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ohwilleke
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- TL;DR Summary
- The latest state of art predictions of the Standard Model value of muon g-2 are just 0.2 sigma from the latest global average experimental results, due to improved calculations of the two QCD parts of the calculation, one of which, the hadronic light-by-light (HLbL) part has gotten less attention.
What Is Muon g-2?
The electron, muon (a heavy electron), and the tau lepton (a really heavy electron), each have a measurable property called a "magnetic moment".
This differs from the naive expected value of two because of myriad possible instance where a photon or weak force boson is emitted, takes various possible paths and is transformed in various ways, and then is reabsorbed. The deviation of the magnetic moment from the value two is called the anomalous magnetic moment of the electron, muon, or tau, as the case may be, and for reasons rooted in tradition, the anomalous magnetic moment of the electron, muon, or tau, is customarily stated as the measured value of the magnetic moment minus two, with the end result divided by two. Thus the raw values show in the chart below (which are somewhat outdated) are converted into g-2 form by subtracting two and then dividing by two.
The dominant effect involves the emission and reabsorption of photons (with the possible creation and annihilation of virtual electrons or muons) which is governed solely by quantum electrodynamics (QED) which is the quantum version of electromagnetism that is part of the Standard Model of Particle Physics. QED alone describes the first five significant digits of muon g-2 to ± 1 in the fifth digit.
Measurements of the anomalous magnetic moment of the electron were one of the first important experimental observations that convinced scientists that QED was more accurate than classical electromagnetism using Maxwell's equations. But, the mass of the electron, muon, or tau as the case may be influences the value of its anomalous magnetic moment, so the values of the magnetic moment of each of these particles is a bit different.
At higher levels of precision, considering more obscure and involved paths of emission and reabsorption of virtual particles, all three of the Standard Model forces, electromagnetism (QED), the weak force (whose contribution is usually abbreviated (EW), and the strong force (whose contribution is usually abbreviated QCD for quantum chromodynamics, the Standard Model theory of the strong force), contribute to the magnetic moment of the particle. So, this makes precision measurements of these magnetic moments global tests to see if the Standard Model is missing something.
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 developments in the calculation or in discussion of the experimentally measured value). The money chart from that paper, reproduced in power point form in a Fermilab presentation when its first measurements were announced (with dated numbers for the calculated value of each component) is as follows:
Electron g-2 v. Muon g-2 v. Tau g-2
As it happens, it is much harder to measure to magnetic moment of the tau (which has a shorter mean lifetime and is much more massive) than the magnetic moment of the muon.
But the magnetic moment of the muon has a more interesting calculation and source than the magnetic moment of the electron because it is more massive so higher mass virtual particles are more relevant to calculating muon g-2.
By comparison, the value of the electron magnetic moment g (not g-2) most precisely measured in 2006 by Gerry Gabrielse, et al., is:
The experimentally measured muon g-2 is greater than the electron g-2 by a factor of approximately:
The tau lepton's anomalous magnetic moment (i.e. tau g-2) has a quite precise (but not quite as precise as the electron g-2 and muon g-2) Standard Model predicted value of 0.00117721 ± 0.00000005, which is a bit higher than muon g-2. But experimental measurements of tau g-2 from the LHC are still very crude: The Particle Data Group value as explained in a power point presentation providing more background on the question from 2020 is -0.018 ± 0.017.
The Experiments
The current experimental value of muon g-2 is based upon three of five runs of the Fermilab data gathering, with two more planned. The next one is expected to make a notable improvement in the precision of the measurement. The last one is expected to make only an incremental tweak to the results from the previous four runs. The final results from Fermilab are currently expected in late 2025.
The the Brookhaven National Laboratory (BNL) Muon E821 experiment that preceded the current Fermilab experiment collected data through the year 2001 and published its final report in 2006. Its final report value is also included in the global average measurement of muon g-2 (with inverse error weighting in the average).
