Differential equations for loop integrals in Baikov representation
Jorrit Bosma, Kasper J. Larsen, Yang Zhang
TL;DR
The paper proves that differential equations for multi-loop Feynman integrals can be derived in Baikov representation without relying on dimension-shift identities, by establishing a fundamental ideal membership for the Baikov polynomial F. It further shows that, for a large class of two- and three-loop diagrams, an enhanced ideal membership allows avoiding squared propagators in intermediate IBP steps, and that the resulting systems can be transformed into canonical ε-forms. Through explicit examples of planar and non-planar massless double-box topologies, the authors demonstrate the method’s practicality and provide constructive procedures for the necessary cofactors and basis changes. They also identify a counterexample where the enhanced membership fails, indicating that classification of diagrams where the property holds remains an open problem. Overall, the approach offers a computationally favorable route to differential equations for loop integrals with reduced reliance on dimension shifts and squared propagators, potentially easing high-precision NNLO calculations.
Abstract
We present a proof that differential equations for Feynman loop integrals can always be derived in Baikov representation without involving dimension-shift identities. We moreover show that in a large class of two- and three-loop diagrams it is possible to avoid squared propagators in the intermediate steps of setting up the differential equations.