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 ${\alpha }^k$, $k\ne 1$, where $\alpha$ 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 $\alpha$) multiplied by a a phase factor. Statement (ii) holds for spherically symmetric potentials $V\left(r\right)\mathrm{}$ for which the first $N$ derivatives (the phase shift is expanded asymptotically up to $p^{-N}$) exist for every real positive value of $r$, including $r=0\mathrm{}$, 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 [$V\left(r\right)=V(-r)\mathrm{}$ for $r\approx 0$]. The upper limit on the coupling constant $\alpha$ is $\alpha \ll p{\mu }_1$ with the additional condition $p{\mu }_1\gg 1$. Here ${\mu }_1$ is a characteristic length of the potential. The lower limit on the scattering angle $\theta$ is given by $\theta \gg {\left(p{\mu }_2\right)}^{-1\mathrm{}}$, where ${\mu }_2$ is another characteristic length of the potential and is usually of the same order of magnitude as ${\mu }_1$. The problem of the model independence and other consequences of the theory are discussed.