Shell model study of using an effective field theory for disentangling several contributions to neutrinoless double-$\beta$ decay

Weak interaction in nuclei represents a well-known venue for testing many of the fundamental symmetries of the Standard Model. In particular, neutrinoless double-$\beta$ decay offers the possibility to test beyond Standard Model theories predicting that neutrinos are Majorana fermions and the lepton number conservation is violated. This paper focuses on an effective field theory approach to neutrinoless double-$\beta$ decay for extracting information regarding the properties of the beyond Standard Model Lagrangian responsible for this process. We use shell model nuclear matrix elements and the latest experimental lower limits for the half-lives to extract 12 lepton-number-violating parameters of five nuclei of experimental interest and lower limits for the energy scales of the new physics. Using the most stringent limits that we obtain for the values of the lepton-number-violating parameters, we predict new half-life limits for the other nuclei of experimental interest, in the case of 12 neutrino double-$\beta$ decay mechanisms. We provide an analysis that could reveal valuable information regarding the dominant neutrinoless double-$\beta$ decay mechanism, if experimental half-life data become available for different isotopes.

Figure 1

The $0\nu \beta \beta$ decay process diagrams: (a) presents the generic description of the process, (b) shows the most studied case in the literature, that of the light left-handed neutrino exchange, (c) is the long-range component of the $0\nu \beta \beta$ decay diagram, while (d) displays the short-range part.

Figure 2

Similar to Fig. 1, we present the nucleon-level diagrams of $0\nu \beta \beta$ decay process: (a) the generic description of the process, (b) the light left-handed neutrino exchange, (c) the long-range component, and (d) the short-range contribution. On the second line, (e) the pion-neutrino component, (f) the one-pion long-range contribution of the ${\mathcal{R}/}_p$ SUSY-induced $0\nu \beta \beta$ diagram, and (g) the two-pion long-range contribution of the ${\mathcal{R}/}_p$ SUSY-induced $0\nu \beta \beta$. The effective couplings ${\eta }_{1\pi }$ and ${\eta }_{2\pi }$ are related to Eq. (16) as ${\eta }_{1\pi }=c_{1\pi }{\eta }_{\pi N}$ and ${\eta }_{2\pi }=c_{2\pi }{\eta }_{\pi N}$.

Figure 3

The ratio between the ${}^{136}\mathrm{Xe}$ half-life and the $T_{\alpha }$ half-lives of several experimentally interesting isotopes, in the case of 12 EFT LNV couplings plus ${\eta }_{0N}$. The left to right order of the bars corresponds to up to down order in the legend. The height of the bars represents the difference between results obtained with different SRC parametrizations. ${\eta }_{0N}$ plays a similar role to ${\varepsilon }_3^{RRz\left(LLz\right)}$.

Figure 4

Same as Fig. 3, but for the Strasbourg-Madrid choice of Hamiltonians.

Figure 5

Same as Fig. 3, but for ${}^{76}\mathrm{Ge}$ instead of ${}^{136}\mathrm{Xe}$, with the CMU choice of Hamiltonians.

Figure 6

Same as Fig. 5, but for the Strasbourg-Madrid effective Hamiltonians.