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}$^2)=-268.77(50)\times 10^{-6}$forresonant$\tau$-pairproductionin$e^+e^-\rightarrow \Upsilon(nS) \rightarrow$ ${\tau }^{+}{\tau }^{-}$, $n=1,2,3$. $\text{Re}{F}_2^{\text{eff}}$ 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 $a_{\tau }$.