Baxter equation beyond wrapping

The Baxter-like functional equation encoding the spectrum of anomalous dimensions of Wilson operators in maximally supersymmetric Yang-Mills theory available to date ceases to work just before the onset of wrapping corrections. In this paper, we work out an improved finitedifference equation by incorporating nonpolynomial effects in the transfer matrix entering as its ingredient. This yields a self-consistent asymptotic finite-difference equation valid at any order of perturbation theory. Its exact solutions for fixed spins and twists at and beyond wrapping order give results coinciding with the ones obtained from the asymptotic Bethe Ansatz. Correcting the asymptotic energy eigenvalues by the Lüscher term, we compute anomalous dimensions for a number of short operators beyond wrapping order. 1. Asymptotic Baxter equation and wrapping. To date [1] there is a significant body of data which suggests that the spectrum of all anomalous dimensions in planar maximally supersymmetric gauge theory can be computed overcoming complicated calculations Feynman diagrams. This finding [2, 3, 4, 5] generalizes previous observations that the spectrum of one-loop maximal-helicity gluon X = F+⊥ operators in pure gauge theory, O = tr{D1 + X(0)D2 + X(0) . . .DL + X(0)} , (1) can be calculated by identifying the dilation operator with the Hamiltonian of a noncompact Heisenberg magnet [6, 7]. The correspondence works by placing the elementary fields X(0) of the Wilson operator on the spin-chain sites and identifying spin generators with the ones of the collinear SL(2) subgroup of the (super)conformal group. The noncompactness of the spin chain is a consequence of the fact that there are infinite towers of covariant derivatives D+ acting on those fields. The sl(2) subsector of the maximally supersymmetric gauge theory, which we study in this paper, is spanned by the Wilson operators (1) with the elementary complex scalar field X = φ + iφ [4]. The one-loop integrable structure was generalized to all orders in ’t Hooft coupling g = g YM Nc/(4π ) and, though one is currently lacking a putative spin-chain picture for the dilatation operator, a set of Bethe Ansatz equations was put forward which survives a number of nontrivial spectral checks [8]. However, these equations allow one to calculate anomalous dimensions for Wilson operators as long as the order of the perturbative expansion does not exceed the length of the operator. Namely, when the interaction of the spins in the chain start to wrap around it, the aforementioned equations start to fail [9, 10]. Thus the true anomalous dimension for the Wilson operators are given a sum of two terms γ(g) = γ(g) + γ(g) . (2) The first contribution is determined by the solution to the asymptotic Bethe Ansatz equations and can be written as [11]