The Vainshtein, Presnyakov and Sobelman approximation in heavy particle collisions

Abstract

Some aspects of the VPS theory for direct excitation in heavy particle collisions are critically examined. It is shown that the integral representation of the VPS amplitudes cited in the literature is in error, and its correct form is given. The theoretical basis for the use of the peaking approximation to simplify the mathematics of the VPS amplitudes is examined and it is shown that the terms thus neglected are related to the second-order and higher-order terms of a perturbative expansion. It is also shown that neglect to the term containing the projectile-core interaction from the VPS theory can not be justified, and this term plays an important role of cancelling a class of singularities which make the post form of the VPS amplitudes computed in the peaking approximation discontinuous. It is also pointed out that the VPS amplitudes for s to s transitions have logarithmic divergence which results from the well known cause of the mean excitation energy being equal to zero in the second-order perturbation theory term. The relationship of the VPS theory with eikonal approximations is also clarified.