Some of the experimental apparatus used at Brookhaven (which is on Long Island in New York State), was transported to Fermilab (which is near Chicago, Illinois), to be used in the current muon g-2 experiment, in an epic moving challenge involving barges and oversized loads on land.
The Brookhaven and Fermilab measurements were consistent with each other, although the Fermilab measurements are more precise.
J-PARC will also be doing an independent muon g-2 measurement around the time the Fermilab's muon g-2 measurement is completed (date taking starts in the year 2028 and the first preliminary results will probably take a couple of years after that to be reported, so around the year 2030).
Of course, in high energy physics, like any major project, timelines are subject to unexpected delays so you shouldn't count on these results being announced on their scheduled dates.
Muon g-2 measurements and predictions
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
The global average of the experimental measurements of muon g-2 compares to the leading Standard Model predictions with an improved muon g-2 hadronic vacuum polarization calculation of:
116,592,019 ± 38 (which is a relative error of 0.37 parts per million).
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 between theory and prediction is only 40 ± 44.9 (about 0.9 sigma, i.e. 0.9 standard deviations a.k.a. 0.9 σ), with the Standard Model prediction having a value that is still a bit lower than the experimental value. Physicists call results that are 2 sigma or less apart "consistent" with each other, and in theory, the average discrepancy between theory and experiment should be 1 sigma.
The QED (i.e. electromagnetic) + EW (i.e. weak force) Standard Model contributions to the 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 resulting in a reduction of about 0.06 in the QED part of the calculation identified in April of 2024, which would be a significant downward shift in that calculation relative to its uncertainty, that has been noted in the literature. But it is so small in absolute terms that it is immaterial for these purposes because the original calculation was already so precise.
The difference, which is the experimentally implied hadronic component value (HVP plus HLbL), is:
7186.47 ± 22.2
The hadronic QCD component (which when combined with the QED and EW parts is the total muon g-2) is the sum of two parts: the hadronic vacuum polarization (HVP) and the hadronic light by light (HLbL) components. Both of these involve paths that include virtual quarks and gluons, divided, somewhat arbitrarily, between one set of paths whose effect is larger and easier to calculate precisely (HVP) and another set of paths who effect is smaller and harder to calculate precisely (HLbL).
HVP, in principle can be broken into LOHVP (leading order hadronic vacuum polarization) and higher order HVP, but the higher order HVP effects in muon g-2 are small enough to be estimated crudely, given the uncertainties already present in the leading order calculation.
In the data driven Theory Initiative analysis (the Theory Initiative was the group chosen by Fermilab to make its initial benchmark Standard Model prediction for muon g-2), the total QCD amount (a.k.a. hadronic contribution) is 6937 ± 44 which is broken out as HVP = 6845 ± 40, which is a 0.6% relative error and HLbL = 98 ± 18, which is an 18% relative error. This is a lot smaller than the total QCD amount implicated by the experimental measurement which prompted theorists to think for a long time that the failure of the Standard Model prediction to match the experimentally measured value was a sign of beyond the Standard Model physics and spawned all sorts of papers to explain the difference.
The latest HVP calculation component, however, 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) suffered from reliance on somewhat older experimental data that wasn't quite precise enough for this parts per million precision calculation:
So, it turned out that the Standard Model prediction for the value of muon g-2 actually does match the experimental measurement of muon g-2 at sub parts per million levels, and that earlier calculations of the Standard Model predicted values were just a bit off, due the reliance in earlier calculations on data to substitute for some of the harder calculations that wasn't sufficiently accurate for this level of precision calculation. Basically, the error estimates in the underlying data relied upon for the earlier calculations was (in good faith) understated.
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 18%. Refining the HLbL calculation wasn't a priority because the HLbL is only 1.3% of the total hadronic contribution and this component has only half the uncertainty of the HVP contribution (which works out to a lot less impact on the total uncertainty when the uncertainty from the different components are combined).
