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

TL;DR

This work advances the analytic and numerical treatment of a two-loop non-planar three-point function relevant to $gg\to t\bar t$ and $gg\to \gamma\gamma$ via a massive top loop. Subtopologies admit canonical differential equations solvable in terms of multiple polylogarithms over an irrational alphabet, while the top topology requires elliptic integrals; the authors develop one-fold integral representations and analytic continuations across all phase-space regions. A maximal-cut analysis yields the homogeneous elliptic solutions, which, through Euler's variation of constants, produce compact inhomogeneous representations that are real-valued in the intended regions and amenable to fast numerical evaluation. The results are cross-validated with SecDec and demonstrate a practical path toward handling two-loop non-planar integrals beyond MPLs, with potential extensions to more intricate three- and four-point functions.

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 $γγ$ 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.

Figures (1)

• Figure 1: Finite top-level master integrals $m^4\, I_{1,1,1,1,1,1,0}$ (left) and $m^6\, I_{1,2,1,1,1,1,0}$ (right). Solid curves were obtained with our representations, dots with SecDec 3.