Searching for $D^0 \to \pi^+\pi^- \nu \bar{\nu}$ decays and measuring angular asymmetries in polarised $\Lambda_c^+ \to p \mu^+\mu^-$ decays at FCC-ee

This note presents sensitivity studies for the rare charm decays $D^0 \to \pi^+\pi^- \nu \bar{\nu}$ and $\Lambda_c^+ \to p \mu^+\mu^-$ at FCC-ee, when operated at the $Z$ pole with $6 \times 10^{12}$ produced $Z$ bosons. The results indicate that an observation of the decay $D^0 \to \pi^+\pi^- \nu \bar{\nu}$  with a significance of 5(3)$\sigma$ could be possible for branching fractions at the level of $[1.8 - 2.1] \times 10^{-7}$ ($[2.9-3.4] \times 10^{-7}$). Angular asymmetries in $\Lambda_c^+ \to p \mu^+\mu^-$ could be measured in decays of polarized $\Lambda_c^+$ baryons with a precision of 0.9%. The precision expected at FCC-ee allows for the parameter space of a large class of motivated BSM models to be constrained significantly.