An analytic superfield formalism for tree superamplitudes in D=10 and D=11

Journal of High Energy Physics, May 2018

Abstract Tree amplitudes of 10D supersymmetric Yang-Mills theory (SYM) and 11D supergravity (SUGRA) are collected in multi-particle counterparts of analytic on-shell superfields. These have essentially the same form as their chiral 4D counterparts describing \( \mathcal{N}=4 \) SYM and \( \mathcal{N}=8 \) SUGRA, but with components dependent on a different set of bosonic variables. These are the D=10 and D=11 spinor helicity variables, the set of which includes the spinor frame variable (Lorentz harmonics) and a scalar density, and generalized homogeneous coordinates of the coset \( \frac{\mathrm{SO}\left(D-2\right)}{\mathrm{SO}\left(D-4\right)\otimes \mathrm{U}(1)} \) (internal harmonics). We present an especially convenient parametrization of the spinor harmonics (Lorentz covariant gauge fixed with the use of an auxiliary gauge symmetry) and use this to find (a gauge fixed version of) the 3-point tree superamplitudes of 10D SYM and 11D SUGRA which generalize the 4 dimensional anti-MHV superamplitudes.

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An analytic superfield formalism for tree superamplitudes in D=10 and D=11

HJE An analytic superfield formalism for tree superamplitudes in D=10 and D=11 Igor Bandos 0 1 2 IKERBASQUE 0 1 2 Basque Foundation for Science 0 1 2 Models, Superspaces 0 48011 , Bilbao , Spain 1 P. O. Box 644, 48080 Bilbao , Spain 2 Department of Theoretical Physics, University of the Basque Country UPV/EHU Tree amplitudes of 10D supersymmetric Yang-Mills theory (SYM) and 11D supergravity (SUGRA) are collected in multi-particle counterparts of analytic on-shell superfields. These have essentially the same form as their chiral 4D counterparts describing Field Theories in Higher Dimensions; Scattering Amplitudes; Supergravity - N = 4 SYM and N = 8 SUGRA, but with components dependent on a different set of bosonic variables. These are the D=10 and D=11 spinor helicity variables, the set of which includes the spinor frame variable (Lorentz harmonics) and a scalar density, and generalized homogeneous coordinates of the coset SO(D−4)⊗U(1) (internal harmonics). SO(D−2) We present an especially convenient parametrization of the spinor harmonics (Lorentz covariant gauge fixed with the use of an auxiliary gauge symmetry) and use this to find (a gauge fixed version of) the 3-point tree superamplitudes of 10D SYM and 11D SUGRA which generalize the 4 dimensional anti-MHV superamplitudes. 1 Introduction 1.1 1.2 1.3 2.1 2.2 2.3 2.4 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 5.1 5.2 5.3 5.4 5.5 6.1 6.2 6.3 6.4 6.5 An analytic on-shell superfield description of 10D SYM – i – linearized 11D SUGRA 5 Spinor helicity formalism and on-shell superfield descriptions of the Spinor helicity formalism in D=11 Spinor frame and spinor helicity formalism in D=11 Linearized D=11 SUGRA in the Lorentz harmonic spinor helicity formalism Constrained on-shell superfield description of 11D SUGRA Analytic on-shell superfields of 11D SUGRA 5.5.1 5.5.2 SO(9) SO(7)×SO(2) harmonic variables Analytic on-shell superfields from constrained on-shell superfields Notation D=4 spinor helicity formalism D=4 superamplitudes and on-shell superfields 2 Spinor helicity formalism in D=10 Vector harmonics Spinor frame in D=10 D=10 spinor helicity formalism D=10 SYM multiplet in the Lorentz harmonic spinor helicity formalism From constrained to unconstrained on-shell superfield formalism SU( 4 ) invariant solution of the constrained superfield equations The on-shell superfields are analytic rather than chiral Analytic superfields and harmonic on-shell superspace Analytical basis and analytic subsuperspace of the harmonic on-shell superspace On (w, w¯)-dependence of the analytic superfields. Complex spinor harmonics 21 4.6.1 4.6.2 Origin of internal harmonics Comment on harmonic integration 5.6 Supersymmetry transformation of the analytic superfields 6 Analytic superamplitudes in D=10 and D=11 Properties of analytic superamplitudes From constrained to analytic superamplitudes. 10D SYM Analytic superamplitudes of 11D SUGRA from constrained superamplitudes 35 Supersymmetry transformations of the analytic superamplitudes Supermomentum in D=10 and D=11 Convenient parametrization of spinor harmonics (convenient gauge fixing of the auxiliary gauge symmetries) Reference spinor frame and minimal parametrization of spinor harmonics Generic parametrization of spinor harmonic variables and K#I = 0 gauge Internal harmonics and reference internal frame Complex spinor frames and reference complex spinor frame 8 3-point analytic superamplitudes in 10D and 11D Three particle kinematics and supermomentum 3-points analytical superamplitudes in 10D SYM and 11D SUGRA 8.2.1 8.2.2 8.2.3 3-points analytical superamplitude of 10D SYM. Gauge fixed form Searching for a gauge covariant form of the 3-points superamplitude Analytical 3-point superamplitude of D = 11 supergravity 9 Conclusion and discussion A On D=4 spinor helicity formalism A.1 Momentum conservation in a 3-point 4D amplitude A.2 3-gluon amplitude and superamplitude in maximal D=4 SYM variables B BCFW-like deformations of complex frame and complex fermionic 36 36 37 39 40 42 42 43 43 44 46 47 50 53 53 55 1 Introduction An impressive recent progress in calculation of multi-loop amplitudes of d=4 supersymmetric Yang-Mills (SYM) and supergravity (SUGRA) theories, especially of their maximally supersymmetric versions N = 4 SYM and N = 8 SUGRA [1–5], was reached in its significant part with the use of spinor helicity formalism and of its superfield generalization [6, 7, 9–13]. This latter works with superamplitudes depending on additional fermionic variables and unifying a number of different amplitudes of the bosonic and fermionic fields from the SYM or SUGRA supermultiplet. The spinor helicity formalism for D=10 SYM was developed by Caron-Huot and O’Connel in [14] and for D=11 supergravity in [15] (more details can be found in [16]). The progress in the latter was reached due to the observation that the 10D spinor helicity variables of [14] can be identified with spinor Lorentz harmonics or spinor moving frame variables used for the description of massless D=10 superparticles in [17–19]. (Similar observation was made and used in D=5 context in [20]). The spinor helicity formalism of [15] uses 11D spinor harmonics of [21–24]. As far as the generalization of D=4 superamplitudes is concerned, in [14] a kind of Clifford superfield representation of the amplitudes of 10D SYM was constructed. However, this later happened to be quite nonminimal and difficult to apply. Then the subsequent papers [25–28] used the D=10 spinor helicity formalism of [14] in the context of type II – 1 – groups’ SO(D − 2)i of the light-like momenta ka(i) of i-th scattered particles and obey a set of differential equations involving fermionic covariant derivatives Dq+(i). This formalism is quite different from the 4D superamplitude approach; some efforts on development of the necessary technique and on deeper understanding of its structure are still required to be accomplished to make possible its efficient application to physically interesting problems. In this paper we develop a simpler analytic superfield formalism for the description HJEP05(218)3 of 11D SUGRA and 10D SYM amplitudes. In it the superamplitudes are multiparticle counterparts of an on-shell analytic superfields, which depend on the fermionic variable in exactly the same manner as the chiral superfields describing N = 8 SUGRA and N = 4 Spin(D−2) SYM. However, the component fields in these analytic superfields depend on another set of bosonic variables including some internal harmonic variables (see [32–34]) wqA, w¯qA parametrizing the coset Spin(D−4)⊗U(1) . These are used to split the set of (2N ) real spinor fermionic coordinates θq− of the natural on-shell superspaces of 11D SUGRA and 10D SYM on the set of N complex spinor coordinates ηA− and its complex conjugate η¯−A. The analytic on-shell superfields describing 11D SUGRA and 10D SYM depend on η A− but not on η¯−A and, in this sense, are similar to the chiral on-shell superfields describing N = 8 SUGRA and N = 4 SYM. However, as in higher dimensional case η A− = θq−w¯qA is formed with the use of harmonic variable w¯qA, we call these superfields analytic rather than chiral. We show how the analytic superamplitudes are constructed from the basic constrained superamplitudes of 10D SYM and 11D SUGRA and the set of complex (D − 2) component null-vectors UI i related to the internal frame associated to i-th scattered particle. We describe the properties of analytic superamplitudes and present a convenient parametrization of the spinor harmonics (gauge fixing with respect to a set of auxiliary symmetries acting on spinor frame variables), which allows to establish relations between D=10, 11 superamplitudes and their 4d counterparts. Using such relation we have found a gauge fixed expressions for the on-shell 3-point tree superamplitudes. These can be used as basic elements of the analytic superamplitude formalism based on a generalization of the BCFW recurrent relations [7]. The derivation and application of these latter, as well as the use of analytic superamplitudes to gain new insight for further development of the constrained superamplitude formalism will be the subject of future papers. The rest of this paper has the following structure. In the remaining part of the Introduction, after a resume of our notation, we briefly review D = 4 spinor helicity and on-shell superfield description of N = 4 SYM and N = 8 SUGRA. In section 2 we describe the D=10 spinor helicity formalism. In section 3 we review briefly the on-shell superfield description of 10D SYM [21]. Analytic on-shell superfield 1An interesting recent analysis of the divergences of higher dimensional maximal SYM theory [29, 30] avoids an explicit use of the 10D spinor helicity formalism but assumes some generic properties of the amplitudes in this formalism. – 2 – approach is developed in section 4. The spinor helicity formalism, constrained on-shell superfield and analytic on-shell superfield descriptions of D=11 SUGRA are presented in section 5. In section 6 we introduce the analytic D=10 and D=11 superamplitudes and describe their properties and their relation with constrained superamplitudes. A real supermomentum, which is supersymmetric invariant due to the momentum conservation, is introduced there. A convenient parametrization of the spinor harmonics is described in section 7. Its study indicated the necessity to impose a relation between internal harmonics corresponding to different scattered particles, which then allowed to associate a complex spinor frames to each of them. In section 7 we also present a convenient gauge fixing of the auxiliary gauge symmetries which leads to a simple gauge fixed form of both real and complex spinor harmonics. This has been used to obtain gauge fixed expressions for 3-point analytic superamplitudes of 10D SYM and 11D SUGRA, which can be found in section 8. We conclude in section 9. Appendix A is devoted to spinor frame re-formulation of 4D spinor helicity formalism, which is useful for comparison of 4D and 10/11D (super)amplitudes. Appendix B shows how to obtain the BCFW-like deformation of the 10/11D spinor helicity and complex fermionic variables from the deformation of real spinor frame and real fermionic variables found in [14, 15]. 