Proton-neutron pairing amplitude as a generator coordinate for double-$\beta$ decay

We treat proton-neutron pairing amplitudes, in addition to the nuclear deformation, as generator coordinates in a calculation of the neutrinoless double-$\beta$ decay of ${}^{76}\mathrm{Ge}$. We work in two oscillator shells, with a Hamiltonian that includes separable terms in the quadrupole, spin-isospin, and pairing (isovector and isoscalar) channels. Our approach allows larger single-particle spaces than the shell model and includes the important physics of the proton-neutron quasiparticle random-phase approximation (QRPA) without instabilities near phase transitions. After comparing the results of a simplified calculation that neglects deformation with those of the QRPA, we present a more realistic calculation with both deformation and proton-neutron pairing amplitudes as generator coordinates. The future should see proton-neutron coordinates used together with energy-density functionals.

Figure 1

Upper panels: Potential energy surfaces for ${\mathrm{SkO}}^{\prime }$ (dotted line) and our corresponding model interaction (solid line) at $\varphi =0$ with no spin-isospin or isoscalar pairing interactions (together denoted by “$pn$” in the lower panels) as functions of $\beta$ for ${}^{76}\mathrm{Ge}$ (left) and ${}^{76}\mathrm{Se}$ (right). Lower panels: Potential energy surface from the model without projection and pn at $\varphi =0$ (solid [red] line, same line as in upper figure), with projection but still without $pn$ at $\varphi =0$ (dotted [blue] line), and with both projection and $pn$ at $\varphi =3$ (dashed black line). Energy is measured from the unprojected value at $\beta =0$.

Figure 2

Dependence of the GCM (solid) and QRPA (dashed) $0\nu \beta \beta$ matrix elements on the strength $g^{T=0}$ of the isoscalar pairing interaction. The red (upper) and blue (lower) lines of each type correspond to the interaction parameters extracted from ${\mathrm{SkO}}^{\prime }$ and SkM*. The divergence in the QRPA near $g^{T=0}/{\overline{g}}^{T=1}=1.5$ is discussed in the text.

Figure 3

Bottom right: ${\mathcal{N}}_{{\varphi }_I}{\mathcal{N}}_{{\varphi }_F}\langle {\varphi }_F|{\mathcal{P}}_F{\widehat{M}}_{0\nu }{\mathcal{P}}_I|{\varphi }_I\rangle$ for projected quasiparticle vacua with different values of the initial and final isoscalar pairing amplitudes ${\varphi }_I$ and ${\varphi }_F$, from the ${\mathrm{SkO}}^{\prime }$-based interaction (see text). Top and bottom left: Square of collective wave functions in ${}^{76}\mathrm{Ge}$ and ${}^{76}\mathrm{Se}$.

Figure 4

Square of the collective wave functions in the calculation that includes deformation, in ${}^{76}\mathrm{Ge}$ (left) and ${}^{76}\mathrm{Se}$ (right), for the ${\mathrm{SkO}}^{\prime }$-based interaction.