Asymptotic Behavior of the Scattering Amplitude and Normal and Abnormal Solutions of the Bethe-Salpeter Equation

Abstract

An integral equation for a scattering amplitude is considered in the ladder approximation. It is assumed that two scalar particles having mass $m$ exchange scalar photons except for the first step, in which a scalar meson having mass $2m$ is exchanged. The exact solution to this equation is found in a compact form in the case of zero energy. Asymptotic behavior of the solution is investigated in the crossed channel. It is shown that the leading term and the second leading term in the asymptotic expansion in $t$ exactly correspond to the normal solutions of the Bethe-Salpeter equation with $n=l+1\mathrm{}$ and those with $n=l+2\mathrm{}$, respectively, where $n$ is the principal quantum number.