  We investigate the renormalization of the radiative decays of the Higgs to
two gauge bosons in the Standard Model Effective Field Theory at mass dimension
eight. Given that these are loop-level processes, their one-loop
renormalization can be phenomenologically important when triggered by operators
generated through the tree-level exchange of heavy particles (assuming a weakly
coupled UV model). By computing the tree-level matching conditions of all
relevant extensions of the Standard Model, we demonstrate that this effect is
indeed present in the $h\to \gamma Z$ decay at dimension eight, even though it
is absent at dimension six. In contrast, the $h\to gg$ and $h\to \gamma\gamma$
decays can only be renormalized by operators generated by one-loop processes.
For UV models with heavy vectors, this conclusion hinges on the specific form
of their interaction with massless gauge bosons which is required for
perturbative unitarity. We study the quantitative impact of the possible
logarithmic enhancement of $h\to \gamma Z$, and we propose an observable to
boost the sensitivity to this effect. Given the expected increased precision of
next-generation high-energy experiments, this dimension-eight contribution
could be large enough to be probed and could therefore give valuable clues
about new physics by revealing some of its structural features manifesting
first at dimension eight.
