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 $a_μ^{\mathrm{hvp,\,nlo}}=-101.69(25)_{\mathrm{stat}}(53)_{\mathrm{syst}}\times10^{-11}$. It lies below the estimate provided by the 2025 White Paper of the Muon $(g-2)$ Theory Initiative by 1.5$σ$ but is two times more precise. It also exhibits a strong tension of 4.8$σ$ with data-driven evaluations based on hadronic cross section measurements excluding the recent result by CMD-3.

Figures (15)

• Figure 1: Feynman diagrams contributing to the hadronic-vacuum polarization to the muon $g-2$ up to $\mathrm{O}(\alpha^3)$. On the left side, we show the LO diagram, contributing at order $\alpha^2$, along with its electromagnetic isospin breaking correction. On the right side we show all the higher-order diagrams to be studied in this work. The diagram set NLOa considers all possible corrections coming from photon-lines and muon-loops. The set NLOb consists of corrections coming from electron- and tau-loops. Finally, diagram NLOc is composed by two QCD insertions.
• Figure 2: Comparison of the time-kernels for the LO and NLO diagrams, the addition NLOa&b$\equiv$NLOa+NLOb is shown as a solid blue line.
• Figure 3: Decomposition of the integrand for NLOb in the isospin basis (left) and in the three characteristic windows for the isovector channel (right). The data used for these plots comes from ensemble E250.
• Figure 4: Modification of the short-distance window applied to the time-kernel for diagram set NLOb following the subtraction in Eq. \ref{['eq:subtraction_def']} for all explored auxiliary virtualities in this work.
• Figure 5: On the left-hand panel we show the continuum extrapolation of the SD window for the isovector channel of the combined NLOa&b contribution. Each line corresponds to a fit, with its opacity proportional to the AIC weight. The leftmost dark vertical line indicates the continuum limit, while the six vertical dotted lines correspond to the lattice spacings available in the CLS ensembles. The black point denotes the final estimate. On the right-hand panel we show a representative chiral-continuum extrapolation of $\Delta_{ls}(a_\mu)$ also for NLOa&b, shown here for the local-conserved (lc) discretization and improvement set 1. The solid vertical line indicates the physical pion mass, and the black point corresponds to the physical-point estimate for this fit.