Planar three-loop QCD helicity amplitudes for V plus jet production at hadron colliders

We compute the planar three-loop Quantum Chromodynamics (QCD) corrections to the helicity amplitudes involving a vector boson v = Z, W-+/-, gamma* two quarks and a gluon. These amplitudes are relevant to vector-boson-plus-jet production at hadron colliders and other precision QCD observables. The planar corrections encompass the leading colour factors N-3, N-2 N-f, NNf2 and N-f(3). We provide the finite remainders of the independent helicity amplitudes in terms of multiple polylogarithms, continued to all kinematic regions and in a form which is compact and lends itself to efficient numerical evaluation. The presented amplitude respects the conjectured symbol-adjacency constraints for amplitudes with three massless and one massive leg.