We derive the two-loop QCD helicity amplitudes for the processes ℓq → ℓqg (ℓ¯q→ℓ¯qg) and ℓg→ℓq¯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 − → q¯qg. 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 The study of the hadronic final state in deep inelastic lepton-nucleon scattering (DIS) allows the determination of aspects of the nucleon structure which are not accessible in inclusive scattering as well as testing predictions of quantum chromodynamics (QCD) to a high accuracy. In particular, the two-plus-one-jet production  rate in DIS can be used both to probe the gluon distribution and to enable a precise determination of th
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