Noncompact Heisenberg spin magnets from high-energy QCD: I. Baxter Q-operator and Separation of Variables

TL;DR

This work solves a completely integrable two-dimensional quantum-mechanical model that arises in high-energy QCD by generalizing the Heisenberg spin magnet to infinite-dimensional SL(2,\mathbb{C}) principal-series representations. The authors develop a nonpolynomial Baxter Q-operator and apply Sklyanin's Separation of Variables to obtain an integral representation for the Hamiltonian's eigenfunctions, leveraging a diagrammatic approach based on uniqueness relations. A central result is the explicit factorized R-matrix and the Q-operator formalism, which relate to the transfer matrices and yield a path to determining the spectrum through Baxter equations, including detailed analysis at N=2 and extensive discussion of analytic properties and asymptotics. The techniques connect high-energy QCD, two-dimensional integrable systems, and conformal-field-theory methods, providing a robust framework for understanding N-gluon compound states through exact, operator-based methods and SoV representations.

Abstract

We analyze a completely integrable two-dimensional quantum-mechanical model that emerged in the recent studies of the compound gluonic states in multi-color QCD at high energy. The model represents a generalization of the well-known homogenous Heisenberg spin magnet to infinite-dimensional representations of the SL(2,C) group and can be reformulated within the Quantum Inverse Scattering Method. Solving the Yang-Baxter equation, we obtain the R-matrix for the SL(2,C) representations of the principal series and discuss its properties. We explicitly construct the Baxter Q-operator for this model and show how it can be used to determine the energy spectrum. We apply Sklyanin's method of the Separated Variables to obtain an integral representation for the eigenfunctions of the Hamiltonian. We demonstrate that the language of Feynman diagrams supplemented with the method of uniqueness provide a powerful technique for analyzing the properties of the model.

Figures (18)

• Figure 1: Feynman diagram representation of the kernel of the $R-$matrix, Eq. (\ref{['R-kernel']}). Each side of the square represents a two-dimensional propagator and the arrow indicates in which order the difference of coordinates is taken.
• Figure 2: Diagrammatical proof of the Yang-Baxter relation based on the "uniqueness" relations. Integration over the position of internal vertices (fat points) is tacitly assumed.
• Figure 3: Diagrammatical proof of Eq. (\ref{['Ru-u']}). Two lines connecting $\vec{w}_1$ and $\vec{w}_2$ cancel each other and the integration over the position of these points is performed using the chain relation (\ref{['chain-rule-delta']}).
• Figure 4: Diagrammatical calculation of the eigenvalues of the $R-$matrix.
• Figure 5: Diagrammatical representation of the transfer matrix $\mathbb{T}_N^{(s_0,\bar{s}_0)}(u,\bar{u})$ defined in (\ref{['t-N']}). The left- and rightmost vertices are located at the same point and integration over its position as well as over the position of the remaining internal vertices is implied.