# Is the Near Integer Muon to Electron Mass Ratio a Coincidence or Significant?

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• Jonathan Scott
Jonathan Scott
Gold Member
I've known for a long time that the muon mass is approximately 3∕2×137 times that of the electron, as in Nambu's empirical mass formula from 1952. I recently wondered what the difference was as a multiple of the electron mass, using current CODATA figures for the muon to electron mass ratio and the fine structure constant:

206.7682830−3∕2×137.035999084 = 1.214284374

The sequence "1428" reminded me of 1/7 so I tried multiplying it by 7, which gave about 8.49999, so I then tried multiplying by 14. The result happens to be equal to 16.999981236 which is surprisingly close to an integer ratio. And the exact value 17/14 is well within the error range for the muon to electron ratio given the uncertainty in the muon mass.

So I'm wondering whether this is just a weird coincidence (like the 999 in the fine structure constant) or something with physical significance. Given that the uncertainty in the mass ratio is given as (46) for the last two digits and the other values are more accurate, then we should have a somewhat more than 5 decimal places of accuracy in the difference, so after multiplying by 14 we should still have 4 or more places. The ratio value that would make the result exactly 17/14 is just over 206.7682843 which increases the last 2 digits by only 13, which is well within the current uncertainty.

The idea behind this expression is that the mass of the electron is α=1/137.035999084 of a basic mass unit (of about 70MeV/c^2, or 2 times 35MeV/c^2) and the muon is 3/2 times that same basic mass unit plus an electromagnetic correction, which I expected to be of the same order as the electron mass, but just some arbitrary number, not a simple rational multiple of it. So I'm inclined to think that it's another weird coincidence, but it was a surprise.

Similarly, again as in Nambu's scheme, the charged pion mass is approximately 4/2×137 times that of the electron, although less well defined, and of course the pion is not a lepton, so there may be other factors involved. In that case, the difference from the exact value is −0.93997, which is again approximately 1 electron mass out, but this time with the opposite sign, and not very accurate given the limited accuracy of the charged pion mass. I have not spotted any significant relationship to a small integer ratio in this case (but it is fairly close to 16/17, 31/33 or more obviously 47/50).

Last edited:
Motore and ohwilleke
Numerology is not science. And approximate numerology is not even numerology.

If I follow you, with one three-digit constant, three operators and six digits you can reproduce a ten digit number (of which only nine are really known). This is not impressive.

Motore and DaveE
Obviously if I made up the formula from scratch with that sort of freedom I could reproduce something to very high accuracy, like those who come up with ideas for the fine structure constant.

However, the formula was already determined by the basic concepts, and the only choice I personally made here was to multiply the "electromagnetic correction" (in units of the electron mass) by 14, which turned an apparently arbitrary value into a plausibly close approximation to an integer, which would be exact somewhere near the middle of the current range of uncertainty. And even if I go to the limits of the current uncertainty, the value still remains close to that integer.

There are many things I could have done to that number, involving pi, e or arbitrary powers, and unless there was a specific theoretical reason for trying some particular formula, I would be very sceptical if I got some apparently simple result from a complex formula. However, in this case, all I did was spot that multiplying by 14 essentially gives an integer to nearly 5 decimal places.

Anyway, I would agree that it's probably just a coincidence, but it was a surprise. It is difficult to analyze the likelihood of such a result in retrospect, but all I did personally was choose the number 14, and that resulted in a number that was an integer to within 2 parts in 100,000.

Motore
It's quite amazing how closely you can approximate a real number with a rational number. Of course "close" depends on how large you allow your integers to be. Plus, sometimes you get lucky. Humans love to assign meaning to luck.

Jonathan Scott said:
However, the formula was already determined by the basic concepts
No it wasn't. And if it were, it would be a personal theory and not appropriate for PF. I don't think you want to go down that path.
DaveE said:
t's quite amazing how closely you can approximate a real number with a rational number
And this is amazing, but not new. Many societies discovered this, thousands of years before decimals.

The approximate formula 3/2*1/alpha for the ratio of the muon and electron masses is taken from Nambu's 1952 paper "An Empirical Mass Spectrum of Elementary Particles", which is of course speculative but well known. All I've done is subtract that from the experimental value of the same ratio.

ohwilleke and arivero
$$m(\mu)/m(e) \approx 2\pi^4 \left(1 + \frac{10}{163} +\frac{1}{100693} \right)$$

ohwilleke and Motore
Jonathan Scott said:
The approximate formula 3/2*1/alpha for the ratio of the muon and electron masses is taken from Nambu's 1952 paper "An Empirical Mass Spectrum of Elementary Particles", which is of course speculative but well known. All I've done is subtract that from the experimental value of the same ratio.
The best known continuation of this idea is a paper from Wilczek - Zee, where they try to get one mass from the other via some electromagnetic loop, so the alpha is justified. Then for the mass of electron alone you have an argument in the first book of Polchinski, where he remarks that the whole mass is of the same order of magnitude that one could expect if it was just coming down from the Planck scale. The equation from Polchinski was surely some numerological lore, well known but not fit for publication in journals; it is also exploited by the guys doing wild speculation on fractal structure.

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