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

Abstract

Abstract:We consider the analytic calculation of a two-loop non-planar three-point function which contributes to the two-loop amplitudes for tt 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 functions, extracting all imaginary parts explicitly. The numerical evaluation of our expressions becomes straightforward in this way.

Elliptic polylogarithms and Feynman parameter integrals

“…(3.1) was studied in ref. [37] by means of the differentialequation method, where it was shown that there are two master integrals for the top topology which satisfy a coupled two-dimensional system. 4 We choose as basis integrals…”

“…[52], the prefactors Ω and Ω (tt) 2 have a natural interpretation in terms of the maximal cuts of the integrals M 1 and M 2 . In order to see why this is the case, let us recall that the two master integrals M 1 and M 2 fulfil a system of two coupled differential equations which, neglecting the subtopologies, read [37]…”

NNLO QCD corrections to γ + η c (η b ) exclusive production in electron-positron collision

“…The calculation of massive multi-loop Feynman integrals is a challenging task due to the appearance of new mathematical structures, which cannot be reduced to the well-known multiple polylogarithms yet. Thanks to the very recent developments in differential equation technique in dealing with multiscale Feynman integrals [26][27][28][29][30], we are now able to calculate those massive multi-loop Feynman integrals of our concern beyond multiple polylogarithms, which are found can be classified into…”

Differential equations for Feynman integrals beyond multiple polylogarithms

“…We are interested in the ones, which are not expressible in in terms of multiple polylogarithms [40,41,[67][68][69][70][71], but are expressible in terms of elliptic generalisations of these functions. Therefore we expect in the differential equation for a given master integral irreducible second-order factors.…”