Lattice determination of the higher-order hadronic vacuum polarization contribution to the muon $g-2$

We present the first lattice QCD calculation of the next-to-leading order (NLO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation combined with the spatially summed vector correlator computed on CLS ensembles with $N_{\mathrm{f}}=2+1$ flavors of $\mathrm{O}(a)$-improved Wilson fermions, spanning six lattice spacings ($0.039$-$0.097$fm) and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain in the continuum limit $a_{\mu }^{\mathrm{h}\mathrm{v}\mathrm{p},\mathrm{n}\mathrm{l}\mathrm{o}}=(-101.57\pm {0.26}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}\pm {0.54}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}})\times {10}^{-11}$, corresponding to a total relative error of 0.6$\mathrm{\%}$. Our result lies 1.4$\sigma$ below the estimate of the 2025 White Paper update and is two times more precise. It also shows a tension of $4.6\sigma$ with data-driven evaluations based on hadronic cross section measurements prior to the CMD-3 result.