Nuclear matrix elements for Majoron-emitting double-$\beta$ decay

A complete calculation of the nuclear matrix elements (NMEs) for Majoron emitting neutrinoless double beta decay within the framework of IBM-2 for spectral indices $n=1,3,7$ is presented. By combining the results of this calculation with previously calculated phase-space factors (PSFs) we give predictions for expected half-lives. By comparing with experimental limits on the half-lives we set limits on the coupling constants $\langle g_{ee}^M\rangle$ of all proposed Majoron-emitting models.

Figure 1

Schematic representation of neutrinoless double-$\beta$ decay accompanied by the emission of one or two Majorons. Adapted from Ref. [9].

Figure 2

Summed electron spectra for the $n=1$, 2, 3, and 7, as well as for the $2\nu \beta \beta$ ($n=5$) decays of ${}^{136}\mathrm{Xe}$.

Figure 3

Single electron spectra for the $n=1$, 2, 3, and 7, as well as for the $2\nu \beta \beta$ ($n=5$) decays of ${}^{136}\mathrm{Xe}$.

Figure 4

Angular correlation between the two emitted electrons for the $n=1$, 2, 3, and 7, as well as for the $2\nu \beta \beta$ ($n=5$) decays of ${}^{136}\mathrm{Xe}$. The calculation for $n=1$, 2, 3, 7 stops at the point where the single electron spectrum goes to zero. Beyond that point it becomes unstable as $G^{\left(0\right)}$ goes to zero faster than $G^{\left(1\right)}$. For $2\nu \beta \beta$ ($n=5$) this is avoided by taking into account the individual energies of the neutrinos, ${\omega }_1$ and ${\omega }_2$, as in Eq. (22) of Ref. [18], instead of the Majoron energy $q$ as in Eq. (5) of Ref. [9].