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Integrability and WKB solution of twist-three evolution equations

A. V. Belitsky

TL;DR

Belitsky demonstrates that twist-three quark-gluon evolution equations in QCD map to an integrable one-dimensional open spin chain with three sites in the large-$N_c$ limit. By exploiting a hidden integral of motion, the diagonalization problem reduces to a master equation solvable via a WKB expansion for large total conformal spin $J$, yielding analytic energy (anomalous dimension) spectra in limiting regimes. The approach combines conformal basis construction, Racah decomposition, and Quantum Inverse Scattering Techniques, with numerical diagonalization validating the analytic results. The framework advances understanding of scale dependence of twist-three distributions and suggests extensions to $1/N_c^2$ effects and to the chiral-even sector, potentially impacting phenomenology of higher-twist observables.

Abstract

We identify an integrable one-dimensional inhomogeneous three-site open spin chain which arises in the problem of diagonalization of twist-three quark-gluon evolution equations in QCD in the chiral-odd sector. Making use of the existence of a non-trivial `hidden' integral of motion the problem of diagonalization of the evolution kernels is reduced to the study of a second order finite-difference equation which is solved in WKB approximation for large conformal spins of the three-particle system. The energies (alias anomalous dimensions) of eigenstates with different scale dependence are found in limiting cases and compared with numerical calculations.

Integrability and WKB solution of twist-three evolution equations

TL;DR

Belitsky demonstrates that twist-three quark-gluon evolution equations in QCD map to an integrable one-dimensional open spin chain with three sites in the large- limit. By exploiting a hidden integral of motion, the diagonalization problem reduces to a master equation solvable via a WKB expansion for large total conformal spin , yielding analytic energy (anomalous dimension) spectra in limiting regimes. The approach combines conformal basis construction, Racah decomposition, and Quantum Inverse Scattering Techniques, with numerical diagonalization validating the analytic results. The framework advances understanding of scale dependence of twist-three distributions and suggests extensions to effects and to the chiral-even sector, potentially impacting phenomenology of higher-twist observables.

Abstract

We identify an integrable one-dimensional inhomogeneous three-site open spin chain which arises in the problem of diagonalization of twist-three quark-gluon evolution equations in QCD in the chiral-odd sector. Making use of the existence of a non-trivial `hidden' integral of motion the problem of diagonalization of the evolution kernels is reduced to the study of a second order finite-difference equation which is solved in WKB approximation for large conformal spins of the three-particle system. The energies (alias anomalous dimensions) of eigenstates with different scale dependence are found in limiting cases and compared with numerical calculations.

Paper Structure

This paper contains 20 sections, 97 equations, 4 figures.

Table of Contents

  1. Brief phenomenological profile and outline.
  2. Approach to scaling violation phenomena.
  3. Evolution equation.
  4. Eigenvalue problem.
  5. Conformal basis: General formalism.
  6. Auxiliary $\theta$-space.
  7. Fundamental basis.
  8. Matrix elements.
  9. Other representations of the basis.
  10. Racah decomposition.
  11. Three-site open spin chain and integrability.
  12. Master equation.
  13. Quantization of integral of motion.
  14. Description from below.
  15. Description from above.
  16. ...and 5 more sections

Figures (4)

  • Figure 1: The diagrammatical representation of the generic form of the leading order evolution equation for the three-parton correlator $Z (x_1, x_2, x_3)$ in the light-cone gauge. The blob $K$ stands for the pair-wise kernel of interaction of $t$-channel particles.
  • Figure 2: The spectrum of the conserved charge $q_T$ and a few trajectories from different sets (see the text).
  • Figure 3: The energy spectrum and the trajectories (in dashed) defined by Eq. (\ref{['TopAfterGap']}). The lowest two eigenvalues (solid curves) are given by (\ref{['ExactE']}).
  • Figure 4: The spectrum of energy and a few (dashed) trajectories described by the analytical formulae of section \ref{['EnergyAbove']}.