Xenon1T Didn't Discover Axions Or An Extraordinary Neutrino Magnetic Moment

The pre-print of a new paper by the XEON1T dark matter detection experiment (press release available here) didn't discovery anything cool and basically admits that in its abstract, which should have read as follows (emphasis added):

We report results from searches for new physics with low-energy electronic recoil data recorded with the XENON1T detector. With an exposure of 0.65 tonne-years and an unprecedentedly low background rate of 76±2stat events/(tonne×year×keV) between 1-30 keV. . . .  

The excess can . . .  be explained by β decays of tritium, which was initially not considered, at 3.2σ significance with a corresponding tritium concentration in xenon of (6.2±2.0)×10^−25 mol/mol. Such a trace amount can be neither confirmed nor excluded with current knowledge of production and reduction mechanisms. . . . 

This analysis also sets the most restrictive direct constraints to date on pseudoscalar and vector bosonic dark matter for most masses between 1 and 210 keV/c2.

The bottom chart in Figure 10 of the paper, below, shows the constraints mention in the last paragraph above:

The necessary level of contamination need to produce this effect with a substance that wasn't tightly screened for because it wasn't pertinent to the primary mission of the experiment to detect WIMP dark matter (which it didn't), is truly tiny. It is roughly one tritium atom (each of which has a mass of 3.016 atomic mass units) per 1,000,000,000,000,000,000,000,000 xenon atoms (each of which has a mass of 131.293 atomic mass units). So, it would take roughly a mere 26 tiny tritium atoms interspersed into each ten kilograms of xenon, to produce the effect observed. This is an irreducible background effect that means that this experimental result is meaningless for telling us about new physics.

What other background effects were accounted for?

But, somebody threw the following language into the abstract (with similar breathless language in the press release), which practically rules out the possibility of new physics in same breath that it announces it, which shouldn't have been more than either a footnote in the body text, or a pre-print by the one or two members of the collaboration who came up with these hypotheses, never to be published, because they don't pass the smell test.

An excess over known backgrounds is observed below 7 keV, rising towards lower energies and prominent between 2-3 keV. The solar axion model has a 3.5σ significance, and a three-dimensional 90% confidence surface is reported for axion couplings to electrons, photons, and nucleons. This surface is inscribed in the cuboid defined by g(ae)<3.7×10^−12, g(ae)g(effan)<4.6×10−18, and g(ae)g(aγ)<7.6×10^−22 GeV^−1, and excludes either g(ae)=0 or g(ae)g(aγ)=g(ae)g(effan)=0. 

The neutrino magnetic moment signal is similarly favored over background at 3.2σ and a confidence interval of μν∈(1.4,2.9)×10^−11μB (90% C.L.) is reported. 

Both results are in tension with stellar constraints.  

. . . The significances of the solar axion and neutrino magnetic moment hypotheses are decreased to 2.1σ and 0.9σ, respectively, if an unconstrained tritium component is included in the fitting.  

Both results contradict the Standard Model. 

The paper also fails to adjust for the "look elsewhere effect", and is engaged in something akin to p hacking.

One of the authors of the paper in a statement made to the New York Times clarifies:

“We want to be very clear that all we are reporting is observation of an excess (a fairly significant one) and not a discovery of any kind,” said Evan Shockley of the University of Chicago in an email.

This very public statement undermining the cymbal crash that is getting attention for this paper, and the self-contradictory language in the abstract and press release, strongly suggests that the 163 scientists in the collaboration were sharply divided over whether such a highly speculative and ill supported claim of new physics should be included in the abstract and press release accompanying the paper. Shockley is doing his best not to have his professional reputation tarnished by this paper by making this statement.

Can we test the most plausible "no new physics" hypothesis of tritium contamination?

No. Why not?

There might have been undetected traces of radioactive tritium (a version of hydrogen with two neutrons) in XENON1T, causing the surrounding liquid to sparkle. The XENON team worked hard to avoid this sort of noise from the beginning, Martens said. Still, he said, the tiny levels of tritium in question here would be impossible to perfectly screen out. And with XENON1T now taken apart to build a bigger future experiment, it’s impossible to go back and check. 

