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Evolution of Color Exchange in QCD Hard Scattering

Nikolaos Kidonakis, Gianluca Oderda, George Sterman

TL;DR

This paper develops a coherent framework for the color evolution of QCD hard scattering due to noncollinear soft gluon emission. By modeling partons with Wilson lines, it derives and diagonalizes soft anomalous dimension matrices Γ_S for all 2→2 light-parton processes at one loop, including qq→qq, q̄q→gg, qg→qg, and gg→gg, with explicit eigenvalues and eigenvectors. The results underpin the resummation of threshold logarithms in dijet production and clarify how color exchange channels control the infrared structure of scattering amplitudes. The systematic color-space diagonalization reveals that, in forward kinematics, singlet exchange is enhanced and higher-dimensional color representations are progressively suppressed, informing practical calculations of hard-scattering cross sections at high energies.

Abstract

In QCD hard scattering cross sections, the color content of the underlying hard scattering evolves with a factorization scale. This evolution is controlled by an anomalous dimension matrix, specific to each hard-scattering reaction. Anomalous dimensions are determined from the renormalization of products of ordered exponentials of the gauge field, which describe the coherent radiation of gluons by incoming hadrons and the observed jets or particles of the final state. The anomalous dimensions depend on the kinematics of the underlying hard scattering, but are free of collinear singularities. A number of these matrices are available in the literature. Here, we exhibit one-loop mixing matrices for the full list of $2 \to 2$ reactions involving light quarks and gluons. The eight-by-eight anomalous dimension matrix for gluon-gluon scattering shows a simplified structure in the basis corresponding to definite color exchange in the $t$-channel.

Evolution of Color Exchange in QCD Hard Scattering

TL;DR

This paper develops a coherent framework for the color evolution of QCD hard scattering due to noncollinear soft gluon emission. By modeling partons with Wilson lines, it derives and diagonalizes soft anomalous dimension matrices Γ_S for all 2→2 light-parton processes at one loop, including qq→qq, q̄q→gg, qg→qg, and gg→gg, with explicit eigenvalues and eigenvectors. The results underpin the resummation of threshold logarithms in dijet production and clarify how color exchange channels control the infrared structure of scattering amplitudes. The systematic color-space diagonalization reveals that, in forward kinematics, singlet exchange is enhanced and higher-dimensional color representations are progressively suppressed, informing practical calculations of hard-scattering cross sections at high energies.

Abstract

In QCD hard scattering cross sections, the color content of the underlying hard scattering evolves with a factorization scale. This evolution is controlled by an anomalous dimension matrix, specific to each hard-scattering reaction. Anomalous dimensions are determined from the renormalization of products of ordered exponentials of the gauge field, which describe the coherent radiation of gluons by incoming hadrons and the observed jets or particles of the final state. The anomalous dimensions depend on the kinematics of the underlying hard scattering, but are free of collinear singularities. A number of these matrices are available in the literature. Here, we exhibit one-loop mixing matrices for the full list of reactions involving light quarks and gluons. The eight-by-eight anomalous dimension matrix for gluon-gluon scattering shows a simplified structure in the basis corresponding to definite color exchange in the -channel.

Paper Structure

This paper contains 19 sections, 128 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Soft Anomalous Dimensions from Wilson Lines
  3. Soft Functions in Dijet Production
  4. Color Evolution in Jet Cross Sections
  5. Evolution for Soft and Hard Functions
  6. Diagonalization of the Color Basis
  7. Anomalous Dimensions for Quark-initiated Scattering
  8. Formalism
  9. Soft anomalous dimension for $q \bar{q}\rightarrow q \bar{q}$
  10. Soft anomalous dimension for $q q\rightarrow q q$
  11. Soft anomalous dimension for $q \bar{q}\rightarrow g g$ and $g g \rightarrow q \bar{q}$
  12. Soft anomalous dimension for $qg \rightarrow qg$ and $\bar{q} g \rightarrow \bar{q} g$
  13. Soft anomalous dimensions for $gg \rightarrow gg$
  14. Initial basis and $\Gamma_{S'}$ at one loop
  15. Block-diagonal form of the anomalous dimension matrix: eigenvalues and eigenvectors
  16. ...and 4 more sections

Figures (7)

  • Figure 1: Eikonal vertex correction and eikonal self-energy for the Drell Yan process.
  • Figure 2: Identity representing factorization in leading regions of corrections to the eikonal vertex $c_I$. ${\bar{S}}$ represents lines of zero momentum, while A, B, 1 and 2 label jet subdiagrams.
  • Figure 3: Reduced diagram representing the leading region for the dijet cross section.
  • Figure 4: Eikonal vertex corrections for partonic processes contributing to the soft anomalous dimension matrices.
  • Figure 5: Graphical representation of the initial color basis for the $gg \rightarrow gg$ process. All lines have color content only.
  • ...and 2 more figures