Testing the importance of collective correlations in neutrinoless $\beta \beta$ decay

We investigate the extent to which theories of collective motion can capture the physics that determines the nuclear matrix elements governing neutrinoless double-$\beta$ decay. To that end we calculate the matrix elements for a series of isotopes in the full $pf$ shell, omitting no spin-orbit partners. With the inclusion of isoscalar pairing, a separable collective Hamiltonian that is derived from the shell model effective interaction reproduces the full shell-model matrix elements with good accuracy. A version of the generator coordinate method that includes the isoscalar pairing amplitude as a coordinate also reproduces the shell model results well, an encouraging result for theories of collective motion, which can include more single-particle orbitals than the shell model. We briefly examine heavier nuclei relevant for experimental double-$\beta$ decay searches, in which shell-model calculations with all spin-orbit partners are not feasible; our estimates suggest that isoscalar pairing also plays a significant role in these nuclei, though one we are less able to quantify precisely.

Figure 1

$B\left(E2\right)$ values for transitions from the first $2^{+}$ state to the ground state in Cr isotopes, as a function of mass number $A$. Experiment [24] (gray circles with error bars, connected by solid line) is compared to calculations with several interactions. Black squares (solid line) show results produced by the KB3G interaction, red squares (dashed line) by the full collective interaction $H_{\mathrm{coll}}$, and purple squares (dotted line) by $H_{\mathrm{coll}}$ without the quadrupole-quadrupole part. Protons have an effective charge of $1.5e$ and neutrons of $0.5e$.

Figure 2

Gamow-Teller part of the $0\nu \beta \beta$ decay matrix elements $M_{0\nu }^{GT}$, for the decay of Ca isotopes into Ti as a function of the neutron number $N_{\mathrm{parent}}$ in the parent nucleus. Results are shown for the KB3G interaction (black circles, solid line), the full collective interaction $H_{\mathrm{coll}}$ (red circles, dashed line), $H_{\mathrm{coll}}$ with the quadrupole-quadrupole term removed (purple squares, dotted line), $H_{\mathrm{coll}}$ with the isoscalar pairing term removed (blue squares, short-dashed line), and $H_{\mathrm{coll}}$ with both the isoscalar-pairing and spin-isospin pieces removed (orange squares, dot-dashed line).

Figure 3

Gamow-Teller part of the $0\nu \beta \beta$ decay matrix elements, $M_{0\nu }^{GT}$, for the decay of Ti isotopes into Cr (top panel), and Cr isotopes into Fe (bottom), as a function of the neutron number $N_{\mathrm{parent}}$ of the parent nucleus. Results are shown for the KB3G interaction (black, solid line), the collective interaction $H_{\mathrm{coll}}$ (red, dashed line), and $H_{\mathrm{coll}}$ without the isoscalar pairing term (blue, short-dashed line).

Figure 4

Top panel: Percentage of ground state in daughter nuclei (Ti isotopes) belonging to $\mathrm{SU}\left(4\right)$ irreducible representations (irreps) that are also present in the corresponding parent nuclei (Ca isotopes), as a function of the neutron number $N_{\mathrm{parent}}$ of the parent nucleus. Results are shown for the KB3G effective interaction (black circles, solid line), the collective Hamiltonian $H_{\mathrm{coll}}$ (red circles, dashed line), $H_{\mathrm{coll}}$ without the isoscalar pairing term (blue circles, short-dashed line), $H_{\mathrm{coll}}$ without both the isoscalar pairing and the spin-isospin terms (orange circles, dot-dashed line), and the KB3G interaction diagonalized in a basis of a seniority-zero states (purple squares, dotted line). Bottom panel: $M_{2\nu }^{GT}\left(\text{cl.}\right)$ (see text) as a function of $N_{\mathrm{parent}}$. Correspondence between results and symbols/lines is the same as in the top panel.

Figure 5

Two-neutrino $\beta \beta$ decay matrix elements $M_{2\nu }^{GT}$, for the decay of Ca isotopes into Ti as a function of the neutron number $N_{\mathrm{parent}}$ in the parent nucleus. Results are shown for the KB3G interaction (black circles, solid line), the full collective interaction $H_{\mathrm{coll}}$ (red circles, dashed line), $H_{\mathrm{coll}}$ with the isoscalar pairing term removed (blue squares, short-dashed line), and $H_{\mathrm{coll}}$ with both isoscalar-pairing and spin-isospin parts removed (orange squares, dot-dashed line).

Figure 6

Gamow-Teller part of $0\nu \beta \beta$ decay matrix elements, $M_{0\nu }^{GT}$, for the decay of Ti isotopes into Cr (top panel) and Cr isotopes into Fe (bottom panel), as a function of the neutron number $N_{\mathrm{parent}}$ in the parent nucleus. Results are shown for the shell model with the collective Hamiltonian $H_{\mathrm{coll}}$ (red, dashed line), the GCM with the same $H_{\mathrm{coll}}$ but without quadrupole-quadrupole interaction, and with the isoscalar pairing amplitude as only coordinate (blue, short-dashed line), and the GCM with the quadrupole-quadrupole interaction and with the axial quadrupole deformation parameter $\beta$ as second coordinate (purple, dotted line).

Figure 7

Estimates of the effect of isoscalar pairing on $0\nu \beta \beta$ decay for nuclei used in or considered for experiments. The Gamow-Teller matrix elements, $M_{0\nu }^{GT}$, are shown for the full shell model effective interaction (black circles), the effective interaction with all $J^{\pi }=1^{+},T=0$ two-body matrix elements set to zero (blue upside-down triangles), and the effective interaction with the isoscalar pairing interaction from $H_{\mathrm{coll}}$ subtracted (purple triangles). For ${}^{48}\mathrm{Ca}$ the result from Fig. 2, obtained from $H_{\mathrm{coll}}$ without the isoscalar pairing term, is also shown (red square).