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We present a lattice calculation of the leading Hadronic Vacuum Polarization(HVP) contribution of the light u- and d-quarks to the anomalous magneticmoment of the muon, $a_\mu^{\rm HVP}(ud)$, adopting the gauge configurationsgenerated by the European Twisted Mass Collaboration with $N_f = 2+1+1$dynamical quarks at three values of the lattice spacing with pion masses in therange 210 - 450 MeV. Thanks to several lattices at fixed values of thelight-quark mass and scale but with different sizes we perform a carefulinvestigation of finite-volume effects (FVEs). In order to remove FVEs wedevelop an analytic representation of the vector correlator, which describesthe lattice data for time distances larger than $\simeq 0.2$ fm. Therepresentation is based on quark-hadron duality at small and intermediate timedistances and on the two-pion contributions in a finite box at larger timedistances. After extrapolation to the physical pion point and to the continuumlimit we obtain $a_\mu^{\rm HVP}(ud) = 619.0~(17.8) \cdot 10^{-10}$. Adding thecontribution of strange and charm quarks, obtained by ETMC, and an estimate ofthe isospin-breaking corrections and quark-disconnected diagrams from theliterature we get $a_\mu^{\rm HVP}(udsc) = 683~(19) \cdot 10^{-10}$, which isconsistent with recent results based on dispersive analyses of the experimentalcross section data for $e^+ e^-$ annihilation into hadrons. Using our analyticrepresentation of the vector correlator, taken at the physical pion mass in thecontinuum and infinite volume limits, we provide the first eleven moments ofthe polarization function and we compare them with recent results of thedispersive analysis of the $\pi^+ \pi^-$ channels. We estimate also thelight-quark contribution to the missing part of $a_\mu^{\rm HVP}$ not coveredin the MUonE experiment.

The light-quark contribution to the leading HVP term of the muon $g - 2$ from twisted-mass fermions