Feynman integral reduction using Gröbner bases

A bstract We investigate the reduction of Feynman integrals to master integrals using Gröbner 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 onc...

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Bibliographic Details
Published in:The journal of high energy physics Vol. 2023; no. 5; pp. 168 - 33
Main Authors: Barakat, Mohamed, Brüser, Robin, Fieker, Claus, Huber, Tobias, Piclum, Jan
Format: Journal Article
Language:English
Published: Berlin/Heidelberg Springer Berlin Heidelberg 19.05.2023
Springer Nature B.V
SpringerOpen
Subjects:
Algorithms
Classical and Quantum Gravitation
Differential and Algebraic Geometry
Elementary Particles
Geometry
High energy physics
Higher-Order Perturbative Calculations
Integrals
Kinematics
Linear algebra
Non-Commutative Geometry
Operators (mathematics)
Partial differential equations
Physics
Physics and Astronomy
Quantum Field Theories
Quantum Field Theory
Quantum Physics
Reduction
Regular Article - Theoretical Physics
Relativity Theory
String Theory
Non-Commutative Geometry
Differential and Algebraic Geometry
Higher-Order Perturbative Calculations
ISSN:1029-8479, 1029-8479
Online Access:Get full text
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Summary:A bstract We investigate the reduction of Feynman integrals to master integrals using Gröbner 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 Gröbner 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 Gröbner basis of the left ideal of IBP relations. For more complicated cases, where the Gröbner basis is computationally expensive, we develop an ansatz based on linear algebra over a function field to obtain the normal-form IBP relations.
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ISSN:1029-8479
1029-8479
DOI:10.1007/JHEP05(2023)168