On the leptonic decays of the $\tau$

Abstract

In view of potential future improvements in the branching fractions of leptonic $\tau$ decays $\tau\to\ell\nu_\tau\bar\nu_\ell$, $\ell=e,\mu$, it is timely to derive the corresponding Standard-Model predictions beyond next-to-leading order. We provide such benchmark values for the fully inclusive decay widths, profiting from higher-order corrections that have been studied in the $\overline{\text{MS}}$ scheme up to three-loop order in the case of muon decay. To obtain complete next-to-next-to-leading-order results, we perform a numerical calculation of the two-loop coefficients in the on-shell scheme, including an evaluation of the hadronic effects, and compare the leptonic results to the known $\overline{\text{MS}}$ coefficients where available. Our final results are presented in the $\overline{\text{MS}}$ scheme, to be able to also include an estimate of the three-loop contribution. In addition, we comment on radiative and rare decays, whose consideration is critical for the conversion between $\overline{\text{MS}}$ and on-shell results.