18 Jun 2024 | Leonardo de la Cruz and Pierre Vanhove
The paper presents an algorithm for determining the minimal order differential equations associated with Feynman integrals in dimensional or analytic regularization. The algorithm extends the Griffiths-Dwork pole reduction to twisted differential forms, which are relevant in dimensional and analytic regularization. The authors demonstrate the applicability of this algorithm by providing explicit examples, including multiloop two-point sunset integrals up to 20 loops for equal-mass cases and generic mass cases at two- and three-loop orders. They also derive differential operators for various infrared-divergent two-loop graphs and apply the algorithm to derive a system of partial differential equations for regulated Witten diagrams in four-dimensional de Sitter space, which arise in the evaluation of cosmological correlators of conformally coupled \(\phi^4\) theory. The paper is organized into sections covering the parametric representation of Feynman integrals, the algorithmic procedure for deriving differential equations, and various examples illustrating the method.The paper presents an algorithm for determining the minimal order differential equations associated with Feynman integrals in dimensional or analytic regularization. The algorithm extends the Griffiths-Dwork pole reduction to twisted differential forms, which are relevant in dimensional and analytic regularization. The authors demonstrate the applicability of this algorithm by providing explicit examples, including multiloop two-point sunset integrals up to 20 loops for equal-mass cases and generic mass cases at two- and three-loop orders. They also derive differential operators for various infrared-divergent two-loop graphs and apply the algorithm to derive a system of partial differential equations for regulated Witten diagrams in four-dimensional de Sitter space, which arise in the evaluation of cosmological correlators of conformally coupled \(\phi^4\) theory. The paper is organized into sections covering the parametric representation of Feynman integrals, the algorithmic procedure for deriving differential equations, and various examples illustrating the method.