Novel and Updated Bounds on Flavor-violating Z Interactions in the Lepton Sector

We investigate the experimental bounds on the Flavor-Violating (FV) couplings of the $Z$ boson to the charged leptons. In addition to the direct LHC searches for FV $Z$ decays to leptons, we investigate indirect bounds from flavor-conserving $Z$ decays to leptons at 1-loop, bounds from LEP searches, Electroweak Precision Observables (EWPO), $\ell_{i}\to\ell_{j}\gamma$ decays, $\ell_{i}\to3\ell_{j}$ decays, $\ell_{i}\to\ell_{j}+\text{inv.}$ decays, FV meson decays to leptons, FV $\tau$ decays to $\mu(e)$ + mesons, muon conversion in nuclei, and from muonium-antimuonium oscillations. For FV $Z$ couplings to $\tau\mu$, we find that $\tau\to\mu\gamma$ yields the strongest bounds, with a level reaching $\mathcal{O}(10^{-5})$, followed by bounds from $\tau\to3\mu(\mu ee)$. For FV $Z$ couplings to $\tau e$, we find that the strongest bounds arise from the decay $\tau\to\mu\mu e$, reaching $\mathcal{O}(10^{-7})$ as well, with bounds from $\tau\to3e$ also yielding strong bounds. For FV $Z$ couplings to $\mu e$, we find that the strongest bounds are obtained from the decay $\mu\to3e$, reaching $\mathcal{O}(10^{-11})$, with bounds from $\mu\to e\gamma$, muon conversion, $K_{L}^{0}\rightarrow\mu e$ and $\mu\to e+\text{inv.}$ also providing strong bounds. We also study projections from future experiments, such as the FCC-ee, Belle II and the Mu2e experiment. For the $Z$ couplings to $\tau\mu$, we find that future experiments could improve the bound to $\mathcal{O}(10^{-6})$, whereas for the $Z$ couplings to $\tau e$, we find that future experiments could improve the bound to $\mathcal{O}(10^{-8})$, and for the $Z$ couplings to $\mu e$, they could improve the bound to $\mathcal{O}(10^{-13})$