When studying particle interaction events in QFT, we usually consider either (a) $2 \to 2$ particle "scattering" events, whose probabilities are quantified by scattering cross-sections, or (b) $1 \to 2$ particle "decay" events, whose probabilities are quantified by decay rates.
I would have naively expected that you could calculate more information than just a decay rate from a $1 \to n'$ process, e.g. a "cross section" for the outgoing particles to have various relative momenta. Similarly, I would have expected that you could calculate a "cross-section" for an $n \to 1$ "merger" interaction, but I've heard that cross-sections aren't defined for $n \to 1$ processes. (Presumably both of these intuitions are wrong for the same reason, since decay and merger processes are related by crossing symmetry.)
For which values of $n$ and $n'$ are $n \to n'$ particle scattering cross-sections well-defined? For those values for which a cross-section isn't defined, is there an equivalent quantity, and what information does it convey?
I remember that this all boils down to a simple counting argument for phase-space degrees of freedom, but I forget the details.