JHEP07(2013)075

Published for SISSA by Springer

Received: May 1, 2013

Accepted: June 19, 2013

Published: July 11, 2013

New expressions for gravitational scattering

amplitudes

Andrew Hodges

Mathematical Institute,

24-29 St Giles, Oxford OX1 3LB, U.K.

E-mail: hodges@maths.ox.ac.uk

Abstract: New methods are introduced for evaluating tree-level gravitational scattering

amplitudes. A new N=7 super-symmetric recursion yields amplitudes free from the spurious

double poles of the N=8 theory. This is illustrated by a new nine-term expression for the

six-graviton NMHV amplitude. The general scheme also implies a simplified recurrence

relation for MHV amplitudes. We show how this relation is satisfied by a new expression

for MHV amplitudes, far simpler than those hitherto identified, and exhibiting manifest

Sn symmetry. This reformulation is related to a new momentum-twistor representation of

the MHV amplitudes.

Keywords: Supersymmetric gauge theory, Scattering Amplitudes

ArXiv ePrint: 1108.2227

c© SISSA 2013 doi:10.1007/JHEP07(2013)075

JHEP07(2013)075

Contents

1 Introduction 1

2 Three-point amplitudes as twistor diagrams 3

3 The BCFW recursion for N=8 super-gravity 6

4 Amplitude identities as contour structures 8

5 N=7 BCFW recursion 10

5.1 Evaluation of supersymmetric diagrams 12

6 All gravitational MHV amplitudes by N=7 BCFW 14

6.1 Comparison of N=7 and N=8 formalism 17

7 NMHV amplitudes 19

8 A new form for the MHV amplitude 21

8.1 A simple recursion for MHV amplitudes 22

8.2 Soft factor expressions 22

8.3 The new formula for MHV amplitudes 24

8.4 Proof of the new formula 25

9 The momentum-twistor numerator for MHV amplitudes 28

1 Introduction

Despite many advances made in the previous decade, it has been found difficult to evaluate

and express gravitational amplitudes in a manner which reflects their symmetries. Thus

the survey of the field in 2012 by Dunbar et al. [1], for instance, showed the daunting

complexity of even the simplest tree amplitudes. The earlier survey of Nguyen, Spradlin,

Volovich and Wen [2] emphasised the lack of an expression comparable to the simplicity

of the Parke-Taylor formula for MHV amplitudes in pure gauge theory. These authors

prescribed certain ‘ideals’ that should be satisfied by such an expression. In this paper,1 at

section 8, we shall find an expression which embodies all these ideals and more, and which

1This paper merges the material of two preprints arXiv:1108.2227, arXiv:1204.1930. This is for two

reasons. (1) It allows a streamlined account of the N=7 recursion, fully integrating the observation of

Henriette Elvang, as noted in arXiv:1108.2227v2. (2) We can go directly to the new formula for MHV

amplitudes reported in arXiv:1204.1930, omitting the computational material in arXiv:1108.2227 which

this later discovery made redundant. The opportunity has also been taken to make minor corrections and

add further references, along with revision of the typography and other matters of presentation.

– 1 –

JHEP07(2013)075

suggests that further enormous simplifications should be possible. This new expression is

indebted to [2] and its emphasis on gravitational soft factors.

First, however, we go through an account of the BCFW recursion procedure [3] as

applied to gravitational scattering. This includes a brief review of the standard N=8

super-symmetric formalism. But the emphasis is on developing an N = 7 super-symmetric

formalism. This greatly improves the effectiveness of the recursion, eradicating spurious

double poles. It yields a direct derivation of the Berends-Giele-Kuijf formula [4] for all MHV

amplitudes, in the form which was more recently confirmed by Mason and Skinner [5]. We

also illustrate the economy of this formalism by deriving a new nine-term expression for

the 6-field NMHV amplitude.

