The uncertainty in the Standard Model prediction of $a_\mu$ is limited entirely by our knowledge of the hadronic contributions. As of the December 2020 publication of the Muon $g-2$ Theory Initiative (TI) white paper (WP), this uncertainty was 0.37 ppm. [QED and electroweak contributions have negligibly small theory uncertainties at 1 and 10 ppb, respectively.]

The two distinct sets of hadronic contributions are hadronic vacuum polarization (HVP) and hadronic light-by-light scattering (HLbL). The former starts at second order in $\alpha$, the latter at third order. In both cases, two complementary approaches are being used to evaluate them: a dispersive method, which takes experimental data as input, and lattice QCD, in which the hadronic corrections are computed ab initio, via numerical simulations in Euclidean spacetime.

For the HLbL contribution, the two approaches yield compatible results with uncertainties at the 20% level, amounting to 0.15 ppm in the total result. Some of the developments since the WP are:

• The short-distance constraints have been improved by the addition of higher orders in the operator product expansion and gluonic corrections to the leading order.
• There are two more lattice calculations published with higher precision than the one available at the time of the WP. Both are compatible within errors with the WP HLbL estimate.
• There is ongoing work on including in a controlled fashion resonances beyond the one- and two-pion states in the dispersive approach.

Outlook: We expect that ongoing work on both approaches will yield reduced uncertainties within the next 1-2 years, down to the 10% level.

For the HVP contribution, the current situation is complicated by several tensions:

• In the dispersive approach, for which experimental measurements of low-energy $e^+e^- \to$ hadrons cross sections serve as input, the predictions of HVP are based on more than twenty years of experimental measurements, performed with quite complementary approaches (based on direct-scan and initial-state-radiation measurements at very different center-of-mass energies $\sqrt{s}$), culminating in the consensus prediction obtained in the Muon $g-2$ TI WP of 2020, which is lower by $5.1\sigma$ compared to the 2023 experimental average for $a_\mu$.
• Since then, new results for the dominant $\pi^+\pi^-$ contribution have become available from the SND experiment that are completely consistent with previous determinations. Also results on sub-dominant channels have been obtained by the BABAR, BESIII, CMD-3, and SND collaborations.
• In February 2023 the CMD-3 collaboration released results on the $\pi^+\pi^-$ cross section that disagree at the (2.5-5)$\sigma$ level with all previous measurements used in the 2020 WP, including those from the previous CMD-2 and SND collaborations, as well as the new SND measurement, all at the same facility. The origin of this discrepancy is currently unknown.
• In view of the puzzle created by the new CMD-3 analysis, the TI organized two scientific seminars and panel discussions, involving experts in these low-energy experiments. Discussions between the panelists and members of CMD-3 are ongoing to scrutinize and hopefully identify possible reasons for the experimental discrepancies.
• Up until 2021, lattice-QCD calculations of HVP were not precise enough to weigh in. The BMW collaboration published the first complete lattice-QCD result with subpercent precision (0.8%) in April 2021. However, their result is closer to the experimental average and in $2.1\sigma$ tension with the data-driven evaluation of the 2020 WP.
• The Lattice community has also focused on the so-called window observables, introduced by the RBC/UKQCD collaboration. Specifically, the intermediate window observable, which comprises about ⅓ of the total HVP, has been adopted as a benchmark quantity. By early 2023, many groups (Aubin et al., BMW, $\chi$QCD, ETM, Fermilab/HPQCD/MILC, Lehner & Meyer, Mainz, RBC/UKQCD) presented results for the dominant light-quark connected contribution to the intermediate window observable with subpercent precision, which are in excellent agreement. Several groups (BMW, ETM, Mainz, RBC/UKQCD) also included subleading (strange, charm, quark-disconnected, and isospin-breaking) contributions in their results, which all agree within the quoted errors. Compared with the data-driven evaluation of the intermediate window observable, there is a discrepancy close to $4 \sigma$. First dispersive studies indicate that the tension in the intermediate window observable is primarily driven by the light-quark connected contribution.
• For the total HVP and the long-distance contribution to HVP (which makes up about ⅔ of the total), independent lattice QCD calculations with subpercent precision are still needed to establish the same level of consolidation for the lattice-QCD methodology as was achieved for the intermediate window. Indeed, contributions from large Euclidean times are more sensitive to certain systematic effects.

