Symmetries and analytic properties of scattering amplitudes in N=4 SYM theory
aa r X i v : . [ h e p - t h ] J un IPhT-T09/076LAPTH-1331/09NSF-KITP-09-95
Symmetries and analytic properties ofscattering amplitudes in N = 4 SYM theory
G.P. Korchemsky ∗ and E. Sokatchev ∗∗∗ Institut de Physique Th´eorique , CEA Saclay,91191 Gif-sur-Yvette C´edex, France ∗∗ LAPTH , Universit´e de Savoie, CNRS,B.P. 110, F-74941 Annecy-le-Vieux, France Abstract
In addition to the superconformal symmetry of the underlying Lagrangian, the scattering ampli-tudes in planar N = 4 super-Yang-Mills theory exhibit a new, dual superconformal symmetry.We address the question of how powerful these symmetries are to completely determine thescattering amplitudes.We use the example of the NMHV superamplitudes to show that the combined action ofconventional and dual superconformal symmetries is not sufficient to fix all the freedom in thetree-level amplitudes. We argue that the additional information needed comes from the study ofthe analytic properties of the amplitudes. The requirement of absence of spurious singularities,together with the correct multi-particle singular behavior, determines the unique linear combina-tion of superinvariants corresponding to the n − particle NMHV superamplitude. The same resultcan be obtained recursively, by relating the n − and ( n − − particle amplitudes in the singularcollinear limit. We also formulate constraints on the loop corrections to the superamplitudes,following from the analytic behavior in the above limits.We then show that, at one-loop level, the holomorphic anomaly of the tree amplitudes leadsto the breakdown of dual Poincar´e supersymmetry (equivalent to ordinary special conformalsupersymmetry) of the ratio of the NMHV and MHV superamplitudes, but this anomaly doesnot affect dual conformal symmetry. On leave from Laboratoire de Physique Th´eorique, Universit´e de Paris XI, 91405 Orsay C´edex, France Unit´e de Recherche Associ´ee au CNRS URA 2306 Laboratoire d’Annecy-le-Vieux de Physique Th´eorique, UMR 5108 ontents s and collinearity in twistor space . . . . . . . . . . . 72.1.2 Grassmann Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Conformal supersymmetry s . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 NMHV tree superamplitude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2.1 Conformal supersymmetry ¯ s and line structure in twistor space . . . . . . . . . . 122.2.2 Grassmann Fourier transform and conformal supersymmetry s . . . . . . . . . . 14 n = 6 NMHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.3.2 n = 7 NMHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.3 General n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4 Spurious poles at loop level . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.4.1 n = 6 NMHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4.2 n = 7 NMHV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.5 Collinear singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.6 Spurious poles cancellation versus collinear factorization . . . . . . . . . . . . . . . . . . 32 N = 4 superconformal symmetry 40B Appendix: Solution of the collinearity conditions 41C Appendix: Conformal s − supersymmetry of the NMHV tree superamplitude 43D Appendix: Collinear limit of the n = 6 NMHV superamplitude 44E Appendix: ¯ Q − anomaly of the n = 6 NMHV superamplitude 46
Introduction
Recently it has been realized that the scattering amplitudes in planar N = 4 super-Yang-Millstheory exhibit a new remarkable symmetry, called dual superconformal symmetry. It appears inaddition to the known classical symmetries of this gauge theory and it is believed to be relatedto the hidden integrability of the N = 4 theory. The dual superconformal symmetry was firstidentified at weak coupling by examining the properties of the tree and one-loop MHV andNMHV scattering amplitudes [1]. It was subsequently shown that at strong coupling this newsymmetry is related, through the AdS/CFT correspondence, to the invariance of the string sigmamodel in AdS × S under bosonic and fermionic T-duality [2, 3].In principle, the combination of this new symmetry with the ordinary N = 4 superconformalsymmetry of the underlying Lagrangian may have far reaching consequences for the scatteringamplitudes and might eventually lead to their exact determination for arbitrary values of the’t Hooft coupling in planar N = 4 SYM theory. However, a lot of work still needs to be donebefore we fully understand all the implications of these symmetries. In the present paper wetry to answer some of the open questions. We first consider the tree-level MHV and NMHVsuperamplitudes, and show that the combined action of the two superconformal symmetries isnot sufficient to completely fix all the freedom in the amplitudes. We demonstrate that theadditional information needed comes from the study of the physical and spurious singularities,or, alternatively, of the collinear singularities of the amplitudes. We then show that the holo-morphic anomaly of the tree amplitudes leads to the breakdown of dual supersymmetry (whichis equivalent to ordinary special conformal supersymmetry) at one-loop level, but this anomalydoes not affect dual conformal symmetry. A convenient framework for discussing the symmetries of scattering amplitudes in N = 4 SYMtheory is on-shell superspace [4, 5, 1, 6], closely related to the light-cone superspace formalism[7, 8]. In this approach, all asymptotic states in N = 4 SYM theory, gluons ( G ± ), gluinos(Γ A , ¯Γ A ) and scalars ( S AB ), are combined into a single on-shell superstate,Φ( p, η ) = G + ( p )+ η A Γ A ( p ) + 12 η A η B S AB ( p )+ 13! η A η B η C ǫ ABCD ¯Γ D ( p ) + 14! η A η B η C η D ǫ ABCD G − ( p ) , (1.1)with the help of Grassmann variables η A carrying helicity 1 / SU (4) index A = 1 . . . p µ (with p = 0) and helicities ranging from +1 ( G + ) to − G − ). Making use ofsuch superstates, we can combine all n − particle color-ordered scattering amplitudes in the N = 4SYM theory into a single on-shell object, the superamplitude A n ( p , η ; . . . ; p n , η n ). It dependson the supermomenta ( p i , η i ) of the particles and its expansion in powers of η ’s generates thescattering amplitudes for the various types of particles from (1.1).Another useful tool for discussing the symmetries of the superamplitudes is the spinor-helicityformalism [9]. In it one resolves the on-shell conditions on the particle momenta p i = 0 byintroducing a pair of commuting spinors for each particle, p α ˙ αi = p µ ( σ µ ) α ˙ α = λ αi ˜ λ ˙ αi , (1.2)1r equivalently p i = | i i [ i | . In Minkowski space-time the two spinors are complex conjugate,˜ λ = ( λ ) ∗ , and are defined up to a phase factor (helicity).The SU (4) invariance of A n ( p , η ; . . . ; p n , η n ) implies that it is expanded in powers of ( η ) k with k = 0 , , . . . , n . On-shell Poincar´e supersymmetry requires the absence of the terms with k = 0 ,
