No-PoleS-Matrix Fit to theΔ(1236)Resonance

Abstract

Recently measurements of $\ensuremath{\pi}\ensuremath{-}N$ 3-3 phase shifts have been made to an accuracy of about 0.1\ifmmode^\circ\else\textdegree\fi{}. We show that the 14 experimental points can be fitted with a 4-parameter no-pole $S$ matrix (having all the usually accepted analytic properties) to a precision of 1\ifmmode^\circ\else\textdegree\fi{}. Whereas this is unacceptable as a fit to the precisely determined ${\ensuremath{\Delta}}^{++}$, it is a better accuracy than that for which all the other elementary particle resonances are known. We conclude that one should associate a resonance with a Breit-Wigner form (implying an $S$-matrix pole) because of the physics in the problem, such as a symmetry model, and not because of the ease in fitting certain experimental data. Thus we stress the need for high-accuracy data to help settle the problem of resonances (such as the ${Z}^{*}$'s) whose dynamic origin might be questionable.