Data-based form factor corrections between the two-pion $\tau$ and $e^+e^-$ spectral functions

The $\tau$ spectral functions are an alternative to $e^+e^-$ cross-sections, where different measurements are not consistent, for computing the hadronic vacuum contribution to the muon magnetic anomaly $a_\mu$. This requires a control of isospin-breaking effects which have to be corrected for. So far these corrections have been evaluated using theoretical models. In this letter, a new approach based only on data is presented for the determination of the most critical correction relating the $e^+e^-$ and $\tau$ pion form factors. An updated evaluation of the total isospin-breaking correction is given and its impact is discussed in the context of $e^+e^-$-based $a_\mu$ predictions and of the direct measurement.