Electromagnetic transitions from isobaric analog states to study nuclear matrix elements for neutrinoless $\beta \beta$ decays and astro-neutrino inverse $\beta$ decays

Experimental studies of nuclear matrix elements (NMEs) for neutrinoless double-$\beta$ decays (DBDs) and astro-neutrino $\left(\nu \right)$ inverse $\beta$ decays (IBDs) are crucial for $\nu$ studies beyond the standard model and the astro-$\nu$ studies since accurate theoretical calculations of the NMEs are hard due to the high sensitivity of the NMEs to the nuclear models and the nuclear parameters used for the models. Some of the important NMEs of electromagnetic transition operators associated with DBD and IBD, including the effective weak couplings, are found to be experimentally obtained by measuring the corresponding electromagnetic gamma (EM:$\gamma )$ transitions from the isobaric analog states (IASs) of the DBD and IBD nuclei. Then the experimental NMEs and the couplings are used for evaluating the DBD and IBD NMEs and for checking the model calculations. The EM-NMEs, the cross sections, and the event rates for the IAS-$\gamma$ transitions are estimated for DBD and IBD nuclei to show the feasibility of the experiments.

Figure 1

Left: GT weak and $\mathrm{IAS}\text{−}M1\gamma$ transition diagrams. Top: DBD with $\nu$ exchange between $n$ and $n^{\prime }$. Middle: Astro-$\nu$ IBD. Bottom: $({}^3\mathrm{He},t)$ CER of the IAS excitation followed by $\mathrm{EM}\left(\gamma \right)$ transition. W, weak boson; M, meson; $\tau$, isospin; $\sigma$, spin. Green circles: Nuclear $\tau \sigma$ vertexes. Green ellipsoids: Nuclear CER $\left(\tau \right)$ and $M1\left(\sigma \right)$ vertexes. In the case of $E1$ the operator $\sigma$ is replaced by $r{Y}_1$. Right: DBD scheme for ${}_Z^A\mathrm{X}\to {}_{Z+2}^A\mathrm{X}$ via the intermediate nucleus ${}_{\mathrm{Z}+1}^A\mathrm{X}$ and the $\gamma$ decay from IAS (red lines) and the $\gamma$ excitation to IAS (blue lines). $(T,T_z)$ are the isospin and its $z$ component.

Figure 2

The energy spectrum of the $({}^3\mathrm{He},t)$ reaction on ${}^{82}\mathrm{Se}$ [34]. The yield in units of ${10}^4/\left(\mathrm{msr}5\mathrm{keV}\right)$ is plotted against the excitation energy in units of MeV. The IAS peak (red) at the forward angle is overscaled. See Ref. [34] for the details. The energy scale below 6 MeV is enlarged. Yields at the $t$ emission angles of 0–0.5, 1–1.5, and 2–2.5, all in degrees, are shown by red, green, and blue, respectively. IAS and GT CERs with no angular momentum transfer are enhanced at the forward angle (red). The blue arrow is $\gamma$ from IAS.

Figure 3

Experimental IAS widths as a function of the isospin $T_z=(N-Z)/2$. Blue squares for IASs in DBD nuclei derived in the present work. Light blue triangles from Ref. [39].

Figure 4

Differential cross sections at $0^{\circ }$ for DBD and astro-$\nu$ nuclei. Green diamonds: IAS excitations. Blue inverse triangles: IAS-$E1\gamma \mathrm{s}$ to $1^{-}$ states. Light blue triangles: $\mathrm{IAS}\text{−}M1\gamma \mathrm{s}$ to $1^{+}$ states.