What are the 2n-1 relations in the Cabibbo and CKM matrices?

[Heidi]

Hi Pfs,
I read a paper about the Cabibbo matrix and the CKM matrix.
The first one is a 2*2 real matrix and the other a 3*3 matrix with complex entries.
In this article i read that a n*n matrix has 2 n*n real degrees or freedom.
The unitarity (orthonormal basis) devides this number by 2.
I read that having n families gives 2n - 1 relations.
We have now (n - 1)(n - 1) real degrees of freedom
So the 2*2 matrix (with 2) families has only 1 real degree of freedom: the
Cabibbo angle.
Could you tell me what are these 2n-1 relations?
thanks

 

[Heidi]

I read this in the CKM matrix article:
2n − 1 of these parameters are not physically significant, because one phase can be absorbed into each quark field (both of the mass eigenstates, and of the weak eigenstates), but the matrix is independent of a common phase. Hence, the total number of free variables independent of the choice of the phases of basis vectors is n²
− (2n − 1) = (n − 1)²
.
Do you understand this sentence?

 

[hutchphd]

Understand is a big word. The sentence is simple
A reformatted version is $$n^2-2n+1=(n-1)^2$$ The physics is not within my teachable knowledge. ?

 

[PeterDonis]

the CKM matrix article

What article? Please give a reference.

 

[PeterDonis]

I read a paper about the Cabibbo matrix and the CKM matrix.

What paper? Please give a reference.

 

[malawi_glenn]

Read this https://www.physicsforums.com/threads/fixing-cp-phases-to-cancel-ckm-phases.1050841/

 

[Heidi]

Look at this article:
https://arxiv.org/abs/0810.2091
paragraph 2.4.
It seems that it is possible to find a basis in which the first column and the first row (2n-1 elements)
are real. (2n-1) more relations.

 

[PeterDonis]

Look at this article:
https://arxiv.org/abs/0810.2091
paragraph 2.4.
It seems that it is possible to find a basis in which the first column and the first row (2n-1 elements)
are real. (2n-1) more relations.

That paper is probably not a good reference if you want to learn about the CKM matrix in the Standard Model. It is more an abstract math paper than a physics paper.

In section 2.4, the "CKM matrix" it is talking about is not even the same one as is in the Standard Model, as far as I can tell. It is a "CKM matrix" for an abstract math toy model whose physical relevance is not even clear.

 

[vanhees71]

A very good introduction to the Standard Model (and thus also about the CKM matrix) is

O. Nachtmann, Elementary Particle Physics - Concepts and
Phenomenology, Springer-Verlag, Berlin, Heidelberg, New
York, London, Paris, Tokyo (1990).

 

[ohwilleke]

Heuristically, here is one way to understand it that makes intuitive sense.

The CKM matrix rows and columns code the probability of transitions from one kind of quark up-type quark to one of three kinds of down-type quarks, and from one of kind down-type quark to one of three kinds of up-type quarks, in a W boson interaction.

In each case the sum of the probabilities of possible transitions has to add up to 100% because if it transitions to a different kind of quark by emitting a W boson it has to end up as some kind of opposite type quark.

The fact that the sum of coded probabilities in every column has to add up to 100% means that when you know two entries in the column you can determine the value of the third entry. Likewise the fact that the sum of coded probabilities in every row has to add up to 100% means that when you know of two entries in the row, you can determine the value of the third entry.

This means that there are four degrees of freedom in the CKM matrix.

Likewise, in the the Cabibbo matrix, you are coding essentially the same things, but before you knew about third-generation quarks and CP-violation. Therefore, theoretically, it should have coded the probability of transitions from one kind of quark up-type quark to one of two kinds of down-type quarks, and from one of kind down-type quark to one of two kinds of up-type quarks, in a W boson interaction (with the entires having real values due to a lack of CP violation). If that had been true knowing one transition probability from one kind of up-type quark to one kind of down type quark would have been enough to deduce the probability of a transition to the other down side quark, and knowing those two values, you could have deducted the rest of the probabilities in the matrix because each row and each column had to code probabilities that add up to 100%. So, there is only one degree of freedom in the theoretical original Cabibbo matrix. Of course, the entries in the original Cabibbo matrix didn't add up to cover all possible outcomes, which is part of how we knew that there were three and not two generations of Standard Model fermions.

It turns out that there are lots of different ways to actually pick those degrees of freedom, and you choose which parameterization you use on the basis of style and convenience. You don't have to simply pick one entry in the Cabibbo matrix, and you don't simply have to pick four entries in the CKM matrix, although you could do that.

Incidentally, this has been known for a long time by particle physics standards. The Cabibbo matrix was introduced in 1963 by Nicola Cabibbo before we knew about third-generation quarks (and even before we actually had a quark model that was firmly established); the CKM matrix, in 1973, was a generalization of this idea to three generations of quarks with a possibility of CP violation, by Makoto Kobayashi and Toshihide Maskawa. The mixing angle of the original Cabibbo matrix (which we now know isn't actually unitary in the probabilities that it codes since it omits transitions to third-generation quarks) is called the Cabibbo angle. The basic concepts were being recapped for a more general audience, for example, in a 1984 article summarizing the concept from the Los Alamos research facility.