Ab initio studies of the double–Gamow-Teller transition and its correlation with neutrinoless double-$\beta$ decay

We use chiral interactions and several ab initio methods to compute the nuclear matrix elements (NMEs) for ground-state-to-ground-state double Gamow-Teller transitions in a range of isotopes and explore the correlation of these NMEs with those for neutrinoless double beta decay produced by the exchange of a light Majorana neutrino. When all the NMEs of both isospin-conserving and isospin-changing transitions from the ab initio calculations are considered, the correlation is strong. For the experimentally relevant isospin-changing transitions by themselves, however, the correlation is weaker and may not be helpful for reducing the uncertainty in the NMEs for neutrinoless double beta decay.

Figure 1

The (a) Fermi and [(b) and (c)] GT-type neutrino potentials (4) as a function of $r_{12}$. The radius parameter $R_A$ is excluded in the neutrino potentials to facilitate comparison with the Coulomb-like potential $1/r_{12}$. Different choices of cutoff values ${\mathrm{\Lambda }}_V$ and ${\mathrm{\Lambda }}_A$ are employed in the dipole form factors. In (c), a different value of the average excitation energy $E_d$ is used. All the Fermi-type neutrino potentials are multiplied with a minus sign to facilitate comparison. In panel (b), the potential $h_{\mathrm{GT},0}^{AA}\left({\mathit{q}}^2\right)=g_A^2\left({\mathit{q}}^2\right)$ is compared to that of the full GT operator (5b) with the cutoff value change to ${\mathrm{\Lambda }}_A=1.09$ GeV.

Figure 2

The function $\varphi \left(x\right)$ defined in Eq. (8) with the limits of $\varphi \left(0\right)=1$ and $\varphi \left(\infty \right)=0$. The shaded area indicates the interval where the average excitation energy takes $E_d\in [7.76,13.71]$ MeV and the relative distance between the two decaying nucleons $r_{12}=1.0$ fm.

Figure 3

The transition densities ${\tilde{C}}^{\alpha }\left(r_{12}\right)$ for the isospin-conserving transitions of DGT [(a)–(c)] and GT-$0\nu \beta \beta$ decay [(d)–(f)] from three ab initio calculations for ${}^6\mathrm{He}$, ${}^{10}\mathrm{Be}$, and ${}^{14}\mathrm{C}$ using the EM1.8/2.0 chiral force. The tilde means that the transition densities are normalized to unity at the first peak position.

Figure 4

Same as Fig. 3 but for isospin-changing transitions of ${}^8\mathrm{He}$, ${}^{22}\mathrm{O}$, and ${}^{48}\mathrm{Ca}$.

Figure 5

Correlation between the NMEs of DGT transitions ($M^{\mathrm{DGT}}$) with those of GT-$0\nu \beta \beta$ decay ($M_{\mathrm{GT}}^{0\nu \beta \beta }$) or the full NMEs of $0\nu \beta \beta$ decay ($M_{\mathrm{Tot}}^{0\nu \beta \beta }$), scaled by a mass-number-dependent factor of either ${\mathrm{A}}^{-1/6}$ or ${\mathrm{A}}^{-1/3}$, respectively. The NMEs are obtained from ab initio calculations with the chiral $\mathrm{NN}+$3N interactions EM1.8/2.0 or $\mathrm{\Delta }{\mathrm{N}}^2{\mathrm{LO}}_{\mathrm{go}}$ for isotopes ranging from $A=6$ to $A=76$. Only the results obtained with EM1.8/2.0 are used in the linear regression with residuals given in the bottom row. The shaded area indicates the confidence interval with a 95% confidence level while the dashed line indicates the prediction interval at a 95% confidence level. See text for details.

Figure 6

Correlation between $M^{\mathrm{DGT}}$ and $M_{\mathrm{GT}}^{0\nu \beta \beta }{A}^{-1/3}$ when considering only isospin-changing transitions (left column) or only isospin-conserving transitions (right column). The green bands show the 95% confidence interval while the dashed lines show the 95% prediction interval. The best fit lines from both cases agree with each other within $1\sigma$.

