N=4 super-Yang-Mills in LHC superspace. Part II: Non-chiral correlation functions of the stress-tensor multiplet

We study the multipoint super-correlation functions of the full non-chiral stress-tensor multiplet in N=4 super-Yang-Mills theory in the Born approximation. We derive effective supergraph Feynman rules for them. Surprisingly, the Feynman rules for the non-chiral correlators are obtained from those for the chiral correlators by a simple Grassmann shift of the space-time variables. We rely on the formulation of the theory in Lorentz harmonic chiral (LHC) superspace elaborated in the twin paper arXiv:1601.06803. In this approach only the chiral half of the supersymmetry is manifest. The other half is realized by nonlinear and nonlocal transformations of the LHC superfields. However, at Born level only the simple linear part of the transformations is relevant. It corresponds to effectively working in the self-dual sector of the theory. Our method is also applicable to a wider class of supermultiplets like all the half-BPS operators and the Konishi multiplet.

The chiral correlators of T have been intensively studied in the past. Their (super)light-cone limits reproduce all the tree-level scattering superamplitudes in N = 4 SYM, as well as the integrands of their loop corrections to all orders in perturbation theory [2][3][4][5][6]. The native superconformal symmetry of the correlators, inherited from the action, is translated into the hidden dual super-conformal symmetry of the amplitudes [7][8][9].
The multi-point chiral supercorrelators of T in the Born-level approximation have been constructed in [10]. There a twistor diagram technique has been developed relying on the twistor approach to N = 4 SYM [11][12][13]. The correlators are built from a special class of nilpotent off-shell superconformal R-invariants. In this approach no space-time integrations are involved, so the Born-level correlators are manifestly rational functions. On shell the nilpotent building blocks are identified with the familiar R-invariants of the superamplitude. Thus, the amplitude-correlator duality becomes manifest.
An independent approach to the chiral correlators of T exploits only their superconformal symmetry and basic analytic properties. This has led to an explicit result for the NMHV-like six-point correlator [14].
Much less is known about the full non-chiral correlators of T . The stress-energy tensor T µν (x), which gives the name to the multiplet, appears in the non-chiral sector. The twoand three-point functions of T are protected [15][16][17][18][19] and so can be easily constructed as free-field theory objects. The four-point function is the first example of a correlator with non-trivial dynamical content. Superconformal symmetry imposes severe restrictions on it [20,21], making it possible to reconstruct the entire super-correlator starting from its bottom component O 20 O 20 O 20 O 20 [22][23][24]. From this non-chiral supercorrelator one can extract many interesting components with conserved currents and Lagrangians.
In this paper we propose a generalization of the chiral diagrammatic technique from [10] to the full (non-chiral) n−point supercorrelators of T in the Born approximation, i.e. at the lowest perturbative order. They are rational functions of the space-time variables x i with i = 1, . . . , n. The R-symmetry SU (4) imposes restrictions on the expansion of the supercorrelators in the Grassmann variables θ i ,θ i . The difference between the numbers of θ andθ for each term in the expansion must be a multiple of four, 4p. We call this number the Grassmann degree of the correlator. In particular, at the Grassmann level (θ) 4p we find p copies of the Lagrangian. According to the Lagrangian insertion procedure, this is the integrand of the p−loop correction to the correlator with n − p points. So, if we can construct all the Born-level correlators, then in principle we know all the loop corrections. For a component of the correlator with Grassmann degree 4p, the Born-level correlator is of order O(g 2p ) in the gauge coupling constant g. This perturbative order corresponds to the number of Lagrangian insertions. Our main result is surprisingly simple. We find very little difference between the full non-chiral and the chiral cases. The off-shell chiral nilpotent invariants need only a slight modification. Let us explain how this works on the example of the six-point Bornlevel supercorrelator T (1) · · · T (6) g 2 of Grassmann degree 4 at order O(g 2 ). It can be evaluated by means of the following effective Feynman rules. They do not follow directly from the N = 4 SYM action, but are the result of partial simplifications of the original ones.
• We draw (connected) vacuum graphs where the vertices correspond to the T 's in the correlator (six in our example). The vertices are of arbitrary valence ≥ 2. They are connected by edges, see Fig. 1. To a vertex of valence p we assign the coupling g p−2 . So, in our example we are allowed to have either one 4-valent vertex or a pair of 3-valent vertices.
• The gauge-fixing parameter ξ . + drops out of the sum of all diagrams.
The chiral Feynman rules of [10] are obtained from the above by setting allθ + = 0. Thus, our main result is that the full non-chiral Born-level correlators of the stress-tensor multiplets are obtained from the chiral ones simply by putting hats on each coordinate difference, x ij →x ij . This is a very non-trivial statement. Indeed, starting with five points, N = 4 superconformal symmetry allows the existence of nilpotent non-chiral invariants, which could alter the above simple result. The fact that this does not happen is likely to reveal some new deep property of the correlators, which remains to be discovered.
We derive these effective Feynman rules by restoring theθ + −dependence of the chiral truncated correlator by means ofQ supersymmetry transformations. The twistor formulation of N = 4 SYM [11][12][13] or the alternative Lorentz-harmonic chiral (LHC) formulation developed in the twin paper [1] are purely chiral, in themQ supersymmetry is not manifest.
The results of the present paper are based on the realization ofQ supersymmetry elaborated in [1]. The theory is formulated in terms of two gauge connections A ++ (x, θ + , u) and A + α (x, θ + , u). They depend on Lorentz harmonics (LHs) u ± α parametrizing the coset SU (2) L /U (1) ∼ S 2 of the chiral half of the Euclidean Lorentz group SO(4) ∼ SU (2) L × SU (2) R . The connections depend on half of the chiral Grassmann variables, θ +A = u + α θ aA , i.e. on a quarter of the odd variables of N = 4 superspace. The LHs allow us to handle efficiently infinite towers of auxiliary fields and pure gauges. Due to them the interaction vertices disappear from the Born-level supercorrelators, but the operator T takes a complicated non-local and non-polynomial form as compared to the conventional component expressions. In this formulation only the Q half of supersymmetry is manifest. TheQ half is non-linear and the algebra closes on shell. TheQ variations of the chiral T give rise to the completeθ + expansion of the full non-chiral T . Fortunately, in the calculation of the Born-level correlators we only need the simple linear part of theQ transformations, corresponding to the self-dual theory. The complicated non-self-dual part ofQ is irrelevant in the Born approximation. Theθ-deviations from the chiral sector of the supercorrelator are mild and they are effectively captured by the self-dual N = 4 SYM theory.
The plan of the paper is as follows. In Section 2 we briefly review the LHC formalism and the main results of [1] which we need here. In Section 3 we introduce the stress-tensor supermultiplet and explain how to construct it from the two LHC gauge connections.
We use chiral 'semi-superfields' since in theLHC framework there are noθ-variables. We successively applyQ-transformations to the chiral T to construct the components in the expansion of T in terms ofθ + . In this way we reconstruct the full supermultiplet, although some of its pieces are irrelevant in the Born-level correlator calculations. In Section 4 we describe the classes of Born-level correlators and reproduce the results of [10] for the chiral correlators in the LHC formalism. In Section 5 we consider the correlators of the various semi-superfield components of T and prove that they are combined together in an object with full supersymmetry by the simple shift x ij →x ij in the chiral Feynman rules for T . In Section 6 we show that this shift corresponds to a change of basis from the chiral to the Grassmann analytic (or half-BPS) basis. We display some technical details and calculations in several Appendices. In Appendix A we show the equivalence of two different definitions of T . In Appendix B we explain how the half-BPS condition is realized on T . In Appendix C we give an example of an LHC supergraph calculation at loop level. We compute the one-loop four-point correlator of O 20 and demonstrate explicitly the absence of the gauge-fixing parameter ξ in the final answer. In Appendix D we discuss the gauge invariance of the effective Feynman rules. In Appendix E we compare some of our results with the conventional Feynman graph calculation of the components of the nonchiral correlators. In Appendix F we present an alternative calculation of the NMHV-like correlators, still in the LHC formalism but this time in a covariant Landau-type gauge. The result is manifestly Lorentz covariant and free from spurious poles, but is much more bulky than the equivalent light-cone gauge result.

