Evolution equations for quark–gluon distributions in multi-color QCD and open spin chains

Abstract

We study the scale dependence of the twist-3 quark–gluon parton distributions using the observation that in the multi-color limit the corresponding QCD evolution equations possess an additional integral of motion and turn out to be effectively equivalent to the Schrödinger equation for integrable open Heisenberg spin chain model. We identify the integral of motion of the spin chain as a new quantum number that separates different components of the twist-3 parton distributions. Each component evolves independently and its scale dependence is governed by anomalous dimension given by the energy of the spin magnet. To find the spectrum of the QCD induced open Heisenberg spin magnet we develop the Bethe ansatz technique based on the Baxter equation. The solutions to the Baxter equation are constructed using different asymptotic methods and their properties are studied in detail. We demonstrate that the obtained solutions provide a good qualitative description of the spectrum of the anomalous dimensions and reveal a number of interesting properties. We show that the few lowest anomalous dimensions are separated from the rest of the spectrum by a finite mass gap and estimate its value.