Mean-field approximation on steroids: exact description of the deuteron

Abstract

The present article demonstrates that the deuteron, i.e. the lightest bound nuclear system made of a single proton and a single neutron, can be accurately described within a mean-field-based framework. Although paradoxical at first glance, the deuteron ground-state binding energy, magnetic dipole moment, electric quadrupole moment and root-mean-square proton radius are indeed reproduced with sub-percent accuracy via a low-dimensional linear combination of non-orthogonal Bogoliubov states, i.e. with a method whose numerical cost scales as $n_{\text{dim}}^4$, where $n_{\text{dim}}$ is the dimension of the basis of the one-body Hilbert space. By further putting the system into a harmonic trap, the neutron-proton scattering length and effective range in the ${}^{3}S_1$ channel are also accurately reproduced. To achieve this task, (i) the inclusion of proton-neutron pairing through the mixing of proton and neutron single-particle states in the Bogoliubov transformation and (ii) the restoration of proton and neutron numbers before variation are shown to be mandatory ingredients. This unexpected result has implications regarding the most efficient way to capture necessary correlations as a function of nuclear mass and regarding the possibility to ensure order-by-order renormalizability of many-body calculations based on chiral or pionless effective field theories beyond light nuclei. In this context, the present study will be extended to $^{3}$H and $^{3,4}$He in the near future as well as to the leading order of pionless effective field theory.