1. So, the first term in the expansion of the superamplitude is of degree 8. It generates all n − particle MHV amplitudes and we shall denote it by A MHV n . The next term of the expansion, A NMHV n , has degree 12 in η and it generates the NMHV amplitudes, etc. The simplest, tree-level n − particle MHV superamplitude takes the form [4] A MHV;0 n = i δ (4) ( P ni =1 λ αi ˜ λ ˙ αi ) δ (8) ( P ni =1 λ αi η Ai ) h ih i . . . h n i , (1.3)where the superscript ‘0’ in the left-hand side denotes the tree approximation and we use thestandard notation for the Lorentz invariant spinor contractions h ij i = λ αi λ j,α = λ αi ǫ αβ λ βj .At tree level, the superamplitudes have neither infrared nor ultraviolet divergences, so theyinherit all the classical symmetries of the N = 4 Lagrangian. Indeed, it can be shown directlythat A MHV;0 n is invariant under N = 4 superconformal transformations, realized non-locally onthe particle momenta [5].In addition to the ordinary superconformal symmetry, A MHV;0 n has another, dual N = 4 superconformal symmetry . To make it manifest, one introduces new dual variables [10, 11, 1]related to the supermomenta ( p i , η i ) as follows p α ˙ αi = λ αi ˜ λ ˙ αi = x α ˙ αi − x α ˙ αi +1 , λ αi η Ai = θ αAi − θ αAi +1 , (1.4)with the periodicity conditions x n +1 ≡ x and θ n +1 ≡ θ . The dual superconformal symmetryacts locally on the dual variables x i and θ i (and, as a consequence, on the spinor variables λ, ˜ λ )as if they were coordinates in some dual superspace. Most remarkably, the superamplitude (1.3),rewritten in terms of the dual coordinates, transforms covariantly under dual superconformalsymmetry with dual conformal weight (+1) at each point, equal to the helicity of the superstate(1.1). We wish to stress that this symmetry is by no means obvious. Its dynamical origin inperturbation theory is still not completely clear. At strong coupling, it can be interpreted viathe AdS/CFT correspondence as a T-duality symmetry of the string sigma model [2, 3].The n -particle NMHV tree superamplitude A NMHV;0 n has a considerably more complicatedform. It was first found in [1] by comparing its gluon components to the known NMHV gluonamplitudes from [12, 13], and later on rederived by the supersymmetric versions of the generalizedunitarity method [14, 15] and of the BCFW recursion relations [16, 17]. It can be represented ina factorized form involving the MHV tree superamplitude (1.3) as a prefactor, A NMHV;0 n = A MHV;0 n X ≤ s +1 8) to thesimplest NMHV-like ‘googly’ five-particle amplitude A MHV;05 .Comparing the two methods of fixing the freedom in the NMHV tree superamplitudes, wemay say that the spurious singularities operate ‘horizontally’, i.e. they establishes a property ofthe n − particle NMHV amplitude, without reference to other amplitudes (putting aside the use ofthe multi-particle factorization, needed to fix the overall normalization). In this sense, A NMHV;0 n is the unique n − point invariant of the two superconformal symmetries, free from spurious sin-gularities. At the same time, the method based on collinear singularities operates ‘vertically’,i.e. it recursively relates the n − particle amplitude to the amplitudes with fewer particles. Sincein both cases one arrives at the same, unique expression for the NMHV tree amplitude, builtout of ordinary+dual superconformal invariants, the two approaches are equivalent at tree level.However, at loop level this may not be the case anymore (see Section 3.6).A comment is due here about the BCFW recursion procedure [24, 16]. It provides a powerfulmechanism for constructing tree-level scattering amplitudes from elementary blocks, the MHVamplitudes (in fact, the starting point of the recursion is the simplest, 3-particle MHV ampli-tude). It consists in deforming two of the particle momenta by a complex parameter z and thenexploiting the resulting poles in z to factorize the amplitude into simpler blocks. An importantingredient in the construction is the assumption about the regular behavior at z → ∞ . Althoughthis approach cannot directly explain the origin of dual conformal symmetry, it can reduce it tothe properties of the elementary building blocks, the MHV amplitudes, as shown in [25]. Re-cently, the complete tree-level superamplitude has been found in [17] through the supersymmetricgeneralization of the BCFW recursion [26, 25, 6, 27]. The reason why we wish to reexamine thetree superamplitudes is not to find an alternative construction (the BCFW one is very efficient),but to understand to what extent the amplitudes are determined by their symmetries (ordinary Recently it has been shown that the closure of the two superconformal symmetries is infinite dimensional andhas a Yangian structure [18]. This has renewed [19] the hope that the scattering amplitudes in N = 4 planarSYM may be integrable in some sense [2, 3, 20]. The role of the spurious singularities has recently been discussed in [21], in the context of a renewed interest[22, 23] in Witten’s twistor transform [5] of scattering amplitudes. Let us now turn to the situation at loop level, where the massless particle scattering amplitudesare infrared divergent and require regularization. This inevitably breaks part of the classicalsymmetries and renders problematic the way they constrain the all-order amplitudes. In par-ticular, to make use of the ordinary and dual superconformal symmetries we should be able tocontrol their breakdown at loop level.Important progress in this direction has been made after the discovery of the MHV am-plitude/Wilson loop duality. It stipulates the equivalence of the loop corrections to the MHVamplitude A MHV n , on the one hand, with a Wilson loop in dual space-time evaluated along a light-like polygon contour with cusps at the points x i from (1.4), on the other hand. In the N = 4SYM theory, this relation was first observed at strong coupling using the string descriptionof the scattering amplitudes in AdS/CFT [29] and it was later confirmed at weak coupling bymatching the perturbative corrections to both quantities [30, 31, 32, 33, 34, 35, 36, 37]. Thescattering amplitude/Wilson loop duality implies that the dual conformal symmetry of the MHVsuperamplitude is equivalent to the conformal symmetry of the light-like Wilson loops in N = 4SYM. The latter is broken locally by the ultraviolet cusp singularities [38]. This allows us todetermine the dual conformal anomaly of the all-loop MHV superamplitudes from an anomalousconformal Ward identity for the light-like Wilson loop [33]. This Ward identity is a powerfulconstraint on the form of the MHV amplitude. In particular, it fixes the finite part of the n = 4and n = 5 amplitudes in accord with the BDS conjecture [39, 40], while for n ≥ a priori not clear how to determine the dual conformal anomaly. We recallhowever that this anomaly is due to the infrared divergences of the scattering amplitudes. Thelatter have a universal, helicity independent form in N = 4 SYM thus suggesting that the dualconformal anomaly might also be universal for MHV, NMHV, N MHV, ... superamplitudes.This property can be formulated in a compact form by introducing the so-called ratio function.In the NMHV case it is defined by A NMHV n = A MHV n (cid:2) R NMHV n + O ( ǫ ) (cid:3) , (1.6)where A MHV n and A NMHV n stand for the complete (all-loop) superamplitudes and ǫ is the parameterof dimensional regularization. The ratio function R NMHV n defined in this way is infrared finiteand, therefore, one would expect that the symmetries, broken by the divergent factor A MHV n ,could be (partially) restored in R NMHV n . In particular, if the dual conformal anomaly is universal,then R NMHV n should be dual conformally