Dependence of two-proton radioactivity on nuclear pairing models

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.