It's helpful to visualize this for connected sums of surfaces. Instead of removing a whole disk from each surface and attaching along the boundary, imagine only removing the two center points, and then deforming the punctured disks into cylinders sticking out of the surface, and then you attach the cylinders together.

Thank you for the references. The Steinmann book is particularly relevant since it includes a discussion of Compton scattering in the last section.

@CarloBeenakker The question is about the former, a mathematically well-defined but still perturbative version of these computations. Perhaps if convergence isn't known my question is to what extent can they be made well-defined rigorously.

@IgorKhavkine I have looked at several questions here but they all seem to be concerned with the construction of rigorous QFTs à la Wightman axioms etc., and not effective field theories.