Scattering Amplitudes/Wilson Loop Duality In ABJM Theory
ABSTRACT For N=6 superconformal Chern-Simons-matter theories in three dimensions, by a
direct superspace Feynman diagram approach, we compute the two-loop four-point
scattering amplitude with external chiral matter fields. We find that the
result is in perfect agreement with the two-loop result for a light-like
four-polygon Wilson loop. This is a nontrivial evidence of the scattering
amplitudes/Wilson loop duality in three dimensions. Moreover, both the IR
divergent and the finite parts of our two-loop result agree with a BDS-like
ansatz for all-loop amplitudes where the scaling function is given in terms of
the N=4 SYM one, according to the conjectured Bethe equations for ABJM.
Consequently, we are able to make a prediction for the four-loop correction to
the amplitude. We also discuss the dual conformal invariance of the two-loop
arXiv:1107.3139v1 [hep-th] 15 Jul 2011
Preprint typeset in JHEP style - PAPER VERSION
LOOP DUALITY IN ABJM THEORY
Marco S. Bianchi∗, Matias Leoni∗, Andrea Mauri†,#, Silvia Penati∗and Alberto
∗Dipartimento di Fisica, Universit` a di Milano–Bicocca and INFN, Sezione di
Milano–Bicocca, Piazza della Scienza 3, I-20126 Milano, Italy
†Dipartimento di Fisica dell’Universit` a degli studi di Milano
#INFN, Sezione di Milano, via Celoria 16, I-20133 Milano, Italy
E-mail: firstname.lastname@example.org, email@example.com,
Abstract: For N = 6 superconformal Chern–Simons–matter theories in three dimen-
sions, by a direct superspace Feynman diagram approach, we compute the two–loop
four–point scattering amplitude with external chiral matter fields. We find that the
result is in perfect agreement with the two–loop result for a light–like four–polygon
Wilson loop. This is a nontrivial evidence of the scattering amplitudes/Wilson loop
duality in three dimensions. Moreover, both the IR divergent and the finite parts of
our two–loop result agree with a BDS–like ansatz for all–loop amplitudes where the
scaling function is given in terms of the N = 4 SYM one, according to the conjectured
Bethe equations for ABJM. Consequently, we are able to make a prediction for the
four–loop correction to the amplitude. We also discuss the dual conformal invariance
of the two–loop result.
Keywords: AdS/CFT, Chern–Simons matter theories, scattering amplitudes,
2. ABJM model in N = 2 superspace4
3. The four point amplitude at two loops6
4. Dual conformal invariance9
5. A conjecture for the all–loop four–point amplitude11
6. Conclusions 12
The development of efficient techniques for computing scattering amplitudes in gauge
theories has led to the discovery of new unexpected properties in the on–shell sector of
In four dimensions, the use of stringy–inspired methods , twistor string theory
 and AdS/CFT correspondence  has allowed to dig out hidden symmetries for
the planar sector of the maximally supersymmetric Yang–Mills theory. In particular,
planar N = 4 SYM has been proved to be integrable  and the related Yangian
symmetry to be responsible for a duality between scattering amplitudes and Wilson
loops (WL). This duality has been checked perturbatively in many cases –, while
at strong coupling it relies on the self-duality of type IIB string on AdS5× S5under a
suitable combination of bosonic and fermionic T–dualities [14, 15].
Another duality has been found at weak coupling which involves WL and correla-
tion functions of BPS operators –.
Since AdS/CFT has been playing a fundamental role in the discovery of these new
hidden properties and at the same time their perturbative confirmation represents a
non–trivial test of the correspondence, it is mandatory to investigate whether similar
properties emerge in other class of theories for which a string dual description is known.
– 1 –
We are interested in the class of three dimensional N = 6 ABJM theories  which
are dual to type IIA string theory on AdS4× CP3. A distinguished feature of these
models compared to the more famous N = 4 SYM in four dimensions, is that they
are not maximally supersymmetric. Moreover, the proof of the amplitudes/WL duality
in type IIA string on AdS4× CP3is complicated by the emergence of singularities in
the fermionic T–transformations –. Therefore, a priori, it is not totally obvious
that we should expect dualities and hidden symmetries to be realized in ABJM models
exactly in the same way as in their four–dimensional counterpart.
