Evolution equations for quark-gluon distributions in multi-color QCD and open spin chains

TL;DR

This work shows that the scale dependence of twist-3 quark-gluon distributions in the multi-color limit maps to an integrable open $SL(2,\mathbb{R})$ spin-chain problem, enabling exact and asymptotic determination of anomalous dimensions via the Baxter equation and Algebraic Bethe Ansatz. The authors identify two conserved charges, $Q_{S^+}$ and $Q_T$, that label independent components of the distributions, with their anomalous dimensions given by spin-magnet energies. They develop a comprehensive spectral analysis, obtaining exact solutions at special parameter choices, and a detailed asymptotic framework (including WKB) that reveals a dispersion curve, quantization conditions, and a finite mass gap separating the lowest levels from the continuum. The results establish a universality class for three-particle evolution in QCD and provide a solid foundation for understanding Regge behavior and higher-twist distributions in the large-$N_c$ limit, with nonplanar corrections discussed as a perturbation that preserves qualitative structure.

Abstract

We study the scale dependence of the twist-3 quark-gluon parton distributions using the observation that in the multi-color limit the corresponding QCD evolution equations possess an additional integral of motion and turn out to be effectively equivalent to the Schrodinger equation for integrable open Heisenberg spin chain model. We identify the integral of motion of the spin chain as a new quantum number that separates different components of the twist-3 parton distributions. Each component evolves independently and its scale dependence is governed by anomalous dimension given by the energy of the spin magnet. To find the spectrum of the QCD induced open Heisenberg spin magnet we develop the Bethe Ansatz technique based on the Baxter equation. The solutions to the Baxter equation are constructed using different asymptotic methods and their properties are studied in detail. We demonstrate that the obtained solutions provide a good qualitative description of the spectrum of the anomalous dimensions and reveal a number of interesting properties. We show that the few lowest anomalous dimensions are separated from the rest of the spectrum by a finite mass gap and estimate its value.

Figures (4)

• Figure 1: The dependence of the quantized values of the charge $q$ and the energy ${\cal E}(N,q)$ on the spin $N$ for the values of the impurity parameters $\omega_1=\omega_3=3/2$ corresponding to the evolution kernel ${\cal H}_T$. The dotted lines represent two trajectories with $\bar{\ell}=0$ (upper curve) and $\ell=3$ (lower curve) described in Sect. 7.4.
• Figure 2: The flow of the energy ${\cal E}(N,q)$ and the conserved charge $q=2N(N+6)(\nu^2+3/4)$ with $\omega_3$ for $N=5$ and $\omega_1=3/2$. Two vertical dotted lines correspond to $\omega_3=1/2$ and $\omega_3=3/2$ and define the spectrum of the evolution kernels ${\cal H}_S$ and ${\cal H}_T$, respectively.
• Figure 3: The dependence on the energy ${\cal E}(N,q)$ on the conserved charge $q$ at $N=10$. Crosses denote the exact values of the energy and the solid line corresponds to the asymptotic expression (\ref{['dis']}).
• Figure 4: The difference ${\cal E}(N,q)-{\cal E}_{\rm vac}(N)$ for the lowest five energy levels of the Hamiltonian ${\cal H}(\omega_1,\omega_3)$: (a) the dependence on the impurity $\omega_3$ for $N=100$ and $\omega_1=3/2$; (b) the dependence on the spin $N$ for $\omega_1=\omega_3=3/2$.