invariant . Indeed, it has been shown in [1] that the ratiofunction is given, to lowest order in the ’t Hooft coupling a = g N , by the following expression(see Eq. (3.57) below) R NMHV n = n X s,t =1 w st R st (cid:2) aV st ( x ) + O ( a ) (cid:3) + cyclic , (1.7) The duality between scattering amplitudes and light-like Wilson loops was first noticed in QCD in the high-energy (Regge) limit [28]. linearly independent superinvariants R st and w st are ( a − independent)rational numbers. The terms needed to make R NMHV n invariant under cyclic shifts of the labelsof the n particles are denoted by ‘cyclic’. Most importantly, the scalar functions V st ( x ), whichencode the loop corrections in (1.7), are dual conformally invariant . Thus, dual conformal sym-metry is a general property of the ratio function [1, 15, 44, 45].We recall that the superinvariants R st , and thus the NMHV tree superamplitude (1.5) havea larger, dual super conformal symmetry. An obvious question is whether the dual conformalsymmetry of V st can also be promoted to dual superconformal symmetry. The latter is obtainedby adding Poincar´e supersymmetry to dual conformal symmetry (the rest follows from the N = 4superconformal algebra). In this paper we show that the dual Poincar´e ¯ Q − supersymmetry ofthe ratio function (which is also equivalent to ordinary special conformal supersymmetry, see[1] and Eq. (2.3) below) is broken at one loop. We trace this one-loop anomaly back to theso-called holomorphic anomaly [46, 47, 48] of the tree superamplitudes. Strictly speaking, thetree superamplitudes like the MHV one (1.3), are invariant under ¯ Q − supersymmetry only up tocontact terms, due to the collinear pole singularities when h i i + 1 i → 0. When loops are madeout of trees via unitarity, such singularities are integrated over and induce an anomalous behaviorunder dual Poincar´e (and ordinary special conformal) supersymmetry. We illustrate this effectby an explicit calculation of the one-loop ¯ Q − anomaly of the multi-particle discontinuity of the n = 6 NMHV superamplitude. At the same time, we show that the holomorphic anomaly doesnot affect the dual conformal symmetry of this amplitude. Section 2 is devoted to proving ordinary superconformal symmetry of the tree superamplitudes.We first show this for the MHV superamplitude (1.3). Compared to the original treatment in[5], we exhibit a new feature, the equivalence of ¯ s − conformal supersymmetry with the conditionfor twistor space collinearity from [5]. We argue that the combination of ordinary and dualsuperconformal symmetry fixes the form of the MHV superamplitude (1.3), up to a normalizationconstant. We then show that each term in the NMHV superamplitude (1.5) is separately invariantunder both ordinary and dual superconformal symmetries. The ¯ s − conformal supersymmetry isequivalent to the condition that the tree NMHV amplitude is supported on three intersectinglines in twistor space.In Section 3 we study the analytic properties of the NMHV superamplitudes in the vari-ous singular limits: multi-particle and two-particle (or collinear) physical singularities and theunphysical spurious singularities. We first recall the factorization property of tree amplitudesat multi-particle poles and the associated notion of discontinuity. We then examine the physi-cal and spurious singularities of the NMHV superinvariants R rst . Afterwards we show how the n = 6 , , n − point superinvariants and imposing the conditions for correct singularbehavior. In doing this, the absence of spurious singularities turns out to be the most powerfulcondition, fixing all relative coefficients of the superinvariants. The multi-particle factorization Very recently, the role of the collinear singularities and the associated holomorphic anomaly has been studiedin [49]. The authors argue that one can deform the generators by including the holomorphic anomaly andmaintaining the superconformal algebra. Requiring exact invariance under the deformed symmetry leads torecursive relations between tree amplitudes with different numbers of particles. However, it is not clear whetherthis approach can be efficiently pursued at loop level. n = 6 , Q − supersymmetry (or ordinary ¯ s − conformal supersymmetry). Insteadof discussing the amplitude itself, where the effect of the holomorphic anomaly is mixed up withinfrared divergences, we consider the multi-particle cut of the one-loop MHV and NMHV super-amplitudes. This quantity is infrared finite and, as a consequence, it inherits the dual conformalsymmetry of the constituent tree superamplitudes. However, dual supersymmetry is broken andwe compute the corresponding anomaly.Section 5 contains concluding remarks. Some technical details are presented in the appendices. In this section, we discuss the constraints imposed on the tree MHV and NMHV superamplitudesby the superconformal symmetry in N = 4 SYM theory. To this end it is sufficient to considerthe action of the odd generators of Poincar´e ( q, ¯ q ) and conformal ( s, ¯ s ) supersymmetry [5], q Aα = n X i =1 λ iα η Ai , ¯ q ˙ αA = n X i =1 ˜ λ ˙ αi ∂∂η Ai ,s αA = n X i =1 ∂ ∂λ iα ∂η Ai , ¯ s A ˙ α = n X i =1 η Ai ∂∂ ˜ λ ˙ αi , (2.1)the rest follows from the superconformal algebra su (2 , | 4) (see Eq. (A.1)). For example, com-puting the anticommutator { s αA , s B ˙ α } = δ BA k α ˙ α we can obtain the well-known expression for thegenerator of special conformal transformations [5], k α ˙ α = n X i =1 ∂ ∂λ αi ∂ ˜ λ ˙ αi . (2.2)It is second-order with respect to the spinor variables, which reflects the fact that the conformalsymmetry acts non-locally in the momentum representation. Comparing (2.2) with (2.1), wenote that the generators (2.1) are at most first-order in the spinor derivatives. Therefore, theiraction on the superamplitudes (after Fourier transforming the η dependence) is linear and local.This is why we will concentrate on the verification of the invariance of the superamplitudes under¯ s and s supersymmetry, the action of q and ¯ q being quite obvious.The realization of dual superconformal symmetry in the dual superspace (1.4) and the proofthat the MHV superamplitudes, as well as each term inside the NMHV superamplitudes arecovariant, has been presented in detail in [1]. Here we only recall the partial overlap betweenthe two superconformal algebras, namely, the following generators of ordinary (denoted by lowercase letters) and dual (upper case) symmetries coincide:¯ q ˙ αA ≡ ¯ S ˙ αA , ¯ s A ˙ α ≡ ¯ Q A ˙ α . (2.3)6lso, the dual Poincar´e supersymmetry Q αA = P i ∂/∂θ αAi is a trivial consequence of the change ofvariables (1.4). Thus, having proven ¯ s ≡ ¯ Q symmetry, the statement about dual superconformalsymmetry of the amplitudes is essentially reduced to their dual conformal K − covariance. As was shown in Ref. [5], the tree-level MHV superamplitude is invariant under the full native N = 4 superconformal algebra. In this section we take a slightly different route. We generalize(1.3) by introducing the possible dependence on the bosonic variables λ, ˜ λ through an arbitraryfunction, A MHV;0 n = δ (4) ( n X i =1 λ i ˜ λ i ) δ (8) ( n X i =1 λ i η i ) f ( λ, ˜ λ ) . (2.4)Then we ask the question to what extent ordinary and dual superconformal symmetries restrictthe function f ( λ, ˜ λ ). Notice that the expression in the