Froissart bound on inelastic cross section without unknown constants

Abstract

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 onefourth 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 highenergy approximation, upper bounds on energy-averaged inelastic cross sections in terms of lowenergy 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 π 0 π 0 scattering, definingπ . This bound is asymptotically one-fourth of the corresponding Martin-Roy bound on the total cross section, and the scale factor s 1 is one-fourth of the scale factor in the total cross section bound. The average over the interval (s,2s) of the inelastic π 0 π 0 cross section has a bound of the same form with 1=s 1 replaced by 1=s 2 ¼ 2=s 1 .

Pomeron pole plus grey disk model: Real parts, inelastic cross sections and LHC data

“…where C, s 0 are unknown constants.It was proved later [20] that C = 4π/(t 0 ), where t 0 is the lowest singularity in the t−channel .This bound has been extremely useful in theoretical investigations [21] [22] and high energy models [23] [24][25] [26][27] [28][29] [30][31] [32]. Analogous bounds on the inelastic cross section have been obtained by Martin [33]and Wu et al [34]; for pion-pion case, Martin and Roy obtained bounds on energy averaged total [35] and inelastic cross sections [36] which also fix the scale factor s 0 in these bounds.…”