Relative importance of the electron continuum intermediate state in single-electron capture into any state of fast protons from helium-like atomic systems

Single-electron capture by fast protons from helium-like atomic targets is investigated at intermediate and high impact energies. The main purpose of the present study is a comprehensive analysis of the relative importance of the electron continuum intermediate state (ionization continua), with respect to direct transfer. To achieve this goal, first- and second-order theories are employed, and their results are thoroughly compared. The prior form of the boundary-corrected continuum intermediate state method (BCIS) is utilized, in both its three-body and four-body formulation, in addition to the four-body boundary-corrected first-Born approximation (CB1-4B), in both its prior and post form. BCIS methods belong to the class of second-order theories, while CB1 methods belong to the class of first-order theories. Relative importance of ionization continua is examined in the example of single-electron capture in collisions of fast protons with ground-state atomic helium. Both differential and total cross sections are analyzed, for single-electron capture into any final state of the projectile. The presented cross sections, aside from their fundamental importance, are relevant in various interdisciplinary applications, such as in astrophysics, thermonuclear fusion and plasma physics, and medical physics.