Graphing Koide's Rule Solutions

If you take three masses M1, M2 and M3, from smallest to greatest, and then redefine them in units of M1 (which you can do without loss of generality) such that you have 1, M2/M1=x^2, M3/M1=y^2, then Koide's rule can be stated as follows:

(1)        x^2+y^2+1/(x+y+1)^2=2/3

This is equivalent to:

(2)      3x^2+3y^2+3=2x^2+2y^2+4x+4y+4xy+2

which is equivalent to:

(3)    x^2+y^2-4x-4y-4xy+1=0

This is an equation for a hyperbola (the determinant of the coefficients of the equation in the general quadratic curve equation called delta is -27, determinant J of some of those coefficients is negative, and the quantity I=2).

In standard form, this would be written:

(4)   (x+2)^2-4(x+2)(y+2)+(y+2)^2=-9

Graphed (at this link), it looks like:

But, since x>0 and y>0, only the portions of the solution in the top right quadrant of the graph are solutions to the Koide relationship, and since we can arbitrarily define x>y or x=y, only half of the solutions in that quadrant apply.

Examination of the graph reveals that there are two basic classes of Koide relationship solutions.  One group of solutions is near the origin where M1>>M2 and M1>>M3 and M2 and M3 are somewhat similar in size.  In the other group of solutions, M3>>M2 and M3>>M1.  This is far out on the x-axis.

An example of a solution of the first type is the charm-bottom-top quark triple, which is a very close approximation of the Koide triple.

Another solution of this type can be constructed using the particle data group values of the up and down quark masses (2.3 MeV and 4.8 MeV respectively).  The positive mass that would complete this triple using the unmodified Koide's formula is 232.14 MeV (holding the down quark mass fixed and considering the range of up quark masses consistent with the 0.38-0.58 up/down mass ratio of PDG implies a range of 210.96 MeV-252.20 MeV; looking at those ratios while keeping the sum of the light quark masses in the PDG range tweaks the outcome 5-8% or so).  This also isn't particularly close to a hypothetical "pole mass" for the strange quark, which is heavier than this in a naive extrapolation, but is somewhat ill defined in any case as perturbative QCD really isn't applicable to this extreme infrared energy scale.

Of course, this isn't a good fit for reality at all.  The low end is twice the muon mass and more than twice the best estimate of the strange quark mass.  The high end is about a fifth of the charm quark mass and a seventh of the tau lepton mass.  But, there is a solution to the formula in this mass range for these light quark masses.

The original electron-muon-tau triple of 0.511 MeV, 105 MeV and 1776 MeV more or less, is an example of the other class of solutions.

It is also worth noting the solution for the special case in which one of the masses is zero.  In that case, the ratio of greater of the other masses to the lesser of the other masses must be approximately 13.92820.  This corresponds to the x-axis intercept on the graph above if it extended that far.

In general, when all three masses are non-zero, the greatest ratio of two masses in the triple must be greater than 13.92820.