$D\to P \ell^+\ell^-$ decays assisted by QCD light-cone sum rules

We suggest a new method to analyse the rare $D_{(s)}\to P \ell^+\ell^-$ decays ($P=\pi,K$), combining QCD light-cone sum rules (LCSR) with hadronic dispersion relations. As our main study case, we consider the $D^+\to\pi^+\ell^+\ell^-$ mode which attracts much of interest from the point of view of GIM cancellation and potential new sources of the FCNC $c\to u$ transitions. The hadronic amplitude of this decay is dominated by the combination of weak annihilation with the emission of a virtual photon. This $D^+\to \pi^+\gamma^*$ amplitude is calculated at spacelike photon virtualities, $-q^2 \gg \Lambda_{QCD}^2$, using LCSR with pion distribution amplitudes. The LCSR results are then fitted to the hadronic dispersion relation in the variable $q^2$. The fit allows us to determine relative phases of the $\rho$-,$\omega$- and $\phi$-meson terms and to estimate the contributions of heavier hadronic states. The latter are parameterized in two different ways: first, with a $z$-expansion and second, using a model with excited vector-meson resonances. The resulting $D^+\to\pi^+\ell^+\ell^-$ width obtained from the LCSR-assisted dispersion relation is not much smaller than the current upper bound measured by LHCb collaboration. In comparison with our result for the dominant $D^+\to \pi^+\gamma^*$ transition, the contribution induced by the short distance $c\to u \ell^+\ell^-$ quark transition generated by the effective $O_9$ operator in Standard Model turns out to be practically invisible. To detect its effect, one readily needs a enhancement of the coefficient $C_9$ from non-standard effects by many orders of magnitude. We also establish new $U$-spin symmetry relations between the amplitudes of $D\to P \ell^+\ell^-$ decays and apply our method to the Cabibbo-favoured modes $D^+_s\to\pi^+\ell^+\ell^-$ and $D^0\to\bar{K}^0\ell^+\ell^-$ which proceed via the same weak annihilation mechanism.