Category morphisms for the CKM parameters (relative weak interaction rates or “mixing angles” between different quarks)

“What determines the masses and mixing angles of the quarks and leptons in the theory? These particles have a pretty random looking pattern of masses, giving nine numbers that the theory doesn’t predict and which have to be put in by hand. The mixing angles are four more parameters that determine precisely how the electroweak forces act on the particles. In the Standard Model, these thirteen parameters appear as the interaction strengths of the Higgs field with the quarks and leptons and are completely arbitrary.”

– Dr Peter Woit, Not Even Wrong, 2006, chapter 8: “Problems of the Standard Model”.

Above: Dr Sheppeard has produced a new blog post with a table of simple formulae for the squares of the relative (normalized) CKM parameter factors, which have a pattern (she gives parameter a a negative sign, presumably a legacy from the matrix analysis she used in deriving the factors, which is irrelevant for our purposes since a is always squared, cancelling out the minus sign). The CKM parameters and the mass morphism patterns were explained in the previous post on this blog. As discussed in the previous post, the weak interaction strength for leptons has a relative value of 1, but between quarks in the same generation it is slightly less than 1, and between quarks of different generations it is greatly less than 1. The reason is that leptons like the muon have only one beta decay route (into electrons), but quarks have various different possible beta decay routes, and only the sum of all the probabilities for the different possibilities is then (in the case of quark decay) equal to 1. Thus, any particular beta decay route in the case of quarks will have a probability less than 1, unlike the case for leptons.

According to orthodox dogma, quarks don’t decay into leptons, although electrons are given off in the beta decay of downquarks into upquarks. There is a paradox here in the way the decay of a lepton and a quark is analyzed in mainstream thinking, because only in the lepton decay case is main decay product considered to occur via the weak boson: in the case of quark decays, mainstream dogma is upheld by arbitarily switching to seeing the weak boson decay products as mere secondary radiations, with the main decay product being considered – inconsistently with lepton beta decay analysis – to be another quark (diagram below), which has very important implications for understanding the Standard Model and quantum gravity (see the previous post):

This is interesting because, as explained in the previous blog post, weak interaction rates depend on masses.


Above CKM parameters from Wikipedia: these are the relative weak interaction strengths between the different quarks (which can alternatively be expressed as mixing “angles”). Looking at the top left to bottom right diagonal in this matrix, we see that weak interaction rates between quarks in the same generation (e.g., d and u are both in the 1st generation, s and c in the second, and b and t in the third) are nearly 1, whereas weak interaction rates between quarks of different generations are appreciably smaller. The relative weak interaction rates between leptons are all exactly 1 on the same scale. Hence, weak interactions between quarks of the same generation have nearly the same strength as weak interactions between leptons.


Above: the pattern in CKM parameters between different quarks is not mimicked by masses, which have a separate pattern. The factors on the arrows between quarks for CKM parameters are squares of relative weak interaction strengths. It may be significant that the ratio (b2 + 1)/a2 ~ 8Pi8/7, while 1/a2 ~ 3Pi3/5, indicating a simple pattern.

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