Relativistic Green's function in complex momentum representation for one-proton emission

We present a novel relativistic Green's function approach formulated in the complex momentum representation (CGF), implemented within three theoretical frameworks: the Dirac equation, the relativistic point-coupling (RMFpc) model, and the relativistic mean-field (RMF) theory. The method provides rigorous treatment of long-range Coulomb potentials through complex momentum representation while employing Land\'e subtraction to remove singularities. Validation via Dirac-CGF calculations of single-proton emissions in $^{37,39}\mathrm{Sc}$ successfully reproduces sharp resonance peaks in the $1{f}_{7/2}$ orbital. Calculations yield precise $Q$ values and lifetimes that show excellent agreement with both experimental data and time-dependent Dirac results. Expanding to the RMF framework, after examining the applicability of the RMF calculation and the validity of $^{37,39}\mathrm{Sc}$ as a magic number plus a proton model, we performed RMFpc-CGF and RMF-CGF calculations. Both clearly displayed resonant peaks in these nuclei. Notably, RMFpc-CGF calculations using PC-PK1 and RMF-CGF calculations using NL3* showed outstanding consistency with experimental $Q$ values and half-lives for $^{37,39}\mathrm{Sc}$, demonstrating the superior capability of these approaches for studying proton emission phenomena.