= Solution
The <tree-level Feynman diagrams> contain an unchanged <spectator quark>, of <up quark> flavour and the two possible <weak charged current> transitions of the <charm antiquark>:
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-305-q3c.png]
{title=Favoured and doubly Cabibbo-suppressed anti-D decays}
For $\bar D^0\to K^+\pi^-$, $\bar c\to\bar sW^-$ and $W^-\to d\bar u$. The <spectator quark> combines with $\bar s$ into the <kaon> $K^+=u\bar s$, while $d\bar u$ forms the <pion> $\pi^-$. The <Cabibbo-Kobayashi-Maskawa matrix> factor is $V_{cs}V_{ud}^*$.
For $\bar D^0\to K^-\pi^+$, $\bar c\to\bar dW^-$ and $W^-\to s\bar u$. The <spectator quark> combines with $\bar d$ into the <pion> $\pi^+=u\bar d$, while $s\bar u$ forms the <kaon> $K^-$. The <Cabibbo-Kobayashi-Maskawa matrix> factor is $V_{cd}V_{us}^*$.
Neglecting <neutral D-meson mixing> and assuming comparable <strong interaction> <matrix elements>, the relative direct <decay widths> are
$$
\boxed{\frac{\Gamma(\bar D^0\to K^-\pi^+)}{\Gamma(\bar D^0\to K^+\pi^-)}
\approx\left|\frac{V_{cd}V_{us}^*}{V_{cs}V_{ud}^*}\right|^2\approx\tan^4\theta_C.}
$$
Here $\theta_C$ is the <Cabibbo angle>. The second process is \b[doubly <Cabibbo suppressed>]: it contains two small <Cabibbo suppression> factors in its amplitude. The <Cabibbo angle> estimate assumes similar <Quantum chromodynamics> <matrix elements>; it is not an exact rate equality.
Back to article page
