Scattering Rates and Lifetime of Exact and Boson Excitons

Although excitons are not exact bosons, they are commonly treated as such provided that their composite nature is included in effective scatterings dressed by exchange. We here prove that, whatever these scatterings are, they cannot give both the exciton scattering rates Tᵢⱼ^-1 and lifetime τ₀ correctly: A striking factor $1/2$ exists between τ₀^-1 and the sum of Tᵢⱼ^-1's, which originates from the composite nature of excitons, irretrievably lost when they are bosonized. This result, which appears as very disturbing at first, casts major doubts on the overall validity of bosonization for problems dealing with interacting excitons.