right-hand side of (2.4) has to be ahomogenous polynomial of degree 8 in the η ’s and, therefore, the function f ( λ, ˜ λ ) is η − indepedent.In addition, f ( λ, ˜ λ ) carries the helicity weights of the scattered (super)particles, i.e. it accountsfor the scaling behavior A MHV;0 n → e i P k χ k A MHV;0 n under λ k → e − iχ k / λ k , ˜ λ k → e iχ k / ˜ λ k and η k → e iχ k / η k . This suggests to use the following ansatz, f ( λ, ˜ λ ) = ϕ ( λ, ˜ λ ) h ih i . . . h n i , (2.5)with ϕ ( λ, ˜ λ ) being a helicity neutral function. There are two ways to construct such functions, byallowing dependence only through the helicity-free momenta (1.2) or through purely holomorphiccombinations like h ij ih kl i / h ik ih jl i . Below we argue that the former are ruled out by the conformalsupersymmetry ¯ s , and the latter by dual conformal symmetry. Thus, the MHV superamplitudecan be fixed in the form (1.3) (up to normalization) by symmetries alone.So, let us demand that all the generators in (2.1) (and hence all generators of su (2 , | A MHV;0 n in (2.4). Obviously, the two Poincar´e supersymmetries q and ¯ q defined in (2.1)do so, due to the fermionic and bosonic delta functions in (2.4). In the following subsections wediscuss the action of the special conformal supersymmetry generators ¯ s and s . ¯ s and collinearity in twistor space When applied to (2.4), the conformal supersymmetry generator ¯ s , Eq. (2.1), acts on the dottedspinor variables ˜ λ . They are present both in the function f ( λ, ˜ λ ) and in the argument of themomentum conservation delta function δ (4) ( P ni =1 λ i ˜ λ i ). The variation of the latter is suppressedby the fermionic delta in (2.4).Thus, acting on A MHV;0 n , the generator ¯ s goes through the delta functions in (2.4) and directlyhits f ( λ, ˜ λ ), giving ¯ s A ˙ α f ( λ, ˜ λ ) = n X i =1 η Ai ∂∂ ˜ λ ˙ αi f ( λ, ˜ λ ) = 0 . (2.6)We should not require each term in this variation to vanish, since not all of the η ’s are lin-early independent. Indeed, the Grassmann delta function in (2.4) imposes a linear relation,7 ni =1 λ αi η Ai = 0. Projecting this relation with, e.g., h n | and h | , we can solve it for η = 1 h n i n − X i =2 h ni i η i , η n = 1 h n i n − X i =2 h i i η i . (2.7)Then the relation (2.6) reduces to¯ s A ˙ α f ( λ, ˜ λ ) = 1 h n i n − X η Ai (cid:18) h n i ∂∂ ˜ λ ˙ αi + h ni i ∂∂ ˜ λ ˙ α + h i i ∂∂ ˜ λ ˙ α (cid:19) f ( λ, ˜ λ ) = 0 . (2.8)Since all the η ’s in this relation are linearly independent, we must impose the constraints F ,i,n f ( λ, ˜ λ ) ≡ (cid:18) h n i ∂∂ ˜ λ i + h ni i ∂∂ ˜ λ + h i i ∂∂ ˜ λ n (cid:19) f ( λ, ˜ λ ) = 0 , (2 ≤ i ≤ n − . (2.9)We recognize that the operator F ,i,n coincides with the well-known operator of collinearity intwistor space [5]. Thus, the conformal supersymmetry ¯ s of the MHV superamplitude (2.4) impliesthat A MHV;0 n has to satisfy a collinearity condition, meaning that A MHV;0 n has support on a singleline in twistor space defined by the points (1 , n ).This is a very strong condition on the ˜ λ − dependence of the function f ( λ, ˜ λ ). The supersym-metry ¯ s does not constrain the λ − dependence, so we could in principle imagine an additionaldependence on λ through purely holomorphic combinations without helicity like h ij ih kl i / h ik ih jl i .However, such combinations are not compatible with dual conformal symmetry (see AppendixA). This implies that the dependence of the function ϕ introduced in (2.5) on λ and ˜ λ mustcome through the helicity-free momenta (1.2), that is ϕ = ϕ ( p , . . . , p n ). In Appendix B we showthat (2.9) yields ϕ = const.We conclude that the requirements of simultaneous invariance of the MHV tree superampli-tude, Eqs. (2.4) and (2.5), under ordinary and dual superconformal symmetry fixes its form upto an overall constant factor.We wish to make an important comment on the discussion in this subsection. The denom-inator in (2.5), although naively holomorphic (a function of λ but not of ˜ λ ) is singular for h i i + 1 i → s up to contact terms. This phenomenon isclosely related to the so-called “holomorphic anomaly” of amplitudes in twistor space [46]. Ofcourse, the above symmetry argument does not take into account the holomorphic anomaly ofthe amplitude. This anomaly becomes important when tree amplitudes are used to form loops,via the unitarity cut technique [46, 47, 48]. As we show in Section 3, it makes the conformalsupersymmetry ¯ s (or the dual Poincar´e supersymmetry ¯ Q ) anomalous at loop level. We still have one more supersymmetry condition to verify, s αA A MHV;0 = 0. This is not so simple,since the generator s αA in (2.1) is given by a second-order differential operator. Although it ispossible to show that s αA A MHV;0 n = 0 directly (see [5]), here we prefer to use another approach,more suitable for generalization to the NMHV case. We first Fourier transform the amplitudewith respect to the odd variables η , which renders the generator s αA first-order and makes thecheck much easier. 8he Grassmann Fourier transform of an N = 4 superamplitude is defined by the n − foldGrassmann integral ˜ A n (¯ η ) = Z n Y i =1 d η i e P ni =1 ¯ η iA η Ai A n ( η ) . (2.10)As explained in [1], [6], the superamplitude ˜ A n (¯ η ) is equivalent to the PCT conjugate of A n ( η )(hence the use of complex conjugate odd variables ¯ η A = ( η A ) ∗ ). Thus, for the n − particle MHVsuperamplitude A MHV containing, e.g., MHV gluon amplitudes with only two negative-helicitygluons, its transformed version ˜ A MHV n contains only two gluons of positive helicity.Let us perform the Fourier transform of the MHV superamplitude (1.3) by taking into accountthat the fermionic delta function in (1.3) can be factorized into two four-dimensional ones, e.g.(cf. (2.7)), δ (8) ( n X λ i η i ) = h n i δ (4) η − h n i n − X h ni i η i ! δ (4) η n − h n i n − X h i i η i ! . (2.11)This relation can be used to do the Fourier integrals with respect to η and η n . The remaining( n − 2) integrals take the form Z n Y d η i e P n ¯ η i η i δ (8) ( n X λ i η i ) = h n i Z n − Y d η i exp ( n − X (cid:18) ¯ η i + h ni ih n i ¯ η + h i ih n i ¯ η n (cid:19) η i ) = h n i n − Y δ (4) (cid:18) ¯ η i + h ni ih n i ¯ η + h i ih n i ¯ η n (cid:19) . (2.12)Collecting the bosonic factors from (1.3), we obtain the Grassmann Fourier transform of the treeMHV superamplitude:˜ A MHV;0 n = i h n i − n ) n Y i =1 h i i + 1 i − δ (4) ( n X i =1 λ i ˜ λ i ) n − Y i =2 δ (4) ( h n i ¯ η i + h ni i ¯ η + h i i ¯ η n ) . (2.13)Note that this expression is a homogenous polynomial in ¯ η ’s of degree 4( n − s Let us proceed to showing that (2.13) is invariant under the action of the supersymmetry genera-tor s αA defined in Eq. (2.1). After the Fourier transform (2.10) it becomes a first-order differentialoperator, s αA = n X i =1 ∂∂λ αi ¯ η iA . (2.14)We have put the ¯ η ’s to the right of the λ derivatives on purpose. When acting on the amplitude(2.13), we will use the Grassmann delta functions in it to express ¯ η i (with i = 2 . . . n − 1) in termsof ¯ η and ¯ η n . This introduces some λ dependence, so we will have to push the λ derivatives in(2.14) through it. The result is s αA ˜ A MHV;0 n = " ¯ η A h n i − n X i =1 h ni i ∂∂λ αi + ( n − λ nα h n i ! − (1 ↔ n ) ˜ A MHV;0 n . (2.15)9ow, we take into account that the amplitude is annihilated by the (chiral) Lorentz