# A non-planar two-loop three-point function beyond multiple polylogarithms

## TL;DR

This paper computes a complex two-loop three-point function relevant for top-quark and photon production, introducing new methods to handle integrals beyond multiple polylogarithms using elliptic integrals.

## Contribution

It presents a novel approach for integrating differential equations without rationalization, and provides explicit integral representations for non-polylogarithmic integrals in two-loop calculations.

## Key findings

- All subtopology integrals expressed via multiple polylogarithms over irrational alphabet.
- Top topology integrals require elliptic integrals, not expressible by multiple polylogarithms.
- Provided one-fold integral representations for all solutions, enabling straightforward numerical evaluation.

## Abstract

We consider the analytic calculation of a two-loop non-planar three-point function which contributes to the two-loop amplitudes for $t \bar{t}$ production and $\gamma \gamma$ production in gluon fusion through a massive top-quark loop. All subtopology integrals can be written in terms of multiple polylogarithms over an irrational alphabet and we employ a new method for the integration of the differential equations which does not rely on the rationalization of the latter. The top topology integrals, instead, in spite of the absence of a massive three-particle cut, cannot be evaluated in terms of multiple polylogarithms and require the introduction of integrals over complete elliptic integrals and polylogarithms. We provide one-fold integral representations for the solutions and continue them analytically to all relevant regions of the phase space in terms of real function, extracting all imaginary parts explicitly. The numerical evaluation of our expressions becomes straightforward in this way.