That fundamental constituent of matter, the electron, has two elder siblings, the muon and the tau.

The electron mass is 0.510998918(44) MeV.

The muon mass is 105.6583692(94) MeV.

The tau mass is 1776.99(+29-26) MeV.

MeV is a Million electron Volts, a particle physicist's unit. For comparison's sake, the hydrogen atom has a mass of about 938.8 MeV. The numbers in brackets are the uncertainties in measurement of the mass. The numbers are taken from links on Tommaso Dorigo's article on this.

What is remarkable about these numbers is that they satisfy the Koide formula with a remarkable precision - a formula with little theoretical justification.

Given the masses of the electron and the muon, this formula predicts the mass of the tau to be

1776.968921( 158) MeV. (from Carl Brannen, linked from Dorigo's page).

How likely is this to be a numerical coincidence?

If we measure everything in terms of the tau mass, then, in the plot below, the red triangle represents the experimental situation and the green curve represents the masses of the electron and muon that would be allowed per the Koide formula. The tau, by definition, has a mass of 1. There are no theoretical constraints on the masses that I'm aware of, and the "theoretically allowed" region of masses is everything between the x-axis and the diagonal black line (and please extend the line to the point {1,1}).

Of course, I could make things look a little less spectacular by measuring everything in terms of the electron mass.

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## 3 comments:

Hi Arun,

indeed, the graphical interpretation gives a different taste to the whole issue. One can imagine a family of curves such as the one drawn, which would span the lower right triangle of the plot, and not cross the data point. The significance of Koide's formula is then that the curve matching the point is obtained with a very simple numerical factor in the RHS.

You rightly turned things around into a prediction of the tau mass. Now, since the tau mass was off the original Koide prediction when he first stumbled into that formula, we now have a reference point to compute a real, genuine significance.

Let me explain: Originally, the tau mass was measured 2-sigma away from the number you write there. Using as a stick the original value of 1-sigma, we are now in a position to measure how likely it is that we are closer to zero sigma in the agreement between tau mass and koide's prediction. To make an example, if at the time of Koide's prediction the tau mass was known with 10 MeV uncertainty, and if now the agreement is to within 10 keV, we can say it is a 1 in a thousand chance, or a little more than 3-sigma evidence that the formula holds water.

Cheers,

T.

Just realized something about the family of curves.

f(x,y) = (1 + x + y)² /(1 + x² + y²)

has a minimum value of 1 (x=0,y=0) and a maximum value of 3 (x=1,y=1) for 0 ≤ x,y ≤ 1.

That still does leave a lot of "simple" numerical factors for the LHS.

In terms of Carl Brannan's parametrization (7) in MASSES2.pdf of the eigenvalues of a circulant matrix, 0 ≤ η² ≤ 1. For greater η² eigenvalues turn negative.

Indeed the equation can be related via a factor 3 to the one measuring the cosine of the angle between (1,1,1) and (1,x,y). Koide's amount to predict 45 degrees.

As for your question on renormalisation groups: mass quotients are not fixed points, but it is usual that a relationship involving mass quotients varies very slowly under renormalisation group. Some publications on Koide's, last year, address this issue.

Dorigo brings a good point recalling that Koide's was actually a prediction when first published. I like to use it to show that the usual claims in blogs and forums about the importance of predictions are overstressed, or that the process of science is oversimplified. Koide had a theory, compositeness, and a prediction, the mass of the tau, and he got it rightly, and nobody really took care of it because nobody was able to fit it in the knowledge corpus.

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