generator m αβ = P ni =1 (cid:0) λ αi ∂ iβ − δ αβ λ γi ∂ iγ (cid:1) , and obtain from (2.15) s αA ˜ A MHV;0 n = ¯ η A λ nα − ¯ η nA λ α h n i n X i =1 λ βi ∂∂λ βi + 2( n − ! ˜ A MHV;0 n . (2.16)The operator in the parentheses counts the degree of homogeneity of ˜ A MHV;0 n in the λ ’s. InEq. (2.13), each h ij i gives 2, and the momentum conservation delta function gives ( − − n cancels against the constant in the parentheses in (2.16), leading to s αA ˜ A MHV;0 n = 0 . (2.17)We remark that the proof of this result can be simplified with the help of the Poincar´esupersymmetry q Aiα = P λ iα ∂/∂ ¯ η iA (after the Fourier transform, see (2.1)). This symmetry haseight fermionic parameters which can be used to ‘gauge away’ eight components of the ¯ η ’s, e.g.,¯ η A = ¯ η An = 0. After this the ( n − 2) Grassmann delta functions in (2.13) imply that all theremaining ¯ η ’s are zero, so the generator (2.14) vanishes when acting on the amplitude. Notethat this gauge fixing is legitimate because the anticommutator { q, s } (see (A.1)) vanishes whenapplied to the amplitude. Let us now extend the analysis to the n -particle NMHV tree superamplitudes. The explicitexpression for A NMHV;0 n was found in [1]: A NMHV;0 n = A MHV;0 n R NMHV;0 n = i δ (4) ( P n λ i ˜ λ i ) δ (8) ( P n λ i η i ) h i . . . h n − n ih n i R NMHV;0 n , (2.18)where the tree-level ratio function admits two equivalent representations, R NMHV;0 n = X ≤ s +1 Arts = t − X r +1 h r | x rs x st | i i η Ai + s − X r +1 h r | x rt x ts | i i η Ai = h r | x rs x st | θ At i + h r | x rt x ts | θ As i + x st h r θ Ar i . (2.22)10t is convenient to use a diagrammatic representation for R rst as a box diagram shown inFig. 1. It has n external legs ordered clockwise. One of the vertices is ‘massless’, i.e. it hasonly one external leg with index r attached to it, while the opposite vertex is always ‘massive’,i.e. it has at least two external legs with indices s, . . . , t − rst ,Eq. (2.22), vanishes if the restrictions (2.20) on the values of the labels r, s, t are not fulfilled.PSfrag replacements R rst = r r − r + 1 tss − t − r − s − sr − trr + 1 t − Diagrammatic representation of the dual superconformal invariant R rst The equivalence between the two representations in (2.19) follows from the identity betweenthe superinvariants [1, 15] X s,t ∈S n ′ R st = X s,t ∈S n ′ R n ′ st , ( n ′ = 5 , . . . , n ) , (2.23)in which the indices satisfy the same conditions (2.20) with r = 1 in the left-hand side sum and r = n ′ in the right-hand side sum. This identity holds without the help of the (super)momentumconservation delta functions δ (4) ( P n λ i ˜ λ i ) δ (8) ( P n λ i η i ). It was found in [15] as a self-consistencycondition for the scattering amplitudes in N = 4 SYM theory within the generalized unitaritycut method. Another identity satisfied by the superinvariants is [1, 15] R rs r − = R r − rs (2.24)(for n = 5 it is equivalent to (2.23), but for n > R ’s. For n = 6and n = 7 the corresponding expressions are R NMHV;06 = 12 R + cyclic ,R NMHV;07 = 17 R + 27 R + 37 R + cyclic . (2.25)Here ‘cyclic’ stands for the terms obtained by cyclic shifts of the indices, i i + 1. They areneeded for the cyclic symmetry of the superamplitude. The superinvariants R rst , and hence their diagrammatic representation, are in one-to-one correspondencewith the three-mass box coefficients from Ref. [13]. each term (2.21) in the sum (2.19) is separatelyinvariant under the full dual superconformal symmetry SU (2 , | Certainly, the NMHV treesuperamplitude, like any divergence-free tree amplitude, should also be invariant under the ordi-nary superconformal symmetry, Eqs. (2.1) and (A.1). What is not obvious, however, is that eachterm (2.21) in the sum (2.19) is separately invariant, as we show in the following subsections. ¯ s and line structure in twistor space As before, checking the two Poincar´e supersymmetries, q A NMHV;0 n = ¯ q A NMHV;0 n = 0 requires noparticular effort. The conformal supersymmetry with generator ¯ s is less trivial to verify. In [1]a proof of ¯ s A NMHV;0 n = 0 was given, based on the equivalence between ¯ s and the dual Poincar´esupersymmetry generator ¯ Q ≡ ¯ s , see (2.3). Here we present an alternative proof which exhibitsthe twistor line structure of the NMHV superamplitude.To this end, it is convenient to rewrite the tree NMHV superamplitude (2.18) as A NMHV;0 n = X ≤ s +1 4) their formbecomes much more complicated. The full expression of the N p MHV tree-level superamplitudeshas been obtained by solving the supersymmetric version of the BCFW recurrence relations [17].Each term in it is manifestly invariant under dual superconformal symmetry. They are also in-variant under ordinary superconformal symmetry, which can be proven along the lines of Section2 (see [63]). One may wonder what is the most general form of the n − point invariants of bothsymmetries and whether the set of invariants entering the tree-level N p MHV superamplitudes iscomplete. The answer will require a systematic classification of all invariants of the two sym-metries. We should mention that a new type of dual superconformal invariant appears forN MHV and more complicated amplitudes at loop level [15]. It corresponds to the four-massscalar boxes which do not contribute to the tree amplitude. In this sense, the NMHV amplitudesplay a special role since the one-loop corrections do not induce new invariants that are not seenat tree level.Even if we know the complete set of invariants, we are still left with the question whatelse is needed to fix the coefficients in the unique linear combination of these invariants, whichcorresponds to the amplitude. In this paper we argued that this additional information comesfrom studying the analytic properties of the amplitude. We demonstrated that two differentapproaches - one based on the cancellation of spurious poles, supplemented with the multi-particle factorization property, and another one relying on the correct collinear limit of the ratiofunction (3.77), lead to the same expression for the NMHV tree superamplitude. It would beinteresting to extend the analysis to the general case of N p MHV superamplitudes.Turning on loop corrections to the scattering amplitudes, we find that both conventionaland dual superconformal symmetries are broken. The anomalous contributions come from twodifferent sources, from the infrared divergences and from the holomorphic anomaly of the treeamplitudes. Further progress in the understanding of the all-loop N = 4 scattering amplitudes isultimately related to our ability to control the anomalies of both symmetries. For the conventionalsymmetry, the above mentioned anomalous contributions affect the conformal boosts ( k ) andspecial conformal supersymmetry ( s and ¯ s ) generators, but their anomalies are not independentdue to { s, ¯ s } = k . For the dual symmetry, the anomaly affects the generators K , ¯ Q and S , butthe latter is not independent due to [ K, ¯ Q ] = S . Therefore, taking into account that ¯ s = ¯ Q , thelist of independent anomalous generators for both symmetries includes dual conformal boosts K ,dual supersymmetry ¯ Q and ordinary conformal supersymmetry s . All of them become anomalousbecause of infrared divergences.We can circumvent the infrared divergences by considering the finite ratio