Two-loop QCD corrections to $ZH$ and off-shell $Z$ boson pair production in gluon fusion

We compute two-loop corrections to the associated production of a Higgs boson with a $Z$ boson and to off-shell $Z$ boson pair production in the gluon fusion channel, mediated by a heavy quark. We perform deep expansions in the high-energy region and around the forward limit and show that their combination covers the whole phase space. Our results constitute the next-to-leading order virtual corrections to these processes. Their numerical evaluation is fast and the dependence on all parameters is maintained, thus a change of parameter values or renormalization scheme is straightforward. As a by-product of our calculation, we also obtain the two-loop heavy quark mediated virtual corrections to the processes of off-shell di-photon and photon-$Z$ production.