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The ratios among the leading-order (LO) hadronic vacuum polarization (HVP)contributions to the anomalous magnetic moments of electron, muon andtau-lepton, $a_{\ell=e,\mu \tau}^{HVP,LO}$, are computed using lattice QCD+QEDsimulations. The results include the effects at order $O(\alpha_{em}^2)$ aswell 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 gaugeconfigurations generated by the Extended Twisted Mass Collaboration with$N_f=2+1+1$ dynamical quarks at three values of the lattice spacing ($a \simeq0.062, 0.082, 0.089$ fm) with pion masses in the range 210 - 450 MeV. We showthat in the case of the electron-muon ratio the hadronic uncertainties in thenumerator and in the denominator largely cancel out, while in the cases of theelectron-tau and muon-tau ratios such a cancellation does not occur. For theelectron-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 $\simeq 0.7 \%$. Ourresult, which represents an accurate Standard Model (SM) prediction, agreesvery well with the estimate obtained using the results of dispersive analysesof the experimental $e^+ e^- \to$ hadrons data. Instead, it differs by $\simeq2.7$ 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 theexperiment 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 theratio $R_{e/\mu }$ at a significance level of $\simeq 5$ standard deviations.

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