  Using the composite boson many-body formalism that takes single-exciton
states rather than free carrier states as a basis, we derive the integral
equation fulfilled by the exciton-exciton effective scattering from which the
role of fermion exchanges can be unraveled. For excitons made of
$(\pm1/2)$-spin electrons and $(\pm3/2)$-spin holes, as in GaAs
heterostructures, one major result is that most spin configurations lead to
brightness-conserving scatterings with equal amplitude $\Delta$, in spite of
the fact that they involve different carrier exchanges. A brightness-changing
channel also exists when two opposite-spin excitons scatter: dark excitons
$(2,-2)$ can end either in the same dark states with an amplitude $\Delta_e$,
or in opposite-spin bright states $(1,-1)$, with a different amplitude
$\Delta_o$, the number of carrier exchanges being even or odd respectively.
Another major result is that these amplitudes are linked by a striking
relation, $\Delta_e+\Delta_o=\Delta$, which has decisive consequence for
exciton Bose-Einstein condensation. Indeed, this relation leads to the
conclusion that the exciton condensate can be optically observed through a
bright part only when excitons have a large dipole, that is, when the electrons
and holes are well separated in two adjacent layers.
