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
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 [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
Hamiltonian.
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
and is determined by demanding
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
. The obtained series in
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
to be compared to the one-loop form of (59). The conjectured asymptotic
all-loop form of the energy density generalizing the one-loop expression (65)
reads
with the total energy being given by
. Note that both expressions manifestly obey BMN
scaling as the quasi-momenta scale like
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
the Equations (94) and (95) yield a prediction for the
energy (or scaling dimension) up to
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 (90) can never enter the asymptotic regime. What happens beyond the
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 [33, 49, 77, 50, 76, 78], 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 (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 [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
. 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 [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.