A few months ago I wrote an article on the Standard Model of particle physics, and made passing reference to the little know Koide relationship of lepton masses. A few of you wrote to ask me about this formula - which proves that some of you did not read the detailed review of it that I wrote a few years ago :-)    (In fairness though, that article was on another website which may have since been de-activated by its managers.)

Let me begin by reviewing the leptons. One of the first subatomic particles to be observed in the laboratory was the electron - a (possibly) point-like particle with tiny mass that carries an electrical charge. It is responsible for electricity, chemistry, and many other everyday phenomena. But it is not alone.

A few years later scientists discovered the muon - a particle which is identical to the electron in every way, except for a mass 200 times greater. A few decades later, physicists found a third member of this group, the tauon or tau-lepton which is nearly 3500 times more massive than the electron. These three particles all have the same electrical charge, the same interactions with weak nuclear forces, and a lack of interaction with the strong nuclear interactions, and so they were given the family name of "charged leptons".

Because of the existence of three nearly identical particles, physicists have often wondered if they are somehow different states of the same particle. Could it be that the muon and tauon are excited states of the electron, rather than different particles. And many experiments have been conducted to search for some form of structure that would create these excited states, but no evidence has ever been found. Except for the Koide formula.

In 1981 Yoshio Koide published an article in which a strange relationship between the lepton masses was presented. Defining a new quantity Q by the formula:


and inserting random numbers for the masses gives 1/3 < Q < 1. Inserting the actual measured masses into this formula gives Q = 2/3, which is exactly halfway between the two limits. And it isn't a minor coincidence either- using the mass values of me = 0.510998910(13)  MeV/c2, mμ = 105.658367(4) MeV/c2, and mτ = 1776.84(17) MeV/c2  gives a value of Q = 0.666659(10), which is only 0.000001 less than being exactly 2/3!

So why should three apparently random masses have such a precise relationship? No one knows.

And it isn't just the leptons either. In 2011/2012 another formula was discovered that has similar properties. If the masses of the three charged leptons are replaced by the masses of the three heaviest quarks, the number is again 2/3! (although in this case the quark masses are not as well measured, and so the agreement is not as strong)

So why should the masses be so related? That is a question that is still being explored. Perhaps there is more to the leptons than we yet know...