# Poster excerpt 11: The Key Role of Nuclear Physics for Neutrinoless Double Beta Decay

*(The following text has been provided by S.Dell’Oro, S.Marcocci, and F. Vissani)*

The discovery of neutrino oscillations and its implication that neutrinos have mass have boosted the importance of neutrinoless double beta decay (0νββ). Indeed, this process proved to be a key tool to investigate the Majorana-Dirac nature of the neutrino, giving also information on the absolute scale and the mass hierarchy.

In the assumption that the 0νββ transition is mediated by the three known neutrino, a fundamental role is played by the *Majorana Effective Mass*, namely

This parameter can be thought as the absolute value of the *ee*-entry of the neutrino mass matrix, where m_i are the masses of the individual v_i, δ and α1,2 are the Dirac and Majorana phases respectively and U_ei are the elements of the mixing matrix that defining the composition of the electron neutrino.

Within this scenario, the theoretical expression of the half-life of the 0νββ can be factorized as:

where m_e is the electron mass, G_0v is the phase space factor and M_0v is the nuclear matrix element describing the transition. In particular, in the recent works, this last term is written emphasizing the axial coupling g_a:

M_0v = g_A^2 M

where M depends mildly on g_A and can be evaluated modeling theoretically the nucleus. An experimental limit on the half-life translates in a limit on the Majorana effective mass by using the above definition.

The main sources of uncertainty in the inference are the nuclear matrix elements. Indeed, despite the fact that independent calculations have assessed a relatively small intrinsic error of about 20%, it also emerged a more important role of g_A than originally thought. It is commonly expected that the value of g_A measured in the weak interactions and decays of nucleons is modified (or *renormalized*) in the nuclear medium toward the value appropriate for quarks. But a further *reduction* might be rather plausible. Therefore, a conservative treatment of the uncertainties should consider at least three cases:

where the last formula includes phenomenologically the effect of the atomic number A, as observed in the two-neutrino double beta decay transition in different nuclei.

Thanks to the knowledge of the oscillation parameters, it is possible to constraint the allowed region for m_bb. In the picture, m_bb is plotted as a function of the *cosmological mass* (i.e. the sum of the three active neutrino masses) which is probed in cosmology. The different colors refer to the possible ordering for the neutrino mass spectra: the normal hierarchy (orange) and the inverted hierarchy (blue). It has to be noticed that the Majorana phases cannot be probed in neutrino oscillation experiments and thus are let free to vary, this resulting in a vertical broadening of the bands.

The horizontal bands show the limit on m_bb obtained with recent experiments using ^{136}Xe and they refer to the three cases of g_A quenching mentioned above. The bands take into account the intrinsic uncertainty of the nuclear matrix elements and phase space factors calculations. As it is clear from the plot, the effect of the uncertainty of g_A is *huge *(almost an order of magnitude)!

Our final message is that nuclear physics is fundamental for neutrinoless double beta investigations. In particular, the issue of the axial vector coupling constant quenching has to be sorted out in the future, either theoretically either experimentally.