From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries

A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N=4 and 6d N=(1,1) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N=4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of N=(1,1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N=(1,1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five- and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N=4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N=(1,1) SYM theory. We exploit this correspondence to construct the super-Poincare and enhanced dual conformal symmetries of massive tree superamplitudes in N=4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.

Abstract: A self-consistent exposition of the theory of tree-level superamplitudes of the 4d N = 4 and 6d N = (1, 1) maximally supersymmetric Yang-Mills theories is provided. In 4d we work in non-chiral superspace and construct the superconformal and dual superconformal symmetry generators of the N = 4 SYM theory using the non-chiral BCFW recursion to prove the latter. In 6d we provide a complete derivation of the standard and hidden symmetries of the tree-level superamplitudes of N = (1, 1) SYM theory, again using the BCFW recursion to prove the dual conformal symmetry. Furthermore, we demonstrate that compact analytical formulae for tree-superamplitudes in N = (1, 1) SYM can be obtained from a numerical implementation of the supersymmetric BCFW recursion relation. We derive compact manifestly dual conformal representations of the five-and six-point superamplitudes as well as arbitrary multiplicity formulae valid for certain classes of superamplitudes related to ultra-helicity-violating massive amplitudes in 4d. We study massive tree superamplitudes on the Coulomb branch of the N = 4 SYM theory from dimensional reduction of the massless superamplitudes of the six-dimensional N = (1, 1) SYM theory. We exploit this correspondence to construct the super-Poincaré and enhanced dual conformal symmetries of massive tree superamplitudes in N = 4 SYM theory which are shown to close into a finite dimensional algebra of Yangian type. Finally, we address the fascinating possibility of uplifting massless 4d superamplitudes to 6d massless superamplitudes proposed by Huang. We confirm the uplift for multiplicities up to eight but show that finding the uplift is highly non-trivial and in fact not of a practical use for multiplicities larger than five.

Introduction
Scattering amplitudes of maximally supersymmetric Yang-Mills theories in 3, 4, 6 and 10 dimensions possess remarkable properties. Next to their constitutional maximally extended super-Poincaré symmetries they all enjoy a hidden dual conformal symmetry -at least at the tree-level [1][2][3][4][5]. The four dimensional N = 4 super Yang-Mills (SYM) theory is distinguished in this series as it also has superconformal symmetry in the standard sense. The standard superconformal symmetry then further enhances the dual conformal symmetry to a dual superconformal symmetry [2,3]. On top the closure of the two sets of superconformal symmetry algebras leads to an infinite dimensional symmetry algebra of Yangian type [6]. It is the manifestation of an underlying integrable structure in planar N = 4 SYM. The key to the discoveries of these rich symmetry structures of maximally supersymmetric Yang-Mills theories in various dimensions is the use of a suitable on-shell superspace formalism along with spinor helicity variables to package the component field amplitudes into superamplitudes, which was pioneered in 4d in [7]. In this work we shall focus on the four and six dimensional maximally theories: The 4d N = 4 SYM and the 6d N = (1, 1) SYM models. While the massless tree amplitudes of 4d N = 4 SYM are very well studied and in fact known analytically [8], not so much is known about the massive amplitudes on the Coulomb branch of this theory. These amplitudes are obtained by giving a vacuum expectation value to the scalar fields and yield -arguably -the simplest massive amplitudes in four dimensions. Alternatively, these massive amplitudes arise from the amplitudes of the maximally supersymmetric 6d N = (1, 1) SYM theory upon dimensional reduction, where the higher dimensional momenta yield the masses in the 4d theory. Indeed, compact arbitrary multiplicity amplitudes for particular subclasses of Coulomb branch amplitudes have been obtained in [9] by making use of modern on-shell techniques. The massive 4d N = 4 SYM amplitudes are invariant under a dual conformal symmetry which is inherited from the 6d N = (1, 1) SYM theory as shown in [4]. Moreover, this symmetry remains intact also at loop-level if one restricts the loop-momentum integrations to a four-dimensional subspace. This prescription is equivalent to the Higgs regularization for infrared divergences in 4d proposed in [10], where such an extended dual conformal invariance was conjectured and tested at the one-loop four-point level. The dimensional reduction of 6d N = (1, 1) SYM to four dimensions yields N = 4 superamplitudes expressed on a non-chiral superspace [11] which is distinct to the usual chiral superspace of [7]. In this work we explicitly construct all generators of the standard and dual (super) conformal symmetry generators acting in the non-chiral N = 4 on-shell superspace as well as in the N = (1, 1) on-shell superspace. We also determine the standard and dual symmetries of massive N = 4 amplitudes as they are induced from an enhanced super-Poincaré and enhanced dual conformal symmetry of the 6d N = (1, 1) SYM theory.
The most efficient method to analytically construct tree-level amplitudes is based on an on-shell recursive technique due to Britto, Cachazo, Feng and Witten (BCFW) [12,13]. In contrast to the earlier Berends-Giele off-shell recursion relations [14], the BCFW relation uses only on-shell lower-point amplitudes, evaluated at complex, shifted momenta. The BCFW recursion relation is easily generalizable to an on-shell recursion for superamplitudes, as was done for N = 4 SYM in [15] (see also [16]). In fact the knowledge of the dual superconformal invariance of superamplitudes motivates an ansatz in terms of dual conformal invariants. Together with the super BCFW recursion this allowed for the complete analytic solution [8]. In fact the variant of the BCFW recursion for 4d N = 4 SYM in non-chiral superspace has not been written down before and we will do so in this work. The BCFW recursion for 6d N = (1, 1) SYM theory was established in [17,18] and tree-level amplitudes of multiplicities up to five were derived. The one loop corrections were obtained in [19]. In this work we point out how a numerical implementation of the BCFW recursion for N = (1, 1) SYM amplitudes in combination with a suitable set of dual conformal invariant basis functions may be used to derive compact five and six-point amplitudes as well as arbitrary multiplicity amplitudes for certain subclasses related to the 4d amplitudes with two neighboring massive legs mentioned above [9]. In fact, the method we propose is very general and could be applied to further cases as well.
A very tempting option to obtain massive 4d amplitudes of N = 4 SYM was introduced by Huang in [11]. He indicated that it should be possible to invert the dimensional reduction of N = (1, 1) to massive N = 4 by uplifting the massless non-chiral superamplitudes of N = 4 SYM to six-dimensional superamplitudes of N = (1, 1) SYM. Non-chiral superamplitudes of N = 4 SYM are straightforward to obtain using the non-chiral BCFW recursion, resulting in an eminent practical relevance of a potential uplift. It is indeed very surprising that in fact the massive Coulomb branch amplitudes or equivalently the six-dimensional amplitudes might not contain any more information than the massless four-dimensional amplitudes of N = 4 SYM.
It is the aim of this paper to provide a self-consistent and detailed exposition of the theory of superamplitudes for 4d N = 4 SYM and 6d N = (1, 1) SYM. The paper is organized as follows. We discuss the needed spinor helicity formalisms in section 2. Section 3 and 4 are devoted to the on-shell superspaces of both theories and the standard and hidden symmetries of the associated superamplitudes. In section 5 we discuss the dimensional reduction from massless 6d to massive 4d amplitudes and establish the inherited (hidden) symmetries of the 4d amplitudes. Section 6 exposes the on-shell BCFW recursion relations for N = 4 SYM in non-chiral superspace as well as for N = (1, 1) SYM. We also provide a proof of dual conformal symmetry of N = (1, 1) superamplitudes thereby correcting some minor mistakes in the literature. Finally in section 8 we analyze in detail the proposal of Huang for uplifting 4d massless N = 4 superamplitudes in non-chiral superspace to 6d N = (1, 1) superamplitudes and point out why this uplift is non-trivial and in fact not of a real practical use for multiplicities larger than five. Notational details and extended formulae are relegated to the appendices.

General remarks
Calculating scattering amplitudes of massless particles, the spinor helicity formalism has become a powerful tool in obtaining compact expressions for tree-level and one-loop amplitudes. The basic idea is to use a set of commuting spinor variables instead of the parton momenta {p i }. These spinors trivialize the on-shell conditions for the momenta (p i ) 2 = 0 .
(2. 1) In what follows we will briefly review the spinor helicity formalism in four and six dimensions. Additional details and conventions can be found in appendix A.

Four dimensions
The starting point of the spinor helicity formalism in four dimensions [20][21][22][23], which we briefly review here, is to express all momenta by (2 × 2) matrices p αα = σ µ αα p µ , pα α =σ µαα p µ , or inversely p µ = 1 2 p αασα α µ = 1 2 pα α σ µ αα , (2.2) where we take σ µ = (1, σ) andσ µ = (1, − σ) with σ being the Pauli matrices. Raising and lowering of the α andα indices may be conveniently defined by left multiplication with the antisymmetric symbol for which we choose the following conventions 12  Besides being related by p αα = αβ αβ pβ β = pα α , these matrices satisfy p 2 = det(p αα ) = det(p αα ), p αα pα β = p 2 δ β α and pα α p αβ = p 2 δβ α . Hence, the matrices pα α and p αα have rank one for massless momenta, implying the existence of chiral spinors λ α and anti-chiral spinorsλα solving the massless Weyl equations These spinors can be normalized such that For complex momenta the spinors λ andλ are independent. However, for real momenta we have the reality condition p * αβ = pα β , implyingλα = c λ * α for some c ∈ R. Hence, the spinors can be normalized such that An explicit representation is with λα = sign(p 0 + p 3 ) λ * α . Obviously, eq. (2.5) is invariant under the SO(2) little group transformations λ α → zλ α ,λα → z −1λα with |z| = 1 . The little group invariant scalar products of massless momenta are then given by a product of two spinor brackets 2p i p j = p i αα pα α j = i j [j i] . (2.10) The spinor helicity formalism allows for a compact treatment of polarizations. Each external gluon carries helicity h i = ±1 and a momentum specified by the spinors λ i andλ i . Given this data the associated polarization vectors are where (q i ) αα = µ α iμα i are auxiliary light-like momenta reflecting the freedom of on-shell gauge transformations. It is straightforward to verify that the polarization vectors fulfill as well as the completeness relation A summary of all our conventions for four dimensional spinors can be found in appendix A.

