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High Energy Evolution - The Wave Function Point of View

Alex Kovner

TL;DR

The work presents a wave-function–based treatment of high-energy QCD evolution, deriving the JIMWLK equation within a Hamiltonian framework and clarifying its connections to the dipole picture, BFKL dynamics, and saturation phenomena. It analyzes how boost-induced Weizsacker–Williams fields drive nonlinear evolution, leading to the BK equation in the appropriate limits and to geometric scaling of scattering amplitudes. The notes critically examine limitations of JIMWLK, discuss potential fixes via density-dependent nonlinearities, and propose a self-duality property that constrains the full evolution kernel, including Pomeron loops. The discussion outlines a path toward a complete understanding of high-energy evolution and its observable consequences, highlighting both theoretical challenges and areas of ongoing progress.

Abstract

These lectures discuss aspects of high energy evolution in QCD. This includes the derivation of the JIMWLK equation, basic physics of its solutions and recent work on inclusion of Pomeron loops. The entire discussion is given in the Hamiltonian framework which gives direct access to the evolution of hadronic wave function under Lorentz boost.

High Energy Evolution - The Wave Function Point of View

TL;DR

The work presents a wave-function–based treatment of high-energy QCD evolution, deriving the JIMWLK equation within a Hamiltonian framework and clarifying its connections to the dipole picture, BFKL dynamics, and saturation phenomena. It analyzes how boost-induced Weizsacker–Williams fields drive nonlinear evolution, leading to the BK equation in the appropriate limits and to geometric scaling of scattering amplitudes. The notes critically examine limitations of JIMWLK, discuss potential fixes via density-dependent nonlinearities, and propose a self-duality property that constrains the full evolution kernel, including Pomeron loops. The discussion outlines a path toward a complete understanding of high-energy evolution and its observable consequences, highlighting both theoretical challenges and areas of ongoing progress.

Abstract

These lectures discuss aspects of high energy evolution in QCD. This includes the derivation of the JIMWLK equation, basic physics of its solutions and recent work on inclusion of Pomeron loops. The entire discussion is given in the Hamiltonian framework which gives direct access to the evolution of hadronic wave function under Lorentz boost.

Paper Structure

This paper contains 14 sections, 116 equations, 8 figures.

Table of Contents

  1. Introduction
  2. JIMWLK as we know it.
  3. The forward amplitude
  4. What of boost?
  5. The Light Cone Hamiltonian.
  6. Forty two
  7. JIMWLK and friends
  8. The dipole model
  9. The BFKL limit
  10. The saturation momentum and the geometric scaling.
  11. What's wrong with JIMWLK?
  12. The simpleminded fix.
  13. The self duality of high energy evolution
  14. So long and thanks for all the fish

Figures (8)

  • Figure 1: Deeply Inelastic Scattering
  • Figure 2: Cartoon of the DGLAP evolution. With improved transverse resolution the number of partons grows but the density decreases.
  • Figure 3: Cartoon of the low x evolution. As the energy increases the partons of fixed transverse size multiply. The density in the transverse plain grows.
  • Figure 4: Schematic portrait of a hadron at high energy. A black central region is surrounded by "gray" disc where the saturation momentum $Q_s$ is of order $\Lambda_{QCD}$.
  • Figure 5: Growth of the longitudinal phase space under boost.
  • ...and 3 more figures