DESY 14-205 Multiple hard scattering and parton correlations in the proton

The standard description of hard processes in proton-proton (and other hadronhadron) collisions uses the concept of factorization. In a standard factorization formula, the proton-proton cross section is given by the convolution of a parton density for each proton with a hard cross section for the interaction of two partons, summed over the relevant combinations of parton species. The physical picture suggested by this formula, represented in the left panel of Fig. 1, is deceptively simple since it suggests that the only interaction taking place in the collision is between the two partons initiating the hard subprocess. This is certainly not the case: the “spectator partons” in each proton are colored and will interact with their counterparts in the other proton, as sketched in the right panel of Fig. 1. To solve this apparent contradiction, we first note that the factorization formula describes semi-inclusive processes of the type pp→ Y +X. Here Y is the set of particles produced by the hard scattering (a lepton pair in Fig. 1), whereas X denotes all other final-state particles. One can specify all details of Y (and compute them from the hard scattering) but must sum over all possible hadronic states in X. The “spectator” interactions just mentioned affect particles in X, and it is a non-trivial statement of the factorization formula that—thanks to unitarity—one can ignore these interactions when computing the inclusive cross section ∑ X σ(pp→ Y +X).