Particle decay rates, CKM matrix etc

[sunrah]

Homework Statement

Calculate following decay rate

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

Use the (Cabibbo–Kobayashi–Maskawa) CKM-matrix.

Homework Equations

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 V_cs would describe this decay? In the denominator decay d-quark couples with c-quark, so would this decay be described by CKM-matrix element V_cd?

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

thank you

 

[vanhees71]

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

 

[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?

 

[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.

 

[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 $\Gamma_{ij} \propto |V_{ij}|^{2}$

 

[mfb]

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.