On the interpretation of Feynman diagrams, or, did the LHC experiments observe H → γγ?

According to the received view Feynman diagrams are a bookkeeping device in complex perturbative calculations. Thus, they do not provide a representation or model of the underlying physical process. This view is in apparent tension with scientific practice in high energy physics, which analyses its data in terms of “channels”. For example the Higgs discovery was based on the observation of the decay H → γγ – a process which can be easily represented by the corresponding Feynman diagrams. I take issue with this tension and show that on closer analysis the story of the Higgs discovery should be told differently.