PreprintarXiv licencearXiv
Feynman Integrals Meet Second-Order Partial Differential Equations
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.
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BibTeX
@misc{chen2026feynman,
title = {{Feynman Integrals Meet Second-Order Partial Differential Equations}},
author = {Xiang Chen and Hantian Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2607.18406},
eprint = {2607.18406},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.18406},
}
RIS
TY - UNPB
TI - Feynman Integrals Meet Second-Order Partial Differential Equations
AU - Xiang Chen
AU - Hantian Zhang
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2607.18406
UR - https://arxiv.org/abs/2607.18406
AB - 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.
ER -