Singularities of a Relativistic Scattering Amplitude in the Complex Angular Momentum Plane

The analytic properties of the partial-wave amplitude in the complex angular momentum plane are investigated in a relativistic scalar-meson theory. Using a $\frac{N}{D}$ decomposition, the numerator and denominator are calculated by perturbative expansions to fourth order in the coupling constant. Higher order poles at negative integer values of $l$ are found in both the numerator and denominator, leading to a breakdown in their perturbation expansions near these singularities. The same breakdown occurs for all but the leading Regge trajectory. It is further found that, to fourth order, due to inelastic processes, the denominator has branch cuts for values of $l$ near negative integers.