Solution of Canonical Differential Equations for Integrals on Arbitrary Geometries

Abstract

A highly successful approach to computing multi-loop scattering amplitudes is to reduce the Feynman integrals that arise to a smaller set of master integrals using integration-by-parts identities. These dimensionally-regulated master integrals can often be determined by solving a system of first-order partial differential equations with respect to masses and external invariants. The application of this method to large classes of problems became much more streamlined thanks to the introduction of $\epsilon$-factorized canonical forms. There is increasing evidence that a canonical form can always be achieved, although the required transformation may involve transcendental functions related to the periods of geometrical objects such as elliptic curves or Calabi-Yau manifolds. Until now, obtaining numerical values for the master integrals in such cases has been difficult in practice, also due to the lack of closed-form expressions for the transcendental functions involved. We show that this obstruction is only apparent. Since the original master integrals satisfy linear differential equations with rational coefficients, any functions appearing in the transformation to a canonical basis satisfy, by construction, rational differential equations as well. By solving these auxiliary equations, the numerical evaluation of the canonical system reduces to solving an enlarged rational system. We implement this strategy in a C\texttt{++} package and apply it to the two-loop master integrals that enter di-jet and $\gamma$+jet hadro-production via a heavy-quark loop.