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Parton distribution functions are distributions of quarks and gluons inside the nucleon, but besides the three valence quarks, they are virtual particles right? So if virtual particles are just a mathematical tool to calculate amplitudes, what exactly are these distributions? Is the quark-gluon sea inside the nucleon actually physical?

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This is a partial answer that may be deleted if a better is given.

Reading this evaluation from data of parton distribution functions may help you understand the complexity of the matter.

Quantum field theory can be used to model the complexity of the proton, described qualitatively here. In the above reference the specific theoretical models used to evaluate the data are referenced. For the specific reactions in the paper . These models are entered into the monte carlo they use to fit for the parton distributions for the proton, from the specific data.

The models are mathematically complicated. The virtual lines exchanged in Feynman diagrams are only calculational and included in the mathematical formulation, but not as particles.

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