  We use the composite boson (coboson) many-body formalism to tackle scattering
lengths for cold fermionic atoms. We show that bound dimers can be taken as
elementary entities provided that fermion exchanges between them are treated
exactly, as can be done through the coboson formalism. This alternative tool
extended to cold atom physics not only makes transparent many-body processes
through Shiva diagrams specific to cobosons, but also simplifies calculations.
Indeed, the integral equation we derive for the atom-dimer scattering length
and solve by restricting the dimer relative motion to the ground state, gives
values in remarkable agreement with the exact scattering length values for all
fermion mass ratios. This remarkable agreement also holds true for the
dimer-dimer scattering length, except for equal fermion masses where our
restricted procedure gives a value slightly larger than the accepted one
($0.64a_d$ instead of $0.60a_d$). All this proves that the scattering of a
cold-atom dimer with an atom or another dimer is essentially controlled by the
dimer relative-motion ground state, a physical result not obvious at first.
