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Light-like polygonal Wilson loops in 3d Chern-Simons and ABJM theory

We study light-like polygonal Wilson loops in three-dimensional Chern-Simons and ABJM theory to two-loop order. For both theories we demonstrate that the one-loop contribution to these correlators cancels. For pure Chern-Simons, we find that specific UV divergences arise from diagrams involving two cusps, implying the loss of finiteness and topological invariance at two-loop order. Studying those UV divergences we derive anomalous conformal Ward identities for n-cusped Wilson loops which restrict the finite part of the latter to conformally invariant functions. We also compute the four-cusp Wilson loop in ABJM theory to two-loop order and find that the result is remarkably similar to that of the corresponding Wilson loop in N=4 SYM. Finally, we speculate about the existence of a Wilson loop/scattering amplitude relation in ABJM theory.


Introduction and conclusions
Wilson loops are the central non-local observables in any gauge theory and thus of intrinsic interest. In 3d Chern-Simons theory they are the principal observables and topologically invariant with exactly known correlation functions [1] in the Euclidean (or Wick rotated) theory. This exact result is analytic in the inverse Chern-Simons parameter k and perturbative studies in a loop-expansion of the effective coupling constant 1/k can reproduce the exact topological and finite result to the first orders [2,3], modulo regularisation subtleties leading to or not leading to an integer shift of k (for a review see [4]). Wilson loops in Minkowski-space with cusps and light-like segments, however, display particularly strong divergences in 4d gauge theories and seem to not have been considered in the 3d Chern-Simons literature before.
In this paper we study such light-like Wilson loops with cusps of polygonal shape in perturbation theory up to the next-to-leading order in 1/k. We do this for both the pure 3d Chern-Simons theory as well as its conformal N = 6 supersymmetric extension known as Aharony, Bergman, Jafferis, Maldacena (ABJM) theory [5]. The N = 6 ABJM theory is built upon an SU(N) × SU(N) gauge symmetry which allows for a planar N → ∞ limit with λ = N/k held fixed, where k is the common absolute value of the Chern-Simons parameters of the two SU(N) subgroups. In this limit the ABJM theory is conjectured to be dual to type IIA string theory on AdS 4 × CP 3 , representing an exact gauge-string duality pair very similar in nature to the well studied 4d N = 4 super Yang Mills/AdS 5 × S 5 string duality pair. Supersymmetric Wilson loops in ABJM theory have been defined in [6][7][8] for the 1/6 BPS and recently in [9] for the 1/2 BPS case. The correlators for these loops of Euclidean, circular geometry are moreover known exactly in terms of a supermatrix model [10] using localisation techniques [11]. Our motivation to consider polygonal light-like Wilson loops in the 3d Chern-Simons ABJM gauge theory stems from the Wilson loop/scattering amplitude duality in N = 4 super Yang-Mills. This duality was discovered in the dual AdS 5 × S 5 string picture at strong gauge coupling in [12] and shown to exist also in the weak coupling regime [13][14][15] with profound consequences on the symmetries of these correlators leading to a dual superconformal [16] respectively Yangian symmetry [17] of scattering amplitudes, for reviews see [18,19]. Moreover, there are many structural similarities of the 3d N = 6 superconformal ABJM theory to N = 4 super Yang-Mills, most notably the emergence of hidden integrability [20][21][22][23], (for reviews see [24][25][26][27][28][29][30][31]) in the planar limit for the spectral problem of determining anomalous scaling dimensions of local operators [32,33].
Given these insights the question arises whether there could also be such a scattering amplitude/Wilson loop duality in the ABJM theory. Scattering amplitudes in the ABJM theory have been analysed by Agarwal, Beisert and McLoughlin [34] who in fact studied more general mass deformed superconformal Chern-Simons theories with extended supersymmetries at the one-loop order. There a vanishing result for the four-point one-loop amplitudes in the ABJM theory was found and the authors speculated whether the two-loop scattering amplitudes in N = 6 Chern Simons (ABJM theory) could be simply related to the one-loop N = 4 Yang-Mills amplitudes. The main result of our paper is that this picture is consistent at least up to the two-loop order: We observe a cancellation of the one-loop graphs for null polygonal loops and find that the fourcusp Wilson loop at the two-loop order is of the same functional form as the four-point MHV amplitude of N = 4 up to constant numerical terms.
Specifically we calculate the expectation value of the n-cusped Wilson loop operator in the planar limit 1 for light-like polygonal contours C in pure Chern-Simons and ABJM theory, see Appendix A for conventions of the Lagrangian and the path ordering.
The contour of the n-sided polygon C is given by n points x i (i = 1, ..., n) where we parametrise each edge C i via where s i ∈ [ 0, 1 ] and C = C 1 ∪...∪ C n . The calculations are performed in 3-dimensional Minkowski space with metric η µν = diag(1, −1, −1) , (1.3) and the segments of the contour are light-like, i.e. p 2 i = 0. Note that due to the light-like contour there is no difference in ABJM theory between the standard loop operator (1.1) above and the 1/6 BPS supersymmetric loop operator of [6], as the terms in the exponential coupling to the scalars drop out.
