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 YangMills theory (SYM) and 11D supergravity (SUGRA) are collected in multiparticle counterparts of analytic onshell 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 3point tree superamplitudes of 10D SYM and 11D SUGRA
which generalize the 4 dimensional antiMHV 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 onshell superfield description of 10D SYM
– i –
linearized 11D SUGRA
5
Spinor helicity formalism and onshell 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 onshell superfield description of 11D SUGRA
Analytic onshell superfields of 11D SUGRA
5.5.1
5.5.2
SO(9)
SO(7)×SO(2) harmonic variables
Analytic onshell superfields from constrained onshell superfields
Notation
D=4 spinor helicity formalism
D=4 superamplitudes and onshell 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 onshell superfield formalism
SU(
4
) invariant solution of the constrained superfield equations
The onshell superfields are analytic rather than chiral
Analytic superfields and harmonic onshell superspace
Analytical basis and analytic subsuperspace of the harmonic onshell
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 3point analytic superamplitudes in 10D and 11D
Three particle kinematics and supermomentum
3points analytical superamplitudes in 10D SYM and 11D SUGRA
8.2.1
8.2.2
8.2.3
3points analytical superamplitude of 10D SYM. Gauge fixed form
Searching for a gauge covariant form of the 3points superamplitude
Analytical 3point superamplitude of D = 11 supergravity
9
Conclusion and discussion A On D=4 spinor helicity formalism
A.1
Momentum conservation in a 3point 4D amplitude
A.2 3gluon amplitude and superamplitude in maximal D=4 SYM
variables
B BCFWlike 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 multiloop amplitudes of d=4
supersymmetric YangMills (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 CaronHuot 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 lightlike momenta ka(i) of ith 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 onshell 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 onshell superspaces of 11D SUGRA and 10D SYM
on the set of N complex spinor coordinates ηA− and its complex conjugate η¯−A. The analytic
onshell 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 onshell 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
nullvectors UI i related to the internal frame associated to ith 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 onshell 3point 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 onshell 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 onshell superfield description of 10D SYM [21]. Analytic onshell 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 onshell
superfield and analytic onshell 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 3point 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 reformulation of 4D spinor helicity formalism,
which is useful for comparison of 4D and 10/11D (super)amplitudes. Appendix B shows
how to obtain the BCFWlike 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 )
Rsymmetry group.
Finally, a, b, c, d = 0, 1, . . . , (D − 1) are Dvector 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
npoint (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 ith particle.
In particular, ith lightlike 4momentum pµ(i) is determined in terms of λ(αi) = (λ¯(α˙i))∗ by
CartanPenrose 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 CartanPenrose representation (1.4)
a
and also allows to show that pa(i)p(i) = 0.
The nparticle 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 ith 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 onshell 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 ) Rsymmetry 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 superhelicity 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 superhelicity 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 socalled
onshell 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(42N ) = {λ, λ¯, η, η¯} obeying eq. (1.13)
describe the onshell states of N = 4 SYM and N = 8 SUGRA. They can be considered as
homogeneous superfields on chiral onshell superspace
Σ(4N ) = {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 onshell superfields can be obtained by quantization of D = 4 BrinkSchwarz
superparticle with N extended supersymmetry in its FerberShirafuji 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 onshell superfield formalism from
D=10 and D=11 superparticle quantization. Here we will not consider superparticle
quantization but describe briefly the resulting constrained onshell superfields and constrained
superamplitude formalism of [15, 16] and use these as a basis to search for the analytic
onshell 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 3point superamplitudes of D=4 N = 4 SYM
theory. These are two: the antiMHV (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 lightcone 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 Ddimensional lightlike momentum ka(i), ka(i)k(i) = 0, by
the condition that one of the lightlike 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
onshell 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 lightlike 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 lightlike 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 lightlike 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 socalled 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 lightlike
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
Ddimensional observer. Notice that this is in agreement with the fact that a lightlike
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
lightlike 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
Ddimensional 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 Ddimensional 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 MajoranaWeyl (MW) spinor representation is
16dimensional, 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, cspinor and sspinor
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(αq−γqIq˙vβ)q˙
+ = ΓaαβuIa ,
– 9 –
where γqIp˙ = γ˜p˙q with I = 1, . . . , 8 are SO(8) ClebshGordan 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
lightlike 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 lightlike 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 CartanPenrose 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 lightlike momentum of the particle by (2.32) (cf. [14]) so that the
onshell 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 onshell states of spinor superpartner
of the gauge field can be described by
When describing the onshell states of the SYM multiplet, it is suggestive to replace ρ# by
its conjugate coordinate and consider the field on the ninedimensional 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) cspinor 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
onshell superfields, which can be considered as oneparticle prototypes of tree
superamplitudes. A constrained onshell 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 onshell
superspace with bosonic coordinates x= and vα−q, and fermionic coordinates θq−
The 10D supersymmetry acts on the coordinates of Σ(98) 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 onshell superspace Σ(98) 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(98) (3.1) contains too many component fields
so that onshell 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 sspinor 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 Fouriertransformed onshell 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 cspinor 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 onshell superfield description of 10D SYM
In this section we present an analytic superfield formalism for the onshell 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 onshell superfields of 10D SYM
from [21] in terms of one analytic onshell superfield. In section 6 we generalize this for
the case of superamplitudes and describe an analytic superamplitude formalism.
