Feynman Diagrams in Four-Dimensional Holomorphic Theories and the Operatope

TL;DR

This paper introduces a novel geometric approach to computing Feynman integrals in four-dimensional holomorphic theories by relating them to the Fourier transform of a polytope called the Operatope, with recursive relations to determine results.

Contribution

It recasts Feynman integrals as Fourier transforms of the Operatope and derives quadratic recursion relations, applicable to a broad class of twisted supersymmetric quantum field theories.

Findings

Feynman integrals expressed as Fourier transforms of the Operatope

Derived quadratic recursion relations for integral evaluation

Applicable to general twisted supersymmetric theories

Abstract

We study a class of universal Feynman integrals which appear in four-dimensional holomorphic theories. We recast the integrals as the Fourier transform of a certain polytope in the space of loop momenta (aka the ``Operatope''). We derive a set of quadratic recursion relations which appear to fully determine the final answer. Our strategy can be applied to a very general class of twisted supersymmetric quantum field theories.