Constraints on two-neutron separation energy in the Borromean ${}^{22}$C nucleus

The recently extracted matter radius of carbon isotope ${}^{22}$C allows us to estimate the mean-square distance of a halo neutron with respect to the center-of-mass of this nucleus. By considering this information, we suggest an energy region for an experimental investigation of the unbound ${}^{21}$C virtual state. Our analysis, in a renormalized zero-ranged three-body model, also indicates that the two-neutron separation energy in ${}^{22}$C is expected to be found below $\sim$0.4~MeV, where the ${}^{22}$C is approximated by a Borromean configuration with a pointlike ${}^{20}$C and two $s$-wave halo neutrons. A virtual-state energy of ${}^{21}$C close to zero, would make the ${}^{22}$C, within Borromean nuclei configurations, the most promising candidate to present an excited bound Efimov state or a continuum three-body resonance.