Neutron-proton pairing and double-<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>β</mml:mi></mml:math> decay in the interacting boson model

Abstract

Background: The interacting boson model has been used extensively to calculate the matrix elements governing neutrinoless double-$\ensuremath{\beta}$ decay. Studies within other models---the shell model, the quasiparticle random-phase approximation, and nuclear energy-density functional theory---indicate that a good description of neutron-proton pairing is essential for accurate calculations of those matrix elements, even though the isotopes used in experiment have significantly more neutrons than protons. The usual interacting boson model is based only on like-particle pairs, however, and the extent to which it captures neutron-proton pairing is not clear.Purpose: To determine whether neutron-proton pairing should be explicitly included as neutron-proton bosons in interacting-boson-model calculations of neutrinoless double-$\ensuremath{\beta}$ decay matrix elements. In this paper we restrict ourselves to nuclei in the lower half of the $pf$ shell, where exact shell model calculations are possible.Method: An isospin-invariant version of the nucleon-pair shell model is applied to carry out shell-model calculations in a large space and in a collective subspace, and to define effective operators in the latter. A democratic mapping is then used to define corresponding boson operators for the interacting boson model, with and without an isoscalar neutron-proton pair boson.Results: Interacting-boson-model calculations with and without the isoscalar boson are carried out for nuclei near the beginning of the $pf$ shell, with a realistic shell-model Hamiltonian and neutrinoless double-$\ensuremath{\beta}$-decay operator as the starting point. Energy spectra and double-$\ensuremath{\beta}$ matrix elements are compared to those obtained in the underlying shell model.Conclusions: The isoscalar boson is not important for energy spectra but improves the results for the double-$\ensuremath{\beta}$ matrix elements. To be useful at the level of precision we need, the mapping procedure must be further developed to better determine the dependence of the boson Hamiltonian and decay operator on particle number and isospin, and extended to heavier nuclei. The benefits provided by the isoscalar boson in the nuclei examined here, however, suggest that through an appropriate combination of mappings and fitting, it would make interacting-boson-model matrix elements more accurate in the heavier nuclei used in experiments.