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 ${\sigma }_{inel}$ which is one-fourth of the corresponding upper bound on ${\sigma }_{tot}$, and Wu, Martin,Roy and Singh improved the bound by adding the constraint of a given ${\sigma }_{tot}$. 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. E.g. for ${\pi }^0{\pi }^0$ scattering, defining ${\sigma }_{inel}={\sigma }_{tot}-({\sigma }^{{\pi }^0{\pi }^0\to {\pi }^0{\pi }^0}+{\sigma }^{{\pi }^0{\pi }^0\to {\pi }^{+}{\pi }^{-}})$,we show that for c.m. energy $\sqrt{s}\to \infty$, ${\bar{\sigma }}_{inel}(s,\infty )\equiv s{\int }_s^{\infty }d{s}^{\prime }{\sigma }_{inel}(s^{\prime })/s^{\prime 2}\le (\pi /4)(m_{\pi }{)}^{-2}[\ln (s/s_1)+(1/2)\ln \ln (s/s_1)+1{]}^2$ where $1/s_1=34\pi \sqrt{2\pi }{m}_{\pi }^{-2}$ . 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 ${\pi }^0{\pi }^0$cross section has a bound of the same form with $1/s_1$ replaced by $1/s_2=2/s_1$.

9 pages. Submitted to Physical Review D

References in corpus (4)

• Measurement of the total cross section from elastic scattering in $pp$ collisions at $\sqrt{s}=7$ TeV with the ATLAS detector
• Hadronic forward scattering: Predictions for the Large Hadron Collider and cosmic rays
• The Froissart bound for inelastic cross-sections
• An upper bound on the total inelastic cross-section as a function of the total cross-section

Cited by in corpus (2)

• Analyticity Properties and Asymptotic Behavior of Scattering Amplitude in Higher Dimensional Theories
• Inelastic diffraction and role of reflective scattering at the LHC