function definedin (1.6). Quite remarkably, the dual conformal anomaly cancels in the ratio of superampli-tudes and the K − symmetry is restored. However, this is not the case for the dual Poincar´e¯ Q − supersymmetry. We demonstrated by an explicit one-loop calculation that the NMHV ratiofunction is not invariant under the action of ¯ Q , and that the breakdown of dual ¯ Q − supersymmetry(but not of dual conformal symmetry) can be traced back to the holomorphic anomaly of thetree amplitudes. Since ¯ Q ≡ ¯ s , this also means that the ordinary special conformal supersym-metry and, as a consequence, ordinary conformal symmetry become anomalous. Such brokensymmetries potentially lose their predictive power, unless we can understand and control theiranomalies. The question if it is also an invariant of ordinary superconformal symmetry has not been investigated yet. 39n example where we have been able to make all-loop predictions, based on an anomaloussymmetry, is dual conformal symmetry. Through the MHV amplitude/Wilson loop duality wecan trace the origin of this anomaly to the cusp singularities of the light-like Wilson loop. Thesesingularities are localized at the cusp points, which makes the analysis of their conformal behaviorrelatively easy. Moreover, the Wilson loop is a correlator in dual space, which is calculated froma conformal Lagrangian. The breaking of conformal symmetry is due to the ultraviolet regulator,and the Lagrangian formalism provides us with the standard tools of the Ward identities. All ofthis resulted in a very simple all-loop dual conformal anomalous Ward identity [33].One possible way to control the anomalies of scattering amplitudes would be to extend theabove mentioned duality to non-MHV amplitudes and to identify the dual object describing theratio function. Such an object should necessarily incorporate the particle helicities and it shouldbe a function of the supercoordinates of the n particles in the dual superspace Z i = ( x i , θ i )(with i = 1 , . . . , n ). In addition, as follows from our analysis, it should also have a number ofunusual features. First of all, due to the ¯ Q − anomaly, this object should not be super-Poincar´einvariant at loop level. Secondly, the collinear limit of the ratio function (3.77) suggests that,viewed as a function of Z i , the object dual to the n − particle ratio function should reduce tothe ( n − − particle one when three neighboring points lie on the same line in dual superspace, Z i +1 = zZ i +2 + (1 − z ) Z i . It is natural to expect that such an object would be defined on an n − gon in the dual superspace with vertices located at the points Z i (with i = 1 , . . . , n ). Webelieve that solving this problem may provide the key to the complete understanding of thesuperamplitudes, with their anomalous symmetries. Acknowledgments We would like to thank Nima Arkani-Hamed, Niklas Beisert, Zvi Bern, Lance Dixon, JamesDrummond, David Gross, David Kosower, Juan Maldacena, Edward Witten for interesting dis-cussions. ES would like to acknowledge the hospitality of the Galileo Galilei Institute (Florence)and of the Kavli Institute for Theoretical Physics (Santa Barbara), where part of this work wasdone. This work was supported in part by the French Agence Nationale de la Recherche undergrant ANR-06-BLAN-0142, by the CNRS/RFFI grant 09-02-00308 and by the National ScienceFoundation under Grant No. PHY05-51164. A Appendix: N = 4 superconformal symmetry Throughout the paper, we denote the generators of conventional ( q, ¯ q, s, ¯ s, . . . ) and dual super-conformal symmetry ( Q, ¯ Q, S, ¯ S, . . . ) by lower-case and upper-case letters, respectively. Theirexplicit expressions can be found in [1]. The nontrivial commutation relations for both symme-tries are { q Aα , q ˙ αB } = δ AB p α ˙ α , { s αA , s B ˙ α } = δ BA k α ˙ α , [ k α ˙ α , p β ˙ β ] = δ βα δ ˙ β ˙ α d + m αβ δ ˙ β ˙ α + m ˙ α ˙ β δ βα , [ p α ˙ α , s βA ] = δ βα q ˙ αA , [ k α ˙ α , q βA ] = δ βα s ˙ αA , { q αA , s βB } = m αβ δ AB + δ αβ r AB + δ αβ δ AB ( d + c ) , [ p α ˙ α , s ˙ βA ] = δ ˙ β ˙ α q Aα , [ k α ˙ α , q ˙ βA ] = δ ˙ β ˙ α s αA , { q ˙ αA , s B ˙ β } = m ˙ α ˙ β δ BA − δ ˙ α ˙ β r BA + δ ˙ α ˙ β δ BA ( d − c ) . (A.1)40ere we find the generators of translations p , conformal boosts k , Lorentz m, ¯ m , SU(4) rotations r , dilatation d and central charge c . As explained in [1], the conformal part of this algebra ( s , ¯ s , k ) can be obtained form thePoincar´e part ( q , ¯ q , p ) by applying the discrete operation of conformal inversion I , namely, k = IpI , s = I ¯ qI , ¯ s = IqI . Thus, it is often more convenient to check dual conformal symmetryby just looking at the properties of the amplitude under inversion in dual superspace. Here welist some of the inversion rules, needed in this paper (the details can be found in [1]): I [ x ij ] = x − i x ij x − j , I [ λ i ] = x i | λ i i p x i x i +1 = x i +1 | λ i i p x i x i +1 , I [˜ λ i ] = [˜ λ i | x i p x i x i +1 = [˜ λ i | x i +1 p x i x i +1 , (A.2)where ( x − ) α ˙ α ≡ x α ˙ α /x and we have used the standard bra/ket notation for spinors. It is easyto see that the contraction of two adjacent spinors h i i + 1 i is covariant, I [ h i i + 1 i ] = h i i + 1 i x i +1 p x i x i +2 , (A.3)while any other contraction h ij i (with j = i ) is not. This explains why the holomorphic helicity-free cross-ratio h ij ih kl i / h ik ih jl i mentioned in Section 2.1.1 cannot be dual conformal. Further,a typical example of a dual conformally covariant expression is the following string: I [ h i | x ij x jk | k i ] = h i | x ij x jk | k i x j q x i x i +1 x k x k +1 . (A.4)Because of the two equivalent form of the transformations of λ in (A.2), the above string remainscovariant if we replace h i | → h i − | or | k i → | k − i . B Appendix: Solution of the collinearity conditions Consider the collinearity condition (a generalization of (2.9)) F a,i,b ϕ ( λ, ˜ λ ) ≡ (cid:18) h ab i ∂∂ ˜ λ i + h bi i ∂∂ ˜ λ a + h ia i ∂∂ ˜ λ b (cid:19) ϕ ( λ, ˜ λ ) = 0 , ( a + 1 ≤ i ≤ b − . (B.1)Here a < b are the end points of a line segment in twistor space, and i = a + 1 , . . . , b − ϕ ( λ k , ˜ λ k ) = ϕ ( p k ), k = 1 , . . . , n .Any additional purely holomorphic dependence on λ does not affect the collinearity condition(here we neglect the possible holomorphic anomalies).For the sake of this argument, let us assume that we are in signature (+ + −− ), where we cantreat λ and ˜ λ as independent variables. Then, the collinearity condition (B.1) can be interpretedas the invariance of ϕ under simultaneous shifts of ˜ λ i , ˜ λ a and ˜ λ b ,˜ λ i → ˜ λ i + ˜ ǫ i h ab i , ˜ λ a → ˜ λ a + ˜ ǫ i h bi i , ˜ λ b → ˜ λ b + ˜ ǫ i h ia i ( a + 1 ≤ i ≤ b − 1) (B.2) The central charge of the superamplitudes vanishes, so we might as well consider the superalgebra psu (2 , | ǫ i . Let us choose ˜ ǫ i = − ˜ λ i / h ab i , so that the shifted ˜ λ i vanish. Then, ϕ ( p , . . . , p a , . . . , p b , . . . , p n ) = ϕ ( p , . . . , p ′ a , , . . . , , p ′ b , . . . , p n ) , (B.3)where p ′ i = 0 for a + 1 ≤ i ≤ b − p ′ a = − | a ih b | x a b +1 h ab i , p ′ b = | b ih a | x a b +1 h ab i , x a b +1 = p a + . . . + p b . (B.4)We verify that p ′ a + p ′ b = p a + . . . + p b , as it should be. In other words, the collinearity condition(B.1) means that the function ϕ can