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Here's a 3-loop diagram for light-by-light scattering in scalar QED (from Schwartz textbook question 9.2):

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The question 9.2 asks approximately how many other diagrams contribute at the same order in perturbation theory and hinted that I do not need to draw the diagrams. There are two types of vertices in scalar QED, coming from $- ie A_{\mu} \left[ \phi^* (\partial^{\mu}\phi) - (\partial^{\mu}\phi^*) \phi \right]$ and $e^2 A_{\mu}^2 |\phi|^2 $ terms in the Lagrangian, respectively. The only way I can approach this problem is by permuting the two kinds of vertices at $O(e^8)$ and then do a bit simple permutations to count the possible diagrams based on the position of each vertex. Is there a simpler/more efficient way I can resolve this?

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I believe that the approach you outline in your question is more-or-less all that Schwartz is looking for. In particular, with 8 vertices of the type $A_\mu \phi^* \partial_\mu \phi$ and $4$ external photon fields, there are $12$ $A_\mu$ fields that need to be contracted in some way, and 8 $\phi^*$ fields with $8 $ $\phi$ fields. To heuristically count the number of contractions, imagine that the $\phi$ fields are contracted in some pattern, and note that if you list the 12 $A_\mu$ fields in a row, the first $A_\mu$ can be contracted together with any of the remaining $11$, the next uncontracted field has $9$ options, then $7$ options, ... leading to $11!! = 11 \times 9 \times \cdots \times 1$ options.

Of course, then there is the scalar contractions, and also the seagull vertex to consider. But I believe that all that Schwartz wants from this problem is to point out the ~factorial growth in the number of diagrams.

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