Six dimensions
Similar to four dimensions, the six-dimensional spinor-helicity formalism [24] provides a solution to the on-shell condition p 2 = 0 for massless momenta by expressing them in terms of spinors. As a first step one uses the six-dimensional analog of the Pauli matrices Σ µ and Σ µ to represent a six-dimensional vector by an antisymmetric 4 × 4 matrix p AB = p µ Σ µ AB , p AB = p µ Σ µ AB , or inversely p µ = 1 4 p AB Σ µ BA = 1 4 p AB Σ µ BA . (2.14) Besides being related by p AB = 1 2 ABCD p CD , these matrices satisfy p AB p BC = δ C A p 2 and det(p AB ) = det(p AB ) = (p 2 ) 2 . Hence, for massless momenta, p AB and p AB have rank 2 and therefore the chiral and anti-chiral part of the Dirac equation have two independent solutions, labeled by their little group indices a = 1, 2 andȧ = 1,2 respectively. Raising and lowering of the SU (2) × SU (2) little group indices may be conveniently defined by contraction with the antisymmetric tensors ab and ȧḃ The anti-symmetry of p AB and p AB together with the on-shell condition p AB p BC = 0 yields the bispinor representation An explicit representation of the chiral and anti-chiral spinors is given by As a consequence of the properties of the six-dimensional Pauli matrices, the spinors are subject to the constraint It is convenient to introduce the bra-ket notation By fully contracting all SU (4) Lorentz indices it is possible to construct little group covariant and Lorentz invariant objects. The simplest Lorentz invariants are the products of chiral and anti-chiral spinors These little group covariant spinor products are related to the little group invariant scalar products by The spinor products are 2 × 2 matrices whose inverse is Each set of four linear independent spinors labeled by i, j, k, l can be contracted with the antisymmetric tensor, to give the Lorentz invariant four brackets [iȧj˙bkċlḋ] = ABCDλ i Aȧλ j Bḃλ k Cċλ l Dḋ = det(λ iȧλ jḃλ kċλ lḋ ) . (2.25) Note that in the above expressions the 4x4 matrix appearing in the determinants is defined through its four columns vectors {λ a i λ b j λ c k λ d l } and similarly for the second expression. The four brackets are related to the spinor products by where I k = (i k ) a k , J k = (j k )ȧ k are multi indices labeling the spinors. Finally, it is convenient to define the following Lorentz invariant objects Similar to the four dimensional case, the polarization vectors of the gluons can be expressed in terms of spinors by introducing some light-like reference momentum q with q · p = 0, where p denotes the gluon momentum. The four polarization states are labeled by SO(4) SU (2) × SU (2) little group indices and can be defined as It is straightforward to verify the properties as well as the completeness relation (2.32) -6 -3 Four-dimensional N = 4 SYM theory

On-shell superspaces and superamplitudes
Dealing with scattering amplitudes of supersymmetric gauge theories is most conveniently done using appropriate on-shell superspaces. Most common for treating N = 4 super Yang-Mills theory are [7,25,26] chiral superspace: The Grassmann variables η A i ,η iA transform in the fundamental, anti-fundamental representation of SU (4) and can be assigned the helicities with h i denoting the helicity operator acting on leg i. With their help it is possible to decode the sixteen on-shell states into a chiral or an anti-chiral superfield Φ (η), Φ (η), defined by As a consequence of eq. (3.2) the super fields carry the helicities The chiral and anti-chiral superfield are related by a Grassmann Fourier transformation Chiral and anti-chiral color ordered superamplitudes A n can be defined as functions of the respective superfields Due to eq. (3.7) both superamplitudes are related by a Grassmann Fourier transformation The superamplitudes are inhomogeneous polynomials in the Grassmann odd variables η A i , η i A , whose coefficients are given by the color ordered component amplitudes. A particular component amplitude can be extracted by projecting upon the relevant term in the η i expansion of the super-amplitude via 11) and similar in anti-chiral superspace. By construction the chiral and anti-chiral superamplitudes have a manifest SU (4) R symmetry. The only SU (4) R invariants are contractions with the epsilon tensor Consequently the appearing powers of the Grassmann variables within the superamplitudes need to be multiples of four. As a consequence of supersymmetry the superamplitudes are proportional to the supermomentum conserving delta function with the chiral q αA = i λ α i η A i or anti-chiral conserved supermomentumqα A = iλα iη i A . Since the Grassmann variables carry helicity, eq. (3.2), their powers keep track of the amount of helicity violation present in the component amplitudes. Hence, decomposing the superamplitudes into homogeneous polynomials is equivalent to categorizing the component amplitudes according to their degree of helicity violation (3.14) The highest amount of helicity violation is present in the maximally helicity violating (MHV) superamplitude or in the MHV superamplitude in anti-chiral superspace. In gen- are the (Next to) p MHV and the (Next to) p MHV superamplitudes . The complexity of the amplitudes is increasing with the degree p of helicity violation, the simplest being the MHV superamplitude in chiral superspace [7] A MHV , (3.17) and the MHV superamplitude in anti-chiral superspace , (3.18) which are supersymmetric versions of the well known Parke-Taylor formula [27]. The increasingly complicated formulae for the amplitudes A N p MHV n have been obtained in reference [8]. Plugging the MHV decomposition, eq. (3.14), into eq. (3.9) we obtain the relation simply stating that A N p MHV n and A N n−4−p MHV n contain the same component amplitudes. Depending on whether p < n − 4 − p or p > n − 4 − p it is therefore more convenient to use the chiral or the anti-chiral description of the amplitudes, e. g. the N n−4 MHV = MHV amplitudes are complicated in chiral superspace whereas they are trivial in anti-chiral superspace. Hence the most complicated amplitudes appearing in an n point chiral or anti-chiral superamplitude are the helicity amplitudes of degree p = n 2 −2, called minimal helicity violating (minHV) amplitudes .

Non-chiral superspace
Besides the well studied chiral and anti-chiral superspaces there is as well the non-chiral superspace 20) which is more natural from the perspective of the massive amplitudes and the six dimensional parent theory that we are interested in. Here the SU (4) indices of the fields get split into two SU (2) indices m and m according to Note that the due antisymmetry the fields φ mn = −φ nm and φ m n = −φ n m represent only one scalar field respectively, whereas the φ mn = −φ n m account for the four remaining scalars. If raising and lowering of the SU (2) indices are defined by left multiplication with = iσ 2 and −1 , the non-chiral superfield reads with the abbreviations η 2 = 1 2 η m η m ,η 2 = 1 2η m η m . The non-chiral superfield is a scalar and has zero helicity. Obviously, the non-chiral superamplitudes will not have a SU (4) R symmetry, but rather will be invariant under SU (2, 2) transformations. With the convention m ∈ {1, 4}, m ∈ {2, 3} the non-chiral superfield is related to the chiral and anti-chiral superfield by the half Grassmann Fourier transformations with the conserved supermomenta q m α = i η m i λ i α andq m α = iη m iλ iα . Since we additionally have h i Υ i = 0, the non-chiral superamplitudes have the general form It should be stressed that the dependence of f n only on the momenta {p i , q i ,q i } is distinct to the situation for the chiral or anti-chiral superamplitudes, where we have a dependence on the super-spinors Analyzing the half Fourier transform (3.23) relating the superfields we see that the non-chiral superamplitudes are homogeneous polynomials in the variables q i andq i of degree 2n and the MHV decomposition (3.14) of the chiral superamplitudes translates to a MHV decomposition of the non-chiral superamplitudes (3.26) where the N p MHV sector corresponds to a fixed degree in the variables q i andq i This reflects the chiral nature of N = 4 SYM theory.
Each of the three superspaces presented above has an associated dual superspace. In general, dual superspaces naturally arise when studying dual conformal properties of color ordered scattering amplitudes. Part of the spinor variables get replaced by the region momenta x i , which are related to the ordinary momenta of the external legs by and a new set of dual fermionic variables θ i orθ i is introduced, related to the fermionic momenta by Obviously, the amplitudes will depend on differences of dual variables , as the dual variables are only defined up to an overall shift. With the identifications x 1 = x n+1 , θ 1 = θ n+1 , andθ 1 =θ n+1 , the dual variables trivialize the momentum and supermomentum conservation. The dual chiral superspace is given by with the constraints Analogously, the dual anti-chiral superspace is given by with the constraints In the case of the dual non-chiral superspace it is possible to completely eliminate all spinor variables and express the superamplitudes solely with the dual variables which are subject to the constraints Note that x 2 i i+1 = 0 is a consequence of eq. (3.35). In fact the Grassmann even dual variables y mm i are not independent as they can be expressed by Hence, the amplitudes will not depend on them. However, the variables y mm i are necessary for the construction of the dual non-chiral superconformal symmetry algebra presented in section 3.3 and appendix B .
A further possibility is to study superamplitudes using the full superspaces obtained by adding the dual variables to the chiral, anti-chiral and non-chiral superspaces. The full chiral superspace is given by with the constraints Analogously, the full anti-chiral superspace has the variables subject to the constraints Finally, the full non-chiral superspace is given by with the constraints

Symmetries of non-chiral superamplitudes
We are going to give a complete derivation of the symmetry generators of the non-chiral superamplitudes at tree level, which has not yet been done in full detail in the literature. Part of the results presented here can be found in reference [11]. For recent textbook treatments of the superconformal and dual superconformal symmetry of the chiral superamplitudes see [28,29]. A detailed presentation of the non-chiral superconformal algebra and its relevant representations is given in appendix B.

Superconformal symmetry of non-chiral superamplitudes
Due to the half Fourier transformation connecting the non-chiral and the chiral superspace, the SU (4) R symmetry is turned into an SU (2, 2) R symmetry. The conformal symmetry does not involve Grassmann variables, hence the tree-level non-chiral superamplitudes are invariant under the conformal algebra su(2, 2), with generators {pα α , m αβ , mαβ, d, k αα } . and their conjugates All other symmetry generators now follow from the non-chiral superconformal symmetry algebra listed in appendix B. Commuting the supersymmetry generators q αn ,qα n , qα n ,q with the conformal boost generator k αα yields the superconformal generators The central charge c and the hypercharge b are given by: (3.46) As already stated at the beginning, the non-chiral superamplitudes have a su(2, 2) R symmetry. Up to the constant in the R-dilatation d and some sign ambiguities, its generators {p nn , m nm , m n m , d , k nn } are related to the conformal generators {pα α , m αβ , mαβ, d, k αα } by the replacements λ ↔ η andλ ↔η  where are just the ordinary Lorentz generators m αβ , mαβ acting in dual non-chiral superspace and we used the abbreviations The dual momentum P αα and the dual supermomenta Q αm , Qα m are the generators of translations with respect to the dual variables x and θ,θ The trivial translation invariance in the dual y variable leads to the dual R-symmetry generator (3.51) The conjugate dual supermomenta Q ṅ α , Q n α are given by the action of the superconformal generators s ṅ α , s n α in dual non-chiral superspace. Hence, we have  and their commutators and anti-commutators immediately follow from the dual super Poincaré algebra and the fact that the inversion is an involution, i. e. I 2 = 1. As we are going to show in section 6.2, using the BCFW recursion, the tree-level non-chiral superamplitudes transform covariantly under inversions if the coordinates of full non-chiral superspace invert as, (3.55) The inversion rules of the Levi-Civita tensors can be deduced from I 2 [λ α i ] = λ α i , and I 2 [λα i ] =λα i since the inversion is an involution. Note that the inversion defined in eq. (3.55) is compatible with the constraints eq. (3.41) in full non-chiral superspace. The simplest purely bosonic dual conformal covariants are With the help of the inversion rules (3.55) and its definition (3.53), the action of the dual conformal boost generator in dual non-chiral superspace can be calculated by applying the chain rule, Applying the Schouten identity (A.4) we obtain immediately leading to e. g.
The final result is (3.63) and the dual R-symmetry boost generator For a complete list of the non-chiral superconformal algebra and its dual representation we refer to appendix B.

