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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.

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    $\begingroup$ Where have you heard that $n\to1$ is not well-defined? It seems well-defined to me... $\endgroup$ – AccidentalFourierTransform Sep 2 '19 at 23:24
  • $\begingroup$ @AccidentalFourierTransform Innisfree makes that claim for the case $n = 2$ in comments for two different answers to physics.stackexchange.com/q/325790/92058. $\endgroup$ – tparker Sep 3 '19 at 0:30
  • $\begingroup$ I wouldn't take Innisfree (or any other PSE user, for that matter) as a reliable source for anything. $\endgroup$ – AccidentalFourierTransform Sep 3 '19 at 0:41
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    $\begingroup$ @AccidentalFourierTransform Hence my question ... $\endgroup$ – tparker Sep 3 '19 at 2:30
  • $\begingroup$ haha, no worries,let's see if there is a clear answer $\endgroup$ – innisfree Sep 4 '19 at 3:48

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