depend on ˜ λ i in the range a ≤ i ≤ b only via the dualcoordinates x a b +1 = x a − x b +1 .In the special case of the collinearity condition (2.9), we set a = 1, b = n and we find p + . . . + p n = 0, hence p ′ = p ′ n = 0. Thus, the only solution to (2.9) is ϕ = const.Now, let us turn to the case considered in Section 2.2.1, where we modified the known bosonichelicity structure f st by a helicity-free and dual conformally invariant function ϕ ( λ, ˜ λ ), see (2.34).As explained in Section 2.1.1, we can restrict it to be a function of the momenta, ϕ ( p , . . . , p n ).This function, like f st itself, should simultaneously satisfy the three collinearity conditions (2.33).The particularity here is that two of the lines intersect at point 1. Repeating the steps aboveand taking care to shift ˜ λ along the lines (1 , s − 1) and ( t, 1) at the same time, we obtain thefollowing general solution: ϕ ( p ′ , , . . . , , p ′ s − , p ′ s , , . . . , , p ′ t − , p ′ t , , . . . , . (B.5)Here the five light-like non-vanishing momenta are defined by p ′ = | ih s − | x s h s − i + | ih t | x t h t i p ′ s − = | s − ih | x s h s − i , p ′ s = | s ih t − | x ts h s t − i (B.6) p ′ t − = | t − ih s | x st h s t − i , p ′ t = | t ih | x t h t i , and they satisfy the conservation condition p ′ + p ′ s − + p ′ s + p ′ t − + p ′ t = 0 . (B.7)From them we can form five independent Lorentz invariant dot products. By inspecting thevarious combinations, we find five such dot products, which are dual conformal (up to irrelevantholomorphic factors depending only on λ ): p ′ · p ′ s − ∼ h | x s x st | t i , p ′ s · p ′ s − ∼ h | x s x st | t − i p ′ t · p ′ t − ∼ h | x t x ts | s i , p ′ · p ′ t ∼ h | x t x ts | s − i p ′ s · p ′ t − ∼ x st . (B.8)All of them appear in the bosonic function f st (2.27) and give it the necessary dual conformalweights.The additional factor ϕ ( p k ) in (2.34) must then be a dual conformally invariant function of thefive variables (B.8). However, the latter involve only five points in dual space, 1 , s − , s, t − , t ,from which it is impossible to build a conformal cross-ratio. Thus, we conclude that ϕ = const. The six dot products of the four independent momenta satisfy a linear relation following from the light-likenessof the fifth vector in (B.7). Appendix: Conformal s − supersymmetry of the NMHVtree superamplitude Let us show that the NMHV tree superamplitude satisfies the relation[∆ α − ( s − λ ˇ2 α − ( t − s − λ ˆ2 α ] ˜ A st = 0 , (C.1)where ˜ A st is given by (2.35) and the operator ∆ α has the form∆ α = s − X h ˇ2 i i ∂∂λ αi + t − X s h ˆ2 i i ∂∂λ αi (C.2)with spinors ˇ2 α and ˆ2 α defined in (2.36). The differential operator ∆ α has the following properties:∆ α h ˇ2 i i = 0 , (1 ≤ i ≤ s − α h ˆ2 i i = 0 , ( s ≤ i ≤ t − α n X λ βi ˜ λ ˙ βi = 0 , (C.3)which allow us to show that all the delta functions in (2.35) are annihilated by ∆ α leading to∆ α ˜ A st = ˜ A st ∆ α ln (cid:0) h n i h i f st (cid:1) , (C.4)with f st given by (2.27). Further properties of ∆ α include∆ α ln h k k + 1 i = − ˇ2 α for 1 ≤ k ≤ s − − ˆ2 α for s ≤ k ≤ t − 20 for t ≤ k ≤ n (C.5)together with ∆ α ln x st = − ∆ α ln h ˆ2 t i = ∆ α ln h | x t x − ts | s − ih i = − ˆ2 α ∆ α ln h | x t x − ts | s ih i = ˇ2 α ∆ α ln h ˆ2 t − i = 0 . (C.6)Applying these relations, we obtain from (C.4)∆ α ln (cid:0) h n i h i f st (cid:1) = ( s − α + ( t − s − α . (C.7)Substituting this result into (C.4) yields (C.1).We conclude by remarking that although the q -supersymmetry gauge ¯ η n = ¯ η = 0 simplifiesthe computation, one can verify (2.38) without fixing the gauge. In this case the operator s αA has an additional term compared to (2.37), s αA = (cid:18) ¯ η A + ¯ η A h n ih n i + ¯ η nA h ih n i (cid:19) [∆ α − ( s − α − ( t − s − α ]+ ¯ η nA λ α − ¯ η A λ nα h n i n X i =1 λ αi ∂∂λ αi + 2( n − ! . (C.8)43he counting operator in the second line gives 4 when applied to h n i h i f st and ( − 4) whenapplied to the momentum conservation delta function, so (2.38) still holds. D Appendix: Collinear limit of the n = 6 NMHV super-amplitude In this Appendix we show that the collinear limit of the six-gluon NMHV amplitude (3.74) followsfrom the analogous relation for the n = 6 NMHV superamplitude (3.76).The tree-level n = 6 NMHV superamplitude has the form [1, 15] A NMHV;06 = A MHV;06 ( R + R + R ) . (D.1)Replacing the superinvariants by their explicit expressions (2.21), we find after some algebra A NMHV;06 = δ (4) ( X λ i ˜ λ i ) δ (8) ( X η i λ i ) (cid:20) δ (4) ( η [56] + η [64] + η [45]) x h ih i [45][56] h | x | h | x | δ (4) ( η [12] + η [26] + η [61]) x h ih i [61][12] h | x | h | x | 2] + δ (4) ( η [34] + η [42] + η [23]) x h ih i [23][34] h | x | h | x | (cid:21) , (D.2)where the three terms inside the square brackets correspond to the three terms in (D.1). Weexpect that in the collinear limit this expression should reduce to the n = 5 NMHV superampli-tude [1, 15] A NMHV;05 = δ (4) ( X λ i ˜ λ i ) δ (8) ( X η i λ i ) δ (4) ( η [34] + η [42] + η [23]) h i [12][23][34][45][51] . (D.3)Let us examine (D.2) in the collinear limit 6 k i = 6. We observe that thefirst term in (D.2) remains regular in this limit, while the two other terms contain the vanishingfactors [61] and h i in the denominator. However, examining the argument of the Grassmanndelta-function in the second term in (D.2) we find that it vanishes in the collinear limit η [12] + η [26] + η [61] k → √ z ¯ z η ℓ [ ℓ 2] + √ z ¯ z η ℓ [2 ℓ ] + √ z ¯ z η [ ℓℓ ] = 0 , (D.4)with ¯ z = 1 − z . Therefore, the dominant contribution to A NMHV;06 only comes from the last termin (D.2), leading to A NMHV;06 6 k → h i√ z ¯ z δ (4) ( X λ i ˜ λ i ) δ (8) ( X η i λ i ) δ (4) ( η [34] + η [42] + η [23]) h ℓ i [ ℓ ℓ ] , (D.5)where the sums inside the delta functions run over i = ℓ, , ..., 5. Comparing this relation with(D.3), we conclude that A tree6 (1 , , , , , k → h i√ z ¯ z A tree5 ( ℓ, , , , , (D.6)in agreement with (3.76). 44et us now apply (D.2) to reproduce the collinear limit of the six-gluon helicity-split amplitude(3.74). Using (1.1), we find that this amplitude appears as a particular term in the expansion ofthe n = 6 NMHV superamplitude, A NMHV6 = A (1 + + + − − − )( η ) ( η ) ( η ) + . . . , (D.7)where ( η ) ≡ ǫ ABCD η A η B η C η D / 4! and the ellipsis denote terms describing the remaining six-pointNMHV amplitudes. The latter include six-gluon NMHV color-ordered amplitudes with differentordering of negative and positive helicities as well as six-particle NMHV amplitudes involvinggluinos and scalars.Let us now expand both sides of (D.2) in powers of η ’s and identify one particular termdisplayed in the right-hand side of (D.7). The first term in the right-hand side of (D.2) is regularin the collinear limit and, therefore, can be discarded. Then, the term ∼ ( η ) ( η ) ( η ) can beeasily extracted from (D.2) by making use of the identities δ (8) ( X η i λ i ) δ (4) ( η [12] + η [26] + η [61]) = ([12] h i ) ( η ) ( η ) ( η ) + . . . ,δ (8) ( X η i λ i ) δ (4) ( η [34] + η [42] + η [23]) = ([23] h i ) ( η ) ( η ) ( η ) + . . . , (D.8)where expressions in the right-hand side are homogenous polynomials in η ’s of degree 12 andthe ellipses denote remaining terms. Substituting (D.8) into (D.2) and making use of (D.7), wefind that the second and third term in the right-hand side of (D.2) provide the