N = 4 SYM in Lorentz harmonic chiral superspace
In this Section we summarize the Lorentz harmonic chiral (LHC) formulation of the N = 4 SYM action developed in [1]. This approach involves infinite towers of auxiliary and pure gauge higher-spin component fields. In order to handle them efficiently we combine them in harmonic superfields having infinite harmonic expansions on the sphere S 2 .
We introduce harmonic variables on the chiral half of the Euclidean Lorentz group SO(4) ∼ SU (2) L ×SU (2) R . The harmonics u ±α are defined as two SU (2) L spinors forming an SU (2) L matrix: u ±α ∈ SU (2) L : u +α u − α ≡ u +α αβ u −β = 1 , (u +α ) * = u − α . (2.1) The index ± refers to the charge of these variables with respect to U (1) L ⊂ SU (2) L . Thus, the harmonic variables defined in this way describe the compact coset S 2 ∼ SU (2) L /U (1) L . The harmonic functions on the coset, homogeneous under the action of U (1) L with charge q, are defined by their harmonic expansion (here q ≥ 0) 2) The coefficients f α 1 ...α 2n+q are higher-spin symmetric tensors of rank 2n + q, i.e. they carry the representation (n + q/2, 0) of the Lorentz group. We extensively apply the rules of harmonic calculus introduced in [25]. We need the covariant harmonic derivatives Together with the charge operator ∂ 0 (which counts the U (1) L charge of the harmonics, ∂ 0 u ±α = ±u ±α ) the harmonic derivatives form the algebra of SU (2) realized on the indices ± of the harmonics, i.e. [∂ ++ , ∂ −− ] = ∂ 0 . The derivatives ∂ ++ and ∂ −− have the meaning of the raising and lowering operators of SU (2) L . Harmonic integration amounts to projecting out the singlet part of a chargeless integrand, according to the rule It is compatible with integration by parts for the harmonic derivatives ∂ ++ and ∂ −− . We would like to describe the theory by superfields. However, the N = 4 extended superspace with coordinates xα α , θ αA ,θα A , (2.5) where A = 1, . . . , 4 is an SU (4) (R-symmetry) index, is too big. We are not attempting to formulate the theory in terms of unconstrained superfields on this space, Φ(xα α , θ αA ,θα A ). Instead, we deal with chiral-analytic 'semi-superfields' which depend only on a quarter of the Grassmann variables, They are also harmonic functions having an infinite expansion on S 2 in terms of ordinary fields on space-time. We call such objects 'semi-superfields' because only Q-supersymmetry is realized linearly on them. The realization ofQ-supersymmetry is non-linear. They are chiral because they do not depend onθ, and they are Lorentz-(or L-)analytic because they are annihilated by the LH-projected derivative ∂ + A ≡ u +α ∂ ∂θ αA , i.e.
We formulate the N = 4 SYM theory in terms of two chiral-analytic semi-superfields, They are defined as the gauge connections for two covariant derivatives, the harmonicprojected space-time derivative ∂ + α ≡ u +α ∂ ∂xα α and the harmonic derivative ∂ ++ : with respect to a gauge group with a chiral-analytic parameter Λ(x, θ + , u). The action of N = 4 SYM can be written as a sum of two terms Here the chiral-analytic Lagrangian L CS is of the Chern-Simons type, In the cubic interaction term we see the gauge coupling constant g. As is customary in gauge theories, it accompanies every gauge group commutator (or equivalently, the structure constant f abc of the gauge group SU (N c )). In other words, the Abelian theory is the free (or g = 0) theory. The second non-polynomial term is chiral but not L-analytic, 1 , This action term is local in (x, θ) space but non-local in the harmonic space (each copy of A ++ depends on its own harmonic variable). The bilinear (n = 2) term in the expansion is part of the kinetic Lagrangian since it does not vanish in the free theory (g = 0). However, it turns out more convenient to treat it as an 'interaction' when deriving the Feynman rules. We leave the constant ω in (2.11) and in what follows unfixed, although there is a canonical choice for it. It measures the deviation from the self-dual sector of the theory described by the Chern-Simons type action (2.10) (see [30]). As we will see in Sects. 4 and 5, ω does not appear in the Born-level MHVlike and non-MHV-like super-correlators. Roughly speaking, it counts the number of loop integrations, which equals zero at Born level. The reader who is only interested in our main result about the non-chiral Born-level super-correlators, can set ω = 0. However, we need to consider all the contributions with ω = 0 to justify this result.
Both terms in (2.9) are invariant under a gauge group with chiral-analytic parameters Λ(x, θ + , u). The invariance of S CS holds up to total derivatives, that of S Z is based on the following property of the gauge variation of L Z : The variations of (2.9) with respect to the super-connections A + α and A ++ yield the equations of motion 14) The first one is kinematical (it expresses the auxiliary fields contained in A + α in terms of the physical fields in A ++ ). The second equation is the dynamical one. In it we see the super-connection for the other harmonic derivative in (2.3), (the charge operator ∂ 0 needs no connection because the gauge parameter is a harmonic function of charge zero, ∂ 0 Λ = 0). The (unique) solution to this nonlinear differential harmonic equation was found by Zupnik [29], . . , n, as in (2.11). It is local in (x, θ) space but nonlocal in the harmonic space (each copy of A ++ depends on its own harmonic variable). Unlike the connection A ++ , this one is not chiral-analytic but only chiral, i.e. it depends on the full chiral θ αA . The non-local expression (2.16) is rather similar to L Z (2.11). So, it is not a surprise that it appears in the variation of L Z , (2.17) From here one easily derives the field equation (2.14) by varying S N =4 with respect to A ++ . The action (2.9) is invariant under Q andQ supersymmetries. Q-supersymmetry is realized linearly, where we used the L-analyticity ∂ + B A = 0. Q supersymmetry is more involved. We writeQ as a sum of two generators, Their action on the super-connections is summarized in the following table, We see that theQ-variation mixes the two connections. The important difference between Q andQ Z is that the former is linear and the latter is non-linear (non-polynomial in A ++ ) due to the presence of A −− (see (2.16)). TheQ-variation of A + α (and the first term in thē Q-variation of A ++ ) is a remnant of the conventionalQ-transformation,Q = ∂θ − θ∂ x (our chiral 'semi-superfields' do not depend onθ). More precisely, it is θ +B ∂ − β for L-analytic superfields (the full shift −θ βB ∂ ββ involves both θ + and θ − ). The generatorQ leaves the Chern-Simons action S CS (2.9) invariant. TheQ Z term in (2.19) is needed because of the presence of S Z in (2.9). Remark thatQ Z is proportional to the constant ω, which measures the deviation from the self-dual sector.
, modulo the equations of motion (2.13), (2.14) and up to a compensating gauge transformation with a field-dependent chiral-analytic parameter Λ whose explicit expression can be found in [1]. The Abelian subalgebras formed by the Q-variations and by theQ-variations close off shell. However, the subalgebra formed by the completeQ-variations closes only on shell (the kinematical constraint (2.13) is needed for the closure on A + α , but the field equations are not required for the closure on A ++ ).
TheQ-variation of the gauge connections will enable us to reintroduce theθ variable in our chiral setup. In the 'semi-superfield' formalism only the Q-supersymmetry is manifest. In order to reconstruct the full supermultiplet of some composite operators (which is not chiral, i.e. it involves both the θ andθ variables), we apply successively theQ-variations. In Section 3, where we work out theseQ-variations for the stress-tensor multiplet, we use the fact that this transformation commutes with the gauge variation, where the gauge parameter transforms as a chiral-analytic semi-superfield, The non-polynomial action (2.9) takes the conventional component field form in the Wess-Zumino gauge. The gauge parameter Λ with its infinite harmonic expansion can eliminate almost all of the higher-spin fields from A ++ . The remainder is where we apply the short-hand notations In this (non-supersymmetric) gauge A ++ contains part of the N = 4 vector multiplet: the six scalars φ AB , the four chiral fermions ψ A α , and the Lagrange multiplier G αβ for the self-dual field strength tensor F αβ . After the elimination of the auxiliary fields by means of the kinematical field equation (2.13), the other connection takes the form It contains the rest of the multiplet: the gauge field A αα and the four anti-chiral fermions ψα A . In the next section the Wess-Zumino gauge will help us reveal the physical content of some composite operators.
In the standard superspace approach to N = 4 SYM (see, e.g., [31]) the scalars of the vector multiplet are identified with the super-curvature W AB = −W BA , which appears in the anticommutator of two chiral spinor covariant derivatives, The absence of a curvature symmetric in the Lorentz indices α, β on the right-hand side is the defining constraint of N = 4 SYM in this standard formulation. In our LHC approach we can construct the curvature W AB directly in terms of the harmonic gauge connection A −− (or effectively, in terms of A ++ , recall (2.16)), This curvature is gauge covariant and is (covariantly) harmonic independent, Finally, we quantize the action (2.9) in the light-cone gauge 2 where the gauge-fixing parameter ξ . −α belongs to a new set of LH variables, this time for the second (anti-chiral) factor of the Euclidean Lorentz group SO(4) ∼ SU (2) L × SU (2) R . These new harmonics satisfy the usual SU (2) conditions (recall (2.1)) With the help of the two sets of harmonics we can project the space-time coordinate xα α onto its light-cone components After fixing the gauge we can derive the propagators for the super-connections. They are singular distributions in both the harmonic and coordinate spaces. We need the harmonic delta function δ(u, v) defined by the property with a test function (2.2). We also need the complex delta function δ 2 (t) ≡ δ(t,t) satisfying the relation ∂ ∂t 1 t = πδ 2 (t). More specifically, we use δ 2 (x The set of propagators is as follows where we have used translation invariance to set x 2 = θ 2 = 0. We do not specify if the projections x± ± (see (2.29)) and θ + are made with the harmonic u 1 or u 2 , in view of the harmonic delta functions above. We also suppress the color structure δ ab for the sake of brevity. In the light-cone gauge (2.27) only the quadratic terms in S CS (see (2.9), (2.10)) survive, and these propagators correspond to them. The bivalent vertex in S Z (the n = 2 term in (2.11)) is treated as an 'interaction'. The genuine interaction vertices come only from the S Z part of the action S N =4 (the terms with n ≥ 3). The gauge superfield A + α is non-interacting in the light-cone gauge and hence can appear only at the external points of the LHC supergraphs.

The stress-tensor supermultiplet
The N = 4 SYM stress-tensor multiplet is a short supermultiplet. 3 Its superconformal primary is the half-BPS scalar operator O AB;CD 20 = tr(φ {AB φ CD} ) (where {} denotes the traceless part) made from the six physical scalars of the theory (recall (2.22)). It belongs to the representation 20 of the R-symmetry group SU (4). The stress-tensor multiplet is most naturally described in N = 4 harmonic (or R-analytic) superspace [27,34,35]. We introduce a new set of harmonics w a +A , w a −A and their conjugatesw A −a ,w A +a on the Rsymmetry group SU (4), projecting the index A = 1, . . . , 4 of the (anti)fundamental irrep onto the subgroup SU (2) × SU (2) × U (1) (indices a, a , ±): So far we have introduced three species of harmonics: chiral LHs u ± α (see (2.1)), which are coordinates of the superspace and 'harmonize' the SU (2) L half of the Euclidean Lorentz group; antichiral LHs ξ . ± α (see (2.28)), which are auxiliary parameters needed to fix the gauge (2.27) and 'harmonize' the SU (2) R half of the Euclidean Lorentz group; RHs (w ± ,w ± ) on the R-symmetry group SU (4), which are a convenient tool for dealing with SU (4) tensors. In order to distinguish the RHs from the LHs u ± α we indicate the Lorentz charge of u ± (and of all projected quantities) upstairs, while the R-symmetry charge of the RHs w ± ,w ± is shown downstairs. The antichiral LHs ξ . ± α have a dot above the charge. With the help of the RHs the half-BPS operator can be projected onto the highestweight state of U (1) charge (+4): The harmonic projection automatically renders the tensor traceless and hence irreducible. The short stress-tensor multiplet is annihilated by half of the N = 4 supercharges, namely, by the RH projections Q α +a =w A +a Q α A andQ a +α = w a +AQ Ȧ α . Equivalently, it satisfies the R-analyticity conditions The supercovariant derivatives verify the N = 4 supersymmetry algebra {D Ȧ α , D βB } = δ A B ∂ βα . Consequently, the projected spinor derivatives in (3.4) anticommute, so the two shortening conditions are compatible. To solve the constraints we decompose the Grassmann variables with the help of the RHs, Further, we define the R-analytic basis 4 In this basis Consequently, the stress-tensor multiplet satisfying the conditions (3.4) is described by an R-analytic superfield 5 depending only on half of the projected Grassmann coordinates. We emphasize that this superfield is not chiral. Among its components we find the stress-energy tensor T µν (x). It is an R-symmetry singlet, hence it does not depend on the RHs. Let us now see how this multiplet can be reformulated in our LHC superspace. As stated in Section 2, our dynamical semi-superfields A ++ , A + α are chiral, i.e. they do not depend onθ. Therefore we need to expand T in the antichiral variablesθ + , The 'semi-superfield' components T , M, . . . ,L are gauge invariant operators depending only on θ αa + . They are summarized in Table 1, where their RH charge and scaling dimension are indicated, and are depicted in Fig. 2. These are semi-superfields in the sense that they transform covariantly under Q supersymmetry, butQ supersymmetry transforms them into where the operator O is defined in (3.3),F is the anti-self-dual YM field strength and In the last line we see the anti-chiral on-shell N = 4 SYM Lagrangian Its chiral counterpart L(x) (see (E.2)) appears as the top component in the expansion of the chiral semi-superfield T above.
Now we want to express the various semi-superfield components of the stress-tensor multiplet in terms of our chiral-analytic super-connections A ++ and A + α . We use simultaneously two types of harmonics, LHs u ± and RHs w ± ,w ± , with which we project the indices of the Grassmann variables and derivatives. The semi-superfield components T , M, . . . ,L depend on θ αa + , whereas the semi-superfields A + α and A ++ depend on θ +A . So the two types of superfields live in different halves of the chiral superspace. We need both types of harmonics to split the chiral superspace into four two-dimensional subspaces, Analogously, we covariantly split the eight chiral Grassmann derivatives ∂α A into four two-dimensional derivatives, We define the maximal powers of the (projected) Grassmann coordinates as follows and analogously for (θ − ) 4 , (θ + − ) 2 , (θ − + ) 2 , (θ − − ) 2 and for the Grassmann derivatives. They decompose as follows 12) and similarly for the spinor derivatives (∂) 8 , etc. Our strategy for constructing the component semi-superfields in the expansion (3.9) will be as follows. Firstly, we propose a natural candidate for the chiral semi-superfield T made out of A ++ and we show that it is gauge invariant and R-analytic. It is the starting point in the construction of the other component semi-superfield that are accompanied byθ + in the expansion (3.9). The dependence onθ + is restored by acting repeatedly with theQ supersymmetry transformations. In our chiral formulation this allows us to construct operators carrying dotted Lorentz indices. These operators, contracted with thē θα + variables, are identified with the components in the expansion (3.9).
We realize the supersymmetry transformations on the semi-superfields A ++ and A + α , so that the superspace coordinates are inert. Alternatively, supersymmetry may act on the superspace coordinates leaving the components of the superfield inert. The two realizations are related to each other by the requirement that the superfield stay invariant. In the R-analytic basis (3.6) we haveQ a −α ≡ w a −AQ transformed by theQ-variations (see (3.19), (3.21), (3.30), (3.34)). In this way we are able to reconstruct recursively the full non-chiral stress-tensor supermultiplet, starting from its chiral truncation T . The procedure is depicted schematically in Fig. 3. The generatorQ mixes A + α and A ++ and acts nonlinearly on A + α (recall (2.19) and (2.20)). So it introduces dependence on A + α in the components M, . . . ,L.

