# For which values of $n$ and $n'$ are $n \to n'$ particle scattering cross-sections well-defined?

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.

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