# Spherical Contours, IR Divergences and the geometry of Feynman parameter integrands at one loop

## TL;DR

This paper introduces spherical contours in Feynman parameter space to analyze IR divergences and develop a new method for determining Feynman integrands without momentum space reference, with applications to $ ext{N}=4$ SYM.

## Contribution

It extends spherical contours to compute IR divergences and introduces a Feynman parameter space analog of leading singularities, offering a novel approach to loop integrand analysis.

## Key findings

- Spherical contours relate to IR divergences in one-loop graphs.
- A new method to determine Feynman integrands without momentum space.
- Insights into Feynman integrands in $ ext{N}=4$ SYM.

## Abstract

Spherical contours introduced in \cite{SphericalContours} translate the concept of "discontinuity across a branch cut" to Feynman parameter space. In this paper, we further explore spherical contours and connect them to the computation of leading IR divergences of 1 loop graphs directly in Feynman parameter space. These spherical contours can be used to develop a Feynman parameter space analog of "Leading Singularities" of loop integrands which allows us to develop a method of determining Feynman parameter integrands with no reference to the momentum space loop integrand. Finally, we explore some interesting features of Feynman parameter integrands in $\mathcal{N}=4$ SYM.