Abstract
Normal-ordering provides an approach to approximate three-body forces as effective two-body operators and it is therefore an important tool in many-body calculations with realistic nuclear interactions. The corresponding neglect of certain three-body terms in the normal-ordered Hamiltonian is known to influence translational invariance, although the magnitude of this effect has not yet been systematically quantified. In this work we study in particular the normal-ordering two-body approximation applied to a single harmonic-oscillator reference state. We explicate the breaking of translational invariance and demonstrate the magnitude of the approximation error as a function of model space parameters for $^4\rm{He}$ and $^{16}\rm{O}$ by performing full no-core shell-model calculations with and without three-nucleon forces. We combine two different diagnostics to better monitor the breaking of translational invariance. While the center-of-mass effect is shown to become potentially very large for $^4\rm{He}$, it is also shown to be much smaller for $^{16}\rm{O}$ although full convergence is not reached. These tools can be easily implemented in studies using other many-body frameworks and bases.
Highlights
The need for an effective three-nucleon force (3NF) to describe the strong nuclear interaction in atomic nuclei is well established [1]
Besides the effective field theories (EFTs) arguments, it has been shown that several experimental findings are difficult to reproduce without the inclusion of a 3NF, such as certain three-nucleon scattering observables [7], the A = 3, 4 binding energies [8,9], and selected light nucleus spectroscopy [10,11,12]
The importance of residual 3NFs was shown to be small by explicit comparison with calculations using full 3NFs. These benchmarks were performed at a fixed oscillator frequency and the dependence on model-space parameters has not been investigated. This is important since we show that the sensitivity of SR-NO2B to the choice of basis frequency could be significant
Summary
The need for an effective three-nucleon force (3NF) to describe the strong nuclear interaction in atomic nuclei is well established [1]. Approximation schemes that will include the most important physics of 3NFs, but at a lower computational cost One such approximation scheme is the single-reference normal-ordering two-body (SR-NO2B) approximation [14,15], which potentially can incorporate the dominant piece of the 3NF as an effective two-nucleon force (2NF) and at significantly reduced computational cost. Such designs exist for reference states with broken particle-number symmetry [30], and rotational symmetry [31].
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