The Curious Formula From Dr Koide

In 1981, Yoshio Koide made an observation that the masses of 3 sub-atomic particles, which appear at first sight to be unrelated, can be tied together with a simple formula.

Electron0.511 keV/c2me
Muon105.7 MeV/c2mμ
Tau1776 MeV/c2mτ


The electron is the most well-known particle in the whole of physics, discovered by J. J. Thomson in 1897.

It forms the basis of electricity and has been used for practical purposes ever since Thomas Edison started testing the first carbon filament light bulb in 1879.

The electron was discovered before the other 2 well-known sub-atomic particles, the proton and the neutron. However, the Koide formula does not relate to the proton and neutron. Instead it relates to 2 other particles that behave a bit like the electron.


The muon was discovered by Carl D. Anderson and Seth Neddermeyer in a study of cosmic radiation in 1936. It appears to behave just like an electron in all situations, apart from being about 200x heavier.

The existence of particles called mesons with masses in that range had been predicted by Hideki Yukawa. However, the properties of the muon were clearly not in agreement with those expected of a meson, so its discovery came as a complete surprise. The Nobel laureate I.I. Rabi famously remarked "Who ordered that?"

Unlike the electron, the muon is unstable, lasting on average just 2 microseconds (which believe-it-or-not is quite a long time in particle terms). It lasts just long enough that a significant fraction of them make it to ground level, having been generated by cosmic ray collisons in the upper atmosphere.


You might think that a third and even heavier version of the electron might be called the "tauon" for consistency, but instead it is referred to simply as the tau particle. Whatever!

The tau particle was discovered between 1974 and 1977 by Martin Lewis Perl at the Stanford Linear Accelerator Center (SLAC).

Unlike the electron, which is everywhere and the muon which arrives constantly via cosmic rays, the tau is not conspicuous in everyday life. To date, particle accelerators are the only places it has ever been observed.

So what exactly did Dr Koide discover? We don't know what prompted him to do so, but he found that there was a way of relating the masses of the electron, muon and tau:

Q = (me + mm + mt) / (sqrt(me) + sqrt(mm) + sqrt(mt))^2  = 2/3

On first inspection, you might think "so what"? Indeed, that is pretty much the considered view of most particle physicists. Even if the numbers are related to a high degree of accuracy in a way that seems unlikely to be just coincidence, nobody has the faintest idea why that should be.

Although some theories have been started on the basis of the Koide formula, they were seemingly not productive and so were soon abandoned.

Without dwelling further on the physical significance of the Koide formula, we will interpret it as a metaphorical signpost from the universe that the electron, muon and tau belong together at a fundamental level.

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