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

We compute the two-loop Quantum Chromodynamics (QCD) corrections to all partonic 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 a vector boson $V$ to three partons $V \to q\bar{q}g$, $V \to ggg$ with a vector and axial vector coupling in both channels, including singlet and non-singlet contributions. For the quark channel, we use a recent tensor decomposition and extend the calculation to $\mathcal{O}(\epsilon^2)$. For the gluonic channel, we define a new tensor decomposition which allows us to compute the vector and the axial vector amplitudes at once and to perform the computation of the amplitudes to $\mathcal{O}(\epsilon^2)$. We provide finite remainders of the helicity amplitudes analytically continued to all relevant scattering regions $q\bar{q} \to V g$, $q g \to V q$ and $gg \to V g$. The axial vector contribution to the gluon-induced channel completes the set of two-loop amplitudes for this process, while the extension to $\mathcal{O}(\epsilon^2)$ represents the first step in the calculation of next-to-next-to-next-to-leading-order (N$^3$LO) QCD corrections to $Z$+jet production at hadron colliders.