Monthly Archives: December 2009

ArXiv and the Wolfenstein Parameterization

A paper appeared on arXiv last week, “On one parametrization of Kobayashi-Maskawa matrix” 0912.0711 by Petre Dita. The abstract:

An analysis of Wolfenstein parametrization for the Kobayashi-Maskawa matrix shows that it has a serious flaw: it depends on three independent parameters instead of four as it should be. Because this approximation is currently used in phenomenological analyzes from the quark sector, the reliability of almost all phenomenological results is called in question. Such an example is the latest PDG fit from \cite{CA}, p. 150. The parametrization cannot be fixed since even when it is brought to an exact form it has the same flaw and its use lead to many inconsistencies.

Among phenomenologists, this is a pretty serious accusation. There are hundreds of papers on arXiv alone that use the Wolfenstein parameterization. It’s the basis for the PDG estimates on the CKM matrix. If it’s true this is really big news in elementary particles.

The Dita paper claims that the Wolfenstein parameterization is defective because its apparent four real degrees of freedom are redundant; instead there are only three. Such a defect would prevent the parameterization from exploring “almost all” of the space of possible 3×3 unitary matrices. Instead of the whole 4-dimensional real manifold of 3×3 unitary matrices (up to multiplication of rows and columns by complex phases), one would obtain only a 3-dimensional submanifold.

In particular, the paper claims that it is impossible to use the Wolfenstein parameterization to obtain a unitary 3×3 matrix with the magnitude of all amplitudes the same (and equal to sqrt(1/3) ). This is the “democratic unitary 3×3 matrix”, a subject Marni Sheppeard and I have explored at length. It took me a few minutes to verify that it is possible to set these parameters (lambda, A, rho, and eta) to obtain a unitary matrix with all magnitudes equal.
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