It seems much longer but it was only two months ago that I wrote a post giving the CKM matrix in magic unitary form. With half the Nobel prize in physics going to Kobayashi and Maskawa, the K and M of the CKM matrix, I should include a quick update.

The CKM matrix is usually written in absolute magnitude form. Recent experimental measurements, after correcting to ensure compatibility with unitarity (from hep-ph/0706.3588), is:

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.

There are a lot of ways we could choose the complex phases. However, there is only one way (except for the sign of the imaginary unit i) of writing the matrix as a unitary matrix whose rows and columns sum to unity. This is called the “magic” property. The MNS matrix is quite simple when written this way. So I wrote a computer program and found the magic unitary form for the CKM matrix, to see if it would also have a simple form. Here it is:

Note that each row and column sums to 1, the rows and columns are orthonormal, and the absolute value of each element is as given for the experimental measurements at the top of the page.

The above is a slight improvement from the data I gave before. This is due to an error in one of the digits of the input data, that is, in the experimental measurements. Of course it is not accurate to all its digits, but I’ve included them because it’s a complicated nonlinear relationship between the absolute values of a unitary matrix and its magic unitary form; I don’t know how to properly scale the errors. (I could get them by varying the input data to the computer program that finds the magic solution. If someone wants this data, ask for it in the comments.) Below the fold, I’ll include the data in LaTeX format so you can copy it more easily:

\left(\begin{array}{ccc}

+0.949908262&+0.050759137&-0.000667400\\

+0.059487123&+0.938327410&+0.002185465\\

-0.009395386&+0.010913451&+0.998481935

\end{array}\right)

\pm i \left(\begin{array}{ccc}

-0.215902315&+0.220148844&-0.004246529\\

+0.217784181&-0.258647309&+0.040863128\\

-0.001881866&+0.038498465&-0.036616598

\end{array}\right)

I should add that the method that the computer program uses is to search using a parameterization of the magic unitary 3×3 matrices that I found. The method is described in the previous post, CKM as a magic unitary matrix.