Rare semileptonic four-body charm decays such as $D^0\to h^-h^+e^-e^+$ (with $h=\pi,K$) can proceed via long-distance (LD) resonant contributions or short-distance (SD) Flavor Changing Neutral Currents (FCNC) $c\to ue^-e^+$, which are highly suppressed in the Standard Model. These modes are therefore sensitive to potential new physics effects and provide a probe of Lepton Flavour Universality through a comparison with their muonic counterparts $D^0\to h^-h^+\mu^-\mu^+$.

The LHCb experiment has searched for the $D^0\to \pi^-\pi^+e^-e^+$ and $D^0\to K^-K^+e^-e^+$ decays using Run 2 data, corresponding to an integrated luminosity of 6 fb$^{-1}$ collected at $\sqrt{s}=13$ TeV. The decay $D^0\to \pi^-\pi^+e^-e^+$ was observed for the first time when the dielectron pair is compatible with coming from $\rho(770)$, $\omega(782)$ or $\phi(1020)$ intermediate resonances. The decay $D^0\to K^-\pi^+[e^-e^+]_{\rho/\omega}$ was used as normalization channel. To account for the very different background levels and invariant-mass shapes, the dataset was split into events with and without reconstructed bremsstrahlung emission associated with the electrons.

The measured branching fractions are
$\mathcal{B}(D^0\to\pi^-\pi^+[e^-e^+]_{\rho/\omega}) = (4.5\pm1.0\pm0.7\pm0.6)\times10^{-7}$ and
$\mathcal{B}(D^0\to\pi^-\pi^+[e^-e^+]_{\phi}) = (3.8\pm0.7\pm0.4\pm0.5)\times10^{-7}$, where the uncertainties are statistical, systematic and related to the normalization Branching Fraction value.

With the enhanced LHCb trigger capabilities and large collected statistics in Run 3, updated searches and branching-fraction measurements are underway. Preliminary studies on the normalization channel indicate a significant increase in signal yields.