Preliminary results can be found in literature, concerning integrability [26, 27, 28,
29] and related Yangian symmetry [30, 31]. Perturbative investigation of these proper-
ties have been performed. At tree level, scattering amplitudes are invariant under dual
superconformal symmetry [32, 33] whose generators are the level–one generators of a
Yangian symmetry . A first indication of the duality between scattering amplitudes
and WL comes from the fact that at one–loop, both the four–point amplitude  and
the light–like four–polygon WL [35, 36] vanish. Recently, n-point correlators of BPS
scalar operators have been proved to vanish at one–loop , so providing first evidence
of a triad correlation functions/WL/amplitudes duality in three dimensions.
However, non–trivial perturbative support to these dualities can come only at or-
ders where these quantities do not vanish.
At two loops, for the ABJM model the light–like four–polygon WL has been com-
puted in the planar limit . In dimensional regularization, taking the light–like limit
(xi− xi+1)2≡ x2
i,i+1→ 0, the result is non–vanishing and given by
+ K + O(ǫ)
where µ is the mass scale of dimensional regularization, λ is the ABJM coupling con-
stant and K = [8ln2 + 12ln2(2) − a6], with a6a numerical constant.
In this paper, using N = 2 superspace description and a direct Feynman diagram
approach, we evaluate the two–loop contribution to the planar scattering superampli-
tude of four chiral superfields which, in components, gives rise to the amplitude for two
scalars and two chiral fermions. The amplitude involves two external particles in the
bifundamental representation of the U(N)×U(N) gauge group and two particles in the
antifundamental. This is what mostly resembles a MHV amplitude in four dimensions.
Defining M4to be the superamplitude divided by its tree level contribution, we find
+ K1+ O(ǫ)
where K1= 4ζ2+ 3log22 is a numerical constant.
– 2 –
This result we obtain has a number of remarkable properties. First of all, as in the
N = 4 SYM case, the two–loop amplitude is proportional to the tree level contribution
times a function of the kinematic invariants. We find that, up to an additive, scheme
dependent constant, and up to the overall sign, this function matches exactly the result
(1.1) once the IR regularization is formally identified with the UV one and the particle
momenta are expressed in terms of dual coordinates, pi= xi,i+1.
Therefore, at least for the four–point amplitude, we find evidence of the following
logM4= log?W4? + const. (1.3)
that should hold order by order in the perturbative expansion of the two objects.
Quite remarkably, the two–loop result we have found has the same functional struc-
ture as the one–loop correction to the four–point scattering amplitude for the four
dimensional N = 4 SYM theory. As proved in , for N = 4 SYM all momentum inte-
grals up to four loops are dual to four dimensional true conformally invariant integrals,
well defined off-shell. As a consequence, the four–point amplitude satisfies anomalous
Ward identities associated to dual conformal transformations , as dual conformal
invariance is broken in the on–shell limit by the appearance of IR divergences which
require introducing a mass regulator.
A natural question arises whether the same pattern is present in three dimensional
ABJM theories. We discuss dual conformal invariance of the momentum integrals
that occur in our two–loop diagrammatic calculation, which does not assume dual
conformal invariance a priori. We find that, even if at the level of the integrands every
single diagram does not appear to be dual to a conformally invariant one, once we
sum over all possible permutations of the external particles, each of them is invariant
when evaluated off-shell and strictly in three dimensions. This means that, off–shell,
where IR divergences are absent, it should be possible to rewrite the amplitude as a
linear combination of scalar integrals associated to dual invariant diagrams in three
In the N = 4 SYM case, an ansatz for all–loop n–point MHV amplitudes has
been proposed [37, 38], where the all–loop amplitudes exponentiate and turn out to
be determined by the one–loop result times the perturbative expansion of the scaling
function fN=4(λ) as a function of the ’t Hooft coupling.
Remarkably, we find that the two–loop four–point function for the ABJM model
can be obtained from the second order expansion of the same BDS–like ansatz where the
four dimensional scaling function is substituted by the three dimensional one, fCS(λ)
as obtained from the conjectured asymptotic Bethe equations .
– 3 –
Therefore, we make the conjecture that the all loop four–point amplitude is given
where now λ is the ABJM coupling and C(λ) is a scheme–dependent constant.
Since fCS(λ) is known up to order λ4[28, 29], we may predict the exact four–loop
contribution to the four–point function (see eq. (5.6)).
