Effect of a Castillejo-Dalitz-Dyson Pole in the $\pi N$ $I=\frac{1}{2}$, $J={\frac{1}{2}}^{+}$ Amplitude

Recent phase-shift analyses of the $I=\frac{1}{2}$, $J={\frac{1}{2}}^{+}$ $\pi N$ amplitude imply that the real part of the phase shift changes sign at about 150-MeV pion lab kinetic energy and either passes through 90° or reaches a maximum slightly below 90° at about 600 MeV. These features of the phase shift strongly contradict the predictions of dispersion-theory calculations of the nucleon mass, which generally yield a negative phase shift decreasing with increasing energy. We assume that the observed phase-shift behavior is due to multichannel-coupling effects which can be accounted for in a single-channel calculation only by including a Castillejo-Dalitz-Dyson pole. A previous determination of the nucleon mass, $m$, and coupling with pions, $\frac{g^2}{4\pi }$, based on the assumed knowledge of the $\pi N$ $I=\frac{3}{2}$, $J={\frac{3}{2}}^{+}$ ($N^{*}$) resonance parameters, is repeated with the inclusion of a CDD pole. A variant of the Balázs method is used. The CDD pole parameters are determined so as to give the phase-shift behavior mentioned above. The Balázs-pole residues are determined by requiring crossing symmetry in a form relating the $\pi N$ amplitude on the nearby left-hand cut to physical $\pi N$ scattering. We find $m=0.97\mathrm{}$ in units of the physical nucleon mass; $\frac{g^2}{4\pi }=10.5\mathrm{}$; and a $\pi N$ $I=\frac{1}{2}$, $J={\frac{1}{2}}^{+}$ scattering length between about -0.038 and -0.048 (in units of the cube of the pion Compton wavelength), depending on the assumed energy at which the phase shift changes sign. These parameters appear to be very insensitive to the choice of Balázs-pole positions.