What the paper is about: The authors report the first lattice quantum chromodynamics (QCD) calculation, with sub‑percent precision, of the next‑to‑leading order (NLO) hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment (called g−2). The muon g−2 is a very precise quantity that can reveal discrepancies with the Standard Model. The HVP part comes from the effects of quarks and gluons on the electromagnetic field seen by the muon. The paper finds aNLO_hvp = (−101.57 ± 0.26_stat ± 0.54_syst) × 10⁻¹¹, for a total relative error of 0.6%.

What the researchers did: The team computed the relevant QCD correlator on 35 gauge ensembles produced by the CLS collaboration. These ensembles include 2+1 flavors of improved Wilson fermions (that is, they simulate the up, down and strange quarks) and cover six lattice spacings between about 0.039 and 0.097 fm, as well as pion masses down to the physical value. They used the time‑momentum representation (TMR), which turns the lattice correlator into the needed contribution via time integrals, and they separated the calculation into three distance windows—short, intermediate and long—so they could treat different systematic effects where they matter most. They also applied finite‑volume and isospin‑breaking corrections to move from the lattice setup to the physical theory.

How the calculation works and why a cancellation matters: The NLO HVP contribution comes from three classes of diagrams. One class (called NLOa) contains extra photon lines and muon loops and gives a negative contribution. Another class (NLOb) contains electron and tau loops and contributes positively, partially canceling NLOa. The third class (NLOc) has two separate QCD insertions and is much smaller (roughly 25 times smaller than the combined NLOa+NLOb). A key feature used by the authors is a strong cancellation between the NLOa and NLOb time‑kernel functions at large Euclidean times; their sum becomes very small beyond about 1.5 fm and crosses zero near 3.6 fm. That suppression reduces sensitivity to the hardest long‑distance uncertainties (statistical noise, finite‑volume effects and isospin breaking) and is essential for reaching sub‑percent precision.