1.1 Notation notation here. As we will use many different types of indices, for reader convenience we resume the index The equations in D=10 and D=11 cases often have similar structure and we use similar notations in these two cases. To describe these in a universal manner and also to stress this similarity, it is convenient to introduce parameters N and s, which take values N = 4, 8 and s = 1, 2 for the case 10D SYM and 11D SUGRA, respectively, 10D SYM : 11D SUGRA : N = 4 , N = 8 , s = 1 , s = 2 . (1.1) (1.2) These characterize the number of supersymmetries and maximal spin of the quanta of the dimensionally reduced theories, N = 4 SYM and N = 8 SUGRA. Clearly, s = N /4. The symbols from the beginning of the Greek alphabet denote Spin(1, D − 1) indices (this is to say, indices of the minimal spinor representation of SO(1, D − 1)) α, β, γ, δ = 1, . . . , 4N . Notice that, when we consider D=4 SYM and SUGRA, we use the complex Weyl spinor indices α, β = 1, 2 and α˙ , β˙ = 1, 2 so that the above equations do not apply. The spinor indices of the small group SO(D −2) (indices of Spin(D −2)) are denoted by The vector indices of SO(D − 2) are denoted by (1.4) (1.5) (1.6) The latter notation also applies to the 4D dimensional reduction of 11D and 10D theories, where A, B, C, D denote the indices of the fundamental representation of SU(N ) R-symmetry group. Finally, a, b, c, d = 0, 1, . . . , (D − 1) are D-vector indices. In D=4 we also use µ, ν, ρ = 0, 1, 2, 3 to stress the difference from D = 10 and D = 11. The symbols i, j = 1, . . . , n are used to enumerate the scattered particles described by n-point (super)amplitude. 1.2 D=4 spinor helicity formalism component bosonic Weyl spinors λ(αi) = (λ¯(α˙i))∗ (α = 1, 2; α˙ = 1, 2), In spinor helicity formalism the scattering amplitudes of n massless particles A(1, . . . , n) := A(p(1), ε(1); . . . , p(n), ε(n)) are considered to be homogeneous functions of n pairs of 2A(1, . . . , n) := A(p(1), ε(1); . . . ; p(n), ε(n)) = A(λ(1), λ¯(1); . . . ; λ(n), λ¯(n)) . (1.3) The spinor λ(αi) carries the information about momentum and polarization of i-th particle. In particular, i-th light-like 4-momentum pµ(i) is determined in terms of λ(αi) = (λ¯(α˙i))∗ by Cartan-Penrose relation (α = 1, 2, α˙ = 1, 2, µ = 0, . . . , 3) [41, 42] µ pAA˙(i) := pµ (i)σαα˙ = 2λα(i)λα˙ (i) ¯ ⇔ pµ (i) = λ(i)σµ λ¯(i). Here σαµα˙ are relativistic Pauli matrices obeying σµ αα˙ σµβ β˙ = 2ǫαβǫα˙ β˙ with ǫαβ = 10 −01 = ǫα˙ β˙ . This identity explains equivalence of two forms of the Cartan-Penrose representation (1.4) a and also allows to show that pa(i)p(i) = 0. The n-particle amplitude is restricted by n helicity constraints where the operator hˆ(i)A(1, . . . , n) = hiA(1, . . . , n) , hˆ(i) := 1 2 ∂ α λ(i) ∂λ(αi) counts the difference between degrees of homogeneity in λ(αi) and λ¯(α˙i). Its eigenvalue hi, the helicity of i-th particle, defines the amplitude homogeneity property with respect to the phase transformations of λ(αi) and λ¯(α˙i), A(. . . , eiβi λ(αi), e−iβi λ¯(α˙i), . . .) = e2ihiβi A(. . . , λ(αi), λ¯(α˙i), . . .) . (1.7) It is quantized: the amplitude is a well defined function of complex variable λ(αi) if and only if βi is equivalent to βi + 2π, and this happens when 2hi ∈ Z. In the case of gluons hi = ±1 and in the case of gravitons hi = ±2. 1.3 D=4 superamplitudes and on-shell superfields A superamplitude of N = 4 SYM or N = 8 supergravity depends, besides n sets of complex bosonic spinors, on n sets of complex fermionic variables η(Ai) ((η(Ai))∗ = η¯A(i)) carrying the index of fundamental representation of the SU(N ) R-symmetry group A, B = 1, . . . , N , A(1; . . . ; n) = A(λ(1), λ¯(1), η(1); . . . ; λ(n), λ¯(n), η(n)) , η(Ai)η(Bj) = −η(Bj)η(Ai) . It obeys n super-helicity constraints, with N 4 hˆ(i)A({λ(i), λ¯(i), η(Ai)}) = A({λ(i), λ¯(i), η(Ai)}) , A = 1, . . . , N ∂ 2hˆ(i) = λ(i) ∂λ(αi) α − λ¯(α˙i) ∂λ¯(α˙i) ∂ A ∂ + ηi ∂ηiA . 1 N ! which obey the super-helicity constraint hˆΦ(λ, λ¯, η) = sΦ(λ, λ¯, η) , s = It is important that the dependence of amplitude on fermionic variables is holomorphic: it depends on ηiA but is independent of η¯A(i) = (η(Ai))∗. Furthermore, according to (1.10), the degrees of homogeneity in these fermionic variables is related to the helicity hi characterizing dependence on bosonic spinors. Hence, decomposition of superamplitude on the fermionic variables involves amplitudes of different helicities. These superamplitudes can be regarded as multiparticle generalizations of the so-called on-shell superfields 2 Φ(λ, λ¯, ηA) = f (+s) +ηAχA + 1 ηBηAsAB +. . .+η∧(N −1)Aχ¯A +η∧N f (−s) , η∧N = ηA1 . . . ηAN ǫA1...AN , η∧(N −1)A = ηB2 . . . ηBN ǫAB2...BN , (1.12) 1 (N −1)! N 4 , 2hˆ = +λα ∂ ∂λα − λ¯α˙ ∂ ∂λ¯α˙ + η A ∂ ∂ηA , A = 1, . . . , N . (1.8) (1.9) (1.10) (1.11) (1.13) (1.14) The chiral superfields on a real superspace Σ(4|2N ) = {λ, λ¯, η, η¯} obeying eq. (1.13) describe the on-shell states of N = 4 SYM and N = 8 SUGRA. They can be considered as homogeneous superfields on chiral on-shell superspace Σ(4|N ) = {λ, λ¯, η} – 5 – satisfying eq. (1.13), which just fixes the charge of superfield with respect to a phase transformations of its arguments.2 Such on-shell superfields can be obtained by quantization of D = 4 Brink-Schwarz superparticle with N -extended supersymmetry in its Ferber-Shirafuji formulation [45, 46] (see also [47, 48] as well as [49] and [50]). This observation has served us as an important guide: in [16] we show how to obtain the 10D and 11D on-shell superfield formalism from D=10 and D=11 superparticle quantization. Here we will not consider superparticle quantization but describe briefly the resulting constrained on-shell superfields and constrained superamplitude formalism of [15, 16] and use these as a basis to search for the analytic on-shell superfields and analytic superamplitude formalism. To conclude our brief review, let us present the expressions for basic building blocks of the 4D superamplitude formalism, the 3-point superamplitudes of D=4 N = 4 SYM theory. These are two: the anti-MHV (MHV) Here we set the SYM coupling constant to unity and use the standard notation for the contraction of 4D Weyl spinors < ij > =< λiλj >= λiαλαj = ǫαβλβiλαj , [ ij ] =< ij >∗= [λ¯i λ¯j ] = λ¯iα˙ λ¯α˙ j = ǫα˙ β˙ λ¯β˙iλ¯α˙ j . 2 Spinor helicity formalism in D=10 As we have already mentioned in the Introduction, the D=10 spinor helicity formalism [14] can be constructed using the spinor (moving) frame or Lorentz harmonic variables. To describe these it is convenient to start with introducing the vector frame variables or vector harmonics (called light-cone harmonics in [51, 52]). 2.1 Vector harmonics The property of vector harmonic variables are universal so that, instead of specifying ourselves to D=10 dimensional case, we write the equations of this section for arbitrary number D of spacetime dimensions. This will allow us to refer on these equations when considering spinor helicity formalism for 11D supergravity. 2The relative charges of bosonic and fermionic coordinates of this phase transformations can be restored from the relation between supertwistors and standard superspace coordinates [45]. In superamplitude context these relations can be found e.g. in [2]. – 6 – Let us consider a vector frame u(abi) = 1 2 # ua(i) + ua=i , uIai , 2 1 ua#i − uai = ∈ SO(1, D − 1) . (2.1) a It can be associated with D-dimensional light-like momentum ka(i), ka(i)k(i) = 0, by the condition that one of the light-like vectors of the frame, say ua=i = u0ai − u(aDi−1), is proportional to this ka(i), k(ai) = ρ(#i)u(ai=) . The additional index i will enumerate particles scattered in the process described by an on-shell amplitude. Below in this section, to lighten the equations, we will omit this index when this does not lead to a confusion. The condition (2.1) implies u(ac)ηabu(d) = η(c)(d), which can be split into [51, 52] b (2.2) (2.3) (2.4) (2.5) (2.6) (2.7) (2.8) ua=ua= = 0 , ua#ua# = 0 , uI ua= = 0 , a ua=ua# = 2 , uI ua# = 0 , a uI uaJ = −δIJ , a and also u(ac)η(c)(d)u(bd) = ηab, which can be written in the form of δab = 12 ua=ub# + 21 ua#ub= − uIaubI . Notice that the sign indices = and # of two light-like elements of the vector frame (see (2.3) and (2.4)) indicate their weights under the transformations of SO(1, 1) subgroup of the Lorentz group SO(1, D − 1), ua= 7→ e−2αua= , ua# 7→ e+2αua# , uIa 7→ uaI . It is convenient to change the basis and to consider the splitting of the vector frame matrix (2.1) on two light-like and (D − 2) orthogonal vectors in the form [51] u(b) = a ua=, ua#, uIa , u(ca)uc(b) = η(a)(b) = 2 0  This is manifestly invariant under the direct product SO(1, 1) ⊗ SO(D − 2) of