A Lower Bound on Inelasticity in Pion-Pion Scattering

TL;DR

This paper establishes a lower bound on the inelastic cross section in pion-pion scattering at high energies, based on analyticity assumptions of the scattering amplitude within a specific domain.

Contribution

It introduces a new analyticity-based lower bound on inelasticity in pion-pion scattering, extending previous bounds by incorporating domain assumptions in the complex t-plane.

Findings

Derived a lower bound on inelastic cross section at high energies.

Bound depends on analyticity domain parameters and energy s.

Provides insights into the asymptotic behavior of scattering amplitudes.

Abstract

Assuming that the pion-pion scattering amplitude and its absorptive part are analytic inside an ellipse in $t$- plane with foci $t=0$, $u=0$ and right extremity $t=4 m_{\pi}^2 +\epsilon $, ($\epsilon > 0$), except for cuts prescribed by Mandelstam representation for $t\geq 4 m_{\pi}^2$, $u\geq 4 m_{\pi}^2$ , and bounded by $s^N$ on the boundary of this domain, we prove that for $s\rightarrow \infty$, \sigma_{inel} (s) > \frac{Const}{s^{5/2} }\exp {[-\frac{\sqrt{s}}{4} (N+5/2) \ln {s} ]}.