Yangian symmetry of superamplitudes
The conventional and dual superconformal algebras present at tree level close into an infinite dimensional symmetry algebra known as the Yangian Y [psu(2, 2|4)] as was shown for the chiral and anti-chiral superamplitudes in [6]. This symmetry algebra is a loopalgebra with a positive integer level structure, whose level zero generators J a i are given by the original superconformal generators where [·, ·} denotes the graded commutator and f c ab are the structure constants of the superconformal algebra. Invariance under the level one Yangian generators J [1] a with the bi-local representation then follows from the covariance under the non-trivial dual superconformal generators K αα , S α A . The level one generators obey the commutation relations as well as the Serre relation, for details we refer to [6]. Similar to the chiral superamplitudes the non-chiral superamplitudes have a Yangian symmetry as well, which has been investigated in [11]. The infinite dimensional Yangian symmetry of the tree-level superamplitudes is a manifestation of the expected integrability of the planar sector of N = 4 SYM. In principle it should be possible to exploit the algebraic constraints, that the Yangian invariance puts on the amplitudes, to determine the amplitudes efficiently. The fact that the Yangian symmetry is obscured by the manifest local and unitary Lagrangian formulation of N = 4 SYM theory led to the development of alternative formulations [30][31][32], that enjoy a manifest Yangian symmetry but lack manifest locality and manifest unitarity.

On-shell superspace and superamplitudes
In this section we introduce the maximal supersymmetric N = (1, 1) SYM theory in six dimensions based on references [4, 11, 17-19, 33, 34]. The N = (1, 1) SYM theory can be obtained by dimensionally reducing the N = 1 SYM theory in ten dimensions and the dimensional reduction of N = (1, 1) SYM to four dimensions is given by N = 4 SYM theory. Hence, without presenting its Lagrangian we can immediately write down its onshell degrees of freedom: gluons: g aȧ scalars: s, s , s , s gluinos: χ a , λ a anti-gluinos:χȧ,λȧ (4.1) The amplitudes of N = (1, 1) SYM theory are most conveniently studied using the six dimensional spinor helicity formalism introduced in section 2.3 and the non-chiral on-shell superspace introduced in [17] whose Grassmann variables ξ a ,ξȧ carry little group indices and can be used to encode all the on-shell degrees of freedom into the scalar superfield Ω = s + χ a ξ a + s ξ 2 +χȧξȧ + g aḃ ξ aξḃ +λ˙bξ˙bξ 2 + s ξ 2 + λ a ξ aξ 2 + s ξ 2ξ2 , (4.3) with the abbreviationsξ 2 = 1 2ξȧξȧ , ξ 2 = 1 2 ξ a ξ a . Superamplitudes can now be defined as functions of the superfields A n = A n (Ω 1 , Ω 2 , . . . , Ω n ) . By construction these superamplitudes are invariant under the SU (2) × SU (2) little group but, as explained in [17], do not have the SU (2) R × SU (2) R symmetry of N = (1, 1) SYM theory. As a consequence of the missing R-symmetry, the superamplitudes can not be decomposed according to the degree of helicity violation as in four dimensions (3.14).
The non-chiral superamplitudes are homogeneous polynomials of degree n + n in the Grassmann variables The tree-level superamplitudes of N = (1, 1) are known only up to five external legs [17]. We now review the known amplitudes starting with n = 3. The special three point kinematics require the introduction [24] of the bosonic spinor variables u a i , w a i ,ũ iȧ andw iȧ , defined in appendix appendix A.2.1. With the definition the three point amplitude reads [24] (4.7) and has a manifest cyclic symmetry, and symmetry under chiral conjugation. The four point amplitude has the nice and simple form with the conserved supermomenta being given by 9) and the Grassmann delta functions The five point amplitude can be computed using the BCFW recursion, presented in section 6.3. The result, obtained in [18], has the form This representation of the five-point superamplitude lacks any manifest non-trivial symmetry apart from supersymmetry and is much more complicated than the four point amplitude eq. (4.8). As the five point amplitude indicates, superamplitudes with more than three partons have the general form Judging from the increase in complexity going from n = 4 to n = 5, any straightforward application of the BCFW recursion, using eq. (4.11) as initial data, cannot be expected to yield reasonable results for amplitudes with more than five external legs. Obviously new strategies are necessary to investigate higher point tree amplitudes of N = (1, 1) SYM theory.

Superpoincaré symmetry
Although a part of the symmetries of tree-level N = (1, 1) SYM theory amplitudes appear in the literature, e. g. in [4,34], a complete list of all generators and their algebra is missing. This section aims to close this gap. We start with the symmetries of the tree level superamplitudes in on-shell superspace In contrast to its four-dimensional daughter theory, N = 4 SYM theory, the six-dimensional N = (1, 1) SYM theory has no conformal symmetry since the gauge coupling constant in six dimensions is not dimensionless. However, we have a super Poincaré symmetry The super Poincaré algebra is given by the supersymmetry algebra and the commutators involving the m A B of the SO(1, 5) Lorentz symmetry with covering group SU * (4) read The translation symmetry is trivially given by momentum conservation 16) and the representation of the (1, 1) supersymmetry generators and their conjugates is The correct form of the su(4) Lorentz generators is a bit more involved since the chiral and anti-chiral spinors are subject to the constraints However, it is straightforward to show that the generators m A B given above commute with these constraints.
Besides the super Poincaré symmetry there are a few additional trivial symmetries. First of all, we have the dilatation symmetry whose generator simply measures the dimension of a generator G The non-zero dimensions are As already mentioned before, the on-shell superfield and consequently the superamplitudes are manifest symmetric under the SO(4) SU (2) × SU (2) little group, whose generators are given by Finally there are two hyper charges that correspond to a U (1) × U (1) subgroup of the SU (2) × SU (2) R-symmetry that we sacrificed for the manifest little group invariance. The action of the hyper charges on some generator G are given by 25) and the non-zero values are Note that the constants in d, b,b are not fixed by the algebra and have been chosen such that they annihilate the superamplitude.

Enhanced dual conformal symmetry
All the symmetries presented up to this point exactly match the expectations. Beautifully there is an additional non-trivial symmetry of the superamplitudes [4]. Similar to N = 4 SYM theory in four dimensions, the N = (1, 1) SYM theory in six dimensions has a treelevel dual conformal symmetry. Due to the lack of a superconformal symmetry, the dual conformal symmetry does get not promoted to a full dual superconformal symmetry.
In analogy to four dimensions we extend the on-shell superspace by dual variables to the full non-chiral superspace The variables are subject to the constraints Similar to the non-chiral superamplitudes of N = 4 SYM theory, it is possible to express the superamplitudes of N = (1, 1) SYM solely using the dual superspace variables {x, θ,θ}.
The amplitudes only depend on differences of dual variables, resulting in translation symmetries with respect to each of the dual variables. Hence, we define the dual translation generator to be 29) and the dual supermomenta are Although it is easy to algebraically construct conjugates Q A , Q A to the dual supermomenta, these conjugates would imply the invariance under the superconformal generators s A = ξ a i ∂ iAa and s A = ξ iȧ ∂ iAȧ , which is not the case. We conclude that the amplitudes have an supersymmetry enhanced dual Poincaré symmetry Though we do not have a full dual super Poincaré symmetry we have a dual conformal symmetry, which we are going to derive in what follows. First we recall that for n > 3 the superamplitudes have the form It is possible to define a dual conformal inversion I of the variables of the full superspace eq. (4.27) such that the function f n inverts covariantly In contrast to four dimensions the product of momentum and supermomentum conserving delta functions is not dual conformal invariant due to the mismatch of the degrees of momentum and supermomentum conserving delta functions The inversion leading to eq. (4.33) is defined as where β is some arbitrary constant. Equations (4.35) and the fact that the inversion needs to be an involution on the dual variables, i. e. I 2 = 1, imply the inversion rules of the sigma matrices Consistency between the inversions of x and the chiral and anti-chiral spinors requires the following inversion of the epsilon tensors of the little group Consequently, we have I 2 = −1 on all variables carrying a little group index. Since the superamplitude is little group invariant this is no obstacle. We note that the inversion defined in eqs. (4.35) to (4.41) differs from the one presented in [4] by some signs which are necessary in order to yield the desired inversion of the amplitudes. The proof of eq. (4.33) is straightforward using the BCFW recursion and will be presented in section 6.4. Similar to the four dimensional case we now define the generators From eq. (4.33) it immediately follows, that f n is annihilated by the dual superconformal generators S A , S A , but is covariant under dual conformal boosts carrying little group indices we have I 2 = −1, the action of K AB on a little group invariant object is given by The coefficients of the derivatives are straightforward to obtain leading to In an analogue calculation or by calculating the commutators of K AB with the dual supermomenta Q A ,Q A we obtain Obviously the dual superconformal generators S A , S A are related to the conformal gener- Adding dual conformal inversions promotes the enhanced Poincaré symmetry to an enhanced dual conformal symmetry The generators M A B of the SU (4) Lorentz symmetry 1 act canonically on all generators carrying SU (4) indices (4.48) 1 We drop the star of SU * (4) from now on.
The remaining non-zero commutation relations are (4.49) The dual dilatation generator is given by and, as a consequence of eqs. (4.43) and (4.49), acts covariantly The dual Lorentz generators M A B are equal to the action of the on-shell Lorentz generators m A B in the full superspace. Their representation can be obtained from the dual conformal algebra eq. (4.49) and is given by

Dimensional reduction to massless N = 4 SYM
In this section we explain how the six dimensional tree-level superamplitudes can be mapped to non-chiral superamplitudes of massless N = 4 SYM. Similar mappings can be found in references [11,17,18,34]. In order to perform the dimensional reduction we restrict the six dimensional momenta to the preferred four dimensional subspace p 4 = p 5 = 0. Because of our special choice of six dimensional Pauli matrices, compare eq. (A.24), one can express the six dimensional spinors in terms of four dimensional ones In the four dimensional subspace the contractions with the six-dimensional Pauli matrices read and the supermomenta are Obviously, both, ξ a and ξȧ have to be mapped to η m andη m . Here we make the choice recall that we are using the convention m ∈ {1, 4} and m ∈ {2, 3} for the non-chiral 4d superspace of section 3.2. This implies the maps of the supermomenta and supermomentum conserving delta functions Applying the map of the Grassmann variables eq. (4.57) to the six dimensional superfield eq. (4.3) and comparing it with the four dimensional non-chiral superfield eq. (3.22) yields the following map of the six and four dimensional on-shell states scalars: gluinos: gluons: With the help of eqs. (4.54), (4.55), (4.57) and (4.58) it is possible to perform the dimensional reduction of any six dimensional superamplitude. For a detailed analysis of the connection between the massless amplitudes in six and four dimensions and an investigation of a potential uplift from four to six dimensions we refer to section 8.
5 From massless 6d to massive 4d superamplitudes 5.1 On-shell massive superspace in 4d from dimensional reduction In section 4.3 we dimensionally reduced the massless six-dimensional amplitudes to massless four-dimensional ones. In analogy, we now want to perform the dimensional reduction of the superamplitudes of N = (1, 1) SYM to the massive Coulomb branch amplitudes of N = 4 SYM. When performing the dimensional reduction we need to choose an appropriate set of massive four-dimensional on-shell variables. For the bosonic part of the on-shell variables we choose two sets of helicity spinors {λ α ,λα} and {µ α ,μα} to write the bispinor representation of a four dimensional massive momentum as We introduce abbreviations for the spinor contractions where the mass parameters m andm are in general complex numbers, related to the physical mass by p 2 = mm.
For the particular representation of the six-dimensional Pauli matrices listed in appendix A, the six-dimensional spinors can be expressed using the two sets of four dimensional spinors introduced above and the six-dimensional momenta and dual momenta are given by Here p αα = p µ σ µ αα , x αα = x µ σ µ αα are the contractions of the first four components of the six-dimensional vectors with the four-dimensional Pauli matrices and m = p 5 − ip 4 , n = x 5 − ix 4 . Our conventions for four dimensional spinors can be found in appendix A.
Since we are interested in massive four dimensional amplitudes in the following, we from now on set the fourth spatial component of all six-dimensional vectors to zero, thereby effectively performing the dimensional reduction from a massless five-dimensional to a massive four dimensional theory. This is equivalent to setting n =n = x 5 and imposing the constraint m =m on the spinor variables, which together with the reality condition for the momenta λ * = ±λ, µ * = ±μ results in the 5 real degrees of freedom of a massive four dimensional momentum and a spin quantization axis 2 .
Inserting the dimensional reduction of the spinors into the definition of the supermomenta we obtain Each helicity spinor starts out with 4 real degrees of freedom, the reality condition λ * = ±λ and the U (1) helicity scaling λ → exp[iα]λ cuts this down to 3 real degrees of freedom. The further condition λ µ = μ|λ brings us to 5=3+3-1 degrees of freedom.
generalizing the four-dimensional massless case of eq. (4.56). It is then convenient to define the Grassmann part of our four-dimensional massive on-shell variables to be leading to the four-dimensional supermomenta related to the six-dimensional ones by The dual fermionic momenta θ a i α ,θ a iα are defined by 11) and are related to the six-dimensional dual fermionic momenta by In conclusion the massive Coulomb branch amplitudes of N = 4 SYM may be expressed either by the on-shell variables In the associated full superspace the constraints on the variables read With the help of the maps eqs. (5.3) to (5.5), (5.7), (5.9) and (5.12) it is straightforward to translate any representation of a six-dimensional superamplitude into our four-dimensional variables. From the general form or the six-dimensional superamplitudes we can deduce the general form of the massive amplitudes to be

