  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 \mu_B << Q and probes the proton prior to the hard
collision. This corresponds to evaluating the PDFs at the scale \mu_B. 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 \mu_B, the beam function factorizes into a
convolution of calculable Wilson coefficients and PDFs. Below \mu_B, the
initial-state evolution is described by the usual PDF evolution which changes
x, while above \mu_B it is governed by a different renormalization group
evolution which sums double logarithms of \mu_B/Q and leaves x fixed. As an
example, we prove a factorization theorem for "isolated Drell-Yan", pp -> Xl+l-
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.
