The Wolfenstein parameterization of the CKM matrix (articulated first in L. Wolfenstein, "Parametrization of the Kobayashi-Maskawa Matrix", 51 (21) Physical Review Letters 1945 (1983)) emphasizes the extent to which the probability of a quark flavor change when it emits a W boson depends upon a change in quark family.
* A transition from the first generation to the second generation (or visa versa) happens with a probability of lambda squared (about 5.07%-5.08%).
* A transition from the second generation to the third generation (or visa verse) happens with a probability of about A squared times lambda to the fourth power (about 0.16%-0.17%).
* A transition from the first generation to the third generation (or visa versa) happens with a probability roughly equal to the probability of a transition from the first generation to the second generation, multiplied by the probability of a transition from the second generation to the third generation, times an adjustment in the form of a complex number an absolute value of a magnitude on the O(1) that includes a CP violating phase. In all, a first to third generation (or visa versa) quark family transition happens with a probability of about 0.0012% to 0.0075%.
* The probability that a quark will remain in the same quark generation is equal to one minus the probability that it will change generations (about 94.9202% in the first generation, about 94.7585 in the second generation, and about 99.8293% in the third generation).
* The Wolfenstein paramterization emphasizes that the slight percentage differences between the probability of CKM matrix entry Vcb and Vts, between Vcd and Vus, flows mostly from compensating from other entries in that row, such as the considerably more significant (roughly 6-1) differences between the tiny Vtd and Vub. There are, in fact, more than the two CP violating parameters shown in a simplified version of the Wolfenstein paramterization, in the Standard Model CKM matrix, but those effects are tiny on a percentage basis relative to the magnitude of the other, much larger CKM matrix entries.
* Of course, the decay of a quark at rest is mass-energy conservation barred from becoming a heavier quark. It can only decay to the lighter quark (something that is possible for all types of quarks except up quarks). Only quarks with sufficient kinetic energy can transition to higher mass quarks.
In the Wolfenstein parameterization, "lambda" is roughly 0.02257 (and is another way of stating the Cabibbo angle), "A" is roughly 0.814, and the "p-in" CP violating term is about 0.135 minus 0.349i.
The point that the Wolfenstein parameterization underscores is that the CKM matrix derives largely from crossing one of two (or both) of the fundamental fermion families, not just from the particular quarks involved in the transition.
The probability of a second to third generation transition is consistent with two-thirds (between 62.7% and 69.7% with a best fit value of 66.3%) of the square of the probability of a first to second generation transition.
Some CKM Matrix Structure Conjectures
It wouldn't be too hard to imagine that in some deeper "within the Standard Model" theory, that before adjusting for a complex numbered CP violating term, that:
(1) the probability of a second to third generation quark flavor change is exactly two-thirds of the probability of a first to second generation quark flavor change squared (before considering a Wolfenstein parameterization style complex numbered CP violating term with both a real and an imaginary part), and
(2) that there is a CP violating term of similar absolute magnitude in all of the CKM matrix terms but that it is such a tiny percentage of the larger terms that it is impossible to discern experimentally in those terms, thus, we notice CP violation only in low probability CKM matrix entries only because the effect is large relatively to the pre-CP violation term signal.and
(3) that there is some deeper reason for both the magnitude of the Cabibbo angle and the magnitude of the CP violating term in the CKM matrix.
Assumption (1), if true, would reduce the number of experimentally measured, parameters in the CKM matrix from four to three.
This also suggests that if the absolute magnitude of the probability of a first to second generation flavor change, or a second to third generation flavor change, is a function of the quark mass matrix, that it depends on mass of both quarks in the first generation, both quarks in the second generation, and both quarks in the third generation, respectively.
But, given this disconnect between individual quark masses and CKM matrix probabilities, that it seems more likely that any causal relationship between the CKM matrix and the quark mass matrix flows from the CKM matrix to the mass matrix and not the other way around.
This implicitly disproves the possibility that I considered recently, that the CKM matrix probabilities and the fermion mass matrix might be a two way street with the parameters resulting from a mutual balance between the two. It could very well be that the CKM matrix probabilities do drive the form of the mass matrix of the fermions in the manner that I had suggested previously. But, if the CKM matrix can be derived almost entirely from the two electroweak gauge couplings of the Standard Model, then the fermion masses do not meaningfully drive the value of its parameters.
Footnote on the Cabibbo Angle and Weinberg Angle
One of the great unsolved problems in physics is to determine any deeper principles that relate the couple dozen or so experimentally measured constants of the Standard Model to each other, so that the model would have fewer experimentally measured parameters.
