Well, I code-named Descartes’ theorem “Kiss precise”, so that at least is related.

To generalize Descartes’ theorem to circles intersecting at an angle, I used a nice proof by Jerzy Kocik: https://arxiv.org/abs/1910.09174

He uses a map from disks with radius (r) and midpoint (x,y) to a (3+1)D Minkowski space. I vaguely see this as a clue. Also, I vaguely see lepton circles tiling event horizons in momentum space…

When I once learned about the Koide formula it reminded me on this https://en.wikipedia.org/wiki/Descartes%27_theorem , but I was unable to formulate anything substantial about it. I liked the relation to geometric curvature though. Maybe this is related to your geometric explanation?

]]>The Koide formula (rewritten a bit):

(0 + m_e + m_mu + m_tau)^2 = 3/2 * (0 + m_e^2 + m_mu^2 + m_tau^2)

For 4 disks to exactly touch, the radii must satisfy:

(1/r0 + 1/r1 + 1/r3 + 1/r3)^2 = 2 * (1/r0^2 + 1/r1^2 + 1/r3^2 + 1/r3^2)

It turns out the factor 2 changes to a factor 3/2 if we demand the disks, instead of touching, intersect at an angle whose cosine is 2/3.

Also, we can set 1/r0 to zeo by making the circle a line (a circle of infinite radius)

So we get:

(0 + 1/r1 + 1/r3 + 1/r3)^2 = 3/2 * (0 + 1/r1^2 + 1/r3^2 + 1/r3^2)

The Koide formula can be visualised as a configuration of disks:

I made a little web sit one this:

https://westy31.home.xs4all.nl/KissPrecise/The%20Koide%20Formula%20and%20the%20Kiss%20Precise.html

I don’t know if this brings us any closer to knowing if this formula is more than pure coincidence.

The value of 3/2 for the 1/cosine is special: 1/cos(alfa) is exactly equal to the factor that apears in the generalised Kiss precise formula.

]]>Yep. There are species of Cauchy-Schwarz and Arithmetic-Geometric mean problems which can be proved more laboriously with Lagrange multipliers.

]]>Neglect this post, it messes up the values 1/3 and 1. The other post corrects this.

]]>The value 2/3 means that the vector h = {√me, √mμ, √mτ} has an angle of exactly 45° with the vector {1,1,1}. Any vector h on the 45° cone gives a value of 2/3.

For an arbitrary value of Q other than 2/3 the general formula for the this angle θ is given:

So for Q=2/3 this formula gives:

For Q=1/3 we get 0°, the minimum angle.

For Q=1 we get an angle of 54.73° which is the angle between {1,1,1} and any of {0,0,1} , {0,1,0} and {1,0,0}. If we go further then at least one of the three components becomes negative.

So the angle of θ=45° with respect to {1,1,1} is the geometric translation of Koïde’s coincidence.

(and yes, we all do know how mass renormalization and all that is done in the textbooks I may suppose)

]]>Nice! Without loss of generality we can normalize the masses so their sum is one; by symmetry if the function assumes a minimum at a unique point on this triangle it must be the center, where the masses are equal, and there the function is 1/3. However, the trouble (as usual) with this argument is showing that there’s a unique minimum; it seems just as easy to use Lagrange multiplier to find all the critical points, discover there’s just one, and then investigate the function on the boundary of the triangle, where one or more of the masses is zero.

]]>.

With we get:

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Squaring we get the LHS of the inequality:

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The RHS is obtained by noting that:

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Hence

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]]>Yes, in the current ‘top-down’ thinking about particle physics, where the fundamental physics appears at high energies and gets tweaked by the renormalization group as we descend to lower energies, it’s very hard to understand why an exact relation like the Koide formula would hold at low energies. I added a tiny bit about this to my blog article: mainly, a link to a paper by Koide where he discusses some theories in which symmetries would prevent the lepton masses from depending on the energy scale. So he recognizes that this issue is important.

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