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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.

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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