Wilson loops in Chern-Simons theory are usually defined with a framing procedure [1][2][3] which may be thought of as a widening of the Wilson line to a ribbon. This is necessary in order to define an integer twisting number of the individual loop and acts as a particular point-splitting regulator for collapsing gauge field propagators in perturbation theory while preserving the topological structure of the theory. Here we refrain from framing our loops as we do not encounter the problem of collapsing gauge field propagators due to the piece-wise linear structure of our loops. Moreover, the ABJM theory is not topological due to metric dependent interactions in the matter sector, so that there is no need for framing from that perspective either. Instead we regulate our correlators by the method of dimensional reduction which has been tested to the three-loop order in pure 3d Chern-Simons to yield a vanishing β-function and to satisfy the Slavnov-Taylor identities [35]. Here the tensor algebra is performed in 3 dimensions to obtain scalar integrands and then the dimension of the integrations are analytically continued.
The outcome of our computations at one-loop order in pure Chern-Simons and ABJM theory is that as claimed Moreover, conformal Ward identities force W n 1-loop to depend only on conformally invariant cross ratios of the (x i − x j ) 2 and for n = 4 and 6 we show that these functions vanish.
At the two-loop order we computed the tetragonal Wilson-loop W 4 in pure Chern-Simons and N = 6 superconformal Chern-Simons (ABJM theory). The result in dimensional reduction regularisation with d = 3 − 2ǫ for the correlator in pure planar Chern-Simons reads We remark that this result displays a breakdown of finiteness and topological nature of the light-like four-cusp Wilson loop in 3d Chern-Simons at the two-loop order due to divergences 2 associated to two cusps at a light-like distance, see section 3.2. For the same correlator in the ABJM theory we find This is indeed of the same functional form as the one-loop result in N = 4 super Yang-Mills theory, where one has [13] (1.7) It would certainly be interesting to explore this relationship also beyond four cusps. Also a two-loop computation of the four-particle scattering amplitudes in ABJM theory would be very desirable in order to compare to our result.
From the string perspective the scattering amplitude/Wilson loop duality in the AdS 5 /CF T 4 system arises from a combination of bosonic and fermionic T-dualities under which the free AdS 5 × S 5 superstring is self-dual [36,37]. Hence, for the existence of an analogue duality in ABJM theory one would require a similar self-duality of the AdS 4 ×CP 3 superstring under a suitable combination of T-dualities. This problem was analysed by Adam, Dekal and Oz in [38] with a negative outcome: The Green-Schwarz σ-model is not self-dual under bosonic T-dualtities in the transverse AdS 4 directions combined with fermionic ones within the framework of the fermionic Buscher dualisation procedure employed in [36]. However, the analysis of [38] started from a partially κ-symmetry gauge fixed formulation of the AdS 4 × CP 3 superstring in terms of a supercoset σ-model [39,40]. This restriction was overcome in [41] where, building upon a complete superspace formulation of AdS 4 × CP 3 [42], again the non-existence of a T-self-duality of the AdS 4 × CP 3 superstring was found. Interestingly however, a very recent paper [43] has uncovered a Yangian symmetry of tree-level amplitudes in ABJM theory pointing towards integrability of the latter. Moreover, the authors argue about the possibility of a self-duality of the AdS 4 × CP 3 superstring upon T-dualizing also along the CP 3 directions, which could provide a loop-hole for a scattering-amplitude/light-like Wilson loop duality of the ABJM theory. In order to settle this question a two-loop scattering amplitude computation in ABJM theory would be very desirable in order to compare to our result (1.6).
Our paper is organised as follows: In section 2 we perform the one-loop computation in pure Chern-Simons and ABJM theory for the tetragon and the hexagon Wilson loop. Section 3 is devoted to the two-loop problem of the tetragon in pure Chern-Simons theory where we compute all relevant graphs. Then, in section 4, we perform an independent check of our results by deriving anomalous conformal Ward identities for the Wilson loop. The latter can be generalised to an arbitrary number of points. In section 5 we include the matter diagrams arising in ABJM theory at the two-loop order and combine the results for the tetragon to obtain our final result (1.6). In the appendix we collect a detailed account of our conventions and give the technical details of the computation of the two-loop graphs using Mellin-Barnes techniques.

Note added
After publication of this paper the works [44], [45] appeared which report on a two loop calculation of four-point scattering amplitudes in ABJM theory. Most interestingly, the results coincide precisely with the divergent and finite pieces of our Wilson loop computation in (1.6) up to a constant. In the published version of this article there was an erroneous sign in (5.11) leading to a spurious sign difference between the Wilson loop and the scattering amplitude. We thank the authors of [45] for pointing this out to us.