Twistor representation of a structure which breaks conformal symmetry, as gravity

does, can make that broken symmetry manifest in a new and useful way. The essential point

is that linearized gravity only breaks conformal invariance in a very weak sense, introducing

non-singular numerator factors, while the underlying singularity structure depends only on

conformally invariant geometry. We shall exploit the simplicity of these numerators, by

using the twistor diagram formalism. Roger Penrose initiated the twistor description of

gravity over 40 years ago, and the gravitational scattering diagrams in [6], since neglected,

embody the essential concepts used in this analysis.2

This approach is thus different from the currently prevailing line of attack on gravita-

tional amplitudes, which is based on extending the insights from string theory which led

to the KLT relations. Recent work (see for instance [7]) shows its great significance for

computing loop integrals in N=8 super-gravity. This approach is less natural in twistor

geometry. Indeed there is a certain complementarity here. String-based relations make

a complete break from conformally invariant geometry but give rise to powerful relations

between amplitudes which are independent of helicity. Twistor-based relations emphasise

the connection with conformal geometry, but at the cost of according what may seem an

excessively dominant role to helicity conservation and violation. Hopefully, further devel-

opments will before long unify these approaches, perhaps by making use of insights from

Witten’s twistor string model.3

Having established the N=7 formalism, we focus on the simple recurrence relation

that it yields in the special case of MHV amplitudes. Although the BGK formula rather

trivially solves this relation, we may observe that there is another quite different solution,

which takes the form of a simple determinant. This remarkable fact supplies the ‘ideal’

description of MHV amplitudes which has hitherto been so elusive.

Finally, we write MHV amplitudes in terms of the momentum-twistors introduced

in [12]. This is a useful innovation even though there is no apparent role for the dual

conformal invariance that momentum twistors were devised to represent. The emergent

feature is that a numerator polynomial in the momentum-twistors captures the content of

2A further reason for the use of the diagrams is that the recent work of Arkani-Hamed et al. [8] has made

great advances in elucidating their combinatorial structure, showing the close connection with the concept

of the positive Grassmannian, so that they are firmly established at the core of on-shell scattering theory.3More recent developments (particularly [9–11]), influenced by the new formula found in section 8, have

enhanced this possibility.

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JHEP07(2013)075

the gravitational amplitude. This feature is closely linked with the existence of the new

determinant formula.

2 Three-point amplitudes as twistor diagrams

We first review the twistor diagram formalism as originated by Penrose [6] and developed by

this author for gauge-field scattering. The material in [13] showed how the original BCFW

recursion is equivalent to a simple and natural composition of twistor diagrams. The super-

symmetric extension of the diagram formalism [14] then established its computational value

in [15].

The diagrams are not essential to the new formulas for amplitudes to be arrived at

in this paper, but they assist visualisation of the various symmetries of the theory, and

cast new light on the resulting expressions. In particular, they make contact with the

significant theme of representing amplitudes as integrals over geometric regions, so that

addition formulas arise from nothing but the addition of regions. This emergent geometric

concept has played a useful part in recent advances, beginning with [12]. Another reason

for interest in the diagram formalism is that it is closely connected the Grassmannian

description of amplitudes.

Indeed Nima Arkani-Hamed, Freddy Cachazo and their group discovered the twistor

Grassmannian structure for the amplitudes after studying the properties of twistor dia-

grams [16, 17]. Development of the Grassmannian calculus has since then inspired entirely

new insights. So far it has been defined only where there is a given ring-ordering of the

fields, as in the colour-ordered sectors of the (planar part of) gauge theory. But current

investigations [8, 18, 19] suggest the existence of a more fundamental picture in which

different orderings can be combined. One would expect this to encompass the various re-

lations (Kleiss-Kuijff, etc.) between amplitudes associated with different colour-orderings,

and perhaps also gravitational amplitudes. Thus formulating gravitational amplitudes as

twistor diagrams should make a useful contribution to such investigations.

The starting-point for BCFW recursion lies in the 3-field amplitudes, and it is the

simple 3-field amplitude for gravity which makes the recursive principle such an attractive

proposition. We shall first fill in some material stated but not given a detailed explanation

in [13, 14]. Twistor diagram structure is based on the twistor-integral representation of the

momentum delta-function on massless fields, i.e. the operation of integrating the product

of some set of massless fields over Minkowski space. For four massless scalar fields, the

representation is particularly simple, being conformally invariant, and can be written as

the diagram:1

3

2

4

0 1

00

01(2.1)

Here the black and white vertices denote twistor (Zα) and dual twistor (Wα) variables,

all to be integrated out with the natural form. Each thick line represents a pole singularity:

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JHEP07(2013)075

the label 0 denotes a simple pole of form (W.Z)−1, and the label 1 denotes a double pole

of form (W.Z)−2. We ignore questions of sign and factors of 2πi in what follows.4 The

external labels 1, 2, 3, 4 denote twistor functions corresponding to massless scalar fields,

with the two ‘ears’ as a reminder that these functions are actually cohomological. The

double poles are examples of twistor transform lines, which do nothing but transform the

twistor representative of a field to the dual twistor representative. The single poles can

be considered as Cauchy poles which reduce the integral to one where all the integrated-

out variables lie on the same line in projective twistor space. The space of such lines is

equivalent to the space of points in Minkowski space, and so the complete twistor integral

is equivalent to a∫d4x integration over Minkowski space.

