Rearrangement of valence neutrons in the neutrinoless double-$\beta$ decay of ${}^{136}\mathrm{Xe}$

A quantitative description of the change in ground-state neutron occupancies between ${}^{136}\mathrm{Xe}$ and ${}^{136}\mathrm{Ba}$, the initial and final state in the neutrinoless double-$\beta$ decay of ${}^{136}\mathrm{Xe}$, has been extracted from precision measurements of the cross sections of single-neutron-adding and -removing reactions. Comparisons are made to recent theoretical calculations of the same properties using various nuclear-structure models. These are the same calculations used to determine the magnitude of the nuclear matrix elements for the process, which at present disagree with each other by factors of 2 or 3. The experimental neutron occupancies show some disagreement with the theoretical calculations.

Figure 1

(a) Outgoing deuteron spectrum measured at ${\theta }_{\mathrm{lab}}=5.9^{\circ }$ following the ${}^{136}\mathrm{Ba}$($p,d$)${}^{135}\mathrm{Ba}$ reaction at an incident proton energy of 23 MeV and (b) the outgoing ${}^4\mathrm{He}$ spectrum measured at the same angle following the ${}^{136}\mathrm{Ba}$(${}^3\mathrm{He},\alpha$)${}^{135}\mathrm{Ba}$ reaction at 32 MeV. Some relevant states are labeled by their energies in keV and spin-parity assignments.

Figure 2

Example angular distributions demonstrating (a) $\ell =4$ and (b) $\ell =5$ discrimination in the (${}^3\mathrm{He},\alpha$) reaction on ${}^{136}\mathrm{Ba}$. The solid black lines show DWBA calculations for the assigned $\ell$ value, while the dashed lines show the distribution associated with the alternative $\ell$ value.

Figure 3

Change in proton occupancies (top) and neutron vacancies (bottom) in the $0\nu 2\beta$ decay of ${}^{136}\mathrm{Xe}$ in the bar charts. The proton data are from Ref. [8] and the neutron data from the present work. The three different theoretical calculations are from the shell model, SM1 [14] and SM2 [50], and the interacting boson model, IBM [51]. The discrepancy between the theoretical calculations and the experimental data is shown on the right where the error bars show the experimental uncertainty.