NOTE: When this work was already completed, a paper  appeared, which has
significant overlap. Although we draw the same conclusions, we stress that our compu-
tation, being based on a direct Feynman diagram approach is completely independent
from that of .
2. ABJM model in N = 2 superspace
An on–shell realization of N = 6 supersymmetric ABJM models can be given in terms
of N = 2 three dimensional superspace . For U(N)×U(N) gauge group, the physical
field content is organized into two vector multiplets (V,ˆV ) in the adjoint representation
of the first and the second U(N)’s, coupled to chiral multiplets Aiand Bicarrying a
fundamental index i = 1,2 of a global SU(2)A×SU(2)Band in the bifundamental and
antibifundamental representations of the gauge group, respectively.
The N = 6 supersymmetric action reads
S = SCS+ Smat
d3xd2θ ǫikǫjlTr(AiBjAkBl) +2πi
?d3xd2¯θ ǫikǫjlTr(¯Ai¯Bj ¯Ak¯Bl)
Here k is an integer, as required by gauge invariance of the effective action. In the
perturbative regime we take λ ≡N
The quantization of the theory can be easily carried on in superspace after per-
forming gauge fixing (for details, see for instance ). In momentum space and using
– 4 –
Landau gauge, this leads to gauge propagators
whereas the matter propagators are
a× D2¯D2δ4(θ1− θ2)
¯ a× D2¯D2δ4(θ1− θ2) (2.4)
where a,b and ¯ a,¯b are indices of the fundamental representation of the first and the
second gauge groups, respectively. The vertices employed in our two–loop calculation
can be easily read from the action (2.2) and they are given by
Tr(¯AiV Ai) − Tr(BiV¯Bi) + Tr(¯BiˆV Bi) − Tr(AiˆV¯Ai)
Tr(A1B1A2B2) − Tr(A1B2A2B1)
+ h.c. (2.5)
We work in euclidean superspace with the effective action defined as eΓ=?eS.
theory are the ones involving matter external particles. In N = 2 superspace language
this means having A, B and their complex conjugates as external superfields. Given the
structure of the vertices, it is straightforward to see that only amplitudes with an even
number of external legs are non–vanishing. This is consistent with the requirement for
the amplitudes to be Lorentz and dilatation invariant .
Each external scalar particle carries an on–shell momentum pαβ(p2= 0), an SU(2)
index and color indices corresponding to the two gauge groups. We classify as particles
the ones carrying (N,¯N) indices and antiparticles the ones carrying (¯ N,N) indices.
Therefore, (Ai,¯Bj) are particles, whereas (Bi,¯Aj) are antiparticles.
We are interested in the simplest non–trivial amplitudes, that is four–point ampli-
tudes. Without loosing generality we consider the (ABAB) superamplitude. All the
other superamplitudes can be obtained from this one by SU(4) R–symmetry transfor-
The color indices can be stripped out, as we can write
Since the vector fields are not propagating, the only non–trivial amplitudes of the
A4(σ(1),··· ,σ(4)) δ
where the sum is over exchanges of even and odd sites between themselves.
– 5 –
3. The four point amplitude at two loops
We study four-point scattering amplitudes of the type (AiBjAkBl), where the external
A,B particles carry outgoing momenta p1,...,p4 (p2
variables are defined by s = (p1+ p2)2,t = (p1+ p4)2,u = (p1+ p3)2.
At tree level the amplitude is simply given by the diagram in Fig. 1 (a) associated
to the classical superpotential in (2.5). Its explicit expression is
i= 0). As usual, Mandelstam
At one loop it has been proved to vanish . In N = 2 superspace language it’s easy
to see that the only diagram that can be constructed (Fig. 1 (b)) leads to an off-shell
trivially vanishing integral.
Figure 1: Diagrams contributing to the tree level and 1–loop four-point scattering amplitude.
At two loops, in the planar sector, the amplitude can be read from the single trace
part of the two–loop effective superpotential
where the sum runs over the first six diagrams in Fig. 2, plus the contribution from
the 1P-reducible graph in Fig. 2(g) where the bubble indicates the two–loop correction
to the chiral propagator.
