Kinematic Flow for Cosmological Loop Integrands

TL;DR

This paper extends a graphical rule-based approach to derive differential equations for cosmological loop integrands, revealing how kinematic flow predicts their structure and singularities.

Contribution

It introduces a method to determine loop integrand differential equations using graphical rules, including the identification of vanishing basis functions.

Findings

Graphical rules predict differential equations for loop integrands.

Basis functions and singularities are represented by tubings of marked graphs.

The method correctly predicts all loop integrand differential equations.

Abstract

Recently, an interesting pattern was found in the differential equations satisfied by the Feynman integrals describing tree-level correlators of conformally coupled scalars in a power-law FRW cosmology [1,2]. It was proven that simple and universal graphical rules predict the equations for arbitrary graphs as a flow in kinematic space. In this note, we show that the same rules$\unicode{x2013}$with one small addition$\unicode{x2013}$also determine the differential equations for loop integrands. We explain that both the basis of master integrals and the singularities of the differential equations can be represented by tubings of marked graphs. An important novelty in the case of loops is that some basis functions can vanish, and we present a graphical rule to identify these vanishing functions. Taking this into account, we then demonstrate that the kinematic flow correctly predicts the differential equations for all loop integrands.