In this contribution we present a lattice calculation of the leading-order electromagnetic and strong isospin-breaking (IB) corrections to the quark-connected hadronic-vacuum-polarization (HVP) contribution to the anomalous magnetic moment of the muon.
The results are obtained adopting the RM123 approach in the quenched-QED approximation and using the QCD gauge configurations generated by the ETM 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), at several lattice volumes and with pion masses between $\simeq 210$ and $\simeq 450$ MeV.
After the extrapolations to the physical pion mass and to the continuum and infinite-volume limits the contributions of the light, strange and charm quarks are respectively equal to $\delta a_\mu^{\rm HVP}(ud) = 7.1 ~ (2.5) \cdot 10^{-10}$, $\delta a_\mu^{\rm HVP}(s) = -0.0053 ~ (33) \cdot 10^{-10}$ and $\delta a_\mu^{\rm HVP}(c) = 0.0182 ~ (36) \cdot 10^{-10}$.
At leading order in $\alpha_{em}$ and $(m_d - m_u) / \Lambda_{QCD}$ we obtain $\delta a_\mu^{\rm HVP}(udsc) = 7.1 ~ (2.9) \cdot 10^{-10}$, which is currently the most accurate determination of the IB corrections to $a_\mu^{\rm HVP}$.