contribution to A NMHV;06 (1 + , + , + , − , − , − ) which coincides, respectively, with the first and second term inthe right-hand side of (3.74).Let us now examine the first relation in (D.8) in the collinear limit 6 k 1. We recall that, invirtue of (D.4), the Grassmann delta function in the left-hand side vanishes in this limit and,as a consequence, the second term in (D.2) does not contribute to the singular behavior of thesuperamplitude (D.5). It is easy to see however that the first term in the right-hand side of thisrelation is different from zero in the limit 6 k δ (4) ( η [12] + η [26] + η [61]) k → [12] ( η ) + [12] [26]( η ) ( η ) + . . . + [26] ( η ) → , (D.9)where we applied (3.75) for i = 6. Here, in the second relation each term represents a particularthree-particle on-shell state which can be identified using (1.1). For instance, the term ( η ) describes a three-gluon state G − G +1 G +2 , the term ( η ) η describes an antigluino-gluino-gluonstate ¯Γ Γ G +2 , and so on. Each of these states contributes to the right-hand side of the firstrelation in (D.8) but their sum vanishes. Thus, the second term in (D.2) produces an MHV-typecontribution to the collinear limit of the six-gluon NMHV amplitude, but it cancels inside theNMHV super amplitude after one takes into account the contribution of the six-point NMHVamplitudes, in which the two collinear gluons G − G +1 are replaced by a pair of two particles(gluons, gluinos, scalars) with the total helicity and the SU(4) charge equal to zero.45 Appendix: ¯ Q − anomaly of the n = 6 NMHV superam-plitude The n = 6 NMHV superamplitude has three-particle cuts and its discontinuity in, say, s =( p + p + p ) has the following form at one loopDisc s A NMHV;16 = A NMHV;05 ( − ℓ , , , , − ℓ ) ⋆ A MHV;05 ( ℓ , , , , ℓ )+ A MHV;05 ( − ℓ , , , , − ℓ ) ⋆ A NMHV;05 ( ℓ , , , , ℓ ) , (E.1)where we used the same notations as in (4.1). Here the second term can be obtained from thefirst one through shift of indices i i + 3. Explicit expressions for 5-particle superamplitudesentering (E.1) are A NMHV;05 ( − ℓ , , , , − ℓ ) = δ (4) ( X λ i ˜ λ i ) δ (8) ( X λ i η i ) δ (4) ( η [23] + η [31] + η [12]) h ℓ ℓ i [ ℓ ℓ ][ ℓ ℓ ] , A MHV;05 ( ℓ , , , , ℓ ) = δ (4) ( X λ i ˜ λ i ) δ (8) ( X λ i η i ) 1 h ℓ ih ih ih ℓ ih ℓ ℓ i , (E.2)where the sum in the argument of delta functions runs over all external particles and the con-ventions are used λ − ℓ i = − λ ℓ i and ˜ λ − ℓ i = ˜ λ ℓ i .The tree-level superamplitudes respect dual superconformal symmetry but for A NMHV6 thissymmetry is broken at loop level by infrared divergences. Invoking the same arguments as forone-loop MHV superamplitude, we can use (E.1) to argue that Disc s A NMHV;16 is free from dualconformal K − anomaly. Going to all loops and to an arbitrary number of external particles, wewrite A NMHV n in a close analogy with (4.5) as A NMHV n = A MHV;0 n W NMHV n . (E.3)In distinction with (4.5) the function W NMHV n is not related to light-like Wilson loop W n . Never-theless, W NMHV n satisfies the same Ward identities as W n , Eq. (4.6) and (4.7). As a consequence,Disc x ,j +1 ln W NMHV n is invariant under dual conformal transformations.Let us now turn to dual supersymmetry and apply the operator ¯ Q A ˙ α = P η Ai ∂ i, ˙ α to both sidesof (E.1)¯ Q A ˙ α (Disc s A NMHV;16 ) = ¯ Q A ˙ α A NMHV;05 ( − ℓ , , , , − ℓ ) ⋆ A MHV;05 ( ℓ , , , , ℓ ) (E.4)+ A NMHV;05 ( − ℓ , , , , − ℓ ) ⋆ ¯ Q A ˙ α A MHV;05 ( ℓ , , , , ℓ ) + ( i → i + 3) . As before, the right-hand side of this relation is different from zero only due to holomorphicanomaly. It is generated by angular brackets h . . . i in the denominator of tree-level superam-plitudes. The superamplitude A NMHV;05 ( − ℓ , , , , − ℓ ), Eq. (E.2), has only one such factor h ℓ ℓ i and it provides the contribution to ¯ Q A ˙ α A NMHV;05 localized at ℓ ∼ ℓ or equivalenltly( ℓ + ℓ ) = s = 0. Similar to MHV case, this contribution is suppressed by the addi-tional factor h ℓ ℓ i coming from integration over η ℓ i , Eq. (4.14). As a result, the first termin the right-hand side of (E.4) does not contribute to the anomaly. The MHV superamplitude A MHV;05 ( − ℓ , , , , − ℓ ), Eq. (E.2), has two factors in the denominator, h ℓ i and h ℓ i , whosecontribution is localized at the kinematical configurations with ℓ ∼ p and ℓ ∼ p , respectively.46e can further simplify the calculation by writing down n = 5 NMHV superamplitude en-tering the second term in (E.4) in the factorized form A NMHV;05 ( − ℓ , , , , − ℓ ) = A MHV;05 ( − ℓ , , , , − ℓ ) × M ( ℓ , ℓ ) δ (4) ( η [23] + η [31] + η [12]) (E.5)with M ( ℓ , ℓ ) = h ℓ ih ih ih ℓ ih ℓ ℓ i [ ℓ ℓ ][ ℓ ℓ ] = 1 P h ih i [12][23] h | ℓ ℓ | i [1 | ℓ ℓ | 3] (E.6)and P = ℓ + ℓ = p + p + p . Substituting this relation into (E.4) and taking into account that M ( ℓ , ℓ ) is free from the holomorphic anomaly, we find that evaluation of (E.4) is analogous tothat for MHV superamplitude (4.9) with the only difference that contribution of each kinematicalconfiguration like (4.16) is multiplied by the additional factor M ( ℓ , ℓ ). For ℓ ∼ p and ℓ ∼ p this factor can be further simplified leading, respectively, to M = 1 x h ih ih i [12][23][34] h | x | | x | i , M = 1 x h ih ih i [61][12][23] h | x | | x | i . (E.7)Combining these relations with (4.18) (evaluated for j = 3 and n = 6) we obtain¯ Q A ˙ α (cid:18) Disc x A NMHV;16 (cid:19) = − πic Γ A MHV;06 δ (4) ( η [23] + η [31] + η [12]) (E.8) × " ˜ λ α Ξ A h iM h ih | x | | x | i − ˜ λ α Ξ A h iM h ih | x | | x | i + ( i → i + 3) . Using the definition (2.22) we replace Ξ − functions by their explicit expressionsΞ A h ih i = Ξ A h ih i = − ( η [56] + η [46] + η [45]) A (E.9)and, finally, obtain the expression (4.19) for the ¯ Q − anomaly of one-loop n = 6 NMHV superam-plitude.Next, we turn to the anomaly of the ratio function. It follows from the definition (1.6) that A NMHV;16 = R NMHV;16 A MHV;06 + R NMHV;06 A MHV;16 . (E.10)We apply ¯ Q A ˙ α to both sides of this relation and take into account ¯ Q − invariance of tree amplitudesto get (cid:16) ¯ Q A ˙ α R NMHV;16 (cid:17) A MHV;06 = ¯ Q A ˙ α A NMHV;16 − R NMHV;06 (cid:16) ¯ Q A ˙ α A MHV;16 (cid:17) . (E.11)Finally, we take discontinuity with respect to x in the both sides of this relation, substitute(4.19) and (4.20) and replace the tree-level ratio function R NMHV;06 by its explicit expression,Eqs. (2.25) and (3.33), to get¯ Q A ˙ α (cid:18) Disc x R NMHV;16 (cid:19) = 2 πic Γ ( η [56] + η [64] + η [45]) A δ (4) ( η [23] + η [31] + η [12]) × h ih ih ih i x [12][23] ˜ λ α (1 + ξ )[1 | x | i [34][45] + ˜ λ α (1 + ξ )[3 | x | i [56][61] ! + ( i → i + 3) . (E.12)47ere the notation was introduced for ξ = x [34] h ih | x | h | x | , ξ = x h i [61] h | x | h | x | , (E.13)such that ξ /ξ = M / M (see Eq. (E.7)). Notice that ξ and ξ are invariant under dualconformal transformations and going through some algebra we find ξ = ξ = ξ = x x x x x x − x x x + x x x − x x x . (E.14)In addition, ξ is invariant under shift of indices i i +3 and, therefore, (1+ ξ ) can be factored outin the right-hand side of (E.12). Then, the remaining expression is proportional to the coefficientin front of A MHV;06 in (4.19). In this way, we arrive at the relation (4.21). References [1] J. M. Drummond, J. Henn, G. P. Korchemsky and E. 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