We study decays of $D^0$, $D^{+}$, and $D_s^{+}$ mesons into two pseudoscalar mesons by expressing the decay amplitudes in terms of topological amplitudes. Including consistently $SU(3{)}_F$ breaking to linear order, we show how the topological-amplitude decomposition can be mapped onto the standard expansion using reduced amplitudes characterized by $SU(3)$ representations. The tree and annihilation amplitudes can be calculated in factorization up to corrections which are quadratic in the color-counting parameter $1/N_c$. We find new sum rules connecting $D^{+}\to K_S{K}^{+}$, $D_s^{+}\to K_S{\pi }^{+}$, and $D^{+}\to K^{+}{\pi }^0$, which test the quality of the $1/N_c$ expansion. Subsequently, we determine the topological amplitudes in a global fit to the data, taking the statistical correlations among the various measurements into account. We carry out likelihood ratio tests in order to quantify the role of specific topological contributions. While the $SU(3{)}_F$ limit is excluded with a significance of more than 5 standard deviations, a good fit (with $\mathrm{\Delta }{\chi }^2<1$) can be obtained with less than 28% of $SU(3{)}_F$ breaking in the decay amplitudes. The magnitude of the penguin amplitude $P_{\text{break}}$, which probes the Glashow-Iliopoulos-Maiani mechanism, is consistent with zero; the hypothesis $P_{\text{break}}=0$ is rejected with a significance of just $0.7\sigma$. We obtain the Standard-Model correlation between $\mathcal{B}(D^0\to K_L{\pi }^0)$ and $\mathcal{B}(D^0\to K_S{\pi }^0)$, which probes doubly Cabibbo-suppressed amplitudes, and find that $\mathcal{B}(D^0\to K_L{\pi }^0)<\mathcal{B}(D^0\to K_S{\pi }^0)$ holds with a significance of more than $4\sigma$. We finally predict $\mathcal{B}(D_s^{+}\to K_L{K}^{+})=0.012_{-0.002}^{+0.006}$ at $3\sigma$ C.L.

• Received 31 March 2015

DOI:https://doi.org/10.1103/PhysRevD.92.014004