  Urgent theoretical progress is needed in order to provide an estimate in the
Standard Model of the recent measurement by LHCb of direct CP violation in
charm-meson two-body decays. Rescattering effects must be taken into account
for a meaningful theoretical description of the amplitudes involved in such
category of observables, as signaled by the presence of large strong phases. We
discuss the computation of the latter effects based on a two-channel coupled
dispersion relation, which exploits isospin-zero phase-shifts and inelasticity
parameterizations of data coming from the rescattering processes $ \pi \pi \to
\pi \pi $, $ \pi K \to \pi K $, and $ \pi \pi \to K \overline{K} $. The
determination of the subtraction constants of the dispersive integrals relies
on the leading contributions to the transition amplitudes from the $ 1/N_C $
counting, where $N_C$ is the number of QCD colours. Furthermore, we use the
measured values of the branching ratios to help in selecting the
non-perturbative inputs in the isospin limit, from which we predict values for
the CP asymmetries. We find that the predicted level of CP violation is much
below the experimental value.
