Extension of Axiomatic Analyticity Properties for Particles with Spin, and Proof of Superconvergence Relations

Abstract

It is shown that any regularized helicity amplitude that is known from axiomatic local field theory to satisfy dispersion relations for $\ensuremath{-}{t}_{0}\ensuremath{\le}t\ensuremath{\le}0$ is in fact analytic in the quasi-topological product $|t|<R\ifmmode\times\else\texttimes\fi{}s$ in the cut plane with cuts $s=C+\ensuremath{\lambda}$, $s=\ensuremath{-}t\ensuremath{-}\ensuremath{\mu}+{C}^{\ensuremath{'}}$, where $\ensuremath{\lambda}$, $\ensuremath{\mu}\ensuremath{\ge}0$ and $R$ is a fixed number. This is the extension to the scattering of nonzero-spin particles of a result obtained in the scalar case. As a first consequence, the Froissart limits are extended to all helicity amplitudes. Furthermore, it is shown that for $\ensuremath{-}{t}_{0}\ensuremath{\le}t\ensuremath{\le}0$ and $s$ going to infinity, the regularized helicity amplitudes in the $t$ channel, with initial (final) helicities ${\ensuremath{\lambda}}_{1}$ and ${\ensuremath{\lambda}}_{2}$ (${\ensuremath{\mu}}_{2}$ and ${\ensuremath{\mu}}_{2}$), are bounded by ${{C}_{s}}^{1\ensuremath{-}max(|\ensuremath{\lambda}|,|\ensuremath{\mu}|)}{(\mathrm{ln}s)}^{2}$ if $\ensuremath{\lambda}+\ensuremath{\mu}$ is even, or by ${{C}_{s}}^{1\ensuremath{-}max(|\ensuremath{\lambda}|,|\ensuremath{\mu}|)}{(\mathrm{ln}s)}^{3}$ if $\ensuremath{\lambda}+\ensuremath{\mu}$ is odd, where $\ensuremath{\lambda}={\ensuremath{\lambda}}_{1}\ensuremath{-}{\ensuremath{\lambda}}_{2}$ and $\ensuremath{\mu}={\ensuremath{\mu}}_{1}\ensuremath{-}{\ensuremath{\mu}}_{2}$. This gives superconvergent amplitudes as soon as one of the spins is larger than 1. The case of spin-0-spin-1 scattering is marginal, and in the absence of any detailed dynamical information, one cannot obtain a superconvergent amplitude in that case.