Higher-loop contributions to the planar dilatation operator in the SU(2) subsector are by now firmly established for the two-loop [21] and three-loop level [12, 53]. In quantum spin chain language they take the explicit forms

In general, the -loop contribution to the dilatation operator involves interactions of neighboring spins, i.e. the full dilatation operator will correspond to a long-range interacting spin chain Hamiltonian. Note also the appearance of novel quartic spin interactions at the three-loop level. Generically even higher interactions of the form are expected at the loop levels.Integrability remains stable up to the three-loop order and acts in a perturbative sense: The conserved charges of the Heisenberg chain receive higher order corrections in of the form as one would expect. The full charges commute with each other () which translates into commutation relations for the various loop contributions upon expansion in , i.e.

However, opposed to the situation for the Heisenberg chain [55], there does not yet exist an algebraic construction of the gauge theory charges at higher loops. Nevertheless, the first few have been constructed manually to higher loop orders [14].An additional key property of these higher-loop corrections is that they obey BMN scaling: The emergence of the effective loop-counting parameter in the limit leads to the scaling dimensions for two magnon states in quantitative agreement with plane-wave superstrings.

Motivated by these findings, Beisert, Dippel and Staudacher [15] turned the logic around and simply assumed integrability, BMN scaling and a Feynman diagrammatic origin of the -loop SU(2) dilatation operator. Interestingly, these assumptions constrain the possible structures of the planar dilatation operator completely up to the five-loop level (and possibly beyond).

How can one now diagonalize the higher-loop corrected dilatation operator? For this, the ansatz for the Bethe wave-functions (57) needs to be adjusted in a perturbative sense in order to accommodate the long-range interactions, leading to a “perturbative asymptotic Bethe ansatz” for the two magnon wave-function [109]

Here, one needs to introduce a perturbative deformation of the S-matrix which is determined by the eigenvalue problem. Moreover, suitable “fudge functions” enter the ansatz which account for a deformation of the plane-wave form of the eigenfunction when two magnons approach each other within the interaction range of the spin chain HamiltonianDoes this mean that the AdS/CFT correspondence does not hold in its strong sense? While this logical possibility certainly exists, an alternative explanation is that one is dealing with an order-of-limits problem as pointed out initially in [15]. While in string theory one works in a limit of with held fixed, in gauge theory one stays in the perturbative regime and thereafter takes the limit, keeping only terms which scale as . These two limits need not commute. Most likely, the above-mentioned “wrapping” interactions must be included into the gauge theory constructions in order to match the string theory energies. On the other hand, the firm finite results at order of the gauge theory are still free of “wrapping” interactions: These only start to set in at the four-loop level (in the considered minimal SU(2) subsector). Moreover, to what extent the integrability is preserved in the presence of these “wrapping” interactions is unclear at the moment. Certainly, the resolution of this discrepancy remains a pressing open problem in the field.

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