The muon g-2 discrepancy is currently about 3.5 sigma (about two parts per million) by my calculation, based on the data below. New experimental results coming soon will tweak that considerably, although no one knows if it will reduce it or increase it. From the introduction:

The Particle Data Group (PDG) gives an updated value for the muon anomaly in the form [6]

a exp = 116 592 091(54)(33) × 10^−11. (2)

The current muon experiment at Fermilab plans to reduce the experimental uncertainty by the factor of four, σfuture ≈ (1.0 ÷ 1.5) × 10^−10 = (10 ÷ 15) × 10^−11 .

The theoretical results for the muon anomaly in the SM are traditionally represented as a sum of three parts, a SM = a QED + a EW + a had (3)

with a QED, a EW being the leptonic and electroweak parts, respectively, and a had includes quarks. In fact, the separation in Eq. (3) is such that quark loops are included also in a EW. The leptonic part is computed in perturbation theory and reads a

QED = 116 584 718.95(0.08) × 10^−11. (4)

The computation extends up to five-loop level, using both analytical and numerical techniques. At present, the numerical results are steadily being checked/refined with powerful analytical methods for Feynman integral evaluation. In view of the experimental uncertainty, the QED part of the theory prediction for the muon anomaly can be considered to be exact, giving a negligible uncertainty.

The electroweak part is known to two loops and reads a EW = 153.6(1.0) × 10^−11. (5)

The absolute value of a EW is small and the uncertainty of this contribution is negligible for comparison with present experiments. The hadronic part a had in the SM is related to quark contributions. To leading order (LO) in the fine structure constant α it is given by the two-point function of hadronic electromagnetic currents through the vacuum polarization of the photon. In order to match the experimental accuracy of the muon anomaly one has to include next-to-leading order (NLO) contributions in α. At this order the four-point function of hadronic electromagnetic currents starts to contribute.

The accurate calculation of light hadronic contributions is impossible in perturbative Quantum Chromodynamics (QCD) as they are represented by (almost) massless light quarks and are infrared (IR) singular in perturbation theory. This is the main obstacle for obtaining high precision SM predictions. Instead, the theoretical estimates for hadronic modes utilize scattering data. The LO hadronic contribution extracted from e +e − data is given by

a(LO; had; e +e −) = 6931(33)(7) × 10^−11. (6)

Other estimates may include also data from hadronic τ lepton decays [4], a(LO; had; τ ) = (689.46 ± 3.25) × 10^−10. (7)

In our estimates we stick to the PDG value in Eq. (6) for definiteness, called a(LO; had). The hadronic contribution is rather large and should be computed with a precision of one or two per mille. This is a challenge for the theory in a situation where there are no appropriate tools for an analytical theoretical computation. Presently, the lattice is emerging as a promising tool for this task. In NLO there are further hadronic contributions. They are extensively discussed in the literature and estimated in various approaches.

The current total SM prediction reads:

a SM = 116 591 823(1)(34)(26) × 10^−11. (8)

The difference ∆aµ = a exp − a SM = 268(63)(43) × 10^−11 (9)

might uncover physics beyond the SM. It is not formally statistically significant yetbut is considered to be rather serious for the prospect of discovering new physics.The main theoretical uncertainties come from the hadronic LO part and from the NLO contribution of the genuine four-point function called light-by-light (LBL).

The paper is here.

The errors in theoretical calculation by component are summarized roughly as follows (in comparable units):

QED 0.08 (i.e. 0.2% of the total)

Weak Force 1.00 (i.e. 2.9% of the total)

QCD 33.73 (i.e. 96.9% of the total).

The error from the components combined is +/- 34 * 10^-11.

The theoretical value also has a +/- 23 * 10^-11 component and a +/ 1.0 * 10^-11 component. I don't know what they stand for but that brings total theoretical prediction error to +/- 43 * 10^-11.

Proportion of total value from each component:

QED 99,994% (116 584 718.95)

Weak Force 0.00013% (153.6)

QCD 0.006% (6931)

Relative error percentage:

QED 0.000 000 0686%

Weak Force 0.65%

QCD 4.88%

Gravity is not considered but would make a negligible contribution if it was considered.

Gravity is not considered but would make a negligible contribution if it was considered.

The error bars on the error are also greatest where the error is greatest.

There are actually two parts of the hadronic component. The larger one, which is more accurately known but still a significant source of error involves hadronic vacuum polarization, the smaller one, which as a 25% margin of error, is the hadronic "light on light" contribution which involves the interactions of light virtual quarks. Broken down the impacts times 10^-11 are about:

HVP 6 850.6 ± 43

HLbL 105 ± 26

Thus, about a third of the total error comes from a component that makes up only about 0.0001% of the total result, and almost all but 3% or so of the remaining error comes from a component that makes up only 0.004% of the total result.

