  Sensitivity of two-proton emitting decay to nuclear pairing correlation is
discussed within a time-dependent three-body model. We focus on the $^6$Be
nucleus assuming $\alpha + p + p$ configuration, and its decay process is
described as a time-evolution of the three-body resonance state. For a
proton-proton subsystem, a schematic density-dependent contact (SDDC) pairing
model is employed. From the time-dependent calculation, we observed the
exponential decay rule of a two-proton emission. It is shown that the density
dependence does not play a major role in determining the decay width, which can
be controlled only by the asymptotic strength of the pairing interaction. This
asymptotic pairing sensitivity can be understood in terms of the dynamics of
the wave function driven by the three-body Hamiltonian, by monitoring the
time-dependent density distribution. With this simple SDDC pairing model, there
remains an impossible trinity problem: it cannot simultaneously reproduce the
empirical $Q$ value, decay width, and the nucleon-nucleon scattering length.
This problem suggests that a further sophistication of the theoretical pairing
model is necessary, utilizing the two-proton radioactivity data as the
reference quantities.
