Lagrangian Insertion in the Light-Like Limit and the Super-Correlators/Super-Amplitudes Duality

In these notes we describe how to formulate the Lagrangian insertion technique in a way that mimics generalized unitarity. We introduce a notion of cuts in real space and show that the cuts of the correlators in the super-correlators/super-amplitudes duality correspond to generalized unitarity cuts of the equivalent amplitudes. The cuts consist of correlation functions of operators in the chiral part of the stress-tensor multiplet as well as other half-BPS operators. We will also discuss the application of the method to other correlators as well as non-planar contributions.


Introduction
Generalized unitarity [1,2,3] is a method that has been tremendously successful in computing loop-level scattering amplitudes (for a review see for instance [4]) so it is natural to attempt to apply a similar method to the computation of correlation functions.
One strategy is to compute form factors, sew them together to momentum space correlation functions and subsequently Fourier transform back into real space [5]. This approach has many merits: form factors of some operators have been shown to have simple structures reminiscent of the ones found in scattering amplitudes [6,7] and although the work cited here deals with N = 4 super-Yang-Mills it could easily be applied to other theories as well. We will employ this approach when dealing with the wholly supersymmetric case as it provides a simple way of dealing with the fermionic variables. Unfortunately, correlation functions are best expressed in real space so some of the symmetries of the expressions may not be apparent until after the Fourier transform.
Our focus will be on a different strategy: since we only consider correlation functions in N = 4 super-Yang-Mills we may employ the Lagrangian insertion procedure [8] which we will reformulate such that it becomes similar to generalized unitarity, introducing a notion of cuts in real space 1 . The advantage of this approach is that we stay in real space the whole time.
We will apply this method to the super-correlators/super-amplitudes duality which relates correlation functions of operators in the chiral part of the stress-tensor multiplet to scattering amplitudes at the level of the integrands in planar N = 4 super-Yang-Mills [10,11,12,13]. The duality was inspired by the duality between amplitudes and Wilson loops [14,15,16,17] whose supersymmetric version was found in [18,19]. The duality between scattering amplitudes and Wilson loops can be complicated to deal with at the quantum level because of the appearance of divergences needing to be regularized 2 and in an attempt at clarifying matters, it was made part of a triality with correlation functions in a special light-like limit being dual to Wilson loops [21] and at the integrand level to scattering amplitudes [10,12,13]. In [22] twistor space methods were used to prove the equivalence between the supersymmetric correlation functions and the Wilson loop introduced in [18].
The duality provides a simple example to try out this approach as one can define generalized unitarity cuts for the dual scattering amplitudes. The cuts of the correlation functions will turn out to be equivalent to the generalized unitarity cuts of the dual scattering amplitudes as long as the duality is correct in the Born approximation. The cuts will consist entirely of correlation functions of half-BPS operators whose form factors we are going to need. The calculations will not depend on the number of operators/external states in the correlation functions/amplitudes.
The duality between correlation functions and Wilson loops has also been expanded to include additional operators [23]. This duality has been discussed using Feynman diagram techniques in [24] and using twistor space methods in [25]. Even though there is no duality with scattering amplitudes, it might still be possible to compute the correlation functions with the cuts introduced here as we will discuss in the last part of the notes.
The notes are structured as follows: section 2 deals with generalized unitarity and lists the form factors we are going to need, section 3 deals with the Lagrangian insertion procedure and introduces the notion of real space cuts, section 4 deals with the duality and how to compute cuts for the correlation functions, section 5 discusses more general correlation functions and section 6 sums up the results. Note that both real space and momentum spinors appear throughout this paper: section 2 uses momentum spinors, section 4 uses real space spinors and section 4.2 uses both types of spinors.

