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Davide Castelvecchi has a news item in Nature summarizing various results about the possibility of a muon g-2 anomaly.

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Davide Castelvecchi has a news item in Nature summarizing various results about the possibility of a muon g-2 anomaly.

High Energy, Nuclear, Particle Physics News on Phys.org

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I did a lengthy write up of the paper (most of the length was to provide context for readers who haven't been following these developments).

The paper and its abstract are as follows:

In addition to being closer to the BMW result, the paper also clarifies more precisely where the discrepancy between the Theory Initiative calculation and the BMW calculation of the expected Standard Model value of muon g-2 comes from, making further research easier.

There are a couple of ways that the Theory Initiative value could be flawed and the BMW model found to be correct, in which case the measured value of muon g-2 and the predicted Standard Model value would be consistent.

The Theory Initiative determination might be flawed because the experimental data it is using to substitute for some lattice QCD calculations is itself flawed. This is something that was found previously to have caused the muonic proton radius problem.

If that is the problem, it could be resolved by redoing the electron collider experiments at the Linear Electron-Positron Collider experiment (LEP) from 1989-2000, upon which the Theory Initiative is mostly relying, with the greater precision and quality control methods that the subsequent two decades of high energy physics has made possible.

On the other hand, if the problem with the Theory Initiative calculation is the way that this experimental data was incorporated into the overall calculation has some subtle flaw, a new theoretical paper could point out the source of the error. This task would be advanced by a better understanding of what part of the Theory Initiative determination is most likely to be flawed allowing scientists to better focus on what kind of methodological error might be involved, which is what this new paper helps to do.

This is the second paper in a month favoring the BMW calculation. The previous one, also noted by Jester had an abstract and citation as follows:

The write up in*Nature* cited by @atyy explains that:

**Abbreviated Background**

The contributions of the QED, EW, and QCD parts to the overall result, and of the two subparts of the QCD part called hadronic vacuum polarization (HVP) and hadronic light by light (HLbL), from Theory Initiative's determination at Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020). can be summarized as follows (multiplied by 10^11 for easier reading):

Muon g-2 = QED+EW+QCD

About 99.5% of the uncertainty in the muon g-2 calculation comes from the QCD part, even though it is the source of only about 0.006% of the absolute value of anomalous component of the muon's magnetic moment (really less if you add two and double it to get the non-manipulated original value of the full magnetic moment of the muon).

Both the HVP and HLbL parts of the QCD calculation make a material contribution to the uncertainty in the muon g-2 calculation, but the HVP part contributes much more to the overall uncertainty than the HLbL part, because when you have multiple sources of uncertainty in a calculation, the bigger uncertainties tend to swamp the smaller ones unless they are very close in magnitude and there are a great many distinct smaller ones.

There are two leading determinations of the Standard Model prediction's hadronic component which is a profoundly more difficult calculation. One is the Theory Initiative value that substitutes experimental data for some lattice QCD computations of the predicted value, which has a significant difference from the experimentally measured value of muon g-2. The other is the BMW group value that is consistent with the experimentally measured value using purely lattice QCD computations.

The experimental results from directly measuring muon g-2, and theoretically calculated Standard Model predictions for the value of muon g-2 can be summarized as follows (multiplied by 10^11 for easier reading, with one standard deviation magnitude in the final digits shown in parenthesis after each result):

Fermilab (2021): 116,592,040(54)

Brookhaven's E821 (2006): 116,592,089(63)

Combined measurement: 116,592,061(41)

Difference between measurements: 59 (0.7 sigma)

Theory Initiative (TI) prediction: 116,591,810(43)

BMW prediction: 116,591,954(55)

Difference between predictions: 144 (2.1 sigma)

Combined measurement v. TI: 251 (4.2 sigma)

Combined measurement v. BMW: 107 (1.6 sigma)

In order to maintain perspective it is also important to note, however, that both experimental measurements of muon g-2, and both leading Standard Model theoretical predictions, are identical to the first six significant digits. Thus, they are in perfect agreement up to the one part per million level. Only at the parts per ten million level do discrepancies emerge.

