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

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.