Level
The first component T (x, θ + , w) in the expansion (3.9) is the so-called chiral stress-tensor multiplet studied extensively in [10], in the twistor framework. 7 Here we repeat the construction, in the LHC framework. We want to find a gauge-invariant operator of dimension 2 which satisfies the chiral half of the R-analyticity conditions (3.4), namely, The other half of (3.4) involves the RH-projected derivative∂ a +α = w a +A∂ Ȧ α , but it is trivial in our chiral setup,∂ Ȧ α T = 0. Both properties, gauge invariance and R-analyticity, can be achieved by acting on the Lagrangian L Z (2.11) with the R-analyticity projection operator or equivalently the integral over the anti-analytic odd variables θ − , . (3.14) Here we have displayed explicitly the gauge coupling constant g. Note that the constant ω drops out from T , so if we wish, we can set ω = 0 and still have a non-trivial T . The R-analyticity of T is manifest because (∂ + ) 5 = 0. Its gauge invariance is based on the property (2.12). Indeed, where we have split the derivatives (∂ + ) 4 = 1 4 (∂ − + ) 2 (∂ + + ) 2 , under the LH integral, by projecting with LHs u ± . The two derivatives (∂ + + ) 2 annihilate the gauge variation ∆L. In fact, the gauge invariance of T suffices for showing that of the action S Z (see (2.9)). Indeed, the Grassmann measure d 8 θ in S Z can be alternatively split up using RHs, (∂) 8 = (∂ − ) 4 (∂ + ) 4 , therefore The operator T is represented schematically in the first picture in Fig. 4. There each line corresponds to a superfield A ++ in the expansion (3.14).
An alternative form of the operator T is given by the standard expression for the stress-tensor multiplet ('supercurrent') in N = 4 harmonic (R-analytic) superspace [27]: Here W ++ (x, θ a + ,θ +a , w) =w A +a a b w B +b W AB and W AB is the SYM curvature defined by the full (non-chiral) constraint (2.24). In our chiral description it is truncated to W ++ (x, θ + , w) + O(θ + ). We then have (see (2.25)) The equivalence of the two forms (3.14) and (3.17) is shown in Appendix A. We remark that the last relation implies ∂ + +a T = 0, i.e. the shortening condition (3.13) projected with the LH u + α . Using the LH independence of T (see Appendix A), we can remove this projection and recover the full constraint (3.13).
Let us now examine the component field content of T . The operator is made from the semi-superfield A ++ only, therefore it is appropriate to use the WZ gauge (2.22). To simplify the discussion we restrict it to the Abelian (linearized or g = 0) case and we drop the fermions, From (3.14) we obtain Figure 4. Schematic representation of the operators T (3.14) and M (3.18), respectively. The diagram corresponds to a term in the infinite expansion with lines representing the superfields A ++ . M is obtained from T by aQ-variation (see (2.20)) of one A ++ .
with (θ α + · θ β + ) ≡ θ αa + ab θ βb + . This calculation uses the definitions (3.12) and the identitiy . At the level of (θ + ) 4 in T we are supposed to find the linearized chiral on-shell Lagrangian (see the first line in (3.10)). There the field G αβ is just a Lagrange multiplier, a priori having nothing to do with the self-dual YM curvature F αβ . The two are identified only after adding the Chern-Simons piece of the action (see (2.10)) and solving the algebraic equation for G αβ , more precisely F αβ = ωG αβ .

Levelθα +a
Let us now examine the firstQ superpartner of T , the fermionic semi-superfield M from the expansion (3.9). It is obtained as theQ−variation of T defined in (3.14). Taking into account (2.17), (2.19) and (2.20), we find On the other hand, from (3.9) we deduceQ Ȧ The new operator is automatically gauge invariant as a consequence of the commutativity of theQ supersymmetry and the gauge transformations (2.21). We emphasize that the prepotential A + α appears for the first time in theQ variation of T , it is not necessary for the description of the chiral sector of the stress-tensor multiplet.
The operator M is depicted in the second picture in Fig. 4. Roughly speaking, it is obtained from T by replacing one A ++ in the expansion (3.14) byQ − A ++ . 8 The other projection of the variation, w a +AQ Ȧ α T is trivial due to the half-BPS nature of the stress-tensor multiplet. For explanations see Appendix B.
Let us examine the expression for the superpartner in the WZ gauge (2.22), (2.23).
Restricting to the first component in M, we need the term ∼ (θ) 4 under the harmonic integral. In the WZ gauge and in the Abelian (g = 0) case Indeed, together with the explicit θ +A in (3.18) the quartic term (θ) 4 under the integral contains φψ and Aψ A . The second product does not survive the projector (∂ + ) 4 because the field ψ A alone cannot carry so manyw + . Finally, the projection with w a +A gives zero (R-analyticity), while that as shown in the second line of (3.10).

Level
Now we implement the secondQ-variation of T . We start with the operator As before, this operator is gauge invariant due to the mechanism (2.21). It consists of two types of terms, In the first type the twoQ act on different A ++ from T (see (3.14)), in the second type bothQ act on the same A ++ . The first term R due to the cyclic property of the trace. After the symmetrization (a b ), the two terms withQ − A ++ are automatically antisymmetric inα,β, as follows from relabeling the points under the trace. The operator R 0 is depicted in the first picture in Fig. 5.
bothQ act on the same A ++ . In this combination the double variationQQ does not appear. Indeed, a simple calculation leads tō This expression vanishes upon the symmetrization (a b ). Therefore we have The operator R Z is depicted in the third picture in Fig. 5. Taking into account (3.12) we rewrite (3.26) as follows In the last line we have used W ++ = (∂ + + ) 2 A −− from (3.17) and its analog W −− = (∂ + − ) 2 A −− . Also, the corollary of (2.26) ∂ ++ tr(W ++ W −− ) = 0 implies that this LHchargeless quantity is independent of u ± , so the harmonic integral becomes trivial (recall (2.4)) and we finally obtain This operator is gauge invariant by itself, which implies that its complement R (a b ) 0 also is. Note that R Z is proportional to ω, thus its occurrence is similar in a sense to generating a loop correction (see the discussion in Section 4.1). We will not need it in the calculation of the Born-level correlators. Another important property of the operator R (a b ) Z , closely related to the previous one, is that it vanishes at the linearized (free or Abelian or g = 0) level, i.e., (3.28) Since the operator is gauge invariant, this can be shown in the WZ gauge. We find Notice the absence of LHs, as follows from the linearized property ∂ ++ W −− = 0. Further, the derivatives ∂ + in (3.27) do not act on W ++ , so we need to consider ( (2.22) to the scalars only, we obtain by means of (2.16) From here we find the projection W ++ of the curvature according to (3.17), and the projection W −− is obtained by replacing w +A → w −A in the previous formula, i.e. by exchanging lower level + and −. Note that W ++ explicitly depends on the LHs through the projected odd variables θ ± and carries zero LH-charge. It is only covariantly harmonic independent, see (2.26). Acting with the R-analyticity projector (∂ + ) 4 in (3.27) we find Indeed, restricting to the bilinear term in (3.23), we see that the necessary (θ) 4 come from the second line in (3.20), while the harmonic integral is trivial.
3.4 Level (θ 2 + ) (αβ) Now we consider the secondQ-variation of T of the type It is obtained as the first variation of (3.18). This variation acts either upon In other words, the two variations from (3.30) act either on two different A ++ in T (see (3.14)) or on the same A ++ . Consequently, the operator S (αβ) consist of terms of two types, The expression for S is similar to R 0 (3.23), The operator S is shown in the first graph in Fig. 5 Due to the symmetrization (αβ),Q Z does not contribute to the variation of A + α (see (2.20)). So, we can replaceQ byQ in the combination a b Q a (3.24). All other terms appear in a symmetrized form. Thus we have The operator S is depicted in the second picture in Fig. 5. The operator S is gauge invariant, but its two constituents S and S are not. Unlike the operator R Z , here both operators (3.32) and (3.33) have linearized (Abelian) terms. In the WZ gauge (3.20) one can easily see that S starts with S (αβ) = 1 4ψ +a (αψ +β)b a b . The calculation is analogous to the one for R 0 .
One can also easily find the bottom component of S in the Abelian approximation. The term ∼ (θ) 4 under the harmonic integral comes from the lowest components in . Together the bottom components of S + S match the expression for S in (3.10).

Levels
At the two top levels in the expansion (3.9) we observe a similar pattern. We define these operators as the triple and quadrupleQ-variations of T , respectively, First of all, let us consider the parts N 0 andL 0 of these operators obtained by acting withQ, i.e. we drop the termQ Z from (2.19) and replaceQ →Q in the previous formula. Let us recall that the Abelian subalgebra of theQ-variations closes off shell, therefore (Q) 5 T = 0 andQ a −αL 0 = 0. The order of the operatorsQ is irrelevant, so in this case the two expressions in (3.34) coincide.
Further,Q cannot hit a given A ++ in (3.14) more than twice, This is due to the presence of θ + − inQ − (see (2.20)). The operator N 0 consists of terms of two types depicted in the first two pictures in Fig. 6. Analogously,L 0 consists of terms of three types depicted in the first three pictures in Fig. 7. They are the higher-order analogs of M (3.19), R 0 (3.23) and S (3.31). In the WZ gauge (2.22), (2.23) they produce the bilinear terms in the bottom components of the superfields N andL shown in the last two lines in (3.10). Namely,Fψ in N 0 comes from the second diagram in Fig. 6, andFF in L 0 (see (3.11)) comes from the third diagram in Fig. 7. We will need only the N 0 andL 0 pieces of (3.34) for the calculation of the Born-level correlators. Q QQQQ QQQ Figure 6. The first two diagrams correspond to the operator N 0 and the last two to N Z (see (3.36)). They are obtained from the pictures in Fig. 5 acting byQ andQ Z (see (2.20)). The blob corresponds to the L-analyticity projector (∂ + ) 4 .  Figure 7. The first three diagrams correspond to the operatorL 0 ; the next four toL Z ; the last two toL ZZ (see (3.36)). They are obtained from the pictures in Fig. 6 acting byQ andQ Z (see (2.20)). Each blob corresponds to an L-analyticity projector (∂ + ) 4 defined with the appropriate harmonic u + . The 7th diagram here is a variation of the 2nd diagram in Fig. 6, so it is different from the 3rd diagram in Fig. 6.
Next, let us turn on theQ Z part ofQ (2.19). It is proportional to the constant ω measuring the deviation from the self-dual sector. In this way we find the higher-order analogs of R Z (3.26). The operators (3.34) can be decomposed as follows, where the subscripts Z and ZZ refer to the number ofQ Z -variations involved. Consequently, Let us recall that theQ-variations form a closed algebra (up to a gauge transformation which is irrelevant since we consider gauge-invariant operators) only on shell. Therefore, we are able to define N Z ,L Z andL ZZ only modulo the field equations (2.13), (2.14). The two terms in (3.34) corresponding to different orderings ofQ are equal on shell. Acting with the fifthQ on T we will not automatically get zero, but a non-trivial operator which vanishes only on shell.
There are two types of contributions to N Z depicted in the last two pictures in Fig. 6. They are obtained in two ways: (i) by theQ-variation of the last picture in Fig. 5 which represents R Z (3.26); (ii) by theQ Z -variation of the first picture in Fig. 5 which represents R 0 (3.23).
At the top level of the expansion (3.9) the situation is more involved. The operator L Z gets contributions from the 4th to 7th diagrams in Fig. 7. They are obtained in two ways: (i) acting byQ on N Z depicted in the last two pictures in Fig. 6; (ii) acting byQ Z on N 0 depicted in the first two pictures in Fig. 6. Then we see a new possibility: We can act on N Z byQ, since it contains the connection A + α . In this way we obtain the last two diagrams in Fig. 7. They correspond to theL ZZ -part ofL (3.36).
In the WZ gauge these operators arising from theQ Z -variations are responsible for the non-Abelian completions of the bottom components of the superfields in the last two lines in (3.10). The terms ωg[φ, φ]ψ and ωgψ[φ,ψ] come from N Z andL Z , respectively. The term The operators N Z ,L Z andL ZZ are purely non-Abelian. Moreover,L ZZ appears only at the second order in the gauge coupling g, The explicit expressions for the N 0 andL 0 parts (see (3.36)) of N andL are clear from Figs. 6,7. However, we show only schematically the non-Abelian completions (3.37) in Figs. 6,7, since they are rather involved and are not needed for our purposes here. They play no role in our study of the Born-level correlation functions of the different components (3.9) of the stress-tensor supermultiplet T in Section 5. Roughly speaking, theQ Z -variation introduces an additional L-analyticity projector (∂ + ) 4 in the game (see (2.20)), which perturbs the match between the Grassmann degree of the Born-level correlation function and its order in the coupling g. We explain this in more detail in Section 5.3.
4 Born-level correlation functions of the stress-tensor multiplet (chiral sector)