An integral form for three massless scalar fields is obtained by taking the fourth scalar

field to be the constant field, i.e. φ4(x) ≡ 1. (A more rigorous treatment would have to

consider the exact sense in which this limit is taken, but this is not necessary at this level

of formality.) In the twistor picture, the constant field is just the elementary state based at

infinity, with representation f(Z4) = {(Z4.I1)(Z4.I2)}−1, where I1α and I2β are two dual

twistors whose skew product is the infinity twistor Iαβ.

Integrating out Z4 leaves the expression

0

1 2

3

0 1

(2.2)

as the twistor-diagram version of the scalar φ3 vertex, i.e. the delta-function 1.δ(∑3

1 pi).

Here the crooked leg between 1 and 3 denotes the factor (IαβZα1 Z

β3 )−1, where IαβZ

α1 Z

β3 is

equivalent to the product of spinors conventionally written 〈13〉. At first sight this is an

asymmetrical expression, but symmetry is restored by a formal integration by parts. One

advantage of the diagram formalism is that identities are readily expressed by operations

of integrating by parts on the integrands, simple because the integrands are so simple.

To translate the standard gauge-theoretic 3-point amplitude,

〈12〉3〈23〉〈31〉δ

(3∑

1

pi

), (2.3)

we therefore apply the operators

(IαβZα1 Z

β2 )3(IαβZ

α2 Z

β3 )−1(IαβZ

α3 Z

β1 )−1 (2.4)

to (2.2). Before doing this, we introduce some further notation. Following Penrose [6, 20]

it is helpful to write the simple pole (W.Z)−1 as [W.Z]1, then to define [W.Z]n in such a

way that ∂∂Zα [W.Z]n = Wα[W.Z]n+1. For positive n this gives a pole of order n together

4We are also over-simplifying: for genuine contours to exist for these twistor integrals, the formalism

requires a shift from W.Z = 0 to W.Z = k, as described in [13]. But for the developments described in this

paper, this refinement can be safely ignored.

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JHEP07(2013)075

with the factor (n − 1)!. Thus [W.Z]n+1 is the factor which in diagrams is indicated by a

thick line labelled with n.

But now we can extend the definition to non-positive integers. For n = 0, the formal

object [W.Z]0 can be defined as a contour with boundary on W.Z = 0.5 Then [W.Z]−1 is

defined as the numerator (W.Z), combined with such a boundary contour, [W.Z]−2 as the

numerator 12(W.Z)2 combined with such a boundary, and so on.

Now we are ready to apply the operators (2.4). The denominators cancel and the

result can be written simply as:

3

1 2

3

(2.5)

All the results in [13] came from applying BCFW composition to this primitive object

and its dual.

The super-symmetric generalization in [14] came from replacing all the lines by N=4

super-boundaries and all the external functions by super-fields. The 3-field super-amplitude

then becomes symmetric in the three external super-fields, as:

1 2

3

N = 4

(2.6)

(The notation in [14] introduced arcs instead of straight lines to distinguish the N=4

super-boundaries from the classical boundaries, but now that we need to introduce N=7

and N=8 as well, this distinction will be dropped, and an indication of the super-symmetry

given explicitly instead.)

The super-symmetric extension of BCFW is automatic in this formalism, and all am-

plitudes become characterized as simply integrals of the external fields over regions with

boundaries specified by these elements.

Having rehearsed this derivation of 3-amplitudes in N=4 gauge theory, it is straight-

forward to adapt it to find the analogous gravitational 3-amplitude

〈12〉6〈23〉2〈31〉2 δ

(3∑

1

pi

). (2.7)

5In the original Penrose notation the object [W.Z]0 was indicated by a line with a (−1) label, and then

later, as in [13], a wavy line was used. We are now adopting a single simple thin straight line to indicate

boundary-defining elements such as [W.Z]0, with a double thin line for [W.Z]−1. The reason for this notation

is that these boundary-defining elements emerge as the fundamental objects of the theory, and so should

have the simplest possible expression. This notation is also consistent with the diagrams of [8].