One loop: Chern-Simons and ABJM theory
In this section we consider the one-loop expectation value of polygonal Wilson loops with n cusps. We would like to consider kinematical configurations for which all non-zero distances satisfy −x 2 ij > 0, such that the result for the Wilson loops will be real (In particular, this allows us to drop the iǫ prescription of the propagators). For n odd, however, it is impossible to find vectors p µ i that lead to such configurations. For this reason, we will only discuss n even.
At one-loop level, we only need terms quadratic in the expansion of the Wilson loop operator, and the free part of the action. Therefore, at one loop order, the expectation value of (1.1) in ABJM theory coincides with the one in pure Chern-Simons theory.
The expectation value at one loop is a sum over all possible diagrams where the propagator stretches between edges i and j, where the domain of integration is given by Here and throughout the paper, we absorb the dimensional regularisation scale (µ 2 ) ǫ into k and only display it explicitly in our final results. Using the Chern-Simons propagator in the Landau gauge 3 and plugging in the expressions (1.2) for z i , we obtain We can immediately see that in this gauge I i,i and I i,i+1 vanish due to the antisymmetry of the ǫ tensor. This corresponds to diagrams where the propagator ends on the same edge or on adjacent edges, as shown in Figures 1(b) and 1(c), respectively. Therefore we only need to keep diagrams of the type shown in Figure 1(a). The latter are manifestly finite in three dimensions and therefore we set d = 3 in the remainder of this section.

Tetragon
As explained above, in the Landau gauge, the only non-vanishing contributions to (2.1) for the tetragon are I 31 and I 42 . Setting d = 3, they are given by and Taking into account that we have a closed contour, i.e. i p i = 0, we can write ǫ(p 2 , p 3 , p 4 ) = −ǫ(p 2 , p 3 , p 1 ) = −ǫ(p 1 , p 2 , p 3 ) and thus the contributions from the two diagrams cancel each other We will see in section 4 that this result is compatible with the restrictions imposed by conformal symmetry.

Hexagon and higher polygons
For the hexagon there are two different non-vanishing types of contributions, I i+2,i and I i+3,i , as shown in Figure 2. The former appears in six orientations, i = 1 . . . 6 (with the convention that i + 6 ≡ i), while the latter appears in three orientations, i = 1, 2, 3. Specialising the general formula (2.3) to these cases we have and We checked numerically for various non-symmetric hexagon configurations that the sum over all diagrams vanishes, Although we do not yet have an analytical proof for generic kinematical configurations, we can show that (2.9) is true for special configurations, as we will see presently. Consider the configuration where opposite edges are anti-parallel, i.e. p i = −p i+3 . From (2.8) we see that I i,i+3 = 0 due to the antisymmetry of the ǫ tensor. Furthermore, taking into account that for this configuration we have x 2 i,i+2 = x 2 i+3,i+5 , it is easy to see from equation (2.7) that the integrands of I i,i+2 and I i+3,i+5 are the same. Finally, using i p i = 0 one can see that the Levi-Civita symbols produce a differing sign, such that i.e. the contributions coming from those diagrams cancel pairwise, and we arrive at equation (2.9), in the specific anti-parallel kinematical configuration p i = −p i+3 . It is tempting to speculate that all n-cusped Wilson loops vanish at one-loop order in Chern-Simons theory.

Two loops: Chern-Simons theory
In the this section we calculate the two-loop contributions to the tetragonal light-like Wilson loop in pure Chern Simons theory. The results are consistent with the anomalous conformal Ward identity to be discussed in section 4.
Expanding the Wilson loop to quartic order, see (A.12), and performing Wick contractions leads to the topologies shown in Figure 3. We are taking the planar limit and therefore drop all non-planar graphs. Moreover, all diagrams where one propagator is connected to a single edge or adjacent edges vanish in our gauge for the same reason as at the one-loop order and are not displayed.

Ladder diagrams
Let us begin by computing diagrams of ladder topology as shown in Figure 3d. There are two different orientations of this diagram, and it is easy to see that they give the same contribution. Taking into account this factor of 2, we have that the contribution of the ladder diagrams is The integral is finite and may be calculated for d = 3 We computed this integral by first carrying out some of the parameter integrals and then deriving a differential equation for it, which could be solved. The result is remarkably simple, Including the prefactors and dropping O(ǫ) terms, the contribution to the Wilson loop is (3.5)

Vertex diagrams
The diagrams with one three-gluon vertex shown in Figures 3a , 3b and 3c are obtained by contracting the cubic term in the expansion of the Wilson loop in (A.12) with the interaction term of the Lagrangian, where and Here the indices of I ijk refer to the edges of the Wilson loop that the propagators attach to. The expression can be shown to be antisymmetric under the exchange of any two indices, and therefore the only non-vanishing contributions are the ones for i = j = k. As a consequence, topologies 3a and 3b can be discarded.
Specialising to the tetragon, we have four contributions which are symmetric under x 2 13 ↔ x 2 24 and thus it is sufficient to compute one of them where the second and third line is obtained by introducing Feynman parameters in the standard way and integrating over w. More details may be found in Appendix B.1. ∆ is given by One might naively think that this diagram should give a finite answer due to the antisymmetry of the ǫ tensors. The result would indeed be finite in the case of smooth contours [2] or contours with a single cusp. However, the presence of two cusps gives rise to a region in the integration space of Feynman parameters where the first summand in the third line of (3.8) induces a divergent contribution. The relevant region of Feynman parameters is s 1 → 1, s 3 → 0 (and β 1 → 0, β 3 → 0), see equation (3.9), and is illustrated in Figure 4. Due to the presence of three independent vectors p µ 1 , p µ 2 and p µ 3 the ǫ tensors do not suppress this region. We find that this term produces a 1/ǫ pole in dimensional reduction. The second summand in the third line of (3.8) is finite.
We separated the divergent and finite pieces using Mellin-Barnes techniques. The details can be found in Appendix B.1. We have not computed the coefficients of the ǫ −1 and ǫ 0 terms analytically, but we have good numerical evidence that they give the following result: where a 6 = 8.354242685 ± 2 · 10 −9 , see (B.27). Taking into account all prefactors and restoring the regularisation scale, k → µ −2ǫ k, we can write the result, up to terms of order ǫ, as

