Coloured Scalars Mediated Rare Charm Meson Decays to Invisible Fermions

We consider effects of coloured scalar mediators in decays $c\to u \, {\it invisibles}$. In particular, in these processes, as invisibles, we consider massive right-handed fermions. The coloured scalar $\bar S_1\equiv (\bar 3, 1, -2/3)$, due to its coupling to weak singlets up-quarks and invisible right-handed fermions ($\chi$), is particularly interesting. Then, we consider $\tilde R_2 \equiv (\bar 3, 2, 1/6)$, which as a weak doublet is a subject of severe low-energy constraints. The $\chi$ mass is considered in the range $(m_K - m_\pi)/2\leq m_\chi \leq (m_D - m_\pi)/2$. We determine branching ratios for $D\to \chi \bar \chi$, $D\to \chi \bar \chi \gamma$ and $D\to \pi \chi \chi$ for several $\chi$ masses, using most constraining bounds. For $\bar S_1$, the most constraining is $D^0 -\bar D^0$ mixing, while in the case of $\tilde R_2$ the strongest constraint comes from $B\to K {\it missing\, energy}$ . We find in decays mediated by $\bar S_1$ that branching ratios can be $\mathcal B(D\to \chi \bar \chi)< 10^{-8}$ for $m_\chi=0.8$ GeV, $\mathcal B(D\to \chi \bar \chi \gamma) \sim 10^{-8}$ for $m_\chi=0.18$ GeV, while $\mathcal B(D^+ \to \pi^+ \chi \bar \chi )$ can reach $ \sim 10^{-8}$ for $m_\chi=0.18$ GeV. In the case of $\tilde R_2$ these decay rates are very suppressed. We find that future tau-charm factories and Belle II experiments offer good opportunities to search for such processes. Both $\bar S_1$ and $\tilde R_2$ might have masses within LHC reach.