the above scaling symmetry (2.7) and the rotation group SO(D − 2) mixing the spacelike vectors uIa, SO(D − 2) : ua= 7→ ua= , ua# 7→ ua# , uIa 7→ uaJ OJI , OOT = I . (2.9) If only one light-like vector ua= of the frame is relevant, as it will be the case in our discussion below, the transformations mixing ua# and uIa can be also considered as a symmetry. These are so-called K(D−2) transformations K(D−2) : ua= 7→ ua= , ua# 7→ ua# + uIa K#I + 41 ua=(K#I K#I ) , uIa 7→ uIa + 21 ua=K#I (2.10) (identified in [17, 18] as conformal boosts of the conformal group of Euclidean space). – 7 – To make the associated momentum (2.2) invariant under SO(1, 1) transformations (2.7), we have to require that ρ# 7→ e+2αρ# , (2.11) and this explains the index # of ρ multiplier in (2.2). Of course, we can use (2.11) to set ρ# = 1. However, it happens to be much more convenient to keep SO(1, 1) unfixed and to use it as identification relation (gauge symmetry acting on) vector harmonics (2.1). The complete expression for light-like momentum (2.2) is invariant under HB = [SO(1, 1) ⊗ SO(D−2)] ⊂×K(D−2) transformations (2.7), (2.9), (2.10). This is the Borel subgroup of SO(1, D − 1) so that SO(1, D − 1)/HB coset is compact; actually it is isomorphic to the sphere S(D−2). If we use H transformations as identification relation on the set of vector harmonics, these can be considered as a kind of homogeneous coordinates of such a sphere [17, 18] ua=, ua#, uIa o = SO(1, D − 1) [SO(1, 1) ⊗ SO(D − 2)] ⊗ K(D−2) = S(D−2) . (2.12) Such a treatment as constrained homogeneous coordinates of the coset makes the vector frame variable similar to the internal coordinate of harmonic superspaces introduced in [32, 33], and stays beyond the name vector harmonics or vector Lorentz harmonics, which we mainly use for them. In the context of (2.2), S(D−2) in (2.12) can be identified with the celestial sphere of a D-dimensional observer. Notice that this is in agreement with the fact that a light-like Dvector defined up to a scale factor can be considered as providing homogeneous coordinates for the S(D−2) sphere {ua=} = S(D−2) . The usefulness of seemingly superficial construction with the complete frame (2.12) becomes clear when we consider spinor frame variables, which provide a kind of square roots of the light-like vectors of the Lorentz frame. 2.2 Spinor frame in D=10 b To each vector frame u(a) we can associate a spinor frame described by Spin(1, D − 1) valued matrix Vα(β) ∈ Spin(1, D − 1) related to u(ba) by the condition of the preservation of D-dimensional Dirac matrices V ΓbV T = u(ba)Γ(a) , V T Γ˜(a)V = Γ˜bu(a) , b and also of the charge conjugation matrix if such exists in the minimal spinor representation of D-dimensional Lorentz group, V CV T = C , if C exists for given D . (2.13) (2.14) (2.15) – 8 – In the case of D=10, where the minimal Majorana-Weyl (MW) spinor representation is 16-dimensional, the SO(1, 1) × SO(8) invariant splitting of vector frame in (2.1) is reflected by splitting the spinor frame matrix on two rectangular blocks, vα+q˙ and vα−q, Vα(β) = vα+q˙, vα−q ∈ Spin(1, D − 1) , which are called spinor frame variables or Lorentz harmonic (spinor Lorentz harmonic). Their sign indices ± indicate their scaling properties with respect to the SO(1, 1) transformations, and their columns are enumerated by indices of different, c-spinor and s-spinor representations of SO(8) group, The set of constraints on 10D Lorentz harmonics are given by eqs. (2.14) in which Γaαβ = σαaβ = σβaα and Γ˜a αβ = σ˜a αβ = σ˜a βα are 16×16 generalized Pauli matrices, which obey σaσ˜b + σbσ˜a = 2ηabI16×16. We prefer to write this relation in the universal form Γaαγ Γ˜b γβ + Γbαγ Γ˜a γβ = 2δαβ , which also describes the properties of symmetric 32×32 11D Dirac matrices introduced below (see section 5.2). The charge conjugation matrix does not exist in 10D Majorana -Weyl spinor representation so that there is no way to rise or to lower the spinor indices. The elements of the inverse of the spinor frame matrix V(β)α = v−α! q˙ v+α q ∈ Spin(1, D − 1) Vα(β)V(β)γ := vα+q˙vq−˙γ + vα−qvq+γ = δαγ v−αvα+p˙ = δq˙p˙ , q˙ v+α q vα+p˙ = 0 , v−αvα−q = 0 , q˙ vq+αvα−p = δqp . (2.16) (2.18) (2.19) (2.20) (2.21) (2.22) (2.23) (2.24) are introduced as additional variables, which obey the constraints and relations For brevity, we will call vq−˙α and vq+α inverse harmonics. The constraints (2.14) can be split on the following set of SO(1, 1) ⊗ SO(8) covariant ua=Γaαβ = 2vαq−vβq− , vq+˙ Γ˜avp+˙ = ua#δq˙p˙ , vq− Γ˜avp+˙ = uIaγqIp˙ , ua=δqp = vq− Γ˜avp− , 2vαq˙+vβq˙ + = Γaαβua# , 2v(α|q−γqIq˙v|β)q˙ + = ΓaαβuIa , – 9 – where γqIp˙ = γ˜p˙q with I = 1, . . . , 8 are SO(8) Clebsh-Gordan coefficients obeying I γI γ˜J + γJ γ˜I = δIJ I8×8 , γ˜I γJ + γ˜J γI = δIJ I8×8 . Although the constraints for the inverse harmonics (2.19) ua=Γ˜a αβ = 2vq−˙αvq−˙β , vq+Γavp+ = ua#δqp , vq−˙Γavp+ = −uIaγpIq˙ , ua=δq˙p˙ = vq−˙Γavp−˙ , 2vq+αv+β = Γ˜aαβua# , q 2vq−˙(αγqIq˙vq+β) = −Γ˜aαβuIa , (2.25) (2.26) (2.27) (2.28) (2.29) (2.30) (2.31) (2.32) (2.33) (2.34) can be obtained from (2.22)–(2.24) and (2.21), it is convenient to keep their form in mind. The constraints (2.22) allow us to treat harmonic vα−q as a kind of square root of the light-like vector ua= of the vector frame. Similar to this latter, vα−q can be also treated as a constrained homogeneous coordinates of the coset isomorphic to the celestial sphere HJEP05(218)3 {vα−q} ∈ S8 . Actually, eq. (2.29) abbreviates the spinorial counterparts of (2.12) and (2.13); the complete form of the first of these is {(vα+q˙, vα−q)} = Spin(1, 9) [SO(1, 1) ⊗ Spin(8)] ⊂×K8 = S8 , harmonics vα+q˙ by where KD−2 (K8 in our 10D case) leaves vα−q invariant and acts on the complementary KD−2 : vα+q˙ 7→ vα+q˙ + K#I vα−pγpIq˙ . 1 2 In a model with [SO(1, 1) ⊗ Spin(D − 2)] ⊂×KD−2 gauge symmetry vα+q˙ does not carry degrees of freedom: any vα+q˙ forming Spin(1, D − 1) matrix with given vα−q can be obtained from some reference solution of this condition, vα+q˙0, by KD−2 transformations (2.31). This justifies the simplified form of (2.29) where only vα−p are presented as the constrained homogeneous coordinates of the sphere. 2.3 D=10 spinor helicity formalism When the vector frame is attached to a light-like momentum as in (2.2), the constraints (2.22) for the associated spinor frame imply that the following D=10 counterparts of the D=4 Cartan-Penrose relations (1.4) hold: kaΓaαβ = 2ρ#vα−qvβ−q , ρ#vq− Γ˜avp− = kaδqp. In D=10 we should also mention the existence of the similar relations for the inverse harmonics (2.19), ka Γ˜a αβ = 2ρ#vq−˙αvq−˙β , ρ#vq−˙Γavp−˙ = kaδq˙p˙ . ka = ρ#ua= , D=10 SYM multiplet in the Lorentz harmonic spinor helicity formalism The polarization vector of the vector field can be identified with spacelike vectors uIa of the frame adapted to the light-like momentum of the particle by (2.32) (cf. [14]) so that the on-shell field strength of the D=10 gauge field can expressed by in terms of one SO(8) vector wI . It is easy to check that both Bianchi identities and Maxwell equations in momentum representations are satisfied, k[aFbc] = 0 = kaF ab. As we have already said, the polarization spinor can be identified with the spinor frame variable vq−˙α. Hence, in the linear approximation, the on-shell states of spinor superpartner of the gauge field can be described by When describing the on-shell states of the SYM multiplet, it is suggestive to replace ρ# by its conjugate coordinate and consider the field on the nine-dimensional space R ⊗ S8: wI = wI (x=, vq−) ψq = ψq(x=, vq−) . (2.35) (2.36) (2.37) (2.39) (2.40) (2.41) (2.42) (2.43) in terms of a fermionic SO(8) c-spinor ψq˙. Indeed, due to (2.35), the field (2.39) solves the free Dirac equation. When the formalism is applied to external particles of scattering amplitudes, the bosonic wI and fermionic ψq˙ are considered to be dependent on ρ# and on spinors harmonics vα−q related to the momentum of the particle through (2.33), wI = wI (ρ#, vq−) ψq = ψq(ρ#, vq−) The supersymmetry acts on these 9d fields by δǫψq˙(x=, vq−) = ǫ−qγqIq˙ wI (x=, vq−) , δǫwI (x=, vq−) = 2iǫ−qγqIq˙∂=ψq˙(x=, vq−) , where 8 component fermionic ǫ−q is the contraction of the constant fermionic spinor ǫα with the spinor frame variable, Contracting the first equations in (2.33) and in (2.34) with v−β and vα−q, and usq˙ ing (2.21) we easily find that these obey the massless Dirac equations (or, better to say, D = 10 Weyl equations) kaΓaαβvq−˙β = 0 , kaΓ˜a αβvβ−q = 0 . Thus, they can be identified, up to a scaling factor, with D=10 spinor helicity variables harmonics vq−˙α: The polarization spinor of the D=10 fermionic fields [14] can be associated with the inverse χα = vq−˙αψq˙ and and ǫ−q = ǫαvα−q . The above described fields of the spinor helicity formalism for 10D SYM can be collected in on-shell superfields, which can be considered as one-particle prototypes of tree superamplitudes. A constrained on-shell superfield formalism for linearized 10D SYM was proposed in [21]. We briefly describe that in this section and, in the next section 4, use it as a starting point to obtain a new analytic superfield description of 10D SYM. In [21] the constrained superfields describing 10D SYM are defined on the real on-shell superspace with bosonic coordinates x= and vα−q, and fermionic coordinates θq− The 10D supersymmetry acts on the coordinates of Σ(9|8) by (3.1) (3.2) (3.4) (3.5) (3.6) δǫx= = 2iθq− ǫαvα−q , δǫθq− = ǫαvα−q , δǫvα−q = 0 . This specific form indicates that our on-shell superspace Σ(9|8) can be regarded as invariant subspace of the D=10 Lorentz harmonic superspace, i.e. of the direct product of standard 10D and 11D superspaces and of the internal sector parametrized by Lorentz harmonics Spin(1,9) (vα+q˙, vα−q) ∈ Spin(1, 9) considered as homogeneous coordinates of the coset Spin(1,1)⊗Spin(8) . The generic unconstrained superfield on Σ(9|8) (3.1) contains too many component fields so that on-shell superfield describing linearized D=10 SYM should obey some superfield equations. Such equations have been proposed in [21]. To write them in a compact form we will need the fermionic derivatives covariant under (3.2) , ∂q+ := ∂ ∂θq− , These carry the s-spinor indices of Spin(8) group and obey d = 1 N = 8 extended supersymmetry algebra A one particle counterpart of a superamplitude is actually given by Fourier images of the superfield on (3.1) with respect to x=. These will depend on the set of coordinates (ρ#, θq−, vα−q), where ρ# is a momentum conjugate to x=. The fermionic covariant derivative acting on such Fourier-transformed on-shell superfields reads and obeys {Dq+, Dp+} = 4iδqp∂= . Dq+ = ∂q+ + 2ρ#θq− , {Dq+, Dp+} = 4ρ#δqp . 