Asymptotic behaviour of pion-pion total cross-sections

Abstract: We derive a sum rule which shows that the Froissart-Martin bound for the asymptotic behaviour of the pipi total cross sections at high energies, if modulated by the Lukaszuk-Martin coefficient of the leading log2 s behaviour, cannot be an optimal bound in QCD. We next compute the total cross sections for pi+pi−, pi±pi0 and pi0pi0 scattering within the framework of the constituent chiral quark model (CχQM) in the limit of a large number of colours Nc and discuss their asymptotic behaviours. The same pipi cross sections are also discussed within the general framework of Large-Nc QCD and we show that it is possible to make an Ansatz for the isospin I = 1 and I = 0 spectrum which satisfy the Froissart-Martin bound with coefficients which, contrary to the Lukaszuk-Martin coefficient, are not singular in the chiral limit and have the correct Large-Nc counting. We finally propose a simple phenomenological model which matches the low energy behaviours of the σtotalpi±pi0(s) cross section predicted by the CχQM with the high energy behaviour predicted by the Large-Nc Ansatz. The magnitude of these cross sections at very high energies is of the order of those observed for the pp and pp ̄ scattering total cross sections