The absolute value of the combined hadronic component has fallen about about 24 * 10^-11, about 0.3%, and the error has fallen from about 50.25 to about 33.73, about a third, since the figures above from a different source that I cite above were reported. I am suspicious, however, that the errors stated in the theoretical value quoted above in the new review article might actually be electro-weak (1), HVP (34) and HLbL (26), even though that is not exactly what the article states, in which the combined QCD error is 43 which is only about a 15% improvement in QCD error from my previous source attributable almost entirely to refinements in the HVP part.

The combined experimental error is +/- 63 * 10^-11, currently about a third greater than the theoretical error. The new Fermi experiment should bring the experimental error down to about +/- 16 * 10^-11.

An experimental value two standard deviation above what was measured would leave a discrepancy of 142 to be explained with improved theoretical estimates. HVP and HLbL minus two standard deviations would be about:

HVP 6764.6 (a reduction of 86 which is about 1.3%).

HLbL 53 (a reduction of 52 which is about 50.5%).

The combined reduction would be 138.

So, realistically, almost all of the muon g-2 anomaly could be explained by a modestly overstated HVP contribution and a greatly overstated HLbL contribution in the QCD part of the theoretical muon g-2 calculation. Indeed, I think it is more fruitful to use the muon g-2 anomaly as a pointer towards what kind of errors appear to be present in the QCD calculations than to use it as a pointer towards new physics.

The combined experimental error is +/- 63 * 10^-11, currently about a third greater than the theoretical error. The new Fermi experiment should bring the experimental error down to about +/- 16 * 10^-11.

An experimental value two standard deviation above what was measured would leave a discrepancy of 142 to be explained with improved theoretical estimates. HVP and HLbL minus two standard deviations would be about:

HVP 6764.6 (a reduction of 86 which is about 1.3%).

HLbL 53 (a reduction of 52 which is about 50.5%).

The combined reduction would be 138.

So, realistically, almost all of the muon g-2 anomaly could be explained by a modestly overstated HVP contribution and a greatly overstated HLbL contribution in the QCD part of the theoretical muon g-2 calculation. Indeed, I think it is more fruitful to use the muon g-2 anomaly as a pointer towards what kind of errors appear to be present in the QCD calculations than to use it as a pointer towards new physics.

Of course, unknown unknowns could increase both theoretical and experimental errors.

This issue is discussed another recent review article that I mention in a post on March 26, 2019. The issue is also discussed in previous blog posts on June 29, 2018, February 1, 2018 (regarding a resolution that turns out to have been obviously flawed upon peer review), January 3, 2018, August 8, 2017, January 13, 2017, November 22, 2013 (humorous), November 21, 2013, and November 12, 2013.

UPDATE September 11, 2019:

A paper from earlier this year lays out the issues well:

This issue is discussed another recent review article that I mention in a post on March 26, 2019. The issue is also discussed in previous blog posts on June 29, 2018, February 1, 2018 (regarding a resolution that turns out to have been obviously flawed upon peer review), January 3, 2018, August 8, 2017, January 13, 2017, November 22, 2013 (humorous), November 21, 2013, and November 12, 2013.

UPDATE September 11, 2019:

A paper from earlier this year lays out the issues well:

# Review of Lattice QCD Studies of Hadronic Vacuum Polarization Contribution to Muon g-2

(Submitted on 25 Jan 2019 (v1), last revised 4 Jun 2019 (this version, v2))

Lattice QCD (LQCD) studies for the hadron vacuum polarization (HVP) and its contribution to the muon anomalous magnetic moment (muon g-2) are reviewed. There currently exists more than 3-sigma deviations in the muon g-2 between the BNL experiment with 0.5 ppm precision and the Standard Model (SM) predictions, where the latter relies on the QCD dispersion relation for the HVP. The LQCD provides an independent crosscheck of the dispersive approaches and important indications for assessing the SM prediction with measurements at ongoing/forthcoming experiments at Fermilab/J-PARC (0.14/0.1 ppm precision). The LQCD has made significant progress, in particular, in the long distance and finite volume control, continuum extrapolations, and QED and strong isospin breaking (SIB) corrections.In the recently published papers, two LQCD estimates for the HVP muon g-2 are consistent with No New Physics while the other three are not. The tension solely originates to the light-quark connected contributions and indicates some under-estimated systematics in the large distance control. The strange and charm connected contributions as well as the disconnected contributions are consistent among all LQCD groups and determined precisely. The total error is at a few percent level. It is still premature by the LQCD to confirm or infirm the deviation between the experiments and the SM predictions. If the LQCD is combined with the dispersive method, the HVP muon g-2 is predicted with 0.4% uncertainty, which is close upon the target precision required by the Fermilab/J-PARC experiments.Continuous and considerable improvements are work in progress, and there are good prospects that the target precision will get achieved within the next few years.

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