This is greater precision, for example, than the theoretically much easier tasks than the empirically determined precision of a first round counting ballots cast in a statewide or national election, or the count of the number of people residing in the United States on a particular day every ten years in the decennial census. The discrepancies are arising at a precision equivalent to one millimeter per ten kilometers.

**What's At Stake?**

The stakes regarding which of these calculations of the Standard Model predicted value are high, because the value of muon g-2 is an experimentally observable quantity that is globally sensitive to the existence and magnitude of most lower energy deviations from the Standard Model of Particle Physics in a single high precision measurement.

If the Theory Initiative value is correct, something about the Standard Model is incorrect at energy scales that can be reached by existing or near future high energy physics experiments.

If the BMW value is correct, then any deviations between Nature and the Standard Model at energy scales that can be reached by existing or near future high energy physics experiments, are limited to those that cancel out in muon g-2 calculations (which as a practical matter, rules out almost all seriously considered experimentally accessible new particle physics theories).

If the BMW calculation is right, we are almost certainly in a "new physics desert" and a next generation particle collider will probably reveal no new physics.

If the Theory Initiative calculation is right, there is a very high likelihood that new physics is right around the corner and will be seen at a next generation particle collider.

Given that the cost of a next generation particle collider is on the order of billions of dollars to build and operate for its full experimental run, the desirability of this very big new purchase for humanity hinges to a great extent on whether the BMW or Theory Initiative calculation is right.

Also, even if we can't resolve that question, knowing precisely what is causing the differences between the calculations can highlight and clarify which kind of next generation collider is most likely to be useful to see new physics if they are out there, and what features it is important for a next generation collider to have to be able to resolve the questions that are the underlying source of the discrepancy.

The paper and its abstract are as follows:

We present a lattice determination of the leading-order hadronic vacuum polarization (HVP) contribution to the muon anomalous magnetic moment, aHVPμ, in the so-called short and intermediate time-distance windows, aSDμ and aWμ, defined by the RBC/UKQCD Collaboration.

We employ a subset of the gauge ensembles produced by the Extended Twisted Mass Collaboration (ETMC) with Nf=2+1+1 flavors of Wilson-clover twisted-mass quarks, which are close to the physical point for the masses of all the dynamical flavors. The simulations are carried out at three values of the lattice spacing ranging from ≃0.057 to ≃0.080 fm with linear lattice sizes up to L≃7.6~fm.

For the short distance window we obtain aSDμ(ETMC)=69.33(29)⋅10^{−10}, which is consistent with the recent dispersive value aSDμ(e+e−)=68.4(5)⋅10^{−10}within ≃1.6σ.

In the case of the intermediate window we get the value aWμ(ETMC)=235.0(1.1)⋅10^{−10}, which is consistent with the result aWμ(BMW)=236.7(1.4)⋅10^{−10}by the BMW collaboration as well as with the recent determination by the CLS/Mainz group of aWμ(CLS)=237.30(1.46)⋅10^{−10}at the ∼1.0−1.3σ level.However, it is larger than the dispersive result aWμ(e+e−)=229.4(1.4)⋅10^{−10}by ≃3.1σ. The tension increases to ≃4.2σ if we average our ETMC result with the BMW and the CLS/Mainz ones.

C. Alexandrou, "Lattice calculation of the short and intermediate time-distance hadronic vacuum polarization contributions to the muon magnetic moment using twisted-mass fermions" arXiv:2206.15084 (June 30, 2022) (82 pages).Our accurate lattice results in the short and intermediate windows hint at possible deviations of the e+e− cross section data with respect to Standard Model (SM) predictions distributed somewhere in the low (and possibly intermediate) energy regions, but not in the high energy region.

In addition to being closer to the BMW result, the paper also clarifies more precisely where the discrepancy between the Theory Initiative calculation and the BMW calculation of the expected Standard Model value of muon g-2 comes from, making further research easier.

There are a couple of ways that the Theory Initiative value could be flawed and the BMW model found to be correct, in which case the measured value of muon g-2 and the predicted Standard Model value would be consistent.

