Copositive geometry of Feynman integrals
ABSTRACT: Copositive matrices and copositive polynomials are objects from optimization. We connect these to the geometry of Feynman integrals in physics. The integral is guaranteed to converge if its kinematic parameters lie in the copositive cone. Pólya’s method makes this manifest. We study the copositive cone for the second Symanzik polynomial of any Feynman graph. Its algebraic boundary is described by Landau discriminants.
We provide a proof-of-concept implementation of the methods described in the paper. These compute certificates for containment of kinematic parameters in the copositive cone.
Our julia package can be downloaded here: CopositiveFeynman.zip.
The basic functionalities of our package is illustrated in a Jupyter notebook, which can be downloaded here: demo_CopositiveFeynman.ipynb.
A more detailed description of the code is contained in the comments in the source code file.
Project page created: 03/04/2025
Project contributors: Bernd Sturmfels, Máté L. Telek
Corresponding author of this page: Máté L. Telek, mate.telek@mis.mpg.de
Software used: Julia (Version 1.11.1)
License for code of this project page: MIT License (https://spdx.org/licenses/MIT.html)
License for all other content of this project page (text, images, …): CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/)