General structure of the correlation functions of the stress-tensor multiplet
In this and the following section we study the n−point correlation functions of the stresstensor multiplet (3.9). The general structure of their Grassmann expansion is as follows: It is determined by the requirement of invariance under the Z 4 center of the R-symmetry group SU (4), which acts on the odd variables as follows: θ → ζθ,θ → ζ −1θ with ζ 4 = 1.
The expansion parameters satisfy the inequalities Indeed, N = 4 superconformal symmetry has 16 + 16 odd generators which can be used to gauge away 16 θ's and 16θ's. This is why the maximal power of θ + in the purely chiral sector (atθ + = 0) is 4n − 16 and not 4n. We classify the correlators G(x, w) according to their Grassmann degree 4p, i.e., the difference between the number of θ + 's andθ + 's. It is known that the (singular) light-like limit of the chiral sector of the correlators corresponds to scattering superamplitudes in N = 4 SYM (see [2]- [5]). We therefore apply the amplitude terminology by dubbing the correlators G r with p = 0 'MHV-like', G p,r with p = 1 are called 'NMHV-like' or more generally, 'non-MHV-like' for any p > 0. Alternatively, the MHV-like correlators may be called non-nilpotent because they have a component at θ =θ = 0. The non-MHV-like correlators are nilpotent (they vanish at θ =θ = 0). The third line in (4.1) corresponds to the conjugate of the first. 9 Our aim in this paper is the calculation of the first two lines in (4.1), using the chiral semi-superfield approach elaborated in the previous Sections, i.e. the correlators for which the number of θ + is greater or equal to the number ofθ + . Our chiral formulation of N = 4 SYM becomes less efficient for the calculation of (mostly) anti-chiral objects (see Section 5.3 for more details), but this is not a problem since the whole information about the nilpotent correlators is contained in the first line in (4.1). The MHV-like correlators in the second line are self-conjugate.
We further classify the purely chiral correlators G nMHV p,0 according to the level in the gauge coupling constant g 2 with ≥ 0. The minimal possible level for a given type of correlator is known as the Born level. For MHV-like correlators it is = 0, i.e. it corresponds to the free theory. Such correlators are made from free propagators in the conventional formulation of the theory. For non-MHV-like correlators the Born level is = p. The easiest way to see this is to recall that the top component in the chiral stresstensor multiplet T in (3.10) is (θ + ) 4 L where L is the (chiral on-shell) Lagrangian of N = 4 SYM. Thus, among the components of G nMHV p,0 we can find the correlator . This corresponds to p Lagrangian insertions in the (n − p)−point correlator of half-BPS operators O. Each Lagrangian insertion is equivalent to a derivative of the generating functional with respect to the coupling g 2 , hence the lowest possible level g 2p . If the level in the coupling is higher than the minimal one, > p, we call such correlators 'loop corrections'. A characteristic feature of the Born level correlators is that they are rational functions of the space-time coordinates x i , whereas the loop corrections involve genuine Feynman integrals of the ( − p)−loop type. 10 In this Section we are interested in the purely chiral sector of the correlator (4.1) obtained by setting allθ = 0, i.e. the correlators G nMHV p,0 and G MHV 0 in (4.1). It corresponds to dropping all semi-superfields but T in the expansion (3.9). Thus, here we deal with the correlators The question how to restore the dependence onθ for such correlators, i.e. how to obtain the sector G nMHV p,r and G MHV r in (4.1), is discussed in Section 5. As mentioned above, we do not consider G nMHV p,r in detail for two reasons: (i) our chiral formalism is not convenient for their study; (ii) they can be obtained from G nMHV p,r by conjugation, or, in other words, using the equivalent antichiral formalism.
In this Section we show several examples of Born-level calculations of chiral correlators. We apply the Feynman rules in LHC superspace outlined at the end of Section 2 (for more details see [1]). The MHV-like case in Section 4.2 corresponds to the free theory and is thus very simple. Nevertheless, it illustrates very well the techniques we use to evaluate the Feynman graphs.
The non-MHV case in Section 4.3 shows the main advantage of the LHC formulation of N = 4 SYM. As explained there, to obtain the Born level of a correlator of Grassmann degree 4p we ought to collect the necessary power g 2p . It can come either from the expansion of T in (3.14) in terms of the dynamical superfield A ++ at the external vertices of the correlator, or from the internal (interaction) vertices in a given Feynman graph. If a power of g 2 comes from an internal vertex from the S Z part of the action (2.9), this automatically lowers the Grassmann degree of the Feynman graph below the required (θ + ) 4p . Such interaction terms appear in the loop corrections to correlators of lower Grassmann degree, but 10 Exceptions from this rule are possible, see [24]. There it is shown that the connected MHV-like correlator , which corresponds to G MHV r with n = 4, r = 4, starts at level g 2 and contains a oneloop integral. Its disconnected part is still rational, however. not at Born level for the given degree 4p. As a consequence, the Born-level Feynman graphs are made from external multilinear vertices, coming from the expansion of T , connected with free propagators (2.31). Thus the chiral Born-level correlators are of order ω 0 in the normalization constant in (2.11). At no step we need to do Feynman integrals d 4 x. This is very different from the usual component field Feynman graphs at Born level. There, to obtain a correlator of Grassmann degree p > 0, one needs to employ genuine interaction vertices for the gluon, gluinos and scalars. This requires the evaluation of certain (not always simple) Feynman integrals. Although the end result is a rational function of the coordinates x i , i = 1, . . . , n, the intermediate steps may be rather intricate. Examples of such component correlators were given in [10], another one is shown in Appendix E. Even if the computation of loop corrections is not the subject of this paper, we give an example of a correlator at one-loop level (involving an interaction vertex) in Appendix C. Alternatively, we can view it as the calculation of the five-point, n = 5, Born-level correlator (4.2) with p = 1. This LHC supergraph calculation reproduces the partial non-renormalization theorem [20] at one-loop level.
The content of this section is closely related to the calculation of chiral nilpotent correlators in [10] whithin the alternative twistor formalism. The new original results on the non-chiral sector, for which the LHC formalism has been developed, appear in Section 5. of the chiral operators T defined in (3.14). This correlator involves only bivalent external vertices from the non-polynomial expansion (3.14) of T = 1 ω (∂ + ) 4 L Z . Indeed, recall that the n-valent vertex is proportional to g n−2 . Each vertex brings in the R-analyticity projector (∂ + ) 4 = d 4 θ − and each propagator contains δ 4 (θ + ) (see (2.31)), therefore the correlator is of Grassmann degree (θ) 0 , i.e. it is not nilpotent. The bivalent 'interaction' vertices induced by the bilinear part of S Z (2.9) are not allowed. Indeed, they bring in eight Grassmann integrations d 8 θ, which would kill any diagram that consists only of bivalent vertices. There is a single contributing Feynman graph shown in (4.4), which gives (we drop the color factor N 2 Here u ij denotes the harmonic at the bivalent vertex i connected with the vertex j and the denominator originates from (3.14). Note that to each external point i corresponds its own set of RHs, e.g. the R-analytic T (i) depends on θ i+ defined by (3.5) with its own w i+ , Here we apply the shorthand notations (2.6) and (2.29)). Then we integrate out the harmonics u ij with i > j by means of the harmonic delta functions (2.30), Next we calculate the Grassmann integrals as in [10]. We decompose each θ A into a pair of θ a + , θ a − with the help of the RHs (see (3.5)), and correspondingly we represent each δ 4 as a pair of δ 2 , where 12 Notice that the argument of each δ 2 in (4.7) is invariant under Q supersymmetry (see It is often convenient to use a particular parametrization of the SU (4) harmonic coset, which amounts to choosing a gauge for the local coset denominator. 13 Then and (4.8) simplifies considerably From (4.6) and (4.7) we see that, e.g., θ 1− appears twice, in the two propagators 1 → 2 and 1 → 3 joining at the bivalent vertex at point 1. Using the Grassmann delta functions δ 2 ((θ 1− +B 12 )·u + 12 ) and δ 2 ((θ 1− +B 13 )·u + 13 ) we can integrate out θ 1− . This produces an LH factor (u + 12 u + 13 ) 2 which cancels the corresponding factor from the denominator. Repeating the procedure at each point we find  The remaining harmonic integrals are easy to calculate. In [1] we discussed the harmonic integrals of the complex delta function and proved the formula Clearly, the same calculation applies to the n−point correlator T (1) . . . T (n) g 0 , also made only from bivalent external vertices. As explained earlier, this correlator may become nilpotent (i.e., ∼ (θ + ) 4p ) starting at the coupling level g 2p .

4.2.2
Two points: T T The two-point MHV-like correlator 13 More precisely, one uses the isomorphism SU is somewhat degenerate and requires care in treating the singularities (on the picture we have split the points in the LH space). The two propagators A ++ A ++ connect the same points, which creates a potential problem with the denominator (u + 12 u + 1 2 ) 4 . Indeed, the delta functions δ(x . ++ ) identify the two harmonics, u + 12 ∼ u + 1 2 ∼ x . + 12 . The resolution is to first do the Grassmann integration using (4.7). This results in a factor of y 4 12 (u + 12 u + 1 2 ) 4 which cancels out the harmonic denominator. Then the harmonic integrals can be safely done by means of (4.12) producing a factor of 1/x 4 12 and we find the familiar result

Non-MHV-like correlators (chiral case)
In the MHV-like example (4.6) we encountered only bivalent vertices. Consequently, the number of Grassmann integrations d 4 θ − matches the number of Grassmann delta functions from the propagators and the final result is not nilpotent. To obtain a non-MHV-like (i.e., nilpotent) correlator we need vertices of higher valency in the expansion of T . Consider, for instance, a 3-valent vertex at point 1 connected via propagators A ++ (1)A ++ (j) with points j = 2, 3, 4: 14 The harmonic delta functions from the propagators have already been used to integrate out half of the LHs. Decomposing each Grassmann δ 4 as in (4.7), we see that θ 1− appears in three different δ 2 : . (4.17) Here (x (4.10)). This result is equivalent to the R-vertex in [10], upon the identification x (w +i − w +j ) bB αB = αa , where we have used the parametrization (4.9).
The variation of the argument of δ 2 in (4.17) vanishes due to the Schouten identity. The rest of the calculation of a complete Feynman graph with the above chiral structure follows exactly Ref. [10]. Any non-MHV-like chiral correlator at Born level can be constructed from the elementary building block (4.17) using the effective Feynman rules: (i) a propagator 1 π 2 y 2 ij /x 2 ij corresponds to a line connecting vertices i and j; (ii) a p-valent vertex i connected with vertices j 1 , . . . , j p factorizes into a product of 3-valent vertices An important question is how the final result (the sum of all Feynman graphs) becomes independent of the gauge-fixing parameter ξ . + (2.27). The argument given in [10] amounts to showing that the sum of graphs is free from spurious singularities. It is adapted (and extended to the non-chiral case) to our formalism in Appendix D. A much more difficult problem is to eliminate ξ . + explicitly from the sum of Feynman graphs without breaking Lorentz invariance. Appendix C contains an example of such a calculation where the LHC supergraph formalism is applied to the simplest non-trivial non-MHV-like correlator, the five-point NMHV-like correlator.
In conclusion of this short review we would like to emphasize the following point about the structure of the chirl correlators at Born level. A p-valent vertex from T is proportional to g p−2 and its contribution is of Grassmann degree (θ + ) 4p . Thus, we see that for Feynman diagrams made out of external vertices from T without interaction vertices from S Z , the order in the coupling and the Grassmann degree match. This is exactly what we expect for Born-level correlators. If we would include an interaction vertex from S Z , then according to (3.15) this is equivalent to integrating out one of the external points. This would take us out of the class of the Born-level correlators. So, the Born-level correlators are given only by free Feynman diagrams. No x-space integrations are needed. The whole information about the interactions is encoded in the operators T .

Restoring theθ dependence at Born level
In this section we work out the full θ andθ dependence of the purely chiral correlators (4.3) from the previous section. The latter were obtained by switching offθ, i.e. by restricting to the operator T in (3.9). What we need to do to restore the dependence onθ is aQ supersymmetry transformation at each point i with parametersθ i+ (see (3.5)). In other words, we need to replace T by itsQ−descendants from (3.9) and calculate a whole new family of correlators. Together they form the full correlator (4.1). Our result is surprisingly simple: The dependence onθ is restored by making simple shifts in the space-time variables x ij , see (5.1). More precisely, the effective Feynman rules of the previous section, modified by the shifts, are valid for the non-chiral correlators from the first and second lines in (4.1).
To show how non-trivial this result is, think of the following possibility allowed by Q andQ supersymmetry alone. From the θ + at three points p, q, r we can construct the linear combination Θ pqr = B pq − B pr invariant under Q supersymmetry (recall the discussion below (4.17)). Similarly, from theθ + at three points (not necessarily different from the first three) we can construct a linear invariant ofQ supersymmetry,Θ lmn . We can then add Θ ijkΘlmn to the hatted variablex ij in (5.1) (after canceling the SU (4) weight with harmonics) and still be consistent with the full supersymmetry. Superconformal symmetry (generators S andS) impose further constraints but it nevertheless remains possible to construct nilpotent invariants involving 5 or more points. The non-trivial claim that we make here is that such arbitrary invariants do not appear, everything is contained in the minimal modificationx ij in (5.1).

MHV-like correlators
We start by considering a simple example, which already shows the main features of how Q supersymmetry helps us to restore the dependence onθ in the correlators.
The Grassmann expansion of the Born-level MHV-like correlators (second line in (4.1)) goes in terms of θθ, in order to preserve the symmetry under the Z 4 center of the Rsymmetry group SU (4). Therefore, restoring theθ dependence of a chiral MHV-like correlator, we will automatically also restore the full θθ dependence. In fact, in this simple case the answer is well known and it is entirely fixed by N = 4 supersymmetry. In the three-point case we have The generalization to the n-point case is straightforward. From our point of view, to restore theθ + dependence in (4.4) we need to act with the supersymmetry shifts This should have the effect of putting hats on x ij , i < j = 1, 2, 3. Let us show how to obtain this result by means of LHC supergraphs. We need to calculate the correlators of the various operators from the expansion (3.9).

