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Consider a calculation of photon vacuum polarization in scalar QED. The 1PI diagram which people calculate is this one,

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But in scalar QED the seagull vertex is also possible

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Then why not include a diagram like this one,

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Which would then make the propagator sum to go something like this,

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and so on.

Now my question is am I correct to add such a diagram? If not why? Also if indeed such terms are added in a full calculation please refer to articles which take these contributions into account.

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1 Answer 1

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That's a completely valid diagram, but it's higher order in perturbation theory. The 3-point vertex comes with a factor of $e$, and the 4-point vertex comes with a factor of $e^2$, so your diagram is order $e^4$, which is higher than the order $e^2$ contributions calculated in standard textbooks. Note that for the purposes of computing corrections to the propagator in this theory, "order $e^2$" is synonymous with "one loop", and "order $e^4$" is synonymous with "two loop".

Two loop corrections can be found in various papers. For example, this paper claims to calculate two-loop corrections in scalar QED using dimensional regularization.

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