Absorptive Parts and the Bethe-Salpeter Equation for Forward Scattering

Abstract

We have developed a Laplace transform approach to the Bethe-Salpeter equation for absorptive parts of forward scattering amplitudes. The method appears direct and unsophisticated and is useful for computation. It is essentially identical to decomposition into four-dimensional partial waves, but the inversion formula is more straightforward. We have obtained the high-energy behavior of sums of several types of φ4 graphs in the weak- and strong-coupling limits. These examples illustrate some general results we obtain for Mth order φ4 kernels. Specifically, the absorptive part behaves as sn0(log s)β for high s; for weak coupling λ, n0 ∼ λ½M, while for strong coupling in the ladder graph approximation, n0/λ½ → 1/2π. We have also proven an interesting inequality related to absorptive parts. One of its corollaries is that uncrossed ladder graphs ``majorize'' crossed ones.