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JHEP07(2013)075

We obtain, for linearized gravity without super-symmetry, the 3-vertex:

1 2

3

5 (2.8)

where the dashed line indicates a simple numerator factor IαβZα1 Z

β2 = 〈12〉. It is simple

to extend this to super-fields with N=8 super-symmetry, obtaining:

1 2

3

N = 8 (2.9)

which, using integration by parts, is readily seen to be symmetric in {123}.At first sight these twistor expressions have lost the simplicity of the statement that

‘gravity is the square of Yang-Mills’. But they convey the more accurate idea that ‘gravity

times φ3 is the square of Yang-Mills’, which correctly accounts for the square of the delta-

function in the equation. In the diagrams the conformal-symmetry-breaking denominator

factor in the φ3 vertex complements the conformal-symmetry-breaking numerator factor

in the gravitational vertex.

3 The BCFW recursion for N=8 super-gravity

The use of N=8 super-symmetric BCFW recursion is well established for the calculation of

tree amplitudes, as discussed in [21]. In particular, it can be employed to find the 4-field

(super-)amplitude from the 3-amplitudes.

Pivoting on (23), we have that M(1234) = M(13k)◦M(24k) +M(34k)◦M(12k). The

BCFW composition may be written diagrammatically as:

3 2

14

+1

3

4

N = 8

2

(3.1)

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JHEP07(2013)075

The first of these diagrams can be simplified by (i) integration by parts and (ii) elimi-

nating a double twistor transform, giving:

1

3 2

4

N = 8

3 2

14

+

(3.2)

This sum is correct for any super-fields, but to check and illustrate the result we shall

specialise to the simplest case, M(1+2+3−4−). In this case the super-symmetry is trivially

integrated out. Equivalently, for this particular helicity and pivoting we could have used

the formalism without any super-symmetry. So the diagrams reduce to the N=0 diagrams:

1

3 2

4

3 2

14

+

5(3.3)

where the external functions are now all of classical homogeneity degree (−6).

These diagrams are equivalent to the terms:

[12]〈34〉7〈24〉〈13〉〈41〉2〈23〉2 −

[12]〈34〉6〈12〉〈41〉2〈23〉2 =

[12]〈34〉6〈24〉〈13〉〈12〉〈41〉〈23〉 (3.4)

which is the correct amplitude.

So far the diagrams have added nothing to the results obtained, and indeed this last

calculation is completely equivalent to that of [22], with their early use of BCFW recursion

without super-symmetry. But some features of the amplitude are well illustrated by the

diagrammatic representation. The breaking of conformal invariance by a numerator factor

is particularly clear. Momentum conservation is built into the diagram, and identities such

as [13]〈24〉−1 = −[34]〈12〉−1 amount to simple steps of integrating by parts. The double

lines code double poles. But the cancellation of the double poles in the sum remains just

as obscure in the diagram formalism as in the standard spinor calculus.

Indeed the diagrams just make more vivid the general inelegance of the N=8 calculus:

they are more complex than the result. This drawback becomes more noticeable in cal-

culating more advanced results, where the cancellation of double poles becomes less and

less tractable with the growth of algebraic complexity, as is shown explicitly in [22]. Field

theory seems to have made a curiously retrograde step: it was the triumph of DeWitt [23]

— in work that established the vital role of the helicity representation, the foundation of

all modern developments — to calculate the 4-graviton amplitude in the form that would

now be written as[12]4〈34〉4s12s13s14

. (3.5)

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JHEP07(2013)075

This formula brings out a simplicity of the amplitude which is so lacking in this recursion

method. What is the origin of the spurious double poles? It is that the momentum

conservation condition s12+s13+s14 = 0 has been used to write (s12s13s14)−1 as −s−214 s

−112 −

s−214 s−113 . This splits the amplitude into two terms which each resemble the gauge-theoretic

4-field analogue, with two kinematical denominators rather than three. Such terms have

natural representations as box-shaped twistor diagrams, such diagrams being equivalent

to the BCFW joining of two 3-vertices. At first sight there might seem to be no other

way of fitting three singularities into a calculus designed to encompass just two, and so it

might appear that these essentially artificial double poles would be ineradicable within any

formalism making use of on-shell recursion methods.

Yet it turns out that these double poles can be eliminated at source by making use

of N = 7 rather than N = 8 supersymmetry. One way of seeing why N = 7 is relevant is

to observe that although gauge-field amplitudes have genuine fourth-power behaviour, and

although gravity is characterized as ‘the square of Yang-Mills’, gravitational amplitudes

never actually contain an eighth power. Seventh powers are the highest that actually

occur, and the eighth powers are, like the double poles, spurious. We shall look at this in

more detail shortly, but as a preparatory step look first at a remarkable contour structure

that emerges from twistor diagrams, and becomes the key to fitting in the third singularity.

