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

Abstract

If physics beyond the Standard Model (BSM) explains the $4.2σ$ difference between the Standard-Model and measured muon anomalous magnetic moment, $a_μ$, minimal flavor violation predicts a shift in the analog quantity for the $τ$ lepton, $a_τ$, 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 $τ$-pair production in $e^+e^-\rightarrow Υ(nS) \rightarrow$ $τ^+τ^-$, $n=1,2,3$. $\text{Re}\, F_2^\text{eff}$ can be measured from $e^-$-helicity-dependent transverse and longitudinal asymmetries in $τ$-pair events, which requires a longitudinally polarized $e^-$ beam. We discuss how Belle II asymmetry measurements could probe $a_τ^\text{BSM}$ 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 $a_τ$.