Gauge field and ghost loops
It is well known [35] that in the dimensional reduction (DRED) scheme the gauge field loop diagrams shown in Figure 3e exactly cancel against the ghost loop diagrams shown in Figure 3f : Details of this cancellation can be found in appendix B.2.

Result for the two-loop tetragon in CS theory
Summing up the results (3.5), (3.11) and (3.12) for the tetragon, interestingly, the ln 2 (x 2 13 /x 2 24 ) terms in the two-gauge-field diagram and the vertex diagram exactly cancel and we obtain (3.13) whereμ 2 = µ 2 πe γ E and we recall that a 6 = 8.354242685 ± 2 · 10 −9 . As we will see in the next section, the cancellation observed here that led to the finite part of (3.13) being a constant is in fact a consequence of the (broken) conformal symmetry of the Wilson loops under consideration.

Anomalous conformal Ward identities
The structure of the above results can be understood from conformal symmetry, by deriving anomalous conformal Ward identities for the Wilson loops. Here we follow very closely reference [46] . We would like to use the specific properties of Wilson loops with light-like contours C under conformal transformations. The key point is that such contours are stable under conformal transformations, i.e. the deformed contour C ′ is also made of n light-like segments. This can be seen as follows. The cusp points x i form a contour with light-like edges, i.e. x 2 i,i+1 = 0. It is obvious that the light-likeness conditions are preserved by translations, rotations, and dilatations. Special conformal transformations are equivalent to an inversion x µ → x µ /x 2 followed by a translation and another inversion. Thus it remains to investigate the transformation under inversions. Since under the latter , it is clear that the light-likeness of the contour is preserved by all conformal transformations.
If the Wilson loop W n were well defined in d = 3 dimensional Minkowski space it would enjoy the conformal invariance of the underlying gauge theory and we would conclude that W (C) = W (C ′ ) . This is indeed the case at one loop order, see section 2. However, as we have seen in section 3, starting from two loops, divergences force us to introduce a regularisation and calculate in d = 3 − 2ǫ dimensions, thereby breaking the conformal invariance of the action. The latter leads to an anomalous term in the conformal Ward identities for the Wilson loops, as we will see presently.
The expectation value of the Wilson loop can be written as a functional integral where µ is the regularisation scale that keeps the action dimensionless in d = 3 − 2ǫ. The pathordered exponential is invariant under dilatations and the Lagrangian is covariant with weight ∆ L = 3, whereas the measure d d x does not match this weight for d = 3 − 2ǫ. This results in a non-vanishing variation of the action with respect to dilatations and special conformal transformations. The conformal Ward identities can be derived by acting on both sides 4 of (4.1) with the generators of conformal transformations, see [46][47][48]. This leads to the Ward identities for dilatations and special conformal transformations. Here the operators on the left hand sides act in the canonical way on the coordinates of the cusp points, We emphasise that thanks to the factor of ǫ on the r.h.s. of (4.2) and (4.3) it is sufficient to know the divergent part of the integrals appearing on the r.h.s. of those equations in order to obtain information about the finite part of W n . The dimensionally regularised Wilson loop W n is a dimensionless scalar function of the cusp points x ν i , which appear paired with the regularisation scale as x 2 ij µ 2 . As a consequence W n This provides a consistency condition for the right hand side of (4.2).

One-loop insertions
At order N/k we have a contribution from the contraction of the kinetic part of the Lagrangian insertion with the second order expansion of the Wilson loop operator, shown in fig. 5, The direct calculation of the right hand sides of (4.2) and (4.3), yields a vanishing result as x Figure 5: Lagrangian insertion contributing to the Ward identities at one loop.
ǫ → 0. Thus we have in other words the conformal symmetry is unbroken for ǫ = 0. As a consequence, the expectation value of the Wilson loop is constrained to be a function of conformally invariant variables. Starting from the Lorentz invariants x 2 ij the most general conformal invariants are the cross-ratios In our case where neighbouring points are light-like separated, x 2 i,i+1 = 0, non-vanishing crossratios can only be written down starting from n = 6. The special conformal Ward identities (4.7) then imply that W n (1) is given by a function of conformal cross-ratios, Since there are no non-vanishing conformal cross-ratios at four points, W 4 (1) must be a constant. Let us now compare against the results of our one-loop computation of section 2. There, the constant on the r.h.s. of the second equation in (4.9) was found to be zero for the tetragon. Moreover, analytical investigations of certain symmetric contours and numerical investigations for non-symmetric contours show that g 6 (u ijkl ) is zero for the hexagon. As mentioned before, we expect that the result remains true for higher polygons, i.e. g n = 0.