The Theory Initiative determination might be flawed because the experimental data it is using to substitute for some lattice QCD calculations is itself flawed. This is something that was found previously to have caused the muonic proton radius problem.

If that is the problem, it could be resolved by redoing the electron collider experiments at the Linear Electron-Positron Collider experiment (LEP) from 1989-2000, upon which the Theory Initiative is mostly relying, with the greater precision and quality control methods that the subsequent two decades of high energy physics has made possible.

On the other hand, if the problem with the Theory Initiative calculation is the way that this experimental data was incorporated into the overall calculation has some subtle flaw, a new theoretical paper could point out the source of the error. This task would be advanced by a better understanding of what part of the Theory Initiative determination is most likely to be flawed allowing scientists to better focus on what kind of methodological error might be involved, which is what this new paper helps to do.

This is the second paper in a month favoring the BMW calculation. The previous one, also noted by Jester had an abstract and citation as follows:

Euclidean time windows in the integral representation ofthe hadronic vacuum polarization contribution to the muon g−2serve to test the consistency of lattice calculations and may help in tracing the origins of a potential tension between lattice and data-driven evaluations.

In this paper, we present results for the intermediate time window observable computed using O(a) improved Wilson fermions at six values of the lattice spacings below 0.1 fm and pion masses down to the physical value. Using two different sets of improvement coefficients in the definitions of the local and conserved vector currents, we perform a detailed scaling study which results in a fully controlled extrapolation to the continuum limit without any additional treatment of the data, except for the inclusion of finite-volume corrections. To determine the latter, we use a combination of the method of Hansen and Patella and the Meyer-Lellouch-Lüscher procedure employing the Gounaris-Sakurai parameterization for the pion form factor. We correct our results for isospin-breaking effects via the perturbative expansion of QCD+QED around the isosymmetric theory.

Marco Cè, et al., "Window observable for the hadronic vacuum polarization contribution to the muon g−2 from lattice QCD" arXiv:2206.06582 (June 14, 2022) (report number MITP-22-038, CERN-TH-2022-098).Our result at the physical point is a^{win}_{μ}=(237.30 ± 0.79 stat ± 1.22 syst) × 10^{−10}, where the systematic error includes an estimate of the uncertainty due to the quenched charm quark in our calculation.Our result displays a tension of 3.8σ with a recent evaluation of a^win(μ) based on the data-driven method.

The write up in

Lattice QCD had not played a prominent part in the consensus paper because at that time the technique’s predictions were not sufficiently precise. State-of-the-art mathematical techniques and sheer supercomputing power subsequently helped the BMW team to give their lattice-QCD simulations enough of a boost to make the grade. Since then, at least eight teams of physicists around the world have been racing to validate or improve on the BMW prediction. They have started by focusing on a limited range of the particle energies that BMW simulated.

Two preliminary results from this energy ‘window’ were posted on the arXiv preprint repository in April 2022: one by Christopher Aubin at Fordham University in New York City, and his collaborators4, and the other by Gen Wang at the University of Aix-Marseille in France5. Earlier this month, two more groups — one led by Hartmut Wittig at Johannes Gutenberg University in Mainz, Germany, the other by Silvano Simula at the National Institute for Nuclear Physics in Rome — announced their own window results at a muon conference in Los Angeles, California. Simula’s group is writing a preprint, and Wittig’s group submitted its preprint on 14 June6. All four calculations validated BMW’s own window results, even though their lattice techniques vary. “Very different ways of approaching the problem are getting a very similar result,” says Aubin.

“As time goes by, the different groups are converging on a result that agrees with BMW’s, at least in the intermediate window,” says Davide Giusti, a physicist at the University of Regensburg, Germany, who is a former member of Simula’s collaboration, and who now works with another lattice-QCD group led by his Regensburg colleague Christoph Lehner.

