An investigation of the Tucson-Melbourne three-nucleon force in the nuclear many-body problem

Abstract

The no-core shell-model approach has proven to be extremely useful for the theoretical determination of the properties of light (A ≤ 12) nuclei. However, at present this method does not accommodate a three-nucleon interaction into the potential that it employs. The problem is introduced with a largely historical development of both the effective interaction formalism and three-nucleon interactions, placing the motivation in context. This work makes a first attempt to incorporate such a three-nucleon interaction into the no-core shell-model ansatz. To this end, a variant of the two-pion-exchange Tucson-Melbourne three-nucleon interaction has been chosen. A three-body translationally-invariant harmonic-oscillator basis is constructed, and matrix elements of the three-nucleon interaction in this basis are calculated. The majority of this is accomplished through standard angular-momentum algebraic techniques, with the most expensive component being the spatial one, as it requires a transformation of the basis set with computationally intensive transformation brackets. Given the ability to determine the matrix elements for the chosen Tucson-Melbourne force, the practicality of employing these in calculations is demonstrated, with calculations on the three-body nuclei ³H and ³He. These are simple calculations, where the Tucson-Melbourne matrix elements are added to those of the two-body effective potential (a slight inconsistency which future studies will aim to fix). The dependence of binding energies on the harmonic-oscillator parameter, hΩ, and the Tucson-Melbourne cutoff parameter, Λ are examined. The former is found to be small in the range of hΩ considered, while the latter is shown to be consistent with previous works that have explored this dependence using other methods. The convergence of the binding energy with increasing model space size is slow, but this is perhaps attributable to the unrenormalized nature of the three-body matrix elements. The ultimate aim of this research is to find a viable method for constructing a three-body effective interaction from a given "realistic" three nucleon interaction, for use in no-core shell-model calculations. The current work demonstrates that such a scheme is feasible, and should yield results more consistent with experiment. Such a three-body effective interaction should also achieve quicker convergence with model space size than shown here, as the three-nucleon matrix elements will be renormalized to account for the geometry of the model space. Thus, one will have constructed an ab initio method for calculations on light nuclei, that includes a three-nucleon interaction, and converges quickly in the determination of nuclear properties.