Factorization at the LHC: From parton distribution functions to initial state jets

We study proton-(anti)proton collisions at the LHC or Tevatron in the presence of experimental restrictions on the hadronic final state and for generic parton momentum fractions. At the scale Q of the hard interaction, factorization does not yield standard parton distribution functions (PDFs) for the initial state. The measurement restricting the hadronic final state introduces a new scale μBQ and probes the proton prior to the hard collision. This corresponds to evaluating the PDFs at the scale μB. After the proton is probed, the incoming hard parton is contained in an initial-state jet, and the hard collision occurs between partons inside these jets rather than inside protons. The proper description of such initial-state jets requires ``beam functions''. At the scale μB, the beam function factorizes into a convolution of calculable Wilson coefficients and PDFs. Below μB, the initial-state evolution is described by the usual PDF evolution which changes x, while above μB it is governed by a different renormalization group evolution that sums double logarithms of μB/Q and leaves x fixed. As an example, we prove a factorization theorem for ``isolated Drell-Yan'', pp→X⁺^- where X is restricted to have no central jets. We comment on the extension to cases where the hadronic final state contains a certain number of isolated central jets.