Towards the exact dilatation operator of N=4 super Yang-Mills theory

TL;DR

This paper explores the structure of the dilatation operator in planar N=4 SYM, revealing a potential new integrable spin chain model with coefficients expressed via hypergeometric functions, indicating a square root scaling of anomalous dimensions at strong coupling.

Contribution

It introduces a novel approach to determine the dilatation operator by splitting it into parts based on pairwise interactions and finds coefficients that suggest a new integrable spin chain model.

Findings

Coefficients of spin interactions are expressible in hypergeometric functions of mbda.

Anomalous dimensions likely scale as mbda^{1/2} at large coupling.

Potential identification of a new integrable spin chain Hamiltonian.

Abstract

We investigate the structure of the dilatation operator D of planar N=4 SYM in the sector of single trace operators built out of two chiral combinations of the 6 scalars. Previous results at low orders in `t Hooft coupling \lambda suggest that D has the form of an SU(2) spin chain Hamiltonian with long range multiple spin interactions. Instead of the usual perturbative expansion in powers of \lambda, we split D into parts D^(n) according to the number n of independent pairwise interactions between spins at different sites. We determine the coefficients of spin-spin interaction terms in D^(1) by imposing the condition of regularity of a BMN-type scaling limit. For long spin chains, these coefficients turn out to be expressible in terms of hypergeometric functions of \lambda, which have regular expansions at both small and large values of \lambda. This suggest that anomalous dimensions of ``long'' operators in the two-scalar sector should generically scale as square root of \lambda at large coupling, i.e. in the same way as energies of semiclassical states in dual AdS5 x S5 string theory. We speculate that D^(1) may be a Hamiltonian of a new integrable spin chain.