Method of Pole Dominance and Applications

Abstract

A method for calculating scattering amplitudes at low to moderate energies in terms of their bound-state and resonant poles is presented. By comparing expressions obtained for $s$-wave scattering lengths with corresponding predictions of current algebra, the widths of the $\rho$, $K^{*}$, $\Delta$, and $\omega$ resonances are calculated in close agreement with experiment. Furthermore, without reference to current algebra, all the detailed features of $\pi N$ elastic scattering (such as the behavior of the $P_{11}$ phase shift) up to an energy of 400 MeV are obtained within an error of about 10 to 20%. The principal advantage of the present method over previous pole-dominance models is that the calculation of amplitudes has been reduced to the evaluation of a number of rapidly convergent integrals. In some cases, one or two undetermined subtraction constants must be introduced, but in the examples of $\pi K$ and $\pi N$ elastic scattering only one unknown constant arises, and it can be evaluated by means of the Adler self-consistency condition or an $M=1\mathrm{} O\left(4\right)\mathrm{}$ assignment for the pion.