Some Comments on the Jin-Martin Lower Bound

Abstract

We present an extremely simple derivation of the Jin-Martin bound in a slightly weakened form. Explicitly, let ${f}^{\mathrm{AB}\ensuremath{\rightarrow}\mathrm{AB}}(s)$ be the spin-averaged forward-scattering amplitude for the process $\mathrm{AB}\ensuremath{\rightarrow}\mathrm{AB}$ [normalized by ${\ensuremath{\sigma}}_{\mathrm{tot}}=4\ensuremath{\pi}\frac{\mathrm{Im}f(s)}{k\ensuremath{\surd}s}$]. Then we prove that there is a constant $C$ and a sequence ${s}_{n}\ensuremath{\rightarrow}\ensuremath{\infty}$ such that either ${f}^{\mathrm{AB}\ensuremath{\rightarrow}\mathrm{AB}}({s}_{n})>C{{s}_{n}}^{\ensuremath{-}2}$ (all $n$) or ${f}^{A\overline{B}\ensuremath{\rightarrow}A\overline{B}}({s}_{n})>C{{s}_{n}}^{\ensuremath{-}2}$ (all $n$). We also clarify the connection between allowable asymptotic behavior and the sign of the scattering length found by Jin and Martin.

Locations

• Physical review. D. Particles, fields, gravitation, and cosmology/Physical review. D. Particles and fields - View
• CaltechAUTHORS (California Institute of Technology) - View - PDF