In (3.2) we have factorized the tree level expression, so that M(t)
contributions to A4/Atree
In order to evaluate the diagrams we fix the convention for the upper-left leg to
carry momentum p1and name the other legs clockwise. The total contribution from
every single graph is then given by summing over all possible permutations of the
external legs accounting for the different scattering channels.
– 6 –
The momentum–dependent contributions in (3.2) are the product of a combina-
torial factor times a sum of ordinary Feynman momentum integrals arising after per-
forming D-algebra on each supergraph (details can be found in [49, 48]).
Massless scattering amplitudes are affected by IR divergences. We deal with them
by dimensional regularization, d = 3−2ǫ, ǫ < 0. All 2-loop 1PI diagrams and the single
Figure 2: Diagrams contributing to the two-loop four-point scattering amplitude.
dark-gray blob represents one-loop corrections and the light-gray blob two-loop ones.
non-1PI diagram are depicted in Fig.2. The simplest graph a has a D-algebra given by
a two-loop factorized Feynman integral. Its s-channel contribution, shown in Fig. 2, is
k2(k + p1+ p2)2l2(l − p3− p4)2= −G[1,1]2
where µ is the mass scale of dimensional regularization and the G function is defined
G[a,b] =Γ(a + b − d/2)Γ(d/2 − a)Γ(d/2 − b)
(4π)d/2Γ(a)Γ(b)Γ(d − a − b)
Taking into account all contributions of this type with color/flavor factors we obtain
= −3ζ2λ2+ O(ǫ). (3.6)
The s-channel contribution shown in the picture of diagram b (there are three other
contributions), produces a D-algebra given by
l2(l + k)2(k − p4)2(k + p3)2=
2G[1,1]Γ(1 + 2ǫ)Γ2(−2ǫ)
(4π)d/2Γ(1/2 − 3ǫ)(s/µ2)2ǫ.
– 7 –
Therefore, summing over all four contributions we obtain
1PI= (4πλ)2G[1,1]Γ(1 + 2ǫ)Γ2(−2ǫ)
(4π)d/2Γ(1/2 − 3ǫ)
which infrared divergent.
Diagram c, on the other hand, is infrared divergent even when considered off-
shell. This unphysical infrared divergence is cured when we add the 2PI diagram
corresponding to two-loop self–energy corrections to the superpotential, depicted in
Fig. 2 g. The contribution from the diagram in the picture, with the correction on the
g= −3G[1,1]G[1,3/2 + ǫ](p2
The first term of this expression is infrared divergent even off–shell, but precisely this
term cancels the infrared divergence of the diagram of type c exactly. The second term
in (3.9) comes from a double factorized bubble and is finite when d → 3, but since
we take the momenta to be on–shell before we expand in ǫ, then this piece vanishes
on–shell. It turns out that after this cancelation between diagrams of topology c and
the 2–loop self–energy corrections to the superpotential, the remainder is proportional
to the same integral as for diagram b; this is
1PI+ M2PI= −3Mb
Diagrams of type d may be calculated using Mellin-Barnes techniques. Specifically,
the D-algebra of the one drawn in the Fig. 2 is
Γ3(1/2 − ǫ)Γ(1 + 2ǫ)Γ2(−2ǫ)
(4π)dΓ2(1 − 2ǫ)Γ(1/2 − 3ǫ)(s/µ2)2ǫ,
(k + p4)2(k − p3)2(k + l)2(l − p4)2l2
4(p3+ p4)ν(k + p4)ρ(l − p4)σ
and taking into account all eight contributions from permutation with flavor/color
factors we obtain
1PI= −(4πλ)22Γ3(1/2 − ǫ)Γ(1 + 2ǫ)Γ2(−2ǫ)
(4π)dΓ2(1 − 2ǫ)Γ(1/2 − 3ǫ)
Using the identities derived in  it is possible to write diagram e as a combination
of diagrams b and d plus a double factorized bubble which we can drop for the same
reasons as mentioned above for diagram c. We thus find
– 8 –
The most complicated contribution comes from diagram f, which involves a non-
trivial function of the ratio s/t of kinematic invariants. Surprisingly, after some can-
celations it turns out to be finite. The D-algebra for the specific choice of external
momenta in the Fig. 2 f results in the Feynman integral
k2(k − p2)2(k + l + p3)2(l − p4)2l2,
which after taking the on–shell limit can be expressed exactly as a single one-fold
Mellin-Barnes integral which is finite in the limit ǫ → 0:
(1 + s/t)Γ3(1/2 − ǫ)
(4π)dΓ2(1 − 2ǫ)Γ(1/2 − 3ǫ)(t/µ2)2ǫ×
2πiΓ(−v)Γ(−2ǫ − v)Γ∗(−1 − 2ǫ − v)Γ2(1 + v)Γ(2 + 2ǫ + v)
Taking into account all four permutations, flavor/color factors and expanding in ǫ we
2log2(s/t) + 3ζ2
?+ O(ǫ) (3.18)
Collecting all partial results, after some algebra we may reduce the result to the
following compact form
+ K1+ O(ǫ)
where µ′2= µ28πe−γ, and K is a constant given by K1= 4ζ2+ 3log22.1