Super-Poincaré symmetry
We now want to investigate the symmetries of the massive amplitudes using the on-shell variables eq. (5.13) introduced in the last section. To be more precisely we are interested in the symmetries of f n , defined in eq. (5.20), on the support of the delta functions. Similar to the massless four-dimensional case we define shorthand notations for derivatives with respect to spinors Judging from the symmetries of the six-dimensional superamplitudes, presented in section 4.2, and the imposed constraint m =m, we expect a five-dimensional super Poincaré symmetry. It remains to show how this symmetry is realized on the on-shell variables eq. (5.13).
Obviously we have translation invariance as well as the Lorentz generators associated to rotations in the first four spatial directions. Lorentz rotations l µ5 involving the fifth spatial dimension correspond to the generator Supersymmetry is realized as Trivially we have a dilatation symmetry with the generator Performing the dimensional reduction of the spinors, eq. (5.3), the independence of λ A andλ A gets lost. As a consequence only one SU (2) factor of the SU (2) × SU (2) little group symmetry survives the dimensional reduction. Indeed we have the SU (2) helicity generators They fulfill the following closing algebra along with the generic [d, j] = dim(j) j for any generator j, all other commutators vanishing. A necessary condition for the generators to be well defined on the massive amplitudes under consideration is that they commute with the constraint m =m. One indeed shows that this is the case, e.g.
Clearly the nice form of the algebra is suggesting the existence of a SU (2) symmetry with respect to the Grassmann label a, introduced in eq. (5.7). However, at this point we see no indication that such a symmetry is realized on the massive superamplitudes (5.20) for multiplicities larger than four and the introduction of the Grassmann variables ζ a ,ζ a and their dual partners θ a ,θ a should be regarded as a very convenient way to compactly write down the algebra. Indeed, the SU (2) symmetry of the algebra will be explicitly broken if we include the generators r 1 , r 2 of U (1) × U (1) R-symmetry realized on the massive superamplitudes (5.20) Invariance under r a follows from the hyper charges b,b of (4.26) of the six-dimensional superamplitudes. We have

Enhanced dual conformal symmetry
We now want to investigate the symmetries in the dual superspace (5.14). Similar to the on-shell case we already know from the the six-dimensional amplitudes that we will have an extended dual conformal symmetry. Obviously the massive amplitudes have an extended dual Poincaré symmetry with generators Translation invariance in the dual variables implies the symmetries The Lorentz generators L αβ ,Lαβ, W αα are simply given by the action of the on-shell Lorentz generators l αβ ,lαβ, w αα in dual superspace and making the relation of W αα to the Lorentz rotations l µ5 more obvious than in on-shell superspace. The dual dilatation is given by (5.38) and acts covariantly on the amplitude From the six-dimensional superamplitude we know that the massive tree amplitudes are covariant under dual conformal inversion and we only need to find the representation of the dual conformal boost generator in the dual variables eq. (5.14). We emphasize that in order to obtain the correct expression for the µ = 0, 1, 2, 3 components of the dual conformal boost generator we cannot simply plug the 4d variables into the expression for K AB given in eq. (4.45) since this leads to the wrong result. The four-dimensional spinor variables solve the constraint (2.19) on the six-dimensional spinors and thus spoil the assumed independence of chiral and anti-chiral spinors ∂λ A ∂λ B = 0 in the six-dimensional representation of the dual conformal boost generator K AB .
Since there is no obstacle in translating the inversion rules of the six-dimensional dual momenta (4.35), one possibility to obtain the action of the dual conformal boost generator K αβ = IP βα I in the full superspace is to start with the inversion rules for the bosonic dual variables and extend the corresponding part of the dual conformal boost generator K αα acting only on the bosonic dual variables such that it commutes with the constraints (5.15) to (5.19). Note that the additional minus sign in the inversion rules for n originates from the six-dimensional mostly minus metric η 55 = −1.
Requiring that the dual conformal generator K αα x,n commutes with the bosonic constraints (5.15) to (5.17) leads to has a non-vanishing commutator with the right hand side of the fermionic constraints (5.18) and (5.19), we have to introduce the following fermionic terms: commutes with all constraints. The part of K αα acting on the on-shell variables The representation of K 5 = IM I in four-dimensional variables may be obtained in a similar way or by Lorentz rotation [W αα , K ββ ] = αβ αβ K 5 of K αα . The representations of K αα and K 5 in dual superspace are (5.47) and the action of K 5 on the on-shell variables is given by The dual superconformal generators (5.49) can be obtained from the commutators of K αα with the dual supermomenta Q β a andQβ a . In full superspace they coincide with the supersymmetry generatorsqα a ,q α ā Sα a =qα a ,Sα a =q α a , (5.50) similar to the massless case. The dual conformal algebra reads along with the generic [D, J] = dim(J)J for all generators J. We omitted all commutators that are either vanishing or equal to the corresponding commutators in the on-shell algebra eq. (5.29). The action of the R-symmetry charges r a in dual superspace are given by Some further remarks are in order here. as we already mentioned, the generator w αα arises from the Lorentz-generators l µ5 , just as m is related to the momentum in the extra dimensional direction p 5 . As has been shown in [4], if the loop momentum is restricted to be four-dimensional, which is equivalent to the Higgs regularization described in [10], the cut constructible parts of the loop amplitudes invert as Due to the four dimensional loop momenta, the five dimensional Lorentz invariance as well as the dual translation invariance in the x 5 direction are lost. Hence, w αα is a manifest symmetry of the tree-superamplitudes but no symmetry of the Higgs regularized loop amplitudes. Since the dual conformal boost generator is given by K µ = IP µ I, the inversion properties (5.54) only imply that (K µ + 2 i x µ i ) is a symmetry of the regularized loop amplitudes for µ = 0, 1, 2, 3, whereas the tree-amplitudes have the full five-dimensional dual conformal symmetry.

Yangian symmetry
The obvious question now arises: Can one reinterpret the dual conformal operator in six dimensions as a level-one Yangian generator in a four dimensional massive theory? To answer this we proceed in great analogy to the work [6] where a Yangian symmetry of tree superamplitudes was established for N = 4 SYM as reviewed insection 3.3.3. We continue by translating the expression for K αα + i x i αα to four dimensional on-shell variables. Inserting into the part of the dual conformal boost generator acting on the on-shell variables eq. (5.45), one finds the non-local result Here we dropped the terms + (x 1 ) β α l αβ + (x 1 )β αlαβ + (x 1 ) αα d + n 1 w αα + (θ 1 ) a αqα a + (θ 1 ) ȧ αq α a + 1 2 p αα (5.58) which annihilate the tree amplitudes on their own because they are each proportional to symmetry generators. Since the tree superamplitude is independent of x 1 , θ 1 , n 1 and K αα + i x i αα annihilates it, one could also apply the reverse logic by concluding from (5.58) that d, l αβ ,lαβ, w αα ,qα a ,q α a are symmetries of the tree amplitudes. The Higgs regularized loop amplitudes explicitly depend on n 1 and are not invariant under w αα . Consequently, the term n 1 w αα cannot be dropped at loop level. Let us proceed by investigating the structure of the dual conformal boost generator in on-shell variables a bit further. Upon adding to (K αα + i x i αα ) of eq. (5.57) the quantity  We note that m (1) can also be obtained from the action of K 5 on the on-shell variables (5.48) in the same way as p (1) has been obtained from K αα in (5.42).
A natural question to be addressed in future work is whether or not there exist the level-one fermionic generators q aα . However, already at this point it is clear that the non-local symmetry generators found will not lift to the complete super Poincaré algebra but rather stay confined to the super-translational piece. In particular there will be no level-one w (1) αα symmetry generator.

General remarks
The BCFW on-shell recursion [12,13] is a valuable tool in calculating color ordered treelevel amplitudes in gauge theories, as it allows to recursively calculate an n point amplitude from lower point amplitudes. As a direct consequence, the knowledge of the three point amplitudes and the BCFW recursion relation are sufficient to obtain all color ordered tree amplitudes of a particular gauge theory. In what follows we will briefly outline the general form of the BCFW recursion, for some more details we refer to to the excellent review [35].
The basic idea is to analytically continue two external momenta by introducing lightlike shifts proportional to the complex parameter z that neither spoil the on-shell condition of the two shifted momenta nor the overall momentum conservation. If the shift vector r has the properties then the shift has the desired properties Using region momenta instead, the shifts in eq. (6.2) can be reproduced by the single shift Color ordered tree amplitudes have simple analytic structure since they only have poles where sums of consecutive momenta go on-shell, i. e. x 2 i j = 0. As a consequence A n (z) is an analytical function that has only the simple poles z j solving the on-shell condition i. e. the poles are given by If the analytically continued amplitude A n is vanishing as |z| → ∞ it is a simple fact that the contour integral of An z over a circle at infinity is vanishing. By virtue of the residue theorem this allows to relate the physical amplitude to the residues of An z at the poles z j Due to the general factorization properties of tree amplitudes, these residues are given by products of lower-point on-shell amplitudes multiplied by the residue − Res . (6.8) Introducing the abbreviationŝ the final form of the BCFW on-shell recursion is where the sum goes over all poles z j and over all helicities of the intermediate states. Note that we assumed the vanishing of A n (z) for large z to derive the recursion relation which is not a general feature for all gauge theories and all possible shifts. For details we refer to [36] and [37].
In the following sections we will derive supersymmetric versions of the BCFW recursion eq. (6.10) for the four-dimensional N = 4 SYM theory and the six-dimensional N = (1, 1) SYM theory.

