Two-loop amplitudes for e+ e- -> q qbar g: the nf-contribution

TL;DR

The paper computes the $n_f$-dependent part of the two-loop amplitude for $e^+ e^- \to q \bar{q} g$ as a key component of NNLO predictions for $e^+ e^- \to 3$-jets. It introduces an efficient nested-sums framework that maps tensor integrals to scalar integrals, yielding results in terms of multiple polylogarithms with simple arguments suitable for analytic continuation. The finite part, including $c_{12}^{(2),fin}$, is given as a function of the kinematic variables $x_1$ and $x_2$, with auxiliary functions $R$ and $R_1$, and its infrared structure is validated against Catani's factorization with cross-checks against prior work (e.g., Garland:2001tf). These nf-contributions provide a robust, transferable building block for full NNLO 3-jet phenomenology and related processes such as DIS and hadron-hadron production; subsequent independent calculations confirmed agreement with the complete two-loop amplitude.

Abstract

We discuss the calculation of the nf-contributions to the two-loop amplitude for e+ e- -> q qbar g. The calculation uses an efficient method based on nested sums. The result is presented in terms of multiple polylogarithms with simple arguments, which allow for analytic continuation in a straightforward manner.