Higher-loop contributions to the planar dilatation operator in the SU(2) subsector are by now firmly established for the two-loop  and three-loop level [12, 53]. In quantum spin chain language they take the explicit forms
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., 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 .
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  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 9. By construction they are invisible in the asymptotic regime (or rather larger than the highest loop order considered) of well separated magnons. The detailed form of these functions is completely irrelevant for the physical spectrum as a consequence of the factorized scattering property of the integrable system. With this perturbative asymptotic Bethe ansatz (90), one shows that the form of the Bethe equations remains unchanged, i.e. the perturbative S-matrix (91) simply appears on the right hand side of the equations  to conjecture an asymptotic all-loop expression for the perturbative S-matrix 10 [33, 49, 77, 50, 76, 78], these effects have been studied explicitly at the four-loop level  where the wrapping effects set in for the first time in the SU(2) subsector. No “natural” way of transforming the generic dilatation operator to the wrapping situation was found. Finally, let us restate that it has not been shown so far that a microscopic long-range spin chain Hamiltonian truly exists, which has a spectrum determined by the conjectured perturbative asymptotic Bethe equations (94) and (95) of Beisert, Dippel and Staudacher. In any case, the proposed all-loop asymptotic Bethe equations (94, 95) may now be studied in the thermodynamic limit, just as we did above for the one-loop case. This was done in  and , and allows for a comparison to the results obtained in Section 3 for the energies of the spinning folded and closed string solutions. Recall that these yield predictions to all-loops in . While the two-loop gauge theory result is in perfect agreement, the three-loop scaling dimensions fail to match with the expected dual string theory result! This three-loop disagreement also arises in the comparison to the near plane-wave string spectrum computed in [46, 45, 44, 43], i.e. the first corrections to the Penrose limit of to the plane-wave background.
Does 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 . 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.
© Max Planck Society and the author(s)