Two-loop insertions
At two loops there are several diagrams that contribute to the insertion of the Lagrangian into the Wilson loop, L(x)W n , that correspond to the kinetic term, the gauge field vertex, the ghost kinetic term and the ghost vertex in L(x). Those diagrams are shown in Figure 6. We do not display diagrams that vanish for kinematical reasons as at one-loop level.
Just as at one-loop level, only diagrams giving rise to divergent integrals will contribute to the anomalous Ward identities.

Insertion into the ladder diagram
Let us consider the insertion of the kinetic term of the action into the ladder diagram as shown in Figure 6a. For the dilatation Ward identity these are exactly the two-gluon diagrams calculated above, which are finite. For the special conformal Ward identity the integration is slightly more complicated, but the finiteness is easy to check for all contributions. Thus, this diagram does not contribute to the anomalous Ward identities.

Insertion of the interaction term
Next, we can contract the cubic order expansion of the Wilson loop with the vertex term of the Lagrangian insertion, as shown in Figure 6b, For the dilatation Ward identity, we trivially have which is just the vertex diagram that was calculated in (3.6), up to a factor of i. The contribution to the special conformal Ward identity is more complicated. We have where the coefficient ln (2)

Insertion of the kinetic term into the vertex diagram
Furthermore, we can contract one gauge field of the kinetic term of the insertion with the Wilson loop and the other one with the 3-gauge-field vertex, leading to a diagram of the type displayed in Figure 6c, Let us Wick-contract the kinetic term with A ν (z j ) (the two other contractions are discussed below.) We obtain The only dependence on the insertion point x is in I νσ . For the dilatation Ward identity the integral d d x I νσ can easily be computed (for details see Appendix B.3.2) and effectively gives a propagator such that we have where a factor of 3 was included since the insertion can be in any of the 3 propagators of the vertex diagram and thus we get three times the same contribution. The contribution to the special conformal Ward identity is more complicated, since the integration d d x x ν I νσ does not just yield a propagator. Performing the calculation, we find where the coefficient ln(2) was computed numerically (details can be found in Appendix B.3.2).

Insertions with gauge field and ghost loops
The gauge-field-ghost-insertions in Figure 6 cancel pairwise. For the dilatation Ward identity the insertions of the three gauge-field and the gauge-field-ghost vertices as shown in diagrams 6d, 6g are identical to the gauge field and ghost loop diagrams (3.12) and thus cancel. It is not necessary to perform the integration over the insertion point to see how the cancellation occurs and thus the contributions to the special conformal Ward identity cancel as well (for details on the cancellation see B.2).
Insertions of the kinetic term into the gauge field propagator as shown in diagrams 6e, 6h cancel as well, since the insertion is the same for both diagrams and thus the algebraic relations responsible for the cancellation in (3.12) remain unchanged.
Inserting gauge field respectively ghost kinetic terms into the propagators inside the loop in diagrams 6f, 6i produce slightly more complicated expressions. Nevertheless they cancel as well as can be seen in a straightforward calculation. For the dilatation Ward identity the cancellation can be seen in an even simpler way by noticing that the integration over the insertion point x effectively yields a gauge field respectively ghost propagator. Thus the diagrams are identical to the ones in (3.12) and cancel.

Anomalous Ward identities and generalisation to higher polygons
Summing up the divergent contributions of (4.11) (4.18), and inserting them into the dilatation Ward identity (4.2), we obtain where we have written the factor 4 as ( 4 i=1 1) to emphasise its origin from the sum of four vertex-type diagrams. Note that only the divergent part of the vertex-diagram was required here. Summing up (4.15) and (4.19), and inserting them into the special conformal Ward identity (4.3), we obtain Let us now explain how these equations can be generalised from n = 4 to arbitrary n. In our two-loop computation, we found that the only diagrams contributing to (4.20) and (4.21) are those producing poles in ǫ. The mechanism for how these poles are generated was described in section 3.2, see in particular Figure 4. It is clear that for n > 4 cusps, the same type of vertex diagram will produce the divergent terms. Although those diagrams will depend on one further kinematical variable w.r.t. the four-point case, this dependence cannot change the (leading) UV pole ǫ −1 of the diagrams. Since there are n diagrams of this type at n points, we expect and We will now proceed to discuss the solution of these Ward identities and compare them to the result of the two-loop computation of the tetragon Wilson loop in section 3.4.

