The partial wave amplitude for high-energy pion-pion scattering (II)

Abstract

The previously formulated approach to the problem of determining the high-energy behaviour of the pion-pion partial wave amplitudes is carried one step further. The approach is based on a bootstrap treatment of the inelastic contributions to elastic scattering, and the asymptotic boundary condition is the diffraction picture. Using this boundary condition the integral equation for the inelasticity R l I ( ν) is approximated in such a way that it involves R l I ( ν) in the high-energy region only. It is argued that the real part of the partial wave amplitude can be calculated in terms of low-energy parameters (in a first approximation), and using the expression obtained in this way for Re A l I ( ν) the integral equation for R l I ( ν) is solved asymptotically. It turns out that R l I ( ν) behaves logarithmic in the asymptotic region, and subtractions are therefore unnecessary in the partial wave dispersion relattions. It seems as if the approach involves forces of a central character. The equations determining the high-energy parameters by a low-energy boostrap requirement are given. In a crude version of the present approach there exists the possibility that all other resonances can be generated from the lowest S-wave resonance. The result obtained for A l I ( ν) in the high-energy region is compared with what one obtains from the Regge hypothesis; if there is agreement, the present approach gives the same result as is obtained by taking into account only the vacuum trajectory.