Figure 7

Correlation between $M^{\mathrm{DGT}}$ and $M_{\mathrm{Tot}}^{0\nu \beta \beta }{A}^{-1/3}$ derived from the results of both ab initio (red) and conventional (blue) nuclear models [45, 50, 51, 57, 82], except for the results of QRPA calculations [57] for isospin-conserving transitions in (a) and isospin-changing in (b).

Figure 8

Same as Fig. 7, but for the correlation between $M^{\mathrm{DGT}}$ and $M_{\mathrm{Tot}}^{0\nu \beta \beta }{A}^{-1/6}$.

Figure 9

The transition densities [cf. Eq. (10)] of isotopes in $p$, $sd$, and $fp$ shells from the VS-IMSRG calculation with the EM1.8/2.0 interaction. Asterisks indicate isospin conserving transitions.

Figure 10

Sampled transition densities ${\tilde{C}}_A^{\kappa }\left(r_{12}\right)$ with the parameters fitted to the results of VS-IMSRG calculations for the isospin-conserving transitions in $p$-, $sd$-, and $fp$-shell nuclei, except for the parameters $c$ and $d$, which vary from 0 to 1 and from 0.5 to 4.5 ${\mathrm{fm}}^2$, respectively.

Figure 11

The NMEs from the integral of the sampled transition densities in Fig. 10. Arrows show the direction of increasing the $d$ value.

Figure 12

The correlation relations are derived from the sampled transition densities for isospin-conserving transitions. The area is originated from the variants of the parameters $c$ and $d$ which control the short-range and long-range behavior, respectively. The NMEs from the VS-IMSRG calculations are added for comparison.

Figure 13

Sampled transition densities with parameters fitted to the VS-IMSRG results for isospin-changing transitions in the (a) $sd$- and (b) $fp$-shell nuclei, with the parameters $c$ and $d$ increasing from 0 to 1 and from 0.5 to 4.5 ${\mathrm{fm}}^2$, respectively.

Figure 14

The NMEs from the integral of the sampled transition densities in Fig. 13. (a) For $sd$-shell nuclei and (b) for $fp$-shell nuclei. Arrows show the direction of increasing the $d$ value.

Figure 15

(a) The transition densities from conventional shell-model calculations; (b) sampled transition densities with parameters fitted to the isospin-changing transition densities in (a). (c) The NMEs from the integral of the sampled transition densities in (b).

Figure 16

Correlation between the NMEs $M^{\mathrm{DGT}}$ of DGT transitions and the $M_{\mathrm{GT}}^{0\nu \beta \beta }$ scaled by $A^{-1/3}$ using a subset of isospin-changing transitions for the $sd$- and $fp$-shell nuclei from the conventional shell-model calculation as well as the results obtained with the VS-IMSRG wave functions and the transition operators in the harmonic oscillator (HO) basis, Hartree-Fock basis (HF), or fully (full) evolved basis. See text for details.

Figure 17

The transition densities of both DGT and the GT part of the $0\nu \beta \beta$ decay from ${}^{48}\mathrm{Ca}$ to ${}^{48}\mathrm{Ti}$ from the conventional nuclear shell-model calculation with the GXPF1A interaction and the VS-IMSRG calculation using the transition operators in three types of basis: HO basis, HF basis, and the fully evolved basis.

Figure 18

Evolution of the transition distribution of the GT part of the $0\nu \beta \beta$ decay and DGT transitions from the VS-ISMSRG in (a) ${}^6\mathrm{He}$ and (b) ${}^8\mathrm{He}$.

Figure 19

The transition densities of the DGT and GT part of the $0\nu \beta \beta$ decay in ${}^{48}\mathrm{Ca}$ from the IM-GCM calculation using either the bare or evolved transition operator, where the same nuclear wave functions but different transition operators are used, respectively.