Noncompact Heisenberg spin magnets from high-energy QCD: II. Quantization conditions and energy spectrum

TL;DR

This work provides a complete solution to the spectrum of color-singlet compound states of N reggeized gluons in multi-color QCD by mapping the problem to a noncompact SL(2, C) Heisenberg spin magnet and solving it with the Baxter Q-operator. The authors derive quantization conditions for the integrals of motion via the Q-operator's pole and asymptotic structure, construct Q-blocks that factorize the eigenvalues, and obtain the energy and quasimomentum from these blocks. The study yields detailed ground-state properties for N up to 8, reveals a rich trajectory-like organization of the higher charges, and shows that the intercepts of even-N states lie above unity while odd-N intercepts lie below unity, both approaching unity as N grows. These results illuminate the Regge-limit dynamics of high-energy QCD and hint at a deep connection between integrable spin chains and reggeon dynamics, with implications for the Odderon/Pomeron sectors and potential stringy interpretations in the large-N_c limit.

Abstract

We present a complete description of the spectrum of compound states of reggeized gluons in QCD in multi-colour limit. The analysis is based on the identification of these states as ground states of noncompact Heisenberg SL(2,C) spin magnet. A unique feature of the magnet, leading to many unusual properties of its spectrum, is that the quantum space is infinite-dimensional and conventional methods, like the Algebraic Bethe Ansatz, are not applicable. Our solution relies on the method of the Baxter Q-operator. Solving the Baxter equations, we obtained the explicit expressions for the eigenvalues of the Q-operator. They allowed us to establish the quantization conditions for the integrals of motion and, finally, reconstruct the spectrum of the model. We found that intercept of the states built from even (odd) number of reggeized gluons, N, is bigger (smaller) than one and it decreases (increases) with N approaching the same unit value for infinitely large N.

Figures (10)

• Figure 1: The spectrum of quantized $q_3^{1/3}$ for the system of $N=3$ particles. The total $SL(2,\mathbb{C})$ spin of the system is equal to $h=1/2$.
• Figure 2: Comparison of the exact spectrum of quantized $q_3^{1/3}$ at $h=1/2$ (crosses) with the WKB expression (\ref{['q3-WKB']}) (circles).
• Figure 3: The dependence of quantized $q_3(\nu_h;\ell_1,\ell_2)$ on the total spin $h=1/2+i\nu_h$. Three curves correspond to the trajectories with $(\ell_1,\ell_2)=(0,2)\,, (2,2)$ and $(4,2)$.
• Figure 4: The energy spectrum corresponding to three trajectories shown in Figure \ref{['Fig-3D']}. The ground state is located on the $(0,2)-$trajectory at $\nu_h=n_h=0$.
• Figure 5: The spectrum of the integrals of motion $q_4$ at $N=4$ and the total spin $h=1/2$. The left and the right panels correspond to the eigenstates with the quasimomentum $\mathop{\rm e}\nolimits^{i\theta_4}=\pm 1$ and $\pm i$, respectively.