Computation of Contour Integrals on ${\cal M}_{0,n}$

TL;DR

This paper introduces an algorithm for analytically computing contour integrals of rational functions over the moduli space of punctured spheres, which are relevant in particle physics scattering amplitudes.

Contribution

It presents a novel iterative algorithm combining KLT operations, Petersen's theorem, and Hamiltonian decompositions to evaluate complex integrals on ${ m M}_{0,n}$.

Findings

Provides a systematic method to compute integrals in particle physics

Reduces complex integrals to simpler building blocks

Connects integrals to bi-adjoint scalar theory amplitudes

Abstract

Contour integrals of rational functions over ${\cal M}_{0,n}$, the moduli space of $n$-punctured spheres, have recently appeared at the core of the tree-level S-matrix of massless particles in arbitrary dimensions. The contour is determined by the critical points of a certain Morse function on ${\cal M}_{0,n}$. The integrand is a general rational function of the puncture locations with poles of arbitrary order as two punctures coincide. In this note we provide an algorithm for the analytic computation of any such integral. The algorithm uses three ingredients: an operation we call general KLT, Petersen's theorem applied to the existence of a 2-factor in any 4-regular graph and Hamiltonian decompositions of certain 4-regular graphs. The procedure is iterative and reduces the computation of a general integral to that of simple building blocks. These are integrals which compute double-color-ordered partial amplitudes in a bi-adjoint cubic scalar theory.