The updated BMW calculation of the hadronic vacuum polarization contribution to the muon anomalous magnetic moment a(μ) in the Standard Model is squarely consistent, at 0.9 sigma with the experimental measurement of muon g-2 (even without one or two additional tweaks to parts of the calculation other than the HVP component that improve the fit further) despite the fact that its uncertainty in this new and improved HVP calculation is 40% smaller (producing a 31% reduction in the uncertainty of the overall theoretical calculation) and the fact that the 2023 experimental measurement has 55% less uncertainty, making agreement of the two values a harder to hit target.
The experimental value is based upon the first three phases of a six phase experiment. Phase four, whose results are tentatively expected late in 2024 (but could easily be delayed until 2025 or even 2026), will significantly reduce the uncertainty in the experimental result. The improvements from phases five and six are expected to be much smaller. The uncertainty in the experimental result is already about 42% smaller than the uncertainty in the calculation of the Standard Model predicted value. The experimentally measured value will probably continue to have less uncertainty than the Standard Model predicted value calculation for several years to come, at least.
The bottom line value for the new 2024 BMW calculation is:
a(µ) = 116,592,019(38) × 10^−11.
Partial calculations confirm the BMW result vis-a-vis the partially experimentally based analysis in the so called "window" part of the calculation (as does the recent CMD-3 experimental measurement of the same experimental inputs that went into the "Theory Initiative" calculation):
The apparent discrepancy between the Theory Initiative calculation of the Standard Model prediction and the experimental results has been the source of countless papers proposing proposals for new physics.
This discrepancy was a result of an inaccurate calculation of the Standard Model prediction for muon g-2 by the Theory Initiative. This was mostly a result of the incorporation of slightly inaccurate experimental data (regarding electron-positron annihilations that produce pion pairs) into their hadronic vacuum polarization calculation. Their source for the experimental data that it used, despite the source's good faith efforts to estimate the uncertainty accurately, understated the uncertainty of that experimental data by roughly a factor of four (as revealed by the spread of the results of subsequent experimental measurements of the same phenomena).
The paper presenting the new lattice QCD calculation of muon g-2 and its abstract are as follows:
We present a new lattice QCD calculation of the leading order hadronic vacuum polarization contribution to the muon anomalous magnetic moment a(μ). We reduce uncertainties compared to our earlier computation by 40%, arXiv:2002.12347. We perform simulations on finer lattices allowing for an even more accurate continuum extrapolation. We also include a small, long-distance contribution obtained using input from experiments in a low-energy regime where they all agree. Combined with other standard model contributions our result leads to a prediction that differs from the measurement of a(μ) by only 0.9 standard deviations. This provides a remarkable validation of the standard model to 0.37ppm.
A. Boccaletti, et al., "High precision calculation of the hadronic vacuum polarisation contribution to the muon anomaly" arXiv:2407.10913 (July 15, 2024).
The new paper also notes that:
In the near future we expect that other lattice collaborations will also provide precise calculations of a(LO-HVP)(µ) that can confirm or refute our results. Also, more data for the e+e− → π+π− cross section is expected soon.
For those of you who aren't visual learners (with values times 10^11 shown below):
New BMW calculation (2024): 116,592,019(38)
BMW calculation (2020): 116,591,954(55)
Theory Initiative calculation (2020): 116,591,810(43)
World Experimental Average (2023) : 116,592,059(22)
Fermilab Run 1+2+3 data (2023): 116,592,055(24)
Combined measurement (2021): 116,592,061(41)
Brookhaven's E821 data (2006): 116,592,089(63)
Why Does This Calculation Matter?
The consistency between the experimentally measured value of muon g-2 and the calculation of the value of muon g-2 predicted by the Standard Model, is important because it a strong global test, at high precision, of the accuracy of all parts of the Standard Model at once.
In this test, the experimentally measured result matches the theoretical calculation of muon g-2's value (at a precision of about 4 parts per ten million).
This rules out many kinds of new physics that a next generation particle collider could detect.
While there are still legitimate reasons to build a next generation particle collider (e.g., to refine the precision of the Standard Model's constants, some of which are known to a less than parts per thousand level), the likelihood that a next generation particle collider will reveal new physics beyond the Standard Model is greatly diminished by the current state of the art muon g-2 experimental results and the current state of the art calculation of the Standard Model predicted value for muon g-2. So, the physics justification for a next generation particle collider (which will cost something on the order of $17 billion U.S. dollars if built) as a means of discovering new physics is greatly undermined by this result.
Muon g-2 is less sensitive to new physics at energy scales higher than those of any of the proposed next generation particle colliders, however, so it doesn't rule out new physics at those energies as strongly.
