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 parametrizations 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 colors. Furthermore, we use the measured values of the branching ratios to help in selecting the nonperturbative 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.

• Received 6 June 2023
• Accepted 14 July 2023

DOI:https://doi.org/10.1103/PhysRevD.108.036026

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Published by the American Physical Society