An achievement like this would both allow for greater precision in fundamental physics, since theoretically determined values of constants that are harder to measure precisely could be determined using experimentally measured values of quantities that are easier to measure precisely. It would also illuminate us by providing a deeper understanding of how a relative kludge of a rule book for the universe really works, much like grand unified theories hope to do.
Given the conjectures above, finding a way to derive the Cabibbo angle from first principle is a tempting prize.
The Cabibbo angle
The Cabibbo angle, also known as Euler angle Theta12 in the standard parameterization of the CKM matrix, is defined as the inverse tangent of the absolute value of CKM matrix element Vus divided by the absolute value of CKM matrix element Vud. The currently measured sine of the Cabibbo angle is 0.2256* with a one standard deviation confidence interval range of 0.22325 to 0.2265.
This assumes the value of quoted at Wikipedia based on a PDG source for the two CKM matrix elements of Vus = 0.22534 and Vud = 0.97427. But, late last year, a new precision measurement of the first element of 0.22290(90) was made which also tweaks the global best fit for the second number in the Cabibbo angle formula. As a result, the sine of the Cabibbo angle is really a bit more than 0.2230.
The Weinberg angle.
The Weinberg angle, also known as the "weak mixing angle" is defined as the inverse cosine of the mass of the W boson over the mass of the Z boson.
In many applications, the quantity actually used and measured is the square of the sine of the weak mixing angle, which runs with the energy scale of the interaction. At the Z boson mass, the sine squared of the weak mixing angle is 0.23120 +/- 0.00015, while at the energy scale of 0.16 GeV, the sine squared of the weak mixing angle is 0.2397 +/- 0.0013.
The Weinberg angle and the Cabibbo angle compared
Using the old measurement, the square of the sine of the Weinberg angle was 2.48% larger than the sine of the Cabibbo angle. These two values are inconsistent with each other at a 6.14 standard deviation level (and slightly less but still clearly different with the new value of the Cabibbo angle).
The electroweak gauge coupling constants and Higgs vev
The Weinberg angle is also defined as the inverse tangent of the bare electromagnetic force gauge coupling constant g' divided by the bare weak force gauge coupling constant g.
The magnitude of the fundamental electric charge "e", in turn, is the bare weak force gauge coupling constant g times the sine of the weak force mixing angle (and thus can be determined solely from g and g').
The Fermi coupling constant is a function of the weak force gauge coupling constants g, the reduced Plank's constant, the speed of light, and the W boson mass, or alternately, directly from the Higgs vacuum expectation value "v", which has been experimentally measured in connection with measurement of the lifetime of the muon to be 246.22 GeV/c.
The mass of the W and Z bosons can be computed from g, g' and v.
The square of the bare electromagnetic force gauge coupling constant g' equals four pi times the fine structure constant times the permittivity of free space. While I don't personally know how to determine the relative magnitudes of the fine structure constant and permissivity of free space exclusively from g, g' and v, I believe that it is possible to do so.
If the supposition that the mass of the Higgs boson is equal simply to the W boson mass plus half of the Z boson mass, then its mass can be computed from g, g' and v as well, i.e. MH=1/2*g*v+1/4*g*sqrt(g2+g'2). The world average measurement of the Higgs boson mass per PDG is currently 125.9+/-0.4 GeV/c and the theoretical value of the Higgs boson mass given this assumption and using PDG world average measurement values for the W and Z boson masses is 125.98 GeV/c.
Thus, if one accepts one experimentally plausible conjecture about the relationship of the W and Z boson masses to the Higgs boson mass, then the masses of all of the massive gauge bosons in the Standard Model, the strength of the fundamental electric charge, and the strength of the strong and weak forces can be computed from first principles using only three physical constants: g, g' and v (and the speed of light and Planck's constant).
If one further accepts the proposition that the sum of the square of the masses of all of the fundamental particles in the Standard Model is equal to the square of the Higgs vacuum expectation value (something that is empirically true using the PDG mass estimates of those particles to a precision well in excess of the margin of error in the underlying measurements and a very beautiful and plausible hypothesis), then one can also computer the aggregate sum of the squared masses of the fundamental fermions using only g, g' and v (plus Plank's constant and the speed of light), although not their masses relative to each other.
In other words, if that is true, one can compute the overall mass scale of the fundamental fermions of the Standard Model from just g, g' and v.
But, there is no current theory that explains how the CKM matrix parameters can be computed from these constants. The CKM matrix parameters in the Standard Model are simply experimentally measured inputs.
Likewise, there is no widely accepted theory to explain the relative masses of the fundamental fermions of the Standard Model from first principles, although extended versions of Koide's formula come pretty close to approximating these relative masses with resort to just one more constant (e.g. the ratio of the muon mass to the electron mass).
Is the Cabibbo angle a function of the Weinberg angle?
