Neutron-proton pairing and double-$\beta$ decay in the interacting boson model

Background: The interacting boson model has been used extensively to calculate the matrix elements governing neutrinoless double-$\beta$ 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-$\beta$ 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-$\beta$-decay operator as the starting point. Energy spectra and double-$\beta$ 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-$\beta$ 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.

Figure 1

Four-nucleon spectra for (a) $T=2$, (b) $T=1$, and (c) $T=0$, corresponding to low-lying levels in the nuclei ${}^{44}\mathrm{Ca},{}^{44}\mathrm{Sc}$, and ${}^{44}\mathrm{Ti}$. The left column (dashed blue) shows all levels obtained with the bare Hamiltonian in the collective subspace constructed from $S$ and $D$ pairs. The middle column (dotted red) shows the same for $S,D$, and $P$ pairs. The right column shows the low-energy levels obtained in the shell model with the KB3G interaction; the ones that are exactly reproduced with an effective Hamiltonian in the $SD$ and $SDP$ subspaces are drawn in dash-dotted purple and those that are reproduced only in the $SDP$ subspace in dotted red.

Figure 2

Spectra of the $A=46$ nuclei (a) ${}^{46}\mathrm{Ca}$ $\left(T=3\right)$ and (b) ${}^{46}\mathrm{Ti}$ $\left(T=1\right)$. The shell-model spectra (SM) are produced by the KB3G interaction with six nucleons in the $pf$ shell. The IBM spectra, with three bosons $(sd$ or $sdp)$, are produced by a Hamiltonian interpolated between the bare ${\widehat{H}}_{\mathrm{b}}^{\mathrm{b}}$ and the effective ${\widehat{H}}_{\mathrm{e}}^{\mathrm{b}}$ (see text).

Figure 3

Spectra of the $A=48$ nuclei (a) ${}^{48}\mathrm{Ca}$ $\left(T=4\right)$, (b) ${}^{48}\mathrm{Ti}$ $\left(T=2\right)$, and (c) ${}^{48}\mathrm{Cr}$ $\left(T=0\right)$. The shell-model spectra (SM) are produced by the KB3G interaction with eight nucleons in the $pf$ shell. The IBM spectra, with four bosons $(sd$ or $sdp)$, are produced by a Hamiltonian interpolated between the bare ${\widehat{H}}_{\mathrm{b}}^{\mathrm{b}}$ and the effective ${\widehat{H}}_{\mathrm{e}}^{\mathrm{b}}$ (see text).

Figure 4

Spectra of the $A=50$ nuclei (a) ${}^{50}\mathrm{Ti}$ $\left(T=3\right)$ and (b) ${}^{50}\mathrm{Cr}$ $\left(T=1\right)$. The shell-model spectra (SM) are produced by the KB3G interaction with ten nucleons in the $pf$ shell. The IBM spectra, with five bosons $(sd$ or $sdp)$, are produced by a Hamiltonian interpolated between the bare ${\widehat{H}}_{\mathrm{b}}^{\mathrm{b}}$ and the effective ${\widehat{H}}_{\mathrm{e}}^{\mathrm{b}}$ (see text).

Figure 5

Eight $0\nu \beta \beta 0_1^{+}\to 0_1^{+}$ matrix elements between $f_{7/2}$-shell nuclei, calculated in the shell model [19] and in the (a) IBM and the (b) $p$-IBM. The full black lines connect the shell-model results and the dashed blue (dotted red) lines the IBM $(p$-IBM) results with an $x$ parameter fit to energy spectra (see text). The shaded areas indicate upper and lower limits defined by $x=1$ (bare operators) and $x=0$ (effective operators).

Figure 6

Same as Fig. 5 for the Gamow-Teller part of eight $0\nu \beta \beta 0_1^{+}\to 0_1^{+}$ matrix elements between $f_{7/2}$-shell nuclei.

Figure 7

Same as Fig. 5 for the collective Hamiltonian without isoscalar pairing in the (a) IBM, without isoscalar pairing in the (b) $p$-IBM with isoscalar pairing in the (c) IBM, and with isoscalar pairing in the (d) $p$-IBM.

Figure 8

Contributions to the Gamow-Teller matrix element $M_{0\nu }^{GT}$ from different terms in the boson Hamiltonian (see text) for the IBM (cross-hatched) and $p$-IBM (solid).