## Tuesday, November 30, 2021

### A New Experimental Challenge To Lepton Universality Overstates Significance

The BESIII experiment reports the first case of strong experimental evidence for lepton flavor universality violation (i.e. electrons, muons and tau leptons having properties other than mass that differ from each other), outside semi-leptonic decays of B mesons.

But, its claim of a strong departure from lepton flavor universality is, in fact, a rookie class misinterpretation of the reported data which the introductory text seems to acknowledge, which is, in fact, not inconsistent with lepton flavor universality at the two sigma level.

Background

Instead of studying semi-leptonic B meson decays, this study looks at the decays of excited J/Psi mesons (J/ψ), a spin-1 (i.e. vector) charmonium meson (i.e. it has a charm quark and an anticharm quark as valence quarks), with a ground state mass of 3097 MeV and a mean lifetime in the ground state of 7.1(2)*10-21 seconds.

In contrast, the B mesons showing apparent lepton universality violations have a b quark and a non-b antiquark (or a non-b quark and a b antiquark) as valence quarks, ground state rest masses of 5297-5415 MeV. Pseudoscalar B mesons have a mean lifetime on the order of 1.5(1)*10-12 seconds.

The Results

The data sample consists of (448.1 ± 2.9) × 106 ψ(3686) events collected with the BESIII detector.

As in the semi-leptonic B meson decay case, in the fully leptonic decays of J/ψ mesons, the decays with muons are significantly less likely than decays with electrons, although, unlike the clean B meson decay case, much of this discrepancy is present in the Standard Model prediction.

The introduction to the paper provides important context:

On the one hand, lepton flavor universality (LFU) is expected to be obeyed in SM. In recent years, however, indications for violation of LFU have been reported in semileptonic decays of the kind b → s ℓ+ℓ −. In 2014, LHCb measured the ratio of branching fractions RK = B(B+ → K+µ +µ −)/B(B+ → K+e +e −), and found a deviation from the SM prediction by 2.6σ. The measurements have continuously been updated by LHCb and Belle. Very recently, LHCb reported their latest result with full Run I and Run II data, which deviates from SM prediction by more than 3σ. . . . It is therefore urgent to investigate the validity of LFU in other experiments.
J/ψ → ℓ +ℓ −, where ℓ may be either e or µ, are two such precisely measured channels, and their measured branching fractions are consistent with Quantum Electrodynamics (QED) calculations.
Other purely leptonic decays, which have never been studied experimentally, are J/ψ → ℓ + 1 ℓ − 1 ℓ + 2 ℓ − 2 ℓ, where ℓ1 = ℓ2 = e, ℓ1 = ℓ2 = µ or ℓ1 = e and ℓ2 = µ. For the first two cases, there is no special order for the four leptons.
Recently, the branching fractions of J/ψ → ℓ + 1 ℓ − 1 ℓ + 2 ℓ − 2 ℓ decays were calculated at the lowest order in nonrelativistic Quantum Chromodynamics (NRQCD) factorization in the SM. Given the collinear enhancement when the lepton mass tends to zero, the predicted branching fraction of J/ψ → e+e−e+e- is (5.288 ± 0.028) × 10−5, significantly greater than that of J/ψ → e+e−µ+µ- ((3.763 ± 0.020) × 10^−5) and two orders of magnitude greater than that of J/ψ → µ+µ−µ+µ- ((0.0974 ± 0.0005) × 10^−5).
Therefore, the ratio Beeee: Beeµµ: Bµµµµ provides a good opportunity to verify the validity of LFU.

(I omit a discussion of the muon g-2 anomaly as a motivation for new lepton coupling particles, which I personally think is due to a flawed theoretical prediction which is contradicted by a methodologically more sound theoretical prediction that matches the experimental result).

The paper and its abstract are as follows:

BESIII Collaboration, "Observation of J/ψ decays to e+e−e+e− and e+e−μ+μ" arXiv:2111.13881 (November 27, 2021).

Analysis

The branching fraction of the four e decay, combining uncertainties in quadrature, is 432 ± 32 (times 10^-7), and for the mixed e and mu decay is 245 ± 32 (times 10^-7).

The data sample consists of (448.1 ± 2.9) × 106 ψ(3686) events collected with the BESIII detector. So, the raw data is about 19,378 ± 1434 four e decays and 10,978 ± 1434 mixed e and mu decays.

The theoretically predicted ratio of the decay branching fractions is 1.405, with the third possibility, which is not seen, predicted to be too small to detect. The ratio of the best fit measurements of the branching fractions experimentally is 1.76.

What is the two sigma range for the ratio of these two values?

The formula for the correct calculation is here with a good shortcut approximation here. Basically, if the denominator of z/w is always positive, you can used the approximation V(R) = V(y)/x^2 + V(x)y^2/x^4 where V equals standard deviation squared and the letter stands for the mean value, if you also assume the independence of the uncertainties (which probably isn't true but is close enough). Applying that very good approximation, the standard deviation of the experimental ratio is about 0.26476 with a two sigma range of about plus or minus 0.53, which is a range from 1.23 to 2.29. So, the experimental result is easily consistent with the lowest order QCD predicted ratio assuming lepton flavor universality of 1.405 at the two sigma level, even without considering the full theoretical uncertainty involved in using the lowest order QCD prediction for that ratio, which is less problematic when looking at ratios of branching fractions than it is in predicting absolute values, but still introduces meaningful uncertainty in the prediction that isn't cleanly quantified in the paper.

Sure, the lowest order QCD prediction was low in both cases, this is unsurprising because the theoretical prediction doesn't include the uncertainty introduced from not including higher order QCD terms.

Lowest order QCD predictions are frequently far from the mark, especially in absolute, rather than relative terms. Higher order QCD predictions frequently differ significantly from lowest order QCD predictions and the tensions between the QCD prediction and the experimental result in this case, no doubt mostly represents shared theory error in the lowest order QCD prediction, rather than anything more profound.

As the introduction to the paper itself observes, it is the ratio of of the two branching fractions that is relevant to testing the lepton flavor universality hypothesis.

Bottom Line

Rather than being a five sigma discovery class evidence in a new decay channel for LFU violation, this result is actually consistent with LFU at the two sigma level, and actually, as a result, if anything, it tends to disfavor the conclusion that LFU violation is present in any context outside of semi-leptonic B meson decays. But given the great uncertainty in the ratio of the two values, this new experiment, honestly, doesn't really tell us anything one way or the other.

To the extent it is a null result, however, contrary to its abstract, this, in turn, casts doubt on whether those LFU violating, new physics supporting tensions are really correct, or simply reflect an error in making a theoretical prediction or implementing a screening of events to consider somehow.