Ratios of the hadronic contributions to the lepton $g-2$ from Lattice QCD+QED simulations

The ratios among the leading-order (LO) hadronic vacuum polarization (HVP) contributions to the anomalous magnetic moments of electron, muon and tau-lepton, $a_{\ell=e,\mu \tau}^{HVP,LO}$, are computed using lattice QCD+QED simulations. The results include the effects at order $O(\alpha_{em}^2)$ as well as the electromagnetic and strong isospin-breaking corrections at orders $O(\alpha_{em}^3)$ and $O(\alpha_{em}^2(m_u-m_d))$, respectively, where $(m_u-m_d)$ is the $u$- and $d$-quark mass difference. We employ the gauge configurations generated by the Extended Twisted Mass Collaboration with $N_f=2+1+1$ dynamical quarks at three values of the lattice spacing ($a \simeq 0.062, 0.082, 0.089$ fm) with pion masses in the range 210 - 450 MeV. We show that in the case of the electron-muon ratio the hadronic uncertainties in the numerator and in the denominator largely cancel out, while in the cases of the electron-tau and muon-tau ratios such a cancellation does not occur. For the electron-muon ratio we get $R_{e/\mu } \equiv (m_\mu/m_e)^2 (a_e^{HVP,LO} / a_\mu^{HVP,LO}) = 1.1456~(83)$ with an uncertainty of $\simeq 0.7 \%$. Our result, which represents an accurate Standard Model (SM) prediction, agrees very well with the estimate obtained using the results of dispersive analyses of the experimental $e^+ e^- \to$ hadrons data. Instead, it differs by $\simeq 2.7$ standard deviations from the value expected from present electron and muon (g - 2) experiments after subtraction of the current estimates of the QED, electro-weak, hadronic light-by-light and higher-order HVP contributions, namely $R_{e/\mu} = 0.575~(213)$. An improvement of the precision of both the experiment and the QED contribution to the electron (g - 2) by a factor of $\simeq 2$ could be sufficient to reach a tension with our SM value of the ratio $R_{e/\mu }$ at a significance level of $\simeq 5$ standard deviations.