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

Abstract

We present the first lattice QCD calculation of the next-to-leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment with sub-percent precision. We employ the time-momentum representation for the space-like kernel, which is 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, covering six lattice spacings between $0.039$ and $0.097\,$fm and a range of pion masses including the physical value. After accounting for finite-size corrections and isospin-breaking effects, we obtain as our final, continuum-extrapolated result $a_\mu^{\mathrm{hvp,\,nlo}}=-101.57(26)_{\mathrm{stat}}(54)_{\mathrm{syst}}\times10^{-11}$. It lies below the estimate provided by the 2025 White Paper of the Muon $(g-2)$ Theory Initiative by $1.4\sigma$ but is two times more precise. It also exhibits a strong tension of $4.6\sigma$ with data-driven evaluations based on hadronic cross section measurements excluding the recent result by CMD-3.