Shape of theN*(1236)Resonance

Abstract

Corrections to the Breit-Wigner shape of the ${N}^{*}(1236)$ resonance are calculated using analyticity and inelastic unitarity incorporating the $N\ensuremath{\pi}$ and ${N}^{*}\ensuremath{\pi}$ channels in a propagator formalism. This method, which includes only a right-hand cut and which evaluates the effect of bubble insertions in the propagator, is motivated by the rigorous results which have been proved for the nucleon. It is argued that even though background and left-hand cuts have been neglected, it is the inclusion of inelasticity that enables the ${P}_{33}$ phase-shift data to be reproduced, even at energies well above resonance. The structureless-vertex decay model with a Breit-Wigner propagator gives an ${N}^{*}$ shape too asymmetric, and inclusion of an inelastic channel with an analytic propagator serves to correct this. Assuming ${N}^{*}\ensuremath{\pi}$ as the only inelastic channel, the ${N}^{*++}{N}^{*++}{\ensuremath{\pi}}^{0}$ coupling is estimated as 170\ifmmode\pm\else\textpm\fi{}50, where the unknown behavior of the vertices far off the mass shell causes the uncertainty. This estimate can be compared with about 75 from relativistic $\mathrm{SU}(6)$, and 136 using Adler-Weissberger techniques. The ${P}_{33}$ partial-wave amplitude constructed on this model has a left-hand pole which simulates the effect of the neglected nucleon-exchange short cut, and which tends to lie too far left and with too large a residue. The application of the method to other resonances and bound states is discussed.