Two body non-leptonic $D^0$ decays from LCSR and implications for $\Delta a^{\rm dir}_{\rm CP}$

Motivated by the recent measurements of CP violating effects in singly Cabibbo suppressed $D^0$ decays, we revisit the theoretical predictions of these channels. Using up-to-date values for the decay constants and form factors, we find already within naive QCD factorisation surprisingly good agreement between the central values of the branching ratios and the corresponding experimental data. We further extend the study of these modes by employing the method of light-cone sum rules (LCSR) with light-meson light-cone distribution amplitudes. Using for the first time this framework to compute the leading contribution to the decay amplitude, we can again describe well the experimental branching ratios for the modes $D^0 \to \pi^+ K^-$, $D^0 \to K^+ K^-$, $D^0 \to \pi^+ \pi^-$ and $D^0 \to K^+ \pi^-$. The combination of our results with known predictions for the penguin contributions, obtained with LCSR, leads to an upper bound for the value of direct CP violation expected in the Standard Model of $|\Delta a_{\rm CP}^{\rm dir}| \leq {2.4} \times 10^{-4} \,,$ which is approximately a factor six smaller than the current measurement.