Helicity Amplitudes for $\mathcal{O}({\alpha }_s)$ Production of $W^{+}{W}^{-},W^{\pm }Z,ZZ,W^{\pm }\gamma ,$ or $Z\gamma$ Pairs at Hadron Colliders

We present the one-loop QCD corrections to the helicity amplitudes for the processes $q\text{qb}\to W^{+}{W}^{-},ZZ,W^{\pm }Z,W^{\pm }\gamma$, or $Z\gamma$, including the subsequent decay of each massive vector boson into a pair of leptons. We also give the corresponding tree-level amplitudes with an additional gluon radiated off the quark line. Together, these amplitudes provide all the necessary input for the calculation of the next-to-leading order QCD corrections to the production of any electroweak vector boson pair at hadron colliders, including the full spin and decay angle correlations.