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We derive the two-loop QCD helicity amplitudes for the processes $l q \to lqg$ ($l \bar q \to l \bar q g$) and $l g \to l q\bar q$, which are the partonicreactions yielding $(2+1)$-jet final states in deep inelastic lepton nucleonscattering. The amplitudes are obtained by analytic continuation of the knownhelicity amplitudes for $e^+e^- \to q\bar q g$. We separate the infrareddivergent and finite parts of the amplitudes using Catani's infraredfactorization formula. The analytic results for the finite parts of theamplitudes are expressed in terms of one- and two-dimensional harmonicpolylogarithms. To evaluate these functions numerically, we list in detail thenon-trivial (and kinematic region dependent) variable transformations one needsto perform.

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