Asymptotic Behaviour of Pion--Pion Total Cross-Sections

21 pages, 5 figuresInternational audienceWe derive a sum rule which shows that the Froissart-Martin bound for the asymptotic behaviour of the ππ total cross sections at high energies, if modulated by the Lukaszuk-Martin coefficient of the leading log2s behaviour, cannot be an optimal bound in QCD. We next compute the total cross sections for π+π−, π±π0 and π0π0 scattering within the framework of the constituent chiral quark model (C$\chi$QM) in the limit of a large number of colours $\Nc$ and discuss their asymptotic behaviours. The same ππ 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 $\sigma_{\pi^{\pm}\pi^0}^{\rm total}(s)$ cross section predicted by the C$\chi$QM 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 $p{\bar p}$ scattering total cross sections