3.2 The basic superfield equations of D=10 SYM [21] are imposed on the fermionic superfield Ψq˙ = Ψq˙(x=, θq−˙, vαq˙−) carrying c-spinor index of SO(8). The superfield V I is defined by eq. (3.7) itself, which also imply that it obeys Dq+V I = 2iγqIq˙∂=Ψq˙ . This equation shows that there are no other independent components in the constrained An analytic on-shell superfield description of 10D SYM In this section we present an analytic superfield formalism for the on-shell D=10 SYM, which is alternative to both the Clifford superfield approach of [14] and to the constrained superfield formalism, which we have described above (more details can be found in [16]). We begin by solving the equations of the constrained on-shell superfields of 10D SYM from [21] in terms of one analytic on-shell superfield. In section 6 we generalize this for the case of superamplitudes and describe an analytic superamplitude formalism. 4.1 From constrained to unconstrained on-shell superfield formalism To arrive at our unconstrained superfield formalism it is convenient to write the superspace equations (3.8) and (3.7) for on-shell superfields describing 10D SYM [21] in the form of Dq+W I = 2iγqIq˙Ψq˙ , Dq+Ψq˙ = γqIq˙ ∂=W I , The superfield V I in (3.7) and (3.8) is related to W I by V I = ∂=W I . After such a redefinition, we can discuss the bosonic superfield W I as fundamental and state that Ψq˙ is defined by the γ-trace part of (4.1). The first terms in its decomposition on fermionic coordinates are W I = wI + 2iθ−γI ψ + iθ−γIJ θ−∂=wI − 3 2 θ−γIJ θ− θ−γI ∂=ψ + . . . . We are going to show that, after breaking SO(8) symmetry down to its SO(6)=SU( 4 ) subgroup, eq. (4.1) splits into a chirality condition for a single complex superfield (Φ = W 7 + iW 8) and other parts which, together with (4.2), allow to determine Ψq˙ and all the remaining components of W I in terms of this single chiral superfield. 4.2 SU(4) invariant solution of the constrained superfield equations Breaking SO(8) 7→ SO(6) ⊗ SO(2) ≈ SU( 4 ) ⊗ U(1), we can split the vector representation 8v of SO(8) on 6+1+1 of SO(6), W I = (W Iˇ, W 7, W 8), Iˇ = 1, . . . , 6 . (4.3) Dq+ Φ¯ = − δqp − i(γ7γ˜8)qp Ψp. 1 2 Pq±p = δqp ± i(γ7γ˜8)qp , are orthogonal projectors and hance that (4.5) implies conditions with P+P+ = P+ , P+ + P− = I , P−P− = P− , (P+)∗ = P− , P+P− = 0 δqp − i(γ7γ˜8)qp Dp+Φ = 0 , δqp + i(γ7γ˜8)qp Dp+Φ¯ = 0. As, according to (4.8), the projectors P+ and P− are complementary and complex conjugate, we can introduce complex 8×4 matrix wqA and its complex conjugate w¯qA such that δqp + i(γ7γ˜)q8p = 2wqAw¯pA , δqp − i(γ7γ˜)q8p = 2w¯qAwpA. In terms of these rectangular blocks eqs. (4.9) can be written as chirality (analyticity) Then introducing we find that (4.1) implies Φ = W 7 − iW 8 2 , Φ¯ = W 7 + iW 8 2 , Ψq = γq8q˙Ψq˙ , It is important to notice that the matrices The remaining parts of eqs. (4.5) determine the fermionic superfield Ψq˙, Ψq˙ = wq˙AΨ¯ +A + w¯q˙AΨA+ , ΨA+ = − DA+Φ , i 4 Ψ¯ +A = − i D¯ A+ Φ¯ . 4 Eq. (4.2) allows us to find also the derivatives of the remaining 6 components W Iˇ of the SO(8) vector superfield W I , 1 8 ˇ ∂=W I = (γIˇγ˜8)qpDq+Dp+(Φ − Φ¯) . To conclude, we have solved the equations for constrained on shell superfields of 10D SYM [21] in terms of one chiral (analytic) on-shell superfield Φ and its c.c. Φ¯ (4.4). Our solution breaks explicitly the manifest SO(D − 2) = SO(8) ‘little group’ invariance of the constrained superfield formalism down to SO(D − 4) = SO(6) (called ‘tiny group’ in [26]). Actually, one can avoid this explicit SO(8) 7→ SO(6) ⊗ SO(2) ≈ SU( 4 ) ⊗ U(1) symmetry breaking by using the method of harmonic superspaces [32, 34]. To this end we must write the general solution of the constrained superfield equations in a formally SO(8) invariant form by introducing a ‘bridge’ coordinates parametrizing SO(8)/[SU( 4 ) ⊗ U(1)] coset: the SO(8) valued matrix I U (J) = UI Jˇ, UI (7), UI (8) = UI Jˇ, 1 2 UI + U¯I , UI − U¯I 1 2i ∈ SO(8) . (4.15) This is transformed by multiplication on SO(8) matrix from the left and by multiplication by SO(6) × SO(2) ⊂ SO(8) matrix from the right. The conditions of orthogonality of the U (J) matrix (4.15), U (J)U (K) = δ(J)(K), imply that the complex vector UI is null and has I I I the norm equal to 2, UI UI = 0 , U¯I U¯I = 0 , UI U¯I = 2 , as well as that it is orthogonal to six mutually orthogonal real vectors UI Iˇ UI UI Jˇ = 0 , U¯I UI Jˇ = 0 , UI JˇUI Kˇ = δJˇKˇ . Now we can easily define SO(8) covariant counterparts of the projectors in (4.6) Pq+p = Pq−p = 1 2 1 2 δqp + i(γI γ˜J )qpUI(7)U (8) J δqp − i(γI γ˜J )qpUI(7)U (8) J = = 4 4 1 γI γ˜J U¯I UJ , 1 γI γ˜J UI U¯J . Furthermore, we can define the 8×8 SO(8) valued matrices wq(p) and wq(˙p˙), which are γqIp˙UI(J) = wq(p)γ((pJ))(q˙)w p(˙q˙) , q q w(p′)w(q′) = δ(p)(q) , p˙ p˙ w(q′˙) w(p′˙) = δ(q˙)(p˙) . The elements of these real matrices can be combined in two rectangular 8 × 4 complex related to (4.17) by conjugate blocks These obey (cf. (4.10)). and factorize the orthogonal projectors (4.18) (4.16) (4.17) (4.18) (4.19) (4.20) (4.21) (4.22) (4.23) wqAw¯pA + w¯qAwpA = δqp , w¯qBwqA = δBA , wqAwqB = 0 , w¯qAw¯qB = 0 , Pq+p = 4 1 γI γ˜J U¯I UJ = wqAw¯pA , Pq−p = 4 1 γI γ˜J UI U¯J = w¯qAwpA wqA = (w¯qA)∗ , wq˙A = (w¯q˙A)∗ , A = 1, 2, 3, 4 . With a suitable choice of representation of 8d Clebsch-Gordan coefficients γqIq˙ = γ˜Iq˙q, the first equation in (4.19) can be split into In (4.24) U/qJp˙ := γqIp˙UIJ = iwqAσAJˇBwpB˙ + iw¯qAσ˜JˇABw¯p˙B , ˇ ˇ U/qp˙ := γqIp˙UI = 2w¯qAwpA˙ , U/¯qp˙ := γqIp˙U¯I = 2wqAw¯p˙A . σAB = −σBIA = −(σ˜IˇAB)∗ = Iˇ ˇ 1 2 ǫABCDσ˜IˇCD , Iˇ = 1, . . . , 6 , A, B, C, D = 1, . . . , 4 are 6d Clebsch-Gordan coefficients which obey σIˇσ˜Jˇ + σJˇσ˜Iˇ = 2δIˇJˇδAB , σAIˇBσ˜IˇCD = −4δ[AC δB]D , ˇ Iˇ I σAB σCD = −2ǫABCD . (4.27) Using (4.25) and (4.22), it is not difficult to check that eqs. (4.23) are satisfied.3 The above bridge coordinates or harmonic variables [32–34] can be used to define the SO(8) invariant version of complex covariant derivatives (4.12), and of complex linear combinations of 8 bosonic superfields W I Φ = W I UI , Φ¯ = W I U¯I , which are analytic and anti-analytic, (4.11). The expression for fermionic superfield Ψq˙ can be written in the form of (4.13), but now with w and w¯ factorizing the covariant projectors (4.23). It is also not difficult to write the covariant counterpart of the expression (4.14) for other 6 projections W Iˇ = W J UJIˇ of the 8-vector superfield W I . However, a more straightforward expression for W Iˇ = W J U I ˇ in terms of Φ reads where DJˇ = , is one of the covariant harmonic derivatives (first introduced in [32] and [33] for SU(2)/U(1) and SU(3)/[U(1)×U(1)] harmonic variables). In our case the other covariant derivatives are W I = −DIˇΦ , ˇ + + 3 In this calculation and below the following identity is useful DJˇ = 1 ∂UIJˇ − U Jˇ ∂ ∂ I ∂U¯I i σ˜JˇABw¯qB ∂wqA − ∂ 2 2i σJAˇBwqB˙ ∂w¯q˙A ∂ (4.24) (4.25) (4.26) conjugate to (4.30), and + i σIˇJˇBA 2 ∂ wqB˙ ∂wqA˙ −w¯q˙A ∂w¯q˙B ! + i σIˇJˇBA 2 ∂ ! , D(0) = UI ∂UI ∂ −U¯I ∂ ∂U¯I + 1 2 ∂ w¯qA ∂w¯qA −wqA ∂w∂ qA + 1 2 wqA˙ ∂wqA˙A −w¯q˙A ∂w¯q˙A ∂ ! , (4.32) DIˇJˇ = 1 2 U KI ˇ ∂ ∂U KJˇ −U KJ ˇ ∂ ˇ ∂U KI ∂ wB q ∂wqA −w¯qA ∂w¯qB + ∂ ∂ providing the differential operator representation of the U(1) and Spin(6) = SU( 4 ) generators on the space of internal harmonics. These covariant derivatives preserve all the constraints on harmonic variables, eqs. (4.21), (4.22), (4.24) and (4.25), and form the so(8) algebra. One can easily check that, by construction, our analytic superfield (4.28) obeys These equations are consistent with the analyticity conditions (4.11) as DJˇΦ = 0 , DIˇJˇΦ = 0 , D(0)Φ = Φ . [DJˇ, D¯ A+] = 0 . internal harmonics. Analytic superfields and harmonic on-shell superspace Thus, we have solved the superfield equations for constrained on-shell superfields of D = 10 SYM in term of one complex analytic superfield Φ obeying the chirality-type equation (4.11) with complex fermionic derivatives (4.12) defined with the use of Spin(6)⊗U(1) = SU( 4 )⊗U(1) coset coordinates (4.21), (4.22), which we, following [32–34], call harmonic variables or Spin(8) SO(8) These analytic superfields are actually defined on a ‘harmonic on-shell superspace’ which can be understood as direct product of the on-shell superspace (3.1) and the Σ(3(D−3)|2N ) = {(x=, vα−q; w¯qA, wqA; θq−)} , {x=} = R1 , {vα−q} = SD−2 , {(w¯qA, wqA)} = Spin(D − 2) Spin(D − 4) ⊗ U(1) . Here and below in (4.42), to exclude the literal repetition of the same equations, we write them in the form applicable both for D = 10 and D = 11 cases, for which q = 1, . . . , 2N , α = 1, . . . , 4N , N = ( 4 8 for D = 10 , for D = 11 . (4.33) HJEP05(218)3 (4.34) (4.35) (4.36) (4.37) (4.38) (4.39) Supersymmetry acts on the coordinates of the harmonic on-shell superspace by (cf. (3.2)) δǫx= = 2iθq− ǫαvα−q , δǫθq− = ǫαvα−q , δǫvα−q = 0 , δǫw¯qA = 0 = δǫwqA , and leaves invariant the covariant derivatives (3.3) as well as D¯ A+ = w¯qADq+ used to define analytic superfields Φ by D¯ A+Φ = 0, (4.11). To see that the analytic superfields are actually functions on a sub-superspace of (4.38), we have to pass to the analytic coordinate basis. Analytical basis and analytic subsuperspace of the harmonic on-shell superspace The presence of additional harmonic variables allows to change the coordinate basis of the harmonic on-shell superspace Σ(3(D−3)|2N ) to the following analytical basis Σ(3(D−3)|2N ) = {(xL=, vα−q; w¯qA, wqA; ηA−, η¯−A)} , xL= : = x = + 2iηA−η¯−A , ηA− := θq−w¯qA , η¯−A = θq−wqA . The supersymmetry acts