Supersymmetric BCFW for N = 4 SYM in non-chiral superspace
As it has not been done in the literature before, we are going to present the BCFW recursion in the non-chiral super space {λ α i ,λα i , η m i ,η m i }, introduced in section 3. Additionally we will use the BCFW recursion to prove the postulated covariance in eq. (3.54) of the nonchiral superamplitudes under the dual conformal inversions (3.55), as well as to calculate the four-, five-and six-point superamplitudes.
Based on the previous section it is straightforward to write down a set of shifts preserving both bosonic and fermionic momentum conservation λ 1 → λ1(z) = λ 1 + zλ n ,λ n →λn(z) =λ n − zλ 1 , (6.11) η n → ηn(z) = η n − zη 1 ,η 1 →η1(z) =η 1 + zη n , (6.12) leading to the poles, eq. (6.6), of the shifted superamplitude. The corresponding dual shifts are According to the same arguments as in chiral and anti-chiral superspace, the BCFW recursion in non-chiral superspace is given by where the two dimensional delta functions of objects χ m ,χ m carrying Grassmann indices have the definition δ 2 (χ m ) = 1 2 χ m χ m , δ 2 (χ m ) = 1 2χ m χ m such that d 2 η δ 2 (η) = d 2η δ 2 (η) = 1. We recall from eq. (3.25) that the superamplitudes with n > 3 partons in non-chiral superspace have the form i. e. the only ηP j ,ηP j dependence of the integrand in the BCFW recursion eq. (6.15) originates from delta functions of the three point amplitudes and the delta functions of the fermionic momenta, making the Grassmann integrations straightforward. For the four point amplitude we obtain In agreement with [11]. Introducing the definitions we present the results of the Grassmann integrations in eq. (6.15) for the three different cases j = 2, 2 < j < n − 2 and j = n − 2. In the case j = 2 the left superamplitude has to be A MHV 3 since A MHV 3 does not exist for the three point kinematics of this case. We obtain A n−1 (P 2 , p 3 , . . . , pn) For practical applications it is convenient to rewrite B 2 as For practical applications it is more convenient to use the following expression for B j (6.24) In the case j = n − 2 the right superamplitude has to be A MHV 3 due to the special three point kinematics and the integration gives x 2 1n−1 P n−2 n − 1 P n−2 n , (6.25) which may be rewritten as .

(6.26)
Now the integrated non-chiral BCFW recursion relation reads In this form it is straightforward to prove the dual conformal symmetry of the non-chiral superamplitudes. Applying the inversion rules eq. (3.55), we find , I P j |θ1 j+1 = P j |θ1 j+1 which proves the covariance (3.54) of the non-chiral superamplitude under the dual conformal inversions (3.55).
In order to obtain useful representations of the non-chiral superamplitudes from the integrated BCFW recursion eq. (6.27) it remains to remove the hats from the the shifted dual point1 by using identities like e. g.
After removing all hats the obtained expression may still contain spinors. However, these spinors can be removed by multiplying and dividing with the chiral conjugate spinor brackets. The final expression will only depend on {x i , θ i ,θ i } and besides x 2 ij it can be expressed by the dual conformal covariant objects where the prefactor of 1 2 has been introduced for convenience. Carrying out the recursion step from four to five points we obtain Dual conformal invariance of these expressions is easy to verify by simply counting the inversion weights on each dual point. In principle all non-chiral amplitudes could be obtained by a half Fourier transform of the known chiral or anti-chiral superamplitudes. However, it is in general nontrivial to carry out these integrations in a way that leads to a useful representation of the amplitude. One exception are the MHV and MHV part of the non-chiral superamplitude, which can be obtained by either solving the BCFW recursion or by performing the half Fourier transform in the way described in [11]. The result we found and also checked numerically is and similar for the MHV part. Note that our result differs from the one presented in [11].

Supersymmetric BCFW for N = (1, 1) SYM
The supersymmetric BCFW recursion of N = (1, 1) SYM theory in six dimensions will play a central role when investigating massive amplitudes in sections 7 and 8. It has been introduced in reference [17]. In what follows we will closely follow the detailed review presented in reference [18]. At the end of this section we will use the BCFW recursion relation to prove the dual conformal covariance, eq. (4.42), of the superamplitudes. As a first step we introduce the shift vector that obviously has the desired properties r · p 1 = 0 = r · p n . The requirement r 2 = 0, implies 0 = ab ȧḃ X aȧ X bḃ = 2 det(X). Hence X aȧ is some arbitrary rank one matrix and has a spinor helicity representation X aȧ = x axȧ . Equation (6.36) implies 37) and the shifts of the momenta p 1 and p n (6.2) can be reinterpreted as shifts of the chiral and anti-chiral spinors. The equations have the simple solutions Or after inserting the definition (6.37) of the shift vector Supermomentum conservation can only be maintained if the Grassmann variables of legs 1 and n are shifted as well ξ1 a = ξ 1a + zX aȧ [1ȧ|q n /s 1n , ξn b = ξ nb + zX aȧ [1ȧ|n b ξ a 1 /s 1n , ξȧ 1 =ξȧ 1 − zX aȧ [q n |1 a /s 1n ,ξ˙b n =ξ˙b n − zX aȧ [n˙b|1 a ξȧ 1 /s 1n , (6.41) resulting in the following shifts of the supermomenta or with the definition of r being inserted The dual shifts are given by Note that the Grassmann shift variables s A ands A can alternatively be obtained by solving the equations The above set of supersymmetry preserving shifts leads to a shifted superamplitude whose residues at the poles eq. (6.6) are given by a product of two lower point superamplitudes. Similar to the supersymmetric BCFW recursions of N = 4 SYM, the sum over intermediate states is realized by an integration with respect to the Grassmann variables of the intermediate leg.
Using the abbreviations introduced in eq. (6.9) the BCFW recursion of N = (1, 1) SYM theory in six dimensions reads A n−j+1 (P j , p j+1 , . . . , pn) z=z j (6.47) Similar to the non-chiral BCFW recursion in four dimensions, eq. (6.15), the explicit minus sign originates from the choice d 2 ξ = 1 2 dξ a dξ a , d 2ξ = 1 2 dξȧdξȧ for the integration measure and can be fixed by projecting the four point function resulting from the six-dimensional BCFW recursion eq. (6.47) to four dimensions and comparing it with eq. (6.19). Starting point for the recursion is the three-point superamplitude of eq. (4.7) [24]. For applications of the BCFW recursion it is more convenient to use the following alternative representation of the three point amplitude As has been shown in [17], the BCFW recursion yields the four point function .

(6.49)
Note that the four-point amplitude is fixed up to a numerical factor by supersymmetry and dual conformal symmetry.
In the remainder of this section we will explicitly carry out the Grassmann integrations in the BCFW recursion eq. (6.47). First of all we recall that for n ≥ 4 an n-point superamplitude has the form In order to consistently treat ingoing and outgoing particles we adopt the prescription Structurally there are the three different cases j = 2, 2 < j < n − 2 and j = n − 2 to be analyzed. Starting with the contribution j = 2 in eq. (6.47), we want to evaluate . , x n (6.52) Taking the representation eq. (6.48) of A 3 , the only dependence on ξP 2 ,ξP 2 is contained in Grassmann delta functions, and the integration boils down to solving the linear equations for ξP 2 ,ξP 2 , with the abbreviation K = −P 2 . The solution is Using eqs. (A.28) to (A.30) it is straightforward to show that on the support of (u1 − u 2 )(ũ1 −ũ 2 ) this implies qP 2 = q1 + q 2 ,qP 2 =q1 +q 2 , (6.56) and therefore The integral of the three-point amplitude has the solution where k 1 and k 2 are some arbitrary reference vectors and u a w a = 1 =ũȧwȧ has been used. The final result is 59) evaluated at z = z 2 . In the case j = n − 2 we need to evaluate B n−2 = −i x 2 1 n−1 δ 4 (q) δ 4 (q) f n−1 x1, . . . , x n−1 d 2 ξP n−2 d 2ξP n−2 A 3 p n−1 , pn,P n−2 . (6.60) Here we already exploited that on the support of the three-point amplitude we have The remaining integral of the three-point amplitude in eq. (6.60) is given by evaluated at z = z n−2 . Similar to the case j = 2, arbitrary reference momenta k n , k n−1 have been introduces in order to get rid of the u,ũ variables. Finally there is the general case 2 < j < n − 2 with no three-point amplitudes involved To carry out the integration we want to rewrite the fermionic delta functions. Due to the algebra eq. (A.12) of the six-dimensional Pauli matrices, we have the identity which implies Consequently the fermionic delta functions can be rewritten as follows (6.69) The two-dimensional Grassmann delta functions are defined as δ 2 (χ a ) = 1 2 χ a χ a and δ 2 (χȧ) = 1 2χȧχȧ such that d 2 ξ δ 2 (ξ a ) = 1 = d 2ξ δ 2 (ξȧ). This allows us to easily carry out the Grassmann integrations and similarly for the anti-chiral integration The full contribution is This expression is straightforward to implement numerically. Unfortunately, it is ill suited to directly obtain reasonable analytical expressions for higher point amplitudes because of the auxiliary variable X aȧ = x axȧ contained in the shift eq. (6.36). In contrast to four dimensions the shift vector is not fixed by requiring r 2 = 0, r · p 1 = 0 = r · p n . This ambiguity is reflected by the presence of X aȧ in the definition of the shift vector. Obviously the amplitudes are independent of the shift vector, i. e. independent of X aȧ . In principle it should be possible to remove the shift vector from the right hand side of eq. (6.73) without inserting its definition eq. (6.37), only using its general properties eq. (6.1). Unfortunately, even in the easiest case of the five point superamplitude this is very hard to achieve. As long as it is not understood how to obtain f n ({x i , θ i ,θ i }) from the output of the BCFW recursion, eq. (6.73) will be limited to numerical applications. Indeed, in sections 7 and 8 we will extensively use a Mathematica implementation of the integrated BCFW recursion (6.73). Independence of X aȧ and the arbitrary reference momenta entering B 2 and B n−2 provides a nontrivial check of the numerical results obtained from the implementation. In fact, taking the four point amplitude (6.49) as initial data, independence of the six-point component amplitudes on X aȧ requires the explicit minus sign appearing in the BCFW recursion relation eq. (6.47).

