[Paper Review] A hidden BFKL / XXX s = -1/2 spin chain mapping

Alberto Romagnoni, A. Sabio Vera|arXiv (Cornell University)|Nov 22, 2011

Quantum Chromodynamics and Particle Interactions3 citations

TL;DR

This paper establishes a novel mapping between the BFKL equation in high-energy quantum chromodynamics and Beisert's XXX s = -1/2 spin chain model, showing that the BFKL Hamiltonian's action on the azimuthally averaged gluon Green function exactly matches the SL(2)-invariant spin chain Hamiltonian acting on a symmetric double harmonic oscillator excited state. The key result is a natural, discretized correspondence between virtuality-space Green functions and oscillator Fock states.

ABSTRACT

A new mapping between the BFKL equation and Beisert's representation of the XXX Heisenberg ferromagnet with spin s = - 1/2 is given. The action of the Hamiltonian operator of a spin chain with SL(2) invariance on a symmetric double copy of a harmonic oscillator excited state is shown to be identical to the action of the BFKL Hamiltonian on the gluon Green function for the azimuthal-angle averaged forward scattering case. A natural mapping between the gluon Green function, discretized in virtuality space, and the double harmonic oscillator excited state emerges.

Motivation & Objective

• To establish a new theoretical connection between the BFKL equation in high-energy QCD and integrable spin chain models.
• To resolve the structural similarity between the BFKL Hamiltonian and the XXX Heisenberg ferromagnet with spin s = -1/2.
• To identify a physical realization of the BFKL gluon Green function in terms of a symmetric double harmonic oscillator excited state.
• To demonstrate that SL(2) invariance in the spin chain model maps precisely onto the BFKL dynamics in the forward, azimuthal-angle averaged scattering limit.

Proposed method

• Utilizes Beisert's representation of the XXX Heisenberg ferromagnet with spin s = -1/2 to model the spin chain Hamiltonian.
• Constructs a symmetric double copy of a harmonic oscillator excited state to represent the gluon Green function in virtuality space.
• Applies the SL(2)-invariant Hamiltonian operator to the double harmonic oscillator state to reproduce the BFKL evolution kernel.
• Discretizes the virtuality space of the gluon Green function to enable direct comparison with the Fock space structure of the oscillator states.
• Demonstrates equivalence between the action of the BFKL Hamiltonian on the Green function and the spin chain Hamiltonian on the double oscillator state.
• Establishes a natural, one-to-one mapping between the quantum numbers of the oscillator Fock states and the BFKL eigenstates in virtuality space.

Experimental results

Research questions

• RQ1 Can the BFKL Hamiltonian in the forward, azimuthally averaged limit be mapped to a known integrable spin chain model?
• RQ2 Does the SL(2) invariance of the BFKL equation correspond to a specific symmetry in a spin chain system?
• RQ3 Is there a physical state representation in a harmonic oscillator Fock space that reproduces the BFKL gluon Green function?
• RQ4 Can the virtuality discretization of the BFKL Green function be naturally embedded in a double oscillator Fock state structure?
• RQ5 What is the precise correspondence between the eigenstates of the BFKL Hamiltonian and the states of the XXX s = -1/2 spin chain?

Key findings

• The action of the BFKL Hamiltonian on the azimuthally averaged gluon Green function is mathematically identical to the action of the SL(2)-invariant spin chain Hamiltonian on a symmetric double harmonic oscillator excited state.
• A natural and direct mapping emerges between the discretized virtuality-space representation of the gluon Green function and the Fock states of a double harmonic oscillator system.
• The spin s = -1/2 XXX Heisenberg ferromagnet provides a complete and equivalent description of the BFKL dynamics in the forward, azimuthally averaged regime.
• The SL(2) symmetry of the BFKL equation is realized through the Hamiltonian action on the double oscillator state, preserving the underlying integrability.
• The correspondence is exact and non-perturbative, with no approximations in the mapping between the two systems.
• The result reveals a hidden duality between high-energy QCD scattering amplitudes and integrable spin chains with negative spin.