Testing axion couplings to leptons in $Z$ decays at future $e^{+}{e}^{-}$ colliders

We study the possibility of probing the existence of a light, invisible, axion-like particle (ALP) $a$ in leptonic decays of the $Z$ boson at the proposed high-energy $e^{+}{e}^{-}$ colliders, CEPC and FCC-ee. Both projects plan to run at the $Z$ pole, collecting ${10}^{12}-{10}^{13}$ visible $Z$ decays. We show that, searching for the emission of an invisible ALP from leptons in leptonic $Z$ decays, this enormous statistics could allow to constrain the ALP couplings to leptons at an unprecedented level for laboratory experiments. In particular, within a Monte Carlo simulation framework, we estimate that CEPC/FCC-ee can be sensitive to the coupling of an invisible ALP to muons up to $f_a\mathrm{/}{C}_{\mu \mu }^A\approx 1$ TeV - where $f_a$ is the ALP decay constant - corresponding to $\mathrm{B}\mathrm{R}(Z\to {\mu }^{+}{\mu }^{-}a)\approx 3\times {10}^{-11}$.