The electric dipole moment of the tau lepton revisited

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