Generalized Unitarity
Generalized unitarity is a method for computing perturbative quantities at loop order, it has been used with great success to calculate scattering amplitudes but can be applied to form factors [6,7,26,27,28,29,30,31,32] and to correlation functions [5] as well by considering Fourier transforms of the operators with some arbitrary momenta flowing in. The method exploits information found at lower loop orders by setting internal propagators on-shell. Formally, this can thought of as replacing specific propagators with delta functions: 2 See [20] for a discussion of some of the anomalies that this can cause This procedure divides the desired scattering amplitude into amplitudes of lower loop order. Since the method specifically refers to propagators it deeply depends on the existence of a Feynman diagram representation but avoids using Feynman rules directly and instead uses on-shell amplitudes as its building blocks which, at least in gauge theories, are a lot simpler.
Generalized unitarity does not seem to be as effective when applied to correlation functions as it is for scattering amplitudes as correlation functions are best formulated in real space whereas generalized unitarity is a method that must be applied in momentum space so many of the symmetries of the expressions will not become apparent before one performs a Fourier transform. Nonetheless we will use this technique when considering the wholly supersymmetric case where it shall prove to be quite useful.
To apply generalized unitarity to correlation functions requires the use of form factors which are quantities in between correlation functions and form factors consisting of both local operators and on-shell external states. The operators that will be relevant to us are of the type: where we have used harmonic variables to make the following projections: of the super space, super charges and scalar fields respectively. Here a, b are SU(2) indices, α is a spinor index and A, B are the usual R-symmetry indices. We will follow the notation and conventions of [12,13] closely with respect to both harmonic variables and spinors some of which can be found in appendix A. The form factors for these types of operators have been dealt with in [7,31]. For our purposes we are only going to need MHV form factors together with the knowledge that the other form factors can be found using MHV rules. For d = 2 the super-Fourier transform of the MHV form factor is given by: This particular operator is part of the stress-tensor multiplet and its highest component is the on-shell chiral Lagrangian that will also appear as part of the Lagrangian insertion procedure: In order to write (4) in terms of the super-space variables one has to do an inverse super-Fourier transform: so the on-shell chiral Lagrangian correspond to the part of (4) proportional to (γ) 0 . For d > 2 MHV form factors will have a fermionic content that in addition to the supermomentum conserving delta function consists of a polynomial of degree 2(d−2) in η −a = (ī) A −a η A . We are not interested in the explicit expressions of the form factors only in the relation between the form factors for an operator T d and form factors for an operator T d−1 . For this purpose we consider the quantity F T d which is the form factor stripped of the fermionic delta function: These quantities satisfy an interesting relation found through BCFW recursion [31]: where the primes on 1 and n − 1 denote that they have been shifted: in order to respect conservation of momentum and super-momentum. Note that it is always possible to use conservation of super-momentum to rewrite F such that it does not depend on the Grassmann variables of two of the external legs.

Lagrangian Insertion in the Light-Like Limit
Lagrangian insertion is a useful method for constructing correlation functions in N = 4 super-Yang-Mills. It exploits the fact that, after a suitable rescaling of fields, differentiation of a correlation function with respect to the coupling will bring down a factor of the on-shell chiral Lagrangian: which is also the operator that appeared in the expansion of the operator, T 2 , see (5). This trick allows one to relate the lth order correction of the correlator: to the l − mth order correction of the correlator: throwing away any contact terms. In addition to being easier than a direct application of Feynman rules it also gives the correlator in a form that mimics more closely the form that scattering amplitudes have in momentum space; notice for instance that computed this way the lth order correction will naturally contain l variables to be integrated over similar to the way that the loop order l of scattering amplitudes contain l loop momenta. Normally one would compute the correlator in (13) using standard Feynman rules but inspired by generalized unitarity we will instead consider different limits of the type: where each limit consists of a set of distances becoming light-like. In the denominator the Lagrangian insertions have to be replaced by other operators since the Lagrangians cannot be connected directly to each other but only by going through vertices so the lowest non-zero correlator would be at some loop level; the relevant operators will be the lowest fermionic components of the operators described in section 2 as we want the denominator to just be a collection of scalar propagators. The light-like distances fall into three different categories: y i − y j , y i − x j and x i − x j though we will mainly be interested in the first two types, the last type being important for a BCFW recursion relation [33,34] 3 . Similarly to generalized unitarity no limit will give the full result but each limit will determine a specific part of the full expression and one will have to compute several different limits until the integrand is completely fixed. It is of course not immediately obvious that these limits will completely determine the integrand, or to borrow an expression from generalized unitarity that the correlation function is cut-constructible, but for the correlators discussed in this paper we will argue that it is indeed the case.

