Probing $τ$ lepton dipole moments at FCC-ee
We study the sensitivity of future high-luminosity $e^+e^-$ colliders to the electromagnetic and weak dipole moments of the $τ$ lepton within the Standard Model Effective Field Theory. We analyse several complementary processes across the FCC-ee operating stages and show that the exceptional luminosity and clean environment of future $e^+e^-$ colliders allow for substantial improvements over current limits from LEP and the LHC. The $e^+e^- \to τ^+τ^-$ process provides excellent sensitivity to weak dipole operators through resonant $Z$ production and $Z$-pole energy scans, while $γγ\to τ^+τ^-$ yields the strongest constraint on the anomalous magnetic moment owing to the dominance of linear interference effects in the near-threshold region. Higher-energy stages improve the sensitivity to the $τ$ electric dipole moment. The radiative processes $e^+e^- \to τ^+τ^-γ$ and $H\toτ^+τ^-γ$ provide complementary, though subleading, probes of $τ$ dipole operators. Combining all channels, we estimate the statistical reach at the 95% CL to be $|Δa_τ|<2.8\times10^{-5}$, $|d_τ|<2.5\times10^{-18}\,e\,{\rm cm}$, $|Δa_τ^Z|<1.0\times10^{-5}$, and $|d_τ^Z|<4.4\times10^{-19}\,e\,{\rm cm}$, improving upon the current bounds by several orders of magnitude. These projections quantify the potential statistical reach of the FCC-ee program, motivating dedicated studies incorporating detector effects, systematic uncertainties, and higher-order theoretical predictions.