4 Amplitude identities as contour structures

The simplest relation between differently colour-ordered amplitudes is that

A(1234) +A(1243) +A(1423) = 0 , (4.1)

which can be read as saying that when a fourth field is added to the ordered three-vertex

A(123), the three possible points of insertion yield colour-stripped amplitudes which sum

to zero. (Hence, if the gauge field is U(1), the amplitude vanishes, so that this equation

can be seen as the simplest case of the ‘U(1) decoupling identity’.)

If we write this identity in the form of N=4 twistor diagrams, we can draw them in

such a way as to emphasise that they give the effect of inserting field 4 in three different

ways into the 3-amplitude A(123):

1 2

3 N = 4

4

+

4

1 2

3

21 2

3 4

+ = 0 .

(4.2)

and this indicates that the zero sum is actually an implication of the more primitive contour

identity:

+ + = 0 .

1 1

4 4 4

22

3 3

(4.3)

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JHEP07(2013)075

We shall call this ‘the triangular contour identity.’ This structure has remarkably wide

implications. It can be used to express the U(1) decoupling identity, and more generally

the Kleiss-Kuijff relations, for all MHV amplitudes, these relations also being related to

the validity of BCFW evaluation with non-adjacent pivots. But this would go beyond the

topic of this note, and here we will confine ourselves to three observations directly relevant

to later claims in this paper.

(1) Note that the second and third terms in (4.2) can be seen as the result of inserting

the 3 field into the dual 3-amplitude for {124}. Hence by applying the dual triangular

identity,

1 2

3

N = 4

1 2

34

4

=

(4.4)

which is the square identity of N=4 gauge theory.

(2) Now consider the MHV amplitude for five gauge fields. We may note that the sym-

metrized sum A(12345)+A(13245) is actually symmetric in {235} and in {14}. That

this is so follows straightforwardly by the calculation

1

〈12〉〈23〉〈34〉〈45〉〈51〉 +1

〈13〉〈32〉〈24〉〈45〉〈51〉 =〈14〉

〈12〉〈13〉〈24〉〈34〉〈45〉〈51〉 . (4.5)

On the other hand, we can see this symmetry directly and graphically by using

diagrams. Attach the 5 field consistently between the 4 and the 1 field in the dual of

the 3-term relation (4.2), using twistor transforms as necessary. We obtain

N = 4

2

1

5 4

3 21

5 4 31 4

2 3 5

+ + = 0 .(4.6)

The first term is A(13245) and the third term is A(12345). The second term mani-

festly has the symmetry stated. We shall find a gravitational analogue of this sym-

metry in section 6 below.

(3) Again starting with the three term identity (4.2), operate with:

(IαβW4α

∂

∂Zβ3

)−1(W4.Z2)I

γδ ∂

∂Zγ2

∂

∂Zδ3.

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JHEP07(2013)075

The first term vanishes (using the fact that (W4.Z2)[W4.Z2]1 = 0), and we obtain a

two-term identity:

1

3

2

4

0

0

1

3

2

4

0

0

=N = 4 (4.7)

We will call this the ‘square symmetry identity’. Clearly, this is equivalent to knowing

that the scalar box diagram (2.1) is completely symmetric in all four fields, so it does

not tell us anything new; but it is remarkable that this symmetry can be seen as a

consequence of the more fundamental triangular contour identity.

These three examples should suffice to show the importance of this additive contour struc-

ture. It is worth noting that such additions of contours naturally take us beyond the scope

of the twistor diagrams representing the colour-stripped amplitudes in N=4 gauge theory.

Such diagrams behave like topological disks, with the colour-trace running round the ex-

terior as in a ribbon diagram, this property being conserved by the BCFW joining rule.

The more general diagrams introduced here are not necessarily planar, as for instance the

third term in (4.6). This is of course appropriate as we are intending to use this more

general type of diagram for gravitational amplitudes, which have no natural planarity, and

correspond to closed rather than open strings.

