From equations in coordinate space to Picard-Fuchs and back

Abstract

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. Firstly, we explain that these functions satisfy an equation of order $2^n$. 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.

References in corpus (17)

• Yangian symmetry of scattering amplitudes in N=4 super Yang-Mills theory
• Integrability and Matrix Models
• Feynman Diagrams and Differential Equations
• A (Bounded) Bestiary of Feynman Integral Calabi-Yau Geometries
• Feynman Integrals in Dimensional Regularization and Extensions of Calabi-Yau Motives
• Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space
• Picard-Fuchs equations for Feynman integrals
• On the evaluation of a certain class of Feynman diagrams in x-space: Sunrise-type topologies at any loop order
• Feynman integrals as A-hypergeometric functions
• FeynGKZ: a Mathematica package for solving Feynman integrals using GKZ hypergeometric systems
• Algorithms for minimal Picard-Fuchs operators of Feynman integrals
• GKZ-hypergeometric systems for Feynman integrals
• Position Space Equations for Banana Feynman Diagrams
• Challenges of Matrix Models
• Feynman Integrals of Grassmannians
• The sunrise integral around two and four space-time dimensions in terms of elliptic polylogarithms
• GKZ-system of the 2-loop self energy with 4 propagators

Cited by in corpus (2)

• Three-loop banana integrals with four unequal masses
• On geometric bases for A-polynomials II: ${\mathfrak{su}}_3$ and Kuberberg bracket