Solution to the anomalous conformal Ward identities
Using D x 2 ij = 2x 2 ij it is clear that the most general solution to the dilatation Ward identity (4.22) is where f is an arbitrary function of dimensionless variables and we recall thatμ 2 = µ 2 πe γ E . Of course, this is exactly what we expect from (4.5).
The result for the special conformal Ward identity is more interesting. Plugging (4.24) into the special conformal Ward identity (4.21) and using K ν ln(x 2 kl ) = 2(x k +x l ) ν , it is easy to see that the function f n is allowed to depend on conformally invariant cross-ratios only, i.e. f n (x 2 ij /x 2 kl ) = g(u abcd ). Therefore we finally have In the four-point case, there are no non-vanishing cross-ratios, and therefore in particular g 4 must be a constant. This is in agreement with (3.13) and thus represents an independent check of the direct perturbative computation, including its finite part (recall that deriving the Ward identity does not rely on the finite parts of the direct perturbative computation). So, even though the result for the vertex diagram (3.11) was obtained numerically, its functional form is an analytical result, since we know the analytical expression for the ladder diagram and the sum of vertex and ladder diagram through the solution to the anomalous conformal Ward identity.

Two loops: ABJM theory
Here we explain how the results are modified in ABJM theory. We use the Wilson loop operator proposed in [6] W (A,Â) = 1 2N Note that the sign(s) in the exponent(s) in (5.1) are correlated to corresponding signs in the Lagrangian by the requirement of gauge invariance, see Appendix A.

Gauge field contributions
In ABJM theory there is a second copy of the gauge fieldÂ µ with opposite sign in the Lagrangian (A.20). Up to a sign, the gauge field contributions for both gauge groups are identical at one loop,

W
(1) due to the different sign of the propagator for the second gauge field, A µ A ν = − Â µÂν . Thus, at one loop the diagrams cancel. This does not differ from the result of pure Chern-Simons theory, since the expectation value at one loop vanishes, as we found in section 2 for n = 4, 6. At two loops, however, the sign has no effect, since the two-gluon diagram contains an even number of propagators and in the vertex diagram we have to take into account the sign of the interaction term as well. Therefore, the two-loop diagrams are identical

Matter contributions
In pure Chern-Simons theory the one-loop correction to the gauge field propagator is zero, since the contributions of gauge fields and ghosts exactly cancel against each other, see (3.12).
In ABJM theory we have to take into account fermionic and bosonic matter in the loop. This gauge field self energy has been calculated in [49,50,6] and the corrected propagator reads 5 for details see Appendix C. We can drop the derivative term in (5.5) (it would not contribute to the gauge-invariant Wilson loop) and instead use the propagator which up to two small differences is the tree level N = 4 SYM gluon propagator. The first difference is a trivial prefactor, and the second is that since we are at two loops, the power of 1/x 2 is 1 − 2ǫ here, as opposed to 1 − ǫ in the one-loop computation in N = 4 SYM. Thus it is clear that the results will be very similar to the expectation value of the Wilson loop in N = 4 SYM. The corresponding calculation in N = 4 SYM was carried out in [13] and we briefly summarise the results. As in N = 4 SYM we have three classes of diagrams shown in figure 7. Diagram 7a vanishes due to the light-likeness of the edges, whereas 7b yields a divergent, and 7c yields a finite contribution. We have There are four diagrams I i+1,i of the type shown in fig. 7b. It is sufficient to compute one of them, as the others can be obtained by the replacement i → i + 1. Setting e.g. i = 2, j = 1, we have Furthermore, there are two finite diagrams I i+2,i of the type shown in fig. 7c. Setting d = 3 and taking i = 3, j = 1 we have Taking the sum over all contributions we obtain and thus the full matter part is where we have restored the regularisation scale µ 2 . The full result is obtained by adding the CS part (3.13) to (5.11) and the result can be rewritten in a form in which the ǫ −1 terms cancel where µ ′ 2 = 8πe γ E µ 2 and const. = 8 ln(2) + 20 ln 2 (2) + 2π 2 − a 6 and where a 6 is given in (B.27) and fits the value a 6 = − 2 3 π 2 + 16 ln(2) + 8 ln 2 (2) such that const. = 8 3 π 2 + 12 ln 2 (2) − 8 ln (2). 6 This is the result quoted in the introduction (1.6).

A Conventions
An n-sided polygon can be defined by n points x i (i = 1, ..., n), with the edge i being the line connecting x i and x i+1 . Defining and parametrising the position z µ i on edge i with the parameter s i ∈ [0, 1] we have Furthermore, we use the notation ǫ(p, q, r) = ǫ µνρ p µ q ν r ρ One can easily check that with the definition We consider 3-dimensional Minkowski space with metric Using (A.4) we can rewrite the scalar products Furthermore, using the definition (A.2), one can easily show that