5 N=7 BCFW recursion

It should be clear that the N=7 formalism does capture all the information in standard

N=8 supergravity. Two N=7 supermultiplets of 128 fields correspond to one N=8 super-

multiplet of 256 fields. The identification can be made by picking out (say) the eighth

super-component. We then divide the N=8 supermultiplet of 256 fields into two classes,

those that do not contain an 8 and those that do. These classes naturally map into two

dual N=7 128-multiplets. Thus for the scalar fields, φ(8)1234 is mapped to φ

(7)1234, whilst φ

(8)1238

is mapped to φ′(7)4567 in the dual N=7 multiplet. For the twistor representation, this means

that for a given super-field, either a Zα or a Wα representation is specified, and there is no

twistor transform available. (The double transform is well-defined, however, and we shall

use it.) In particular, the gravitons have to be represented by functions of homogeneity +2

in either Zα or Wα, depending on helicity.

The original preprint of this material gave an N=7 scheme without proving its validity.

The author is indebted to Henriette Elvang, who immediately referred to [24] for a simple

proof that it is valid. Section 4 of that paper establishes that a shifted gauge-theoretic

amplitude, using any shift defined by adjacent pivots, will exhibit z3−N behaviour. Hence

N=3 super-symmetry will fail to give rise to a valid BCFW recursion. Henriette Elvang

points out that the argument used in that paper applies equally to super-gravity and shows

a z6−N behaviour for any shifted amplitude. Hence any N=7 super-gravity amplitude will

necessarily have z−1 behaviour, sufficient to justify the BCFW decomposition.

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JHEP07(2013)075

The shifted super-momenta, as employed in these arguments, give rise to BCFW ex-

pressions which correspond directly to the super-twistor diagrams of this paper. This

connection is explained in [16]. It has a simple origin: the original non-super-symmetric

diagram theory [13] showed how the twistor diagrams and their composition rule were

equivalent to the non-super-symmetric BCFW terms and their composition rule. This

connection is preserved when super-symmetric extension is applied to both formalisms.

Thus the results of [24] apply directly to the N=7 diagrams used in this paper. A simple

induction on n, starting with n = 3, then shows that BCFW decomposition rule using N=7

diagrams is valid for all n-graviton tree amplitudes of all helicities.

In section 6 and section 7 we shall go on to find tree-level amplitudes by such an N=7

super-symmetric BCFW recursion. Here we shall just set out the 3-amplitudes which form

the starting-point for the recursion, and check that the 4-amplitude emerges correctly.

To express the 3-amplitudes as diagrams requires care since there are no twistor trans-

forms within the N=7 formalism; an external field is defined as being either a twistor or

a dual twistor (super-)function. But double transforms are well-defined, and allow us to

write the three-amplitudes as:

N = 7

1 3

2

(5.1)

and its conjugate. Here 1, 2, 3, are super-functions of degree (+2). For gravitons, i.e. with

bosonic homogeneity (+2) outside and hence all the supersymmetry on the internal lines,

this diagram correctly corresponds to amplitude 〈12〉6〈23〉−2〈31〉−2.This expression can be seen to be symmetric in {12} by taking a double twistor

transform on the 2 field and moving the numerator factor across using integration by

parts. BCFW composition, applied to these 3-amplitudes, gives the correct four-field

amplitude. Pivoting on (23) there is now only one term for M(1+2−3+4−), namely

M(1+2−k) ◦M(3+4−k), and that gives

N = 7

1 2

4 31

2 3

4

=(5.2)

This is the simplest possible twistor diagram that there could be for the amplitude.

It amounts to integration of the simple numerator IαβZα1 Z

β3 I

γδW2γW4δ = 〈13〉[24] over a

super-volume. No double poles now arise, the numerator seventh power is natural, and the

expression has manifest symmetry in {13} and {24}, which is the best that can be attained

within the N=7 formalism.

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JHEP07(2013)075

5.1 Evaluation of supersymmetric diagrams

Note that the four gravitons are being considered as the seven-fold super-symmetrized

partners of spin-32 fields of the opposite helicity type. From this point of view, the four-

graviton interaction is being treated as the super-symmetrized version of a process in

which four spin-32 fields exchange gravitons. The amplitude for this process comes from

treating (5.2) as an N=0 diagram, i.e. with four twistor functions of degree (−5) on the

outside, and is:〈13〉5[24]

〈12〉〈23〉〈34〉〈41〉 . (5.3)

We will now study the N=7 diagram (5.2) and its evaluation in an efficient manner.

