Quasiclassical QCD Pomeron

TL;DR

This work develops a quasiclassical framework for the Regge limit of perturbative QCD, solving the Baxter equation for n Reggeons by mapping to the XXX Heisenberg magnet and invoking a 1/h expansion tied to inverse conformal weight. It reveals a rich structure of holomorphic and antiholomorphic conserved charges, quantization conditions, and root-density distributions that determine the Reggeon spectrum, including the BFKL Pomeron and perturbative Odderon. Through leading and beyond-leading order analyses, the paper provides practical resummation schemes, Euler transformations, and analytic continuation strategies, yielding estimates for intercepts and illuminating the analytic properties (including non-Borel-summability for n≥3) of the energy functions. These results advance understanding of hard Pomeron dynamics in multi-Reggeon QCD and offer computational tools for assessing high-energy scattering amplitudes in the perturbative regime. The methods connect integrable spin chains, conformal symmetry, and Regge theory in a coherent, calculable framework with potential applications to small-x structure functions and high-energy phenomenology.

Abstract

The Regge behaviour of the scattering amplitudes in perturbative QCD is governed in the generalized leading logarithmic approximation by the contribution of the color--singlet compound states of Reggeized gluons. The interaction between Reggeons is described by the effective hamiltonian, which in the multi--color limit turns out to be identical to the hamiltonian of the completely integrable one--dimensional XXX Heisenberg magnet of noncompact spin $s=0$. The spectrum of the color singlet Reggeon compound states - perturbative Pomerons and Odderons, is expressed by means of the Bethe Ansatz in terms of the fundamental $Q-$function, which satisfies the Baxter equation for the XXX Heisenberg magnet. The exact solution of the Baxter equation is known only in the simplest case of the compound state of two Reggeons, the BFKL Pomeron. For higher Reggeon states the method is developed which allows to find its general solution as an asymptotic series in powers of the inverse conformal weight of the Reggeon states. The quantization conditions for the conserved charges for interacting Reggeons are established and an agreement with the results of numerical solutions is observed. The asymptotic approximation of the energy of the Reggeon states is defined based on the properties of the asymptotic series, and the intercept of the three--Reggeon states, perturbative Odderon, is estimated.

Figures (10)

• Figure 1: The histogram of the normalized roots of the solution of the $n=2$ Baxter equation for $h=50$ versus the distribution density $\rho(\sigma)$ in the leading large$-h$ limit.
• Figure 2: Possible forms of the potential $V_0(\lambda)$ entering into $n=3$ Baxter equation which lead to two different solutions of (\ref{['crit']}): (a) $\hat{q}_3 = 0.18 < q_3^*$ and (b) $\hat{q}_3 = 0.25 > q_3^*$. The shadow area represents the support ${\cal S}$.
• Figure 3: Two different forms of the distribution density for the $n=3$ Baxter equation: (a) $\hat{q}_3 < q_3^*$ and (b) $\hat{q}_3 \ge q_3^*$, corresponding to the potentials of fig. \ref{['poten']} (a) and (b), respectively.
• Figure 4: The distribution of quantized $q_3$ for different values of the conformal weight $3 \le h \le 40$: (a) The results of the numerical solution of the $n=3$ Baxter equation. The dashed line represents the upper and lower limits for $q_3$ in (\ref{['crit-q3']}). The solid line indicates one of the curves to which quantized $q_3$ belong. (b) One--parametric families of curves defined in (\ref{['Apm']}) and (\ref{['Bpm']}).
• Figure 5: The distribution of the quantized $q_3$ and $q_4$ corresponding to the polynomial solutions of the $n=4$ Baxter equation. The quantum numbers lie inside the generalized triangle defined in (\ref{['crit-q4']}).