Mini-review of charmonium weak decays at BESIII

The weak decays of charmonium states such as $J/\psi$ and $\psi(2S)$ are instrumental in probing both nonperturbative QCD dynamics and the flavor structure of the Standard Model (SM). The extreme rarity of charmonium weak decays renders them highly sensitive to physics beyond the SM, particularly in channels that are heavily suppressed in the SM, such as flavor-changing neutral-current (FCNC) decays. This review highlights the critical role of the BESIII experiment, which leverages an unprecedented data sample of over $10^{10}$ $J/\psi$ and $2.7\times10^{9}$ $\psi(2S)$ events to achieve leading sensitivity in searches for charmonium weak decays. We present the latest and most stringent upper limits established by BESIII on various semileptonic, nonleptonic, and FCNC charmonium weak decay channels.