We propose to use rich informations on πp total cross sections below N (∼ 10 GeV) in addition to high-energy data in order to discriminate whether these cross sections increase like log ν or (log ν) 2 at high energies, since it is difficult to discriminate between asymptotic log ν and (log ν) 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 σ (+) tot above 70 GeV in terms of highenergy parameters constrained by these two FESR. We can show from this analysis that the (log ν) 2 behaviours is preferred to the log ν behaviours.

We set up a general framework to describe ππ scattering below 1 GeV based on chiral low-energy expansion with possible spin-0 and 1 resonances. Partial wave amplitudes are obtained with the N/D method, which satisfy unitarity, analyticity and approximate crossing symmetry. Comparison with the phase shift data in the J = 0 channel favors a scalar resonance near the ρ mass.

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