The Cabibbo-Kobayashi-Maskawa (CKM) matrix is a three by three unitary matrix, each element of which contains the square root of the probability that the up-type quark for that row will be converted into a down-type quark of that column in the event that it emits a W+ boson.
Seven of the nine elements can be explained by an utterly transparent theory.
The ratio of the probability that up-type quarks will transition into strange quarks to the probability that up-type quarks will transition into down quarks is always within the margin of experimental error of the ratio of the mass of the strange quark to the mass of the down quark, for up quarks, for charm quarks and for top quarks.
Likewise, the ratio of the probability than an up quark will transition into a bottom quark to the probability than an up quark will transition into a strange quark is within the margin of experimental error of the ratio of the mass of the bottom quark to the mass of the strange quark.
Alas, this very simple relationship between the quark masses and the CKM transition probabilities breaks down in two instances. The ratio of the probability that a top quark will transition into a bottom quark is about fourteen times greater than the ratio of the probability that a top quark will transition into a strange quark (the ratio is about 611-1 and should be about 44-1).
Likewise the ratio of the probability that a charm quark will transition into a bottom quark is about 588 times the probability that it will transition into a strange quark, about 13 times as great as the 44-1 ratio that would be expected in a simple formula.
The top to bottom transition probability, and the charm to bottom transition probability, are both about what one would expect if the bottom quark had a mass of 51-63 GeV (notably, more than 1/2 of the Z boson mass but less than 1/2 of the Higgs boson mass), rather than the canonical 4.2 GeV.
These discrepancies are too great to address with various combinations of the running of quark masses with energy scales.
One almost imagines that there are really two bottom quarks, one with a mass of 4.2 GeV that couples to up quarks, but not to charm and top quarks, and another with a mass of 55 GeV that couples to charm and top quarks, but not to up quarks, that are otherwise identical, although this is ridiculous and would have been discovered long ago if it were true, one would suppose.
While this transparent theory about CKM matrix elements clearly doesn't work unmodified, it is worth noting how well it fits seven of the nine elements in manner that suggests that the CKM matrix elements flow very directly from the quark masses alone, before looking for an elusive fix to this flawed theory.
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