Landau and leading singularities in arbitrary space-time dimensions

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.