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I need to find all possible contractions (in the sense of Wick contractions) for 5 fields. One can of course start drawing randomly, but I'm sure there is some kind of algorithm to do this systematically... So, does this algorithm have a name, and is there software that can do this for me?

Also, is there an analytic expression to give me the number of possible contractions $C$, given a number of fields $n$? This question suggests that the formula should take the form $C(n)=n!!$ or something similar, but unfortunately I wasn't able to find a reference for this statement and it seems to already fail for $n=4$, i.e. $C(4)=4!!=8\neq 10$. This would at least provide a good orientation if one is finished with the calculation or not...

Qmechanic
  • 201,751
Marius Jaeger
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  • related/possible duplicate: https://physics.stackexchange.com/q/304869/84967 – AccidentalFourierTransform Apr 26 '21 at 21:05
  • a good starting point for both questions is to compute the path integral (to the relevant order) since all possible contractions just drop out of it automatically. a baby version of this is to familiarise yourself with multiple derivatives of Gaussian exponentials (assuming you have canonically normalised the kinetic term, there is a map between the two procedures). – Wakabaloola Apr 27 '21 at 14:00

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