Estimate top quark branching ratios without calculating

[ClaraBS]

Homework Statement

Hi, I need to get an estimate of the branching ratios of the processes [/B]

t ---> b + W ---> b + c anti-s

t ---> b + W ---> b + anti-tau nu(tau)

without doing the calculations, just a qualitative estimation.
I know I have to take into account the way W decays leptonically or hadronically, but I cannot arrive to an answer.
Thanks a lot for any help.

Homework Equations

The Attempt at a Solution

 

[mfb]

I know I have to take into account the way W decays leptonically or hadronically

Well that is obvious from the problem statement.
What could lead to a deviation from a ratio of 1?
Is "c quark" a full and precise description of a quark, especially in QCD?

 

[ClaraBS]

Is "c quark" a full and precise description of a quark, especially in QCD?

No, it is just a flavour eigenstate. I have to consider the CKM matrix relating flavour to mass eigenstates... So, given the CKM element $V_{cs} = 0.974$, the probability for W to decay into c anti-s should be its square, $0.974^2 \approx 0.95$ ?
But what about the leptonic channel?

 

[mfb]

Well, that is a contribution, but that effect is small.
What about colors?

The decay probability W->c $\bar s$ is certainly not 95%. That is just a relative value (relative to a diagonal CKM matrix).

 

[ClaraBS]

It seemed to me really strange to get 95%, in fact, but I am definitely not getting the point

 

[mfb]

Concerning the CKM matrix: if the matrix would be diagonal (no transitions between generations), then the matrix element would be 1 for $u \bar d$, $c \bar s$ and even for $t \bar b$. The last one is impossible due to energy/momentum conservation of course. The "1" would just tell you that the two former decays have approximately the same probability. It does not tell you the absolute probability of any decay.
Now the matrix element does not give exactly 1. So $c \bar s$ is a bit less likely, because $c \bar d$ and $u \bar s$ can also happen.
This is a tiny effect, however - 5% difference.

What about colors? Quarks can have three different colors. For $W \to c \bar s$, do you know which color the c quark gets? If not, do you care about it? If not, you should consider all possible cases as decay modes, and add their individual probabilities.