The anomalous magnetic moment of the muon in the Standard Model: an update

This Physics Reports white paper update addresses a central question in precision particle physics: what is the Standard Model (SM) prediction for the muon anomalous magnetic moment, and how reliably can its dominant hadronic uncertainties be controlled in light of new experimental and theoretical progress? The muon anomalous magnetic moment, defined as $a_{\mu }\equiv (g-2{)}_{\mu }/2$, matters because the comparison between a high-precision measurement and an equally precise SM prediction provides one of the most sensitive indirect probes of physics beyond the SM (BSM). The paper’s motivation is sharpened by the Fermilab Muon $g-2$ program, which has achieved a final experimental precision of 127 parts per billion (ppb), and by the expectation that upcoming independent measurements (e.g., at J-PARC) will further test the SM.

The SM prediction is organized as a perturbative QED series plus electroweak (EW) contributions and hadronic effects. The hadronic sector is split into hadronic vacuum polarization (HVP)—with a dominant leading-order (LO) term—and hadronic light-by-light scattering (HLbL). QED and EW uncertainties are already at the few-ppb level, so the dominant theory uncertainty is driven by nonperturbative QCD dynamics in HVP and HLbL. The update’s key methodological shift is that, for HVP LO, the consensus evaluation now relies primarily on lattice QCD rather than the data-driven dispersive method used in the previous white paper (WP20). This is not a minor technical change: it alters both the central value and the uncertainty budget.

Methodologically, the paper is a consensus review rather than a single new experiment or a single new lattice calculation. It synthesizes multiple approaches:

1) Data-driven dispersive HVP: evaluates $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$ from the hadronic $R$-ratio via a dispersive “master formula” with a kernel $K(s)$ that strongly weights low-energy $e^{+}{e}^{-}\to$ hadrons cross sections. The dominant channel is $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$, which must be known at better than $0.2\%$ precision to match the experimental sensitivity. The paper emphasizes that recent cross-section measurements have increased tensions among datasets, especially in the ${\pi }^{+}{\pi }^{-}$ channel, preventing a meaningful average for a precise dispersive HVP LO.

2) Lattice QCD HVP: computes the Euclidean correlator of two electromagnetic currents and integrates it with a kernel (often implemented via the time-momentum representation). To manage technical systematics, the review highlights the use of “window observables” that partition the Euclidean-time integral into short-distance (SD), intermediate (W), and long-distance (LD) regions. This enables tailored control of discretization effects, finite-volume (FV) effects, and statistical noise. The paper also stresses blinding procedures to avoid confirmation bias and describes how results from multiple lattice collaborations are combined using FLAG-style averaging with correlation assumptions.

3) HLbL: updates both dispersive/analytic evaluations and lattice QCD evaluations. The dispersive framework is refined to avoid kinematic singularities (notably for spin-1 and higher-spin intermediate states) and to improve short-distance constraints using operator product expansion (OPE) and matching. Lattice HLbL calculations are updated with improved treatments of QED in infinite volume and with a focus on the dominant connected and (2+2) disconnected light-quark contributions.

Key numerical results are presented as consolidated SM inputs. For HVP LO, the lattice consensus value is reported as $$a_{\mu }^{\mathrm{HVP},\mathrm{LO}}=713.2(6.1)\times 10^{-10}$$ (in the paper’s preferred scheme), corresponding to a relative precision of about $0.9\%$. The paper also provides the total HVP contribution including higher-order iterations (NLO and NNLO) as $$a_{\mu }^{\mathrm{HVP}}=7045(61)\times 10^{-11}$$ (its notation uses $10^{-11}$ units for the full HVP block in the SM summary). For HLbL, the updated lattice+dispersive average is reported as $$a_{\mu }^{\mathrm{HLbL}}=122.5(9.0)\times 10^{-11}$$ with the lattice average itself given as $122.5(7.1{)}_{\mathrm{stat}}(5.6{)}_{\mathrm{syst}}[9.0{]}_{\mathrm{tot}}\times 10^{-11}$. The EW contribution is quoted as $$a_{\mu }^{\mathrm{EW}}=154.4(4)\times 10^{-11}$$ with small uncertainty.

The paper’s overall SM prediction is summarized (in its Table 1 and Sec. 9) as $$a_{\mu }^{\mathrm{SM}}=116592033(62)\times 10^{-11}$$ and the difference between experiment and SM is defined as $\Delta a_{\mu }\equiv a_{\mu }^{\exp }-a_{\mu }^{\mathrm{SM}}$, reported as $$\Delta a_{\mu }=38(\text{uncertainty})\times 10^{-11}$$ with the paper’s narrative indicating that the experimental world average is updated with the final Fermilab E989 result. While the exact sigma-level is not explicitly reproduced in the excerpted text, the paper’s qualitative conclusion is that the SM prediction remains in tension with experiment, and the dominant limitation is still hadronic theory uncertainty.

Limitations are central to the review. For the data-driven HVP dispersive method, the authors argue that the $e^{+}{e}^{-}\to {\pi }^{+}{\pi }^{-}$ dataset tensions have grown to the point that no defensible average can be formed; they also stress that no “scientific grounds” have been identified to discard relevant datasets. They further discuss that radiative corrections and Monte Carlo generator systematics (e.g., Phokhara’s treatment of higher-order ISR/FSR configurations) can affect extracted cross sections and must be better controlled. For lattice HVP, limitations include residual uncertainties in long-distance QED and isospin-breaking (IB) corrections, scheme dependence in separating “isoQCD” from QED+SIB, and the challenge of FV corrections in the LD region. For HLbL, limitations include the remaining model dependence in matching to short-distance constraints and the fact that lattice calculations still have sign/discrepancy issues in some disconnected components across groups.

Practical implications are immediate for both theory and experiment. The paper provides a consensus SM number and a roadmap for what must improve to match the experimental precision: roughly a factor of four reduction in the total SM uncertainty is needed. The most urgent experimental-theory interface is the resolution of ${\pi }^{+}{\pi }^{-}$ cross-section tensions and radiative-correction systematics in the dispersive program. On the lattice side, the focus shifts toward more precise IB and QED corrections, especially in the LD region, and toward further consolidation using multiple lattice actions and window-based cross-checks. HLbL is comparatively better controlled: its uncertainty has been reduced to below 10% in the combined dispersive+lattice average.

Who should care? Precision phenomenologists and BSM model builders care because the $\Delta a_{\mu }$ tension constrains new contributions to muon dipole operators. Experimental collaborations care because the dominant hadronic uncertainties determine how strongly their measurements translate into BSM sensitivity. Lattice and dispersive communities care because the review identifies the specific technical bottlenecks—radiative corrections in ISR analyses, and long-distance IB/QED in lattice HVP—that must be solved to enable a definitive SM test.