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Small x Phenomenology: Summary and Status

The Small x Collaboration, Bo Andersson, S. Baranov, J. Bartels, M. Ciafaloni, J. Collins, M. Davidsson, G. Gustafson, H. Jung, L. Joensson, M. Karlsson, M. Kimber, A. Kotikov, J. Kwiecinski, L. Loennblad, G. Miu, G. Salam, M. H. Seymour, T. Sjoestrand, N. Zotov

TL;DR

The paper surveys the status of small-$x$ phenomenology, arguing that high-energy QCD requires resummation of large logarithms beyond traditional DGLAP. It foregrounds $k_\perp$-factorization with unintegrated gluon densities and the CCFM/LDC formalisms as unifying frameworks bridging BFKL and DGLAP, with multiple UPD parameterizations and dedicated Monte Carlo generators. Major contributions include the development of off-shell matrix elements, advances toward NLL corrections, and the demonstration that forward-jet and $F_2$ data can be described within this small-$x$ formalism. The work also identifies critical theoretical challenges—gauge invariance beyond LO, NL corrections, and UPD constraints—and argues for coordinated collaboration to constrain UPDs via more exclusive measurements.

Abstract

The aim of this paper is to summarize the general status of our understanding of small x physics. It is based on presentations and discussions at an informal meeting on this topic held in Lund, Sweden, in March 2001. This document also marks the founding of an informal collaboration between experimentalists and theoreticians with a special interest in small x physics. This paper is dedicated to the memory of Bo Andersson, who died unexpectedly from a heart attack on March 4th, 2002.

Small x Phenomenology: Summary and Status

TL;DR

The paper surveys the status of small- phenomenology, arguing that high-energy QCD requires resummation of large logarithms beyond traditional DGLAP. It foregrounds -factorization with unintegrated gluon densities and the CCFM/LDC formalisms as unifying frameworks bridging BFKL and DGLAP, with multiple UPD parameterizations and dedicated Monte Carlo generators. Major contributions include the development of off-shell matrix elements, advances toward NLL corrections, and the demonstration that forward-jet and data can be described within this small- formalism. The work also identifies critical theoretical challenges—gauge invariance beyond LO, NL corrections, and UPD constraints—and argues for coordinated collaboration to constrain UPDs via more exclusive measurements.

Abstract

The aim of this paper is to summarize the general status of our understanding of small x physics. It is based on presentations and discussions at an informal meeting on this topic held in Lund, Sweden, in March 2001. This document also marks the founding of an informal collaboration between experimentalists and theoreticians with a special interest in small x physics. This paper is dedicated to the memory of Bo Andersson, who died unexpectedly from a heart attack on March 4th, 2002.

Paper Structure

This paper contains 21 sections, 65 equations, 10 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The $k_\perp$-factorization approach
  3. Off-shell matrix elements
  4. Order $\alpha_s$ off-shell matrix elements
  5. Next-to-Leading corrections to $\alpha_s$ off-shell matrix elements
  6. Unintegrated parton distributions
  7. Introduction
  8. The CCFM evolution
  9. LDC and the transition between BFKL and DGLAP
  10. The CCFM equation in the transverse coordinate representation
  11. Available parameterizations of unintegrated gluon distributions
  12. Beyond leading logarithms
  13. General next-to-leading order investigations
  14. Anomalous dimensions.
  15. CCFM and problems beyond leading logs
  16. ...and 6 more sections

Figures (10)

  • Figure 1: Schematic picture of a typical unitarity diagram for deep inelastic scattering. An incoming proton with a large positive light-cone momentum, $P^+$, is being probed by a photon with a large virtuality and a large negative light-cone momentum. The photon scatters on a parton from the proton with space-like momentum $k$.
  • Figure 2: Kinematic variables for multi-gluon emission. The $t$-channel gluon momenta are given by $k_i$ and the gluons emitted in the initial state cascade have momenta $p_i$. The upper angle for any emission is obtained from the quark box, as indicated with $\Xi$. We define $z_{\pm i}=k_{\pm i}/k_{\pm (i\mp 1)}$ and $q_i=p_{\perp i}/(1-z_{+i})$.
  • Figure 3: Diagrammatic representation of LO, NLO and resolved photon processes in the collinear approach (top row) and compared to the $k_\perp$-factorization approach.
  • Figure 4: Schematic diagrams for $k_\perp$-factorization: $(a)$ shows the general case for hadroproduction of (heavy) quarks. $(b)$ shows the one-loop correction to the Born diagram for photoproduction $(c)$ shows the all-loop improved correction with the factorized structure function ${\cal F}(x,k_\perp^2)$ or ${\cal A}(x,k_\perp^2,\bar{q}^2)$.
  • Figure 5: Schematic diagrams for next-to-leading contributions: $(a)$ virtual corrections, $(b)$ real corrections, $(c)$ the process $\gamma^* +q \to (q \bar{q})+q$, $(d)$ the process $\gamma^* +q \to (q \bar{q} g)+q$, $(e)$ diagrams showing the reggeization of the gluon
  • ...and 5 more figures