Electric dipole moment of the tau lepton revisited

We reconsider the issue of search for a nonzero electric dipole form factor (EDM) ${d}_{\ensuremath{\tau}}(s)$ using optimal observables in ${\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}}$ production by ${e}^{+}{e}^{\ensuremath{-}}$ collisions center-of-mass energy range from $\ensuremath{\tau}$-pair threshold to about $\sqrt{s}\ensuremath{\sim}15\text{ }\text{ }\mathrm{GeV}$. discuss general formalism and apply it two $CP$-odd that are sensitive real imaginary part ${d}_{\ensuremath{\tau}}(s)$, respectively. compute expectation values covariances these $CP$ at $\sqrt{s}=10.58\text{ }\mathrm{GeV}$ with subsequent decays ${\ensuremath{\tau}}^{\ifmmode\pm\else\textpm\fi{}}$ into major leptonic or semihadronic modes. For $\ensuremath{\tau}$ pions three charged we take full kinematic information hadronic system account. Assuming Belle II experiment KEKB accelerator will eventually analyze data corresponding an integrated luminosity $50\text{ }{\mathrm{ab}}^{\ensuremath{-}1}$ applying acceptance cuts on final-state pions, find 1 s.d. sensitivities $\ensuremath{\delta}\mathrm{Re}{d}_{\ensuremath{\tau}}=6.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }e\text{ }\mathrm{cm}$ $\ensuremath{\delta}\mathrm{Im}{d}_{\ensuremath{\tau}}=4.0\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ can be obtained events where both $\ensuremath{\tau}$'s decay semihadronically. consider also ideal case no particles applied. With $4.5\ifmmode\times\else\texttimes\fi{}{10}^{10}$ $\ensuremath{\delta}\mathrm{Re}{d}_{\ensuremath{\tau}}=5.8\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ $\ensuremath{\delta}\mathrm{Im}{d}_{\ensuremath{\tau}}=3.2\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}20}\text{ }\mathrm{cm}$, again leptons Furthermore, potential magnitude EDM type-II two-Higgs doublet extension scalar leptoquark extensions Standard Model, taking account phenomenological constraints.