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 calculation, make 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 (the more correct in hindsight alternative to the Theory Initiative calculation that was a first principles lattice QCD calculation of HVP, which was incredibly processor and scientist time intensive, in contrast to a data driven one which was somewhat less calculation intensive) and increased the HLbL contribution from 92 ± 18 to 106.8 ± 14.7. That paper stated:
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 half from the Theory Initiative relative uncertainty). 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 HVP and HLbL calculations of the Standard Model value of muon g-2 (both from 2024), and world average experimental value for muon g-2 (from 2023), is just 6.5 ± 41.3 (less than 0.2 sigma).
This gap is so small compared to the one sigma gap that would be expected from random chance, that it suggests (although it does not prove) that it is likely that the experimental uncertainties, or the calculation uncertainties, or both, may be overestimated. This is pretty common in electroweak physics where the sub-disciplinary ethos favors erring on the high side when estimating experimental and calculation uncertainties, and the sources of potential uncertainties are well understood.
Other Recent HLbL work
As the introduction to the new paper explains in a nice overview of the HLbL calculation:
Another new paper calculates the neutral pion contribution to HLbL which is the single largest component of the HLbL contribution, which accounts for about half of the HLbL contribution:
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):
The estimate for the neutral pion contribution is about 3.5 lower than in that part of the HLbL calculation in the Theory Initiative White Paper.
This also implies that the uncertainty from the charged pion, the eta, the eta prime, charged kaon, and quark loop contributions to HLbL, while modest in absolute magnitude (about 29-45 from all of them combined) have combined uncertainties on the order of 13-17. 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
The calculation of the Standard Model expected value of muon g-2 is one of the most precise predictions ever confirmed in the history of science.
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 much below 33-34, 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.
Estimating the energy scale at which the confirmation of the Standard Model predicted value of muon g-2 strongly disfavors new physics is more art than science. The physics blogger "Jester" has estimated it at something on the order of ten times of energy scale of the LHC (in a comment to a blog post that I do not have right at hand).
The electron, muon (a heavy electron), and the tau lepton (a really heavy electron), each have a measurable property called a "magnetic moment".
This differs from the naive expected value of two because of myriad possible instance where a photon or weak force boson is emitted, takes various possible paths and is transformed in various ways, and then is reabsorbed. The deviation of the magnetic moment from the value two is called the anomalous magnetic moment of the electron, muon, or tau, as the case may be, and for reasons rooted in tradition, the anomalous magnetic moment of the electron, muon, or tau, is customarily stated as the measured value of the magnetic moment minus two, with the end result divided by two. Thus the raw values show in the chart below (which are somewhat outdated) are converted into g-2 form by subtracting two and then dividing by two.
The dominant effect involves the emission and reabsorption of photons (with the possible creation and annihilation of virtual electrons or muons) which is governed solely by quantum electrodynamics (QED) which is the quantum version of electromagnetism that is part of the Standard Model of Particle Physics. QED alone describes the first five significant digits of muon g-2 to ± 1 in the fifth digit.
Measurements of the anomalous magnetic moment of the electron were one of the first important experimental observations that convinced scientists that QED was more accurate than classical electromagnetism using Maxwell's equations. But, the mass of the electron, muon, or tau as the case may be influences the value of its anomalous magnetic moment, so the values of the magnetic moment of each of these particles is a bit different.
At higher levels of precision, considering more obscure and involved paths of emission and reabsorption of virtual particles, all three of the Standard Model forces, electromagnetism (QED), the weak force (whose contribution is usually abbreviated (EW), and the strong force (whose contribution is usually abbreviated QCD for quantum chromodynamics, the Standard Model theory of the strong force), contribute to the magnetic moment of the particle. So, this makes precision measurements of these magnetic moments global tests to see if the Standard Model is missing something.
