We investigated a non-trivial fixed-point potential originating from the neutron–deuteron contact interaction using a basic form known as the Skornyakov–Ter-Martyrosian (S–TM) equation. The Wilsonian renormalisation group (RG) equation was employed to derive a non-trivial fixed-point potential of the short-range interaction in either the presence or the absence of the three-body force. In the meantime, we compared our computations to the uniqueness of the non-trivial fixed-point potential that describes nucleon–deuteron scattering with a bound state at zero energy. Then, at low energy, we introduced an effective potential that comes from a perturbative connection of the kernel of the S–TM equation to the relevant fixed-point potential. We chose the best sets of local regulators to clarify the efficiency of the perturbative framework within the development of the neutron–deuteron contact interaction. Our results showed that the main sharp momentum cut-offs in phase shift were removed, especially when the two-body scattering length was physically larger. Also, this development reduces the phase shift’s dependence on different schemes of regulators
