Bethe Ansatz for QCD Pomeron,

TL;DR

Korchemsky demonstrates that high-energy QCD in the generalized leading logarithmic approximation can be mapped to a two-dimensional transverse gluon dynamics described by reggeized gluons forming color-singlet states (Pomeron, Odderon) whose Hamiltonian coincides with the XXX Heisenberg magnet for spin $s=0$ in the large-$N$ limit. The spectrum of multi-Reggeon states is obtained via a generalized Bethe Ansatz built from a Baxter $Q$-operator, linking holomorphic and antiholomorphic sectors through conformal $SL(2,\mathbb{C})$ symmetry and yielding a complete set of commuting integrals of motion. For $n=2$ (BFKL Pomeron) the Baxter equation reduces to a hypergeometric form with energies expressible in terms of digamma functions; for $n\ge 3$ the spectrum is accessed via a Legendre-basis expansion and quantization conditions that render the problem algebraic. The results illuminate how Regge behavior of total cross sections and small-$x$ structure functions emerges from integrable Reggeon dynamics, offering a principled framework to address unitarity and high-energy QCD phenomenology. The work also uncovers deep connections between Baxter solutions and families of orthogonal polynomials, highlighting rich mathematical structure underlying perturbative Reggeon dynamics.

Abstract

The equivalence is found between high-energy QCD in the generalized leading logarithmic approximation and the one-dimensional Heisenberg magnet. According to Regge theory, the high energy asymptotics of hadronic scattering amplitudes are related to singularities of partial waves in the complex angular momentum plane. In QCD, the partial waves are determined by nontrivial two-dimensional dynamics of the transverse gluonic degrees of freedom. The "bare" gluons interact with each other to form a collective excitation, the Reggeon. The partial waves of the scattering amplitude satisfy the Bethe-Salpeter equation whose solutions describe the color singlet compound states of Reggeons -- Pomeron, Odderon and higher Reggeon states. We show that the QCD Hamiltonian for reggeized gluons coincides in the multi-color limit with the Hamiltonian of XXX Heisenberg magnet for spin s=0 and spin operators the generators of the conformal SL(2,C) group. As a result, the Schrodinger equation for the compound states of Reggeons has a sufficient number of conservation laws to be completely integrable. A generalized Bethe ansatz is developed for the diagonalization of the QCD Hamiltonian and for the calculation of hadron-hadron scattering. Using the Bethe Ansatz solution of high-energy QCD we investigate the properties of the Reggeon compound states which govern the Regge behavior of the total hadron-hadron cross sections and the small-x behavior of the structure functions of deep inelastic scattering.

Figures (10)

• Figure 1: Unitary Feynman diagrams contributing to the structure function of deep inelastic scattering in the generalized leading logarithmic approximation in the multicolor limit, $N\to\infty$. Solid lines represent $n$ reggeized gluons propagating in the $t-$channel. Dotted lines denote interaction of reggeized gluons with their nearest neighbours. Upper and lower blobs describe the coupling of gluons to virtual photon $\gamma^*$ and proton state $\rm p$, respectively. For finite $N$ one has to add the diagrams with pair--wise interactions between $n$ reggeized gluons.
• Figure 2: The Bethe--Salpeter equation for the transition operator $T_n(\omega)$, describing $n \to n$ elastic scattering of reggeized gluons in the $t-$channel. Iterations of this equation reproduce the ladder diagrams of fig. \ref{['ladder']}.
• Figure 3: The energy of the holomorphic Reggeon hamiltonian for $n=3$ corresponding to the polynomial solutions of the Baxter equation. The solid line represents the energy for $n=2$ defined in (\ref{['fin']}). Points on this line correspond to the degenerate solutions, $q_3=0$, of the Baxter equation for $n=3$.
• Figure 4: The holomophic energy for $n=4$ corresponding to the polynomial solutions of the Baxter equation. The solid line represents $\varepsilon_2(h)$.
• Figure 5: The holomorphic energy $\varepsilon_3$ for $n=3$ as a function of quantized $q_3$. The maximum value of the energy $\varepsilon_3=-6$ corresponds to the degenerate solution $q_3=0$.