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Jonathan Scott

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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).

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).

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