Threshold Behavior of Partial-Wave Dispersion Relation and Short-Range Force

Abstract

In a paper on the existence of a ghost-free solution of the unsubtracted partial-wave despersion relation, Frye and Warnock conjectured that the positive-definite value at threshold of the unitarity integral on the physical region must be canceled by the short-range force (i.e., multiparticle-exchange contribution) arising from the first and second double spectral functions in order to guarantee the required $p$ wave and higher threshold behavior. We examined this conjecture explicitly in an unsubtracted dispersion relation for the $I=l=1$ pion-pion scattering amplitude. With certain choices of the $\ensuremath{\rho}$ Regge parameters, we show in our approximation scheme that this conjecture is nearly satisfied, and we can conclude that the so-called subtraction constant or threshold factor which is usually arbitrarily introduced to correct the threshold behavior could be expressed in terms of known physical quantities. However, with some other choice of $\ensuremath{\rho}$ Regge parameters, our numerical result shows that this conjecture is not satisfied by our one-$\ensuremath{\rho}$-Regge-pole approximation and indicates a need for additional Regge poles or Castillejo-Dalitz-Dyson poles.