Toward testing the magnetic moment of the tau at one part per million
Andreas Crivellin, Martin Hoferichter, J. M. Roney
Abstract
If physics beyond the Standard Model (BSM) explains the $4.2\ensuremath{\sigma}$ difference between the Standard Model and measured muon anomalous magnetic moment ${a}_{\ensuremath{\mu}}$, minimal flavor violation predicts a shift in the analog quantity for the $\ensuremath{\tau}$ lepton ${a}_{\ensuremath{\tau}}$ at the ${10}^{\ensuremath{-}6}$ 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 $s=(10\text{ }\text{ }\mathrm{GeV}{)}^{2}$ and report the first complete two-loop prediction of $\mathrm{Re}\text{ }{F}_{2}^{\mathrm{eff}}(100\text{ }\text{ }{\mathrm{GeV}}^{2})=\ensuremath{-}268.77(50)\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}6}$ for resonant $\ensuremath{\tau}$-pair production in ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}\mathrm{\ensuremath{\Upsilon}}(nS)\ensuremath{\rightarrow}{\ensuremath{\tau}}^{+}{\ensuremath{\tau}}^{\ensuremath{-}}$, $n=1$, 2, 3. $\mathrm{Re}\text{ }{F}_{2}^{\mathrm{eff}}$ can be measured from ${e}^{\ensuremath{-}}$-helicity-dependent transverse and longitudinal asymmetries in $\ensuremath{\tau}$-pair events, which requires a longitudinally polarized ${e}^{\ensuremath{-}}$ beam. We discuss how Belle II asymmetry measurements could probe ${a}_{\ensuremath{\tau}}^{\mathrm{BSM}}$ at ${10}^{\ensuremath{-}6}$, assuming such a polarization upgrade of the SuperKEKB ${e}^{+}{e}^{\ensuremath{-}}$ 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 ${a}_{\ensuremath{\tau}}$.