= Ds nonleptonic tree amplitudes
{c}
For <Ds meson> spectator decays, $c\to sW^+$ followed by $W^+\to u\bar d$ carries CKM magnitude $|V_{cs}V_{ud}|$ and can form either $\phi\pi^+$ or $K^+\bar K^0$. Replacing $c\to s$ by $c\to d$ carries $|V_{cd}V_{ud}|$ and can form $K^+\pi^0$ through the $-d\bar d/\sqrt2$ pion component. With otherwise equal reduced hadronic strengths, the last channel's rate is suppressed relative to the neutral-antikaon channel by $|V_{cd}/V_{cs}|^2/2$. Actual <branching fractions> also depend on phase space, colour contractions and strong final-state interactions; the CKM factors do not determine exact exclusive rates.
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