  The Majorana nature of massive neutrinos will be crucially probed in the
next-generation experiments of the neutrinoless double-beta ($0\nu 2\beta$)
decay. The effective mass term of this process, $\langle m\rangle^{}_{ee}$, may
be contaminated by new physics. So how to interpret a discovery or null result
of the $0\nu 2\beta$ decay in the foreseeable future is highly nontrivial. In
this paper we introduce a novel three-dimensional description of $|\langle
m\rangle_{ee}^{}|$, which allows us to see its sensitivity to the lightest
neutrino mass and two Majorana phases in a transparent way. We take a look at
to what extent the free parameters of $|\langle m\rangle_{ee}^{}|$ can be well
constrained provided a signal of the $0\nu 2\beta$ decay is observed someday.
To fully explore lepton number violation, all the six effective Majorana mass
terms $\langle m\rangle_{\alpha\beta}^{}$ (for $\alpha, \beta = e, \mu, \tau$)
are calculated and their lower bounds are illustrated with the two-dimensional
contour figures. The effect of possible new physics on the $0\nu 2\beta$ decay
is also discussed in a model-independent way. We find that the result of
$|\langle m\rangle_{ee}^{}|$ in the normal (or inverted) neutrino mass ordering
case modified by the new physics effect may somewhat mimic that in the inverted
(or normal) mass ordering case in the standard three-flavor scheme. Hence a
proper interpretation of a discovery or null result of the $0\nu 2\beta$ decay
may demand extra information from some other measurements.