The contributions of the QED, EW, and QCD parts to the overall result, and of the two subparts of the QCD part called hadronic vacuum polarization (HVP) and hadronic light by light (HLbL), from Theory Initiative's determination at Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020). can be summarized as follows (multiplied by 10^11 for easier reading):

Muon g-2 = QED+EW+QCD

QED = 116 584 718.931 ± 0.104

EW = 153.6 ± 0.1

QCD = HVP+HLbL = 6937 ± 44

The QED and EW components are both profoundly easier to calculate, and are much more precisely determinable, than the QCD component, and are undisputed.HVP = 6845 ± 40HLbL = 92 ± 18

About 99.5% of the uncertainty in the muon g-2 calculation comes from the QCD part, even though it is the source of only about 0.006% of the absolute value of anomalous component of the muon's magnetic moment (really less if you add two and double it to get the non-manipulated original value of the full magnetic moment of the muon).

Both the HVP and HLbL parts of the QCD calculation make a material contribution to the uncertainty in the muon g-2 calculation, but the HVP part contributes much more to the overall uncertainty than the HLbL part, because when you have multiple sources of uncertainty in a calculation, the bigger uncertainties tend to swamp the smaller ones unless they are very close in magnitude and there are a great many distinct smaller ones.

There are two leading determinations of the Standard Model prediction's hadronic component which is a profoundly more difficult calculation. One is the Theory Initiative value that substitutes experimental data for some lattice QCD computations of the predicted value, which has a significant difference from the experimentally measured value of muon g-2. The other is the BMW group value that is consistent with the experimentally measured value using purely lattice QCD computations.

The experimental results from directly measuring muon g-2, and theoretically calculated Standard Model predictions for the value of muon g-2 can be summarized as follows (multiplied by 10^11 for easier reading, with one standard deviation magnitude in the final digits shown in parenthesis after each result):

Fermilab (2021): 116,592,040(54)

Brookhaven's E821 (2006): 116,592,089(63)

Combined measurement: 116,592,061(41)

Difference between measurements: 59 (0.7 sigma)

Theory Initiative (TI) prediction: 116,591,810(43)

BMW prediction: 116,591,954(55)

Difference between predictions: 144 (2.1 sigma)

Combined measurement v. TI: 251 (4.2 sigma)

Combined measurement v. BMW: 107 (1.6 sigma)

In order to maintain perspective it is also important to note, however, that both experimental measurements of muon g-2, and both leading Standard Model theoretical predictions, are identical to the first six significant digits. Thus, they are in perfect agreement up to the one part per million level. Only at the parts per ten million level do discrepancies emerge.

This is greater precision, for example, than the theoretically much easier tasks than the empirically determined precision of a first round counting ballots cast in a statewide or national election, or the count of the number of people residing in the United States on a particular day every ten years in the decennial census. The discrepancies are arising at a precision equivalent to one millimeter per ten kilometers.

The stakes regarding which of these calculations of the Standard Model predicted value are high, because the value of muon g-2 is an experimentally observable quantity that is globally sensitive to the existence and magnitude of most lower energy deviations from the Standard Model of Particle Physics in a single high precision measurement.

If the Theory Initiative value is correct, something about the Standard Model is incorrect at energy scales that can be reached by existing or near future high energy physics experiments.

If the BMW value is correct, then any deviations between Nature and the Standard Model at energy scales that can be reached by existing or near future high energy physics experiments, are limited to those that cancel out in muon g-2 calculations (which as a practical matter, rules out almost all seriously considered experimentally accessible new particle physics theories).

If the BMW calculation is right, we are almost certainly in a "new physics desert" and a next generation particle collider will probably reveal no new physics.

If the Theory Initiative calculation is right, there is a very high likelihood that new physics is right around the corner and will be seen at a next generation particle collider.

Given that the cost of a next generation particle collider is on the order of billions of dollars to build and operate for its full experimental run, the desirability of this very big new purchase for humanity hinges to a great extent on whether the BMW or Theory Initiative calculation is right.

Also, even if we can't resolve that question, knowing precisely what is causing the differences between the calculations can highlight and clarify which kind of next generation collider is most likely to be useful to see new physics if they are out there, and what features it is important for a next generation collider to have to be able to resolve the questions that are the underlying source of the discrepancy.

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Theory predictions plotted (figure 8 in the paper):

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