Outlook:

• The current status and next steps will be discussed at the 6th plenary workshop of the Muon $g-2$ TI in Bern, September 4-8, 2023.
• A new analysis of the $\pi^+\pi^-$ cross section based on the full statistics collected by the BABAR experiment is underway. Using a new analysis technique, independent of particle identification, it is complementary to the previous method with good prospects for reducing systematic uncertainties. A specific experimental study of radiative corrections is close to publication with possible impact on some of the analyses by other experiments. Final results are expected in 2024.
• The SND 2020 analysis of $e^+e^-\to\pi^+\pi^-$ was based on 10% of the available statistics. An update using the full data set is in preparation.
• The published analysis of the $e^+e^-\to\pi^+\pi^-$ cross section of the BESIII experiment was based on a data sample of 2.9/fb, collected at $\sqrt{s}=3.77$ GeV. BESIII are currently finalizing a new large data sample of 17.1/fb at this center-of-mass energy. First results from an analysis exploiting the extended data set are expected in the near future. In contrast to the published analysis, this will allow for the normalization of the di-pion event yield by di-muon events, which will reduce the uncertainty in the cross section measurement significantly. Besides the data sample taken at 3.77 GeV, more than 20/fb of additional data at higher $\sqrt{s}$ will be available for ISR analyses as well. Furthermore, BESIII plan to publish the analyses of the existing energy scan data in the range between 2 and 5 GeV. This will provide the world's most precise measurements of the inclusive $R$ ratio as well as the $R_c$ ratio.
• The existing $e^+e^- \to \pi^+\pi^-(\gamma)$ measurements, KLOE-2008, KLOE-2010, KLOE-2012, and their combination are based on data taken at $\sqrt{s} = 1.02$ GeV in 2002 and at $\sqrt{s} = 1.00$ GeV in 2006. A larger data set, taken in 2004/2005 at $\sqrt{s} = 1.02$ GeV, has so far not been analyzed for the measurement of the hadronic contributions to the muon $g-2$. The 7 times larger statistics of the 2004/2005 data set will allow, together with improved versions of the Monte Carlo generators for radiative corrections and modern analysis techniques, to halve the uncertainty of the KLOE published measurement (~0.4%). To avoid possible biases with published KLOE or other experiments the analysis will be blinded.
• Radiative corrections and Monte Carlo generators, in particular for the crucial $e^+e^-\to\pi^+\pi^-(\gamma)$ channel, are being scrutinized. This includes the calculation and implementation of higher-order and structure-dependent corrections.
• Given the ongoing efforts by lattice groups worldwide, we expect to see new lattice-QCD results for the total HVP contribution and the long-distance window observable with a precision comparable to BMW and the data-driven approach, as well as complete subleading hadronic contributions, in the near future. The lattice community is committed to providing a consolidated result for the total HVP that matches the final precision of the Fermilab E989 experiment.
• In order to better understand the possible origins of the tensions between data-driven and lattice HVP evaluations, other window quantities and related observables have been proposed. First studies comparing lattice QCD and data-driven evaluations of these and other observables have been presented and more are expected. Detailed comparisons will require precise knowledge of the covariance matrices in both approaches.
• Measurements of hadronic cross sections to refine theoretical predictions of the muon $g-2$ are key elements of the Belle II physics program. The Belle II experiment has collected a sample corresponding to 362/fb to date and will resume data acquisition in winter 2023. A measurement of the $e^+e^- \to \pi^+\pi^-\pi^0$ cross section using half of the current data set is planned for public release in 2023. A follow-up measurement of the $e^+e^- \to \pi^+\pi^-$ cross section, based on the current full data set, is expected by 2025 with a precision comparable to the BABAR 2009 result. Further studies will follow to include other final states and also to reduce systematic errors.
• The MUonE experiment at CERN will provide an independent and competitive method to compute the HVP contribution to the muon $g-2$, based on the high-precision measurement of the shape of the differential cross section of muon-electron elastic scattering as a function of the space-like squared momentum transfer. A three-week test run will be performed at CERN in summer 2023 to assess the detector performance and validate the overall method. A possible measurement of the leptonic vacuum polarization contribution to the effective electromagnetic coupling at few-percent accuracy is also envisaged. A technical proposal for the full experiment, with a first run before 2026 with a scaled detector, will be prepared in 2024 with the final goal to measure the HVP contribution at 0.3% statistical accuracy and comparable systematic error after CERN’s Long Shutdown 3 (LS3).