Isospin breaking corrections in $2\pi$ production in tau decays and $e^{+}{e}^{-}$ annihilation: consequences for the muon $g-2$ and CVC tests

Abstract

We revisit the isospin-breaking corrections relating the $e^{+}{e}^{-}$ hadronic cross-section and the tau decay spectral function, focusing on the di-pion channel, that gives the dominant contribution to the hadronic vacuum polarization piece of the muon $g-2$. We test different types of electromagnetic and weak form factors and show that both, the Gounaris-Sakurai and a dispersive-based approach, describe accurately $\tau$ lepton and $e^{+}{e}^{-}$ data (less when KLOE measurements are included in the fits) and comply reasonably well with analyticity constraints. From these results we obtain the isospin-breaking contribution to the conserved vector current (CVC) prediction of the $\mathrm{BR}(\tau \to \pi \pi {\nu }_{\tau })$ and to the $2\pi$ hadronic vacuum polarization (HVP) contribution to the muon $g-2$, in agreement with previous determinations and with similar precision. Our results abound in the convenience of using tau data-based results in the updated data-driven prediction of the muon $g-2$ in the Standard Model.

Comments: 32 pages, 10 figures, 14 tables. Minor changes. Version accepted for publication in PRD