The Super-Correlators/Super-Amplitudes Duality
The duality between correlation functions and scattering amplitudes considers operators of the type (5) placed at points x 1 to x n with neighbouring points being light-like separated but otherwise generic thereby creating a polygon. The sides of the polygon are identified with on-shell momenta: while the superspace variables are identified with the fermionic parts of the supertwistor: The duality then states that the ratio of the correlation function over its Born-level expression is equal to the square of a color-ordered amplitude divided by its tree-level MHV formula: where g is the coupling constant. An N k MHV amplitude will correspond to 4k factors of the super-space variables on the correlator side of the duality but the lowest non-trivial order of a correlation function with that many super-space variables is proportional to g 2k which is the reason behind the factor dependent on the coupling constant. Let us consider the correlation function with several Lagrangian insertions and see what happens when they become light-like separated from other operators. As mentioned in [24] the divergence of a side of a light-like polygon is related to the number of derivatives minus the number of propagators between the two operators. From this perspective one might expect that the fermions and field strengths present in (5) would create something more divergent than the scalars. However due to the chirality of the operator this is not the case: at the Born level the correlation function is given completely in terms of the scalar components of the operators while at higher loop orders the fermions and field strengths can only connect through vertices that will lower the divergences to that of simple scalar propagators.
When inserting Lagrangians this conclusion will still hold as the chiral on-shell Lagrangian is simply the highest component of the operator (5) and the divergences found in the limit (14) should be only of the type that simple scalar propagators would give. This is important as the scattering amplitudes of N = 4 super-Yang-Mills do not have any internal propagators squared for generic external momenta.
We will use the approach of [24] in the case of a purely scalar polygon where it will provide some clear insight but we will not use it for the supersymmetric case because it becomes rather cumbersome, especially finding the correct fields that sit at the corners of the polygon, and although the sides of the polygon do seem to act like the supersymmetric Wilson loops of [18,19] the appearance of ghosts at higher loop orders complicates matters.

