Iterative Non-iterative Integrals in Quantum Field Theory

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 decouple to linear differential equations which factorize to first-order. For them complete solution algorithms exist. The next interesting level is formed by those cases that decouple to linear differential equations in which also irreducible second-order factors 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 forms, 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.