Systematic study of two-proton radioactivity half-lives within the generalized liquid drop model with various versions of proximity energies

Two-proton ($2p$) radioactivity constitutes a pivotal probe for studying the nuclear structure of extreme proton-rich nuclei. To improve the calculated accuracy and predictive power of the generalized liquid drop model (GLDM) for describing the $2p$ radioactivity half-lives, this work performs a systematic study within the GLDM framework using distinct proximity energies. It is found that the proximity energy can counterbalance the Coulomb repulsion, lowering the barrier height and simultaneously shifting its peak to larger radial separations. Consequently, the specific proximity energy employed governs both the height and shape of the interaction barrier, thereby dictating the penetration probability, and ultimately, the $2p$ radioactivity half-life. Among all considered proximity energy formalisms, the Prox. 77-Set 13 yields the smallest standard deviation when reproducing experimental data for known true $2p$ emitters, representing an improvement over the original proximity potential. The spectroscopic factor of $2p$ radioactivity ${S}_{2p}$ is found to be essential for reducing systematic deviations. Using the GLDM with Prox. 77-Set 13, we predict $2p$ radioactivity half-lives for candidate nuclei. The predictions agree well with those from other established models and validate the newly proposed Geiger-Nuttall law for $2p$ radioactivity. This work would provide reference for future experimental investigations of $2p$ radioactivity in proton-rich nuclei.