Two-Loop QCD Helicity Amplitudes for (2+1)-Jet Production in Deep Inelastic Scattering

TL;DR

The paper derives two-loop QCD helicity amplitudes for DIS (2+1)-jet production by analytically continuing from $\gamma^* \to q\bar q g$, and uses Catani's infrared factorization to separate poles from finite parts. The finite parts are expressed in terms of harmonic polylogarithms (HPLs) and two-dimensional HPLs (2dHPLs), with a sectorized kinematic treatment to handle the DIS-specific, space-like momentum regions. Renormalized, color-decomposed coefficients are computed, including explicit infrared structure and a finite remainder decomposed into multiple color and flavor classes; numerical evaluation is facilitated by sector mappings and provided in code form. These results supply the essential virtual NNLO ingredients for DIS (2+1)-jet cross sections and polarized observables, contingent on a consistent NNLO subtraction scheme to combine with real-emission contributions for finite predictions.

Abstract

We derive the two-loop QCD helicity amplitudes for the processes $l q \to l qg$ ($l \bar q \to l \bar q g$) and $l g \to l q\bar q$, which are the partonic reactions yielding $(2+1)$-jet final states in deep inelastic lepton nucleon scattering. The amplitudes are obtained by analytic continuation of the known helicity amplitudes for $e^+e^- \to q\bar q g$. We separate the infrared divergent and finite parts of the amplitudes using Catani's infrared factorization formula. The analytic results for the finite parts of the amplitudes are expressed in terms of one- and two-dimensional harmonic polylogarithms. To evaluate these functions numerically, we list in detail the non-trivial (and kinematic region dependent) variable transformations one needs to perform.