Approximate Solution of Partial-Wave Dispersion Relations

Abstract

An exact expressed is derived for the solution of a partial-wave dispersion relation. This expression has the property that, when the zero-width-resonance approximation is made in all integrals where its use is valid, the solution of the dispersion relation is given by simple integrals and no further approximation is required. The result is shown to be independent of the subtraction point and symmetric when applied to a many-channel dispersion relation. Although it utilizes the inverse amplitude, the approximation can be used in some cases of zeros of the amplitude. As a sample calculation, the approximation is applied to the simple $\pi -\pi -\rho$ bootstrap and a self-consistent solution is obtained.