Latest Tau Lepton Measurement Consistent With Koide's Rule

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ohwilleke
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Summary:

Koide predicted the tau lepton mass as a function of the electron and muon masses in 1981 when all three masses were known less accurately. In 2020, the rule still holds true within 1 sigma as measurement accuracy increases.
The latest tau lepton mass measurement, from Belle II is 1777.28 ± 0.75 (stat.) ± 0.33 (sys.) MeV/c^2. The combined error is ± 0.82 MeV/c^2 (which is 0.38 sigma greater the the Koide's rule prediction). This is consistent at a one sigma level with the current Particle Data Group world average measurement of the tau lepton mass.

The current Particle Data Group value for the tau lepton mass is 1776.86 ± 0.12 MeV/c^2 (which is 0.91 sigma below the Koide's rule prediction). This is a relative error of one part per 14,807. In 2014, immediately before the most recent update of the PDG value, it was 1776.82 +/- 0.16 MeV/c^2.

If error is Gaussian (i.e. has a normal distribution) and systemic error estimates are "conservative" then the difference between the true value and the measured value should average a little less than 1 sigma. In real life statistical error is almost always close to Gaussian, but systemic error usually has a distribution best fit by a t-test distribution that has somewhat fatter tails than a Gaussian distribution.

Since the new Belle II measurement is higher than the PDG value, it will nudge the global PDG value towards the Koide's rule value, although not by much, since the significant margin of error means it is weighted only lightly in the world average.

Koide's rule was proposed in 1981 by Yoshio Koide, six years after the tau lepton was discovered, when its mass was known much less accurately. It is a hypothesis about the relationship between the masses of the charged leptons. It predicts that the sum of the three charged lepton masses, divided by the square of the sum of the square roots of the charged lepton masses, is equal to exactly 2/3rds. Since the electron and muon masses are known much more precisely than the tau lepton mass, it is possible to use the original Koide's rule to very precisely predict the tau lepton mass. This prediction using current electron and muon mass measurements is:

1776.96894 ± 0.00007 MeV/c^2.

In 1983, using the then best available measurements of the electron mass and muon mass, the original Koide's rule predicted a tau lepton mass of 1786.66 MeV/c^2. But by 1994 (and probably somewhat sooner than that), the prediction of the original Koide's rule had shifted to 1776.97 MeV/c^2. Thus, the prediction of the original Koide's rule has been essentially unchanged for more than twenty-six years.
 
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The Belle II measurement has such a large uncertainty that I don't expect PDG to ever include it. It will likely be superseded by a measurement with a larger dataset soon. The systematic uncertainty will likely stay, however - they use a different method, which provides an independent cross check, but it can't reach the precision of measurements at the threshold.

A naive weighted average gives 1776.87, i.e. the last digit goes up by 1.
 
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ohwilleke
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A naive weighted average gives 1776.87, i.e. the last digit goes up by 1.
It is worth observing, in the same vein, that the combined eight measurements made prior to 2014 pull the global average down by 5 in the last digit from the value measured in 2014 at BESIII.

The Belle II measurement has such a large uncertainty that I don't expect PDG to ever include it. It will likely be superseded by a measurement with a larger dataset soon. The systematic uncertainty will likely stay, however - they use a different method, which provides an independent cross check, but it can't reach the precision of measurements at the threshold.
Certainly, if Belle II were to produce a new value this year with a larger dataset and hence less statistical uncertainty, I agree that it wouldn't make sense to include this result. And, the systemic uncertainty at Belle II is high enough that it will never be a world's most accurate measurement in isolation. But it is also the first and only published tau lepton mass measurement in the last six years.

The Belle II measurements do have a fairly large uncertainty, but four of the nine several measurements in the current PDG average for the tau lepton mass (shown in bold) have great uncertainty, there are no data points from Belle II so including it would make the average more robust in a difficult to quantify way since it comes from more independent replicating experiments. These old results with high uncertainties, however, like this result, make only a modest tweak to the overall value (and they are all at least 20 years old too).

It is also worth observing that the 2014 measurement, which is the most precise and makes the largest contribution to the overall world average as a result is also the result with the value closest to the Koide's Rule prediction, after the 1996 BES result which has larger error bars (but still very small for 1996). BESIII took more data in April 2018 to update its 2014 measurement according to a December 2018 preprint, but for some reason, hasn't published a new result from that two year old data. The new BESIII result should have a precision of less than ± 0.1 MeV.

The data for the July 2014 paper from BESIII was collected in December 2011, so it took 31 months to publish it. Thirty-one months from April 2018 when the data for its second round to tau lepton mass measurements were collected would be November 2020, so we could get a new record high precision tau lepton mass measurement from BESIII as soon as sometime this fall. A new result from BESIII would replace the 2014 measurement and would dominant the world average even more strongly than its best in the world 2014 measurement does already.

The results that contribute to the currently tau lepton mass world average of the Particle Data Group are:

1776.91 ±0.12+0.10−0.131171
1
ABLIKIM
2014​
D
BES323.3 pb−1, Ecmee= 3.54−3.60 GeV
1776.68 ±0.12 ±0.41682k
2
AUBERT
2009​
AK
BABR423 fb−1, Ecmee=10.6 GeV
1776.81 +0.25−0.23 ±0.1581
ANASHIN
2007​
KEDR6.7 pb−1, Ecmee= 3.54−3.78 GeV
1776.61 ±0.13 ±0.35
2
BELOUS
2007​
BELL414 fb−1Ecmee=10.6 GeV
1775.1 ±1.6 ±1.013.3k
3
ABBIENDI
2000​
A
OPAL1990−1995 LEP runs
1778.2 ±0.8 ±1.2
ANASTASSOV
1997​
CLEOEcmee= 10.6 GeV
1776.96 +0.18−0.21−+0.25−0.1765
4
BAI
1996​
BESEcmee= 3.54−3.57 GeV
1776.3 ±2.4 ±1.411k
5
ALBRECHT
1992​
M
ARGEcmee= 9.4−10.6 GeV
1783 +3−4692
6
BACINO
1978​
B
DLCOEcmee= 3.1−7.4 GeV

One resource that I wish was more easily available would be a compilation of the PDG values for the various data points is reports over time. So, for example, you could watch the evolution of the world average measurements of the tau lepton mass, or the strong force coupling constant, over the past few decades.
 
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I have heard rumors that the BES III update isn't that far into the future.

Belle II now has about 10 times as much data recorded as used for the tau mass measurement, but as it's not competitive I don't expect an update with that alone. Maybe with the fall dataset, at that time the systematic uncertainty will be larger and future work would need to focus on them instead of statistics.
 
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