Integrability and WKB solution of twist-three evolution equations

TL;DR

This paper demonstrates the integrability of a specific spin chain model related to twist-three quark-gluon evolution in QCD, and solves the associated equations using WKB approximation to find anomalous dimensions.

Contribution

It identifies an integrable spin chain model in QCD and applies WKB methods to solve the evolution equations for large conformal spins.

Findings

Derived explicit solutions for eigenvalues in limiting cases

Compared analytical results with numerical calculations

Established integrability of the twist-three evolution equations

Abstract

We identify an integrable one-dimensional inhomogeneous three-site open spin chain which arises in the problem of diagonalization of twist-three quark-gluon evolution equations in QCD in the chiral-odd sector. Making use of the existence of a non-trivial `hidden' integral of motion the problem of diagonalization of the evolution kernels is reduced to the study of a second order finite-difference equation which is solved in WKB approximation for large conformal spins of the three-particle system. The energies (alias anomalous dimensions) of eigenstates with different scale dependence are found in limiting cases and compared with numerical calculations.