4.1
From constrained to unconstrained onshell superfield formalism
To arrive at our unconstrained superfield formalism it is convenient to write the superspace
equations (3.8) and (3.7) for onshell 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 γtrace 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) onshell 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 ClebschGordan 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 ClebschGordan 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 antianalytic, (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 8vector 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 onshell superspace
Thus, we have solved the superfield equations for constrained onshell superfields of D = 10
SYM in term of one complex analytic superfield Φ obeying the chiralitytype 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 onshell superspace’
which can be understood as direct product of the onshell 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 onshell 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 subsuperspace of (4.38),
we have to pass to the analytic coordinate basis.
Analytical basis and analytic subsuperspace of the harmonic onshell
superspace
The presence of additional harmonic variables allows to change the coordinate basis of the
harmonic onshell 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 ith and jth, 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 ith 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 3point 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 ith particle.
3point analytic superamplitudes in 10D and 11D
Three particle kinematics and supermomentum
and
8
8.1
Let us study 3particle 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 onshell
3particle amplitude can be defined only for complexified Kj=iI which implies that the lightlike
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.
3points 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
Ddimensional space, they should reproduce the known answer for 4dimensional 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 3point 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
3points analytical superamplitude of 10D SYM. Gauge fixed form
We chose as D=4 reference point the antiMHV 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 nullvector 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 3points 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 3point superamplitude as a
basis of a gauge fixed superamplitude formalism. In it all the K8i symmetries acting on
ith 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 lightcone gauge [51].
Analytical 3point superamplitude of D = 11 supergravity
Similarly, the form of 3point 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 3point 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 lightcone 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 3point 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 onshell 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 npoint 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 lightlike 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 npoint superamplitudes with the use of onshell
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 3point analytic superamplitudes, in particular the search for its more convenient,
parametrization independent form, as well as the derivation of the BCFWtype recurrent
relations for the analytic superamplitudes. These should be more closely related to the
relations for D=4 superamplitudes [7, 11] than the BCFWtype 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 superBCFW 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 oneparticle 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
Heisenberglike 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 oneparticle
−
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 oneparticle 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 nonminimal: in the 10D case, 9
components of constrained superfield, all nonvanishing, are constructed of two fields describing
the onshell 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 onshell 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 3point 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
3point 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 NSRtype and had been formulated in D=10, quite
soon [63] it was appreciated its relation with nullsuperstring [50] (see [64, 65] for related
results and [66] for more references on nullstring) 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 GreenSchwarz 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 201566793P, partially financed with FEDER/ERDF
(European Regional Development Fund of the European Union), by the Basque Government
Grant IT97916, 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 AlvarezGaume 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 lightlike 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 lightlike
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 ith lightlike 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 NewmanPenrose
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 ith frame counterparts of the above described
complex nullvectors 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 3gluon 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 3particle
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 3particle 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 3particle
kinematics by K(=1,2,3). We will also use the solution of (A.23) in terms of complex nonvanishing 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
3gluon amplitude and superamplitude in maximal D=4 SYM
The standard expression for the 3point 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 antiMHV (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 PenroseFerber 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 Rsymmetry 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 3point antiMHV 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
BCFWlike deformations of complex frame and complex fermionic
variables
An important tool to reconstruct tree D = 4 (super)amplitudes from the basic 3point
(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 BCFWtype 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 nth particle,
provided we choose
kd(a1) = k(a1) − zqa ,
kd(an) = k(n) + zqa ,
a
on a lightlike 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 lightlikeness of qa (B.9) implies the nilpotency of the matrix M,
We can also write the expression for lightlike 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 lightlike momenta k(1)
and k(n) (B.8) or, equivalently, (2.22) with shifted lightlike 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 superBCFW 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 (CCBY 4.0), which permits any use, distribution and reproduction in
any medium, provided the original author(s) and source are credited.
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