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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.

tparker
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    Where have you heard that $n\to1$ is not well-defined? It seems well-defined to me... – AccidentalFourierTransform Sep 02 '19 at 23:24
  • @AccidentalFourierTransform Innisfree makes that claim for the case $n = 2$ in comments for two different answers to https://physics.stackexchange.com/q/325790/92058. – tparker Sep 03 '19 at 00:30
  • I wouldn't take Innisfree (or any other PSE user, for that matter) as a reliable source for anything. – AccidentalFourierTransform Sep 03 '19 at 00:41
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    @AccidentalFourierTransform Hence my question ... – tparker Sep 03 '19 at 02:30
  • haha, no worries,let's see if there is a clear answer – innisfree Sep 04 '19 at 03:48
  • @AccidentalFourierTransform so who would you take as a reliable source? someone with a Th.D perhaps? https://www.youtube.com/watch?v=DxrlcLktcxU – MathematicalPhysicist Sep 04 '19 at 11:59
  • @mathematical I don’t know much about this issue. I just naively observed that it seems tricky to calculate a cross section for an final state of a single on-shell particle, as there don’t appear to be sufficient phase space integrals to kill four Dirac functions for energy momentum conservation – innisfree Sep 04 '19 at 14:19
  • Another thing I don’t know much about: n -> n’ processes don’t in general have cross sections in the sense that the equivalent quantity doesn’t have dimension of an area unless n=2. But this is just terminology I guess – innisfree Sep 04 '19 at 14:22
  • @innisfree I'm afraid I don't understand your counting. If there are $n$ external particles in $d$-dimensional spacetime, then aren't there $n$ different $d$-dimensional integrals (one for each external particle's envelope function), $n$ delta functions from each external particle's mass shell condition, and $d$ delta functions for energy-momentum conservation? So $nd$ integrals and $n+d$ constraints gives $nd-n-d$ degrees of freedom (not incorporating Lorentz invariance)? Could you spell out your counting? – tparker Sep 07 '19 at 05:06
  • I agree with your counting. Your final formula is $n(d-1) -d$, so for $n=1$, it's less than zero (-1 to be precise), right? – innisfree Sep 09 '19 at 03:00
  • @innisfree But $n$ is the total number of external particles, incoming and outgoing, so for a $2 \to 1$ merger process we have $n = 3$ and plenty of integrals to spare. – tparker Sep 09 '19 at 06:06
  • Oh sorry, I indeed misunderstood you. No, I think $n$ should be the number of final state particles. We don't integrate over the initial state - that's fixed in the calculation. – innisfree Sep 09 '19 at 08:13

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