The tritium hypothesis fits the data to a confidence level of 3.2 sigma. Joey Neilsen, a physicist at Villanova University in Pennsylvania, who is not involved in XENON, said that corresponds to about a 1 in 700 chance that random fluctuations would have produced the signal.

The half-life of tritium is 12.3 years. The experiment ran for roughly two years.

Are there other problems with the solar axion hypothesis?

Yes.

One telltale clue would suggest whether the solar axions hypothesis should be taken seriously: seasonal changes in the data, Yu said.  

“If the signal were indeed from solar axions, one would expect a modulation in the signal due to the relative position of the sun to the Earth,” she said.  

As our planet gets a bit more distant from the star it orbits, the solar axion stream should weaken. As Earth gets closer to the sun, Yu said, the signal should get stronger.  

Martens said that no seasonal variation is visible in the XENON1T signal. The signal is too faint, and the experiment ran too briefly at just two years, for XENON1T to have picked it up.

The argument that axions are "well motivated" is also overstated.

Axions are a hypothesis proposed to explain why the strong force doesn't have CP violation (i.e., to oversimplify, the strong force of the Standard Model, like the electromagnetic force, but unlike the weak force, has no arrow of time and treats particles and antiparticles with the same electric charge in the same way). 

But, the "strong CP problem" isn't a problem in any meaningful sense of the word, with anything but a preconceived notion about how Nature should have chosen its physical constants that it is not obliged to honor. 

Also, the lack of CP violation in the strong force already has a natural and reasonable explanation, which is that gluons which transmit the strong force are massless in the Standard Model. And, massless particles (the photon, the gluon and the hypothetical graviton) because they travel at the speed of light, don't experience time. But, since CP violation is equivalent to time direction violation (if CPT is perfectly conserved as there is every reason to believe it is), you wouldn't expect a particle that doesn't experience time to work differently going forward in time and going backward in time. 

Moreover, the axions needed to explain this experimental result does not have the properties of the axion needed to explain the "strong CP problem" if it was a problem. It would instead be a theoretical variation on the original axion concept with no real physical or mathematical motivation.

Jester (a.k.a. Adam Falkowski) at his Resonances blog is likewise unconvinced (emphasis added):

The XENON collaboration was operating a 1-ton xenon detector in an underground lab in Italy. Originally, this line of experiments was devised to search for hypothetical heavy particles constituting dark matter, so called WIMPs. For that they offer a basically background-free environment, where a signal of dark matter colliding with xenon nuclei would stand out like a lighthouse. However all WIMP searches so far have returned zero, null, and nada. 

Partly out of boredom and despair, the xenon-based collaborations began thinking out-of-the-box to find out what else their shiny instruments could be good for. One idea was to search for axions. These are hypothetical superlight and superweakly interacting particles, originally devised to plug a certain theoretical hole in the Standard Model of particle physics. If they exist, they should be copiously produced in the core of the Sun with energies of order a keV. This is too little to perceptibly knock an atomic nucleus, as xenon weighs over a hundred GeV. However, many variants of the axion scenario, in particular the popular DFSZ model, predicts axions interacting with electrons. Then a keV axion may occasionally hit the cloud of electrons orbiting xenon atoms, sending one to an excited level or ionizing the atom. These electron-recoil events can be identified principally by the ratio of ionization and scintillation signals, which is totally different than for WIMP-like nuclear recoils. This is no longer a background-free search, as radioactive isotopes present inside the detector may lead to the same signal. Therefore collaboration have to search for a peak of electron-recoil events at keV energies.

This is what they saw in the XENON1t data

Energy spectrum of electron-recoil events measured by the XENON1T experiment. 

The expected background is approximately flat from 30 keV down to the detection threshold at 1 keV, below which it falls off abruptly. On the other hand, the data seem to show a signal component growing towards low energies, and possibly peaking at 1-2 keV. Concentrating on the 1-7 keV range (so with a bit of cherry-picking), 285 events is observed in the data compared to an expected 232 events from the background-only fit. In purely statistical terms, this is a 3.5 sigma excess. . . . 

