Two-Loop QCD Helicity Amplitudes for $e^+e^- \to 3$~Jets
L. W. Garland, T. Gehrmann, E. W. N. Glover, A. Koukoutsakis, E. Remiddi
TL;DR
The paper develops a projection-based framework to derive analytic one- and two-loop QCD helicity amplitudes for $e^+e^- \to q\bar{q}g$ by decomposing the hadronic current into a gauge-invariant tensor basis and extracting coefficients with explicit $d$-dimensional projectors. Loop integrals are reduced to a master set of planar and non-planar topologies, with ultraviolet renormalization in the $\overline{MS}$ scheme and infrared structure organized via Catani’s factorization, yielding finite remainders expressed in harmonic polylogarithms and two-dimensional HPLs. Helicity amplitudes, implemented in the Weyl–van der Waerden formalism, are extended to include $Z$ and $\gamma$ exchange and initial-state polarization, with leading-color finite pieces provided to high weight. This work provides essential ingredients for NNLO predictions of three-jet observables and polarization-sensitive event shapes, and connects with prior results while enabling more complete, polarization-aware analyses beyond unpolarized cross sections.
Abstract
We compute the two-loop QCD helicity amplitudes for the process e^+e^- --> q bar{q} g. The amplitudes are extracted in a scheme-independent manner from the coefficients appearing in the general tensorial structure for this process. The tensor coefficients are derived from the Feynman graph amplitudes by means of projectors, within the conventional dimensional regularization scheme. The actual calculation of the loop integrals is then performed by reducing all of them to a small set of known master integrals. The infrared pole structure of the renormalized helicity amplitudes agrees with the prediction made by Catani using an infrared factorization formula. We use this formula to structure our results for the finite part into terms arising from the expansion of the pole coefficients and a genuine finite remainder, which is independent of the scheme used to define the helicity amplitudes. The analytic result for the finite parts of the amplitudes is expressed in terms of one- and two-dimensional harmonic polylogarithms.