One of the momentum space Feynman rules in $\phi^{4}$ theory (for correlation functions) is that for an external point with 4-momentum $p$ (with direction headed towards the external point), we need a factor $e^{-ip\cdot x}$. See for example p95 of Peskin and Schroeder.

I am not sure if this means the when the direction of the momentum is headed away from the external point, I should instead use a factor $e^{ip\cdot x}$.

This would seem to make sense to me due to the following argument. Suppose we have some Feynman diagram and have written down its corresponding amplitude. Then:

  1. Reverse one external leg momentum,

  2. Calculate the new amplitude using Feynman rules,

One difference is that the momentum conserving delta function at some vertex changes - we flip the sign of the reversed momentum. You could think of this as a sign flip in the exponent of the Feynman propagator - however this would also flip the sign of the external leg phase factor. Hence I conclude that we should also flip the sign of this.

It would be good to get some confirmation though, since my understanding isn't too great.


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