In principle, every super-boundary has to be expanded as a formal series and then the

resulting fermionic numerators combined. In the case of N=4 theory, the diagram

4

1 2

3

N = 4

(5.4)

with fields of classical homogeneity 0 on the outside, expands to a five-term series of N=0

diagrams

−43 +6 −4 +3

3

3

0

0002 2

2

2

1

11 1

2 22221 1 1 1 1

3 3 3 3 344444

where the binomial coefficients (and the alternating signs) are the result of formal inte-

gration of the fermionic parts of the super-twistors. Likewise, the N=7 diagram expands

to an eight-term binomial sum of N=0 diagrams. As an alternative to following through

this super-algebraic computation, we may use the essential property of supersymmetric

algebra, that it mimicks the effect of derivative operations. This is most easily shown for

N=1. We have that

4

1 2

3

N = 1

(5.5)

expands to the two N=0 diagrams (with an essential minus sign from the anticommuting

elements):

4

1 2

3 4

1 2

3

−0 0

0

0 (5.6)

– 12 –

JHEP07(2013)075

again assuming that there are no super-components in the exterior fields, i.e. that these are

all of classical homogeneity (−3). This can be thought of as representing the equivalent of

Moller scattering for (massless) quarks. Now we observe that this anti-symmetrized sum

is exactly the same as is obtained by the operation 〈24〉/〈13〉 applied to the all-boundary

N=0 diagram, so that the amplitude is

〈24〉〈13〉

〈13〉4〈12〉〈23〉〈34〉〈41〉 =

〈24〉〈13〉3〈12〉〈23〉〈34〉〈41〉 (5.7)

agreeing with the sum of the two separate Moller terms: 〈13〉[24]s−114 + 〈13〉[24]s−112 .

It is worth noting that from this point of view, the anti-commutation in the formal

super-algebra stems from and expresses the anti-symmetry of the infinity twistor Iαβ, which

is in turn a consequence of the representation of space-time points by lines in projective

twistor space.

For N=4 this occurs four times over, so that the N=4 diagram (5.4) is given by

〈24〉4/〈13〉4 times the corresponding N=0 diagram, correctly giving an amplitude

〈24〉4〈13〉4

〈13〉4〈12〉〈23〉〈34〉〈41〉 =

〈24〉4〈12〉〈23〉〈34〉〈41〉 . (5.8)

For the N=7 diagram (5.2) we may likewise simply read off the result (taking into

account the two numerator factors 〈13〉[24]) as:

〈24〉7〈13〉7

〈13〉[24] 〈13〉4〈12〉〈23〉〈34〉〈41〉 =

〈24〉7[24]

〈13〉2〈12〉〈23〉〈34〉〈41〉 , (5.9)

which is correct.

A slightly different evaluation comes from relating the N=7 diagram to the corre-

sponding N=4 diagram, which we can identify with A(1+2−3+4−), and then considering

the numerator factors as arising from applying the operator s12. This gives:

〈24〉3〈13〉3 s12

〈13〉2〈12〉〈34〉 A(1+2−3+4−) = s12A(1+2−4−3+)A(1+2−3+4−) (5.10)

which is the KLT form of the gravitational amplitude.

Similarly, the 5-field N=4 gauge-theoretic MHV amplitude A(1+2−3+4+5−) is repre-

sented by

N = 4

2 1 5

43

(5.11)

with the external fields all of classical homogeneity 0. This would, if expanded out in

individual terms, require extensive super-algebra and then the summation of a trinomial

expression. This is all cut short by observing that the effect of the supersymmetric algebra

– 13 –

JHEP07(2013)075

is exactly the same as applying the operator (〈25〉/〈34〉)4 to the N=0 diagram:

2 1 5

43

3(5.12)

Thus we may correctly write down in one step:

A(1−2+3−4−5+) =〈25〉4

〈12〉〈23〉〈34〉〈45〉〈51〉 . (5.13)

Such methods lie behind the claim made in [15] that all the eight-gauge-field amplitudes

could simply be written down directly from N=4 twistor diagrams. They will be used freely

in the ensuing sections.

6 All gravitational MHV amplitudes by N=7 BCFW

We have already considered the 3-amplitudes and the 4-field amplitude. We continue by

considering the 5-amplitude, using 5 and 1 as pivot fields. There are only two terms,

arising from M(4+5−k) ◦M(1+2−3+k) and M(3+5−k) ◦M(1+2−4+k). The four-field sub-

amplitudes need no simplification before being substituted in, for they are already natural

single-term expressions. The diagrammatic form of the BCFW joining process then yields

the complete amplitude as the sum of two diagrams:

1

3

2

4

+

3

2 1

4

N = 7

55

(6.1)

Remarkably, these diagrams correspond directly to the two terms in the BGK formula.

