determined by matching the amplitude and its derivative with those calculated with a fixed energy dispersion relation. In calculating the latter the contribution of only the $N^{*}$ and $\rho$, together with those of appropriate nucleon-pole terms, are retained. The calculated scattering length in the $T=\frac{3}{2}$ state is in excellent agreement with experimental result, while for the $T=\frac{1}{2}$ state, the calculated value, though of the right sign, is about twice the experimental scattering length.