on the coordinates of this basis by δǫxL= = 4iηA−ǫ¯−A , δǫηA− = ǫA− , δǫη¯−A = ǫ¯−A , where It is generated by the differential operators and leaves invariant the covariant derivatives4 ǫA− = ǫαvα−qw¯qA , ǫ¯−A = ǫαvα−qwqA . Q¯A+ = ∂¯A+ + 4iηA−∂=L , Q+A = ∂+ A D¯ A+ = ∂¯A+ ≡ ∂ ∂η¯−A , D+A = ∂A+ + 4iη¯−A∂=L , D= = ∂=L ≡ ∂ ∂xL= . The harmonic covariant derivatives in the analytical basis have the form DJ = DJˇ − ˇ DJ = DJˇ − ˇ D(0) = D(0) + DIˇJˇ = DIˇJˇ + 2i ηA−σ˜JˇAB ∂ ∂η¯−B i η¯−AσJAˇB ∂ηB− , ∂ 2 2 1 − ∂ 2 ηA ∂ηA− − i η¯−BσIˇJˇBA 1 η¯−A 2 ∂ ∂η¯−A , ∂η¯−A − 2 , ∂ i σIˇJˇBAη− ∂ A ∂ηB− , where DJˇ, DJˇ, D(0) and DIˇJˇ formally coincides with (4.31), (4.30), (4.32) and (4.33). bosonic derivative ∂=L only. 4To be rigorous, one might want to write the L symbol also on the fermionic derivatives in (4.46), ∂¯A+ 7→ ∂¯A+L, ∂+A 7→ ∂+LA. We, however, prefer to make the formulae lighter and write this symbol on the (4.40) (4.41) (4.42) (4.43) (4.44) (4.45) (4.46) (4.47) (4.48) (4.49) (4.50) This implies that the analytic superamplitude of D=10 SYM obeying (6.14) have to be constructed with the use of w¯qA i, wqA˙i and UI i variables only. Similarly, the analytic 11D superamplitudes (6.21) are constructed with the use of w¯qA i and UI i variables. To motivate this identification, let us recall that the only role of the internal harmonics is to split the real fermionic variables θq−i on a pair of complex conjugate ηA−i and η¯i−A thus introducing a complex structure (see discussion in section 4.6). Our choice implies that we induce all the complex structures, for all i = 1, . . . , n, from a single complex structure. This latter is introduced with the reference internal frame (w¯qA, wqA) which serves as a compensator for Spin(D − 2) gauge symmetry of the reference spinor frame. Complex spinor frames and reference complex spinor frame The identification of all the sets of internal harmonics through (7.21) and (7.23) automatically implies that the SO(D − 2) symmetry transformations of the reference spinor frame acts also on the reference internal frame. This allows to introduce a complex reference spinor frame (cf. (4.69)) vα−A := vα−qw¯qA , v¯α−A := vα−pwpA , vα+A := vα+p˙w¯p˙A , v¯α+A := vα+p˙wp˙A , vA−α := vq−˙αw¯q˙A, v¯−Aα := vq−˙αwq˙A, vA+α := vq+αw¯qA, v¯+Aα := vq+αw A q and to express the complex spinor harmonics vαA i := vα−q iw¯qA i , − vA−iα := vq˙i −αw¯q˙Ai , v¯α−iA := vα−qiwqAi , vα+Ai := vα+q˙iw¯q˙Ai , v¯α+i A := vα+q˙iwq˙Ai , v¯−Aα := vq˙ i i −αwq˙Ai , vA+iα := vq+iαw¯qiA , v¯+αA := vq+iαwqAi i in terms of that. In particular, one finds and and vαAi = e−αi−iβi U A†Bi − − vαB + i v¯−αA = e−αi−iβi v¯−αB − 1 2 Ki=I UI vα+B + i 2 Ki=I UIJˇσBJˇC v¯α+C 1 2 Ki=I UI v¯+αB − i 2 Ki=I UIJˇvC+ασ˜JˇCB v¯α−iA = e−αi+iβi v−B + α vA−iα = e−αi+iβi U A†Bi B v−α − 1 2 Ki=I U¯I vα+B + i 2 Ki=I UIJˇσ˜JˇBC v¯α+C 1 2 Ki=I U¯I vB+α + i 2 Ki=I UIJˇσBJˇC v¯+Cα , UBAi , UBAi , . The complex spinor harmonics (7.28) and (7.29) obey vA−αv−B = 0 , α vA+αv−B = δAB , α vA+αv+B = 0, α vA−αv+B = δAB . α v±Aαv±B = 0 , α vA±αvα±B = 0 , ˇ DjJ UI i = 0 . (7.27) (7.28) (7.29) (7.30) (7.31) (7.32) (7.33) (7.34) (7.35) (7.36) HJEP05(218)3 The product of harmonics from different frames, say i-th and j-th, can be calculated using (7.32), (7.34) and (7.36). Clearly for j = i the relation of the form of (7.36) are reproduced for i-th set of complex spinor harmonics. In particular, eqs. (7.32) and (7.33) imply < i−BjA− >:= v¯i−αBvα−Aj = UEBi e−αi−iβi 1 2 Kj=iI UI U A†Ej e−αj−iβj = 1 2 Kj=iI UI U A†Bjie−αi−αj−iβi−iβj . The expression in the first line of (7.37) is convenient to calculate products of bracket matrices, while the second is more compact due to the use of notation U A†Bji := U A†Cj UCBi = UABij . When deriving eqs. (7.32)–(7.37) the following consequences of (4.21), (4.24) and (4.25) are useful w¯qAγqIp˙ = w¯p˙AUI + iσAJˇBwpB˙UIJ , ˇ γqIp˙wp˙A = UI wqA + iw¯qBσ˜JˇBAUIJ , ˇ wqAγqIp˙ = wpA˙U¯I + iσ˜JˇABw¯p˙BUIJ , ˇ γqIp˙w¯p˙A = w¯qAU¯I + iwqBσJBˇAUIJ . ˇ One can also calculate the expressions for < i−j−A >:= vB−iαv¯α−jA. However, in our B perspective the contraction (7.37) is much more interesting as far as it obeys ˇ DlJ < i−BjA− >= 0 , ∀l = 1, . . . , n . The expressions for complimentary harmonics in terms of complex reference spinor frame simplify essentially in the gauge (7.17), K#I = 0, where (7.20) and (7.21), (7.23) i η˜A−i := eαi+iβi UABi ηB−i , (7.37) (7.38) (7.39) (7.40) (7.41) (7.42) (7.43) (7.44) (7.45) (7.46) result in and This allows to find vα+Ai = U A†Bivα+Beαi+iβi , v¯+Aα = v¯+BαUBAieαi+iβi , i v¯α+i A = v¯α+BUBAieαi−iβi , vi+Aα = U A†BivB+αeαi−iβi . < i+BjA− >:= v¯i+αBvα−Aj = U A†Bjie−αji−iβji , < i−BjA+ >:= v¯i−αBvα+Aj = U A†Bjie+αji+iβji = UABij e−αij−iβij . Let us stress that these are gauge fixed expressions: when K#I 6= 0 the r.h.s.-s will acquire the contributions proportional to (7.11). ηA−i carrying SU( 4 )i index, U(1)i charge and SO(1, 1)i weight to Using the bridges eαi , eiβi and UABi we can transform the complex fermionic variable which is inert under SU( 4 )i ⊗ U(1)i ⊗ SO(1, 1)i but transforms nontrivially under the gauge symmetry SU( 4 ) ⊗ U(1) ⊗ SO(1, 1) of the reference complex spinor frame. The advantage of such variables is that η˜A−i + η˜A−j is covariant for any values of i and j. The expressions in the r.h.s.-s of eqs. (7.44) and (7.45), as well as of their counterparts with j = 0, corresponding to the reference complex spinor frame, < i+B .A− >:= v¯i+αBvα−A = UABieαi+iβi , < i−B .A+ >:= v¯i−αBvα+A = UABi e−αi−iβi , (7.47) < .+BjA− >:= v¯+αBvα−Aj = U A†Bje−αj−iβj , < .−BjA+ >:= v¯−αBvα+Aj = U A†Bje+αj+iβj , (7.48) can be used as covariant counterparts of the above SU( 4 )i ⊗ U(1)i ⊗ SO(1, 1)i bridges. They will be useful below in the discussion on 3-point superamplitude of 10D SYM. In particular, it will be important that < i+B .C− >< .−C iA+ > = δABe2αi+2iβi , < i−B .C+ >< .+C iA− > = δABe−2αi−2iβi , < .−BiC+ >< i+C .A− > = δABe+2αi+2iβi , (7.49) < .+BiC− >< i−C .A+ > = δABe−2αi−2iβi , (7.50) det < i−B .A+ >= e−4αi−4iβi , det < i+B .A− >= e4αi+4iβi , (7.51) Then, using (7.2)–(7.4) we split (8.1) into Eq. (8.2) makes (8.3) equivalent to represent the scale and phase factors corresponding to i-th particle. 3-point analytic superamplitudes in 10D and 11D Three particle kinematics and supermomentum and 8 8.1 Let us study 3-particle kinematics in the vector frame formalism. With (2.2) we can write the momentum conservation as where Kj=iI ≡ K[=jiI] = Kj=I − Ki=I (7.12). Using (8.5) and (8.2) we find that (8.4) implies −→ (K 21)2 = 0 ⇒ −→ (K 13)2 = 0 , −→ (K 32)2 = 0 . (8.1) (8.2) (8.3) (8.4) (8.5) (8.6) ρ1#u1=a + ρ2#u2=a + ρ3#u3=a = 0 . ρ 1# + ρ2# + ρ3# = 0 , ρ1#K1=I + ρ2#K2=I + ρ3#K3=I = 0 , # −→ ρ1 (K 1)2 + ρ2 (K 2)2 + ρ3 (K 3)2 = 0 . # −→ # −→ ρ # 1 K1=3I = ρ # 2 K#2=1I , ρ 3 momenta kia of the scattered particles are complex. The solution of eqs. (8.6) for real vectors Kj=iI are trivial. Thus a nontrivial on-shell 3particle amplitude can be defined only for complexified Kj=iI which implies that the light-like The general solution of the momentum conservation conditions can be written in terms of say K1=I and complex null vector KI as K2=I = K1=I + K , I K3=I = K1=I + K I KI KI = 0. (8.7) # 2 1# + ρ2 # , Notice that, to make the above equations valid for arbitrary parametrization (7.14), it is sufficient just to rescale the scalar densities as in (7.16) ρ # −→ ρ˜ ρ˜ # 1 ρ˜ # 3 ρ˜ # 2 K2=1I = K1=3I =: K==I , K==I K==I = 0 are valid for a generic parametrization of the spinor harmonics. 3-points analytical superamplitudes in 10D SYM and 11D SUGRA A suggestion about the structure of 10D and 11D tree superamplitudes may be gained from the observation that, when the external momenta belong to a 4d subspace of the D-dimensional space, they should reproduce the known answer for 4-dimensional tree superamplitudes of N = 4 SYM and N = 8 SUGRA, respectively. Due to the momentum conservation, this is always the case for a three point amplitude and superamplitude. In this section we find the gauge fixed form of the 10D 3-point superamplitude in the gauge (7.17) and also present its 11D cousin. We also describe the first stages in search for covariant form of the three point superamplitudes, which, although have not allowed to succeed yet, might be suggestive for further study. 8.2.1 3-points analytical superamplitude of 10D SYM. Gauge fixed form We chose as D=4 reference point the anti-MHV superamplitude of N = 4 SYM (1.15). As we show in appendix A, using an explicit parametrization of 4D helicity spinors in terms of reference spinor frame we can write it in the following form (see (A.38)) (8.8) (8.9) AMHV(1, 2, 3) = K== e−2i(β1+β2+β3) δ4 ρ˜1#η˜1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A # − = − K2=1 # − (8.10) Here η˜A−i = ηAi/qρ˜i# and ρ˜i# are D=4 counterparts of the rescaled 10D variables (7.46) and (8.8) (see (A.35) and (A.25) in appendix A), K== = K2=1/ρ3# (see (A.27)) and K2=1 is a complex number, which can be associated through K2=1 = K2=11 + iK2=12 with a real in (7.12). It is tempting to identify K2=1 with K2=1I UI of the previous section: D=4 K2=1 K2=1I UI D=10 (and D=11) . (8.11) The argument of the fermionic delta function in (8.10) also has the straightforward # # 10D counterpart ρ˜1 η˜A−1 + ρ˜2 η˜ A3 where η˜A−i and ρ˜i# are defined in (7.46) and (8.8) − in such a way that all of them carry indices, charges and weights with respect to the same SO(1, 1) ⊗ SU( 4 ) ⊗ U(1) acting on the reference complex frame, ρ˜ Thus, the straightforward generalization of (8.10) to the case of D = 10 SYM theory − # − [23]A reads A3D=10 SYM = − where the complex null-vector K==I is defined in (8.9). The multiplier e−2i(β1+β2+β3) makes the superamplitude invariant under U(1) symmetry acting on the reference internal frame variables and supplies it instead with charges +1 with respect to all U(1)i groups, i = 1, 2, 3, related to scattered particles. All other variables in (8.13) are redefined in such a way that they are inert under Q[SO(D − 2)i ⊗ SO(1, 1)i ⊗ U(1)i ⊗ SO(D − 4)i] and are transformed only by SO(D − 2) ⊗ SO(1, 1) ⊗ U(1) ⊗ SO(D − 4) acting on the reference spinor frame and reference internal frame. 