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Higgs-boson production in association with a Dark Photon in $e^+ e^-$ collisions

We study the production of a Higgs boson recoiling from a massless invisible system in $e^ + e^ -$ collisions. This is a quite distinctive signature that can arise when the Higgs boson is produced in association with a massless dark photon, which can happen in BSM scenarios foreseeing an extra unbroken $U(1)$ gauge group. Dark photons can indeed acquire effective couplings to the Higgs boson as occurs in models recently proposed to generate exponentially-spread Yukawa couplings. We analyze the signal and corresponding backgrounds for $H\to b\bar{b}$, and estimate ILC and FCC-ee sensitivities in a model-independent way.