Pinning down $|\Delta c|=|\Delta u|=1$ couplings with rare charm baryon decays

We study the full angular distribution of semileptonic rare charm baryon decays in which the secondary baryon undergoes weak decay with sizable polarization parameter, $\Xi_c^+\to\Sigma^+\,(\to p\pi^0)\ell^+\ell^-$, $\Xi_c^0\to\Lambda^0\,(\to p \pi^-)\ell^+\ell^-$ and $\Omega_c^0\to\Xi^0\,(\to \Lambda^{0} \pi^0)\ell^+\ell^-$. Such self-analyzing decay chains allow for seven additional observables compared to three-body decays such as $\Lambda_c \to p \ell^+ \ell^-$, with different sensitivities to the $|\Delta c|=|\Delta u|=1$ weak couplings. Opportunities to test the standard model in $c \to u$ transitions with standard model null tests and other angular observables are worked out. We show that a joint model-independent analysis of the leptonic $A_{\text{FB}}^\ell$, hadronic $A_{\text{FB}}^{\text{H}}$, and combined forward-backward asymmetry $A_{\text{FB}}^{\ell\text{H}}$ together with the fraction of longitudinally produced leptons, $F_L$, is able to pin down the dipole couplings $C_{7},C^\prime_{7}$ and the semileptonic (axial-) vector ones $C_{10},C^\prime_{9},C^\prime_{10}$. $A_{\text{FB}}^{\text{H}}$ is also accessible with dineutrino $c \to u \nu \bar \nu$ modes and probes right-handed currents.