18 Jun 2024 | Leonardo de la Cruz and Pierre Vanhove
This 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 used in the parametric representation of Feynman integrals. In dimensional regularization, the algorithm is applied to derive inhomogeneous differential equations for multiloop two-point sunset integrals, including cases with equal and different masses. In analytic regularization, it is used to derive partial differential equations for regulated Witten diagrams in de Sitter space. The algorithm works by iteratively reducing the pole order of the differential forms, leading to a system of differential equations that can be solved using linear algebra techniques. The method is demonstrated for various Feynman graphs, including one-loop box graphs, two-loop sunset graphs, and three-point and four-point graphs. The algorithm is also applied to cosmological correlators of conformally coupled φ⁴ theory in four-dimensional de Sitter space. The results show that the minimal order of the differential operator is related to the number of master integrals and the regulator dependence. The algorithm is efficient and can handle large systems of equations, making it a valuable tool for computing differential equations for Feynman integrals in general dimensions.This 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 used in the parametric representation of Feynman integrals. In dimensional regularization, the algorithm is applied to derive inhomogeneous differential equations for multiloop two-point sunset integrals, including cases with equal and different masses. In analytic regularization, it is used to derive partial differential equations for regulated Witten diagrams in de Sitter space. The algorithm works by iteratively reducing the pole order of the differential forms, leading to a system of differential equations that can be solved using linear algebra techniques. The method is demonstrated for various Feynman graphs, including one-loop box graphs, two-loop sunset graphs, and three-point and four-point graphs. The algorithm is also applied to cosmological correlators of conformally coupled φ⁴ theory in four-dimensional de Sitter space. The results show that the minimal order of the differential operator is related to the number of master integrals and the regulator dependence. The algorithm is efficient and can handle large systems of equations, making it a valuable tool for computing differential equations for Feynman integrals in general dimensions.