  Based on the leading-order covariant pionless effective field theory, a
relativistic nuclear Hamiltonian is derived and solved using the variational
Monte Carlo approach for $A\le 4$ nuclei by representing the nuclear many-body
wave functions with a symmetry-based artificial neural network. It is found
that the relativistic effects rescue the renormalizability of the theory, and
overcome the energy collapse problem for $^3$H and $^4$He without promoting a
repulsive three-nucleon interaction to leading order as in nonrelativistic
calculations. Nevertheless, to exactly reproduce the experimental ground-state
energies, a three-nucleon interaction is needed and its interplay with the
relativistic effects plays a crucial role. The strongly repulsive relativistic
effects suppress the energy contribution given by the three-nucleon
interactions, so a strong strength for the three-nucleon interaction could be
required to reproduce the experimental energies. These results shed light on a
consistent understanding of relativistic effects and three-body interactions in
atomic nuclei.
