The same post notes that the apparent top quark, antitop quark mass assymmetry reported at Tevatron by the CDF group there was swiftly contradicted by a D0 group result at same facility.

**The Strange Formula of Dr. Koide**

This is also as good a place as any to link to a wonderful little paper on Koide's formula from 2005 with the title found in the heading of this part of the post. Koide's formula (set forth before the tau mass was determined with nearly the same certainty) states that the sum of the masses of the three charged lepton masses divided by the square of the sum of the square roots of the charged lepton masses is exactly two-thirds. This phenomenological and not theoretically grounded rule has held with extreme precision.

Two significant generalizations of the rule have been made, a generalization for neutrino masses which is correct within measurement error (it changes the sign of one of the terms in the charged lepton equation), and a generalization for all quarks other than the top quark (called Barut's formula after A.O. Barut who published versions of it in 1965, 1978 and 1979), which is accurate to within 5% or so of measured values (some of which aren't known very precisely) but is greatly off the mark for the top quark. The generalized formula for quarks is that the mass of quark N (for N=0, 1, 2, 3, 4 for the u, d, s, c and b quarks respectively) is equal to the mass of the electron, times (1+3/(2*alpha)*sum((for n=0 though n=N)) of n^4), where alpha (roughly equal to 1/137) is the electromagnetic coupling constant. The precision is 2% for the first three charged leptons and charge quarks (related by a constant), and 2.7% for the down type quarks, but only 28% for the up type quarks (the predicted value is about 101,000 MeV which also happens to be roughly the Higgs boson mass most perferred by electroweak measures although no resonnances show up there in particle accellerator experiments rather than the experimental value of 174,000 MeV), with the top quark mass being most grossly inaccurately determined. The predicted value of the generalized value for quarks is 3/2 and in fact it is very nearly pi/2, so perhaps it isn't an error at all.

Also, interestingly, the square root of the difference between the sum of all six quark masses and the sum of the square root of the first first five quark masses is very close to the square root of the average of the W+, W- and Z boson masses (or to within the accuracy of the model, the average of the W and Z boson masses), suggesting that perhaps that a linear combination of these bosons, rather than the top quark, belongs at the sixth position in the hierarchy of mass particles.

The article also discusses the apparent similarities between the ratios of some of the fermion masses to each other and the coupling constants of the Standard Model forces.

The square roots of the mass matrix for quarks and the square roots of the mass matrix for leptons can also be used to recover the Cabibbo angle (13.04 degrees) of the canonical parameterization of the CKM matrix, suggesting a deep connection between the CKM matrix and the square roots of the masses of the fermions.

Amateur physicist and physics blogger Carl Brannen gets some credit for his explorations on this front (see also this 2005 paper), in this article by Rivero and Gsponer. Leonardo Chiatti has published the most recent scholarly article along the same lines as Barut's formula in 2009 (another 2008 paper is here) Gsponer and Hurni in a 2002 paper, suggested from these formulas that the top quark may be a different animal than is less massive quark companions (indeed, since it has not been observed to form mesons or baryons, possibly due to its rapid decay rate associated with its great mass, one can arguably not say with confidence that it really interacts with the strong force).

Other interesting literature on the subject includes a 2006 paper by Kyriakos (inspired by the methods of de Broglie).

While it is all to some extent numerology, these are also empirically established relationships that any theory that would explain them from first principles would have to somehow replicate.

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