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Today it was announced that the Fermilab measurements of the muon's gyromagnetic ratio are in disagreement with the Standard Model with a statistical certainty reported at 4.2 sigma [1, 2], raising the distinct possibility that the discrepancy measured originally at Brookhaven two decades ago is real. On the other hand, a paper published today in Nature claims that through state of the art lattice calculations, they have obtained a new theoretical estimate of the muon's $g$-factor which, to my understanding, they claim reduce the discrepancy between theory and previous experiments.

From what I have read and understood, it seems to me that (1) the new Fermilab sigma figure is presumably being compared to older theoretical calculations and (2) the new theoretical calculation is presumably being compared to older experimental measurements. This seems rather incredible that these two seemingly conflicting announcements come on the same day.

So my question is, what gives? Do the new theoretical calculations (assuming they can be verified) lower the new Fermilab sigma figure?

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This seems rather incredible that these two seemingly conflicting announcements come on the same day.

The pre-print for the Nature paper by the BMW group was placed on arXiv in 2020 around the same time as the muon g-2 Theory Initiative paper (submitted on 8 Jun 2020 and last revised 13 Nov 2020, published here) that the Fermilab collaboration referenced in its announcement. The BMW pre-print was posted February 27, 2020, and last revised on August 18, 2020. (The 14-person BMW team is named after Budapest, Marseille and Wuppertal, the three European cities where most team members were originally based.) So, insiders weren't surprised.

Fermilab announced when its experimental results would be shared. Nature intentionally and strategically released the paper that had been in their offices awaiting publication for about eight months on the same day to maximize its impact. The theory presentation on Zoom this morning from Fermilab actually mentioned the BMW paper in passing although it didn't emphasize the point.

The BMW paper and the Theory Initiative paper use quite different methodologies to calculate the leading order Hadronic Vacuum Polarization (LO-HVP) contribution to the Standard Model muon g-2 calculation, which is the dominant source of theoretical uncertainty in the calculation. The BMW paper makes some refinements in the calculation that are very computation intensive (it took several hundred million "core hours" on seven sets of supercomputers to calculate) and is a straight up theory calculation (Quanta Magazine has a nice discussion of what this involved with a bit more depth). The Theory Initiative paper uses data on electron-positron annihilation products experimentally as a substitute for doing some key calculations that BMW does "on paper" (with BMW criticizing the Theory Initiative and others using similar methods for not transitioning the experimental data into the theoretical calculation in the manner that they think is correct).

The new Fermilab experimental results largely confirmed the previous Brookhaven measurement of muon g-2; the two experimental results were consistent at the 0.6 sigma level. The new Fermilab results moved slightly from the previous measurement in favor of the theoretically predicted values. The results were (all multiplied by a factor of 10-11):

source result
Brookhaven's E821 (2006) 116,592,089(63)
Combined Experimental Measurement 116,592,061(41)
Fermilab (2021) 116,592,040(54)
BMW Standard Model Prediction 116,591,954(55)
Theory Initiative Standard Model Prediction 116,591,810(43)
BMW - Theory Initiative 144

(It is also worth noting that all of the experimental results and all of the theoretical predictions are all identical when rounded to the first six significant digits, i.e. at a parts per million level.)

Compared to the combined results used for the 4.2 sigma compared to the Theory Initiative paper, the BMW number is consistent with the combined experimental result at the 1.6 sigma level. The BMW paper claims 27% more uncertainty in its theoretical result than the Theory Initiative paper and the Theory Initiative result has been replicated (something much more difficult to do in the case of the much more computation resource intensive BMW calculation), which is part of the reasons that Fermilab decided to go with the Theory Initiative benchmark (another is that the Fermilab theory director is part of the Theory Initiative collaboration, while the BMW collaborators aren't affiliated with Fermilab).

A new pre-print making a more precise calculation of the hadronic light by light contribution to muon g-2 (reducing the relative error in that component of the calculation from 20% to 14%) was also strategically released on arXiv today. It adds 14.8 * 10-11 to the BMW prediction. Combined with the BMW prediction this is 158.8 * 10-11 more than the Theory Initiative Standard Model Prediction. So, including this result as well as the BMW prediction would further reduce the experiment-SM prediction discrepancy to 1.3 sigma (the two innovations do not overlap).

It will be interesting to see if the Lattice QCD/SM prediction theorists will be able to reach a consensus on this issue, which was a simmering behind the scenes controversy for months before this announcement. Multiple conferences on the muon g-2 calculation are planned for the next year (as the theory presentation from Fermilab this morning mentioned) and this disparity will be at the top of the agenda at all of them.

The stakes are pretty high. If the Theory Initiative is right, new physics beyond the Standard Model almost certainly exist and are right around the corner. If the BMW collaboration is correct, the prospect for new beyond the Standard Model High Energy Physics discoveries at experiments that are possible to conduct in our lifetimes is dismal.

An intermediate possibility is that neither calculation secures consensus support, but that the QCD community concludes, based upon the disparity between the legitimately calculated Theory Initiative result and the legitimately calculated BMW collaboration result, that both papers have grossly underestimated the amount of theoretical uncertainty present in their SM prediction calculations. This would leave us in the same limbo we've been in on muon g-2 for the last two decades.

If you are a visual leaner, you can read the comic strip explanation.

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    $\begingroup$ Thanks for such deep insight, I'm glad I asked the question! $\endgroup$
    – Kai
    Apr 8, 2021 at 1:27
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    $\begingroup$ Great answer, it's better than the press releases and articles! Except, this is way too pessimistic: "If the BMW collaboration is correct, the prospect for new High Energy Physics discoveries at experiments that are possible to conduct in our lifetimes is dismal." $\endgroup$
    – knzhou
    Apr 8, 2021 at 3:20
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    $\begingroup$ @knzhou Slightly tweaked the language to be less pessimistic without deviating for the point that muon g-2 is a global measure of the extent to which the Standard Model is a complete description of fundamental physics at low to moderate energies. $\endgroup$
    – ohwilleke
    Apr 8, 2021 at 5:47
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    $\begingroup$ Excellent answer. "...the Theory Initiative result has been replicated (something not possible in the case of the much more computation resource intensive BMW calculation)" - I would say, more difficult, but clearly warranted in light of this heightened interest, or in your words: "the stakes are pretty high" ;) As it stands, one can use the two methods as an estimate of the systematic theory uncertainty, unfortunately nullifying the significance of the g-2 measurement :( $\endgroup$
    – rfl
    Apr 8, 2021 at 7:49
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    $\begingroup$ Greetings! I have typeset your results in a table instead of a list, but in the last row I think I may have preserved a typo. $\endgroup$
    – rob
    Jun 9, 2021 at 23:26

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