Feynman integral reduction using Gr\"obner bases
Abstract
We investigate the reduction of Feynman integrals to master integrals using Gr\"obner bases in a rational double-shift algebra Y in which the integration-by-parts (IBP) relations form a left ideal. The problem of reducing a given family of integrals to master integrals can then be solved once and for all by computing the Gr\"obner basis of the left ideal formed by the IBP relations. We demonstrate this explicitly for several examples. We introduce so-called first-order normal-form IBP relations which we obtain by reducing the shift operators in Y modulo the Gr\"obner basis of the left ideal of IBP relations. For more complicated cases, where the Gr\"obner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations.
Keywords
Cite
@article{arxiv.2210.05347,
title = {Feynman integral reduction using Gr\"obner bases},
author = {Mohamed Barakat and Robin Brüser and Claus Fieker and Tobias Huber and Jan Piclum},
journal= {arXiv preprint arXiv:2210.05347},
year = {2023}
}
Comments
25 pages, 4 figures, 12 ancillary files; v2: clarified text and improved notation