Scalar Polygon
The scalar polygon will interact like a Wilson loop separating the space into two parts. This explains the origin of the appearance of the amplitude squared: the inside of the polygon will give one factor the amplitude and the outside another. Our goal will then be to show that cuts with all Lagrangians inside the polygon correspond to the generalized unitarity cuts of the corresponding amplitude; the generalization to cuts with Lagrangian insertions both inside and outside will then be straightforward.
It is important that the cuts separate the inside of the polygon into parts that do interact except through the shared internal lines. As an example consider the cut in figure 1 lines represents distances that have been made light-like 4 . This cut will turn out to correspond to the generalized unitarity cut in figure 1(b) so there should not be any direct interaction between the sides x 2 − x 3 and x 3 − x 4 5 just like there are no explicit factors of 23 or [23] in the generalized unitarity cut. It is not immediately obvious that this is satisfied: for instance the diagram in 2(a) where a scalar polygon interacts through gluons with a single Lagrangian insertion will contribute to the cut while the diagram figure 2(b) which is the same but with an additional gluonic interaction between the two sides of the polygon would ruin this property and so should not contribute to the cut.
To see that the cuts do in fact separate the polygon into two parts only interacting through shared internal lines consider the following: a side of the polygon spanned between the points x i and x i+1 connect through m vertices to m different Lagrangian insertions as shown in figure  3. To more easily distinguish between the insertion points and the points on the polygon we will use tildes when enumerating the insertion points and their spinors, harmonic variables and fermionic variables. Each Lagrangian insertion will supply a single derivative so the diagram will be proportional to: 4 We will be more specific about what we mean by these diagrams later 5 Except of course through the outside of the polygon but as mentioned this will be interpreted as part of the other amplitude in the duality where the propagators are given by: If we focus on the right-most integral we see that it is given by: This clearly becomes divergent when the distance between x i and y1 become light-like. From the point of view of the integral this divergences arises because the integral becomes proportional to (1 − t 1 ) −1 which diverges in the upper limit. Notice also that if x i and y1 are not light-like separated (20) contributes with a factor of (1 − t 2 ) which would ruin the divergences for the subsequent Lagrangian and indeed for all subsequent Lagrangians since the addition of one propagator and one derivative cannot raise the divergence only maintain it. If x i and y1 are light-like separated the integral will not influence the subsequent integrals and one can do the same analysis for the second right-most integral 6 .
Reiterating this argument leads to the conclusion that the diagram in figure 3 only contributes to the cut where a specific y becomes light-like separated from x i if all the Lagrangians to the right of y are also light-like separated from that point and that the diagram only contributes to the cut where y becomes light-like separated from x i+1 if all the Lagrangians to the left of y are also light-like separated from that point.
This shows that the necessary separation does appear. Let us consider what happens when y1 become light-like separated from x i and include the spinor structure from the Lagrangian insertion in (20). Defining (x i − y1) αα to be λ α 1λα 1 we get: . (21) This contribution would then have to be added to the one with the Wilson loop vertex on the other side of x i which can be found through an equivalent calculation though the sign will be opposite 7 : Additional vertices can be added on the gluon line connecting the scalar polygon with y1 and one can show that they will act like the Wilson line vertices. This point is slightly non-trivial as the counting arguments from [24] do not remove all of the unwanted terms. There will be some remaining terms that cannot be described as simple Wilson line vertices times the quantities in (18), (20) and (21) but these terms will not depend on the point x i+1 and when adding the contributions from having a Wilson loop vertex on either side of x i these terms cancel out. This means that in the limit where a Lagrangian insertion point y1 is made light-like separated from x i the diagram acts as if the operator at x i is of the type T 3 and the operator at y1 has its fermionic degree lowered by two which can be put diagrammatically in the form: where full lines represent distances made light-like after dividing by a scalar propagator and vertices where d lines meet correspond to local operators of the type T d 8 and we have suppressed a numerical factor including the coupling constant 9 . Because the line connecting x i and y1 acts like a regular Wilson line it is straightforward to generalize to the case where x i is lightlike separated from more than one Lagrangian insertion. For a generalization to the wholly supersymmetric case we will however need more effective methods.

