Factorization at the LHC: From PDFs to Initial State Jets

TL;DR

This paper introduces beam functions to accurately describe initial-state jets in proton collisions at the LHC, accounting for experimental restrictions and new scales, thus extending the traditional PDF framework.

Contribution

It develops a factorization framework involving beam functions and their evolution, bridging the gap between PDFs and initial-state jets in collider processes.

Findings

Beam functions factorize into Wilson coefficients and PDFs at the scale .

Initial-state evolution differs above and below the scale .

Proven a factorization theorem for isolated Drell-Yan process.

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 \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.