Past exam of the particle physics course of the University of Oxford / 2022 / 2 / a / Solution

The favoured weak charged current transition is $\bar{c}\to \bar{s}{W}^{-}$ followed by $W^{-}\to d\bar{u}$. The spectator quark $u$ combines with $\bar{u}$ to form the $u\bar{u}/\sqrt{2}$ component of the neutral pion, while $d\bar{s}$ forms $K^0$. The product of CKM matrix factors is $V_{cs}{V}_{ud}^{\ast }$. The doubly Cabibbo suppressed transition is $\bar{c}\to \bar{d}{W}^{-}$ followed by $W^{-}\to s\bar{u}$, with factors $V_{cd}{V}_{us}^{\ast }$; the final $s\bar{d}$ is the neutral antikaon. Both are colour-suppressed internal-emission Feynman diagrams, with the same spectator routing.

In the two-generation approximation to the CKM matrix, $V_{ud}=V_{cs}=\cos {\theta }_C$, $V_{us}=\sin {\theta }_C$ and $V_{cd}=-\sin {\theta }_C$. For equal magnitudes of the hadronic matrix elements,The rate ratio is about $2.8\times 10^{-3}$ with the supplied CKM matrix magnitudes. The minus sign of a particular product of CKM matrix elements must be combined with the phases of the hadronic matrix elements and kaon basis states; it is not by itself the observable interference sign.