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4.3 Higher Loops in the SU(2) sector and discrepancies

The connection to an integrable spin chain at one-loop raises the question whether integrability is merely an artifact of the one-loop approximation or a genuine property of planar N = 4 gauge theory. Remarkably, all present gauge theory data points towards the latter being the case.

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 [1253]. In s = 1/2 quantum spin chain language they take the explicit forms

sum L D2 -loop = - sl .sl+2 + 4sl .sl+1- 3 .1 i=1 sum L D = - s .s + (s .s )(s .s )- (s .s )(s .s ) 3- loop l l+3 l l+2 l+1 l+3 l l+3 l+1 l+2 i=1 +10 sl .sl+2 - 29 sl .sl+1 + 20 .1 .
In general, the k-loop contribution to the dilatation operator involves interactions of k + 1 neighboring spins, i.e. the full dilatation operator D = sum oo D k=1 k- loop will correspond to a long-range interacting spin chain Hamiltonian. Note also the appearance of novel quartic spin interactions (si .sj)(sk .sl) at the three-loop level. Generically even higher interactions of the form k (s• .s •) are expected at the k [2] + 1 loop levels.

Integrability remains stable up to the three-loop order and acts in a perturbative sense: The conserved charges of the Heisenberg XXX1/2 chain receive higher order corrections in c of the form (1) (2) (3) Qk = Q k + cQ k + c2 Q k + ... as one would expect. The full charges Ok commute with each other ([Ok, Ol] = 0) which translates into commutation relations for the various loop contributions O(r) k upon expansion in c, i.e.

(1) (1) [Q k ,Q l ] = 0 [Q(1),Q(2)] + [Q(2),Q(1)] = 0 k l k l [Q(1k),Q(3l)] + [Q(2k),Q(2l)] + [Q(3k),Q(1l)] = 0, ... (89)
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 Qk 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 c':= c/J 2 in the J --> oo limit leads to the scaling dimensions V~ --------- D ~ 1 + c'n2 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 k-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 (57View Equation) 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]

i(p1 x1+p2x2) y(x1,x2) = e f (x2 - x1, p1,p2) +S(p2, p1)ei(p2x1+p1x2)f(L - x2 + x1, p1,p2). (90)
Here, one needs to introduce a perturbative deformation of the S-matrix
sum oo n S(p1,p2) = S0(p1,p2) + c Sn(p1, p2) (91) n=1
which is determined by the eigenvalue problem. Moreover, suitable “fudge functions” enter the ansatz
sum oo x»1 f(x,p1,p2) = 1 + cn+|x|fn(x,p1,p2)- --> 1 (92) n=0
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 Hamiltonian9. By construction they are invisible in the asymptotic regime x » 1 (or rather x 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 (90View Equation), one shows that the form of the Bethe equations remains unchanged, i.e. the perturbative S-matrix (91View Equation) simply appears on the right hand side of the equations
prod M eipkL = S(pk, pi) , (93) i=1,i/=k
and is determined by demanding y(x1, x2) to be an eigenfunction of the dilatation operator, just as we did in Section 4.1. Based on the constructed five-loop form of the dilatation operator, the S-matrix is then determined up to O(c4). The obtained series in c turns out to be of a remarkably simple structure, which enabled the authors of [15] to conjecture an asymptotic all-loop expression for the perturbative S-matrix
V~ ------------- f(p1)---f(p2)-+-i 1 p 2 p S(p1, p2) = f(p1)- f(p2)- i with f(p) = 2 cot(2) 1 + c sin ( 2), (94)
to be compared to the one-loop form of (59View Equation). The conjectured asymptotic all-loop form of the energy density generalizing the one-loop expression (65View Equation) reads10
V~ --------------- 2 p c q2(p) = 1 + 8c sin (2)- 1 (95)
with the total energy being given by E = sum M q(p ) 2 i=1 2 i. Note that both expressions manifestly obey BMN scaling as the quasi-momenta scale like -1 p ~ L in the thermodynamic limit as we discussed in Section 4.2. It is important to stress that these Bethe equations only make sense asymptotically: For a chain (or gauge theory operator) of length L the Equations (94View Equation) and (95View Equation) yield a prediction for the energy (or scaling dimension) up to L - 1 loops. This is the case, as the interaction range of the Hamiltonian will reach the length of the spin chain beyond this point, and the multi-magnon wave-functions of (90View Equation) can never enter the asymptotic regime. What happens beyond the L loop level is still a mystery. At this point, the “wrapping” interactions start to set in: The interaction range of the spin chain Hamiltonian cannot spread any further and starts to “wrap” around the chain. In the dimensionally reduced model of plane-wave matrix theory [334977507678], these effects have been studied explicitly at the four-loop level [58] 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 (94View Equation) and (95View Equation) of Beisert, Dippel and Staudacher. In any case, the proposed all-loop asymptotic Bethe equations (94View Equation, 95View Equation) may now be studied in the thermodynamic limit, just as we did above for the one-loop case. This was done in [107] and [15], 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 c'. 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 [46454443], i.e. the first 1/J corrections to the Penrose limit of 5 AdS5 × S 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 [15]. While in string theory one works in a limit of c --> oo with J2/c held fixed, in gauge theory one stays in the perturbative regime c « 1 and thereafter takes the J --> oo limit, keeping only terms which scale as 2 c/J. 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 L results at order c3 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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