Renormalization of twist-three operators and integrable lattice models

TL;DR

The paper tackles the renormalization group evolution of twist-three quark–gluon–quark and three-gluon operators in QCD at leading order, focusing on their Q^2 dependence and transverse-spin observables like g2.It exploits an integrability structure by mapping the gluonic sector to an almost-integrable 1D lattice model (a generalized XXX chain with s = -3/2) plus a perturbation, and the quark–gluon sector to an open inhomogeneous spin chain in the large-Nc limit, enabling analytical control.Using conformal symmetry, Bethe ansatz/Baxter techniques, and a quasiclassical (WKB) expansion in total conformal spin J, the authors obtain explicit spectra, recursion relations for eigenfunctions, and quantization conditions, showing good agreement with numerical diagonalization and offering practical estimates for scale-dependent observables.The work provides a framework to describe twist-three evolution with high precision, clarifying how integrability underpins the structure of anomalous dimensions and suggesting pathways for phenomenological applications in polarized scattering and g2 analyses.

Abstract

We address the problem of solution of the QCD three-particle evolution equations which govern the Q-dependence of the chiral-even quark-gluon-quark and three-gluon correlators contributing to a number of asymmetries at leading order and the transversely polarized structure function g_2(x). The quark-gluon-quark case is completely integrable in multicolour limit and corresponds to a spin chain with non-periodic boundary conditions, while the pure gluonic sector contains, apart from a piece in the Hamiltonian equivalent to XXX Heisenberg magnet of spin s = - 3/2, a non-integrable addendum which can be treated perturbatively for a bulk of the spectrum except for a few lowest energy levels. We construct a quasiclassical expansion with respect to the total conformal spin of the system and describe fairly well the energy spectra of quark-gluon-quark and three-gluon systems.

Figures (12)

• Figure 1: Diagrams (in the light-cone gauge) giving rise to the gluon-gluon two-particle evolution kernel. The graph with the crossed gluon line corresponds to the contact-type contribution arising from the use of the equation of motion.
• Figure 2: Structure of the total evolution kernel for the three-gluon correlation functions $G (x_1, x_3)$. The symbol m.c. stands for the mirror symmetrical contributions of the last two diagrams.
• Figure 3: The exact energy spectrum (\ref{['ADmatrixG']}) versus the selected trajectories generated by the approximate formula (\ref{['SimpleADmatrix']}).
• Figure 4: The log of the absolute value of the numerically diagonalized conserved charge $q$ with $q = 0$ trajectory being removed. The two possibilities of the description are clearly seen: from the top by the trajectories which behave at large $J$ as $q \propto J^3$; and from the bottom with the behaviour $q \propto J^2$.
• Figure 5: The numerical quantized values of the charge (crosses) and WKB approximation (\ref{['WKBexp']},\ref{['QfewWKBcorr']}).