It is also tempting to wonder why these two fundamental parameters of the electroweak portion of the Standard Model are so close to each other.
Is there is some way that the Cabibbo angle could be computed from g and g' as well?
If the sine of the Cabbibo angle is simply the square of the sine of the weak mixing angle, this would certainly be true.
If this was possible, and if my supposition about the role of the Cabibbo angle in setting the absolute magnitude of the weak force flavor changing probabilities apart from the p-in term of the Wolfenstein parameterization of the CKM matrix discussed above is correct (i.e. if the square of Wolfenstein parameter "A" is exactly two-thirds), then the two dominant parameters of the four CKM matrix parameters in the Standard Model could also be derived from first principles solely from g and g'.
Could the two angles be reconciled at some energy scale? No.
Is there perhaps something significant about the energy scale at which the square of the sine of the Weinberg angle is equal to the sine of the Cabibbo angle, i.e. the energy scale at which the square of the sine of the Weinberg angle is equal to 0.2256 or so of the old measurement?
The short answer is no.
The Weinberg angle, consistent with the Standard Model prediction fell by 0.0085 from 0.16 GeV to the Z boson mass, and would need to fall another 0.0056 (about 65.9% of the decline in the previous mass shift range) to drop to the sine of the Cabibbo angle.
But, as the linked paper above measuring the weak mixing angle at 0.16 GeV illustrates, the Standard Model expectation is that the Z boson mass is at or near the minimum value of the weak mixing angle which grows larger again at higher energies, so there is no energy scale at which these to key electroweak parameters of the Standard Model would coincide.
Could the Cabibbo angle be redefined to make the two consistent? Yes.
Could there be, for example, some other natural way of defining an angle similar to the Cabibbo angle so that the sine of the Cabibbo angle would be equal to the square of the sine of the Weinberg angle?
In principle, the answer is yes.
For example, one could imagine redefining it as the inverse tangent of (the absolute value of CKM matrix element Vus plus the absolute value of CKM matrix element Vub) divided by the absolute value of CKM matrix element Vud which would increase the sine of the Cabibbo angle to about 0.22867, and then multiplying this time one plus the fine structure constant (which is roughly 1/137), which would bring it to 0.23034. This would be within one standard deviation of the square of the sine of the Weinberg angle at the Z boson energy scale given the precision of current experimental measurements (the precision of the Weinberg angle measurement is about six times greater than the precision of the Cabibbo angle measurement).
The extension of the definition of the Cabibbo angle to include the addition of CKM matrix element Vub is very natural. The Cabibbo angle was originally defined before the third generation of Standard Model fermions was discovered. In a two fermion generation Standard Model that Cabibbo angle was simultaneous the probability of a transition to a non-first generation quark and the probability of a transition from a first to a second generation quark. Including the CKM matrix element for a transition to a third generation would generalize it using the latter interpretation of its meaning, rather than the former, which were both identical in the two generation case.
The inclusion of a factor of one plus the fine structure constant is less obvious and somewhat arbitrary. But, given that we are talking about an electroweak process that always involves a W boson with has both a weak force coupling and an electromagnetic coupling, it would hardly be stunning that a formula to derive from first principles a probability of quark generation transitions from one generation to another might involve both the weak mixing angle and the electromagnetic coupling constant.
After all, electroweak theory is full of equations that involve various combinations of both the weak and electromagnetic coupling constants g and g' respectively, such as the mass of the Z boson and the fundamental electric charge "e" of the positron and the proton.
If the Cabibbo angle could be determined from first principles using a modified definition such as this one, it would follow that all of the CKM matrix elements, other than the "p-in" parameters of the Wolfenstein parameterization, could be computed from first principles using only the two electroweak coupling constants.
It would then be possible to calculate the CKM matrix elements without any reference to the non-electroweak constants of the Standard Model or any of the mass constants of the Standard Model.
To be clear, I am not actually arguing that I have come up with the correct formula to determine a redefined Cabibbo angle of the CKM matrix from first principles using only g and g'. I am merely arguing that it is well within the realm of possibility to do so and proving the concept by demonstrating how this could be accomplished. This particular example is probably just numerology, but it doesn't seem like an impossible task to come up with a formula that is better justified theoretically, or even to better justify the use of the fine structure constant in that way that I do above in that derivation. But, this example does seem to indicate that it would be possible to come up with such a formula using only quantities that could be plausible related to electroweak interactions.
Some relevant comments were inadvertently posted to the previous R&D post.
Relevant data on this which is up to date at https://arxiv.org/abs/1903.05062
"A global fit to the flavour observables gives 
A = 0:825(9); lambda = 0:2251(3); p = 0:160(7); n = 0:350(6)."
p and n are on the order of lambda to the fourth power numerically.
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