In another question, I was asking about the interaction between photons and electrons. It was suggested that in compton scattering there is no particle to "mediate" the force. The tree-level diagrams in compton scattering of photon and electron are
My questions are:
Why does one not consider the electron to mediate a force between photon and electron? Clearly there is a force, as the two particles scatter, therefore they interact.
Why do force-mediators have to be bosons in general?
If possible I would really prefer an answer which is not just heuristics/storytelling and is grounded in the theory itself. Edit: A helpful link to essentially the same question has been shared. However, I don't see how the answers are satisfactory to rule out electrons as the force-carrier in the diagrams above. I will explain for each answer.
In the first answer, by Ben Crowell, I see two issues: The spins can add up: $1 - 1/2 = 1/2$, but more importantly, Feynman diagrams are a counting technique to get the combinatorics right. They are strongly suggestive of interactions, but there is no inconsistency if angular momentum is not conserved in a diagram. Otherwise, supposing my spin addition is somehow wrong (which I suspect) then naively, we could use this to predict that electrons and photons do not interact because their mediator would have to be an electron, which would violate angular momentum conservation at tree level, and behold! that prediction would be wrong.
In the second answer by Adam, it suggests that the opposite is true, namely that electrons can be thought of as force carriers.
In the third answer by Trimok, it only addresses interactions between fermions so that's just not the case here.