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

TL;DR

This work provides a comprehensive analytic algorithm to compute contour integrals on the moduli space ${\cal M}_{0,n}$ that appear in CHY representations of tree-level scattering. It builds general integrals from elementary building blocks ${m}(\alpha|\beta)$ via a generalized KLT construction, aided by Petersen’s theorem for 4-regular graphs and Hamiltonian-decomposition ideas to ensure compatible Parke-Taylor bases. The method is implemented through a structured reduction: represent the integrand with cross ratios, decompose using 2-factorizations, and express the result as sums over compatible Parke-Taylor pairs with explicit inverses of $m^{\cal L|\cal R}$. The paper substantiates the approach with six-point examples, detailing how to construct left and right bases and perform the final assembly, thereby enabling explicit rational-function expressions for a broad class of integrals. This framework advances analytic access to CHY amplitudes and related field-theory relations, with potential connections to string theory and novel amplitude identities.

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.