Effective-range function methods for charged particle collisions

Different versions of the effective-range function method for charged particle collisions are studied and compared. In addition, a novel derivation of the standard effective-range function is presented from the analysis of Coulomb wave functions in the complex plane of the energy. The recently proposed effective-range function denoted as Delta(l) [Ramirez Suarez and Sparenberg, Phys. Rev. C 96, 034601 (2017)] and an earlier variant [Hamilton et al., Nucl. Phys. B 60, 443 (1973)] are related to the standard function. The potential interest of Delta(l) for the study of low-energy cross sections and weakly bound states is discussed in the framework of the proton-proton S-1(0) collision. The resonant state of the proton-proton collision is successfully computed from the extrapolation of Delta(l) instead of the standard function. It is shown that interpolating Delta(l) can lead to useful extrapolation to negative energies, provided scattering data are known below one nuclear Rydberg energy (12.5 keV for the proton-proton system). This property is due to the connection between Delta(l) and the effective-range function by Hamilton et al. that is discussed in detail. Nevertheless, such extrapolations to negative energies should be used with caution because Delta(l) is not analytic at zero energy. The expected analytic properties of the main functions are verified in the complex energy plane by graphical color-based representations.