3 i 8.2.2 Searching for a gauge covariant form of the 3-points superamplitude Let us try to search for a covariant expression for amplitude which, upon gauge fixing, reproduce (8.13). Again, a guideline can be found in 4D expression (1.15). Counterparts of < ij > blocks are given by the matrices in (7.37) so that a possible 10D cousin of the denominator in (1.15) is given by the trace of the product of three such matrices, q ρi ρ # j# < i−BjA− > with < i−BjA− > defined ρ1#ρ2#ρ3# < 1−A2B− >< 2−B3C− >< 3−C 1A− >= = = 1 1 # 23 ρ˜1 ρ˜ 23 (ρ˜1#ρ˜ 2#ρ˜3# K2=1I UI K3=2J UJ K2=1K UK e−2i(β1+β2+β3) 2 ρ˜3#)2 (K==I UI )3 e−2i(β1+β2+β3) . # The next problem is to search for a counterpart of ηA1 < 23 > expression in the argument of fermionic delta function in (1.15). Here the straightforward generalization ∝ ηB1 < 2−B3A− > does not work: it is not covariant under SU( 4 )1 and SU( 4 )2. The − covariance may be restored by using the matrices (7.45): the matrix ηC−1 < 1−C 2B+ >< 2−B3A− >= η˜B−1e−2α1−2iβ1 U A†B3e−α3−iβ3 (8.14) (8.15) is transformed in ( 1, 1, 4 ) of SU( 4 )1 ⊗ SU( 4 )2 ⊗ SU( 4 )3. However its nontrivial weights (−2, 0, −1) and charges (+1, 0, +1/2), indicated by multipliers e−2α1−2iβ1 and e−α3−iβ3 , do not allow to sum it with its η2− and η3− counterparts without breaking the gauge symmetries. To compensate the above multipliers one can use the matrices (7.49). In such a way we arrive at the expression − − − ≫A: = ηB−1 < 1−B2C+ >< 2−C 3−D >< 3+D.E− >< .−E1F+ >< 1+F .A− >= 21 η˜i−AK3=2I UI = 12 1#η˜i−K==I UI , ρ˜ 3 which is invariant under Q SU( 4 )i ⊗ U(1)i ⊗ SO(1, 1)i and carries the nontrivial representations of only SU( 4 ) ⊗ U(1) ⊗ SO(1, 1) group acting on the reference complex spinor frame. Then, as ≪ ηi−j−k− ≫A with i, j, k given by arbitrary permutation of 123 has the same transformation properties, we can sum them and write the 10D counterpart of the fermionic delta function in (1.15), δ 4 (ρ1#ρ2#ρ3#)1/2 ≪ η1 2 3 − − − ≫A + ≪ η2 3 1 = = (ρ1#ρ2#ρ3#)2δ4 ≪ η1 2 3 − − − ≫A + ≪ η2 3 1 − − − ≫A + ≪ η3 1 2 − − − ≫A = (K==I UI ) 4 (K==I UI )4 24 24 (ρ1#ρ2#ρ3#)2 δ4 ρ˜1#η˜1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A # − = (ρ˜1#ρ˜ # 2 ρ˜3 ) # 2 e−4i(β1+β2+β3) Qi3=1 det < i−B.A+ > δ 4 ρ˜1#η˜1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A , (8.17) # − where in the last lines we have used (8.16) and (7.51). of (8.14) and multiplied by three determinants (7.51), Then, the covariant candidate amplitude is given by (8.17) divided by the product A3D=10 SYM = ? δ 4 (ρ1#ρ2#ρ3#)1/2 ≪ η1−2−3− ≫A + ≪ η2−3−1− ≫A + ≪ η3−1−2− ≫A ρ1#ρ2#ρ3# < 1−A2B− >< 2−B3C− >< 3−C 1A− > 3 i=1 Y det < i−B.A+ > = ρ1#ρ2#ρ3# δ4 ≪ η1 2 3 − − − ≫A + ≪ η2 3 1 − − − ≫A + ≪ η3 1 2 × det < 1−B.A+ > det < 2−C .+D > det < 3−E.F+ > < 1−A2B− >< 2−B3C− >< 3−C 1A− > . One can easily check that in the gauge (7.17) (see (7.9), (7.10) with explicit parametrization (7.32), (7.33), (7.42)) this expressions reduces to (8.13). However, the main problem of the above covariant expression (besides that it depends explicitly on reference complex spinor frame) is that apparently it does not obey (6.14), DjJˇA3D=10 SYM of eq. (8.18)({ρi#, vα−qi; wi, w¯i; ηAi}) 6= 0 . Indeed, it is constructed with the use of blocks (7.45) and (7.47) and, if we consider the complex spinor frames as composed from spinor and internal harmonics as in (7.30) and (8.16) (8.18) (8.19) K2=1I UI !2 2 1 2 × use (7.25), we find, for instance, 2 i Jˇ Thus we should find either a different covariant representation for the gauge fixed amplitude (8.13), or a way to relax/to modify the condition (6.14) for the analytic superamplitudes. Alternatively, one can use the gauge fixed form of the 3-point superamplitude as a basis of a gauge fixed superamplitude formalism. In it all the K8i symmetries acting on i-th spinor frame variables are gauge fixed by the conditions (7.17). This gauge fixing is performed with respect to a symmetry acting on the auxiliary variables, complementary spinor harmonics. The gauge fixed expressions (7.42) and (7.43), as well as the expressions for the physically relevant spinor harmonics (7.30)–(7.29), use the reference spinor frame. This makes our gauge fixed superamplitude formalism manifestly Lorentz covariant. Such a role of reference spinor frame is in consonance with the original idea of introducing Lorentz harmonics to covariantize the light-cone gauge [51]. Analytical 3-point superamplitude of D = 11 supergravity Similarly, the form of 3-point N = 8 4D supergravity superamplitude, which is essentially the square of the N = 4 4D SYM one (see e.g. [11, 12]), suggests the following gauge fixed expression for the basic 3-point superamplitude of 11D supergravity, A3D=11 SUGRA = # [23]A e−2i(β1+β2+β3) δ8 ρ˜1#η˜1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A # − , (8.21) Eq. (8.21) can be obtained by gauge fixing from (ρ1#ρ2#ρ3#)2 δ8 ≪ η1 2 3 − − − ≫A + ≪ η2 3 1 det < 1−B.A+ > det < 2−C .+D > det < 3−E.F+ > < 1−A2B− >< 2−B3C− >< 3−C 1A− > 2 . (8.22) However, as (8.18) in the case of 10D SYM, this expression does not obey eq. (6.14), so that we should find either the reason to relax/to modify these equations, or to search for a different covariant expression reproducing (8.21) upon gauge fixing. Another interesting possibility is to use the gauge fixed spinor frame variables, obeying (7.17) for all sets of spinor harmonics. As we have already stressed above, in distinction with light-cone gauge, such a gauge fixed superamplitude formalism possesses manifest Lorentz invariance and supersymmetry. This possibility is also under study now. In this paper we have constructed the basis of the analytic superfield formalism to calculate (super)amplitudes of 10D SYM and 11D SUGRA theories. This is alternative to the constrained superamplitude formalism of [15, 16] and also to the ‘Clifford superfield’ approach of [14]. The fact that it has more similarities with D=4 superamplitude calculus of N = 4 SYM and N = 8 SUGRA promises to allow us to use more efficiently the D=4 suggestions for its further development. In particular, such a suggestion was used to find the gauge fixed form of the 3-point analytic superamplitude of 10D SYM and 11D SUGRA, eqs. (8.13) and (8.21). n i=1 We have begun by solving the equations of the constrained on-shell superfield formalism of 10D SYM and 11D SUGRA [15, 16, 21] in terms of single analytic superfield depending holomorphically on N = 4 and N = 8 complex coordinates, respectively. These N complex coordinates ηA− are related to 2N real fermionic coordinates θq− of the constrained superfield formalism by complex rectangular matrix w¯qA (= (wqA)∗). This and its conjugate wqA = (w¯qA)∗ obey some constraints which allow us to consider them as homogeneous coordinates of the coset Spin(D−4)⊗U(1) and to call them internal harmonic variables. Similarly, the constrained n-point superamplitudes of the Q SO(D − 2)i covariant constrained superfield formalism can be expressed in terms of analytic superamplitudes which depend, besides the n sets of 10D or 11D spinor helicity variables, also on n sets Spin(D−2)i (w¯qA i, wqAi) of Spin(D−4)i⊗Spin(2)i internal harmonic variables. The sets of 10D and 11D spinor helicity variables include Lorentz harmonics or spinor frame variables vα−q i which, after the constraints and gauge symmetries are taken into account, parametrize the celestial sphere S(D−2). Together with scalar densities ρi#, they describe the light-like momenta and the “polarizations” (SO(D−2)i small group representations) of the scattered particles. The constrained superamplitudes, which depend on these spinor helicity variables and (2N )component real fermionic variables θq−i, carry indices of the small groups SO(D − 2)i. In contrast, the analytic superamplitudes do not carry indices but only charges s = N /4 of U(1)i which act on the internal frame variables (w¯qA i, wqAi) and on the complex fermionic ηA−i = θq−iw¯qA i. They may be constructed from the basic constrained superamplitudes by contracting their SO(D − 2)i vector indices with complex null vectors UIi constructed from bilinear combinations of (w¯i, wi). The dependence of the analytic superamplitudes on internal harmonics is restricted by the equations in terms of harmonic covariant derivatives which reflect the fact that the original constrained superamplitudes are independent of (w¯i, wi). Moreover, the internal harmonics (w¯i, wi) are pure gauge with respect to the SO(D − 2)i symmetry which acts also on the spinor harmonics (vα−q i, vα+q˙ i). We have shown that internal harmonics can be defined in such a way that analytic superamplitudes actually depend only on complex Spin(1,D−1) spinor harmonics (vα∓A i, v¯αA∓i) (7.30) parametrizing the coset [SO(1,1)⊗Spin(D−4)⊗U(1)]⊂×KD−2 , {(vα∓A i, v¯αA∓i)} = Spin(1, D − 1) [SO(1, 1) ⊗ Spin(D − 4) ⊗ U(1)] ⊂×KD−2 (9.1) However, we find convenient to consider these complex spinor harmonics to be composed from the real spinor harmonics, parametrizing the coset isomorphic to the celestial sphere Spin(1,D−1) S(D−2) = [SO(1,1)⊗Spin(D−2)]⊂×KD−2 (6.2), and the above mentioned internal harmonics (w¯i, wi), in spite of that these latter are pure gauge with respect to Spin(D − 2)i symmetry (see (7.21) and (7.7)). We have found a parametrization of the spinor frame variables and of the internal frame which is especially convenient for the analysis of the analytic superamplitudes. This has allowed us to establish the correspondence of higher dimensional quantities with basic building blocks of 4D superamplitudes and to use it to find the expressions for analytic 3point superamplitudes of D=10 SYM and D=11 SUGRA theories. These are the necessary basic ingredients for calculation of the n-point superamplitudes with the use of on-shell recurrent relations, the problem which we intend to address in a forthcoming paper. The first stages in this direction should include a better understanding of the structure of the 3-point analytic superamplitudes, in particular the search for its more convenient, parametrization independent form, as