Singularities and Discontinuities of Feynman Amplitudes

TLDR

The paper studies Landau singularities of amplitudes from arbitrary Feynman graphs, presenting a general formula that extends the unitarity condition. The authors illustrate the general results on single‑loop graphs and analyze non‑Landau singularities using a third‑order vertex part. They show that the discontinuity across a branch cut from any Landau singularity is obtained by replacing propagators with delta functions for lines in the Landau diagram, that this discontinuity’s singularities form a subclass of the original amplitude’s singularities, and that the formula yields the Mandelstam spectral function for fourth‑order scattering.

Abstract

The Landau singularities of the amplitude calculated from an arbitrary Feynman graph are considered. It is shown that the discontinuity across a branch cut starting from any Landau singularity is obtained by replacing Feynman propagators by delta functions for those lines which appear in the Landau diagram. The general formula is a simple generalization of the unitarity condition. The discontinuity is then considered as an analytic function of the momenta and masses; it is shown that its singularities are a subclass of the singularities of the original amplitude which corresponds to Landau diagrams with additional lines. The general results are illustrated by application to some single loop graphs. In particular, the general formula gives an immediate calculation of the Mandelstam spectral function for fourth-order scattering. Singularities not of the Landau type are discussed and illustrated by the third-order vertex part.