# Unitarity and the CKM Matrix

I’m busily working on writing the CKM matrix in Kea’s form, that is, as a unitary matrix that is the sum of 1-circulant and 2-circulant matrices. With the MNS matrix this was easier because I could begin with a unitary matrix, but the CKM matrix is usually given in absolute value form.

Looking through the literature, I’ve found a beautiful paper that digs to the core of the unitarity problem for the CKM matrix: A new type fit for the CKM matrix elements, Petre Dita, hep-ph/arXiv:0706.3588:

Abstract: The aim of the paper is to propose a new type of fits in terms of invariant quantities for finding the entries of the CKM matrix from the quark sector, by using the mathematical solution to the reconstruction problem of 3 x 3 unitary matrices from experimental data, recently found. The necessity of this type of fit comes from the compatibility conditions between the data and the theoretical model formalised by the CKM matrix, which imply many strong nonlinear conditions on moduli which all have to be satisfied in order to obtain a unitary matrix.

Dita’s CKM matrix estimate is:

This needs a little explaining, probably.

Of course the above matrix is not unitary, however, as the paper shows, it is at least compatible with being unitary, that is, there is a unitary matrix whose absolute values are given above. For the problem of writing the CKM matrix in unitary doubly magic form, one would like to begin with a form that is at least unitary. So this is a better place to start than the raw data from the experiments.

The hope from all this is that I will find an equivalent to the tribimaximal matrix for the quarks. I hope that the result will be related to the beautiful and simple MNS matrix decomposition into 1-circulant and 2-circulant parts. While the MNS had two portions, a 1-circulant and a 2-circulant, each real but multiplied by a complex phase, I would think that the CKM would have four portions, two 1-circulants and two 2-circulants, again each multiplied by a complex phase.

So, optimistically, I look forward to good hunting! If it is possible to write the CKM matrix in a form similar to the MNS I think it will be a great victory.

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### 2 responses to “Unitarity and the CKM Matrix”

1. Heh! I’ve just set up an internet connection in Tekapo, so now I have a whole week of free time to play!!!!

2. carlbrannen

Okay, I think I have a parameterization of the unitary 3×3 matrices that happen to be the sum of a 1-circulant and a 2-circulant.

It turns out that any matrix that is “magic” in that its rows and columns all sum to the same number, must be the sum of a 1-circulant and a 2-circulant, and vice versa, in essentially a unique way (subject to an overall constant). Since “magic” is shorter than “the sum of …”, I’m going to use that notation.

I end up with 7 real numbers to parameterize the magic unitary matrices (I’ll check this by computer program later tonight, I hope).

To get all possible unitary matrices requires a standard parameterization of 9 parameters. In the standard model, the 9 parameters that define a unitary matrix consist of 3 mixing angles $\theta_{jk}$, one CP violating phase usually written as $\delta$, and 5 arbitrary phases applied to rows or columns.

If this is correct, it means that adding “magic” as a restriction to unitary removes 2 dimensions to the manifold.