First-orderQ variation
The linear term in the finiteQ 1 transformation (5.2) corresponds to the three-point function of two chiral operators T and the first-levelQ descendant M a α defined in (3.14) and (3.19), respectively. This correlator involves only bivalent external vertices and no interaction vertices S Z (2.9). The extra θ − in the definition of M makes the correlator (5.3) nilpotent, ∼ (θ) 1 . There are two contributing Feynman graphs (see Fig. 8) which differ by the permutation 2 3: In these pictures eachQ represents the transformation of the corresponding A ++ field, (2.20)). The first diagram contributes (compare with (4.5)) Then we integrate out the half of the harmonics u ji with j > i (see (2.30)) and the odd variables θ − using (4.7), which gives − 1 π 3 y 2 12 y 2 13 y 2 This expression almost coincides with its chiral counterpart (4.11), except for the differential operator acting on δ 2 (x . ++ 12 ). We rewrite the last factor in (5.6) by means of the identity (here u stands for the harmonic u 12 ) It follows from the completeness relation for LHs u +α u − β − u −α u + β = δ α β (see (2.1)), the property ∂ ++ u − β = u + β (see (2.3)) and the relation between the space-time and the harmonic derivatives 15 (2.29)). In our case the role of f is played by the complex delta function δ(x . ++ , x . −− ). Since nothing else under the harmonic integral in (5.6) depends on u 12 , the total harmonic derivative term in (5.7) vanishes. In this way the gauge fixing parameter ξ . The only new feature of this calculation, compared to the chiral case considered in Section 4.2, is the identity (5.7) for the propagator connecting the fieldsQ a +α A ++ and A ++ . This trick is essentially local and it applies to any Feynman diagram with a line connecting these two fields. For example, the last diagram in Fig. 8 (5.9), it corresponds to the expansion of the supersymmetrized propagators in the correlator (5.1) with respect tox 12 ,x 23 andx 13 to the first order inθ 1+ ,θ 2+ andθ 3+ , respectively, i.e. to the three-linear termθ 1+θ2+θ3+ in the expansion of (5.1).

Second-orderQ variations
The next step in the restoration of theθ 1+ dependence (see (5.2)) requires doing the second Q variation at point 1. This amounts to inserting the operators R and S from (3.9), Recall that R and S are sums of two terms (see (3.22) and (3.31)). Let us first consider the correlators with the insertions R 0 and S defined in (3.23) and (3.32), respectively. In these operators the twoQ variations are distributed over two superfields A ++ . At order g 0 the operators R 0 and S are restricted to their bilinear terms withQA ++ . Both correlators correspond to the first picture in Fig. 9. The difference between the two operators comes only from the (anti)symmetrization of their indices. We already know how to evaluate this diagram (compare to the third diagram in Fig. 8). The same manipulations as in Section 5.1.1 give rise to two shifts of the type (5.9), one for each line connectingQ + A ++ and A ++ , The first term originates from R 0 and the second from S . The double derivative term (5.12) corresponds to the expansion of the supersymmetrized propagators in the correlator (5.1) with respect tox 12 andx 13 to the first order inθ 1 for each of them.
At the quadratic level inθ 1+ we also need to study the insertion of the operator S (αβ) defined in (3.33). The difference with the previous case is that now bothQ variations act on the same superfield A ++ or equivalently on the same line of the Feynman graph (see the second picture in Fig. 9). After the integration over u i1 with i > 1 and θ 1− we obtain the analog of (5.6), but now with a second-order differential operator acting on δ 2 (x . ++ 12 ): which is contracted with B a γ a b B b δ . Here u stands for the harmonic u 12 . Repeating the manipulations from (5.7), it is easy to recast this expression in the form The second term is a total harmonic derivative, which vanishes under the harmonic integral du 12 . We then carry out the remaining integrations in the Feynman graph (recall the step from (5.8) to (5.9)) and complete it with (θ 1 ) 2 . The result is the second-order term in the expansion of the scalar propagator y 2 12 /x 2 12 from (5.1), which is to be contracted with (θ 1+ ) 2(αβ) . Putting together (5.12) and (5.15) (and the symmetric graphs with points 2 and 3 exchanged), we obtain the second-order term (θ 1+ ) 2 in the expansion of the exponential in (5.2). Finally, let us consider the bilinear terms in theθ + expansion of the mixed typeθ 1+θ2+ . In our sample correlator this corresponds to M insertions (see (3.19)) at points 1 and 2, see the third picture in Fig. 9. The novelty is that a propagator is stretched between Q a −α A ++ andQ b −β A ++ . This diagram, compared to the chiral calculation, contains the factors (θ a 1− u + 12 ) and (θ b 2− u + 21 ) at the vertices 1 and 2, respectively, and the propagator where we use the absence of the propagator A + α A + β (see (2.33)) and the antisymmetry of the propagator A + α A ++ (see (2.32)). The integration over θ − turns (θ a 1− u + 12 )(θ b 2− u + 12 ) into (B a 12 u + 12 )(B b 12 u + 12 ), so we can effectively make the replacement (see (3.24 Thus, the two first-orderQ-variations at both ends of a propagator are equivalent to a (Q) 2 -variation at one end that corresponds to the insertion of S (3.33). The result is (5.15), which is to be contracted with 2(θα 1+ ·θβ 2+ ). This is the bilinear termθ 1+θ2+ in the expansion of the propagator y 2 12 /x 2 12 from (5.1). Remark that the expansion of the propagator y 2 12 /x 2 12 does not contain (θ + ) 2 (a b ) . Indeed, it is accompanied by αβ ∂ αα ∂ ββ (1/x 2 ) = − 1 4 αβ x (1/x 2 ) ∼ 0, up to a contact term. This symmetrization of indices corresponds to R Z (3.26) which does not contribute (see below).
The steps so far allowed us to obtain the desired result (5.1) to the second level in theθ + expansion with respect to each point. However, we have not yet considered the diagrams with insertions of M, R 0 , S, where propagators are stretched betweenQ −Q− A ++ andQ − A ++ , and between a pair ofQ −Q− A ++ . Examples of such diagrams are depicted in Fig. 10. One can immediately see that they vanish. Indeed, after integrating out θ − , both θ 1+ and θ 2+ turn into B 12 , but (B 12 ) 3 = 0 (see (4.10)).
We have not yet considered the second term in (3.22), the operator R Z defined in (3.26). It corresponds to theQ Z -part of theQ transformation (2.19). It contains eight Grassmann derivatives (or integrals), contrary to T , R 0 and S with four Grassmann derivatives (or integrals) each. Consequently, an R Z vertex lowers the Grassmann degree of the whole diagram, just like the vertices from the action S Z (2.9). So in the MHV-like setting considered here such diagrams vanish. Let us also recall the property (3.28) which forbids R Z in the Born-level MHV-like correlators (order O(g 0 )).

Higher-orderQ variations
In our simple example (5.1) the hat expansion does not go beyond the second order for each propagator. Indeed, at the third order the hat shift produces the operator (θy −1θ ) 3 ∂ x x (1/x 2 ) ∼ 0, up to a contact term. The analogous property is (3.35), i.e. the triple action of theQ-variations on the same A ++ is trivial. Therefore, the correlators with insertions of the operators N 0 andL (see Figs. 6 and 7) do not produce essentially new Feynman graphs. In order to restore the full dependence onθ we need to repeat the steps described in Sects. 5.1.1 and 5.1.2 until we reach the maximal level inθ. The operators N Z ,L Z andL ZZ do not contribute due to (3.37).
Let us emphasize once more that the crucial simplification of the above calculation was due to the possibility to discard theQ Z -transformation (2.20). It produces operators in the expansion (3.9) that do not contribute at Born level.

Non-MHV-like correlators
Here we show that the entire procedure used in the restoration of theθ + dependence of the MHV-like correlator (5.1) applies, without any modification, to the Born-level non-MHVlike correlators (first line in (4.1)). The final result is surprisingly simple: We just need to replace all variables x ij in each R-vertex (4.17) and each propagator by the corresponding hatted variablesx ij from (5.1).
We illustrate the procedure on the G nMHV In order to be more specific let us consider a part of a Feynman graph with a 3-valent vertex (5.18), where one of the propagators A ++ A ++ has been replaced by its firstQ variation at point 1. The rest of the diagram is not relevant. As before, we apply the Feynman rules and do the θ − and half of the harmonic integrations. Compared to the analogous chiral calculation in Section 4.3, an additional differential operator as in (5.6) is present. As before in the MHV case, we apply the identity (5.7). The second term in (5.7), which is a total harmonic derivative, vanishes upon integration by parts in du 12 . Indeed, the other factors in the diagram contain u + 12 but not u − 12 . 16 Therefore we can repeat the 16 When integrating the total harmonic ∂ ++ in (5.7) by parts we have to take care of the singularities in the denominator like in (4.15). For example, ∂ ++ u 12 (u + 12 u + 13 ) −1 = δ(u12, u13), but then the product δ 2 (x . ++ 12 )δ 2 (x . ++ 13 )δ(u12, u13) implies that x12 is collinear with x13. We rule out such singular configurations. where we formally treat x 12 , x 13 and x 23 as independent variables, i.e. the derivative acts only on x 12 . Completing withθ 1+ , we can interpret this result as the shift (at the linear level in the θθ expansion) The other diagrams contributing to ( in (4.1)). They contain higher valence R-vertices (4.18) in which x ij undergo hat shifts.
Going beyond the linear level in the θ +θ+ expansion is not different from what we did in the MHV-like case in Section 5.1.2. At the quadratic level (the correlators G nMHV p,2 in (4.1)) we need to consider: (i) a single insertion of the operators R or S, (ii) a double insertion of M. Once again, we discard the part R Z of R because it takes us out of the Born-level approximation. The operators R 0 and S (and a pair of M) amount to a pair of propagators stretched betweenQA ++ and A ++ , while S generates a propagator between QQA ++ and A ++ . The latter is equivalent to a propagator between a pair ofQA ++ that is generated by the double M insertion. The calculations from Section 5.1.2 go through without modifications. We are still able to apply the identity (5.13) = (5.14) and to drop the term with a total harmonic derivative. At this stage we reproduce the hat shifts at the quadratic level. The higher levels (G nMHV p,r with r > 2) involve also the operators N 0 and L 0 , which do not introduce new structures in the graphs. We thus reconstruct the full hat shift x ij →x ij in the effective superdiagrams (R-vertices and scalar propagators) for the chiral correlator (4.3).
However, there is a subtlety at the nMHV-level as compared to the MHV-like correlators, which we need to clarify. As described earlier, going on with theθ + expansion we will never apply more than two generatorsQ to the same line. If we wish to reproduce this behavior from the expansion of the hatted variablesx ij , it should terminate at the quadratic level (θ ij+ ) 2 . In the case of the free propagators in (5.1) this followed from the property (1/x 2 ) = 0 (up to a contact term), but why is it true for the R-vertices (4.17)?
The answer is simple: In the R-vertex we encounter only the projections x . + ij . Hence, when we put hats on such variables, we are dealing only with the projected odd variables (θ . + ij ) +a . They cannot appear more than twice in the hat expansion. Consequently, the expansion of the 'hatted' R-vertex does not produce (θ ij+ ) 2 (a b ) . We have seen in Section 5.1.2 that this holds for the propagator y 2 /x 2 as well. In other words, the R-vertex depends on half of the space-time coordinates and hence is annihilated by the directional d'Alembertian Finally, the mechanism of insertion of M, R 0 , S, N 0 andL 0 described above cannot generate terms of the type (θ ij+ ) 2 (a b ) . Is this true for the expansion of the hatted chiral correlator? As we have just seen, such terms do not arise from the expansion of the propagators y 2 /x 2 ij or of the shifted R-vertices. The higher valence R-vertex (4.18) can always be rewritten so thatx ij appears only in one of its R-vertex factors. So, we do not need to consider cross-terms coming from theθ + expansion of the product of two Rvertices. The only possible source of (θ ij+ ) 2 (a b ) terms are the cross-terms with a linear θ + term from a propagator factor y 2 /x 2 and a linear term from the accompanying shifted R-vertex: Here x stands for x ij and the derivative acts only on x ij in the expression of the R-vertex (4.17). The latter is independent of x . − , so ∂ βα R = ξ .
− B) and the invariance of the R-vertex (4.17) under local rescaling Therefore eq. (5.20) is proportional to (x . + B (a )(x . + B b ) ) = 0. In summary, we can say that the operator (B ij ) ij annihilates the effective supergraphs representing the chiral correlator (4.3). To avoid any misunderstanding let us stress once more that this operator is to be applied to the effective Feynman graphs (not to the chiral correlator itself!) where we treat all variables x ij as independent. A similar argument shows that the expansion of the hatted variables cannot give rise to terms like (θ ij+ ) 3 and (θ ij+ ) 4 . We conclude that the insertion of T , M, R 0 , S, N 0 andL 0 completely reproduces the hat expansion of the effective supergraphs.
This reinterpretation of the modified Feynman rules explains why the non-MHV-like correlators with restoredθ−dependence remain gauge invariant, i.e., independent of ξ . ± . The hat-shift in the propagators,x ij = x ij + θ ij+ y −1 ijθ ij+ , is independent of ξ. After the replacement x ij →x ij the R-vertex (4.17) becomeŝ . Formally, this is the same as the expression in [10], in the gauge Z * = (0, ξ . + α , 0) and with the identification σ ij =x . + ij /x 2 ij . Then we can apply the argument from [10] showing that the dependence on ξ . + cancels out in the sum of all Feynman graphs (for more details see Appendix D).
An important remark about the above effective Feynman rules (putting hats on all variables x ij ) is that this should be done before any attempt to simplify the R-vertices like, for instance, the explicit elimination of the ξ−dependence. In particular, the simplification may make use of algebraic identities of the type x ij + x jk = x ik . Clearly, this is not true for the hatted variables! Our hatting procedure works for all higher valence R-vertices (4.18) needed in an N p MHV-like correlators in the Born approximation (at order O(g 2p )). They are characterized by the difference between the total numbers of θ's andθ's being equal to 4p. Knowing the Born-level correlators, we can find loop corrections to correlators with fewer points by means of the Lagrangian insertion formula In view of (3.9) and (3.15) this is equivalent to including the interaction vertices S Z (2.9) in the Feynman graphs.
In order to collect more evidence for the validity of our effective non-chiral Feynman rules, we performed an independent check. We calculated the component θ 1+θ2+ (θ 3+ ) 4 of the five-and six-point correlators G nMHV 1,1 (4.1), using conventional Feynman rules, see (E.7) and (E.8), respectively. Then we compared them with the same component extracted from T (1)M(2)T (3) . . . T (n) (n = 5, 6), calculated by means of the effective non-chiral supergraphs, and found full agreement. More details can be found in Appendix E.

