Nuclear matrix elements for neutrinoless $\beta \beta$ decays and spin-dipole giant resonances

Nuclear matrix element (NME) for neutrinoless $\beta \beta$ decay (DBD) is required for studying neutrino physics beyond the standard model by using DBD. Experimental information on nuclear excitation and decay associated with DBD is crucial for theoretical calculations of the DBD-NME. The spin-dipole (SD) NME for DBD via the intermediate SD state is one of the major components of the DBD-NME. The experimental SD giant-resonance energy and the SD strength in the intermediate nucleus are shown for the first time to be closely related to the DBD-NME and are used for studying the spin-isospin correlation and the quenching of the axial-vector coupling, which are involved in the NME. So they are used to help the theoretical model calculation of the DBD-NME. Impact of the SD giant resonance and the SD strength on the DBD study is discussed.

Figure 1

DBD transition scheme for ${}_Z{}^A\mathrm{X}_1\to {}_{Z+2}{}^A\mathrm{X}_3$ with the Majorana $\nu$ exchange in the intermediate nucleus ${}_{Z+1}{}^A\mathrm{X}_2$. ${\tau }^{+}$ and ${\tau }^{-}$: Isospin raising and lowering operators. $M^{-}\left(\mathrm{K}\right)$ and $M^{+}\left(\mathrm{K}\right)$: ${\tau }^{-}$ and ${\tau }^{+}$ K-mode NMEs associated with the K-mode DBD NME. k: Intermediate state. (${}^3\mathrm{He},t$): Charge-exchange reaction. 2QP: Two-quasiparticle state. GR: Giant resonance. DGR: Double giant resonance. See text.

Figure 2

The energy spectrum of the (${}^3\mathrm{He},t$) reaction on ${}^{100}\mathrm{Mo}$ [26]. F, GT, and SD are Fermi $0^{+}$, GT $1^{+}$, and SD $2^{-}$ giant resonances, respectively. The energy scale below 4 MeV is enlarged to make the sharp peaks visible. The $1^{-}$ and $0^{-}$ giant resonances are at the higher-energy region of SD. 2QP: Two-quasiparticle states. Yields at the $t$ emission angles of $0^{\circ }-0.5^{\circ }$, $0.5^{\circ }-1^{\circ }$, $1^{\circ }-1.5^{\circ }$, $1.5^{\circ }-2^{\circ }$, $2^{\circ }-2.5^{\circ }$, and $2.5^{\circ }-3^{\circ }$, each in degrees, are shown by red, yellow, pink, light blue, blue, and green, respectively. F and GT transitions are enhanced at the forward angles (red, yellow), while the SD ones at larger angles (blue, green).

Figure 3

(a) Experimental GT $1^{+}$ and SD $2^{-}$ giant-resonance energies of $E_G\left(\mathrm{GT}\right)$ and $E_G\left(\mathrm{SD}\right)$. Solid lines: Fits by Eq. (4). Squares: The observed energies. Inverse triangles: The energies from the experimental $E_G\left(\mathrm{GT}\right)\approx 0.4T_z+9\mathrm{MeV}$ and $E_G\left(\mathrm{SD}\right)\approx 0.4T_z+16.5\mathrm{MeV}$ [6] as used in the pnQRPA. (b) The GT strengths in logarithmic scale. GT $1^{+}$ strength $B^{-}\left(\mathrm{GT}\right)$. S: $B_S^{-}$ (GT) for the nucleon-based sum-rule limit. T: $B_T^{-}\left(\mathrm{GT}\right)$ for the observed total strength up to $E_G\left(\mathrm{GT}\right)+10\mathrm{MeV}$. The strength beyond $E_G$ is corrected for the small contribution from the quasifree scattering. 2QP: $B_{\mathrm{QP}}^{-}\left(\mathrm{GT}\right)$ for the sum of the observed 2QP strengths up to 6 MeV. Straight lines are fits to the data.

Figure 4

$M^{0\nu }$ (blue squares) and $M_A^{0\nu }$ (light blue square) with $g_A^{\mathrm{eff}}/g_A=0.74$ are plotted against (a) $A$ and (b) $N-Z$. Solid lines are fits. See text.