The peak of the excess corresponds to the temperature in the core of the Sun (15 million kelvin = 1.3 keV), so our star is a natural source of these particles (but at this point XENON cannot prove they arrive from the Sun). Furthermore, the particles must couple to electrons, because they can knock xenon's electrons off their orbits 

. . . For QCD axions the defining feature is their coupling to gluons, but in generic constructions one also finds the interaction between the axion and electrons. . . .

in the standard picture, the interactions of neutrinos with electrons are too weak to explain the excess. To that end one has to either increase their flux (so fiddle with the solar model), or to increase their interaction strength with matter (so go beyond the Standard Model). . . . 

How confident should we be that it's new physics? 

Experience has shown again and again that anomalies in new physics searches have, with a very large confidence, a mundane origin that does not involve exotic particles or interactions. In this case, possible explanations are, in order of likelihood, 1) small contamination of the detector, 2) some other instrumental effect that the collaboration hasn't thought of, 3) the ghost of Roberto Peccei, 4) a genuine signal of new physics. 

In fact, the collaboration itself is hedging for the first option, as they cannot exclude the presence of a small amount of tritium in the detector, which would produce a signal similar to the observed excess. Moreover, there are a few orange flags for the new physics interpretation:

1. Simplest models explaining the excess are excluded by astrophysical observations. If axions can be produced in the Sun at the rate suggested by the XENON result, they can be produced at even larger rates in hotter stars, e.g. in red giants or white dwarfs. This would lead to excessive cooling of these stars, in conflict with observations. The upper limit on the axion-electron coupling from red giants is 3*10^-13, which is an order of magnitude less than what is needed for the XENON excess. The neutrino magnetic moment explanations faces a similar difficulty. Of course, astrophysical limits reside in a different epistemological reality; it is not unheard of that they are relaxed by an order of magnitude or disappear completely. But certainly this is something to worry about.  

2. At a more psychological level, a small excess over a large background near a detection threshold.... sounds familiar. We've seen that before in the case of the DAMA and CoGeNT dark matter experiments, at it didn't turn out well.  

3. The bump is at 1.5 keV, which is *twice* 750 eV.  

So, as usual, more data, time, and patience is needed to verify the new physics hypothesis.

Roberto Peccei, whose ghost is mentioned, was the physicist at UCLA who first proposed the axion as a hypothetic particle to explain the lack of CP violation in the strong force back in 1977. He died on June 1, 2020 at age 78 of non-COVID related causes. It isn't implausible that the collaboration allowed the axion claim to be highlighted despite only marginal support for it in this experiment, to honor and call attention to his recent passing.

The third point about 750 eV is also a bit of an inside joke, referencing a 750 GeV diphoton excess of events noted in late 2015 and early 2016 that turned out to be nothing more than a statistical fluke after much undeserved hype.

Lubos Motl is more neutral in his assessment and provides additional links in a blog post on the topic at The Reference Frame, also provides a humanizing lede to the story:

This result has been known for a year or so. Grad students were tortured by the requirement of their silence. But the news got out today.

A New Approach To Measuring Hubble's Constant

Hubble's constant is an observable quantity that quantifies the rate at which the universe is expanding. As Wikipedia in the previous sentence explains (omissions from the source not indicated editorially for ease of reading):

It is often expressed by the equation v = D, with  the constant of proportionality—Hubble constant—between the "proper distance" D to a galaxy, which can change over time, and its speed of separation v, i.e. the derivative of proper distance with respect to cosmological time coordinate. The Hubble constant can also be interpreted as the relative rate of expansion. In this form  = 7%/Gyr, meaning that at the current rate of expansion it takes a billion years for an unbound structure to grow by 7%. Though the Hubble constant is roughly constant in the velocity-distance space at any given moment in time, the Hubble parameter , which the Hubble constant is the current value of, varies with time, so the term 'constant' is sometimes thought of as somewhat of a misnomer.

The value of is about 70 (km/s)/Mpc. "Late universe" measurements using calibrated distance ladder techniques have converged on a value of approximately 73 km/s/Mpc. Since 2000, "early universe" techniques based on measurements of the cosmic microwave background have become available, and these agree on a value near 67.7 km/s/Mpc. (This is accounting for the change in the expansion rate since the early universe, so is comparable to the first number.) As techniques have improved, the estimated measurement uncertainties have shrunk, but the range of measured values has not, to the point that the disagreement is now statistically significant. This discrepancy is called the Hubble tension.