A.1 Cherns-Simons theory and the Wilson loop operator
The Chern-Simons action and the Wilson loop operator if the path ordering 8 is defined as for s < s ′ and s ∈ (0, 1) parametrises the path along the curve C. The path ordered exponential in the Wilson loop operator then has the expansion Quantising the theory with the standard Fadeev-Popov procedure yields the gauge fixing and ghost action In Landau gauge (ξ = 0) the gauge field propagator reads (A.14) 7 More precisely, the action is invariant under infinitesimal transformations g(x) = 1 + iα(x) and transforms like S → S ′ +(2πk)δS, where δS = − 1 24π 2 d d x ǫ µνρ Tr (∂ µ g −1 )g(∂ ν g −1 )g(∂ ρ g −1 )g under finite transformations. Since δS takes integer values, exp(iS) is invariant under large gauge transformations for k ∈ N. 8 There are different conventions on the path ordering. They are equivalent, it is however important to choose the sign in the Wilson loop such that it is gauge invariant. Reversing the sign in the exponent of the Wilson loop does not just reverse the integration contour, due to the path ordering.
where we have rescaled the coupling constant k → µ −2ǫ k and restore the dependence on the regularisation scale µ only in the final results. The ghost propagator is Note that the gauge field propagator is related to the ghost propagator by which can be used to see the cancellation of gauge field and ghost loop contributions to the one-loop gauge field propagator in a simple way. The different conventions found in the literature on Chern-Simons theory deserve a short comment. One can show, that are invariant (In the sense mentioned above) under gauge transformations where s is a real or imaginary parameter. I.e. the sign in the Wilson loop and the Lagrangian are correlated through gauge invariance. Taking a hermitian gauge field (A µ ) † = A µ we can choose s = i, which corresponds to the choice used throughout this document.
All other conventions found in the literature can be obtained by rescaling the gauge field A µ → −A µ , A µ → iA µ etc. Note however, that this changes factors in the Lagrangian, the gauge transformation, the covariant derivative and the Wilson loop. A sign difference in the Wilson loop only may also be due to a different definition of the path ordering (A.11).

A.2 Lagrangian of ABJM theory
The action of ABJM theory is whereS is obtained from S by replacing A µ with the gauge field in the anti-fundamental repre-sentationÂ µ and letting k → −k. Explicitly, we have The field content consists of two U(N) gauge fields (A µ ) ij and (Â µ )ˆiĵ, the complex fields (C I ) iî and (C I )ˆi i as well as the fermions (ψ I )ˆi i and (ψ I ) iî in the (N,N) and (N, N) of U(N) respectively, I = 1, 2, 3, 4 is the SU(4) R index. We employ the covariant gauge fixing function ∂ µ A µ for both gauge fields and have two sets of ghosts (c, c) and (c,ĉ). S int are the sextic scalar potential and Ψ 2 C 2 Yukawa type potentials spelled out in [5]. The covariant derivative is given by ]. It's action on Ψ I , C I is given in [6] and not needed here.
Performing the change of variables with Jacobian y we can rewrite (B.38) in a form where all integrations range from 0 to 1, where ∆ y = − (sxs 1 (s 3ȳ + s 2x y) + tȳs 3 (xs 1 +s 2x )) , In this section, we switch from the Feynman parametrization in equation (B.9) to a Mellin-Barnes representation, as the latter is very convenient to perform a systematic expansion in ǫ. An introduction to the Mellin Barnes technique can be found in [51].
In the first step the sum in the denominator is transformed into an integral over a product of terms. Since the denominator in (B.38) consists of a sum of four terms, we will introduce 3 Mellin parameters z 1 , z 2 , z 3 . By repeated use of the Mellin-Barnes representation where −Re(λ) < β < 0, one obtains where the real parts β i of the integration contour have to be chosen such that the arguments in all Γ functions have positive real part. Applying (B.12) to the denominator of I A (B.9), we can rewrite I A as where dz = (2πi) −1 dz. The integrals over s 1 , s 2 , s 3 , x, y can be carried out using and we arrive at Recall that we investigate the kinematical region where s, t < 0.
One can see that this integral is divergent as ǫ → 0 by noticing that for ǫ = 0 it is impossible to choose the integration contours such that all poles of Γ (...+z 1 ) are to the left of the integration contour and all poles of Γ (... − z 1 ) are to the right of the contour. 9 The reason is that the poles of Γ (z 1 + z 2 + 1) and Γ (−d − z 1 − z 2 − z 3 + 2) = Γ (−z 1 − z 2 − 1 + 2ǫ) "glue together" at z 1 = −z 2 − 1 for ǫ = 0. However, one can find allowed contours for ǫ = 0.
By shifting the contour left to the pole at z 1 = −z 2 − 1 we pick up a residue. The factor of Γ (−z 1 − z 2 − 1 + 2ǫ) evaluated at the residue, results in a divergent factor of Γ (2ǫ). The remaining integral over the shifted contour yields a finite contribution.
The steps of shifting contours and taking residues have been automatised in [52] and we used this package to systematically extract the pole terms. Applying this procedure to (B.15) and expanding in ǫ yields 3 integrals: We do not specify the values of the real parts β i as well as the functions f i , g i here, which are lengthy expressions of products of Γ functions and can be obtained automatically by expanding (B.15) with the help of [52]. Adding up the divergent part of (B.16) and (B.17) we get where by numerical integration one finds A , I A yields where a 1 has the same value as above and The constant fits the value a 6 ≈ − 2 3 π 2 + 16 ln(2) + 8 ln 2 (2) = 8.35424273... . The result can be rewritten in the form

B.2 Gauge field and ghost loops
It is well known [35], that the contributions of ghost and gauge field loops to the gauge field self energy cancel. We briefly review the cancellation of the gauge field and ghost loop corrections in the Wilson loop, since from this it is easy to see, how the cancellation for the insertions in the conformal Ward identities takes place.

