I've been seeing many feynman diagrams lately that I can't quite interpret yet. I've heard a basic Quantum Field Theory lecture and so to me, a Feynman diagram is simply a mnemonic picture to quickly write down (and remember) all of the possible terms in the perturbation series of the matrix element in scattering.
But they seem to be much more than that.
Consider for example this diagram of proton-neutron scattering via pion exchange. The picture seems to have an intuitive meaning, but how can it be a valid Feynman diagram? Wouldn't that imply that there is an underlying theory with a proper Lagrangian from which Feynman rules can be derived that would assign a number to this diagram?
Question: When describing nucleon-nucleon interactions in this way, do people actually write down a Lagrangian and derive those rules? (I am confused because I've seen diagrams like that in several books and they usually write down the cross sections simply from analogy arguments to other theories.)
Any clarification will be greatly appreciated,