Mass-Energy Equivalence in Bound Three-Nucleon Systems

The mass defect formula reflects the equivalence of mass and energy for bound nuclear systems. We study three-nucleon systems $^3$H and $^3$He, considering the neutron and proton as indistinguishable particles ($AAA$ model) or taking into account the real masses of neutrons and protons ($AAB$ model). We have focused on conceptual problems of the $AAA$ model, which is widely used for $3N$ calculations. In particular, the $AAA$ model is incompatible with the mass defect formula, which naturally corresponds to the $AAB$ model. In addition, the $AAA$ model has a cyclic permutation symmetry, which is breaking in the natural $AAB$ model. The latter problem cannot be eliminated within the perturbative $AAA$ approach, in which the mass difference effect is simulated by correcting the kinetic energy operator. Earlier it was reported that the accuracy of such $AAA$ calculations is 1~keV. An example of the $AAB$ calculation, we numerically estimate the effect of the difference between the neutron and proton masses on the energy calculated without any approximation with the accuracy of 0.1~keV. Another manifestation of the equivalence of mass $m$ and energy $E$ can be expressed by the formula $dE/dm=Const$. To show this dependence of the three-body energy on the nucleon mass, we performed realistic calculations within the $AAA$ approximation, varying the averaged nucleon mass. The mass-energy compensation effect for the three-body Hamiltonian is shown. According to this, we have determined the effective nucleon mass required to compensate for the perturbative effect of a three-body potential.