WKB quantization of reggeon compound states in high-energy QCD

TL;DR

This work develops a nonlinear WKB framework to quantize the conserved charges of N-reggeon states in high-energy QCD, exploiting SL(2,C) symmetry and the Baxter equation in separated variables. The quantization conditions are recast as Whitham flows on the spectral curve Γ_N, yielding explicit leading-order spectra for N=3 and revealing universal moduli behavior in two limiting regimes. The N=3 analysis gives analytic and numerical forms for q_2 and q_3, including the energy E3 in the WKB limit, and connects to exact polynomial solutions and odderon-related intercepts. The results provide a coherent classical-quantum correspondence for reggeon dynamics and offer a pathway to incorporate nonleading corrections and higher-N generalizations.

Abstract

We study the spectrum of the reggeized gluon compound states in high-energy QCD. The energy of the states depends on the total set of quantum numbers whose values are constrained by the quantization conditions. To establish their explicit form we apply the methods of nonlinear WKB analysis to the Schrödinger equation for the $N$ reggeon state in the separated coordinates. We solve the quantization conditions for the N=3 reggeon states in the leading order of the WKB expansion and observe a good agreement of the obtained spectrum of quantum numbers with available exact solutions.

Figures (5)

• Figure 1: The definition of the $\alpha-$cycles for real positive moduli: (a) $0 \le u\le \frac{1}{\sqrt{27}}$ and (b) $u > \frac{1}{\sqrt{27}}$.
• Figure 2: Quantized moduli $u$ for real values of the flow parameter $\delta$. The dots indicate the positions of three singularities of the elliptic curve $\Gamma_3$ located at $\delta-\frac{1}{2}=\pm \frac{1}{2}$ and $0$. The behaviour of the function $u=u(\delta)$ around these points and at infinity is described by (\ref{['delta-1/2']}), (\ref{['delta-1']}) and (\ref{['del+']}).
• Figure 3: Two different branches of the function $u(1/x)$ entering into (\ref{['q(x)']}). Dotted lines represent $u=\pm\frac{1}{\sqrt{27}}$ and two dots indicate the singular points $x=1$, $u=\frac{1}{\sqrt{27}}$ and $x=2$, $u=0$. The part of the curve $u^2(x)\le \frac{1}{27}$ corresponds to the polynomial solutions of the Baxter equation and it should be compared with Fig. 4 in Qua.
• Figure 4: The function $q=q(x)$ defining the quantum numbers (\ref{['q(x)']}) of the $N=3$ reggeon compound states. Its behaviour around $x=0_\pm$, $x=1$, $x=2$ and $x=\pm\infty$ is given by (\ref{['F1']}), (\ref{['F2']}), (\ref{['F4']}), (\ref{['F3']}) and (\ref{['h-infty']}), respectively.
• Figure 5: The energy spectrum of the $N=3$ reggeon compound states evaluated in the leading order of the WKB quantization for type-II quantization conditions, $E_3=E_3(h;n_1)$. Three different curves correspond to $n_1=4$, $8$ and $12$. The dotted line represents the BFKL energy of the $N=2$ reggeon state, $E_2(h)=4\left[{\Psi(1)-\Psi(\frac{1}{2}+|h-\frac{1}{2}|)}\right]$.