Other correlators
When we reconstructed the Born-level non-chiral correlators G MHV r and G nMHV p,r in (4.1), we only used part of the operators considered in Section 3. For example, we never used R Z (3.26). Now we wish to discuss the correlators for which these operators are relevant. It is convenient to split the operators in three classes: (i) degree 4, (ii) degree 8, (iii) degree 12 operators, according to the number of Grassmann derivatives in their expression in terms of the gauge connections A ++ and A + α , see Table 2. For example, T (3.14) involves the projector (∂ + ) 4 , i.e. four Grassmann derivatives, whereas R Z (3.26) involves a pair of projectors (∂ + ) 4 and (∂ + ) 4 , i.e. eight Grassmann derivatives. We also treat the S Z part of the action (2.9) as a degree-8 operator because of its Grassmann measure (∂) 8 . Previously we only needed degree-4 operators, which are generated byQ-variations of T . The degree-8 operators R Z , N Z andL Z are generated by a singleQ Z -variation, and the degree-12 operatorL ZZ is generated by a doubleQ Z -variation. So, we can equivalently identify them by counting the powers of ω.
Let us now consider the component (θ 2 1+ ) (a b ) (completed by two θ + -variables) of the MHV-like correlator G MHV 2 (4.1). It does not appear in theθ-expansion of the supersymmetrized free correlator (order O(g 0 )), e.g. (5.1), only the (θ 2 1+ ) (αβ) component arises there at the quadratic level. To see such a component, we need to go to order O(g 2 ) in the perturbative expansion. It is generated by the degree-8 operator R Z (3.26). As an example, degree-4 degree-8 degree-12 Table 2. Classification of the operators from the expansion (3.9) of the stress-tensor supermultiplet T (θ + ,θ + ) according to the number of Grassmann derivatives they bring in Feynman graphs, see (3.14), (3.19), The numbers in the picture denote the degrees of the operators (see Table 2). This is obviously a rational function, since it does not involve space-time integrations. Counting the Grassmann degree of the diagrams, more precisely the difference between the number of θ + andθ + variables, is easy. Each propagator contributes (+4) units, each vertex contributes minus its degree ((−4) or (−8) units for the vertices in the above diagram). Thus for (5.22) we get zero, i.e. it has an equal number of θ + andθ + (equal to two), as it should be for an MHV-like correlator. Finally, we would like to comment on the 'mostly antichiral' supercorrelators G nMHV belongs to this class. We can obtain it by conjugating L(1) O(2) · · · O(5) which we already know (see (C.1) and footnote 19). However, the question is how to reproduce it, in principal, in the chiral formalism.
There are a number of contributing supergraphs, some of which are shown in Fig. 11. They can be split in three classes according to the number of space-time integrations. All of them are proportional to ω 2 . The diagrams involve non-trivial space-time integrations, although their sum is a rational function -the Born-level correlator. Thus, the chiral semisuperfield approach to the correlators G nMHV p,r is analogous to the conventional Feynman graph calculation of the components of the Born-level supercorrelator (an example of such a calculation is given in Appendix E).
This variety of Feynman graphs and especially the presence of non-trivial space-time integrals shows that it would be rather hard to calculate G nMHV p,r in the chiral framework.
However, if we adopt the equivalent antichiral framework, the correlators G nMHV p,r are given -39 -  Figure 11. Graphs contributing to the correlator L (1) O(2) · · · O(5) . They come in three types: (i) graphs with two interaction vertices S Z denoted by gray blobs, (ii) graphs with one interaction vertex S Z and one degree-8 operator, (iii) graphs without interaction vertices but with one degree-12 operator.
by free Feynman rules (just like the non-MHV-like correlators considered in Section 5.2), without any space-time integrals. This means that the complicated expression for a mostly antichiral correlator, computed in the chiral framework, should drastically simplify due to some intricate algebraic and integral identities.
In conclusion, we would like to recall that the crucial simplification in our treatment of mostly chiral Born-level supercorrelators was due to the possibility to discard the second, non-linear term in theQ transformation (2.20) related to the interaction Lagrangian. We have shown that the linearized contribution of this term vanishes, while the non-linear terms are of higher order in the coupling g and hence irrelevant at Born level.

Chiral and R-analytic bases in superspace
In this section we show that the main result of Section 5, namely that theθ dependence is restored simply by putting hats on x ij , has a natural interpretation as a change of basis from chiral to R-analytic superspace. This is indeed the simplest way to render the chiral Feynman rules compatible withQ supersymmetry. However, it is by no means the only way, so the question remains why our final answer for the non-chiral correlators is so simple.

Chiral and R-analytic basis
The dependence onθ for chiral or R-analytic functions needs to be considered in the appropriate basis in superspace. Let us illustrate this on the familiar example of chiral superspace. In the real basis of the full superspace the covariant derivativeD has the form D = ∂/∂θ − 1 2 θ∂/∂x. A chiral superfield satisfies the conditionDΦ(x, θ,θ) = 0. In order to solve it, we define the chiral basis in whichD = ∂/∂θ and the chirality condition can be easily solved: Similarly, an R-analytic superfield is defined by two conditions in terms of RH-projected covariant derivatives (recall (3.4)): To solve these constraints we define the R-analytic basis with coordinates xα α an , θ a + , θ a − , θα +a ,θα −a (3.5), (3.6). In this basis the covariant derivativesw A +a D A and w a +AD A are short (see (3.7)), so the R-analyticity constraints (6.2) simply imply Φ = Φ(x an , θ a + ,θ +a , w). Of particular interest for us is the change of variables from the chiral basis (6.1) to the R-analytic one (3.6), where the completeness condition for the RHs (3.2) is taken into account. The different x's transform according to the type of basis, real, chiral or R-analytic: Let us verify that the change of basis (6.3) reproduces the correct transformation of x ch (recall (3.2)): A . Now, let us turn to our propagators A ++ A ++ and A + α A ++ (see (2.31), (2.32)) containing δ 2 (x . ++ )δ 4 (θ + ). Originally, the quantization was done in terms of 'semi-superfields' A(x, θ, u), no anti-chiral variablesθ were used. Does this mean that our propagators are defined in the chiral basis? The answer is affirmative and follows from the way Q supersymmetry is realized on them. Indeed, the propagators are invariant under simple shifts Qθ = , without touching the space-time variables x. This corresponds to the chiral basis in (6.4). The propagators satisfy the chirality condition Then it is natural to do the R-analyticity projection in (4.5) with the covariant spinor derivatives, This projection satisfies both constraints (6.2), the first due to the projector in (6.6), the second as a corollary of the chirality (6.5) (recall that {D a + , D +a } = 0). If we want the covariant derivatives in (6.6) to become partial derivatives and hence (D + ) 4 = d 4 θ − , we need to change the basis: as stated in (6.6). In the first line in (6.7) theθ dependence is present in the arguments x an = x ch − θ −θ+ . We claim that this simple change of basis restores theQ symmetry of any Feynman graph.

6.2Q invariance of the R-vertices
TheQ invariance of our Feynman graphs is not immediately obvious because the propagator A ++ A ++ ∼ δ 2 (x . ++ )δ 4 (θ + ) is not invariant. 17 Indeed, under the shift x → x +¯ A θ A the propagator transforms as follows (here we use the expanded notation δ 2 (z) = δ(z,z)) and the projection¯ − remains. Nevertheless, as we show below, in the effective Feynman rules the fullQ supersymmetry is restored simply by making the change of basis (6.3). Let us consider the part of a Feynman graph where vertex i is connected with vertices j, k, l. In particular, the propagator stretched between i and j contributes (after integrating out δ(u ij , u ji ); recall the calculation in Section 4.2) The Grassmann delta functions from this and the other propagators allow us to find θ − as functions of θ + , After the change of basis (6.3) with θ − from (6.9), the bosonic delta function in (6.8) has support on (we drop the subscript 'an'; recall (4.10)) and similarly for the conjugate variable x . − ij ch · u − ij = 0. Notice that the projections of θ i− , θ j− involving the other vertices k, l have dropped out. Eq. (6.10) allows us to solve for u + ij = ρ ijx . + ij where ρ ij is an arbitrary normalization factor. The rest of the Feynman diagram (propagators and LH denominators from S Z (2.9), (3.14)) is homogeneous in u + ij of degree zero, so this factor is irrelevant.
Then we consider the second condition following from the support of the bosonic delta function 11) 17 In the twistor formulation of Ref. [13] the chiral propagator δ 4|4 (Z * + σijZi + σjiZj) is manifestlyQ invariant. In particular, with our choice Z * = (0, ξ, 0) the propagator amounts to δ 2 (σij + σji) δ 2 (ξ + σijxi + σjixj) δ 4 (σijθi + σjiθj). The variationQxi = θi is then compensated by the Grassmann delta function. Roughly speaking, the analog of σ for us is u + , but the twistor formalism does not employ u − . This leads to some differences in the wayQ supersymmetry is realized, as we explain in this subsection.
where θ i− and θ j− are now expressed in terms ofx . + ∼ u + according to (6.10) and its analogs for the other propagators: Again the normalization factors ρ drop out due to homogeneity. Eq. (6.11) enables us to find u − ij up to normalization. The only place where u − appears is the Jacobian produced by the harmonic integral π du δ(x . (4.12)), or after the change of basis, with θ i− and θ j− from (6.12). We have (x This result, multiplied by the RH factor y 2 ij from (6.8), produces the expected propagator factor, see (5.1). Together with the replacement u + ij →x . + ij this completes the effective Feynman rules with manifestQ invariance.
In conclusion, the simple form of the modified Feynman rules can be interpreted as the basis change needed to make compatible the initial formulation of the propagators as chiral objects with the R-analytic nature of the operators T , etc. This of course does not explain why the fully supersymmetric correlators do not involve more sophisticated nilpotent anti-chiral invariants. This questions requires further study.

