High-Energy Potential Scattering

Abstract

The asymptotic expansion of the phase shift in inverse powers of the momentum $p$ and the asymptotic expansion of the scattering amplitude in inverse powers of $p$ and the momentum transfer $q$ have been derived for a Dirac and Klein-Gordon particle. It is shown that at high energy (i) the higher-order phase shifts (proportional to ${\ensuremath{\alpha}}^{k}$, $k\ensuremath{\ne}1$, where $\ensuremath{\alpha}$ is the coupling constant) are very small and negligible in the calculation of the amplitude, (ii) the amplitude approaches the first Born approximation amplitude (linear in $\ensuremath{\alpha}$) multiplied by a a phase factor. Statement (ii) holds for spherically symmetric potentials $V(r)$ for which the first $N$ derivatives (the phase shift is expanded asymptotically up to ${p}^{\ensuremath{-}N}$) exist for every real positive value of $r$, including $r=0$, and for which at least one derivative of odd order does not vanish at the origin. Statement (i) is probably correct also for potentials even at the origin [$V(r)=V(\ensuremath{-}r)$ for $r\ensuremath{\approx}0$]. The upper limit on the coupling constant $\ensuremath{\alpha}$ is $\ensuremath{\alpha}\ensuremath{\ll}p{\ensuremath{\mu}}_{1}$ with the additional condition $p{\ensuremath{\mu}}_{1}\ensuremath{\gg}1$. Here ${\ensuremath{\mu}}_{1}$ is a characteristic length of the potential. The lower limit on the scattering angle $\ensuremath{\theta}$ is given by $\ensuremath{\theta}\ensuremath{\gg}{(p{\ensuremath{\mu}}_{2})}^{\ensuremath{-}1}$, where ${\ensuremath{\mu}}_{2}$ is another characteristic length of the potential and is usually of the same order of magnitude as ${\ensuremath{\mu}}_{1}$. The problem of the model independence and other consequences of the theory are discussed.