Rare charm $\boldsymbol{c\to u\,\nu\bar{\nu}}$ dineutrino null tests for $\boldsymbol{e^+e^-}$-machines

Rare $|\Delta c|=|\Delta u|=1$ transitions into dineutrinos are strongly GIM-suppressed and constitute excellent null tests of the standard model. While branching ratios of $D \to P \,\nu \bar \nu$, $D \to P^+ P^- \, \nu \bar \nu$, $P=\pi, K$, baryonic $\Lambda_c^+ \to p \,\nu \bar \nu$, and $\Xi_c^+ \to \Sigma^+ \, \nu \bar \nu$ and inclusive $D \to X \nu \bar \nu$ decays are experimentally unconstrained, signals of new physics can be just around the corner. We provide model-independent upper limits on branching ratios reaching few $\times 10^{-5}$ in the most general case of arbitrary lepton flavor structure, $\sim 10^{-5}$ for scenarios with charged lepton conservation and few $\times 10^{-6}$ assuming lepton universality. We also give upper limits in $Z^\prime$ and leptoquark models. The presence of light right-handed neutrinos can affect these limits, a possibility that can occur for lepton number violation at a TeV, and that can be excluded with an improved bound on $\mathcal{B}(D^0 \to \text{invisibles})$ at the level of $\sim 10^{-6}$, about two orders of magnitude better than the present one. Signatures of $c \to u \nu \bar \nu$ modes contain missing energy and are suited for experimental searches at $e^+ e^-$-facilities, such as BES III, Belle II and future $e^+ e^-$-colliders, such as the FCC-ee running at the $Z$.