Given my final success in writing the CKM matrix as the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix, the next step is to write down the parameterization that allows any 3×3 unitary matrix to be written in this elegant and natural form. Following the method used earlier to parameterize unitary 3×3 magic matrices, and correcting for a few typos (but the description of the method is correct).
We will use four real angles, , as parameters. First, define 6 real numbers I, J, K, R, G, and B as follows:
The the following matrix is unitary:
This is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. Note that the sum of the 3 elements of any row or column is equal to the complex phase .
The choice of the variable names I, J, K, R, G, B is to remind of the permutation group on three colors. I is the identity. J and K are the cyclic (even) permutations. And R, G, and B are the swaps that leave red, green, and blue unchanged, respectively. Earlier, we wrote the weak hypercharge and weak isospin quantum numbers as solutions to the idempotency problem for matrices of a similar form. See Electroweak Unification, Quarks for the rather difficult algebra involved in solving the resulting 6 quadratic equations in 6 unknowns.
The angles and show up three times each in the definitions of I, J, K, R, G, B. They are angles and have added to them constants of 0, 120, and 240 degrees. Equivalent angles show up in the Koide mass equations. As Marni Sheppeard says, this is the result of taking discrete Fourier transforms on three elements.
Of course one wonders what values of the four parameters define the CKM and MNS matrices.
CKM parameters, theta, alpha, beta, gamma:
+0.207642790, -0.006257792, -1.199063520, -0.273088590 (radians)
+11.89705551, -0.35854507, -68.701279, -15.64682364 (degrees)
The CKM parameter angle gamma = -15.65 degrees, defines the split between the average real part and average imaginary part. In the Koide formulas, this was the “valence” or “v” parameter. The angle alpha is close to zero. With gamma being relatively small, this makes I dominate over J and K. Finally, the angle theta defines the ratio of the “sea” (or “s” in the Koide mass formulas) sizes of I,J,K relative to R,G,B, that is, the part that is contributed with three angles alpha, alpha+120, and alpha+240 degrees. Since theta = 11.89 degrees, the CKM matrix is about 5x larger.
MNS parameters, theta, alpha, beta, gamma:
+1.030839502, +0.758336406, -0.424863234, +0.169918454 (radians)
+59.06275282, +43.44947551, -24.34287018, +9.735610275 (degrees)
As you can see, the MNS parameters seem to be pretty much random numbers. This is despite the fact that the MNS unitary matrix, when written in this parameterization, is remarkably simple:
So the arbitrariness of the CKM parameters (which are the result of approximate experimental measurement and consequently have error in them) does not give any sort of evidence that the CKM matrix cannot be written in a simple, elegant, and clean form as the sum of a real 1-circulant and an imaginary 2-circulant matrix.
What I’d like to do next is to rewrite the parameterization in such a way that the MNS parameterization becomes nice. That is, I’d like to see the elegance of the MNS matrix in its parameterization. This might give a clue as to how to analyze the CKM values, and this might allow us to derive a theoretical CKM matrix that is as successful as the tribimaximal theoretical MNS matrix.