There are three charged leptons: the electron, the muon and the tau. Let and be their masses. Then the Koide formula says
There’s no known reason for this formula to be true! But if you plug in the experimentally measured values of the electron, muon and tau masses, it’s accurate within the current experimental error bars:
Is this significant or just a coincidence? Will it fall apart when we measure the masses more accurately? Nobody knows.
Here’s something fun, though:
Puzzle. Show that no matter what the electron, muon and tau masses might be—that is, any positive numbers whatsoever—we must have
For some reason this ratio turns out to be almost exactly halfway between the lower bound and upper bound!
Koide came up with his formula in 1982 before the tau’s mass was measured very accurately. At the time, using the observed electron and muon masses, his formula predicted the tau’s mass was
while the observed mass was
Not very good.
In 1992 the tau’s mass was measured much more accurately and found to be
Koide has some more recent thoughts about his formula:
• Yoshio Koide, What physics does the charged lepton mass relation tell us?, 2018.
He points out how difficult it is to explain a formula like this, given how masses depend on an energy scale in quantum field theory.