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

Abstract

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 ${\ensuremath{\mu}}_{B}\ensuremath{\ll}Q$ and probes the proton prior to the hard collision. This corresponds to evaluating the PDFs at the scale ${\ensuremath{\mu}}_{B}$. 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 ${\ensuremath{\mu}}_{B}$, the beam function factorizes into a convolution of calculable Wilson coefficients and PDFs. Below ${\ensuremath{\mu}}_{B}$, the initial-state evolution is described by the usual PDF evolution which changes $x$, while above ${\ensuremath{\mu}}_{B}$ it is governed by a different renormalization group evolution that sums double logarithms of ${\ensuremath{\mu}}_{B}/Q$ and leaves $x$ fixed. As an example, we prove a factorization theorem for ``isolated Drell-Yan'', $pp\ensuremath{\rightarrow}X{\ensuremath{\ell}}^{+}{\ensuremath{\ell}}^{\ensuremath{-}}$ 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.