Improved calculation of radiative corrections to $\tau\to\pi\pi\nu_\tau$ decays

Abstract

A reliable calculation of radiative corrections to $\tau\to\pi\pi\nu_\tau$ decays is an important prerequisite for using hadronic $\tau$ decays for a data-driven evaluation of the hadronic-vacuum-polarization contribution to the anomalous magnetic moment of the muon, $a_\mu^\text{HVP, LO}[\pi\pi,\tau]$. In this Letter, we present an improved model-independent analysis of these radiative corrections, including, for the first time, effects beyond point-like pions in the evaluation of the loop diagrams. These structure-dependent corrections, implemented via a dispersive representation of the pion form factor, lead to significant changes compared to previous calculations due to enhancements near the $\rho(770)$ resonance. We also devise strategies for the matching to chiral perturbation theory and a stable implementation of the real corrections down to the two-pion threshold, which shows that some higher-order isospin-breaking corrections need to be kept due to a strong threshold enhancement. Finally, we perform dispersive fits to the currently available $\tau\to\pi\pi\nu_\tau$ spectra and discuss the consequences for isospin-breaking corrections in the evaluation of $a_\mu^\text{HVP, LO}[\pi\pi,\tau]$.