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A precise measurement of the Higgs $h \to Z \gamma$ decay is very challengingat the LHC, due to the very low %SM $h \to Z \gamma\, (Z \to \ell \ell)$branching fraction and the shortage of kinematic handles to suppress the largeSM $Z \gamma$ background. We show how such a measurement would be significantlyimproved by considering Higgs production in association with a hard jet. Wecompare the prospective HL-LHC sensitivity in this channel with other Higgsproduction modes where $h$ is fairly boosted, e.g.~weak boson fusion, and alsoto the potential $h \to Z \gamma$ measurement achievable with a future$e^{+}e^{-}$ circular collider ({\sl fcc-ee}). Finally, we discuss new physicsimplications of a precision measurement of $h \to Z \gamma$.

A Boost to $h \to Z \gamma$: from LHC to Future $e^+ e^-$ Colliders