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Effective Hamiltonian for QCD evolution at high energy

Y. Hatta, E. Iancu, L. McLerran, A. Stasto, D. N. Triantafyllopoulos

TL;DR

The paper derives a self-dual effective Hamiltonian for high-energy QCD evolution in the leading logarithmic approximation, formulated as a two-dimensional theory in the transverse plane with two kinds of Wilson lines, $V$ (longitudinal) and $W$ (temporal). The renormalization-group analysis integrates out semi-fast gluons to produce a one-loop effective action, which can be rewritten as a Hamiltonian acting on Wilson-line observables; in the dense limit it reduces to the JIMWLK Hamiltonian, while in the dilute limit it becomes the Bremsstrahlung (BREM) Hamiltonian, dual to JIMWLK. This framework unifies gluon merging and splitting (and their fluctuations) into a boost-invariant description and exhibits a self-duality under exchange of the plus and minus Wilson lines, consistent with Pomeron-loop physics. The work connects and extends prior CGC, Balitsky–JIMWLK, and reggeized-gluon formalisms, and lays groundwork for a complete two-dimensional Hamiltonian theory of high-energy QCD evolution.

Abstract

We construct the effective Hamiltonian which governs the renormalization group flow of the gluon distribution with increasing energy and in the leading logarithmic approximation. This Hamiltonian defines a two-dimensional field theory which involves two types of Wilson lines: longitudinal Wilson lines which describe gluon recombination (or merging) and temporal Wilson lines which account for gluon bremsstrahlung (or splitting). The Hamiltonian is self-dual, i.e., it is invariant under the exchange of the two types of Wilson lines. In the high density regime where one can neglect gluon number fluctuations, the general Hamiltonian reduces to that for the JIMWLK evolution. In the dilute regime where gluon recombination becomes unimportant, it reduces to the dual partner of the JIMWLK Hamiltonian, which describes bremsstrahlung.

Effective Hamiltonian for QCD evolution at high energy

TL;DR

The paper derives a self-dual effective Hamiltonian for high-energy QCD evolution in the leading logarithmic approximation, formulated as a two-dimensional theory in the transverse plane with two kinds of Wilson lines, (longitudinal) and (temporal). The renormalization-group analysis integrates out semi-fast gluons to produce a one-loop effective action, which can be rewritten as a Hamiltonian acting on Wilson-line observables; in the dense limit it reduces to the JIMWLK Hamiltonian, while in the dilute limit it becomes the Bremsstrahlung (BREM) Hamiltonian, dual to JIMWLK. This framework unifies gluon merging and splitting (and their fluctuations) into a boost-invariant description and exhibits a self-duality under exchange of the plus and minus Wilson lines, consistent with Pomeron-loop physics. The work connects and extends prior CGC, Balitsky–JIMWLK, and reggeized-gluon formalisms, and lays groundwork for a complete two-dimensional Hamiltonian theory of high-energy QCD evolution.

Abstract

We construct the effective Hamiltonian which governs the renormalization group flow of the gluon distribution with increasing energy and in the leading logarithmic approximation. This Hamiltonian defines a two-dimensional field theory which involves two types of Wilson lines: longitudinal Wilson lines which describe gluon recombination (or merging) and temporal Wilson lines which account for gluon bremsstrahlung (or splitting). The Hamiltonian is self-dual, i.e., it is invariant under the exchange of the two types of Wilson lines. In the high density regime where one can neglect gluon number fluctuations, the general Hamiltonian reduces to that for the JIMWLK evolution. In the dilute regime where gluon recombination becomes unimportant, it reduces to the dual partner of the JIMWLK Hamiltonian, which describes bremsstrahlung.

Paper Structure

This paper contains 9 sections, 118 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The scattering problem in the CGC formalism
  3. Quantum evolution: The general philosophy
  4. Quantum evolution: The effective action
  5. The background field propagator
  6. The high density regime: JIMWLK Hamiltonian
  7. The general effective action
  8. The low density regime: Bremsstrahlung Hamiltonian
  9. Conclusions

Figures (3)

  • Figure 1: The 2--point function in the effective theory (a) and its BFKL evolution in one step (b).
  • Figure 2: One--step quantum evolution with gluon number changing vertices: (a) a $n \rightarrow 2$ merging process which contributes to the JIMWLK Hamiltonian; (b) a $2 \rightarrow n$ splitting process as included in the BREM Hamiltonian; (c) a general $m\rightarrow n$ process.
  • Figure 3: The distribution of the various background fields in the $x^+ - x^-$ plane. The overlap region around the tip of the light--cone is the interaction region relevant for quantum evolution (see Sect. \ref{['S_prop']}). The contour surrounding the interaction region will be explained in Sect. \ref{['S_GEN']}.