# Particle decay rates, CKM matrix etc

1. Jul 24, 2013

### sunrah

1. The problem statement, all variables and given/known data
Calculate following decay rate

$\frac{\Gamma(D_{0} \rightarrow K^{-} \pi^{+})}{\Gamma(D_{0} \rightarrow \pi^{+} \pi^{-})}$

2. Relevant equations
3. The attempt at a solution

$D^{0} = |cu\rangle , D^{+} = |cd\rangle , B^{0} = |bd\rangle , D_{s}^{+} = |cs\rangle , K^{-} = |su\rangle , π^{+} = |ud\rangle$

I know only that in the numerator decay the charm quark couples with a strange quark, so the CKM matrix element Vcs would describe this decay? In the denominator decay d-quark couples with c-quark, so would this decay be described by CKM-matrix element Vcd?

EDITED:
Is then the ratio: $\left(\frac{V_{cs}}{V_{cd}}\right)^{2}$

thank you

Last edited: Jul 24, 2013
2. Jul 24, 2013

### vanhees71

Think about, how the decay rate is related with the S-matrix element!

3. Jul 24, 2013

### sunrah

hello,

sorry I don't know what an S-matrix is, this isn't covered in our syllabus

is decay rate not proportional to coupling CKM-matrix element?

Last edited: Jul 24, 2013
4. Jul 24, 2013

### vanhees71

Ok, so how have you made the connection between the CKM-matrix elements to the decay rate?

The S matrix contains the observable results of computations in quantum field theory. It gives the transition probability rates for scattering (or in your case decay) processes. It's in fact a pretty delicate issue, related to the socalled LSZ-reduction formalism. After the qft dust has settled, the S-matrix elements are what you calculate perturbatively with help of the Feynman-diagram rules. Take any good book on QFT to learn about it. Peskin/Schroeder is pretty good in explaining it.

5. Jul 24, 2013

### sunrah

Thanks, but it must be possible to work out Gamma, at least proportionally, just from CKM otherwise our tutors would not have set the question. I think rates are amplitude squared so it should be $\Gamma_{ij} \propto |V_{ij}|^{2}$

6. Jul 24, 2013

### Staff: Mentor

I think the answer you got is the intended way to answer the problem.

If you compare it to the actual fraction, it will not match (it is even worse if you compare $D^0 \ \to KK$ with $D^0 \to \pi \pi$). This comes from other processes (the tree-level decay is not the only option), phase space and probably some other effects.