Integrability of Three-Particle Evolution Equations in QCD

Abstract

Abstract:We show that the Brodsky-Lepage evolution equation for the spin 3/2 baryon distribution amplitude is completely integrable and reduces to the three-particle XXX s=−1 Heisenberg spin chain. Trajectories of the anomalous dimensions are identified and calculated using the 1/N expansion. Extending this result, we prove integrability of the evolution equations for twist 3 quark-gluon operators in the large N c limit.

Spin chains and gauge?string duality

“…Actually the phenomena of hidden integrability of the RG flows at large N at least at one loop level seems to be quite general. It has been found for low twist light-cone operators [11] and arbitrary twist quasipartonic operators in SL(2,R) sector [7], operators with large R charge in N=4 SYM theory [13] as well as for evolution in the Regge limit [12]. Precisely, it appeared that the dilatation operators at one loop level in these cases can be identified with Hamiltonians of SL(2,R), SO (6) and SL(2,C) spin chains respectively.…”

Nucleon distribution amplitudes and proton decay matrix elements on the lattice

“…The theory of nucleon DAs has reached a certain degree of maturity. In particular the scale dependence is well understood [12,13] and it reveals important symmetries of the quantum theory that are not seen at the level of the QCD Lagrangian [14]. At the same time, they are much less studied as compared to the usual parton distributions.…”

“…l (and similarly for " m and " n) denotes the Lorentz structure given by the covariant derivatives D ¼ @ À igA on the right-hand side of Eqs. (12)- (14). The indices l, m, n (without bars) are the total number of derivatives acting on the first, second, and third quark, respectively.…”

New integrable structures in large-NQCD

“…They were first found in the analysis of the parton evolution in the Regge regime [30,31]. In a subsequent development, the mixing matrices of Wilson operators with various quantum numbers were identified with Hamiltonians of non-compact spin chains [32,33,34], which at large N turn out to be integrable for maximalhelicity operators.…”

“…1. In more general operators containing covariant derivatives, D λ 1 · · · D λn F µ l ν l should be identified with a single site of the lattice (as in [32,33,34]), which is clear from the structure of the diagrams (D λ 1 · · · D λn F µ l ν l emits a single gluon to the lowest order in perturbation theory).…”