Feynman Integrals Meet Second-Order Partial Differential Equations
High Energy Physics - Phenomenology
2026-07-20 v1 High Energy Physics - Theory
Abstract
We propose a second-order partial differential equation method to solve multi-loop Feynman integrals as an equilibrium problem. As a proof-of-concept demonstration, we perform a Galerkin discretization of the corresponding variational form and employ the finite element method to compute two-loop four-point Feynman integrals. This method can solve the integral over a broad region of multi-dimensional phase space once and for all. This work establishes a new connection between perturbative quantum field theory and modern partial differential equation methods, with the potential to streamline a wide range of phenomenological applications.
Cite
@article{arxiv.2607.18406,
title = {Feynman Integrals Meet Second-Order Partial Differential Equations},
author = {Xiang Chen and Hantian Zhang},
journal= {arXiv preprint arXiv:2607.18406},
year = {2026}
}