QCD v. Non-QCD Calculations In Muon g-2
The QED + Electroweak (EW) predicted value (omitting the Hadronic Vacuum Polarization (HVP) and Hadronic Light by Light (HLbL) components, and before the latest QED component tweak) is: 116,584,872.53 (101). Thus, it accounts for only about 3% of the total uncertainty in the value of muon g-2. Given the experimental results, the implied combined hadronic (i.e. QCD) contribution to the total is about 7186.47 times 10^-11. This is 0.00616% of the total value, despite the fact that it is the source of 97% of the uncertainty in the calculation, because QCD calculations are vastly less precise than QED calculations.
More specifically, in theoretical calculation, the QED calculation is the source of 0.2% of the total uncertainty, the weak force calculation is the source of 2.9% of the total uncertainty, and the QCD calculation is the source of 96.9% of the total uncertainty (with about 10% of the total muon g-2 calculation uncertainty coming from the Hadronic light by light calculation and about 87% of the total muon g-2 calculation uncertainty coming from the Hadronic vacuum polarization calculation).
The QED calculation is the source of 99.994% of the total value of muon g-2, the weak force calculation is the source of 0.00013% (153.6 x 10^-11 ± 1 x 10^-11), and the QCD calculation is the source of about 0.006% (about 99% of which is from the hadronic vacuum polarization calculation and about 1% of which is from the hadronic light by light calculation).
The relative error in the QED calculation is 0.000 000 0686%, the relative error in the weak force calculation is about 0.65%, and the relative error in the overall QCD calculation is about 0.5%. But since the weak force contribution to the calculation is only about 2% of the magnitude of the QCD contribution to the calculation, greater precision in weak force calculation isn't a priority in improving the precision of the total calculation.
Within the two QCD calculations, the Hadronic Vacuum Polarization calculation has a relative uncertainty of a little less than 0.5% and accounts for the bulk of the QCD contribution to muon g-2 (about 99% of the QCD contribution), while the Hadronic Light by Light calculation has a relative uncertainty of about 20%, but involves a much smaller total contribution to the total value of muon g-2 (about 1.0-1.3% of the QCD contribution and 0.000 08% of the total muon g-2 calculation). The Hadronic Light by Light calculation is the source of a little more than 10% of the uncertainty in the QCD contribution, and of about 10% of the uncertainty in the overall muon g-2 calculation.
These subcomponent breakdowns are spelled out in detail, for example, in T. Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020) (which obviously doesn't include the post-2020 develops in the calculation or in discussion of the experimentally measured value).
Other Tweaks To The Muon g-2 Calculation
Two other tweaks in parts of the muon g-2 calculation have also been made and aren't included in the new paper. They reduce the difference between the theoretical calculation and the experimentally measured calculation to about 0.6 sigma, which is a precision of about 2.2 parts per million.
Hadronic Light By Light Calculation (2021)
One is an increase of 14.8 x 10^-11, in the Hadronic light-by-light calculation from 2021. The relative uncertainty in this component of the calculation is reduced from 20% to 14%. Including this tweak also reduces the uncertainty in the latest BMW calculation of the total muon g-2 value (including its new HVP calculation) from ± 38 x 10^-11 to ± 37 x 10^-11.
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 presentation doesn't note it, but there was also an adjustment bringing the result closer to the experimental result in the hadronic light-by-light calculation (which is the smaller of two QCD contributions to the total value of muon g-2 and wasn't included in the BMW calculation) which was announced on the same day as the previous data announcement. The new calculation of the hadronic light by light contribution to the muon g-2 calculation increases the contribution from that component from 92(18) x 10^-11 to 106.8(14.7) x 10^-11.
As the precision of the measurements and the calculations of the Standard Model Prediction improves, a 14.8 x 10^-11 discrepancy in the hadronic light by light portion of the calculation becomes more material.
QED Calculation (2024)
The other is a reduction of about 0.06 x 10^-11 in the QED part of the calculation from April of 2024, which is a five sigma downward shift in that calculation. But it is immaterial to the overall muon g-2 calculation because the original calculation was already so precise.
My prior posts on the subject were summarized on August 10, 2023.
Bottom Line:
With values time 10^11 shown:
New BMW calculation + new HLbL + new QED calculation (2024+2021+2024): 116,692,033.74 ±37.00
World Experimental Average (2023) : 116,592,059 ± 22
Considering the HLbL and QED tweaks as well, this is a bit less than a 0.6 sigma difference from the experimental value. This means that the prediction and measured value (as improved in 2023) are basically perfectly consistent with each other despite a 33% reduction in the uncertainty of the overall theoretical prediction and a 55% reduction in the uncertainty of the experimental measurement. The relative difference is 0.22 parts per million.
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