Higher-order hadronic-vacuum-polarization contribution to the muon g-2 from lattice QCD

We introduce a new method for calculating the $\mathrm{O}({\alpha }^3)$ hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment from $ab-initio$ lattice QCD. We first derive expressions suitable for computing the higher-order contributions either from the renormalized vacuum polarization function $\widehat{\mathrm{\Pi }}(q^2)$, or directly from the lattice vector-current correlator in Euclidean space. We then demonstrate the approach using previously-published results for the Taylor coefficients of $\widehat{\mathrm{\Pi }}(q^2)$ that were obtained on four-flavor QCD gauge-field configurations with physical light-quark masses. We obtain $10^{10} a_\mu^{\rm HVP,HO} = -9.3(1.3)$, in agreement with, but with a larger uncertainty than, determinations from $e^{+}{e}^{-}\to \mathrm{h}\mathrm{a}\mathrm{d}\mathrm{r}\mathrm{o}\mathrm{n}\mathrm{s}$ data plus dispersion relations.