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 developments in the calculation or in discussion of the experimentally measured value). The money chart from that paper, reproduced in power point form in a Fermilab presentation when its first measurements were announced (with dated numbers for the calculated value of each component) is as follows:
Electron g-2 v. Muon g-2 v. Tau g-2
As it happens, it is much harder to measure to magnetic moment of the tau (which has a shorter mean lifetime and is much more massive) than the magnetic moment of the muon.
But the magnetic moment of the muon has a more interesting calculation and source than the magnetic moment of the electron because it is more massive so higher mass virtual particles are more relevant to calculating muon g-2.
By comparison, the value of the electron magnetic moment g (not g-2) most precisely measured in 2006 by Gerry Gabrielse, et al., is:
This translates to an electron g-2 of:
The experimentally measured value of electron g-2 is in mild 2.5 sigma tension with the Standard Model predicted value value measured at a parts per hundred million level, with an experimentally measured value that is slightly lower than the predicted value.
The experimentally measured muon g-2 is greater than the electron g-2 by a factor of approximately:
The tau lepton's anomalous magnetic moment (i.e. tau g-2) has a quite precise (but not quite as precise as the electron g-2 and muon g-2) Standard Model predicted value of 0.00117721 ± 0.00000005, which is a bit higher than muon g-2. But experimental measurements of tau g-2 from the LHC are still very crude: The Particle Data Group value as explained in a power point presentation providing more background on the question from 2020 is -0.018 ± 0.017.
The Experiments
The current experimental value of muon g-2 is based upon three of five runs of the Fermilab data gathering, with two more planned. The next one is expected to make a notable improvement in the precision of the measurement. The last one is expected to make only an incremental tweak to the results from the previous four runs. The final results from Fermilab are currently expected in late 2025.
The the Brookhaven National Laboratory (BNL) Muon E821 experiment that preceded the current Fermilab experiment collected data through the year 2001 and published its final report in 2006. Its final report value is also included in the global average measurement of muon g-2 (with inverse error weighting in the average).
Some of the experimental apparatus used at Brookhaven (which is on Long Island in New York State), was transported to Fermilab (which is near Chicago, Illinois), to be used in the current muon g-2 experiment, in an epic moving challenge involving barges and oversized loads on land.
The Brookhaven and Fermilab measurements were consistent with each other, although the Fermilab measurements are more precise.
J-PARC will also be doing an independent muon g-2 measurement around the time the Fermilab's muon g-2 measurement is completed (date taking starts in the year 2028 and the first preliminary results will probably take a couple of years after that to be reported, so around the year 2030).
Of course, in high energy physics, like any major project, timelines are subject to unexpected delays so you shouldn't count on these results being announced on their scheduled dates.
Muon g-2 measurements and predictions
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
The global average of the experimental measurements of muon g-2 compares to the leading Standard Model predictions with an improved muon g-2 hadronic vacuum polarization calculation of:
116,592,019 ± 38 (which is a relative error of 0.37 parts per million).
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 between theory and prediction is only 40 ± 44.9 (about 0.9 sigma, i.e. 0.9 standard deviations a.k.a. 0.9 σ), with the Standard Model prediction having a value that is still a bit lower than the experimental value. Physicists call results that are 2 sigma or less apart "consistent" with each other, and in theory, the average discrepancy between theory and experiment should be 1 sigma.
The QED (i.e. electromagnetic) + EW (i.e. weak force) Standard Model contributions to the 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 resulting in a reduction of about 0.06 in the QED part of the calculation identified in April of 2024, which would be a significant downward shift in that calculation relative to its uncertainty, that has been noted in the literature. But it is so small in absolute terms that it is immaterial for these purposes because the original calculation was already so precise.
The difference, which is the experimentally implied hadronic component value (HVP plus HLbL), is:
7186.47 ± 22.2
The hadronic QCD component (which when combined with the QED and EW parts is the total muon g-2) is the sum of two parts: the hadronic vacuum polarization (HVP) and the hadronic light by light (HLbL) components. Both of these involve paths that include virtual quarks and gluons, divided, somewhat arbitrarily, between one set of paths whose effect is larger and easier to calculate precisely (HVP) and another set of paths who effect is smaller and harder to calculate precisely (HLbL).
