measured -decay half life-time and the uncertainty due to the evaluation of the nuclear matrix elements. Condition (\[suffcondinv01\]) depends crucially on the value of $( \cos 2 \theta_{\odot} )_{\mbox{}_{\rm MAX}}$ and it is less stringent for smaller values of $( \cos 2 \theta_{\odot} )_{\mbox{}_{\rm MAX}}$  [@PPW].

We can parametrize the uncertainty in due to the poor knowledge of the relevant nuclear matrix elements — we will use the term “theoretical uncertainty” for the latter, through a parameter $\zeta$, $\zeta \geq 1$, defined as: $${\mbox{$\left|  < \! m  \! > \right| \ $}}\!\! = \zeta \Big( ({\mbox{$\left|  < \! m  \! > \right| \ $}}_{\rm \!\!exp})_{\mbox{}_{\rm MIN}} \pm
\Delta \Big)
\label{meffpar01}$$ where $({\mbox{$\left|  < \! m  \! > \right| \ $}}_{\rm \!\!exp})_{\mbox{}_{\rm MIN}}$ is the value of obtained from the measured -decay half life-time of a given nucleus using [*the largest nuclear matrix element*]{} and $\Delta$ is the experimental error. The necessary condition permitting to establish, in principle, that the CP-symmetry is violated due to the Majorana CP-violating phases is: $$1 \leq \zeta <  \frac{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MIN}}} \ $}}}}
{\sqrt{{\mbox{$(\Delta  m^2_{\mathrm{atm}})_{ \! \mbox{}_{\mathrm{MAX}}} \ $}}}(\cos 2 \theta_{\odot})_{\mbox{}_{\rm MAX}} + 2 \Delta}~.
\label{invh:cond1a}$$ Obviously, the smaller $(\cos 2 \theta_{\odot})_{\mbox{}_{\rm MAX}}$ and 