As the chart above illustrates the cosmic microwave background estimates in red are quite consistent and have a small uncertainty, while "late universe" measurements, in blue, have more scatter and larger uncertainties. 

The extent to which this tension is simply experimental error or is instead a clue to the nature of the universe's expansion that is at odds with the simple "Hubble's law" model that it codifies is a major unresolved issue in observational astronomy and cosmology.

A new pre-print measures that quantity in a less commonly used way that is partially independent of the most past methods to measure it using the baryonic Tully-Fisher relation between the mass of the ordinary matter in a galaxy and its rotation rate. The Tully-Fischer relation is as a phenomenological relationship that holds in all observed galaxies, from which MOND (a toy model modified gravity law used to explain dark matter phenomena at galaxy scales and below) was derived, and which dark matter particle theories seek to explain. 

The new measurement is consistent with, but on the high side of, other recent "late universe" methods that determine proper distance by other methods, and almost identical to the value determined in two prior studies using the Tully-Fischer relationship, all of which are summarized here. 

The new paper's measurement of 75.1 ± 3.6 is just barely consistent at the two sigma level with early universe based Dark Energy Survey (DES) measurement, and not quite consistent with the cosmology based measurements of Planck 2018.

We explore the use of the baryonic Tully-Fisher relation (bTFR) as a new distance indicator. Advances in near-IR imaging and stellar population models, plus precise rotation curves, have reduced the scatter in the bTFR such that distance is the dominant source of uncertainty. Using 50 galaxies with accurate distances from Cepheids or tip magnitude of the red giant branch, we calibrate the bTFR on a scale independent of Ho. We then apply this calibrated bTFR to 95 independent galaxies from the SPARC sample, using CosmicFlows-3 velocities, to deduce the local value of Ho. We find Ho = 75.1 +/- 2.3 (stat) +/- 1.5 (sys) km s−1 Mpc−1.

James Schombert, Stacy McGaugh, Federico Lelli, "Using The Baryonic Tully-Fisher Relation to Measure H₀" arXiv: 2006.08615 (June 15, 2020).

McGaugh discusses the paper at his blog with background material in a preceding post summing up the methodological difficulties involved and the approximations that have to be used.

McGaugh notes that historically, the Hubble tension between different theoretical means of calculating the Hubble constant was much greater than it is today, in absolute terms, basically cautioning against reading too much into the discrepancy yet.

What The Harappan Language Was Not

None of the conclusions in this paper are something that I haven't argued for before at this blog.

But, this open access paper by a professor from India writing outside his own discipline (he has published in peer reviewed journals in the field, however), recaps these conclusions and the basis for them at length. It is from 2011 or 2012 and does not appear to have been published in a journal or in book form.

This paper argues against the Dravidian, Vedic and Paramunda Indus theories, and shows why Dravidian languages, Sanskrit or Paramunda languages could not have been candidates for the Indus Valley Civilization which flourished from 2600 BC to 1900 BC in the North-West of India and Pakistan. Supporters of these three hypotheses are welcome to provide a systematic refutation of all the points raised in this paper. This paper adopts a multi-disciplinary approach, drawing conclusions from many different fields of science. Quotes of several mainstream scholars of repute are presented in support of the conclusions arrived at in this paper. An alternative hypothesis of the identity of the Harappans is also presented towards the end of the paper.

Sujay Rao Mandavilli, 'The Demise of the Dravidian, Vedic and Paramunda Indus Hypotheses: A brief explanation as to why these three Hypotheses are no longer tenable" (Open access, self-published online).

Muon g-2 Predictions Recapped

A new 192 page paper prepared by a huge collaboration of authors and led by Fermilab exhaustively reviews the latest efforts to theoretically calculate the property of anomalous magnetic moment of the muon, called muon g-2 reports on the latest state of that effort. Another big review came out earlier this year in March.

The tension between experiment and theoretical prediction of this quantity is one of the leading unexplained problems in physics today. A September 9, 2019 post at this blog also examined the discrepancy in detail. As I summarized then, using slightly different source numbers from this paper (which, if anything assigns a larger share of the error to the QCD component):

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). . . .

