Lets suppose I know the branching fraction of a decay. How can I calculate the branching fraction in a different final state with the Cabibbo angle?
For example:
$$D^+ \rightarrow \bar K^0 + e^+ + \nu_e$$
The branching fraction would be 9%. What would be the branching fraction in a $\pi^0$ final state if the corresponding Cabibbo angle is 0.2?
This is actually a nice problem, steering you to do these estimates in your mind, Fermi-style, aggressively ignoring the inessential, a crucial skill.
You realize that, compared to the mass of the mother particle, the D, ~1.87 GeV, the masses of the two mesons , K ~ 0.5 GeV and π ~ 0.14 GeV are similar, so no phase-space disparities should matter, and the ratio of the respective branching fractions should only be controlled by the squares of the respective amplitudes!
The respective amplitudes are the same c-decay diagrams, one to d, and the other to s, whose ratio goes as $$ |V_{cd}/V_{cs}|\approx \tan\theta_c\approx 0.2, $$ consequently $$ \frac{\Gamma(D\to \pi e \nu)}{\Gamma(D\to \bar K e \nu)}\sim 0.04, $$ amounting to a branching fraction for the π mode of $\sim 4\cdot 10^{-3}$, quite close to what you see in the PDG.
