Three-body problem with velocity-dependent optical potentials: a case of $(d,p)$ reactions

The change in mass of a nucleon, arising from its interactions with other nucleons inside the target, results in velocity-dependent terms in the Schr\"odinger equation that describes nucleon scattering. It has recently been suggested in a number of publications that introducing and fitting velocity-dependent terms improves the quality of the description of nucleon scattering data for various nuclei. The present paper discusses velocity-dependent optical potentials in a context of a three-body problem used to account for deuteron breakup in the entrance channel of $(d,p)$ reactions. Such potentials form a particular class of nonlocal optical potentials which are a popular object of modern studies. It is shown here that because of a particular structure of the velocity-dependent terms the three-body problem can be formulated in two different ways. Solving this problem within an adiabatic approximation results in a significant difference between the two approaches caused by contributions from the high $n$-$p$ momenta in deuteron in one of them. Solving the three-body problem beyond the adiabatic approximation may remove such contributions, which is indirectly confirmed by replacing the adiabatic approximation by the folding Watanabe model where such contributions are suppressed. Discussion of numerical results is carried out for the $^{40}$Ca($d,p)^{41}$Ca reaction where experimental data both on elastic scattering in entrance and exit channels and on nucleon transfer are available.