well as the derivation of the BCFW-type recurrent relations for the analytic superamplitudes. These should be more closely related to the relations for D=4 superamplitudes [7, 11] than the BCFW-type recurrent relations for real constrained 11D and 10D superamplitudes presented in [15, 16]. In particular, one expects the BCFW deformations used in such recurrent relations to have an intrinsic complex structure, similar to the one in D=4 equations of [7]. As we show in appendix B, starting from BCFW deformations of spinor frame and fermionic variables in [15], which are essentially real, this is indeed the case. The resulting BCFWlike deformations of the complex spinor frame variables (7.30) and of the complex fermionic variables (4.42) \ vα−A(n) = vα−A(n) + z vα−A(1) qρ1#/ρn# , v¯[αA(n−) = v¯αA(−n) , v¯[αA(1−) = v¯αA(−1) − z v¯αA(−n) qρn#/ρ1# , ηdAn = ηA n + z ηA−1 − − qρ1#/ρn# , − ηdA1 = ηA 1 − (9.2) (9.3) (9.4) have the structure quite similar to that of the 4D super-BCFW deformations from [11] (see (B.1)–(B.3) in appendix B). Thus presently their exist three alternative superamplitude formalisms for 10D SYM, two of which have been also generalized for the case of 11D supergravity. These are Clifford superfield approach of [14], constrained superamplitude approach of [15, 16] and the analytic superamplitude formalism of the present paper. As discussed in [16], and also briefly commented in section 4.6.1, the one particle counterparts of all three types of superamplitudes can be obtained by different ways of covariant quantization of 10D and 11D massless superparticles. In short, the separation point is how to deal with the Poisson brackets of the fermionic second class constraints, (4.72). The formalism of [14] and the analytic superfield approach of the present paper imply ‘solving’ the constraints by passing to the Dirac brackets (4.73) and quantizing these. In such a way we obtain the Clifford algebra like anticommutation relation (4.74) for 8 (16 in D=11 case) real fermionic variables θˆq−. To arrive at the one-particle counterpart of the superamplitudes from [14], one should consider the superparticle ‘wavefunction’ to be dependent on the whole set of Clifford algebra valued variables θˆq−, i.e. to be a ‘Clifford superfield’. In contrast, to obtain an analytic superfield as superparticle wavefunction, we need to split 8 real θˆq− on 4 complex η A− and its complex conjugate η¯−A, which obey the Heisenberg-like algebra. This implies that η¯−A can be considered as creation operator or complex momentum conjugate to the annihilation operator ηA . Then in the ηA−-coordinate − (or holomorphic) representation the superparticle quantum state vector depends on η , but not on η¯−A. In other words, it will be described by analytic superfield, the one-particle − counterpart of our analytic superamplitudes. From this perspective, one can arrive at doubts in consistency of the Clifford superamplitude approach of [14]. Indeed, in terms of complex fermionic variables the above described appearance of an unconstrained Clifford superfield in superparticle quantization requires to allow the wavefunction to depend on both coordinate η variables in an arbitrary manner. Then such a Clifford superfield wavefunction is not allowed in quantum mechanics in its generic form and some conditions need to be imposed to restrict its dependence on η¯−A and/or η−. The analytic superfields and superamplitudes can be obtained on this way: by imposing on Clifford superfields/superamplitudes just the A− and momentum η¯−A conditions to be independent of η¯−A. The constrained superfields, the one-particle counterparts of the constrained superamplitudes, appear as a result of superparticle quantization if, instead of passing to Dirac brackets (4.73), we realize the fermionic second class constrains as differential operators ∂ Dq+ = ∂θq− + . . . obeying the quantum counterpart (3.4) of (4.72). The ‘imposing’ of the quantum second class constraint is then achieved by considering a θq−-dependent multicomponent state vectors ΨQ (= (Ψq˙, W I ) in D=10) and requiring them to obey a set of linear differential equations Dq+ΨQ = ΔqQP ΨP ((3.7) and (3.8) in D=10; see [16] for details of this procedure). The advantages of this approach is the use of Grassmann fermionic coordinates (rather than Clifford algebra valued ones) as well as its manifest covariance under the ‘small group’ SO(8) (SO(9)) symmetry. The disadvantage is that superfields and superamplitudes are subject to the above mentioned set of quite complicated equations, which have no clear counterpart in D=4 case. This makes the calculations in the constrained superamplitude framework quite involving (in comparative terms) and creates difficulties for the (straightforward) use of the experience gained in D=4. Also the decomposition of constrained superfields on components looks quite non-minimal: in the 10D case, 9 components of constrained superfield, all nonvanishing, are constructed of two fields describing the on-shell degrees of freedom of SYM, bosonic wI and fermionic ψq˙, appearing already in first two terms of the decomposition. In contrast, the components of the analytic superfields include different components of wI = (φ(+), φAB, φ(−)) and ψq˙ = (ψ+1/2 A, ψA−1/2) only ones. Thus the great advantage of the analytic superamplitude formalism is its minimality. It is also much more similar to the on-shell superfield and superamplitude description used for maximal D=4 SYM and SUGRA theories. In particular, this similarity helped us to find the gauge fixed expression for the 3-point analytic superamplitudes of 10D SYM and 11D SUGRA. The price to be paid for these advantages is the harmonic superspace type realization of the SO(8) (SO(9)) symmetry and, consequently, dependence on additional set of harmonic variables w¯qA, wqA parametrizing Spin(8)/[SU( 4 )⊗U(1)] coset. Presently the analytic superemplitude formalism is under further development which, as we hope, will result in a significant progress in 10D and 11D amplitude calculations. An alternative direction we are also working out is to use the structure of the analytic 3-point superamplitude for deriving the expression for its cousin from the real constrained superamplitude formalism [15, 16], and to use the interplay of the constrained and analytic superamplitude approaches for their mutual development. It will be also interesting to reproduce the analytic superamplitudes from an appropriate formulation of the ambitwistor string [60–62]. Notice that, although original ambitwistor string model [60] had been of NSR-type and had been formulated in D=10, quite soon [63] it was appreciated its relation with null-superstring [50] (see [64, 65] for related results and [66] for more references on null-string) and with twistor string [6, 66–68]. This suggested its existence in spacetime of arbitrary dimension, including D=11 and D=4, and the last possibility was intensively elaborated in [69–73]. An approach to derive the analytic superamplitudes from the Green-Schwarz type spinor moving frame formulation of D=10 and D=11 ambitwistor superstring [63] looks promising and we plan to address it in the future publications. Acknowledgments This work has been supported in part by the Spanish Ministry of Economy, Industry and Competitiveness grants FPA 2015-66793-P, partially financed with FEDER/ERDF (European Regional Development Fund of the European Union), by the Basque Government Grant IT-979-16, and the Basque Country University program UFI 11/55. The author is thankful to Theoretical Department of CERN (Geneva, Switzerland), to the Galileo Galilei Institute for Theoretical Physics and INFN (Florence, Italy), as well as to the Simons Center for Geometry and Physics, Stony Brook University (New York, US) for the hospitality and partial support of his visits at certain stages of this work. He is grateful to Dima Sorokin for the interest to this work and reading the draft, to Emeri Sokatchev for useful discussions and suggestions, and to Luis Alvarez-Gaume and Paolo Di Vecchia for useful discussions on related topics. A On D=4 spinor helicity formalism In D=4 Spin(1, 3) = SL(2, C) and the spinor frame or Lorentz harmonic variables vα± = (vα±˙)∗ [49] are restricted by the only condition v−αvα+ = 1, (vα+, vα−) ∈ SL(2, C) ⇔ v−αvα+ = 1 . (A.1) vα+ 7→ ea+ib(vα+ + k#vα−) , vα− 7→ e−a−ibvα− , v¯α+˙ 7→ ea−ib(v¯α+˙ + k¯#v¯α−˙) , v¯α−˙ 7→ e−a+ibv¯α−˙ , the set of such harmonic variables parametrize the sphere S2 [17, 18], {(vα+, vα−)} = Spin(1, 3) SL(2, C) = [SO(1, 1) ⊗ Spin(2)] ⊂×K2 [SO(1, 1) ⊗ U(1)] ⊂×K2 = S2 . When the spinor frame is associated with a light-like momenta by the generalized Cartan Penrose relation pαα˙ = ρ#vα−v¯α˙− (cf. (1.4)), S2 in (A.4) is the celestial sphere. In the scattering problem we can associate the spinor frame to each of n light-like momenta and to express the corresponding helicity spinors of (1.4) in terms of the spinor harmonics λα(i) = qρ(#i)vα−(i) , − λ¯α˙ (i) = qρ(#i)v¯α˙−(i) , pαα˙ (i) = ρ(#i)vα−(i)v¯α˙−(i) . As we have used only vα(i), the complementary spinor harmonic vα(i) remains arbitrary up + to the constraint (A.1), v(−i)α + vα(i) = 1 . Actually, this is the statement of K2 symmetry (parametrized by k# and k¯# in (A.2), (A.3)), which can be used as an identification relation on the set of harmonic variables (as indicated in (A.4)), and in this sense is the gauge symmetry. We can fix these K2(i) gauge symmetries by identifying (up to a complex multipliers) all the complementary spinors of the spinor frames associated to the momenta of the scattered particles (v(+i)v(+j)) ≡ v(i) +α + vα(j) = 0 + + vα(i) ∝ vα(j) ∀ i, j = 1, . . . , n . (A.8) It is convenient to reformulate this statement by introducing an auxiliary spinor frame spinor frames (vα±(i)) is related to that by (cf. (A.2), (A.3)) (vα±), which is not associated to any of the scattered particles, and to state that any of the vα+(i) = eαi+iβi vα+ , v¯α˙+(i) = eαi−iβi v¯α+˙ , − vα(i) = e−αi−iβi (vα− + Ki=vα+) , v¯α˙−(i) = e−αi+iβi (v¯α˙− + K¯i=v¯α+˙) . In this gauge the contractions of the spinors