Majorly confused over the cabibbo matrix?

[jeebs]

Hi,
I'm having some trouble with the Cabibbo matrix. I get vaguely what it's about but there are things I don't understand, and there seems to be so little information explaining about this out there. I know that before the top and bottom quarks were discovered, it was just a 2x2 matrix that looked like this:

$$\left(\begin{array}{cc} |d'> \\ |s'> \end{array}\right) = \left(\begin{array}{cc}V_u_d&V_c_d\\V_u_s&V_c_s\end{array}\right)\left(\begin{array}{cc}|d>\\|s>\end{array}\right)$$

My first problem is that wikipedia states that "the Cabibbo angle is related to the relative probability that down and strange quarks decay into up quarks ($$|V_u_d|^2$$ and $$|V_u_s|^2$$ respectively)." That's fair enough. It then goes on to state that "the various $$|V_i_j|^2$$ represent the probability that the quark of i flavor decays into a quark of j flavor."
Surely it has contradicted itself here, since, for example, $$|V_u_d|^2$$ would be the probability of an up decaying to a down, not the other way round like it says in the first quote?

Anyway, I see that when the right hand side of the matrix is multiplied out it gives
$$|d'> = V_u_d|d> + V_u_s|s>$$
$$|s'> = V_c_d|d> + V_c_s|s>$$
It then replaces the various V_ij as follows:
$$|d'> = cos\theta_c|d> + sin\theta_c|s>$$
$$|s'> = -sin\theta_c|d> + cos\theta_c|s>$$
I am aware that only the weak interaction can change flavour. I am aware that apparently changes within a generation happen much more readily than changes to different ones. Apparently the frequency of strange -> up decays and down -> up decays have been compared to give a cabibbo angle of $$\theta_c = 13.04\degrees$$
Was this measurement comparing the number of times something like $$\Sigma^- (dds) \rightarrow \Sigma^0 (uds)$$ and $$\Sigma^- (dds) \rightarrow n (dds)$$ happened?

Two things bother me here. One is that I cannot think of a reason why $$V_u_s = -V_c_d$$. Why should the probability of a strange turning into an up be equal to the probability of a charm turning into a down? (This is where the confusion over the indices comes in, because if I treated this as down to charm like it says, the quark mass would be increasing, which makes no sense energetically). Also, why is there a minus sign there?

The page has a diagram, shown here: http://en.wikipedia.org/wiki/File:Cabibbo_angle.svg . I don't really know what to make of this. We've got some orthogonal pure quark eigenstates on the horizontal axis & vertical, and at an angle to each axis we've apparently got a "weak eigenstate" (|d'> or |s'>) which are clearly combinations of the |d> and |s> quark eigenstates, the amount of each being determined by $$\theta_c$$. So, the mixing matrix applies a rotation to the vector space of the pure eigenstates to become the space of the weak eigenstates, or something like that, right?

I'm not sure it makes sense, but here is what I've been grappling with:
We are dealing with some particle undergoing the weak interaction. During the interaction the weak eigenstate quark mixture turns into a pure quark eigenstate, since this is what we detect - we find some particle with some definite quark composition, right?.
So, that must mean that for this initial decaying particle, since it's in a weak eigenstate, we don't really know it's definite quark composition?
So, that must mean this initial particle could be one of two different particles?
But, this conflicts with what I was asking about earlier with the comparison $$\Sigma^-$$ decays, since the $$\Sigma^-$$ would have an known composition to start with.
So which one is right? perhaps neither?

I don't even know where I'm going with this. I've been trying to write this question for about 2 hours now. I haven't been this confused in years...

If anyone can decipher any of what I've just attempted to ask, I'll be eternally grateful.

 

[fzero]

Two things bother me here. One is that I cannot think of a reason why $$V_u_s = -V_c_d$$. Why should the probability of a strange turning into an up be equal to the probability of a charm turning into a down? (This is where the confusion over the indices comes in, because if I treated this as down to charm like it says, the quark mass would be increasing, which makes no sense energetically). Also, why is there a minus sign there?

The page has a diagram, shown here: http://en.wikipedia.org/wiki/File:Cabibbo_angle.svg . I don't really know what to make of this. We've got some orthogonal pure quark eigenstates on the horizontal axis & vertical, and at an angle to each axis we've apparently got a "weak eigenstate" (|d'> or |s'>) which are clearly combinations of the |d> and |s> quark eigenstates, the amount of each being determined by $$\theta_c$$. So, the mixing matrix applies a rotation to the vector space of the pure eigenstates to become the space of the weak eigenstates, or something like that, right?

The CKM matrix is a unitary matrix, so there are at most $$N^2$$ components, where $$N$$ is the number of flavors. However, an overall common phase for all states can't be measured, so that removes one degree of freedom for $$N^2-1$$ components (the CKM matrix is an $$SU(N)$$ matrix). We can also absorb $$2N$$ parameters by rescaling all fields, one into each quark field, because the kinetic and mass terms in the Lagrangian involve the product of a field and its conjugate, so are invariant under a global phase. So the total number of free parameters of the CKM matrix is $$(N-1)^2$$. For $$N=2$$, there is one parameter, which is just the Cabibbo angle, and the CKM matrix is a rotation by that angle.

I'm not sure it makes sense, but here is what I've been grappling with:
We are dealing with some particle undergoing the weak interaction. During the interaction the weak eigenstate quark mixture turns into a pure quark eigenstate, since this is what we detect - we find some particle with some definite quark composition, right?.
So, that must mean that for this initial decaying particle, since it's in a weak eigenstate, we don't really know it's definite quark composition?
So, that must mean this initial particle could be one of two different particles?
But, this conflicts with what I was asking about earlier with the comparison $$\Sigma^-$$ decays, since the $$\Sigma^-$$ would have an known composition to start with.
So which one is right? perhaps neither?

I think your confusion is because the mass eigenstates are not the same as the weak eigenstates. A given particle has a definite composition in terms of quark mass eigenstates, but when you consider the weak interaction, these mass eigenstates are mixed into the weak eigenstates. It's really just a basis change and you can still write down amplitudes for processes involving the mass eigenstates as long as you incorporate the CKM matrix elements properly.

 

[tom.stoer]

Please have a look at the definition of your matrix. I think it does not agree with the standard definition found in the literature. E.g. the matrix elements in the first row should have the indices ud us (ub) whereas you wrote ud cd (??); (..) refers to the CKM matrix for three generations.