Abstract
We 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 log2 s behaviour, cannot be an optimal bound in QCD. We next compute the total cross sections for π + π −, π ± π 0 and π 0 π 0 scattering within the framework of the constituent chiral quark model (CχQM) in the limit of a large number of colours N c and discuss their asymptotic behaviours. The same ππ cross sections are also discussed within the general framework of Large-N c 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-N c counting. We finally propose a simple phenomenological model which matches the low energy behaviours of the \( \sigma_{{{\pi^{\pm }}{\pi^0}}}^{\mathrm{total}}(s) \) cross section predicted by the CχQM with the high energy behaviour predicted by the Large-N c Ansatz. The magnitude of these cross sections at very high energies is of the order of those observed for the pp and \( p\overline{p} \) scattering total cross sections.
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Greynat, D., de Rafael, E. & Vulvert, G. Asymptotic behaviour of pion-pion total cross-sections. J. High Energ. Phys. 2014, 107 (2014). https://doi.org/10.1007/JHEP03(2014)107
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DOI: https://doi.org/10.1007/JHEP03(2014)107