Neutron$-^{19}$C scattering: emergence of universal properties in a finite range potential

The low-energy properties of the elastic $s-$wave scattering for the $n-^{19}$C are studied near the critical condition for the occurrence of an excited Efimov state in $n-n-^{18}$C. It is established to which extent the universal scaling laws, strictly valid in the zero-range limit, survive when finite range potentials are considered. By fixing the two-neutrons separation energy in $^{20}$C with available experimental data, it is studied the scaling of the real ($\delta_0^R$) and imaginary parts of the $s-$wave phase-shift with the variation of the $n-^{18}$C binding energy. We obtain some universal characteristics given by the pole-position of $k\cot(\delta_0^R)$ and effective-range parameters. By increasing the $n-^{18}$C binding energy, it was verified that the excited state of $^{20}$C goes to a virtual state, resembling the neutron-deuteron behavior in the triton. It is confirmed that the analytical structure of the unitary cut is not affected by the range of the potential or mass asymmetry of the three-body system.