From the Bethe-Salpeter equation to nonrelativistic approaches with effective two-body interactions

It is known that binding energies calculated from the Bethe-Salpeter equati on in ladder approximation can be reasonably well accounted for by an energ y-dependent interaction, at least for the lowest states. It is also known t hat none of these approaches gives results close to what is obtained by usi ng the same interaction in the so-called instantaneous approximation, which is often employed in nonrelativistic calculations. However, a recently pro posed effective interaction was shown to account for the main features of b oth the Bethe-Salpeter equation and the energy-dependent approach. In the p resent work, a detailed comparison of these different methods for calculati ng binding energies of a two-particle system is made. Some improvement, pre viously incorporated for the zero-mass boson case in the derivation of the effective interaction, is also employed for massive bosons. The constituent particles are taken to be distinguishable and spinless. Different masses o f the exchanged boson (including a zero mass) as well as states with differ ent angular momenta axe considered and the contribution of the crossed two- boson-exchange diagram is discussed. With this respect, the role played by the charge of the exchanged boson is emphasized. It is shown that the main difference between the Bethe-Salpeter results and the instantaneous approxi mation ones are not due to relativity as often conjectured. (C) 2001 Elsevi er Science B.V. All rights reserved.