Landau and leading singularities in arbitrary space-time dimensions

TL;DR

This paper explores the relationship between Landau and leading singularities in one-loop Feynman integrals across arbitrary space-time dimensions, providing new insights and methods for analyzing these singularities systematically.

Contribution

It establishes a novel connection between Landau and leading singularities using multi-dimensional residue theory and extends this understanding to multi-loop integrals.

Findings

Landau and leading singularities are related through inverse square root in specific dimensions.

Systematic derivation of differential equations in canonical form for Feynman integrals.

Extension of singularity connection to multi-loop level using loop-by-loop approach.

Abstract

Using the decomposition of the $D$-dimensional space-time into parallel and perpendicular subspaces, we study and prove a connection between Landau and leading singularities for $N$-point one-loop Feynman integrals by applying multi-dimensional theory of residues. We show that if $D=N$ and $D=N+1$, the leading singularity corresponds to the inverse of the square root of the leading Landau singularity of the first and second type, respectively. We make use of this outcome to systematically provide differential equations of Feynman integrals in canonical forms and the extension of the connection of these singularities at multi-loop level by exploiting the loop-by-loop approach. Illustrative examples with the calculation of Landau and leading singularities are provided to supplement our results.