Froissart bound on inelastic cross section without unknown constants

Assuming that axiomatic local field theory results hold for hadron scattering, André Martin and S. M. Roy recently obtained absolute bounds on the D wave below threshold for pion-pion scattering and thereby determined the scale of the logarithm in the Froissart bound on total cross sections in terms of pion mass only. Previously, Martin proved a rigorous upper bound on the inelastic cross-section σ_inel which is one-fourth of the corresponding upper bound on σ_tot, and Wu, Martin, Roy and Singh improved the bound by adding the constraint of a given σ_tot. Here we use unitarity and analyticity to determine, without any high-energy approximation, upper bounds on energy-averaged inelastic cross sections in terms of low-energy data in the crossed channel. These are Froissart-type bounds without any unknown coefficient or unknown scale factors and can be tested experimentally. Alternatively, their asymptotic forms, together with the Martin-Roy absolute bounds on pion-pion D waves below threshold, yield absolute bounds on energy-averaged inelastic cross sections. For example, for π⁰π⁰ scattering, defining σ_inel=σ_tot-(σ^{π0π0→π0π0}+σ^{π0π0→π+π-}) , we show that for c.m. energy √{s }→∞, σ_{¯ inel}(s ,∞)≡s ∫_s^∞d s^'σ_inel(s^')/s^'2≤(π /4 )(m_π)^-2[ln (s /s₁)+(1 /2 )ln ln (s /s₁)+1 ]² where 1 /s₁=34 π √{2 π }m_π^-2 . This bound is asymptotically one-fourth of the corresponding Martin-Roy bound on the total cross section, and the scale factor s₁ is one-fourth of the scale factor in the total cross section bound. The average over the interval (s,2s) of the inelastic π⁰π⁰ cross section has a bound of the same form with 1 /s₁ replaced by 1 /s₂=2 /s₁.