Muon g-2 HLbL Developments

Background (mostly, but not entirely, from an October 27, 2022 post and a July 18, 2024 post)

The combined result of the experimental measurements of muon g-2 (all of the numbers that follow are in the conventional -2 and divided by two form times 10^-11) is:

116,592,059 ± 22 

This compares to the leading Standard Model predictions of: 

116,592,019 ± 38 (which is a relative error of 370 parts per billion). This is from A. Boccaletti et al., "High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly." arXiv:2407.10913 (July 15, 2024).

The gap is only 40 ± 44.9, with the Standard Model prediction still a bit lower than the experimental value.

The QED + EW predicted value is:

116,584,872.53 ± 1.1

About 90% of the combined uncertainty in this  QED + EW value is from the EW component (there may be an error in the standard QED prediction but it is so small that it is immaterial for these purposes).

The difference, which is the experimentally implied hadronic component value (HVP plus HLbL), is:

7186.47 ± 22.02

This has a plus or minus two sigma range of:

7,142.32 to 7,230.51

The hadronic QCD component is the sum of two parts: the hadronic vacuum polarization (HVP) and the hadronic light by light (HLbL) components.

In the Theory Initiative analysis the QCD amount is 6937(44) which is broken out as HVP = 6845(40), which is a 0.6% relative error and HLbL = 98(18), which is a 20% relative error.

The latest LO-HVP calculation component is 7141 ± 33 (a relative error of just 0.46%).

As of November 1, 2024, it was clear that the Theory Initiative calculation of the Standard Model value of the HVP contribution to muon g-2 (which differs from 5.1 sigma from the experimental value) was the flawed one:

Fermilab/HPQCD/MILC lattice QCD results from 2019 strongly favour the CMD-3 cross-section data for e+e−→π+π− over a combination of earlier experimental results for this channel. Further, the resulting total LOHVP contribution obtained is consistent with the result obtained by BMW/DMZ, and supports the scenario in which there is no significant discrepancy between the experimental value for aμ and that expected in the Standard Model.

Similarly, the introduction to a new paper that is the motivation for this post notes that:

estimates based on τ data-driven approaches or lattice QCD calculations significantly reduce the tension between theoretical and experimental values to 2.0σ and 1.5σ, respectively (less than one σ in [44]). The latest CMD-3 measurement of σ(e+e− → π+π−) also points in this direction.

Reference [44] cited in the block quote above is the current state of the art calculation cited above.

The HLbL Contribution

The Theory Initiative HLbL calculation

None of the refinements of the muon g-2 HVP contribution discussed above tweak the Theory Initiative value of the Hadronic Light by Light (HLbL) contribution of 92 ± 18, even though it has the highest relative error of any of the components of the muon g-2 calculation of nearly 20%, because the HLbL is only 1.3% of the total hadronic contribution and still has only half the uncertainty of the HVP contribution.

But, progress has been made on the HLbL component as well, which is now getting more attention as the experimental result's increased precision and the progress on the HVP contribution makes it relevant.

The Chao (April 2021) HLbL Calculation 

On the day that the first new muon g-2 experimental results from Fermilab were released a "new calculation of the hadronic light by light contribution to the muon g-2 calculation was also released on arXiv." This wasn't part of the BMW calculation and increased the HLbL contribution from 92 ± 18 to 106.8 ± 14.7. That paper stated:

We compute the hadronic light-by-light scattering contribution to the muon g−2 from the up, down, and strange-quark sector directly using lattice QCD. Our calculation features evaluations of all possible Wick-contractions of the relevant hadronic four-point function and incorporates several different pion masses, volumes, and lattice-spacings. We obtain a value of aHlblμ = 106.8(14.7) × 10^−11 (adding statistical and systematic errors in quadrature), which is consistent with current phenomenological estimates and a previous lattice determination. It now appears conclusive that the hadronic light-by-light contribution cannot explain the current tension between theory and experiment for the muon g−2.

En-Hung Chao, et al., "Hadronic light-by-light contribution to (g−2)μ from lattice QCD: a complete calculation" arXiv:2104.02632 (April 6, 2021) (the failure of this pre-print to be published, three and a half years later, however, is somewhat concerning, as there is no obvious flaw in the paper from the eyes of an educated layman).

This would increase the Standard Model prediction's value and lower the uncertainty to:

116,592,033.8 ± 36

This would reduce the gap between this combined theoretical prediction and the world average experimental value to 25.2 ± 42.2 (just 0.6 sigma).

The Zimmerman (October 2024) HLbL Calculation

The most recent total HLbL calculation, from October 2024 reached value of 125.5 ± 11.6, which would reduce the HLbL relative uncertainty to 9% (cutting it in more than half from the Theory Initiative value). This would make the state of the art combined prediction of muon g-2:

116,592,052.5 ± 35

The gap between this combined state of the art calculations of the Standard Model value of muon g-2, and world average experimental value for muon g-2, would be 6.5 ± 41.3 (less than 0.2 sigma).