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%

On one hand, muon g-2 is predicted to exquisite seven significant digit accuracy, suggesting the general correctness of the Standard Model used to calculate it. On the other hand, a strong tension with the most recent precise experimental measurement of that quantity, remains, dashing the hope that much of the discrepancy might have been due to uncertainties in the most uncertain part of the calculation. This review is timely because a new, four times more precise experimental measurement will be available in a year or two.  

Improvements of the precision in the theoretical estimate, which are predominantly from the quantum chromodynamics component of the calculation that make the smallest contribution to the absolute value of this measurable quantity are also expected in the near term.

Conventional wisdom is that the tension will ease, producing an experimentally measured value closer to the theoretically predicted one, despite the much greater precision of the new measurement. 

But, if this does not happen, the quantity muon g-2 is a good global indicator of the existence and magnitude of beyond the Standard Model physics, although it is not a very good tool for determining the precise nature of the new physics because it is affected by essentially all parts of the Standard Model when measured with the precision that is now possible.

A third possibility is that scientists are making some shared conceptual error in how they do their theoretical predictions of the value of muon g-2 that doesn't really amount to new physics. Gravitational corrections, if any, however, should be negligible in magnitude compared to other uncertainties. 

Cheat Sheet For When Experimental Results Are Announced

The current state of the art Brookhaven measurement was:

116,592,089(63) x 10^−11.

The headline that will be used when the experimentally measured value announced later this year or next year can be determined by referring the the cheat sheet below where the number listed is the new experimental result.

Less than or equal to 116,591,582 x 10^-11:

New physics has been discovered contradicting the Standard Model more definitively than any prior measurement in physics. The Brookhaven measurement was deeply flawed.

116,591,581 to 116,591,718 x 10^-11:

There is tension with the Standard Model prediction in the opposite direction from the Brookhaven measurement. The Brookhaven measurement was deeply flawed. At the high end of this range it will be called a "slight tension", at the low end of this range it will be called a "strong tension."

116,591,718 to 116,591,901 x 10^11:

The new measurement has confirmed the Standard Model prediction. Any new physics that impact muon g-2 are too small to observe experimentally.

116,591,902 to 116,592,037 x 10^11:

There is tension with the Standard Model prediction. At the low end of this range it will be called a "slight tension", at the high end of this range it will be called a "strong tension."

Greater than or equal to 116,592,038 x 10^11:

New physics has been discovered contradicting the Standard Model more definitively than any prior measurement in physics.

The Facts

The new paper and its abstract are as follows:

We review the present status of the Standard Model calculation of the anomalous magnetic moment of the muon. 

This is performed in a perturbative expansion in the fine-structure constant 

α and is broken down into pure QED, electroweak, and hadronic contributions. 

The pure QED contribution is by far the largest and has been evaluated up to and including (α5) with negligible numerical uncertainty. The electroweak contribution is suppressed by (mμ/MW)2 and only shows up at the level of the seventh significant digit. It has been evaluated up to two loops and is known to better than one percent. Hadronic contributions are the most difficult to calculate and are responsible for almost all of the theoretical uncertainty. The leading hadronic contribution appears at (α2) and is due to hadronic vacuum polarization, whereas at (α3) the hadronic light-by-light scattering contribution appears. Given the low characteristic scale of this observable, these contributions have to be calculated with nonperturbative methods, in particular, dispersion relations and the lattice approach to QCD. The largest part of this review is dedicated to a detailed account of recent efforts to improve the calculation of these two contributions with either a data-driven, dispersive approach, or a first-principle, lattice-QCD approach. The final result reads aSMμ=116591810(43)×10−11 and is smaller than the Brookhaven measurement by 3.7σ. 

The experimental uncertainty will soon be reduced by up to a factor four by the new experiment currently running at Fermilab, and also by the future J-PARC experiment. This and the prospects to further reduce the theoretical uncertainty in the near future-which are also discussed here-make this quantity one of the most promising places to look for evidence of new physics.

T. Aoyama, et al., "The anomalous magnetic moment of the muon in the Standard Model" arXiv (June 8, 2020).

The conclusion is as follows:

In this paper we provide a detailed analysis and review of the SM calculation of the muon anomalous magnetic moment aµ. The emphasis is on the hadronic contributions, since they dominate the final uncertainty, but the QED and electroweak contributions are also discussed in detail and up-to-date numbers are provided. 

