Towards testing the magnetic moment of the tau at one part per million

If physics beyond the Standard Model (BSM) explains the $4.2\sigma$ difference between the Standard-Model and measured muon anomalous magnetic moment, $a_{\mu}$, minimal flavor violation predicts a shift in the analog quantity for the $\tau$ lepton, $a_{\tau}$, at the $10^{-6}$ level, and even larger effects are possible in generic BSM scenarios such as leptoquarks. We show that this produces equivalent BSM deviations in the Pauli form factor, $F_2(s)$, at $s=(10\,\text{GeV})^2$, and report the first complete two-loop prediction of $\text{Re}\, F_2^\text{eff}(100\,\text{GeV}^2)=-268.77(50)\times 10^{-6}$ for resonant $\tau$-pair production in $e^+e^-\rightarrow \Upsilon(nS) \rightarrow$ $\tau^+\tau^-$, $n=1,2,3$. $\text{Re}\, F_2^\text{eff}$ can be measured from $e^-$-helicity-dependent transverse and longitudinal asymmetries in $\tau$-pair events, which requires a longitudinally polarized $e^-$ beam. We discuss how Belle II asymmetry measurements could probe $a_{\tau}^\text{BSM}$ at $10^{-6}$, assuming such a polarization upgrade of the SuperKEKB $e^+e^-$ collider, and conclude by outlining the next steps to be taken in theory and experiment along this new avenue for exploring realistic BSM effects in $a_\tau$.