Carl Brannen On Mixing Matrix Parameterization

Carl Brannen has an interesting guest post at Quantum Diaries Survivor about his effort to parameterize the weak force mixing matrix (i.e. the probabilities that quarks of one generation produce quarks of a different generation in the course of weak force interactions like the ones involved in nuclear beta decay) in an unconventional way.

Regardless of what you think of his ultimate result (he is now in a PhD program, after years as a working engineer who published physics ideas as a hobby, and his paper is undergoing peer review), the lengthy post is exceptional for doing what most physics papers targeted at a professional audience do not. He spells out the gist of the technical jargon and notation in terms that an educated layman can understand. For example, if you are mystified about what a "Lie group" is, or what bracket notation means, he tells you.

The big and ongoing issue in parameterizing the mixing matrixes, such as the CKM matrix (for quarks) and PMNS matrix (for leptons), are (1) that they contain CP violations, which is to say that the probabilities are not the same for particles and antiparticles (see for example, the latest experimental data on CP violations in B mesons), and (2) that it isn't clear what the minimum number of parameters necessary to specify the matrixes is in nature. He goes at the problem by looking at variations of combinations of the real and complex parts of the matrix in these unitary democratic matrixes.

Since the three by three matrixes are "unitary" (i.e. all probabilities from any given starting point add up to one), the nine entry matrix can be described with at least no more than four parameters. But, is nature really that complicated?

Brannen notes that we can actually use a parameterize the "democratic unitary quark mixing matrix" which the real one approximates, with simple variations of matrixes with a single parameter (which he calls "w"=e^2i(pi)/3 with the diagonal entries (in which a weak force interaction starts and finishes with the same generation of quark) equal to (1+2w)/3, and the six other entries equal to (1-w)/3.

The essential piece of who this fits together (as the mixing matrixes don't have three components that are identical to each other on the diagonal and don't have non-diagonal entries that are quite identical, although entries with flipped indexes are quite close to each other) is as follows:

The traditional parameterization is designed to separate CP-violation from the mixing angles but in fact CP-violation depends on all the parameters. In addition, the traditional mixing angles determine the mixing correctly only in the infinitesimal limit. We show that the parameters can be chosen as defining representations of the permutation group on three elements and this makes CP-violation a secondary consequence of the mixing. Thus, in our view, theoretical attempts to derive CP-violation from a deeper theory should be focused on all the mixing angles rather than just on the 4th parameter of the traditional representation.

This follows from the insight that while CP violation is eliminated if the parameter usually used to quantify CP violation in the CKM matrix is set to zero, that it is also eliminated if any of the other three mixing angles are set to zero.