Supersymmetrization
To find the correct supersymmetrization of the cuts we are going to use generalized unitarity cuts dividing the real space cuts into products of form factors each with exactly one operator. Note that diagrams where the propagator between two operators is completely canceled will give something proportional to a real space delta function; those involving two points on the polygon are excluded because all those points are assumed to be distinct in the duality and the Lagrangian insertion procedure specifically throws away any contact terms so the generalized unitarity cuts with only single-operator form factors completely determine the cuts. In fact the arguments are sufficiently general that we may conclude that the correlation functions in the light-like limit with any number of Lagrangian insertions can be determined completely by generalized unitarity cuts consisting of only single-operator form factors.
The relevant form factors will be the ones found in section 2. We are not going to compute the full generalized unitarity cuts only draw certain conclusions about the fermionic structure of the real space cuts.
It will continue to make sense to divide the correlation function into a part inside and a part outside of the polygon because when connecting the form factors into a polygon there is going to be some number of propagators, r, and twice as many fermionic delta functions, 2r, on the polygon. Upon integration we will be left with r fermionic delta functions all depending on spinor products where one of the spinors correspond to momentum flowing along a side of the polygon; in the light-like limit these spinors will become proportional to the real space spinors (15) and so the fermionic delta functions will correspond to either the outside or the inside of the polygon interacting with the sides of the polygon without any direct interactions between the inside and the outside of the polygon. Planarity ensures that the denominators on the polygon as well as factors not part of the polygon will not give such direct interactions either.
In order to generalize (23) we are going to start with an ansatz and use generalized unitarity to confirm it. Our ansatz will be that the two fermionic delta functions get replaced by: where (1) +a A are the harmonic variables associated with the Lagrangian insertion at y1 and we use χ ã 1/1 to denote 1 θ +ã 1 similar to the notation in (16). One should note that θ A iα does not appear freely in the construction of the correlation functions as that would give twice as many Grassmann variables as for the scattering amplitudes, it only appears as part of very specific products with spinors and harmonic variables so the second term should be interpreted in terms of the following: The factors (ıj) −1 a ′ a are the inverse matrices of (ī) A −a ′ (j) +a A . In order to find the effect of this fermionic delta function on the super-Fourier transform of the form factors we write it in terms of an integral: When multiplied by the form factors these exponents can be removed by shifting the fermionic integration variables for the form factors as follows: The delta function (24) is thereby replaced by imposing the invariance under a specific shift of the fermionic variables. We may write this as: where the γ α 's are now functions of theγ α 's and γ a . To see the effect of this on the generalized unitarity cuts we only need to consider the MHV form factor of an operator T d placed at the point x i and the super-momentum conserving delta functions from the inserted Lagrangian and the neighbouring points on the polygon; the specific details of the generalized unitarity cuts will of course differ a lot but since the N k MHV form factors can be computed from MHV rules these elements will always be present. The form factor for T d will have to share at least one leg with each of these other operators which will give the divergences in the light-like limit, the momenta of these leg will be denoted P1, P i−1 and P i and since they are responsible for the light-like divergences we may replace the real space spinors λ1, λ i−1 and λ i by the momentum spinors λ P1 , λ P i−1 and λ P i at the cost of rescaling γ a so the quantity we are interested in is: , · · · , P i−1 , · · · , P i , · · · , P1, · · · , n .
Notice that the sum γ (27) and that for the first three delta functions integrating over the Grassmann variables of the respective cut legs will also make them invariant under the shift. This first of all means that for d = 2 the shift is in fact a symmetry of the expression, because F M HV T 2 do not depend on any Grassmann variables, as expected. For d > 2 there will some additional Grassmann variables in F M HV T d ; we will write this function without any explicit dependence on either η P i−1 or η P i using conservation of super-momentum and subsequently find the term proportional to η P1−a ǫ ab η P1−b as well as similar factors for all other directions that has been made light-like as part of the cut, from a Feynman diagram perspective we know that such a term should always be present. The integration over the variables γ a can be used to remove this factor: and similar considerations for the other light-like distances will remove all the other Grassmann variables from F M HV The form factor is thus reduced to that of an operator T 2 times some spinor products which can be found by comparing the remnant left from imposing the shift symmetries with F M HV T 2 . It is important to note that (29) is not the entire generalized unitarity cut only a small part but the rest do not interfere with this calculation nor is it directly affected by it and whereas the rest will change depending on the exact generalized unitarity cut considered this part will always be present. Potentially it should be possible to use generalized unitarity to find the correct spinor factor in a systematic way by exploiting relations like (8) however we will instead use that we already found this factor for the scalar polygon in section 4.1. Combining the information gained from the two approaches we get: . . .
. Again full lines represent distances made light-like after dividing out scalar propagators, vertices where d lines meet correspond to an operator T d . As we used that all of the Grassmann variables from F M HV T d in (29) were removed by imposing shift symmetries it is implied that there are delta functions similar to (24) for all but two of the lines meeting at x i .
We will now apply all this to a full cut where a string of m Lagrangian insertions are made light-like separated from each other and the polygon such that the inside of the polygon is split into two with x i being light-like separated from y1 and x j from ym. In terms of the diagram in figure 4(a) the cut can be defined as: We define spinors λ α r with r going from 1 to m + 1 ordered such that λ α 1 is a spinor corresponding to the light-like distance x i − y1 and λ α m+1 is a spinor corresponding to the distance ym − x j and introduce the factor: Finally we use (31) to write the cut in terms of the diagram in figure 4(b) and phrase it in variables common to scattering amplitudes using some of the identities found in appendix B: When reconstructing the part of the correlation function with the propagators corresponding to the cut, the products of the harmonic variables are removed; from a generalized unitarity perspective they correspond to normalizations of the external states. The rest of (34) is exactly equivalent to a generalized unitarity cut with m + 1 cut propagators as shown in figure 5 with the spinor products corresponding to the generalized unitarity cut of an MHV amplitude while everything beyond MHV lies in figure 4(b).
The general nature of (31) allows us to make cuts that also divide the two parts of this cut into smaller pieces just like a generalized unitarity cut can separate the loop amplitude into more than two lower level amplitudes; the calculation is not going to be different from the one above and it is straightforward to show that it will correspond to the correct generalized unitarity cuts.
For the sake of completeness let us point out that nowhere in the calculation leading up to the supersymmetric generalization in (31) did we use that the operator at y1 was the highest fermionic component of the multiplet and the calculation can easily be generalized to limits where the distance x i − x j becomes light-like; in this case the relevant delta functions will be: where as in (25) some of the Grassmann variables should be interpreted in terms of the specific products that appear in the construction of the correlation function.

