The Cabibbo-Kobayashi-Mashkawa (CKM) matrix is the matrix that sets forth the empirically determined square root of the probability of a quark of one type changing into a quark of another type when it emits a W boson.
Any up type quark can give rise to any of the three down type quarks. Any down type quark can give rise to any of the three up type quarks. It is trust as a matter of definition, that the sum of the probabilities of a particular quark, say, a charm quark, changing into each of the possible down type quarks is exactly 100%. It is empirically true, however, that the matrix is unitary. Thus, the sum of the squares of the entries in the columns of the matrix as well as the sum of the squares of the entries in rows of the matrix equal 100%.
The probability of a particular up type quark transforming into a particular down type quark is the same as the probability of that down type quark transforming into that up type quark, except to the extent of CP violation in the matrix, which has complex number valued entries.
Like any unitary matrix, the CKM matrix can be parameterized in an infinite number of ways using four parameters (because a unitary three by three matrix has four degrees of freedom).
The magnitude of the entries is as follows:*
Using the values of the previous section for the CKM matrix, the best determination of the Wolfenstein parameters is:
- λ = 0.2257+0.0009 −0.0010,
- A = 0.814+0.021 −0.022,
- ρ = 0.135+0.031 −0.016, and
- η = 0.349+0.015 −0.017.
* The material from the * to this explanatory note is from Wikipedia.
Up to adjustments for CP violation, this parameterization suggests that the probability of a first to second generation transition (or second to first generation transition) is λ, that the probability of a second to third generation (or third to second generation) transition is Aλ2, and that the probability of a first to third generation (or third to first generation) transition is λ*Aλ2 (i.e. product of the probability of making first one of the single generation step transitions and then the second). The probability of transitioning to a quark of the same generation is the residual probability after the probability of the other two options is subtracted out.
Higher order and variant Wolfenstein parameterizations are discussed in a 2011 paper. This suggests that some tweaks to the original Wolfenstein parameterization are necessary to fit the data, while being motivated by the same principles.
A 1994 paper, expands the Wolfenstein parameterization to a higher order of lambda and pointing out that a symmetric CKM matrix of the kind originally envisioned is almost ruled out by the data. A 2014 paper attempted to generalize this parameterization to leptons.
Nothing mathematically requires that it be possible to parameterize the CKM matrix with three rather than four empirically determined constants. But, there is a way to do so that is consistent with empirical evidence.
Aλ2 is equal to (2λ)4 at the 0.1 sigma level of precision, and there is no place in the Wolfenstein parameterization of the CKM matrix where this substitution cannot be made.
Thus, Vcb becomes approximately (2λ)4, Vts becomes approximately -(2λ)4, Vub becomes approximately (2λ)4*λ*(ρ-iη) and Vtd becomes approximately (2λ)4*λ*(1-ρ-iη).
Moreover if we take ρ-iη to be a single complex number C, and C* to be the complex conjugate of C, then we can state that: Vub becomes approximately (2λ)4*λ*C and Vtd becomes approximately (2λ)4*λ*(1-C*).
Thus, the entire CKM matrix is a function of one empirically determined real number, λ, and one complex number pertinent only to CP violation, C. This approach thus suggests that there is even more method to the apparent randomness of the CKM matrix than the Wolfenstein parameterization would already suggest.
This variation on the Wolfenstein parameterization of the CKM matrix suggests that we should be looking for physics explaining the second to third generation transition that are basically a power of four different than the first to second generation transition, rather than a power of two different. For example, the power of four might have some physical or geometrical relationship to the four dimensions of space-time.
To be clear, I'm not by any means the first person to see the possibility of expressing the CKM matrix with fewer than four parameters. A very different proposed single constant parameterization of both the CKM matrix and PMNS matrix, for example, can be found here.