Proof of dual conformal symmetry of N = (1, 1) superamplitudes
With the help of the BCFW recursion and the inversion rules (4.35) to (4.41) it is straightforward to inductively prove the dual conformal covariant inversion of the N = (1, 1) superamplitudes by showing that each term B i in the integrated BCFW recursion eq. (6.73) inverts as Since the BCFW diagrams involving three-point amplitudes B 2 , B n−2 are related by cyclic relabeling of the indices, we only need to consider one of them as well as the general diagram B j without three-point functions.
We start out with B 2 , eq. (6.58), and investigate the inversion of u1 − u 2 ũ1 −ũ 2 . Simply plugging in the inversion rules yields ] the inhomogeneous term can be rewritten and leads to the result Similarly we find which together with proves the desired inversion of B 2 . What remains is to check the inversion of B j given in eq. (6.72). Again inserting the inversion rules we obtain where we have used x 2 1 j+1 = 0. The inversion of [θ j+11 |x1 j+1 |θ j+11 ] can be obtained by chiral conjugation 3 of (6.80) and together with this concludes the proof of the dual conformal symmetry of the tree superamplitudes.
7 Tree-level superamplitudes of N = (1, 1) SYM theory In four dimensions the supersymmetric BCFW recursion together with the dual conformal invariance allowed for the construction of analytical formulae for all superamplitudes of N = 4 SYM theory [8]. The key to this remarkable result was the use of dual conformal invariant functions for the construction of a manifest dual conformal covariant solution to the BCFW recursion. Of similar importance was the MHV decomposition (3.14) of the superamplitudes, allowing to successively solve the recursion for the increasingly complex N p MHV superamplitudes. Although the non-chiral superamplitudes of N = (1, 1) SYM do not possess a conformal symmetry and an analogue of the helicity violation decomposition of the 4d theory, they still have a dual conformal symmetry and obey a supersymmetric BCFW recursion relation. Hence, it is natural to try to find dual conformal invariant functions suitable to construct a solution to the super-BCFW recursion of N = (1, 1) SYM. Unfortunately, the six-dimensional BCFW recursion, as reviewed in section 6.3, is ill suited to produce compact analytical expressions. In contrast to four dimensions the shift (6.36) is not uniquely fixed and contains auxiliary spinor variables x a , xȧ. Although the amplitudes are independent of these variables, their removal is non-trivial. The main obstacle is that the individual BCFW diagrams are in general not independent of x a , xȧ but only their sum, denying any obvious elimination of the auxiliary variables. In spite of its limitations the six-dimensional BCFW recursion is a powerful tool to obtain numerical values for arbitrary tree amplitudes of N = (1, 1) SYM theory. As we will explain in what follows, this can be exploited to determine manifest dual conformal covariant representations of superamplitudes. On the support of the momentum and supermomentum conserving delta functions, the Ω n,i possess all continuous symmetries of f n . Note that the invariance under the supersymmetry generators q A and q A follows from the invariance under Q A ,Q A and the covariance under dual conformal boosts K AB , compare eqs. (4.46) and (4.49). Besides chiral symmetry, we could equally enforce the other discrete symmetries, which are cyclic invariance and the reflection symmetry. As will become clear in what follows enforcing symmetry under chiral conjugation is essential. Given a set of functions {Ω n,j }, we can make the ansatz

Analytical superamplitudes from numerical BCFW
By construction, the coefficients α i are dimensionless, dual conformal invariant functions of differences x ij of the region momenta x i . The only dual conformal covariant objects that can be built from the x ij are the traces can appear in the coefficients. These traces are given by We draw the important conclusion that the coefficients α i are rational functions of dual conformal invariant cross ratios At multiplicity n, only ν n = 1 2 n(n − 5) of these cross ratios are independent. Since there are no cross ratios at four and five points, the α i will be rational numbers in these cases.
Unless the choice of the Ω n,i has been extremely good, the α i will depend on the cross-ratios for multiplicities greater than five. Nevertheless, it is straightforward to determine them using a numerical implementation of the BCFW recursion relation. Evaluating both sides of eq. (7.2) for a given phase space point π j on a sufficiently large number of component amplitudes, the resulting linear equations can be solved for α i (π j ). Numbering the cross ratios {u 1 , u 2 , . . . , u νn } we make an ansatz for each of the coefficients where {n j } k are all different distributions of k powers among the cross ratios. Inserting the values of the cross ratios and the calculated values of the coefficients α i (π j ) for a sufficiently large number of phase space points, the resulting linear equations can be solved for {a I , b I }. Some remarks are in order here. It is very important to randomly choose the set of component amplitudes used to calculate the α i (π j ). As will be demonstrated later, picking only amplitudes of a particular sector, like e. g. only gluon amplitudes, can lead to dual conformal extensions of this particular sector that are not equal to the full superamplitude. In practice one will successively increase the rank k of the polynomials in eq. (7.7) until a solution is found. In order to not have to worry about numerical uncertainties or instabilities, we chose to use rational phase space points. Using momentum twistors it is straightforward to generating four-dimensional rational phase space points which can be used to obtain rational six-dimensional phase space points of the form Although these phase space points only have four non-zero components, they are sufficiently complex to yield non-zero results for all massive amplitudes 4 . The obvious benefit of the rational phase space points is that all found solutions to the ansatz eq. (7.2) are exact. An important property of the described method for the determination of the superamplitudes is that the obtained representations will contain only linear independent subsets of the basis functions Ω n,i . This may become an obstacle when looking for nice solutions with very simple coefficients α i or ultimately for master formulae valid for arbitrary multiplicities since these not necessarily consist only of linear independent Ω n,i .
Essential for making the ansatz eq. (7.2) is the knowledge of the possible dual conformal covariant objects involving dual fermionic momenta θ i ,θ i . Therefore we recall the inversion of the dual coordinates, compare (4.35)-(4.41), Clearly the objects The only flaw in using them would have been the ruled out six-dimensional Levi-Civita tensors.
have inversion weight minus one on each of the appearing dual points but lack a translation invariance in θ andθ. Fortunately there is a unique way to obtain manifest dual translation invariant objects from the dual conformal covariants eq. (7.10). We define the dual translation invariant objects Hence, the dual conformal covariant, dual translation invariant building blocks for the superamplitudes are and They all have inversion weight minus one on every appearing dual point, e. g.
Keeping in mind that the degree in both θ andθ always increases by one if we successively increase the multiplicity, the last of the building blocks appears most natural. The first two building blocks necessarily appear in pairs and lead to a partial decoupling of the chiral and anti-chiral supermomenta. Consequently the building blocks eqs. (7.13) and (7.14) alone cannot be sufficient to construct an even multiplicity amplitude. Furthermore they are very unfavorable from the four-dimensional perspective as the massless projection of amplitudes containing them has an obscured R symmetry, for details we refer to section 8. Although we found solutions to eq. (7.2) containing all three types of building blocks, we will neglect the building blocks eqs. (7.13) and (7.14) in what follows.
To be more precisely we will try to find representations of the superamplitudes with the general form where the coefficients β IJK are functions of the dual conformal covariants x 2 ij with the correct mass dimension and the correct inversion weights on each of the dual points in the multi-indices I, J, K. Manifest symmetry under chiral conjugation implies β IJK = (−1) n−4 β KJI .
Clearly not all of the building blocks (7.15) are independent. All simple relations follow from

The four and five-point amplitudes
As an instructive illustration of the severe restrictions the dual conformal covariance, eq. (4.33), puts on the functional form of the superamplitudes, we consider the four point amplitude. Indeed, dual conformal covariance fixes the four point amplitude up to a constant and the only possible ansatz is The constant can be fixed by performing the dimensional reduction onto any massless four-dimensional amplitude. For the MHV gluon amplitude with negative helicity gluons at positions three and four we obtain Comparison with the well known Parke-Taylor formula yields α = −i. This trivial calculation should be compared to the comparably complicated calculation using the BCFW recursion in references [17,24].
Recalling the known result for the five point amplitude, eq. (4.11), we want to find the most simple representation of f 5 that is manifest dual conformal covariant. Hence we are searching for dual translation invariant functions of mass dimension minus five, that are of degree one in both θ andθ and invert as The most simple dual conformal covariant building blocks invariant under chiral conjugation are given by Obviously Ω ijklm is zero if less than three of its indices are distinct. From the properties of B ijk |, eq. (7.18), and its definition above follow the properties Ω i j k l m = Ω i l k j m , Ω i j k l m = −Ω i k j m l , Ω i i+1 k l m = −Ω i+1 i k l m , Since there are no dual conformal invariant cross ratios at five point level, we know that eq. (7.27) is either up to a constant equal to f 5 or we need to make a more complicated ansatz including the building blocks B ijk |x il x lk | B kmn ]. Comparing this ansatz with the numerical BCFW recursion we indeed find the beautiful result and the five point amplitude is given by This is the most compact dual conformal covariant expression of the five point amplitude available and should be compared to the form (4.11) of [18]. Making the dual conformal properties manifest led to a significant simplification. Another manifest dual conformal covariant representation has been reported in [11] by uplifting the four-dimensional five point amplitude of non-chiral superspace. We will discuss the potential uplift of massless four-dimensional amplitudes in section 8.

The six-point amplitude
As it turned out, the four and also the five point amplitudes were trivial examples of our general ansatz eq. (7.2), since the coefficients α i were constants. At six points they will in general no longer be constant but rational functions of the three dual conformal invariant cross ratios Similar to the five point case we try to find a representation of the six point amplitude using only the simplest of the building blocks of eq. (7.15). To further reduce the resulting basis, we require chiral symmetry of the building blocks. Hence we only use the Ω i j k l m defined in eq. (7.23). In contrast to five points the objects Ω i j j l m are not all zero at multiplicity six. Nevertheless, we neglect them and stick to the Ω i j k l m with distinct indices. What we are left with are the six building blocks The basis of fifteen terms that we built from the Ω i is where the β ij cancel out the inversion weights of the four overlapping indices present in Ω i Ω j . Because of the existence of the three cross ratios, β ij are not uniquely fixed. One possible choice is We exclude terms of the form (Ω i ) 2 and make the following ansatz for the five point amplitude with α ij = α ij (u 1 , u 2 , u 3 ) being a rational function of the cross ratios. Making an ansatz of the form eq. (7.7) it is straightforward to determine the α ij . The first observation is that out of our fifteen basis elements only eleven are linearly independent, leading to a large number of different representations of the form (7.35). The highly nontrivial linear relations between the Ω i j are only valid on the support of the momentum and supermomentum conserving delta functions and can be determined in the same way as the amplitude. The two ten-term and two eleven-term identities involving complicated functions of the cross ratios can be used to transform a particular solution to eq. (7.35) to any other solution of this form. The complexity of the coefficients α ij varies largely with the choice of linear independent Ω i j in the solution, e. g. some solutions involve rational functions of degree twelve in the cross rations u i . The three simplest of the solutions involve nine Ω i j and rational functions of degrees less than three. One particular of these simple solutions is Inserting the coefficients, the definitions of Ω ij and the cross rations u i , as well as the identity Tr (1 2 3 5 6 4) = x 2 14 x 2 25 Albeit all continuous symmetries and the symmetry under chiral conjugation of the six point amplitude are manifest in the solutions to eq. (7.35), the cyclic 6 and reflection symmetry are not obvious. However, there is no obstacle in finding manifest cyclically symmetric representations by constructing manifest cyclically symmetric basis elements from the Ω i . As a consequence of the manifest cyclic invariance of the basis, the coefficients in the general ansatz eq. (7.2) are cyclically symmetric as well, i. e. are rational functions of symmetric polynomials of the cross ratios.
There are three types of such manifest cyclically symmetric basis elements g 1 (u 1 , u 2 , u 3 )Ω 12 + five cyclic rotations g 2 (u 1 , u 2 , u 3 )Ω 13 + five cyclic rotations The functions g i are arbitrary rational functions of the cross ratios leaving a lot of freedom to define a cyclic basis. Looking at the solution eq. (7.36), reasonable choices are g 1 ∈ {u 1 u 2 , u 1 u 3 , u 2 u 3 }, g 2 ∈ {u 1 , u 2 , u 3 }, and g 3 ∈ {u 1 (u 2 ± u 3 ), u 2 (u 3 ± u 1 ), u 3 (u 1 ± u 2 )}. Indeed, this leads to a solution involving only three cyclically symmetric basis elements. Choosing g 1 = u 2 u 3 , g 2 = u 3 , and g 3 = u 2 (u 1 + u 3 ) we find or equivalently Clearly this representation is not minimal as it consists of all fifteen Ω ij . The contained unphysical pole at u 1 + u 2 + u 3 = 3, might be expressed by the traces As emphasized in section 7.1 it is very important to randomly choose the component amplitudes which are used to calculate the coefficients α i in the general ansatz eq. (7.2). Since we are dealing with a maximally supersymmetric theory one might wonder if it would not be sufficient to consider e. g. only gluon amplitudes and let supersymmetry care for all other amplitudes. Indeed this is a widespread claim within the literature which can be easily disproved. In fact, only eight of the fifteen Ω ij are linear independent on gluon amplitudes compared to eleven on all component amplitudes. Consequently, supersymmetrizing gluon amplitudes as has been done in reference [17] for the three, four and five point amplitudes will not yield the correct superamplitude for multiplicities greater than five. Having said that, it is nevertheless interesting to investigate how such a supersymmetrization of the gluon amplitudes looks like. Therefore we try to find a dual conformal invariant extension of the gluon amplitudes, that is a solution to eq. (7.2) valid on all gluon amplitudes. At six points we do not have to worry about six-dimensional Levi-Civita tensors and it is not necessary to use chiral self-conjugate building blocks. Instead of the Ω i we use the building blocks where the label j indicates that the indices {i, i ± 1, i ± 2, ±3, i ± 4} in Ω u/d i,j have to be taken modulo j. Whenever the label j is equal to the multiplicity n, we will usually drop it. The Ω u/d i are related to the chiral self-conjugate Ω i by The resulting ansatz for the dual conformal extension of the gluon sector is Since the gluon sector is not closed under dual conformal symmetry, the massless coefficients α ij , β ij , γ ij are in general rational functions of the Lorentz invariants x 2 kl . As expected not all of the Ω u/d i Ω u/d j are linear independent on the gluon amplitudes. A good indication that we will find dual conformal covariant solutions to eq. (7.45) is the fact that all two term identities that the Ω u/d i Ω u/d j fulfill on the gluon amplitudes are in fact dual conformal covariant. On the support of the momentum and supermomentum conserving delta functions we have for example the six identities Indeed there are 24 nice three term solutions to eq. (7.45) that are all dual conformal covariant. One of these solutions is Unfortunately, none of the found dual conformal extensions of the gluon sector were equal to the superamplitude. However, they all gave the correct ultra helicity violating (UHV) amplitudes.

