Is the Pole Approximation to the Unphysical Part of the Left Hand Cut Sufficient ?

Abstract

In any application of dispersion relations to kaon-nucleon and nucleon-nucleon scattering the existence of an unphysical part of the left hand cut (LHC) is a challenge. If one is, for example, interested in predictions of real parts of scattering amplitudes or more generally if one uses dispersion relations as constraints in phase shift or amplitude analyses in order to resolve ambiguities, one has to assume some thing about the contributions from the unphysical part of the LHC which clearly weakens the power of the dispersion relations. On the other hand, if an amplitude, that is real as well as imaginary part, for a given t value and for a given energy range is known one may turn around the procedure and use the dispersion relation as a tool to obtain information on the discontinuity of the LHC. The easiest and often used method to exploit this idea is to approximate the discontinuity by a sum of poles and to determine their coupling constants and eventually also their masses. In nucleon-nucleon scattering this method has been applied for example by Bugg (l) and by Verwest et al. (2). With the advent of total cross-section data in pure spin states, however, it turned out by a detailed study performed by us (3) that this pole approximation is not sufficient. We found clear evidence for contributions from the 3π continuum. In the following I will report on that investigation. I will start with a description of the available information on the discrepancy functions and will proceed with a qualitative discussion of the physics of the LHC. Thereafter the discontinuity of the LHC will be discussed quantitatively. I will end up with some conclusions.