Transverse Tau Spin Correlations and the Weak Electric Dipole Moment at Future $Z$ Factories

Abstract

Tau spin correlations at the $Z$ pole probe CP violation with a sensitivity that depends strongly on the spin projection. For a fixed $Z$ sample, transverse moments avoid the chirality suppression of longitudinal and triple-product observables, enhancing the response by a factor $m_Z/2m_τ=26$. Two transverse moments also select the axial tau charge, gaining $|a_τ/v_τ|=13.5$; the sign-weighted normal moment has a response $521$ times that of the triple product. These two transverse moments give single-parameter uncertainties no more than $14\%$ above our calculated Cramer-Rao bound for independent single-pion measurements, approaching the best precision within this analysis. A five-moment fit with $5\%$ off-peak luminosity yields statistical uncertainties of $4.5 (5.5)\times10^{-7}$ and $2.7 (3.2)\times10^{-6}$ on $\operatorname{Re}\widetilde{d}_τ^W$ and $\operatorname{Im}\widetilde{d}_τ^W$, respectively, at FCC-ee (CEPC), improving on ALEPH by three orders of magnitude. The electromagnetic-dipole uncertainty is $9.6 (12)\times10^{-6}$ on $\operatorname{Re}\widetilde{d}_τ^γ$, two orders beyond Belle. These projections assume ideal angular reconstruction and constant form factors across the scan.