The Two-Loop QCD Matrix Element for $e^+e^- \to 3$ Jets
L. W. Garland, T. Gehrmann, E. W. N. Glover, A. Koukoutsakis, E. Remiddi
TL;DR
This work delivers the analytic ${\cal O}(\alpha_s^3)$ virtual QCD corrections to the ${\gamma^*}\to q\bar q g$ matrix element, a key ingredient for NNLO predictions of the e+e- to 3 jets channel. By reducing a large set of two-loop integrals to a master basis of 24 massless four-point integrals and expanding them in dimensional regularization, the authors provide explicit infrared poles consistent with Catani’s factorization and complete finite terms in terms of one- and two-dimensional harmonic polylogarithms. The analysis relies on two independent IBP/LI reduction strategies for reliability, and the master integrals are expressed in a representation that handles all kinematic configurations through a unified set of 2dHPLs. These results constitute a crucial step toward a full NNLO treatment of three-jet production, including necessary real-emission and PDF contributions, and they set the stage for extension to ep and hadron-collider processes.
Abstract
We compute the ${\cal O}(α_s^3)$ virtual QCD corrections to the $γ^*\to q\bar q g$ matrix element arising from the interference of the two-loop with the tree-level amplitude and from the self-interference of the one-loop amplitude. The calculation is performed by reducing all loop integrals appearing in the two-loop amplitude to a small set of known master integrals. Infrared and ultraviolet divergences are both regularized using conventional dimensional regularization, and the ultraviolet renormalization is performed in the $\bar{MS}$ scheme. The infrared pole structure of the matrix elements agrees with the prediction made by Catani using an infrared factorization formula. The analytic result for the finite terms of both matrix elements is expressed in terms of one- and two-dimensional harmonic polylogarithms.