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We present a detailed study of the exclusive radiative decays$Z\to\eta^{(\prime)}\gamma$ employing the QCD factorization approach. We derivea factorization formula for the decay amplitudes valid at leading power in anexpansion in $(\Lambda_{QCD}/m_Z)^2$, which includes convolutions of calculablehard-scattering kernels with the leading-twist quark and gluon light-conedistribution amplitudes of the mesons. Large logarithms arising in theevolution from the high scale $m_Z$ down to hadronic scales are resummed usingthe renormalization group, carefully accounting for the effects of the heavybottom and charm quarks. Our results for the branching ratios are verysensitive to hadronic input parameters, such as the decay constants and mixingangle characterizing the $\eta-\eta'$ system. Using the most recent estimatesof these parameters, we obtain the branching ratios $Br(Z\to\eta\gamma)\sim1.6\cdot 10^{-10}$ and $Br(Z\to\eta'\gamma)\sim 4.7\cdot 10^{-9}$. Ameasurement of these processes at a future high-luminosity Z factory couldprovide interesting information on the gluon distribution amplitude.

Exclusive Radiative Z-Boson Decays to Mesons with Flavor-Singlet Components