HVP, in principle can be broken into LOHVP (leading order hadronic vacuum polarization) and higher order HVP, but the higher order HVP effects in muon g-2 are small enough to be estimated crudely, given the uncertainties already present in the leading order calculation.
In the data driven Theory Initiative analysis (the Theory Initiative was the group chosen by Fermilab to make its initial benchmark Standard Model prediction for muon g-2), the total QCD amount (a.k.a. hadronic contribution) is 6937 ± 44 which is broken out as HVP = 6845 ± 40, which is a 0.6% relative error and HLbL = 98 ± 18, which is an 18% relative error. This is a lot smaller than the total QCD amount implicated by the experimental measurement which prompted theorists to think for a long time that the failure of the Standard Model prediction to match the experimentally measured value was a sign of beyond the Standard Model physics and spawned all sorts of papers to explain the difference.
The latest HVP calculation component, however, 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) suffered from reliance on somewhat older experimental data that wasn't quite precise enough for this parts per million precision calculation:
Similarly, the introduction to a new paper that was the original motivation for this post notes that:
Reference [44] cited in the block quote above is the current state of the art calculation cited above.
So, it turned out that the Standard Model prediction for the value of muon g-2 actually does match the experimental measurement of muon g-2 at sub parts per million levels, and that earlier calculations of the Standard Model predicted values were just a bit off, due the reliance in earlier calculations on data to substitute for some of the harder calculations that wasn't sufficiently accurate for this level of precision calculation. Basically, the error estimates in the underlying data relied upon for the earlier calculations was (in good faith) understated.
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 18%. Refining the HLbL calculation wasn't a priority because the HLbL is only 1.3% of the total hadronic contribution and this component has only half the uncertainty of the HVP contribution (which works out to a lot less impact on the total uncertainty when the uncertainty from the different components are combined).
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 calculation, make 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 (the more correct in hindsight alternative to the Theory Initiative calculation that was a first principles lattice QCD calculation of HVP, which was incredibly processor and scientist time intensive, in contrast to a data driven one which was somewhat less calculation intensive) and increased the HLbL contribution from 92 ± 18 to 106.8 ± 14.7. That paper stated:
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 half from the Theory Initiative relative uncertainty). 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 HVP and HLbL calculations of the Standard Model value of muon g-2 (both from 2024), and world average experimental value for muon g-2 (from 2023), is just 6.5 ± 41.3 (less than 0.2 sigma).
This gap is so small compared to the one sigma gap that would be expected from random chance, that it suggests (although it does not prove) that it is likely that the experimental uncertainties, or the calculation uncertainties, or both, may be overestimated. This is pretty common in electroweak physics where the sub-disciplinary ethos favors erring on the high side when estimating experimental and calculation uncertainties, and the sources of potential uncertainties are well understood.
Other Recent HLbL work
As the introduction to the new paper explains in a nice overview of the HLbL calculation:
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 about half of the HLbL contribution:
Tian Lin, et al., "Lattice QCD calculation of the π0-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):
The estimate for the neutral pion contribution is about 3.5 lower than in that part of the HLbL calculation in the Theory Initiative White Paper.
This also implies that the uncertainty from the charged pion, the eta, the eta prime, charged kaon, and quark loop contributions to HLbL, while modest in absolute magnitude (about 29-45 from all of them combined) have combined uncertainties on the order of 13-17. 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
The calculation of the Standard Model expected value of muon g-2 is one of the most precise predictions ever confirmed in the history of science.
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 much below 33-34, 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.
Estimating the energy scale at which the confirmation of the Standard Model predicted value of muon g-2 strongly disfavors new physics is more art than science. The physics blogger "Jester" has estimated it at something on the order of ten times of energy scale of the LHC (in a comment to a blog post that I do not have right at hand).
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