Daily Archives: August 3, 2008

CKM as a Magic Unitary Matrix

Since writing the MNS as a magic unitary matrix, of course I’ve been working on writing the CKM matrix the same way. This involved learning a lot more about 3×3 unitary and 3×3 magic matrices, and writing a Java program to do the heavy lifting.

The first thing one must do to deal with magic unitary matrices is to define a parameterization of these matrices. A full parameterization of all unitary matrices requires 9 real variables. Five of these define the arbitrary complex phases that can be applied to any row or column (yes there are 3 rows and 3 columns, but one of them is redundant). The remaining 4 variables are usually written in the Wolfenstein parameterization. In this parameterization, three variables are mixing angles and the fourth variable defines the CP violation.

If one is given a magic unitary matrix, the effect of multiplying any row or column by a complex phase would be to destroy the magic. Consequently, putting a unitary matrix into magic form (if this can be done) amounts to choosing a set of unique phases. Therefore, the largest number of free parameters we can expect to need to define magic unitary matrices is the 4 used in the Wolfenstein parameterization (which also amounts to choosing a unique set of phases). In fact, I’ve found a parameterization of the unitary magic matrices on 4 real parameters so one supposes that any unitary matrix can be put into a (more or less unique) magic form.

First, let’s write some reduced notation. We will abbreviate the 1-circ and 2-circ matrices as follows:
Abbreviated circulant matrices
Note that that “E” and “F” are reversed in the above 2-circ definition. This is to make the Fourier transforms easier to deal with. In our abbreviated notation, we are to solve:
1-circulant + 2-circulant matrix as unitary
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