High-precision measurements of the lepton’s electromagnetic dipole moments provide a powerful probe for testing the Standard Model. A nonzero value of the $\tau$ lepton’s electric(weak) dipole moment ($d_{\tau }^{\gamma }$, $d_{\tau }^Z$) would serve as a smoking gun of the new physics. On the one hand, current and future high energy colliders offer an ideal environment for such measurements. On the other hand, it is essential to investigate the optimal measurement method for extracting not only $d_{\tau }^{\gamma }$ and $d_{\tau }^Z$, but also the anomalous magnetic dipole moment ($a_{\tau }$). In this work, we analyze the precision of observables, particularly optimal observables, for determining these physics quantities through the process $e^{+}{e}^{-}\to {\gamma }^{*}/Z\to {\tau }^{+}{\tau }^{-}$ with $\tau$ leptons undergoing (semi)leptonic and hadronic decays at the center-of-mass energies at the Z pole mass and significantly below $m_Z$. By considering the full kinematic information of the decay products, we find that the sensitivities to the weak dipole moments, $\mathrm{Im}[d_{\tau }^Z]$ and $\mathrm{Re}[d_{\tau }^Z]$, can reach ${10}^{-21}e\mathrm{cm}$ at CEPC with $\sqrt{s}=m_Z$. This compares to sensitivities for the electric dipole moments, $\mathrm{Im}[d_{\tau }^{\gamma }]$ and $\mathrm{Re}[d_{\tau }^{\gamma }]$, of ${10}^{-20}e\mathrm{cm}$ at Belle-II $\mathrm{\Upsilon }(4S)$ resonance and ${10}^{-17}e\mathrm{cm}$ at BEPCII $\psi (2S)$ resonance. For $a_{\tau }$, we find that precisions of ${10}^{-5}$ for $\mathrm{Im}[a_{\tau }]$ and $\mathrm{Re}[a_{\tau }]$ can be attained at Belle-II $\sqrt{s}=10.58\mathrm{GeV}$.

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• Electric moment
• Lepton production
• Magnetic moment
• Phenomenology