Other Recent HLbL work

As the introduction to the new paper explains in a nice overview of the HLbL calculation:

The HVP data-driven computation is directly related to the experimental input from σ(e+e− → hadrons) data. HLbL in contrast, requires a decomposition in all possible intermediate states. Recently, a rigorous framework, based on the fundamental principles of unitarity, analyticity, crossing symmetry, and gauge invariance has been developed, providing a clear and precise methodology for defining and evaluating the various low energy contributions to HLbL scattering. The most significant among these are the pseudoscalar-pole (π(0), η and η′) contributions. Nevertheless, subleading pieces, such as the π± and K± box diagrams, along with quark loops, have also been reported, with the proton-box representing an intriguing follow-up calculation. Specifically, a preliminary result obtained from the Heavy Mass Expansion (HME) method —which does not consider the form factors contributions— for a mass of M ≡Mp=938 MeV, yields an approximate mean value of ap−box µ = 9.7 ×10−11. This result is comparable in magnitude to several of the previously discussed contributions, thereby motivating a more realistic and precise analysis that incorporates the main effects of the relevant form factors. In this work, we focus on the proton-box HLbL contribution. We apply the master formula and the perturbative quark loop scalar functions, . . . (which we verified independently), together with a complete analysis of different proton form factors descriptions, which are essential inputs for the numerical integration required in the calculations.

The new paper concludes the proton box contribution to HLbL which was preliminarily estimated at 9.7 is actually 0.182 ± 0.007, which is about 50 times smaller than the preliminary result and immaterial in the total, making the neutral and charged pion, the eta, the eta prime, charged kaon, and quark loop contributions as the primary components of the HLbL contribution to muon g-2.

Another new paper calculates the neutral pion contribution to HLbL which is the single largest component of the HLbL contribution, which accounts for more than half (almost two-thirds) of the HLbL contribution:

We develop a method to compute the pion transition form factors directly at arbitrary photon momenta and use it to determine the 

π0

-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon. The calculation is performed using eight gauge ensembles generated with 2+1 flavor domain wall fermions, incorporating multiple pion masses, lattice spacings, and volumes. By introducing a pion structure function and performing a Gegenbauer expansion, we demonstrate that about 98% of the π0-pole contribution can be extracted in a model-independent manner, thereby ensuring that systematic effects are well controlled. After applying finite-volume corrections, as well as performing chiral and continuum extrapolations, we obtain the final result for the π0-pole contribution to the hadronic light-by-light scatterintg in the muon's anomalous magnetic moment, aπ0−poleμ=59.6(2.2)×10−11, and the π0 decay width, Γπ0→γγ=7.20(35)eV.

Tian Lin, et al., "Lattice QCD calculation of the π(0)-pole contribution to the hadronic light-by-light scattering in the anomalous magnetic moment of the muon" arXiv:2411.06349 (November 10, 2024).

The relative uncertainty in the neutral pion contribution is 3.7%, which is a much larger relative uncertainty than in the EM, weak force, or HVP components, but much smaller than the relative uncertainty in the HLbL calculation as a whole.

This neutral pion contribution calculaton is an incremental improvement in the precision of this estimate, compared to most other recent attempts, and produces in value in the same ballpark as previous attempts (i.e. it is statistically consistent with them):

This also implies that the uncertainty from the charged pion, the eta, the eta prime, charged kaon, and quark loop contributions to HLbL, while small in magnitude (about 29-45 * 10^-11 from all of them combined) have combined uncertainties on the order of 13-17 * 10^-11. This is on the order of 35-45% relative uncertainty, which is far more than any other part of the muon g-2 calculation. 

Future Prospects

As the uncertainty in the HVP calculation falls (and this calculation currently approaches the maximum relative precision possible in QCD), this becomes more material in the overall accuracy of the muon g-2 calculation, and the greater precision will be important as the precision of the experimental measurement continues to improve. QCD calculations definitely can get more precise than the HLbL calculations are today, and especially more precise than the HLbL calculations other than the neutral pion contribution. 

But, it will be quite challenging, and may require a major breakthrough in QCD calculations generally, to get the uncertainty in the muon g-2 calculation to below 33-34 * 10^-11, which would be only about a 3-6% improvement from the best available combination of calculations so far. Therefore, the experimental result will probably be more precise than the QCD calculation for the foreseeable future.

Still, the bottom line, which has been clear since the BMW calculation was published at the time of the first Fermilab muon g-2 measurement, is that there is no muon g-2 anomaly since the predicted value and the measured value are consistent at the 0.2 sigma level. 

This global test of beyond the Standard Model physics at relatively low energies reveals that the Standard Model physics is complete and accurate at sub-parts per million levels, at least at relatively low energies of on the order of low GeVs or less.