CKM Matrix Coefficients: Same for All Interactions?

[Magister]

Are the coefficients of the CKM matrix the same for all interactions?

If yes, why does the CP violation occur only in very specific interactions and not in all of then?

If no, why does the coupling between family quarks interaction dependent?

Thanks

 

[Jim Kata]

One proposed model claims it has to do with the bottom and top quark doublet. According to Weinberg Vol. II because of the appearance of the third generation of quarks it is no longer possible to make the CKM matrix completely real and somehow the fact that the CKM matrix is no longer real can cause T violation and hence CP violation.

 

[drpsycho]

> Are the coefficients of the CKM matrix the same for all interactions?

Yes.

> If yes, why does the CP violation occur only in very specific interactions and not in all of then?

The CP-violating effects are visible only in processes (and for physical observables) where the complex phases of the CKM elements do not cancel away.
I mean, if you have a complex number of the form A*exp(-iB) but when calculating some cross section or other physical quantity it gets multiplied by its own conjugate, i.e. A*exp(+iB), the fact that it is complex makes absolutely no difference with respect to a real number.
Instead, the processes where you can observe CP violation are those where this cancellation does not occur.
I cannot be more precise by hearth, but I suggest you to look for the "CP violation primer" on the webpage of the BaBar experiment, where all this is explained in some detail.

Note: I assumed that you know something about complex numbers. If not, I will have to explain in a more basic way.

 

[drpsycho]

> One proposed model claims it has to do with the bottom and top quark doublet. According to Weinberg Vol. II because of the appearance of the third generation of quarks it is no longer possible to make the CKM matrix completely real and somehow the fact that the CKM matrix is no longer real can cause T violation and hence CP violation.

Not true.
Even with three families of quarks, nothing prevents the CKM matrix from being real. After all, nobody knows why the complex phase has the value that it has: in the Standard Model it is a free parameter, which could have had any other value, including zero. (If zero, the CKM matrix would be real.)

On the other hand, a generic unitary matrix has to be at least 3x3 in order to accommodate a complex phase. So, the discovery of CP violation led Kobayashi and Maskawa (the K and M in "CKM") to postulate a third generation of quarks.

 

[Magister]

But is the CKM matrix in the the $B^0 bar-B^0$ system the same as in the $k^0 bar-k^0$ system?

 

[drpsycho]

But is the CKM matrix in the the $B^0 bar-B^0$ system the same as in the $k^0 bar-k^0$ system?

Yes!

I'll try to explain why your question doesn't make so much sense :)
When you consider an elementary interaction involving a quark q, another quark q', and a W, in the calculation of the amplitude you will have to multiply for g (the weak interaction coupling, which is *universal*, i.e. does not depend on the process), and a factor which depends on the quarks, let's call it Vqq'. This factor is the element in the q-th row and the q'-th column of the CKM matrix.
The $K^0$ is composed of a strange and a down quark (one is a quark and the other an antiquark, and which is which depends on whether it is $K^0$ or $\bar K^0$). So the coefficients of CKM involved will be Vus, Vcs,Vts and Vud, Vcd,Vtd (because the weak interaction can couple the "low" quarks only to "high" quarks, I mean that d,s,b can only couple to u,c,t and not to other d,s,b).
In the $B^0$ the quark composition is one bottom and one down quark, in the $B^0_s$ the composition is one bottom and one strange. So the coefficients will be different coefficients in general (Vub, Vcb, etc.) but the matrix is always the same good CKM matrix.

 

[Magister]

My question in fact is related with the 3 unitary triangles in page 20 of the CP violation Primer. If the CKM matrix is the same for every process why are the 3 triangles different?

Note: I assumed that you know something about complex numbers. If not, I will have to explain in a more basic way.

Yes, I do know something about complex numbers.

 

[drpsycho]

My question in fact is related with the 3 unitary triangles in page 20 of the CP violation Primer. If the CKM matrix is the same for every process why are the 3 triangles different?

Oh, that's a different story.
The property of "unitarity" for a matrix implies that its columns, when treated as vectors, behave as a hortonormal set. And the same for its rows.
Hortonormality, for a set of complex vectors $\vector{v_i}$, means that $\vector{v_i^*}\vector{v_i}= 1$ and $\vector{v_i^*}\vector{v_j}= 0$.
If you multiply first and second rows, for example, you get one of those triangles. But another triangles comes from multiplying first and third rows, and the last triangle from multiplying second and third rows. (Or were they columns instead of rows? I don't remember.)

Now let's go to your question: the K system and the B system explore different rows/columns of the CKM matrix, simply because in one case there are strange and down quarks, in the other bottom and down.

Yes, I do know something about complex numbers.

good :)
I didn't want to sound offensive, it's just that I didn't know what to assume about your background knowledge :)

 

[Magister]

Ok, I got it. Thanks for your help.

By the way, one more question. Are the angles of the unitarity triangle that are experimentally measured?

 

[drpsycho]

Both the angles and the lengths of the sides are experimentally accessible.
But I'm not an expert in B or K physics, so trust more the Primer than me... :)

 

[Magister]

Well, you are certainly more expert than me.
Thanks a lot.