Self-Consistent Determination of the Position and Width of the 33πNResonance

Abstract

Using the procedure discussed in the preceding paper, we determine the approximate position ${M}^{*}$ and width $\ensuremath{\Gamma}$ of the 33 $\ensuremath{\pi}N$ resonance. The $\frac{N}{D}$ formalism is used to generate a unitary $I=J=\frac{3}{2} \ensuremath{\pi}N$ $p$-wave amplitude. The nucleon pole parameters are taken from experiment and the nearby left-hand-cut contribution is assumed to be accurately given by crossing symmetry with only the 33 physical-scattering contribution to the crossing relation retained. Distant left-hand singularities are accounted for by two Bal\'azs poles whose residues are determined by requiring the crossing relation for the real part of the amplitude to be satisfied on the nearby left-hand cut. The real part crossing relation is also assumed to be dominated by the 33 contribution. Self-consistency between the 33 amplitude determined by the $\frac{N}{D}$ calculation and that used to evaluate the crossing relations is obtained by an iterative procedure. We find ${M}^{*}=1.22$ and $\ensuremath{\Gamma}=0.11$ in units of the physical-nucleon mass, compared with the experimental values of ${M}^{*}=1.32$ and $\ensuremath{\Gamma}=0.11$, respectively.