Examination of the consistency of the quasiparticle random-phase approximation approach to double-$\beta$ decay of ${}^{48}\mathrm{Ca}$

The nuclear matrix elements (NMEs) of the neutrinoless and two-neutrino double-$\beta$ decays of ${}^{48}\mathrm{Ca}$ are calculated by the quasiparticle random-phase approximation (QRPA) with emphasis on the consistency examinations of this calculation method. The main new examination points are the consistency of two ways to treat the intermediate-state energies in the two-neutrino double-$\beta$ NME and comparison with the experimental charge-exchange strength functions obtained from ${}^{48}\mathrm{Ca}(p,n)$ and ${}^{48}\mathrm{Ti}(n,p)$ reactions. No decisive problem preventing use of the QRPA approach is found. The obtained neutrinoless double-$\beta$ NME adjusted by the ratio of the effective and bare axial-vector current couplings is lowest in those calculated by different groups and close to one of the QRPA values obtained by another group.

Figure 1

Illustrative example of vanishing component of overlap of states obtained by charge change from $(Z+2,N-2)$ and $(Z,N)$ nuclei.

Figure 2

Calculated half-life of ${}^{48}\mathrm{Ca}$ to $2\nu \beta \beta$ decay as functions of $-g_{T=0}^{\mathrm{pair}}$. The points connected by solid (dotted) lines were obtained using $E_{aF}^{K=0}$'s $(E_{aI}^{K=0}$'s). The parameters $g_A=1.0$ and $g_{(T,T_z)=(1,0)}^{\mathrm{pair}}=-241.43\mathrm{MeV}{\mathrm{fm}}^3$ were used.

Figure 3

Dependence of $M^{\left(0\nu \right)},M_{GT}^{\left(0\nu \right)}$, and $-M_F^{\left(0\nu \right)}$ on $g_{T=0}^{\mathrm{pair}}$.

Figure 4

Strength functions of the GT transition from ${}^{48}\mathrm{Ca}$ to ${}^{48}\mathrm{Sc}$ measured (isolated points with error bars) and calculated by the QRPA (solid line). The origin of the excitation energy $E$ is at the ground state of ${}^{48}\mathrm{Sc}$. The measured one [6] is obtained using Eq. (28) $\left[dB\right(GT)/dE]$. The inset is a magnification of the high-energy region.

Figure 5

The same as the solid line in Fig. 4 but for the transition operator $r^2\sigma \tau$.

Figure 6

The same as Fig. 5 but for the transition operator $(r^2-{\langle r^2\rangle }_{n1f7/2})\sigma \tau$.

Figure 7

The same as Fig. 4 but for ${}^{48}\mathrm{Ti}\to {}^{48}\mathrm{Sc}$. The width parameter of 1.0 MeV is used for excitation energies larger than 4 MeV to simulate the experimental width.

Figure 8

The same as Fig. 5 but for ${}^{48}\mathrm{Ti}\to {}^{48}\mathrm{Sc}$.

Figure 9

The same as Fig. 6 but for ${}^{48}\mathrm{Ti}\to {}^{48}\mathrm{Sc}(\langle r^2\rangle$ of a $1f_{7/2}$ proton is used).

Figure 10

Strength functions measured (isolated points with error bars) and calculated using $O_{\mathrm{mix}}$ [Eq.(30)] for ${}^{48}\mathrm{Ca}\to {}^{48}\mathrm{Sc}$ with $\alpha =-0.03{\mathrm{fm}}^{-2}$ (solid lines). The inset is a magnification.

Figure 11

The same as Fig. 10 but for ${}^{48}\mathrm{Ti}\to {}^{48}\mathrm{Sc}$ with $\alpha =-0.0253{\mathrm{fm}}^{-2}$.

Figure 12

The same as Fig. 4 but for the quenched GT transition operator $\sqrt{0.5}\sigma \tau$.

Figure 13

The same as Fig. 7 but for the quenched GT transition operator $\sqrt{0.38}\sigma \tau$.