The QED contribution, which has been calculated up to tenth order in the perturbative expansion, i.e., O(α 5 ), is reviewed in Sec. 6. The final number depends on the input used for the fine-structure constant α and at present there are two independent determinations that differ by about 2.4 standard deviations. The impact of this discrepancy on the final number for aµ is however well below the uncertainty of the QED contribution itself, which is dominated by the estimated effect of the O(α 6 ) contribution. As final number we take the one based on the value of α obtained from atom-interferometry measurements of the Cs atom [117], see Eq. (6.30), and the latest QED calculations from Refs. [33, 34]: 

a QED µ (α(Cs)) = 116 584 718.931(104) × 10−11 . (8.1) 

Electroweak contributions are reviewed in Sec. 7: they have been calculated up to two loops and an estimate of the leading logarithmic contribution beyond two-loop level is also included in the final estimate. The hadronic loops, which appear at two-loop level, are also included and dominate the uncertainty of the EW contribution. The final result Eq. (7.16) (mainly based on Refs. [35, 36]) reads 

a EW µ = 153.6(1.0) × 10−11 , (8.2) 

with an uncertainty ten times larger than the QED one, but still negligible with respect to the hadronic uncertainties. 

In the section on data-driven evaluations of HVP we reviewed both the available data sets for the e + e − → hadrons cross section and the techniques applied for the evaluation of the HVP dispersive integral. In particular, we provide a detailed discussion of the differences between these approaches and the current limitations of the dispersive HVP evaluation, as they arise from the published experimental uncertainties as well as, crucially, from unresolved tensions among the data sets, especially in the dominant ππ channel. As the main result, Eq. (2.33), we devised a merging procedure that adequately takes into account these tensions, which also drive the differences between the available HVP evaluations. The resulting estimate, based on Refs. [2–7] as well as the main experimental input from Refs. [37– 89], 

a HVP, LO µ = 6931(40) × 10−11 (8.3) 

should provide a conservative but realistic assessment of the current precision of data-driven HVP evaluations. In the same framework, the LO result is complemented by NLO [7] and NNLO [8] HVP iterations, see Eq. (2.34) and Eq. (2.35), 

a HVP, NLO µ = −98.3(7) × 10−11 , 

a HVP, NNLO µ = 12.4(1) × 10−11 , (8.4) 

leading to the sum 

a HVP, LO µ + a HVP, NLO µ + a HVP, NNLO µ = 6845(40) × 10−11 . (8.5) 

Finally, we discussed the prospects for future improvements, including new data from several e + e − experiments as well as the possibility to measure HVP independently in electron–muon scattering. 

The status of lattice QCD+QED calculations of HVP is reviewed in Sec. 3. While lattice calculations can, in principle, provide an alternate, ab initio determination of the HVP contribution, they are, at present, not precise enough to confront the data-driven evaluations. The current “lattice world average,” obtained in Sec. 3.5.1 from a conservative combination of current, published lattice QCD+QED results, is consistent with the data-driven result of Eq. (8.3) but with a large enough uncertainty to also cover the “no new physics” scenario: 

a HVP, LO µ = 7116(184) × 10−11 , (8.6) 

based on Refs. [9–17]. 

The phenomenological estimate of HLbL scattering as reviewed in Sec. 4 is essentially based on a dispersive approach, in analogy to HVP. The various contributions to HLbL can be collected into three main pieces depending on how they have been estimated: (1) the numerically dominant contributions from the single-pseudoscalar poles and large parts of the two-pion intermediate states, both of which rely on data-driven approaches and are under good control; (2) the model-dependent estimates for the sum of scalar, tensor, and axial-vector contributions, as well as the impact of short-distance constraints; all of these still suffer from significant uncertainties, which in the total have been added linearly; (3) the c-quark contribution, which can be estimated using perturbative QCD, with a conservative uncertainty estimate in view of the low scale and potential nonperturbative effects. The final estimates for HLbL from Table 15 (mainly based on Refs. [18–30] and, in addition to e + e − → hadrons cross sections, the experimental input from Refs. [90–109]) and HLbL at NLO [31] from Eq. (4.91) read as follows: 

a HLbL µ = (69.3(4.1) + 20(19) + 3(1)) × 10−11 = 92(19) × 10−11 , (8.7) 

a HLbL, NLO µ = 2(1) × 10−11 , (8.8) 

where the first line gives the three pieces in the same order as discussed above and the total in the second line is obtained by adding the central values of the three contributions and combining the errors in quadrature. The final error is about 20% and is completely dominated by the model estimates of a numerically subdominant part of the total. 

