Indirect searches for sterile neutrinos at a high-luminosity Z-factory

A future high-luminosity $Z$-factory will offer the possibility to study rare $Z$ decays, as those leading to lepton flavour violating final states. Processes such as $Z \to \ell_1^\mp \ell_2^\pm$ are potentially complementary to low-energy (high-intensity) observables of lepton flavour violation. In this work we address the impact of new sterile fermions on lepton flavour violating $Z$ decays, focusing on potential searches at FCC-ee (TLEP), and taking into account experimental and observational constraints on the sterile states. We consider a minimal extension of the Standard Model by one sterile fermion state, and two well-motivated frameworks of neutrino mass generation, the Inverse Seesaw embedded into the Standard Model, and the $\nu$MSM. Our study shows that sterile neutrinos can give rise to contributions to BR($Z \to \ell_1^\mp \ell_2^\pm$) within reach of the FCC-ee. We also discuss the complementarity between a high-luminosity $Z$-factory and low-energy charged lepton flavour violation facilities.