Topological amplitudes in $D$ decays to two pseudoscalars: a global analysis with linear $SU(3)_F$ breaking

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 $D^+\rightarrow K_SK^+$, $D_s^+\rightarrow K_S\pi^+$ and $D^+\rightarrow K^+\pi^0$, 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 five standard deviations, a good fit (with $\Delta \chi^2 <1$) can be obtained with less than $28\%$ of SU(3)$_F$ breaking in the decay amplitudes. The magnitude of the penguin amplitude $P_{\mathrm{break}}$, which probes the Glashow Iliopoulos Maiani (GIM) mechanism, is consistent with zero; the hypothesis $P_{\mathrm{break}}=0$ is rejected with a significance of just $0.7\sigma$. We obtain the Standard-Model correlation between $\mathcal{B}(D^0\rightarrow K_L \pi^0)$ and $\mathcal{B}(D^0\rightarrow K_S\pi^0)$, which probes doubly Cabibbo-suppressed amplitudes, and find that $\mathcal{B}(D^0\rightarrow K_L \pi^0)<\mathcal{B}(D^0\rightarrow K_S\pi^0)$ holds with a significance of more than $4\sigma$. We finally predict $\mathcal{B}(D_s^+\rightarrow K_L K^+)= 0.012^{+0.006}_{-0.002}$ at $3\sigma$ CL.