If we rotate to Minkowski spacetime with mostly minus signature and write the
Mandelstam variables in terms of the dual ones, s = −x2
dependent) constant, and an overall sign, our result matches that (1.1) for the two–loop
expansion of a light–like Wilson loop, once we have identified the IR and UV rescaled
regulators of the Wilson loop and scattering amplitude, respectively, as 1/µ′2
13,t = −x2
24, up to a (scheme–
4. Dual conformal invariance
The two–loop result (3.19) for the four–point amplitude in ABJM theories has the
same functional structure as the one–loop correction to the four–point amplitude in
four dimensional N = 4 SYM theory .
1We note that the analytical value of the constant term matches the numerical result of .
– 9 –
In the N = 4 SYM case, the perturbative results for planar MHV scattering
amplitudes can be expressed as linear combinations of scalar integrals that are off–shell
finite in four dimensions and dual conformal invariant . Precisely, once written in
terms of dual variables, pi= xi+1− xi, the integrands times the measure are invariant
under translations, rotations, dilatations and special conformal transformations. In
particular, invariance under inversion, xµ→ xµ/x2, rules out bubbles and triangles and
up to two loops, only square–type diagrams appear.
Dual conformal invariance is broken on–shell by IR divergences that require intro-
ducing a mass regulator. Therefore, conformal Ward identities acquire an anomalous
A natural question which arises is whether the two–loop result (3.19) for three
dimensional ABJM models exhibits dual conformal invariance.
In order to answer this question, we concentrate on the momentum integrals associ-
ated to the four diagrams in Fig. 2 which are the ones that eventually combine to bring
to the final result (3.19). We study their behavior under dual conformal transformations
when evaluated off–shell and in three dimensions.
We first rewrite their expressions in terms of dual variables and then perform
conformal transformations, the only non–trivial one being the inversion.
Since under inversion x2
four dimensions the elementary invariant building block integrands are squares, in three
dimensions they should be triangles. Therefore, it is immediate to conclude that the
integrands associated to diagrams 2(a) − 2(b) cannot be invariant, since they contain
bubbles. On the other hand, diagrams 2(d)−2(f) contain triangles but also non–trivial
numerators which concur to make the integrand non–invariant under inversion.
Nevertheless, what happens is that every single integral, when evaluated off-shell
in three dimensions, exhibits invariance under conformal transformations.
As an example, we consider the double bubble diagram 2(a). In dual coordinates,
the corresponding integral reads
i)d, it is easy to realize that, while in
B = x2
Performing inversion, this integral gets mapped into a double triangle integral
T = x2
If we evaluate them off-shell, we obtain 
B = x2
T = x2
– 10 –
Therefore, thanks to the off–shell identity B = x2
invariant under inversion at the level of the integral, even if it is not invariant at the
level of the integrand.
A similar analysis can be performed for the other contributing diagrams. For each
diagram separately we have checked numerically that a single contribution correspond-
ing to a particular configuration of the external momenta is not invariant, but the sum
over cyclic permutations of the external momenta leads to an invariant result.
In conclusion, every single topology of diagrams is not manifestly dual conformal
invariant but, once we evaluate the corresponding integral and sum over permutations
of external legs, we obtain a dual conformal invariant result. Therefore, at this stage,
dual conformal invariance works at the level of the integrals but not at the level of the
After all, this does not come as a surprise, since what we learn from the four
dimensional case is that no matter which are the topologies of Feynman diagrams
contributing to the scattering amplitudes, dual conformal invariance should allow to
rewrite the result as a linear combination of scalar integrals which are off-shell finite in
three dimensions and manifestly dual conformal invariant at the level of the integrands2.
