Sum Rules and Bounds on Scattering Amplitudes

Abstract

Using sum rules obtained from crossing and analyticity, and unitarity bounds on scattering amplitudes, we show how new relations between low-energy and high-energy scattering can be derived. These relations can provide tests of a wide range of theoretical ideas. As examples, we discuss several inequalities obtained for $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ and $\ensuremath{\pi}\ensuremath{-}N$ scattering. For $\ensuremath{\pi}\ensuremath{-}\ensuremath{\pi}$ scattering, a number of relations involving the asymptotic behavior of total cross sections are presented, including bounds limiting the size of violations of the Pomeranchuk theorem. Using finite-energy sum rules for $\ensuremath{\pi}\ensuremath{-}N$ scattering, we derive new types of bounds and show how they can be used to probe such things as the nature of the Pomeranchuk trajectory and the assumption of $s$-channel helicity conservation. Finally, we introduce inequality constraints between partial-wave amplitudes of different isospin, and indicate how they can be used to explore the nature of exchange degeneracy, absence of exotics, and duality.