Factorization at the LHC: From PDFs 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 \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.

• The paper presents a new factorization theorem for isolated Drell-Yan processes by incorporating beam functions instead of traditional PDFs.
• The methodology utilizes Soft-Collinear Effective Theory to segregate hard, beam, and soft scale contributions for enhanced precision.
• The approach aligns with Monte Carlo simulations, providing a more accurate analysis of experimental data from proton-proton collisions at the LHC.

Analysis of Factorization in Proton-Proton Collisions

The paper provides a comprehensive study of the factorization in proton-(anti)proton collisions with a focus on the isolated Drell-Yan process at the Large Hadron Collider (LHC). The authors, Stewart, Tackmann, and Waalewijn from MIT, introduce a framework to analyze these collisions by accounting for experimental restrictions on the hadronic final state and general parton momentum fractions. The paper explores the complexities of handling initial-state jets and introduces the concept of beam functions.

In the context of high-energy particle physics, factorization is crucial for separating perturbative QCD calculations from non-perturbative effects described by parton distribution functions (PDFs). Traditionally, factorization theorems have primarily focused on inclusive processes or scenarios like threshold Drell-Yan processes. The study presented here expands on these by defining the isolated Drell-Yan process, which involves specific restrictions on the final state, preventing central jet production. This is particularly relevant for the LHC, where experimental analyses often need to isolate certain signals from the background by imposing constraints.

The paper puts forth a new type of factorization theorem for processes with such restrictions, providing a detailed calculation for the Drell-Yan process where the lepton pair production is accompanied by no central jets. The research indicates that standard PDFs at the hard interaction scale are insufficient to describe the initial state due to low-energy measurements. Instead, beam functions, which account for the transverse virtuality (or off-shellness) of the colliding partons, are essential. This innovative approach significantly modifies the factorization landscape by introducing these functions, which account for the jet-like structure within the initial-state protons formed just before the hard collision.

The authors describe a robust factorization framework in SCET (Soft-Collinear Effective Theory) that separates the dependencies on the hard scale, the beam scale, and the soft scale. The beam functions replace PDFs in the factorized cross section and are evaluated at an intermediate perturbative scale. They are convoluted with a soft function, adding a level of detail that refines the description of the initial states. Such a framework is necessary to accurately account for the large double logarithms that arise due to phase space constraints in an experiment at a collider, allowing the cross section to be resummed.

Moreover, the paper discusses the correspondence of their approach with Monte Carlo simulations, specifically on how initial-state parton showers could potentially incorporate the beam function effects. This represents a step toward a more precise theoretical foundation for comparing simulations with actual collision data, which is critical for validating the predictions of the Standard Model of particle physics or uncovering signs of new physics.

The implications of this research are far-reaching, influencing both theoretical and experimental methods at colliders. Practically, the introduction of beam functions provides a more accurate toolset for analyzing collider data where experimental cuts on the final state are imposed. Theoretically, this research opens up pathways for exploring other processes beyond isolated Drell-Yan, where differential cross sections are pivotal, suggesting that the factorization approach utilized here can be generalized. Future work in this area could further refine our understanding of QCD at hadron colliders and enhance the precision of phenomenological analyses in high-energy physics.

To summarize, the authors provide a pioneering approach for factorizing cross sections with realistic experimental constraints. They introduce and validate a novel methodology through the incorporation of beam functions, significantly advancing the field's ability to handle complex measurements at collider experiments like those at the LHC.