Conclusions
In this paper we use the LHC superspace approach to obtain effective Feynman rules leading to the multipoint Born-level correlation functions of the stress-tensor multiplet. The chiral sector of the multiplet is naturally formulated in terms of LHC superfields. So, the derivation of the Feynman rules for the chiral correlators is straightforward. The description of the non-chiral part of the multiplet is more involved. In our chiral approach the multiplet splits up into several chiral layers corresponding to the expansion in terms ofθ + . Nevertheless, at the level of the effective Feynman rules we succeeded to put the layers together in a manifestly supersymmetric expression for the non-chiral correlator.
The same method is applicable to other supermultiplets as well. Indeed, since we know the expression for the gauge super-curvature in terms of LHC superfields, W AB = ∂ + A ∂ + B A −− (see (2.25), (2.16)), we can construct many other multiplets. For example, projecting W AB with RHs we find the (chiral truncation of the) half-BPS multiplet of weight k: O (k) = tr (W ++ ) k . The (chiral truncation of the) Konishi multiplet is K = ABCD W AB W CD . Clearly, the Feynman rules elaborated for O (2) (stress-tensor multiplet) are straightforward to generalized to O (k) and K. The only complication is that they do not result in concise R-invariants like (4.17), i.e. the Grassmann integrations may be more laborious. Nevertheless, this is a purely algebraic task but not a fundamental obstacle. In [36] the twistor Feynman rules were applied to the calculation of the one-loop dilatation operator with manifest supersymmetry. Our formalism can be equally useful in such calculations of the dilatation operator, perhaps also at higher perturbative levels.
The non-chiral completions of the supermultiplets and the corresponding correlators can also be constructed. For the R-analytic half-BPS O (k) the result is rather obvious. One just needs to take up the chiral Feynman rules and replace x ij byx ij (see (5.1)). For other correlators the restoration ofθ effectively leads to using the corresponding supersymmetrizationx ij of x ij . If two Konishi operators appear at the vertices i and j, we should use the supersymmetrization x ij →x ij = x ij + 1 2 θ iθi − 1 2 θ jθj . If points i and j correspond to a half-BPS and a Konishi operators, respectively, then x ij →x ij = x ij + θ ij+ y −1 ijθ i+ − 1 2 θ jθj . In conclusion, we emphasize once more the surprising simplicity of our main result. The reason why the non-chiral supersymmetrization of the multipoint Born-level correlators amounts to a simple shift of the space-time coordinates remains mysterious. This could be a manifestation of some hidden symmetry of the correlators, perhaps an off-shell analog of the famous Yangian symmetry of the superamplitudes [7][8][9]37]. Or it could follow from the analytic properties of the correlators, which would be incompatible with any other kind of nilpotent superconformal invariant. We hope to come back to these interesting questions in the future.

Appendices A Proof of the alternative form of T
Here we show the equivalence of the two forms (3.14) and (3.17) of the chiral truncation T of the stress-tensor multiplet (see (3.9)). This proof follows closely the analogous equivalence of the traditional chiral form of the action of N = 2 SYM and its harmonic superspace version given in [38].
For simplicity we restrict ourselves to the Abelian (free case) where L Z is given by the bilinear term in (3.14), (2.16)). We start with the form (3.14) and rewrite it as follows: (3.17)). We then write (∂ − + ) 2 = ∂ −− (∂ − + ∂ + + ) and integrate the harmonic derivative by parts. It annihilates the curvature W ++ . To see this, we first remark that the linearized version of (2.26) implies ∂ ++ W ++ = 0. Since W ++ has zero LH charge (scalar), the two properties ∂ ++ W ++ = 0 and ∂ −− W ++ = 0 are equivalent to each other and to the fact that W ++ does not depend on the LHs. So, after the integration by parts ∂ −− can hit only A ++ and turn it into ∂ ++ A −− with the help of (2.15). We thus get (using ∂ + At the last step we used the fact that W ++ is independent of u ± , so du 1 = 1 (see (2.4)).
We have obtained the alternative form (3.17).

B Realization of the half-BPS condition on T
The stress-tensor multiplet (3.9) is half-BPS. This is translated into the shortening conditions (3.4). The shortening becomes manifest in the R-analytic basis (3.6) where the spinor derivatives (3.7), which define the R-analytic superfield, do not have terms with space-time derivatives (see Section 6.1). The situation is the opposite with the supersymmetry generators in the same basis: The generatorQ a − is what we used in Section 3 to relate the various semi-superfield components in (3.9) to each other. The generatorQ a + should, according to the standard formulation, "annihilate the half-BPS primary". In reality, due to the space-time derivative term in it, we expect a relation of the typē for the bottom component in theθ + expansion (3.9). Let us show that this is the case. We project relation (3.18) with the RH w a +A and obtain Q a +α T = −(∂ + ) 4 tr du θ +a Here we have split (∂ + ) 4 = 1 4 (∇ − + ) 2 (∂ + + ) 2 under the trace, used the L-analyticity ∂ + A + α = ∂ + A ++ = 0 and the definition (3.17). The super-curvature W ++ has the property ∇ − + W ++ = 0, which follows from ∂ + + W ++ = 0 by differentiation with ∇ −− and from [∇ −− , ∂ + + ] = ∇ − + and ∇ −− W ++ = 0 (see Appendix A). We have also used the commutation Next, we pull the harmonic derivative ∇ ++ out of the first factor in the second relation in (B.2), integrate it by parts and use ∇ ++ W ++ = 0. On the way ∇ ++ changes one of the spinor derivatives, [∇ ++ , ∇ − + ] = ∂ + + and we obtain [∇ ++ , (∇ − Consequently, in view of (3.17) we have ∂ + + A − + = −W ++ . This does not contribute due to the trace tr ([W ++ , A − α ]W ++ ) = 0, thus we havē To proceed, we need to show that This relation has overall LH charge −1, therefore, if we can show that it holds after differentiation with ∇ ++ , it will also hold as it is (the operator ∂ ++ is invertible on harmonic functions with negative charge). Thus, we act on (B.4) with ∇ ++ , use again ∇ ++ A − α = A + α + ∂ − α A ++ and the L-analyticity of A + α , A ++ to obtain Hitting this with (∂ + + ) 2 , the term with ∂ −− A + α vanishes and we get the right-hand side of (B.5). Finally, with the help of (B.4) the variation (B.3) becomes (3.17)) is independent of the LHs and the integral picks the Lorentz singlet in θ +a + ∂ − α . So, we have confirmed the property (B.1).

C One-loop four-point chiral correlator
In this section we calculate the one-loop correction to the four-point chiral MHV-like correlator T (1)T (2)T (3)T (4) by means of the Feynman rules in LHC superspace. This result has been obtained long ago via Feynman graphs in N = 2 harmonic superspace [39] and then generalized to N = 4 harmonic (or R-analytic) superspace in [40]. With four points there are no nilpotent invariants, so the purely chiral four-point correlator is restricted to its θ = 0 component. It is given by a universal polynomial in the variables x, y, which captures the SU (4)  This supersymmetric calculation will allow us to illustrate the LHC Feynman rules from Section 2 at work. Unlike the Born-level calculations presented in Section 4, this one requires an interaction vertex and consequently integration over space-time. 19 There are several graphs that can possibly contribute. They are indicated below up to permutations of the points 1, 2, 3, 4. The black blobs represent the external points, and the gray blobs are the interaction vertices S Z (2.9). A space-time integral is assigned to each gray blob. In the following we will omit the factors g 2 and ω produced by each graph. 18 The normalization factor is chosen to agree with the LHC Feynman graph calculation presented below.
In fact, by comparing (C.1) to the standard result of Ref. [40] we can fix the conventional value of ω. 19 In reality, this calculation is equivalent to taking the Born-level result for the NMHV-like correlator T (1) . . . T (5) ∼ g 2 (θ) 4 and integrating it over the fifth space-time point.
Here we do not display the Lorentz spinor indices. The second RH j + on the right-hand side of (C.2) is at our disposal since the left-hand side depends only on i + . We profit from the relations (x . + jk ) as well. After the Grassmann integration it is easy to see that the diagrams in Figs. C.5,6,7 do not contribute. The principal contribution comes from the graph in Fig. C  Together with the other permutations of I 1 we reproduce the last line in (C.1). The remaining graphs contribute only to the y-structure y 2 12 y 2 23 y 2 34 y 2 14 and its permutations. In order to complete the calculation we only need to identify the y-structure y 2 12 y 2 23 y 2 34 y 2 14 within the diagrams. We find that this structure originates from I 1 (1, 2, 3, 4) and its permutations I 1 (1, 3, 2, 4), I 1 (1, 4, 3, 2), I 1 (1, 3, 4, 2), contributing the factor .

D Independence of the gauge-fixing parameter
Here we adapt the argument given in [10] to our formalism and, in particular, to the nonchiral modification of the effective Feynman rules by the substitution x ij →x ij . We show that the gauge-fixing spinor ξ . + drops out from the sum of all Feynman graphs for the NMHV-like multipoint correlator.
Let us start with the chiral case (allθ = 0). The argument of [10] amounts to proving the absence of spurious poles of the type Notice the independence of the residues from the fourth points i, j, k. Next, we remark that for the singular configuration specified in (D.3), (D.4) the variables x 2 12 , x 2 13 , x 2 23 become linearly dependent. This can be seen by writing − and using the relations (D.3) and (D.4): This result is equivalent to eq. (4.35) in [10]. The rest of the argument follows [10]. Now, what happens in the non-chiral case? The effective Feynman rules amount to putting hats on the variablesx At first sight, this may invalidate the argument above, in particular the identity x 12 + x 23 = x 13 . However, this does not happen. Due to the full Q andQ supersymmetry, in addition to the gauge (D.5) we can fix theQ SUSY gauge:θ 1+ =θ 2+ = 0 . (D.9) The gauges (D.5) and (D.9) remove the hat fromx . + 23 also loose their hats. 21 The hats survive inx 1i andx 2i (with i = 1, 2, 3) but this does not affect the argument. Then we can repeat all the steps form the chiral case. We have to be careful when taking the limit y 3 → ∞ since y −1 3 appears in the R-vertices and y 3 in the propagator factors, giving non-vanishing limits. We obtain the same residues as above and the same final result (D.8). 22

E Component calculation
In order to support our general argument about the form of the non-chiral correlator, we carried out a calculation in the conventional Feynman graph approach and compared it with the LHC superspace results. The full non-chiral correlator contains plenty of components. We will stay at the level of the firstQ-variation of the chiral NMHV-like correlator and we will consider only one of its components. Specifically we deal with the component θ αa in (4.1)), i.e. we calculate the correlators (n = 5, 6) in the Born approximation (at order g 2 ). Here (recall (3.3), (3.10)) The R-symmetry SU (4) acts on the coordinates of the R-analytic basis of superspace as follows. SU (4) inversion acts upon θ+,θ+, y in the familiar way, I[θ+] = θ+y −1 , I[θ+] = y −1θ +, I[y] = y −1 . What is unusual is the transformation of xan defined in (3.6). Due to the presence of RHs in this definition, xan is not inert under SU (4) inversion, I[xan] = xan + θ+y −1θ +. With this property one can show that the hatted variablexij = xij + θij+y −1 ijθ ij+ is invariant under SU (4). 22 We have checked with Mathematica that the residues cancel in the sum without the gauge (D.10).
L is the on-shell chiral Lagrangian of N = 4 SYM. Let us emphasize that the interaction vertices are induced by the full Lagrangian of N = 4 SYM, not by L. In [10] we performed a similar calculation for all the components of the chiral NMHV correlator, so here we will skip most of the details.
In principle, the relevant Feynman graphs produce a number of complicated Feynman integrals resulting in non-rational contributions (they are logarithms and dilogarithms since we are effectively at one-loop level). The sum of all graphs is supposed to be a rational function, so the non-rational pieces should sum up to zero. Fortunately, we found the correlators G n without such laborious integrations. The only integrals needed are the socalled T -blocks which are rational. They are the building blocks of some of the diagrams. In the chiral calculation [10] we used T -blocks representing the interaction of a gluon with a pair of scalars and the Yukawa interaction of a scalar with a pair chiral fermions: Now we need the interaction of a gluon with a chiral and an anti-chiral fermions as well.
This T -block naturally decomposes as follows (a, b, c are color indices) .

(E.4)
Here we singled out the gluon T -block from (E.3) multiplied by the propagator of fermions. This observation yields a substantial simplification of the calculation (n = 4, 5), using the known chiral correlators There are many gluon diagrams contributing to G n (E.1). Each of them separately is not conformal, so we need to organize them in conformal combinations. The decomposition (E.4) allows us to do it. Indeed, a diagram containing the T -block (E.4) can be split in a sum of two terms, one of which is a fermionic propagator times the corresponding diagram from the chiral correlator (E.5). Thus the chiral correlator (E.5) captures the essential part of almost all gluon diagrams and provides a conformal expression for them. The rest of the gluon contributions independently sums up to a conformal combination. Among such diagrams we encounter the one-loop correction of the fermionic propagator, i.e. ψ α +a (1)ψα −a (2) L(3) . It is a sum of several complicated Feynman integrals. In order not to perform these integration explicitly, we use the second trick that exploits the protectedness of the correlator of two half-BPS operators. It implies that the following three-point correlator calculated in the Born approximation vanishes, Considering the Feynman graphs that contribute to (E.6), we can express the one-loop correction of the fermionic propagator in terms of simple Feynman graphs which are rational and are given by products of T -blocks. 23 The results are as follows: manifestly conformal. We would like to emphasize that the chiral correlator (E.5) captures a substantial part of G n , as it should be since the latter is theQ-variation of the former. We compared the correlators G 5 , G 6 with the relevant component of the non-chiral supercorrelator T (1)M(2)T (3) . . . T (n) , calculated by means of effective supergraphs according to Section 5.2, and found perfect agreement. The complexity of the expression produced by the LHC superspace Feynman graphs and its artificial dependence on the gauge-fixing spinor ξ . + makes the comparison possible only by means of Mathematica. We point out that only the agreement between the two expression for G 6 provides independent support for the LHC superspace results since the five-point correlator (and correspondingly G 5 ) is fixed uniquely by superconformal symmetry.

