Iterative Non-iterative Integrals in Quantum Field Theory

TL;DR

This paper surveys the solutions of second-order differential equations in quantum field theory Feynman integrals, highlighting their connection to elliptic functions, modular forms, and applications in high-order QCD calculations.

Contribution

It introduces the concept of iterative non-iterative integrals for second-order systems and discusses their representations using elliptic functions and modular forms.

Findings

Solutions can be expressed as $_2F_1$ functions.

Elliptic integrals and modular functions are relevant for certain Feynman integrals.

Applications include 3-loop QCD corrections and massive form factors.

Abstract

Single scale Feynman integrals in quantum field theories obey difference or differential equations with respect to their discrete parameter $N$ or continuous parameter $x$. The analysis of these equations reveals to which order they factorize, which can be different in both cases. The simplest systems are the ones which factorize to first order. For them complete solution algorithms exist. The next interesting level is formed by those cases in which also irreducible second order systems emerge. We give a survey on the latter case. The solutions can be obtained as general $_2F_1$ solutions. The corresponding solutions of the associated inhomogeneous differential equations form so-called iterative non-iterative integrals. There are known conditions under which one may represent the solutions by complete elliptic integrals. In this case one may find representations in terms of meromorphic modular functions, out of which special cases allow representations in the framework of elliptic polylogarithms with generalized parameters. These are in general weighted by a power of $1/\eta(\tau)$, where $\eta(\tau)$ is Dedekind's $\eta$-function. Single scale elliptic solutions emerge in the $\rho$-parameter, which we use as an illustrative example. They also occur in the 3-loop QCD corrections to massive operator matrix elements and the massive 3-loop form factors.