Bootstrap Studies of theρMeson and the "Generalized Potential" of the Modified Cheng Representation

Abstract

A bootstrap calculation for the width of the $\rho$ resonance is performed using a simplified version of a Reggeized bootstrap theory proposed recently by the authors. A phenomenological Pomeranchuk input trajectory has been assumed. The $I=2\mathrm{}$ channel is eliminated from the appropriate crossing relation and no statements about this channel are necessary. The $2-\pi$ continuum states are assumed to be dominated by the $\rho$ for the $I=1\mathrm{}$ and by the $f^0$ for the $I=0\mathrm{}$. The input $\rho$ trajectory is parametrized to produce the $\rho$ resonance at the observed mass. The $\rho$ width is then determined by the maximum satisfaction of the crossing relations. The calculation yields a "best" width of 125 MeV for the $\rho$. The problem concerning the simultaneous bootstrap of the $\rho$ mass and width is briefly discussed, and a systematic procedure for obtaining the "generalized potential" of the modified Cheng representation is given.