Position space equations for generic Feynman graphs

TL;DR

This paper extends position space methods to general Feynman diagrams, deriving differential equations for propagator products, and discusses their relation to Picard-Fuchs equations after Fourier transform.

Contribution

It introduces a generalized approach to Feynman integrals in position space for arbitrary graphs using differential equations and 'bananization' techniques.

Findings

Derived differential equations for generic Feynman graphs.

Connected position space equations to Picard-Fuchs equations.

Analyzed Fourier transform challenges in the approach.

Abstract

We propose the extension of the position space approach to Feynman integrals from the banana family to generic Feynman diagrams. Our approach is based on getting rid of integration in position space and then writing differential equations for the products of propagators defined for any graph. We employ the so-called ''bananization'' to start with simple Feynman graphs and further substituting each edge with a multiple one. We explain how the previously developed theory of banana diagrams can be used to describe what happens to the differential equations (Ward identities) on Feynman diagrams after this transformation. Our approach works for generic enough (large enough) dimension and masses. We expect that after Fourier transform our equations should be related to the Picard-Fuchs equations. Therefore, we describe the challenges of Fourier transform that arise in our approach.