The Koide relation

In 1982 Yoshio Koide noticed that the masses of the three charged leptons, the electron, muon and tau, satisfy a simple formula: their sum, divided by the square of the sum of their square roots, equals 2/3. With today's measurements the value is 0.666661, within 9 parts per million of 2/3, although the formula allows anything between 1/3 and 1. When Koide proposed it, the tau's mass was poorly known. From the electron and muon alone, the relation implies a tau mass of 1776.97 MeV, and the precise measurement of 1992 found 1776.9 MeV. Standard physics has no explanation.

The relation has a geometric meaning. Treat the square roots of the three masses as a direction in three-dimensional space. Koide's relation says this direction sits exactly 45 degrees from the diagonal that runs equally along all three axes. The part of the masses shared by all three generations and the part particular to each carry exactly equal weight.

RSI reads the three generations as one 720-degree loop wound about the three independent axes of space. Seen along the diagonal that joins those axes, the three windings sit 120 degrees apart, the same view in which the proton's three loops sit 120 degrees apart and charge comes in thirds. Koide's 45 degrees is then the balance between the winding shared by all three axes and the winding particular to each.

The relation can be written as a common value times 1 + √2 cos(δ + 2πn/3), with the three generations 120 degrees apart. In 2006 Carl Brannen observed that the phase δ equals 2/9 to within one part in a million. With √2 and 2/9, the electron's mass alone gives the muon's to within 10 parts per million and the tau's to within 0.007 percent. The whole charged lepton family reduces to one mass and two geometric numbers, and RSI's task is to derive those two numbers from the medium.