Four-fermion operators, $Z$-boson exchange, and $\tau$ lepton dipole moments

Asymmetry measurements in $e^+e^-\to\tau^+\tau^-$ constitute a promising avenue to obtain competitive constraints on the $\tau$ dipole moments, the anomalous magnetic moment $a_\tau$ and the electric dipole moment $d_\tau$, especially, once a polarized electron beam becomes available, as possible at a future polarization upgrade of the SuperKEKB collider. While the main challenges concern the measurement of these asymmetries and the calculation of radiative corrections at the relevant level of precision, at subleading orders also electroweak effects and the potential impact of four-fermion operators parameterizing other beyond-the-Standard-Model scenarios besides those described by dipole operators need to be taken into consideration. Here, we show that $Z$-boson contributions arise at the level of $\simeq 3\times 10^{-6}$, while we estimate the largest possible effect from four-fermion operators as $\simeq 10^{-5} C \, v^2/\Lambda^2$. In addition, we observe that four-fermion-operator insertions at the loop level can probe Wilson coefficients that are otherwise not constrained directly, and that the imaginary part generated by insertions of the dipole operator at loop level opens another potential avenue towards a determination of $a_\tau$ without the need for a polarized electron beam. Despite the inherent loop suppression, a measurement of the required normal asymmetry $A_N^\pm$ with a precision of $\lesssim 10^{-5}$ would allow one to probe the Schwinger term, which could define an intermediate goal to be realized in the current setting at Belle II.