Quantum integrability in (super) Yang-Mills theory on the light-cone

TL;DR

This work shows that the one-loop dilatation operator governing the scale dependence of single-trace, light-cone Wilson operators built from chiral superfields in ${\cal N}$-extended YM theory is universal and can be mapped to the Hamiltonian of a noncompact Heisenberg $SL(2|\mathcal{N})$ spin chain. Using the light-cone formalism, the authors construct an $SL(2|\mathcal{N})$-invariant two-particle kernel and its $R$-matrix, with a projector that isolates the physical Wilson sector from nonlocal spurions. They demonstrate the universality of the kernel across ${\cal N}=0,1,2,4$ and connect the resulting mixing matrices for various operator sectors to known spin-chain integrable models, thereby uncovering integrability as a general feature of YM in the multi-color limit at one loop. The analysis provides a framework for calculating anomalous dimensions via spin-chain techniques and outlines a path toward solving the spectrum with Bethe Ansatz methods. The results also bridge the gauge theory integrability found in QCD and ${\cal N}=4$ SYM, suggesting broader applicability across supersymmetric extensions and potential implications for understanding operator mixing and spectral properties in gauge theories.

Abstract

We employ the light-cone formalism to construct in the (super) Yang-Mills theories in the multi-color limit the one-loop dilatation operator acting on single trace products of chiral superfields separated by light-like distances. In the N=4 Yang-Mills theory it exhausts all Wilson operators of the maximal Lorentz spin while in nonsupersymmetric Yang-Mills theory it is restricted to the sector of maximal helicity gluonic operators. We show that the dilatation operator in all N-extended super Yang-Mills theories is given by the same integral operator which acts on the (N+1)-dimensional superspace and is invariant under the SL(2|N) superconformal transformations. We construct the R-matrix on this space and identify the dilatation operator as the Hamiltonian of the Heisenberg SL(2|N) spin chain.

Figures (1)

• Figure 1: Feynman supergraphs defining one-loop contribution to the two-particle evolution kernel $\mathbb{H}_{k,k+1}$ in the multi-color limit. The dashed line denotes the '+'--direction on the light-cone and the open circles define the position of the superfields.