Coupling Constants and Unsubtracted Dispersion Relations

Abstract

It is argued that the common occurrence of low-energy resonances and the fact that the cross sections for strongly interacting particles seem to approach nonzero high-energy limits may be related to a physical principle which determines the values of the strong-interaction coupling constants. It is conjectured that, in some cases, these constants may be determined approximately by neglecting inelastic processes and requiring that certain of the low-angular-momentum phase shifts approach $\pm \frac{\pi }{2}$ at high energies. Such a calculation may be made by using unsubtracted dispersion relations for the inverse partial-wave amplitudes. This prescription is illustrated in a few simple models. Our present knowledge of the various pion-nucleon forces is insufficient for a realistic calculation of the pion-nucleon coupling constant $f^2$. However, a simple calculation involving drastic approximations predicts a value of $f^2$ close to the experimental value and a low-energy resonance in the (3,3) state. The relationship of our prescription to that of Albright and McGlinn is discussed briefly.