Feynman amplitudes and Landau singularities for one-loop graphs

Abstract

The subject of Feynman amplitudes with variable momenta and non-zero masses has been studied by physicists since the 1950’s. In the interim, new mathematical methods involving Hodge structures and variations of Hodge structures have been developed. The purpose of this paper is to apply these techniques to the study of amplitudes and Landau singularities in momentum space. While the techniques we develop bear on the general case here, we will mainly focus on the 1-loop case. In this case, for general values of masses and external momenta, the polar locus of the integrand (written in Feynman coordinates) is a smooth quadric. (Exceptionally, in the “triangle case”, the polar locus is a union of a hyperplane and a quadric.) Mathematically, the polar loci form a degenerating family of such objects, which is a familiar and well-studied situation in algebraic geometry. Our objective is firstly to explain motivically the known fact [4] that dilogarithms are ubiquitous in this situation, and secondly to show how the motivic and Hodge-theoretic framework is a powerful way to study thresholds and Landau singularities.