The MNS matrix (in tribimaximal form, which is compatible with all experimental measurements) can be written in elegant form as a unitary matrix which is the sum of a real 1-circulant matrix and an imaginary 2-circulant matrix. See Doubly Magic Matrices and the MNS. Now I’ve got the CKM matrix in the same form. And here it is:

The cool thing about the above is that it only involves 6 real numbers. Three reals define the 1-circulant matrix, +0.973313178, -0.008576543, and +0.000466480, while three more reals define the 2-circulant, +0.225761835, +0.040012680, and -0.004273188.

I’ve marked the larger contributions with red to help you see the symmetry. The real matrix has each of its rows shifted one to the right (i.e. 1-circulant), while the imaginary matrix has them shifted two to the right (i.e. 2-circulant). Note that the above matrix is unitary, which you will have to verify by taking dot products of its rows with the complex conjugatges of its rows, and the same for the columns. (You should get 1s and 0s.)

The experimental values I’m using for the CKM matrix come from hep-ph/0706.3588) and are:

If you square the elements of a row or column of the above, and add them up, you get 1, but it is not unitary. To put it into unitary form, we supplement each of the above real numbers by multiplying by a complex phase. The matrix is unitary when the resulting rows and columns are orthonormal.

The experimental measurement has 9 real numbers. Multiplying them by complex phases so as to make the matrix unitary is going to give you 18 real numbers. And yet I’ve been able to do this, put the experimental numbers into unitary form, and ended up with only 6 real numbers, and those are used in an elegant and symmetrical manner. This is by far the most elegant version of the CKM matrix around. It is the number crunched result of an observation by Marni Sheppeard that the CKM matrix is approximately the sum of a 1-circulant and a 2-circulant matrix. Well, now we have it exactly that way. Victory! But more to come.

To get an experimental measurement from the new magic form, compute the magnitude of the complex number. For instance, to get the top right value of 0.0042982, compute

|+0.000466480 – 0.004273188 i| = sqrt(0.000466480^2 + 0.004273188^2)

Uh, I get 0.0042986, which is within rounding errors on the experimental measurement number.

More generally, the nine elements of the experimental CKM matrix can be found by taking one element from the set {+0.973313178, -0.008576543, +0.000466480} along with one element from the set {+0.225761835, +0.040012680, -0.004273188} and computing the RMS value.

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