Investigations of the pi N total cross sections at high energies using new FESR: log nu or (log nu)^2

We propose to use rich informations on pi p total cross sections below N= 10 GeV in addition to high-energy data in order to discriminate whether these cross sections increase like log nu or (log nu)^2 at high energies, since it is difficult to discriminate between asymptotic log nu and (log nu)^2 fits from high-energy data alone. A finite-energy sum rule (FESR) which is derived in the spirit of the P’ sum rule as well as the n=1 moment FESR have been required to constrain the high-energy parameters. We then searched for the best fit of pi p total cross sections above 70 GeV in terms of high-energy parameters constrained by these two FESR. We can show from this analysis that the (log nu)^2 behaviours is preferred to the log nu behaviours.