Fourier transforms of UD integrals

TL;DR

This paper investigates the position space representation of UD integrals, showing that their Fourier transforms relate to UD functions of space-time intervals, with implications for their role in supersymmetric Yang-Mills theory.

Contribution

It demonstrates that Fourier transforms of UD functions correspond to UD functions of space-time intervals, revealing an indirect relationship in position space.

Findings

Fourier transforms of UD functions are related to UD functions of space-time intervals.

The correspondence between Fourier transforms and UD functions is indirect.

The second UD integral's Fourier transform is itself a UD integral.

Abstract

UD integrals published by N. Usyukina and A. Davydychev in 1992-1993 are integrals corresponding to ladder-type Feynman diagrams. The results are UD functions $\Phi^{(L)},$ where $L$ is the number of loops. They play an important role in N=4 supersymmetic Yang-Mills theory. The integrals were defined and calculated in the momentum space. In this paper the position space representation of UD functions is investigated. We show that Fourier transforms of UD functions are UD functions of space-time intervals but this correspondence is indirect. For example, the Fourier transform of the second UD integral is the second UD integral.