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From equations in coordinate space to Picard-Fuchs and back
Nordita SU; Stockholm Univ, KTH Royal Inst Technol, Nordita, Hannes Alfvens vag 12, SE-10691 Stockholm, Sweden.
Moscow Inst Phys & Technol, Dolgoprudnyi 141701, Russia; Natl Res Ctr, Kurchatov Inst, Moscow 123182, Russia; Inst Informat Transmiss Problems, Moscow 127994, Russia.
Moscow Inst Phys & Technol, Dolgoprudnyi 141701, Russia; Natl Res Ctr, Kurchatov Inst, Moscow 123182, Russia.
Moscow Inst Phys & Technol, Dolgoprudnyi 141701, Russia.
2025 (English)In: International Journal of Modern Physics A, ISSN 0217-751X, E-ISSN 1793-656X, Vol. 40, no 07, article id 2550017Article in journal (Refereed) Published
Abstract [en]

In this paper, we continue the development of a position space approach to equations for Feynman multi-loop integrals. The key idea of the approach is that unintegrated products of Greens functions in position space are still loop integral in momentum space. The natural place to start are the famous banana diagrams, which we explore in this paper. In position space, these are just products of n propagators. First, we explain that these functions satisfy an equation of order 2n. These should be compared with Picard-Fuchs equations derived for the momentum space integral. We find that the Fourier transform of the position space operator contains the Picard-Fuchs one as a rightmost factor. The order of these operators is a special issue, especially since the order in momentum space is governed by degree in x in position space. For the generic mass case, this factorization pattern is complicated and it seems like the order of the Fourier transformed position space operators is much bigger than that of the Picard-Fuchs. Furthermore, one may ask what happens if after factorization we take the Picard-Fuchs operators back into position space. We discover that the result is again factorized, with the rightmost factor being the original position space equation. We demonstrate how this works in examples and discuss implications for more sophisticated Feynman integrals.

Place, publisher, year, edition, pages
World Scientific Pub Co Pte Ltd , 2025. Vol. 40, no 07, article id 2550017
Keywords [en]
Feynman integrals, Picard-Fuchs equation, Green functions in coordinate space
National Category
Subatomic Physics
Identifiers
URN: urn:nbn:se:kth:diva-362992DOI: 10.1142/S0217751X25500174ISI: 001453631700002Scopus ID: 2-s2.0-105000408877OAI: oai:DiVA.org:kth-362992DiVA, id: diva2:1956456
Note

QC 20250506

Available from: 2025-05-06 Created: 2025-05-06 Last updated: 2025-05-06Bibliographically approved

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