3T , the double bubble diagram is
5. A conjecture for the all–loop four–point amplitude
Our result in (3.19) provides the first non–trivial contribution to the four–point scatter-
ing amplitude in the ABJM theory. The analogue quantity in four dimensional N = 4
SYM has been extensively studied and an all–loop iteration conjecture for it has been
given in [37, 38]. The result may be schematically written as
where fN=4(λ) is the scaling function of N = 4 SYM in terms of the ’t Hooft coupling
λ = g2N, the constant C(λ) is independent from the kinematic variables and the IR
divergent contributions are grouped in the first term.
It would be interesting to check whether a similar resummed expression may hold
for scattering amplitudes in the three–dimensional case. Although we only computed
the first non-trivial perturbative order for the amplitude, still we have some indications
that this could be the case.
At first, comparing the conjectured form of the asymptotic all loop Bethe equations
for N = 4 SYM and ABJM theory, Gromov and Vieira noticed  that the scaling
2Note added: This task has been actually accomplished in , where an explicit basis of dual
conformal integrals has been determined on which the amplitude can be expanded.
– 11 –
functions of the two theories should be related as
where h(λ) is the interpolating function of the magnon energy dispersion relation. The
first perturbative orders of h(λ) have been computed at both weak [28, 29] and strong
coupling –. The weak coupling expansion
h2(λ) = λ2− 4ζ2λ4+ O(λ6)λ ≪ 1 (5.3)
can be combined, using (5.2), with the known expansion of the 4d scaling function
fN=4(λ) = λ/2π2− 1/96π2λ2+ O(λ3) . We are then able to write explicitly the 3d
scaling function up to order λ4
fCS(λ) = 4λ2− 4π2λ4+ O(λ6) (5.4)
Assuming (5.1) to hold also in the three dimensional case with the very same constant
coefficients and plugging (5.4) in it, after expanding at order λ2, we curiously find an
exact correspondence with the result we explicitly computed in (3.19). This suggests
that for the three dimensional case, provided we use the correct scaling function, a com-
pletely analogous resummation may take place to give an expression for the amplitude
of the form
If this is the case, since the h2(λ) is known up to order λ4[28, 29], we may predict
the next non trivial order for the 3d four-point scattering amplitude
+ Div + Consts (5.6)
A direct check of this prediction, either with a 4–loop scattering amplitude computation
or using the duality with Wilson loops, could confirm the conjectured exact expression
We briefly summarize the main results of this paper and discuss future developments.
For three dimensional ABJM superconformal models, in a N = 2 superspace setup,
we have computed the planar, two–loop corrections to the chiral (ABAB) four–point
superamplitude. We performed the calculation by a direct Feynman diagram approach,
– 12 –
in a manifestly supersymmetric formalism. We have found a non–vanishing result which
perfectly agrees with the two–loop result for a light–like four–polygon Wilson loop. This
result represents the first non–trivial evidence of an amplitude/WL duality working in
three dimensional superconformal theories and confirms the conjectured duality which
seemed to arise trivially at one–loop.
Its functional structure resembles the one–loop planar four–point amplitude for
N = 4 SYM theory in four dimensions. As in that case, it can be obtained from a BDS–
like ansatz for the all–loop amplitude where the scaling function of four dimensions
is substituted by the three dimensional one, as predicted by the conjectured Bethe
For N = 4 SYM theory the structure of the four point BDS ansatz has been
verified also at strong coupling . It would be interesting to check whether applying
the recipe of  for computing scattering amplitudes at strong coupling to the ABJM
case, the result agrees with a three dimensional version of the BDS ansatz. From our
weak coupling computation we expect this to be the case, at least at four point. Details
on this will be given in .
An important question to be addressed is whether and how dual conformal invari-
ance plays a role in three dimensional models. By explicit calculations, which do not
assume dual conformal invariance a priori, we have verified that every single Feynman
diagram entering the two–loop calculation, when suitably summed over cyclic permu-
tations of the external momenta, is off–shell dual conformal invariant. This shows
nontrivial evidence for dual conformal invariance to hold also for the ABJM model,
despite a strong coupling explanation analogous to the N = 4 case is still unclear
– 13 –
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