B.2.1 Gauge field loop
The gauge field loop-diagram arises at second order in perturbation theory Taking into account that A µ , A ν give 3 identical contractions with one of the vertex terms and that we can contract them either with the x-or y-dependent vertex, we get a symmetry factor of 3 · 3 · 2. The remaining contractions of the gauge fields are dictated by taking into account only planar diagrams. Thus we get To proceed, we recall the relation (A.16) between gauge field and ghost propagator and write where we used ǫ αβγ ǫ δστ ǫ βτ κ ǫ γσρ = − η α κ η δ ρ + η α ρ η δ κ and ∂ x F (x − y) = −∂ y F (x − y) in the last step.

B.2.2 Ghost loop
The ghost loop diagram arises from contraction of the second order perturbation theory expansion of the gauge-field-ghost vertex term where c 8 ist the same as defined above and the factor of 2 is due to the fact that the evaluation of the first line yields two identical planar diagrams that are kept and two identical non-planar diagrams that we drop. Contracting A µ either with the x-or y-dependent vertex, we get a symmetry factor of 2. There is only one way for the remaining contractions and thus we get where a factor of −1 due to the anti-commuting ghost fields in the loop was taken into account.

B.3 Conformal Ward identity B.3.1 Insertion of the interaction term
We can rewrite (4.13) as Introducing Feynman parameters, changing the integration variable to l = x−β 1 z 12 −β 3 z 32 , using the same notation as in app. B.1, integrating over l and evaluating the action of the derivatives yields The last term can be shown to be finite and the first term is very similar to the vertex diagram. Evaluation of the derivatives as in B.1 yields It can be shown, e.g. using the Mellin Barnes technique as in B.1, that all divergent contributions are due to the first term. We have the following divergent contributions: Summarising I ′ 321 then reads and 44) and the two other contractions are contained in cyclic(µ, ν, ρ; z i , z j , z k ). For the dilatation Ward identity the integration over x can be performed by introducing two Feynman parameters. The result simply yields a propagator In the case of the special conformal Ward identity the integration is a little more involved. The integral over d d x can be solved by introducing Feynman parameters and after some algebra one finds (c) = 2c 3 c 4 dz µ,ν,ρ i,j,k d d wǫ δστ ǫ αβγ ǫ ανξ ǫ γσϕ (B.46) where all derivatives are taken with respect to z j and c 4 is a constant obtained through integration over d d x. Inserting the propagators we can write this in a form convenient to solve the integral over d d w 2c 3 c 4 dz µ,ν,ρ i,j,k ǫ δστ ǫ αβγ ǫ ανξ ǫ γσϕ ǫ ρδχ ǫ µτ θ ∂ θ i ∂ ξ j ∂ χ k (2η ϕλ ∂ j,β + η λ β ∂ ϕ j )J j − ∂ ϕ j ∂ β,j J λ j + cyclic (B.47) where the integrals read The integrations over w can be performed by introducing three Feynman parameters β i and we get All three contributions can then be written as 2c 3 c 4 c 5 dz µ,ν,ρ i,j,k ǫ δστ ǫ αβγ ǫ γσϕ ǫ αµξ ǫ νδχ ǫ ρτ θ ∂ θ k ∂ ξ i ∂ χ j (2η ϕλ ∂ i,β + η λ β ∂ ϕ i )J i − ∂ ϕ i ∂ β,i J λ i (B.51) ǫ ανξ ǫ ρδχ ǫ µτ θ ∂ θ i ∂ ξ j ∂ χ k (2η ϕλ ∂ j,β + η λ β ∂ ϕ j )J j − ∂ ϕ j ∂ β,j J λ j ǫ αρξ ǫ µδχ ǫ ντ θ ∂ θ j ∂ ξ k ∂ χ i (2η ϕλ ∂ k,β + η λ β ∂ ϕ k )J k − ∂ ϕ k ∂ β,k J λ where I ijk,p are lengthy terms proportional to 1/∆ p . For the conformal Ward identity we are only interested in the divergent part of the above quantities, which can be automatically extracted with the Mellin-Barnes technique. We find that all terms vanish except for i = j = k. Specialising to the case i = 3, j = 2, k = 1 we find Numerical evaluation of the integrals yields

C One loop gauge field propagator in ABJM theory
Here we review the calculation of the one-loop correction to the gauge field propagator, see also [6]. We have fermionic and bosonic contributions in the loop and thus G (1) µν (p) = G (F,1) µν (p) + G (B,1) µν (p) (C.1) The standard formula 10 leads to the Fourier transform of (C.9)