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*R.K. Ellis et al.: QCD and Collider Physics, pp. 99*to understand how to arrive at the parton density functions starting from the matrix elements in electron-proton deep inelastic scattering (see figure below). But there seems to be a very fundamental concept that I don't understand at all.

In the book mentioned above, the matrix element is split up into a leptonic part ##M_\nu## and a hadronic part ##M^\prime_\mu## connected by the photon propagator (see figure below). From the hadronic part (without the photon propagator attached) they then calculate a matrix element, square it, average over spin, polarisations and colours and then calculate a differential cross section from it.

There are several points, which confuse me about this.

- Matrix elements have external states attached to it. Here, however the leptonic part with its spinors is missing. How is it possible to define a physically meaningful S-matrix element from the hadronic part only?
- How do the differential cross section of the entire Feynman graph and of the hadronic subgraph relate to each other?
- How can I justify throwing out the entire leptonic subgraph from all my calculations, even when squaring the matrix element? After all ##|M|^2=|M_\nu\frac{-ig^{\mu\nu}}{q^2}M^\prime_\mu|^2\neq|M_\nu\frac{-ig^{\mu\nu}}{q^2}|^2\times|M^\prime_\mu|^2##.