Towards higher multiplicities
Inspired by the compact representations eqs. (7.28), (7.38) and (7.41) the next logical step is to try to find a nice representation of the seven point amplitude with the ultimate goal of a master formula valid for arbitrary multiplicities. The main difficulty at higher multiplicities is to make a good choice for the basis Ω n,i used to express the amplitude, compare eq. (7.2). Going from five to six partons the number of terms in the amplitudes increased roughly by a factor of ten. Hence, the number of terms in the seven point amplitude is expected to be of order 100, making systematic studies of the solutions to eq. (7.2) impossible for multiplicities n > 6. Furthermore, the generic solution α i to eq. (7.2) contains complicated rational functions of the ν n = 1 2 n(n − 5) cross ratios which require a huge calculational effort to be obtained from the BCFW recursion using eq. (7.7).
At seven points, a natural starting point is to use a basis constructed from products of the chiral self-conjugate Ω i j k l m defined in eq. (7.23). Hence, the ansatz for the seven point amplitude reads where the coefficient β IJK are functions of the covariants x 2 ij compensating the negative inversion weights in the dual points present in {I, J, K}. The β IJK have mass dimension -22 and are straightforward to obtain by counting the inversion weights in {I, J, K}, compare eq. (7.34). The dimensionless α IJK are rational functions of the seven cross ratios . which implied the reduction for the supermomenta It is instructive to translate the MHV decomposition of the massless four-dimensional superamplitudes into six-dimensional language. Because of the SU (4) R symmetry the N p MHV superamplitude in chiral superspace has the Grassmann dependence 4d chiral superspace: According to eq. (3.23), the chiral super field A n (Φ 1 , . . . , Φ 1 ) is related to the non-chiral superfield A n (Υ 1 , . . . , Υ n ) by the half Fourier transformation Consequently, the N p MHV superamplitude in non-chiral superspace has the Grassmann dependence 4d non-chiral superspace: With the help of the map eq. (7.54) between the four-dimensional and six-dimensional Grassmann variables we can deduce which of the six-dimensional component amplitudes A n i a 1 1 , . . . , i an n , j 1ḃ 1 , . . . , j nḃn , defined in eq. (7.53), correspond to massless four-dimensional N p MHV amplitudes 6d non-chiral superspace: A n N p MHV = O (ξ 1 ) n−p−2 (ξ 2 ) p+2 (ξ1) n−p−2 (ξ2) p+2 . (7.59) Hence, the SU (4) R symmetry of the massless chiral superamplitudes in four dimensions leads to a Grassmann dependence of the form (ξ 1 ) n−a (ξ 2 ) a (ξ1) n−a (ξ2) a in six dimensions. From the six-dimensional perspective the Grassmann dependence of the superamplitudes in the massless four-dimensional limit is a consequence of breaking the SU (2)×SU (2) little group to a U (1) little group in four dimensions because on the four-dimensional subspace the chiral and anti-chiral spinors λ A andλ A are equal.
In the case of the massive four dimensional amplitudes the SU (4) R symmetry is broken and the Grassmann dependence of the corresponding six-dimensional superamplitude is no longer restricted, i.e. all terms of the form (ξ 1 ) n−a (ξ 2 ) a (ξ1) n−b (ξ2) b are appearing except the ones with a, b ∈ {0, n}. We then propose the following little group decomposition of the superamplitudes of N = (1, 1) SYM This decomposition can be further motivated by translating the Grassmann dependence of A a×b n back to chiral superspace using eqs. (7.54) and (7.57) 6d: Hence the little group decomposition in six dimensions corresponds to breaking the fourdimensional SU (4) R symmetry to a SU (2) R × SU (2) R symmetry.  where {n j } k are all different distributions of k powers among the Lorentz invariants. In contrast to the dual conformal invariant case, eq. (7.7), numerator and denominator need to be homogeneous polynomials of equal degree k.

Six-point case
To get an idea of the complexity of the UHV amplitudes we turn to the six point case and make the same ansatz as in eq. (7.45) for the gluon sector where at each recursion step we only use the 2n dual conformal covariant building blocks Ω u i,n defined in eq. (7.43). Due to the special kinematics eq. (7.66) we do not have to worry about six-dimensional Levi-Civita tensors for multiplicities larger than six, hence there is no need for chiral self-conjugate building blocks. The coefficients α i , β i have mass dimension minus six and their functional dependence on the Lorentz invariants x 2 ij can be obtained by modifying the ansatz eq. (7.63) accordingly. We successively determine the solutions to eq. (7.67) and at each multiplicity we keep all one term solutions and feed them back into the recursive ansatz eq. (7.67). As initial data we take the ten equivalent representations of the full five point amplitude following from eq. (7.30), eq. (7.29) and the cyclic invariance of the amplitude Note that the discrete symmetries making the above 10 representations identical only hold within five-point kinematics.
Only the two f n of this set proportional to Ω u 1,5 or Ω d 5,5 yield one term solutions in the recursive construction of f 6 and out of the four one term solutions they produce again only two, namely it is straightforward to generalize them to arbitrary multiplicities. We conjecture the formulae to be valid representations for UHV amplitudes of multiplicities greater than four. Up to multiplicity n = 13 both formulae have been checked by determining the solutions to the recursive ansatz eq. (7.67) which seems sufficient to us to consider eqs. (7.73) and (7.74) to be proven. With regard to the three term solutions (7.65) for all gluon and UHV amplitudes on general kinematics, we expect the formulae eqs. (7.73) and (7.74) to be valid for other sectors as well. The natural guess is of course that the dual conformal extensions of the UHV amplitudes on the special kinematics eq. (7.66) produce the correct gluon amplitudes. However, this is not the case. The reason might be that the gluon sector does not undergo the same significant simplifications as the UHV sector if we specialize the kinematics. Fortunately the found dual conformal extensions of eqs. (7.73) and (7.74) yield an even bigger class of amplitudes. We find the remarkable results that eq. (7.73) is equal to the superamplitude on all little group sectors of the form 1 × a, (n − 1) × a, whereas eq. (7.74) is correct for the chiral conjugate little group sectors a × 1, a × (n − 1). We indicate this by writing and f a×1 a×(n−1) Clearly the chiral conjugate of the formula for f 1×a (n−1)×a n is an alternative representation of f a×1 a×(n−1) n and vice versa.

From massless 4d to massless 6d superamplitudes
A very exciting question, first discussed in [11], is whether or not it is possible to obtain the massless tree-level superamplitudes of the six-dimensional N = (1, 1) SYM by uplifting the massless non-chiral superamplitudes of the four-dimensional N = 4 SYM. If so, as claimed by the author of [11], this implies that the massive four-dimensional amplitudes of N = 4 SYM can be obtained from the massless ones. Since the non-chiral superamplitudes of N = 4 are straightforward to obtain using the well behaved non-chiral BCFW recursion, described in section 6.2, such a correspondence could provide an easy way to obtain the tree amplitudes of N = (1, 1) SYM.
In this section we want to thoroughly investigate the potential uplift of the massless four-dimensional amplitudes and thereby clarify some points in [11].  1). The symmetries of the six-dimensional and four-dimensional superamplitudes have been discussed in detail in sections 3.3 and 4.2. The most relevant in this discussion are the discrete symmetry under chiral conjugation and the R-symmetry of the four-dimensional superamplitudes. In particular the invariance under the R-symmetry generators m nm and m n m implies that all R-symmetry indices within a superamplitude are contracted.
With the help of the maps between the six-dimensional on-shell variables {λ A i ,λ i A , ξ a i ,ξȧ i } and the massless four-dimensional on-shell variables {λ α i ,λα i , η m i ,η m i } eqs. (4.54) and (4.57) it is straightforward to obtain the projection of every six-dimensional object.
Since there is a one-to-one map between the supermomentum conserving delta functions (4.59) we neglect them straight away and investigate the correspondence The tree-level amplitudes of N = (1, 1) SYM theory consist of Lorentz invariant contractions of momenta p i and supermomenta q i ,q i . The only purely bosonic Lorentz invariants are traces of an even number of momenta (k i ) AB , (k i ) AB . However chiral conjugate traces project to the same four-dimensional traces where / k i denotes the contraction of the momentum k i with either the six-dimensional or the four-dimensional gamma matrices and Γ ± = 1 2 (1 ± γ 7 ). Hence, the presence of traces in f 6d n that are not chiral self-conjugate would already spoil the uplift. The chiral conjugate traces differ by terms containing the six-dimensional Levi-Civita tensor. Since N = (1, 1) SYM is a non-chiral theory it is symmetric under chiral conjugation (p i ) AB ↔ (p i ) AB , q i ↔q i and therefore free of six-dimensional Levi-Civita tensor. In conclusion, the only purely bosonic invariants in f 6d n are chiral self-conjugate traces whose projections can be uniquely uplifted from four dimensions Inserting the definition of the gamma matrices, the four-dimensional trace may be written as the sum of two chiral conjugate traces of four-dimensional Pauli matrices There are three possible Lorentz invariants containing supermomenta. All of them have a unique projection to four dimensions Non-chirality of the four-dimensional superamplitudes implies their invariance under the exchanges q 1 i ↔q i3 and q 4 i ↔q i2 . Since Lorentz invariants of the last two types, eqs. (8.6) and (8.7), can only occur pairwise in a six-dimensional superamplitude, it follows that the projection of a six-dimensional superamplitude has always a manifest chiral symmetry in four dimensions. Apparently none of these three six-dimensional Lorentz invariants leads to a manifest R-symmetry in four dimensions. However, any reasonable representation of f 4d n has a manifest R-symmetry. In conclusion, a potential uplift of f 4d n to six-dimensions can only consist of building blocks whose projection to four dimensions is R-symmetric. From the investigation of the three types of six-dimensional Lorentz invariants and their projections, eqs. (8.5) to (8.7), it immediately follows that there is only one such object (8.8) Unlike the claim in [11] there is no combination of six-dimensional Lorentz invariants of the second and third type, eqs. (8.6) and (8.7), that has a R invariant projection to four dimensions. For further details see appendix C. We conclude that if a correspondence of the form eq. (8.1) exists, then the involved representations of f 6d n and f 4d n only contain the building blocks eq. (8.8). As will be explained in the next section, for multiplicities larger than five this is a severe constraint on the representations of f 6d/4d n .

