Complex Angular Momentum in Perturbation Theory

Abstract

The leading singularity in the complex angular momentum plane is studied for certain sets of Feynman graphs. Two models are considered: (a) the ladder graphs in the $\lambda {\varphi }^4$ theory in which bubbles are exchanged, and (b) the ladder graphs for the scattering of two scalar mesons by vector meson exchange. The method used is the summation of the most singular term in every order of perturbation theory. In both models the leading singularity is a branch point on the real $l$ axis to the right of $l=0\mathrm{}$. As the coupling constant is decreased, this branch point approaches $l=0\mathrm{}$. The nature of the branch point is very similar to that of the branch point (near $l=-\frac{1}{2}$ for weak coupling) in the case of scattering from a potential with a $r^{-2\mathrm{}}$ singularity.