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We compute the two-loop Quantum Chromodynamics (QCD) corrections to allpartonic channels relevant for the production of an electroweak boson$V=Z,W^\pm,\gamma^*$ and a jet at hadron colliders. We consider the decay of avector boson $V$ to three partons $ V \to q\bar{q}g$, $ V \to ggg$ with avector and axial vector coupling in both channels, including singlet andnon-singlet contributions. For the quark channel, we use a recent tensordecomposition and extend the calculation to $\mathcal{O}(\epsilon^2)$. For thegluonic channel, we define a new tensor decomposition which allows us tocompute the vector and the axial vector amplitudes at once and to perform thecomputation of the amplitudes to $\mathcal{O}(\epsilon^2)$. We provide finiteremainders of the helicity amplitudes analytically continued to all relevantscattering regions $q\bar{q} \to V g$, $q g \to V q$ and $gg \to V g$. Theaxial vector contribution to the gluon-induced channel completes the set oftwo-loop amplitudes for this process, while the extension to$\mathcal{O}(\epsilon^2)$ represents the first step in the calculation ofnext-to-next-to-next-to-leading-order (N$^3$LO) QCD corrections to $Z$+jetproduction at hadron colliders.

Two-loop helicity amplitudes for $V+$jet production including axial vector couplings to higher orders in $\epsilon$