# Differential equations on unitarity cut surfaces

## TL;DR

This paper introduces a novel approach to deriving differential equations for Feynman integrals that avoids doubled propagators and simplifies the process, especially for complex multi-loop integrals, using unitarity-inspired methods and computational techniques.

## Contribution

It presents a new method to formulate differential equations on unitarity cut surfaces, eliminating the need for IBP reduction in certain cases, and extends the approach to two-loop nonplanar five-point integrals.

## Key findings

- Direct derivation of DEs for one-loop box without IBP reduction.
- Application to two-loop nonplanar five-point integrals at maximal cut.
- Use of finite field techniques for computational efficiency.

## Abstract

We reformulate differential equations (DEs) for Feynman integrals to avoid doubled propagators in intermediate steps. External momentum derivatives are dressed with loop momentum derivatives to form tangent vectors to unitarity cut surfaces, in a way inspired by unitarity-compatible IBP reduction. For the one-loop box, our method directly produces the final DEs without any integration-by-parts reduction. We further illustrate the method by deriving maximal-cut level differential equations for two-loop nonplanar five-point integrals, whose exact expressions are yet unknown. We speed up the computation using finite field techniques and rational function reconstruction.