  The contribution to the binding energy of a two-body system due to the
crossed two-boson exchange contribution is calculated, using the Bethe-Salpeter
equation. This is done for distinguishable, scalar particles interacting via
the exchange of scalar massive bosons. The sensitivity of the results to the
off-shell behavior of the operator accounting for this contribution is
discussed. Large corrections to the Bethe-Salpeter results in the ladder
approximation are found. For neutral scalar bosons, the mass obtained for the
two-body system is close to what has been calculated with various forms of the
instantaneous approximation, including the standard non-relativistic approach.
The specific character of this result is demonstrated by a calculation
involving charged bosons, which evidences a quite different pattern. Our
results explain for some part those obtained by Nieuwenhuis and Tjon on a
different basis. Some discrepancy appears with increasing coupling constants,
suggesting the existence of sizeable contributions involving more than
two-boson exchanges.
