Three-point Green's function in position space

Three-point Green's function in position space

In this post I'm trying to calculate the 3-point interaction function for quantum field theory in position space.
It sounds simple enough. It is just calculating the 4 dimensional integral over w of:
 Δ(x-w)Δ(y-w)Δ(z-w) 
which should produce a symmetric function in three variables: A(|x-y|,|y-z|,|z-x|) however it turns out to be extremely complicated to get a nice formula in terms of symmetric variables.

It would be nice to relate this function to geometric properties such as the area of the triangle but that doesn't seem to be the case. Only the sums of the powers of the lengths of the sides which are symmetric forms. I have a series solution for the case when m is not zero but can't find a solution for when m=0.


If anyone can help let me know!




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