Two-loop amplitudes with nested sums: Fermionic contributions to e+ e- --> q qbar g

TL;DR

This work computes the fermionic ($n_f$) contributions to the two-loop amplitude for $e^+e^- \to q \bar{q} g$ and provides the full one-loop amplitude to ${\cal O}(\varepsilon^2)$, framed within dimensional regularization. The authors introduce and apply a nested-sums method to evaluate two-loop integrals with arbitrary propagator powers, yielding results in terms of multiple polylogarithms that admit straightforward analytic continuation. They verify their nf two-loop results against an independent calculation by Garland et al. and ensure consistency with ultraviolet renormalization and infrared factorization (Catani) structures. The approach offers a fast, general pathway to NNLO predictions for $e^+e^- \rightarrow 3\text{ jets}$ and related processes, with potential applications to deep-inelastic scattering and vector-boson production, and sets the stage for completing the non-$n_f$ two-loop contributions.

Abstract

We present the calculation of the nf-contributions to the two-loop amplitude for e+ e- --> q qbar g and give results for the full one-loop amplitude to order eps^2 in the dimensional regularization parameter. Our results agree with those recently obtained by Garland et al.. The calculation makes extensive use of an efficient method based on nested sums to calculate two-loop integrals with arbitrary powers of the propagators. The use of nested sums leads in a natural way to multiple polylogarithms with simple arguments, which allow a straightforward analytic continuation.

Figures (3)

• Figure 1: Labelling of the external legs for the amplitude. The lepton pair is denoted by $p_4$ and $p_5$.
• Figure 2: The PentaBox represents a typical diagram where the triangle relation can be applied. In the present calculation, the lower left leg is massive.
• Figure 3: The generic $\hbox{CBox}$. For the $\hbox{CBox}_2$, the upper right external leg is massive.