Uplifting massless superamplitudes from four to six dimensions
We want to discuss the implications of eq. (8.8). At four point level f 4d 4 is purely bosonic and the uplift is trivial At five points, any representation of f 4d 5 that has a manifest R-symmetry and a manifest symmetry under chiral conjugation automatically only consists of the building blocks eq. (8.8). Since any reasonable representation of f 4d 5 has a manifest R-symmetry and the chiral symmetry can be made manifest by replacing e. g. the MHV part by the chiral conjugate of the MHV part, any representation of f 4d 5 can be uplifted to six dimensions. By uplifting the representation, eq. (6.33), where the factor of 1 2 originates from the definition (6.32) and we inserted the definition of Ω I;J;K , eq. (7.51). We checked numerically that eq. (8.11) is indeed equal to the five-point amplitude in six dimensions.
Unfortunately the uplift starts to be non-trivial already at multiplicity six. Let {Ω i } denote a set of the chiral self-conjugate building blocks (8.8) for the six-dimensional superamplitudes where ω i = O(q 2 ) andω i = O(q 2 ) are the chiral conjugates in the projection of Ω i . As a consequence of eq. (8.8) an uplift able representation of the six-point amplitudes has the form f 4d and uplifts to f 6d From eq. (8.14) it follows Comparing this with the representation eq. (6.34) obtained for f 4d 6 from the BCFW recursion it is apparent that a generic representation of f 4d 6 does not have the form eq. (8.14) required for an uplift. In contrast to the five point case, making the chiral symmetry manifest does not solve the problem because the minimal helicity violating (minHV) NMHV amplitudes are independent of the MHV and MHV amplitudes. As a consequence, it is straightforward to turn a generic representation into the form but in general the coefficients β ij and γ ij are unrelated. This is the key issue, that to our mind has been overlooked in reference [11]. As a result, finding any representation of f 4d n is not sufficient to obtain the six-dimensional amplitude. In fact, under the assumption that the uplift works, obtaining f 6d n is equivalent to finding a representation of the form obtaining it is non-trivial and a rigorous proof that eq. (8.19) is always a valid representation of the six-dimensional superamplitude is still missing. Of course we could use a numerical implementation of the non-chiral BCFW recursion relation to determine a solution to an ansatz of the form eq. (8.18) but this is not easier than determining f 6d n directly, using the methods described in section 7.1.
Albeit it seems save to say that the uplift is of no practical relevance for the determination of the six-dimensional superamplitudes, it is still very fascinating from the theoretical point of view. It is intriguing that the correct representation of the MHV superamplitude might be sufficient to get the whole six-dimensional superamplitude, or equivalently all massive four-dimensional amplitudes. One thing that would immediately invalidate the uplift are identities of the ω i +ω i that do not uplift to identities of the Ω i . Though we do not have a concrete counterexample for the uplift, there are indeed four-dimensional identities of strings of momenta k i that do not have a six-dimensional counterpart, i. e. 4d: At this point we do not see how such identities could not spoil the uplift without restricting the allowed four-dimensional building blocks.
Using the numerical implementation of the six-dimensional BCFW recursion it is possible to numerically check the uplift. The easiest way to do so is to make an ansatz (7.2) for f 6d n using only the minimal building blocks Ω ijkl defined in eq. (7.15) and determine a solution α i (π) for a massless phase space point with momenta of the form {p 1 i , p 2 i , p 3 i , p 4 i , 0, 0}. Since the coefficients are functions of the Lorentz invariants x 2 ij they have identical numerical values on the 'massive' phase space point {p 1 i , 0, p 3 i , p 4 i , 0, p 2 i } and we can check whether the obtained coefficients α i (π) provide a solution to the massive amplitudes as well. In fact, we checked that up to multiplicity eight that representation of the massless non-chiral amplitudes containing only the minimal building blocks B ijk |B ilm + [B ijk |B ilm ] did always uplift to six dimensions. Since the eight-point amplitude is already very complicated, there is no reason to believe that the uplift of a representation containing only the minimal building blocks will fail beyond eight points. In case of more complicated building blocks the identities (8.21) might become an issue even at multiplicities lower than eight.

Conclusion and outlook
A central motivation for this work was to take first steps towards a generalization of the analytic construction of massless QCD amplitudes from N = 4 SYM ones of [38][39][40] to massive QCD amplitudes by employing N = 4 SYM superamplitudes on the Coulomb branch. For this we constructed all standard and hidden symmetries of the massless sixdimensional superamplitudes of N = (1, 1) SYM theory thereby correcting small mistakes in the proof of the dual conformal symmetry given in [4]. We exploited the symmetries of the six-dimensional amplitudes to derive the symmetries of massive tree amplitudes in N = 4 SYM theory and showed that the five dimensional dual conformal symmetry of the massive amplitudes leads to the presence of non-local Yangian-like generators m (1) , p (1) associated to the masses and momenta in on-shell superspace. An interesting open question is whether or not there exist level-one supermomenta as well.
Furthermore, we explained how analytical formulae for tree-level superamplitudes of N = (1, 1) SYM can be obtained from a numerical implementation of the BCFW recursion relation. The developed method is very general and can be applied to other theories as well. We used it to derive compact manifest dual conformally covariant representations of the five-and six-point superamplitudes. To facilitate the investigation of the six-dimensional superamplitudes we proposed a little group decomposition of them. The little group decomposition is the six-dimensional analog of the MHV-band decomposition in 4d introduced in [9]. It allows a separation into parts of varying complexity as well as the identification of those pieces of the superamplitude that survive in the massless limit to four-dimensions. We exploited the little group decomposition to study UHV amplitudes leading to arbitrary multiplicity formulae valid for large classes of component amplitudes with two consecutive massive legs.
We demonstrated that within a maximally supersymmetric theory it is not always sufficient to consider only gluon amplitudes and the remaining amplitudes follow by supersymmetry. Indeed, the supersymmetrization of the six-dimensional gluon amplitudes, as has been done in reference [17] for the three, four and five point amplitudes, will not necessarily yield the correct superamplitude for multiplicities greater than five. We derived examples of supersymmetric, dual conformally covariant representations of the gluon sector which do not coincide with the superamplitude. Nevertheless, we observed that dual conformal extensions and consequently supersymmetrizations of subsets of amplitudes reproduce at least part of the other component amplitudes. It would be interesting to investigate this in more detail in the future since finding dual conformal extensions of subsets of amplitudes is much simpler than finding the whole superamplitude.
In [11] it has been claimed that all superamplitudes of N = (1, 1) SYM can be obtained by uplifting massless tree-level superamplitudes of N = 4 SYM in non-chiral superspace. In our work we derived the superconformal and dual superconformal symmetries of the non-chiral superamplitudes and used the non-chiral BCFW recursion to prove the dual conformal symmetry as well as to derive the five and six-point superamplitudes. We thoroughly investigated the implications of a potential uplift by identifying the correct fourand six-dimensional Lorentz invariants that should appear in such a correspondence. By performing numerical checks we confirmed the uplift of representations containing only a restricted set of dual conformal covariant and chiral self-conjugate building blocks up to multiplicity eight. However, we proved that finding a representation of the massless non-chiral superamplitudes of N = 4 SYM that can be uplifted is non-trivial for multiplicities larger than five. One possible flaw of the uplift are identities of the four-dimensional building blocks that do not uplift to identities of the corresponding six-dimensional building blocks.
We gave examples of such identities that need to be avoided by restricting the allowed building blocks in order to not spoil the uplift. Despite being of no practical relevance for the determination of the six-dimensional superamplitudes or the massive four-dimensional amplitudes at this point, it is still very fascinating to note that the correct representation of the non-chiral MHV superamplitudes in four dimensions could be sufficient to obtain all six-dimensional superamplitudes, or equivalently all massive four-dimensional amplitudes on the Coulomb branch of N = 4 SYM theory.

A Spinor Conventions
In this appendix we summarize our convention for the four-and six-dimensional spinors and provide the identities relevant for calculations within the spinor helicity formalism.

A.2 Six dimensional Spinors
The six-dimensional Pauli matrices fulfill the algebra Σ µ Σ ν + Σ ν Σ µ = 2η µν . (A.12) We choose the antisymmetric representation They satisfy the following identities The six dimensional Shouten identity reads and contractions of epsilon tensors may be deduced from The first four of the six dimensional sigma matrices are simply related to the Weyl representation of the four dimensional gamma matrices Σ µ = 1 ⊗ · γ µ = 0 −σ µ αβ

C Connection between 4d and 6d Lorentz invariants containing supermomenta
Similar to the 6d Lorentz invariants (8.5), we try to find a combination of the invarnants (8.6) and (8.7) whose four dimensional projection is manifestly R-symmetry invariant.
Because of the non-chiral nature of the six-dimensional amplitudes, the number of chiral and anti-chiral supermomenta are equal and the invariants (8.6) and (8.7) can only appear in the pairs 6d : q i |k 1 . . . k 2r+1 |q j [q k |p 1 . . . p 2s+1 |q l ] (C.1) leading to the following four-dimensional projection However, from eqs. (8.6) and (8.7) it follows that such a term cannot appear from a six dimensional projection. Even for the chiral self-conjugate case, with momenta being the same (k = p, n = m) and supermomenta being conjugate of each other (i = k, j = l), contributions of the form (C.5) do not cancel. Consequently the blocks in eq. (C.1) are irrelevant for a connection between the superamplitudes in six and four dimensions, since the latter are manifestly R-symmetry invariant. Therefore only the invariants of the type eq. (8.8) are natural objects for establishing such a bridge.