Cut-Constructibility
In the previous sections it was shown that the real space cuts of the super-correlators correspond to generalized unitarity cuts of the equivalent super-amplitudes. The only issue that remains is whether or not the correlators are completely determined by the cuts or put differently whether or not for every diagram there exists some sequence of propagators not canceled by any numerator factors that divide the diagram into separate patches. For this purpose we will consider the lth order correction to a super-correlator for which the total number of superspace variables sum up to 4k. This specific loop order can be computed by using l Lagrangian insertions and it should be proportional to the coupling constant to the power 2l + 2k. As argued in the beginning of section 4.2 it can be completely determined by generalized unitarity cuts involving only single-operator form factors and we know that the form factors for the operator T 2 with 2c i external legs are proportional to the coupling constant to the power 2(c i − 1) at tree level. Combining these two ways of counting the power of the coupling constant gives us: Since there are n + l operators we can write this as: and because there are no external legs this sum will be equal to the number of cut propagators. Every cut propagator comes with an integration over the Grassmann variables and so the form factors should have exactly 4(n + k + 2l) Grassmann variables which can only be accomplished if they are all MHV. Using the expression from (4) and the counting arguments from [24] we get that two form factors connected through a cut propagator has exactly the right number of momentum factors to give the divergence of a scalar propagator when the distance between the two corresponding points is made light-like. Of course as shown in section 4.1 having the right number of momentum factors do not necessarily imply that the divergence appears. Still because of the existence of an operator product expansion the correlator should be a function of real space scalar propagators connecting two points of the local operators. Anything not captured by the real space cuts would then be terms where a group of Lagrangian insertion points only connect among themselves or only once to a point on the polygon. We would expect that such terms correspond to unconnected L O L Figure 6: Full lines represent distances made light-like after dividing by the appropriate propagators. O represents some arbitrary operator diagrams and diagrams proportional to a group structure constant with two identical indices and so are not part of the correlation functions used in the duality. To see that this expectation holds: consider any of the unwanted terms, take the light-like limit of all the propagators present and try to construct the generalized unitarity cuts corresponding to these limits. By considering the limit of all the available propagators the arguments of section 4.1 are no longer relevant and all the generalized unitarity cuts one can write down for the unwanted terms, that are consistent with the above considerations, will conform to our expectations.
Note that these arguments did not explicitly use that the operators of the correlation function were light-like separated from each other though we did use that they were placed at distinct points.

