We derive a sum rule which shows that the Froissart-Martin bound for the asymptotic
behaviour of the
total cross sections at high energies, if modulated by the Lukaszuk-Martin coefficient of the
leading
behaviour, cannot be an optimal bound in QCD. We next compute the total cross sections
for ,
and
scattering within the framework of the constituent chiral quark model
(CQM) in the limit of a
large number of colours
and discuss their asymptotic behaviours. The same
cross sections are also discussed within the general framework of
Large- QCD
and we show that it is possible to make an Ansatz for the isospin
and
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- counting.
We finally propose a simple phenomenological model which matches the low energy behaviours of
the cross section
predicted by the CQM
with the high energy behaviour predicted by the
Large- Ansatz.
The magnitude of these cross sections at very high energies is of the order of those observed
for the
and
scattering total cross sections.