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I've seen (Cf. Griffiths' book) many diagrams of weak interaction of the type $q_1 \rightarrow q_2 + W^{\pm}$ (which are justified ultimately in terms of CKM transition values). But I don't understand what are the prescriptions for a charged flavoured meson.

Take for example the $D^-$, which consists of a $d\bar c$ pair, decaying with a $W$ boson to leptonic modes (the hadron cases are very well explained here, for example). Could anyone explain the flavor+charm violation in these cases? As far as I understand, the CKM matrix deals with flavor oscillations only.

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The CKM matrix describes oscillations but that doesn't mean that it doesn't matter in any other process. The CKM matrix matters pretty much in every process involving quarks.

In particular, $d\bar c$ primarily contains two mass eigenstates of quarks. And they are eigenstates from different families. $d$ is the first generation of quarks, $\bar c$ is the second. So if there were no mixing between quarks, as given by the CKM matrix, $d$ and $\bar c$ simply couldn't annihilate into a $W^-$ boson. A quark and an antiquark may only annihilate if they are of the same flavor – or, in the case of the $W$-bosons, if they come from the same generation.

However, the CKM implies that $\bar c$, the mass eigenstate, contains a mixture of the $SU(2)$ partners of the antiquark (mass eigenstates) $\bar d,\bar s,\bar b$ and the part associated with $\bar d$ – mostly an $\bar u$ but not just that – is capable of annihilating with $d$ into the $W^-$ boson that may continue its decay to leptons.

This description makes it clear that the decay only exists to the extent to which the 1st and 2nd generations of quarks mix. So the amplitude will be proportional to $\theta_{12}$, some mixing angle (Cabibbo angle), and the decay rate will go as its second power.

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