The lattice determination of HLbL scattering is reviewed in Sec. 5. The lattice methodology for this quantity has advanced significantly in the last years [110–116] and has now reached a mature stage, resulting in a calculation [32] with reliable estimates of both statistical and systematic uncertainties (Eq. (5.49)): 

a HLbL µ = 78.7(30.6)stat(17.7)sys × 10−11 . (8.9) 

There have been extensive checks between different groups working on the lattice HLbL as well as internal checks of the calculations such as the regression against the leptonic loop or pion-pole contributions. These checks are explained in detail in Sec. 5. 

To obtain a recommendation for the full SM prediction we proceed as follows: for HLbL scattering, there is excellent agreement between phenomenology and lattice QCD, to the extent that it is justified to consider a weighted average. Taking into account that the lattice-QCD value does not include the c-quark loop, we first average the light-quark contribution and add the c quark as estimated phenomenologically in the end. This produces 

a HLbL µ (phenomenology + lattice QCD) = 90(17) × 10−11 , (8.10) 

and, using Eq. (8.8), 

a HLbL µ (phenomenology + lattice QCD) + a HLbL, NLO µ = 92(18) × 10−11 . (8.11) 

For HVP, the current uncertainties in lattice calculations are too large to perform a similar average and the future confrontation of phenomenology and lattice QCD crucially depends on the outcome of forthcoming lattice studies. For this reason, we adopt Eq. (8.3) as our final estimate, emphasizing that the uncertainty estimate already accounts for the tensions in the e + e − data base. Combined with the QED and EW contributions, we obtain 

a SM µ = a QED µ + a EW µ + a HVP, LO µ + a HVP, NLO µ + a HVP, NNLO µ + a HLbL µ + a HLbL, NLO µ = 116 591 810(43) × 10−11 . (8.12) 

This value is mainly based on Refs. [2–8, 18–24, 31–36], which should be cited in any work that uses or quotes Eq. (8.12). It differs from the Brookhaven measurement [1] 

a exp µ = 116 592 089(63) × 10−11 , (8.13) 

where the central value is adjusted to the latest value of λ = µµ/µp = 3.183345142(71) [751], by 

∆aµ := a exp µ − a SM µ = 279(76) × 10−11 , (8.14) 

corresponding to a 3.7σ discrepancy. In constructing Eqs. (8.5), (8.11), and (8.12), we have taken into account the correlations between the uncertainties in the leading and subleading HVP contributions as well as the partial correlation in the case of HLbL, with numbers rounded including subleading digits from the individual contributions. 

The prospects for near-term and long-term improvements of the uncertainties in the SM prediction are excellent. As discussed in Sec. 2, a new measurement of the crucial 2π channel by SND is currently under review, and more measurements of the 2π channel and others are forthcoming, leading to the realistic prospects of reducing the dispersive HVP error by a factor of 2. In addition, independent data-driven input could be provided by the MUonE project. The past five years have seen great progress in the development of methods to address the challenges associated with lattice determinations of a HVP, LO µ at the target precision, as discussed in detail in Sec. 3. This is also evident in the recent high-precision lattice result for a HVP, LO µ [392], which, however, still needs to be scrutinized in detail. With these methods now in place, and with sustained, dedicated effort, lattice results with permil-level precision will be forthcoming. The phenomenological determination of HLbL scattering has been consolidated at a level well below the Glasgow consensus, see Sec. 4, with the dominant contributions derived using data-driven methods in analogy to the dispersive HVP approach. With expected progress on the subleading contributions, a 10% calculation of HLbL scattering now appears feasible. Finally, we expect more independent lattice calculations of the HLbL to appear in the next years. Building on the newly developed methodologies, a 10% lattice calculation of the HLbL also appears feasible by the end of the Fermilab experiment.