Iterative Noniterative Integrals in Quantum Field Theory
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 socalled iterative noniterative 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 3loop QCD corrections to massive operator matrix elements and the massive 3loop form factors.
 Publication:

arXiv eprints
 Pub Date:
 August 2018
 arXiv:
 arXiv:1808.08128
 Bibcode:
 2018arXiv180808128B
 Keywords:

 High Energy Physics  Theory;
 Mathematical Physics
 EPrint:
 27 pages Latex, 3 figures