The hadronic vacuum polarization contribution to the muon $g-2$ at long distances

Abstract: We present our lattice QCD result for the long-distance part of the hadronic vacuum polarization contribution, $(a_\mu^{\rm hvp})^{\rm LD}$, to the muon $g-2$ in the time-momentum representation. This is the numerically dominant, and at the same time the most challenging part regarding statistical precision. Our calculation is based on ensembles with dynamical up, down and strange quarks, employing the O($a$)-improved Wilson fermion action with lattice spacings ranging from $0.035-0.099$ fm. In order to reduce statistical noise in the long-distance part of the correlator to the per-mille level, we apply low-mode averaging and combine it with an explicit spectral reconstruction. Our result is $(a_\mu^{\rm hvp})^{\rm LD} = 423.2(4.2){\rm stat}(3.4){\rm syst}\times 10^{-10}$ in isospin-symmetric QCD, where the pion decay constant is used to set the energy scale. When combined with our previous results for the short- and intermediate-distance window observables and after including all sub-dominant contributions as well as isospin-breaking corrections, we obtain the total leading-order hadronic vacuum polarization contribution as $a_\mu^{\rm hvp} = 724.5(4.9){\rm stat}(5.2){\rm syst}\times 10^{-10}$. Our result displays a tension of 3.9 standard deviations with the data-driven estimate published in the 2020 White Paper, but leads to a SM prediction for the total muon anomalous magnetic moment that agrees with the current experimental average.