Predicting the neutrinoless double-beta decay matrix element of $^{136}$Xe using a statistical approach

Calculation of the nuclear matrix elements (NMEs) for double-beta decay is of paramount importance for guiding experiments and for analyzing and interpreting the experimental data, especially for the search of the neutrinoless double beta decay mode ($0\nu\beta\beta$). However, there are currently still large differences between the NME values calculated by different methods, hence a quantification of their uncertainties is very much required. In this paper we propose a statistical analysis of $0\nu\beta\beta$ NME for the $^{136}Xe$ isotope, based on the interacting shell model, but using three independent effective Hamiltonians, emphasizing the range of the NMEs' most probable values and its correlations with observables that can be obtained from the existing nuclear data. Consequently, we propose a common probability distribution function for the $0\nu\beta\beta$ NME, which has a range of (1.55 - 2.65) at 90\% confidence level, with a mean value of 1.99 and a standard deviation of 0.37.