Threshold and asymptotic behavior of the N D equations

Abstract

Two important problems involved in obtaining solutions of partial wave dispersion relations (by the N D method) are having (i) the correct threshold behavior, and (ii) an acceptable high energy behavior. Various physical and numerical approximations have been made to insure (i) and (ii). We numerically investigate the sensitivity of the solutions of the N D equations to these approximations. For this purpose, we consider J = 1 π-π scattering, employing elastic unitarity and assuming that the left hand cut is dominated by the exchange of the ρ{variant} resonance. Two significant features we find are: (a) The values of the cutoffs needed to product a resonance are quite sensitive to the input "strength" of the left hand cut, e.g., a change of the input width of the ρ{variant} by a factor of two changed the value for a "straight cutoff" to produce a resonance at a given energy by a factor of ten. Due to the results of (a) we wish to emphasize the possible danger in employing a single cutoff in the calculations of SU3 multiplets. (b) If one introduces a pole on the left hand cut in order to insure the threshold behavior (i), then the ranges in values for the cutoffs (to insure (ii)) for which any resonance occurs are extremely narrow. On the other hand, a solution in which the phase shift does not become large is insensitive to the position of this pole. © 1965