from different frames read < v(−i)v(−j) >≡ v(i) −α − vα(j) = e−(αi+αj)−i(βi+βj)Kj=i , < v(−i)v(+j) > = e(αj−αi)+i(βj−βi), α˙ (j) = e−(αi+αj)+i(βi+βj)K¯ j=i , [v¯(−i)v¯(+j)] = e(αj−αi)−i(βj−βi), (A.2) (A.3) (A.4) (A.5) (A.6) (A.7) (A.9) (A.10) (A.11) (A.12) where Kj=i := Kj= − Ki= . (A.13) Of course, we can use the SO(1, 1)i × SO(2)i gauge symmetries to fix also αi = 0 and βi = 0 ∀i = 1, . . . , n, but the multipliers βi might be useful as they actually indicate the helicity of the field or amplitude, while αi can be ‘eaten’ by the ‘energy’ variables ρi . # Indeed, the i-th light-like momentum (A.6) can be now written as pαα˙ (i) = ρe(#i)(vα− + K(=i)vα+) (v¯α˙− + K¯ = (i)v¯α˙ ) + # K¯ (=i)uα−α+˙ + ρe(i) (i) # K= K¯(=i)uα#α˙ , where ρ˜i# = e−2αi ρi# (8.8) and uαα˙ = vα−v¯α˙− , = # uαα˙ = vα+v¯α˙+ , uα±α∓˙ = vα±v¯α˙∓ , are two real and two complex conjugate (ua+− = (ua−+)∗) vectors of Newman-Penrose lightlike tetrade (see [43, 44] and refs. therein). Using the complementary harmonics vα+i, v¯α+˙i of the auxiliary frame as reference spinors, we can identify polarization vectors with the i-th frame counterparts of the above described complex null-vectors ua−+ and ua+− = (ua−+)∗: ε(α+α˙)(i) = uαα˙ (i) ≡ vα−(i)v¯α˙+(i) , −+ ε(α−α˙)(i) = uαα˙ (i) ≡ vα+(i)v¯α˙−(i) . +− In the gauge (A.8) these identification implies that ε((i+)) · ε((j+)) := 1 ε(+) 2 αα˙ (i) (j) ε (+)αα˙ = 0 . Using (A.14) and (A.16) we can easily find # 2 ε((i+))k(j) = ρ(i) (v(−j)v(−i))(v¯(−j)v¯(+i)) , and then, for instance, 1 ε((i+))ε((j−)) = − 2 (v(−i)v(+j))(v¯(−j)v¯(+i)) , 4 # # 4 (ε((1+))k(2)) (ε((2+))ε((3−))) = − ρ(2) (v(−2)v(−1)) (v(−2)v(+3)) (v¯(−2)v¯(+1)) (v¯(−3)v¯(+2)) = − ρ˜(2) K2=1 e2i(β3−β2−β1), This allows us to calculate 3-gluon amplitude of N = 4 4D SYM, M(1+, 2+, 3−) = gǫ((1+))aǫ((2+))bǫ(−)c = g(ε((1+))k(2) ε((2+))ε((3−)) + ε((2+))k(3) ε((3−))ε((1+)) + ε((3−))k(1) ε((1+))ε((2+))) = g 4 = − g e2i(β3−β2−β1) ρ˜(2)K2=1 + ρ˜(3)K3=2 # # = # 4 ρ˜(3)K2=1 e2i(β3−β2−β1) (A.14) (A.15) (A.16) (A.17) (A.18) (A.19) (A.20) (see [58, 59] for the definition of tabc tensor). Notice that the last term in the second line of this equation vanishes as a result of (A.17) and that at the last stage of transformations of this equation we have used the consequence of the momentum conservation in 3-particle process which we are going to discuss now. This implies as well as ρ˜1# + ρ˜2# + ρ˜ 3# = 0 , ρ˜1#K1= + ρ˜2#K2= + ρ˜3#K3= = 0 , Here we have used the notation (A.13) and In our notation the momentum conservation in the 3-particle process is expressed by ρ(#1)vα−(1)v¯α˙−(1) + ρ(#2)vα−(2)v¯α˙−(2) + ρ(#3)vα−(3)v¯α˙−(3) = 0 . 3 K3=2 = 1# K2=1 = 1# K1=3 K1=3 = 2# K2=1 Eq. (A.26) implies (A.21) (A.22) (A.23) (A.24) (A.25) (A.26) (A.27) (A.28) (A.29) The solution of eq. (A.24) is nontrivial only if (K¯(=32))∗ 6= K(=32). In this case one of two branches of the general solution is described by # while K(=1,2,3) can be different but obeying (A.23) with ρ˜(1,2,3) restricted by (A.22). From restricted by 3-particle kinematics by K(=1,2,3). We will also use the solution of (A.23) in terms of complex non-vanishing K== ρ˜i# := e−2αi ρi# . K¯3= = K¯2= = K¯1= ρ˜ 1 3 K#3=2 = K#2=1 = K#1=3 =: K== . ρ˜ 2 v¯α˙−(1) ∝ v¯α˙−(2) ∝ v¯α˙−(3) while vα−(1), vα−(2) and vα−(3) are different. A.2 3-gluon amplitude and superamplitude in maximal D=4 SYM The standard expression for the 3-point amplitude in D=4 SYM is written in terms of < ij >=< λiλj >= λiαλαj = qρi#ρj# < vi−vj− > = q # ρ˜i ρ˜j#e−i(βi+βj)Kj=i . If we were trying to guess the corresponding expression starting from (A.20), the βi dependence indicates that this should be (up to a coefficient) M(1+, 2+, 3−) = < 12 >3 < 12 >4 < 23 >< 31 > < 12 >< 23 >< 31 > (A.30) Using (A.29) and (A.23) one can easily check that this expression indeed reproduce (A.20), = ρ˜3#K2=1 e2i(β3−β2−β1) = (ρ˜3#)2K== e2i(β3−β2−β1) . (A.31) In our notation the anti-MHV (MHV) type superamplitude reads (see (1.15)) while the MHV amplitude is δ 8 λ¯α˙ 1ηA1 + λ¯α˙ 2ηA2 + λ¯α˙ 3ηA3 = 1 δ4 (η1 < 23 > +η2 < 31 > +η3 < 12 >) , (A.32) 1 A=1 i,j=1 X [ij] ηAiηAj . The covariance of δ function under the phase transformations of the bosonic spinors holds when the fermionic variables ηAi have the same phase transformation property as λαi. This reflects its origin in Penrose-Ferber incidence type relation ηAi = θAαiλαi [45] which in terms of our Lorentz harmonic notation reads ηAi = q ρi#ηA−i := qρi#θAαivαi. − Notice also that the indices A of all the fermionic coordinates are transformed by the same SU( 4 ), which is the R-symmetry group of N = 4 D=4 SYM. Using (A.10), (A.9) and (A.22), (A.27), we can write the Grassmann delta function of (A.33) in the form (A.33) (A.34) (A.35) (A.36) where12 ρ˜i# is defined in (A.25). δ4 (η1 < 23 > +η2 < 31 > +η3 < 12 >) = = ρ˜1 ρ˜ # # 2 e−4i(β1+β2+β3) δ4 η˜1−AK3=2 + η˜2−AK1=3 + η˜3−AK2=1 2 ρ˜3 = ρ˜1 ρ˜ 2 ρ˜3 # # 2 ρ˜1 ρ˜2 # # 2 (K==)4 e−4i(β1+β2+β3) δ4 ρ˜1#η1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A # − # − # e−4i(β1+β2+β3)δ4 ρ˜1 η[13]A + ρ˜2 η˜ − η˜A−i := eαi+iβi ηA−i , η˜A[ji] = η˜A−j − η˜A−i , − Similarly, the fermionic delta function in (1.16) can be written as δ 8 λ¯α˙ 1ηA1 + λ¯α˙ 2ηA2 + λ¯α˙ 3ηA3 = δ8 ρ(#1)v¯α˙ (1)ηA(1) + ρ(#2)v¯α−˙2ηA−2 + ρ3#v¯α−˙3ηA−3 − − = δ8 v¯ # α−˙ ρ˜1 η˜ A[13] + ρ˜2 η˜ A[23] + v¯α˙ K¯ 1=ρ˜ 1 η˜ − A[13] + K¯ 2=ρ˜ # 2 η˜ − A[23] v¯α−˙ + K¯1=v¯α+˙ ρ˜1 η˜ A[13] + v¯ α−˙ + K¯2=v¯α+˙ ρ˜2 η˜ A[23] . 12One can check that η˜A−i = θAαi(vα− +Ki=vα+) which makes transparent that all η˜A−i are transformed by the common U(1) ⊗ SO(1, 1) group, but are inert under all the U(1)j ⊗ SO(1, 1)j gauge symmetries, including the one with i = j. In this notation, the multiplier in the MHV superamplitude (1.15) reads 1 K2=1K3=2K1=3 = (K2=1)3 1 (ρ˜1#ρ˜ 2 ρ˜3#)2 (K==)3 # . (A.37) Using (A.37) and (A.34), we can write the 3-point anti-MHV superamplitude (1.15) in the form HJEP05(218)3 AMHV(1, 2, 3) = (K==) e−2i(β1+β2+β3) δ4 ρ˜1#η˜1−A + ρ˜2#η˜2−A + ρ˜3 η˜3A # − = − K2=1 [23]A (A.38) , . B BCFW-like deformations of complex frame and complex fermionic variables An important tool to reconstruct tree D = 4 (super)amplitudes from the basic 3-point (super)amplitude is given by BCFW recurrent relation [7] and their superfield generalization [11]. The counterparts of these latter 4D relations for constrained superamplitudes of 11D SUGRA and 10D SYM have been presented in [15, 16]. They use the real BCFW deformations of real bosonic and fermionic variables of the constrained superamplitude formalism. In contrast, in the case of the BCFW-type recurrent relations for analytic superamplitudes (which are still to be derived), one expects the BCFW deformations used in such recurrent relations to have an intrinsic complex structure, similar to the one of the D=4 relations [7, 11] λ(An) 7→ λd(An) = λ(n) + zλ(A1), A λ(A1) 7→ λd(A1) = λ(A1) , ηdAn = ηA n + z ηA−1 , λ¯(A˙n) 7→ λ¯dA˙ (n) = λ¯(A˙n), λ¯(A˙1) 7→ λ¯d(A˙1) = λ¯(A˙1) − zλ¯(A˙n), ηdA1 = ηA 1 . Let us show how this can be reached starting from the BCFW deformations of real spinor frame variables [15, 16] vα−q(n) = vα−q(n) + z tuu ρ(1) vα−p(1) \ # Mpq , \ vα−q(1) = vα−q(1) − z tu v # u ρ(n) Mqp vα−p(n) , # – 55 – (B.1) (B.2) (B.3) (B.4) (B.5) amplitude calculations. the first and of the n-th particle, provided we choose kd(a1) = k(a1) − zqa , kd(an) = k(n) + zqa , a on a light-like vector qa orthogonal to both k(a1) and k(an), (B.6) (B.7) (B.8) (B.9) (B.10) (B.11) (B.12) (B.13) (B.14) (B.15) qaqa = 0 , qak(a1) = 0 , qak(an) = 0 , 1 Mqp = − qρ(#1)ρ(#n)(u(=1)u(=n)) /q˜αβ : = qa Γ˜aαβ , /qαβ := qaΓaαβ . (vq−(1) /q˜vp−(n)) , MrpMrq = 0 , MqrMpr = 0 . qa = N 1 qρ1#ρn# vq−(1)Γ˜aMqpvp−(n) . The light-likeness of qa (B.9) implies the nilpotency of the matrix M, We can also write the expression for light-like complex vector in terms of deformation matrix, and of the real fermionic variables θdp(n) = θp(n) + z θq−(1) Mqp t − − θdq(1) = θq(1) − z tu − − uu ρ#(1) , ρ(n) ρ(1) v # u ρ(n) Mqp θp(n) . − # Here α = 1, . . . , 4N and q, p = 1, . . . , 4N (we should set N = 8 and 4 for 11D SUGRA and 10D SYM, respectively) and z is an arbitrary number. In principle this can be considered to be real z ∈ R [14], although z ∈ C is neither forbidden and actually more convenient in The above shift of spinor moving frame variables results in shifting the momentum of The nilpotency condition (B.12) guarantees that the shifted spinor moving frame variables obey the characteristic constraints, eqs. (2.33) with shifted light-like momenta k(1) and k(n) (B.8) or, equivalently, (2.22) with shifted light-like u(=1a) and u(=na), zqa ρ(1) ud(=1a) = u(=1a) − # , ud(=na) = u(=na) + zqa # . ρ(n) Notice that (B.4) and (B.5) imply kc1a + kcna = k1a + kna . The complex structure similar to the one of D=4 BCFW deformations can be reproduced after passing to the complex spinor harmonics (7.30)—(7.33) composed from the spinor harmonics and the internal harmonic variables. The internal harmonics can be used to solve the nilpotency conditions (B.12) for the matrix Mqp in (B.4)–(B.7). The solution Mqp = w¯qA 1MABwpBn , with an arbitrary hermitian N × N matrix MAB, results in the following deformation of the complex spinor frame variables (7.28) and of the complex fermionic variables: and vα−A(n) = vα−q(A) + z vα−B(1) MB A qρ(#1)/ρ(#n) , v¯[αA(n−) = v¯αA(−n) , vα−A(1) = vα−A(1) , v¯[αA(1−) = v¯αA(−1) − z MA B v¯αB(n−) qρ(#n)/ρ(#1) ηdAn = ηA n + z ηB 1 ηdA1 = ηA−1 . − (B.16) (B.18) (B.19) These are already quite similar to the 4D super-BCFW transformations (B.1), (B.2), (B.3). To make the similarity even closer, we can choose MB A = δBA. In such a way we arrive at (9.2), (9.3), and (9.4). Open Access. Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited. [1] Z. Bern, J.J. Carrasco, L.J. Dixon, H. Johansson and R. Roiban, Amplitudes and Ultraviolet Behavior of N = 8 Supergravity, Fortsch. Phys. 59 (2011) 561 [arXiv:1103.1848] [INSPIRE]. [2] J.M. Drummond, J. 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Igor Bandos. An analytic superfield formalism for tree superamplitudes in D=10 and D=11, Journal of High Energy Physics, 2018, 103, DOI: 10.1007/JHEP05(2018)103