More General Correlators
Let us finally turn towards other correlators as well as non-planar contributions and discuss how they can be computed.
One type to consider is correlation functions with both operators from the stress-tensor multiplet arranged in a light-like polygon and other operators not part of the polygon. These correlators are a natural extension to the duality between correlation functions and Wilson loops as the light-like limit simply gives the correlation function of a Wilson loop and the additional operators [23]. The additional operators can also be arranged to form a second Wilson loop.
It is still possible to define real space cuts for such correlators even though there is no duality with amplitudes: these cuts will include diagrams where Lagrangian insertion points are made light-like separated from the additional operators as shown in figure 6. It is not clear if such correlators will be cut-constructible, something that may well depend on the specific choice of operators. Adding a single operator to a light-like polygon could be a good starting point for considering correlation functions of other operators as many cuts will be similar to those of the light-like polygon. In the case of the additional operators forming a second Wilson loop the cuts will be similar to the ones used for the duality and can be computed from (31).  One way to construct other correlators that are cut-constructible and also dual to scattering amplitudes is to consider operators of the type half-BPS operators as in (2) with d > 2 as shown in figure 7. This sort of diagram will appear as part of the cuts used in section 4 but one could also use this as the starting point and since equations (28) and (29) do not rely on integration over the super-space variables it should be dual to three different four-point amplitudes and a single six-point amplitude provided we introduce some additional fermionic delta functions like the ones in (24).

Operators at Generic Points
For the duality we considered operators in the stress-tensor multiplet in a specific light-like limit but it would be interesting to compute the cuts for the correlator with operators at generic points. As long as all operators are placed at distinct points the arguments given in section 4.3 are sufficient to argue that the correlation function is cut-constructible.
For the correlation function of only four purely scalar operators the integrand is known to a high loop order [35,36,37,38,39]. We can use the results to check whether the correlators can be constructed from cuts and what those cuts are. The one-loop integrand consists of two types of terms: those that contribute to the duality between scattering amplitudes and correlation functions and some additional terms that can be captured by cuts where all distances to the If this is to make sense as a cut we require that also for similar limits at higher loop orders the Lagrangian insertion should still act like four scalars. This is obviously true for the part of the on-shell Lagrangian proportional to four scalars but the arguments from section 4.1 can be used to argue that it will also be the case for the other parts. Consider for instance the diagrams in figure 9 and the limit where the Lagrangian in the center of the diagrams is made light-like separated from the four operators at the corners of the diagrams: the diagram in 9(a) contribute to the mentioned limit but the diagram in 9(b) do not without some additional light-like limit involving two of the original five operators whereas 9(c) do contribute without any additional limits. The systematic continues with more interactions: only as long as the additional interactions are with the scalar lines will the diagrams contribute to the aforementioned limit without the need for any additional light-like limits involving two or more of the original five operators. For this reason we may conclude that in this limit the Lagrangian insertion acts like four scalars and the cut can be described in terms of the correlation function where the Lagrangian insertion has been replaced by the lowest fermionic component of T 4 .
By inspecting the results from the literature we see that at higher loop orders it is always possible to make the insertion points light-like separated from four other points and let the points of the original operators be light-like separated from each other in pairs so the correlation function should be determined by the cuts if we include cuts where the Lagrangian insertions get replaced by T 4 . These new cuts may appear identical to applying (31) twice, indeed if a Lagrangian insertion is made light-like separated from two purely scalar operators using this relation the result would be proportional to a correlation function with T 4 in place of the on-shell Lagrangian. The difference lies in the fact that for these new cuts the operators made light-like from the Lagrangian insertion may be light-like separated from only one other operator and the factor being pulled out in front of the correlation function when doing the cut seems different. For the relation (31) this factor consists of spinor products and could be found through a fairly simple Feynman diagram calculation but the computation needed for the additional cuts do not seem to be as simple and the factor would involve tensors connecting the SU(4) indices of the harmonic variables.

Non-Planar Diagrams
The duality discussed in setion 4 considers only planar diagrams so we only formulated cuts for the planar theory. However, it is also possible to consider non-planar cuts; equations (28) and (29) do in fact not rely on planarity so the Grassmann structure of (31) will be the same in the non-planar case. The kinematical factor can again be found from considering purely scalar operators and the calculations will be very similar to the one leading to (23). For the sake of clarity we only display the result with a limited number of operators though the generalization is straightforward. We have introduced an additional point x j and defined spinors such that (x i − x j ) αα = λ α jλα j , the limit needed for the cuts is then given by: One should note that outside the planar limit the cuts no longer separate the correlation functions into separate patches as in section 4 though the cuts may still simplify the expression.

