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Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals
Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

2470-0010
Böhm, Janko
50e25a6a-b3c8-40a7-96cd-ad37111d1c00
Georgoudis, Alessandro
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Larsen, Kasper J.
49008353-d8ca-4de6-a377-e34ba737a3e7
Schulze, Mathias
0c14618e-4a67-446a-96a5-e04c6ee84517
Zhang, Yang
b165d56f-015b-4295-bbf4-72438baec173
Böhm, Janko
50e25a6a-b3c8-40a7-96cd-ad37111d1c00
Georgoudis, Alessandro
8df046a2-987e-4b85-a5ed-e1db0e66c4fa
Larsen, Kasper J.
49008353-d8ca-4de6-a377-e34ba737a3e7
Schulze, Mathias
0c14618e-4a67-446a-96a5-e04c6ee84517
Zhang, Yang
b165d56f-015b-4295-bbf4-72438baec173

Böhm, Janko, Georgoudis, Alessandro, Larsen, Kasper J., Schulze, Mathias and Zhang, Yang (2018) Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals. Physical Review D, 98 (2). (doi:10.1103/PhysRevD.98.025023).

Record type: Article

Abstract

Integration-by-parts identities between loop integrals arise from the vanishing integration of total derivatives in dimensional regularization. Generic choices of total derivatives in the Baikov or parametric representations lead to identities which involve dimension shifts. These dimension shifts can be avoided by imposing a certain constraint on the total derivatives. The solutions of this constraint turn out to be a specific type of syzygies which correspond to logarithmic vector fields along the Gram determinant formed of the independent external and loop momenta. We present an explicit generating set of solutions in Baikov representation, valid for any number of loops and external momenta, obtained from the Laplace expansion of the Gram determinant. We provide a rigorous mathematical proof that this set of solutions is complete. This proof relates the logarithmic vector fields in question to ideals of submaximal minors of the Gram matrix and makes use of classical resolutions of such ideals.

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PhysRevD.98.025023 - Version of Record
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Accepted/In Press date: 15 July 2018
e-pub ahead of print date: 27 July 2018

Identifiers

Local EPrints ID: 424639
URI: https://eprints.soton.ac.uk/id/eprint/424639
ISSN: 2470-0010
PURE UUID: a8c02676-23b3-4031-9200-392b0f21b4fb

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Date deposited: 05 Oct 2018 11:39
Last modified: 13 Mar 2019 18:08

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Contributors

Author: Janko Böhm
Author: Alessandro Georgoudis
Author: Mathias Schulze
Author: Yang Zhang

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