F Quantization in a Lorentz-covariant gauge
The light-cone gauge, eq. (2.27), is not the only possible gauge that yields simple propagators in the LHC formulation of N = 4 SYM. An essential drawback of the light-cone gauge is that it breaks the Lorentz and conformal symmetries of the theory, still preserving the SU (4) R-symmetry and Q-supersymmetry. Now we are going to impose a gauge which also preserves Lorentz invariance, This Lorentz-covariant gauge is reminiscent of the usual Lorentz gauge in pure YM. Roughly speaking, the gauge field A αα lives in the LH superfield A + α , eq. (2.23), (in fact this statement is meaningful only in the Wess-Zumino gauge (2.22)). Then the constraint (F.1) implies u − α u +β ∂α α A βα = 0 where A αα (x, u) has an infinite harmonic expansion. Extracting the singlet part of the product of LHs, we obtain the Lorentz gauge ∂α α A αα = 0.
The Lorentz-covariant gauge (F.1) is possible at least locally. Indeed, we can find an Abelian gauge transformation with an L-analytic parameter Λ(x, θ + , u) transforming to this gauge: ∂ −α ∂ + α Λ = Λ = ∂ −α A + α since the d'Alembertian is invertible. We neglect the zero modes of , so the gauge freedom is fixed completely by the condition (F.1).
As in the light-cone gauge, we define the propagators using only the self-dual sector of the theory, which is described by the Chern-Simons Lagrangian L CS , eq. (2.10). The bivalent vertex from the completion to the full theory L Z , eq. (2.11), is treated as an interaction. We impose the gauge (F.1) by means of a Lagrangian multiplier ρ +4 (x, θ + , u), which is an L-analytic superfield carrying U (1) charge (+4). Then the kinetic Lagrangian becomes invertible, allowing us to find the propagators. Unlike the light-cone gauge, we have to introduce ghostsc ++ (x, θ + , u) and c ++ (x, θ + , u) as L-analytic superfields of U (1) charge (+2) with Fermi statistics. Another important difference from the light-cone gauge is that the cubic interaction vertex in the Chern-Simons Lagrangian L CS does not vanish. Moreover, it is supplemented by a cubic interaction vertex for the ghost fields and A + α , So, the Chern-Simons Lagrangian L CS (2.10) is replaced by the sum L kin +L CS,int . Inverting the bilinear form in eq. (F.2) we find the set of propagators The propagators (F.6) and (F.7) are even and (F.5) is odd with respect to swapping the points. In contrast with the propagators in the light-cone gauge, eqs. (2.31)-(2.33), the new ones have a more familiar form. Recalling that the LH analog φ AB (x, u) of the scalar field lives in the superfield A ++ in front of (θ + ) 2 , eq. (2.22), we extract from eq. (F.7) the familiar scalar propagator φ AB (x, u 1 )φ CD (0, u 2 ) = ABCD x −2 δ(u 1 , u 2 ). The spacetime part of A + α A ++ , eq. (F.5), is the usual fermion propagator ∂ αα x −2 . This is not a surprise since we can extract from there ψ −A (x, u 1 )ψα B (0, u 2 ) , eqs. (2.22), (2.23). The propagators satisfy the gauge-fixing condition (F.1) as a corollary of the identity ∂ −α ∂ − α = 0. In particular, ∂ −α A + α A ++ = 0, suggesting to call this gauge a Landau gauge. The propagators involving the Lagrangian multiplier, do not participate in the calculations of gauge-invariant quantities.
The Feynman rules for this new Landau gauge are similar to those in the light-cone gauge (see [1]), with the addition of ghosts appearing only in loops. The main difference is that the propagators are not contact terms in space-time.
As a simple application of these Feynman rules we show that there are no one-loop corrections to the bare propagators induced by L CS,int , eq. (F.2). We do not consider quantum corrections coming from L Z since they have a lower Grassmann degree. There are two diagrams at order O(g 2 ) giving corrections to the propagator A ++ A ++ . The first diagram contains two cubic vertices from L CS , and the second diagram contains a ghost loop. They sum up to zero as a consequence of several facts: (i) the form of the propagators, i.e. A + α A ++ = ∂ − α c ++ c ++ , (ii) the cubic vertices from L CS,int have a similar form, (iii) the ghost fields satisfy the Fermi statistics.
Then, there are no loop diagrams at order O(g 2 ) made of the L CS,int vertices and corresponding to quantum corrections of A + α A ++ and A + α A + β . The reason is the absence of the bare propagator A + α A + β , eq. (F.4).

F.1 Calculations in the Landau gauge
Here we show a nontrivial application of the Feynman rules in the Landau gauge (F.1). The MHV-like correlator G n;0 (4.13), which is given by graphs containing only bivalent vertices from S Z , is immediately reproduced in this formalism. The NMHV-like n-point Born-level correlator G n;1 is much more involved. Compared to the light-cone gauge Feynman rules in Section 4.3, now there are more Feynman graphs and they have a more complicated form. This is the price of the explicit Lorentz invariance of the result.
Like before, we add to the Feynman rules the external vertex T (x, θ + , w), eq. (3.14). It involves the R-analyticity projector d 4 θ − and an LH factor without ω. In the Born approximation we do not consider graphs with internal (interaction) vertices from S Z . They involve eight Grassmann integrations, which shifts the balance between the order O(g 2p ) in the coupling and the Grassmann degree 4p of the Born-level graphs. The interaction vertices from S Z correspond to loop integrals.
The Born-level n-point NMHV-like supercorrelator G n;1 is of order O(g 2 ). For Bornlevel graphs we use the cubic internal vertices from L CS,int and the k-valent external vertices from T . They can be chosen in four ways: • two interaction vertices from S CS and n bivalent vertices from T ; • one internal vertex from S CS , one trivalent vertex and n − 1 bivalent vertices from T ; • no internal vertices, two trivalent vertices and n − 2 bivalent vertices from T ; • no internal vertices, one four-valent vertex and n − 1 bivalent vertices from T .
In the following we omit the quantum corrections to the propagator A ++ A ++ , shown to vanish in eq. (F.8). At order O(g 2 ) the ghost loop can appear only in diagrams of this type. So, ghosts will not appear in our calculation at all.
In drastic contrast with the light-cone gauge, now the Born-level calculation is not a free field theory calculation. We have to consider diagrams with internal (interaction) vertices from S CS which involve genuine Feynman integrals d 4 x. At least for NMHV-like correlators these integrals can be evaluated explicitly and the result is rational. We will need two Feynman integrals of this type. The one-loop integral (known as the star-triangle relation) where we use the shorthand notation (x 132 ) αβ = (x 13 ) αα (x 32 )α β , describes the Yukawa interaction of two fermions of the same chirality and a scalar, Fig. 12, eq. (E.3). The sum of two two-loop H-shape integrals also has a simple rational expression The right hand side of eq. (F.10) is rational due to the symmetrization. One of the loop integrals in eq. (F.10) can be evaluated with the help of eq. (F.9). The remaining one-loop integral can be reduced to scalar integrals (the four-point cross-integral and its three-point truncation, which are known in terms of logs and dilogs) acted upon by two derivatives. After simplification all transcendental pieces cancel out. At higher non-MHV levels the situation is much more involved. The Born-level correlators are also rational, but certain individual Feynman graphs can produce highly transcendental expressions. The sum of all these intricate Feynman integrals has to simplify to a rational expression. The coupling dependence of the correlator is due to the nonpolynomial form of the operator T (3.14) and to the cubic interaction vertex from S CS . Thus we see that the calculation in the Landau gauge is more in the spirit of the usual perturbative calculations in terms of component fields of N = 4 SYM in Appendix E. The advantage of the present approach is the manifest Q-supersymmetry.
There are several subgraphs in an NMHV graph that can make the whole graph vanish, Fig. 13. The first diagram in Fig. 13 contains a pair of interaction cubic vertices from S CS linked by two propagators. The vertices are local in the LH space. The diagram vanishes due to the identity δ 4 (θ 00 · u + )δ 4 (θ 00 · u + ) = 0, eq. (F.7). The second diagram in Fig. 13 contains two cubic vertices from S CS and one external bivalent vertex from T (i) linked to form a triangle. It is proportional to du 1 1 (u + 1 u + 2 ) 2 δ 4 (θ 10 u + 1 )δ 4 (θ 10 u + 2 )δ 4 (θ 00 u + 1 )δ(u 1 , u 2 ) = 0 . (F.11) In the third diagram in Fig. 13 the interaction vertex from S CS and the external vertex T (i) are linked by two propagators, so again the vanishing is due to the Grassmann delta functions, δ 4 (θ 10 u + 1 )δ 4 (θ 10 u + 2 )δ(u 1 , u 2 ) = 0 . (F.12) In the following we omit the topologies containing the subgraphs from Fig. 13. In particular one can easily see that all the supergraphs representing the two-point NMHV-like Bornlevel correlator G 2;1 contain these subgraphs.

F.1.1 Three-point NMHV-like correlator
The three-point NMHV-like Born-level correlator G 3;1 has to vanish. This is obvious in the light-cone gauge (2.27), however in the Landau gauge there are four topologies giving nontrivial contributions, Fig. 14. As a simple illustration of the new rules in this gauge we show that they sum up to zero. In the light-cone gauge we are not allowed to draw more that one propagator between a pair of external vertices because of the mixed delta functions in the propagator A ++ A ++ , eq. (2.31). In the Landau gauge we can link a pair of external vertices by two propagators A ++ A ++ , eq. (F.7). Firstly, let us show that topology (d) in Fig. 14 vanishes because of the harmonic integrations. The Grassmann integration d 4 θ + 0 produces the harmonic factor (u + 1 u + 2 ) 4 (u + 1 u + 3 ) 4 . Then we collect the harmonic factors from the trivalent vertex T (i 1 ), from the bivalent vertices at points i 2 , i 3 , and the harmonics u − 1α u − 1β coming from the two propagators A + α A ++ , and we perform the harmonic integrations, The remaining three topologies (a), (b), (c) are nontrivial. The integral d 4 θ + 0 d 4 θ + 0 of the product of six Grassmann delta functions in the diagram T (a) results in (u + 1 u + 2 ) 8 δ 8 (θ 12 )δ 8 (θ 13 ). To implement the R-analytic projections at points 1, 2, 3 we decompose the Grassmann delta function by means of RHs according to eq. (4.7), d 4 θ 1− d 4 θ 2− d 4 θ 3− δ 8 (θ 12 )δ 8 (θ 13 ) = y 4 12 y 4 13 δ 4 (A 1;23 ) , (F.14) where we introduced the three-point Q-supersymmetry invariant A i;j,k . It is made of two two-point Q-supersymmetry covariants B ij and B ik , eq. (4.10),  Fig. 14 do not involve Feynman integrals, however they require nontrivial harmonic integrations, for example [38] du So, the contributions of the two topologies are Lorentz invariance is also respected, but each particular diagram is not conformally covariant. There are no spurious poles like those in the light-cone gauge produced by the gauge-fixing spinor ξ . − . However, T (a) , T (b) , T (c) contain higher order poles x −4 ij . The allowed singularities of the correlator are simple poles x −2 ij , so the higher order poles have to disappear in the sum of all diagrams.

F.1.2 n-point NMHV-like correlator
The four-point NMHV-like Born-level correlator G 4;1 has to vanish as a consequence of supersymmetry, eq. (4.1). In the light-cone gauge we have six contributing diagrams summing up to zero due to complicated algebraic identities. Working in the Landau gauge also requires a nontrivial calculation to show that G 4;1 = 0. This calculation is very helpful, it allows us to get rid of the higher order poles in the n-point correlator.
After the elimination of the higher-order poles one can show that the sum over all topologies (a) − (g) along with their S 4 permutations equals zero.
The NMHV-like Born-level correlators are nontrivial quantities starting with n ≥ 5 points. In the harmonic supergraph approach the case of the six-point correlator can be viewed as generic. Adding new external points to the six-point correlator but staying at NMHV level corresponds to the insertion of spectator scalar propagators d ij in the harmonic supergraphs, which modify the result rather trivially. We have already observed this phenomenon in the light-cone gauge calculation. The topologies contributing to the n-point correlator are depicted in Fig. 16. The corresponding expressions are rather similar to the four-point case. They are constructed out of the three-and four-point Q-supersymmetry invariants, eq. (F.21).
Let us first consider the topologies that produce higher order poles x −4 ij . The cancellation mechanism is the same as in the four-point correlator example. The relevant diagrams