Discussion
In these notes we have introduced a notion of cuts in real space and shown how this type of cuts on a specific set of limits of correlation functions correspond to generalized unitarity cuts of scattering amplitudes confirming the super-correlators/super-amplitudes duality on a cut-by-cut basis. We also checked that the super-correlators considered in the duality are in fact completely determined by the real space cuts. The results are hardly surprising as the supersymmetric correlation functions have been found to be dual to the supersymmetric Wilson loop of [18] and since the duality between scattering amplitudes and correlation functions is at the integrand level no regularization issues should arise. Nonetheless it provided a simple example to try out this reformulation of the Lagrangian insertion technique.
The real space cuts are written in terms of correlation functions of other half-BPS operators but there is a non-trivial factor that emerges from doing the cut unlike for generalized unitarity where all the non-trivial information lies in the product of amplitudes. This might be a problem for more general correlation functions like the ones considered in section 5.1 where we identified a second type of cuts still written in terms of correlators of half-BPS operators but with factors that do not follow as easily as for the cuts used in the duality.
In general it would be interesting to extend this approach to other operators. It will certainly be possible to define the cuts but it is not clear if the correlators will be cut-constructible nor whether the cuts will be simple. The extension to non-planar diagrams is more straightforward though the cuts will no longer divide the correlation functions into separate patches.
Since the generalized unitarity methods are related the twistor space methods for scattering amplitudes we expect that this approach is related to the twistor space methods used in [22,40] and it would be interesting to find the direct relation.

Acknowledgments
I have benefited from discussions with Henrik Johansson and Radu Roiban and am grateful for useful comments on an early draft by Henrik Johansson and Gregory Korchemsky. This work is supported by the Knut and Alice Wallenberg Foundation under grant KAW 2013.0235. Figures in these notes were drawn using JaxoDraw [41].

A Harmonic Variables and Spinors
In this appendix we briefly sum up some of the conventions and notations used. The harmonic variables are matrices satisfying the following relations: Upper-case Latin indices are SU(4) indices while lower-case Latin indices are SU(2) indices. Since we will be dealing with operators at many different points it is convenient to use a notation that makes for an easy identification of the corresponding harmonic variables for each operator: we choose to denote the harmonic variables of the operator at point x i on the polygon by (i) +a A and the harmonic variables of the Lagrangian insertion at point ym by (m) +a A . It is also useful to introduce this product of harmonic variables: The product correspond to the determinant of the matrix: (ıj) a a ′ = (ı) A −a ′ (j) +a A .
As long as the product (41) is non-zero it is possible to define the inverse matrix (ıj) −1 a a and by expressing SU(4) vectors in terms of (i) +a A and (j) +a A it is possible to show that the following is the identity matrix: We use greek letters from the beginning of the alphabet for spinor indices while µ and ν are reserved for regular Lorentz indices. The spinor indices are raised and lowered as follows: while the spinor product is defined to be: The Levi-Civita symbols are chosen to be: Lorentz vectors can be written in spinor notation by using Pauli matrices: x αα =σ µ αα x µ .

B Jacobians and Useful Identities
Changing from a measure for the fermionic variables θ +ã rα into a measure for the variables χ ã r/r = rθ ã r and χ a r+1/r = r + 1θ ã r is going to introduce the Jacobian: m r=1 r r + 1 2 .
The duality gives the scattering amplitudes in terms of the fermionic parts of the supertwistors, they can be related to the Grassmann variables, η A i , where (η i ) 0 indicates a positive helicity gluon of momentum p i and (η i ) 4 indicates a negative helicity gluon, in the following way: For the Grassmann variables of the internal states there are different possible definitions, we choose: which gives the following super-momentum conserving delta function: The factor in front of the delta function cancels part the Jacobian that arises when changing the measure for the χ A variables into the measure for the η A variables which is given by: ij i1 12 · · · m + 1j 4 . (53)