Possible test of asymptotic behaviours from sum rules

In 1987, the UA4 collaboration measured the ratio ρ(p p-bar) of real to imaginary part of the (pp) forward scattering amplitude at 546GeV centre-of-mass energy. Their result ρ(p p-bar) = 0.24±0.04 is surprisingly large. Normally one would expect ρ(p p-bar) to lie between 0.11 and 0.15 from the dispersion approach. The present report describes how to test the crossing-odd part of high energy total cross sections of several models (including the odderon) using sum rules from a new stand point. The P' sum rule was proposed to test the J-plane singularities with vacuum quantum number, and P'(f) Regge trajectory was discovered a long time ago. Here, this idea is followed to the crossing-odd amplitude but in a somewhat modern way. The crossing-odd (spin averaged) forward amplitude for pp and (p p-bar) scattering is defined and an unsubtracted dispersion relation of a form suitable for numerical calculation is derived. Theoretical calculations of ReF(-)(ω) at high energies are compared with experimental values. A study is made to check the validity of the unsubtracted dispersion relation which indeed is closely consistent with Regge asymptotics. This does not give such a large value for ρ(p p-bar) = 0.24±0.04 as measured by UA4 at 540GeV. Other asymptotic behaviours such as the odderon or a real constant or combinations are not ruled out completely but the strength of these terms has to be small. (Nogami, K.)