M(1+2−3+4+5−) = 〈25〉7 [45]〈35〉[23]〈42〉 − [35]〈45〉[42]〈23〉〈12〉〈13〉〈14〉〈15〉〈23〉〈24〉〈34〉〈35〉〈45〉 . (6.2)

To establish this, consider the first diagram (the second is the same with 3↔ 4). We may

first show that the N=0 diagram

2 1 5

4

6

3

(6.3)

corresponds to〈34〉6[23][45]

〈23〉〈12〉〈13〉〈14〉〈15〉〈45〉 , (6.4)

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JHEP07(2013)075

This is straightforward. Then we use the expansion principle of N=7 diagrams, described

in section 5, and so multiply this by (〈25〉/〈34〉)7. The result follows.

If we relate the N=7 diagrams (6.1) to the corresponding N=4 diagrams, we find that

the two diagrams correspond directly to the two terms in the KLT formula [25]:

M(1+2−3+4+5−) = s23s45A(1+2−3+4+5−)A(1+4+5−2−3+)

+s24s35A(1+2−4+3+5−)A(1+3+5−2−4+) . (6.5)

None of the above actually depended on the use of the twistor diagrams, and could

have been written out in standard terms using shifted (super-)momenta. However, one

feature of the twistor diagram expressions which cannot be seen in standard terms is the

structure of pure numerators which capture the effect of the gravitational interaction. Even

more strikingly, the twistor diagrams (6.1) can naturally be added to give a total geometric

object analogously to the discussion of the five-field amplitude in gauge theory in section

4. We can use this fact to show an emergent feature of the N=7 formalism: this 5-graviton

amplitude, regarded as a single total object, actually has S3 × S2 symmetry structure,

being symmetric in {134} and in {25}.The argument is similar to the analogous result for gauge theory at (4.6), but now

that twistor transforms do not exist we must go back to the more fundamental triangular

contour identity. In fact we need two rules: the first follows from allowing integration by

parts (equivalent to the addition of a total derivative to the integrand), which means that

we can always move numerator factors thus:

= (6.6)

The second rule is the triangular contour identity 4.3, which we can write as:

+ + = 0 .

(6.7)

These two rules can be used, rather as in graph theory or knot theory, to define classes

of equivalent diagram sums.

In the case of the 5-graviton amplitude, with diagrammatic representation (6.1), adding

and subtracting the diagram

3

N = 7

1

2

4

5

(6.8)

– 15 –

JHEP07(2013)075

has the effect of transposing 1 and 3. It is easy to see that this simply corresponds to

adding and subtracting the expression

[24][54]〈25〉7〈12〉〈13〉〈15〉〈23〉〈35〉〈14〉〈34〉 (6.9)

from (6.2), but the effect of the diagram formalism is to make this algebraic step into

something geometrical.

The 6-graviton MHV amplitude M(1+2−3+4+5+6−) gives a straightforward contin-

uation of these features. Using (6−1+) as pivot, there are only three terms in the N=7

BCFW recursion, namely

M(6−3+k) ◦M(1+2−4+5+k) + M(6−4+k) ◦M(1+2−5+3+k)

+ M(6−5+k) ◦M(1+2−3+4+k) , (6.10)

and each of the 5-field sub-amplitudes only needs two terms. Thus the whole amplitude

may be written as the sum of six diagrams:

�

σ(345)

2 6

5

1

3

4

(6.11)

This is directly equivalent to

∑

σ(345)

〈26〉7[23][56][4|5 + 6|1〉〈23〉〈34〉〈45〉〈56〉〈12〉〈13〉〈14〉〈15〉〈16〉 , (6.12)

which is exactly the BGK expression, as restated and proved by Mason and Skinner [5].

Its key feature is the factor [4|5 + 6|1〉 = −[4|2 + 3|1〉, which corresponds directly to the

completely internal numerator factor in the centre of the diagram.

Again, by relating the N=7 diagrams to analogous N=4 diagrams, and expressing the

numerators in terms of sij operators, it is straightforward to re-express this six-term result

as the KLT formula. It would be of interest to see whether this correspondence could be

interpreted in terms of string vertex operators, thus showing some connection between the

twistor diagrams and closed string structure.

It is also the case that this sum, with its manifest S3 symmetry, is actually a geometrical

object with S4 symmetry, provided we are allowed to use the graphical transformation

rules. The explicit proof of symmetry in {1345} is omitted here, as it does not show

any particularly new or interesting feature. Clearly the more important next step is to

demonstrate that the symmetry Sn−2 holds in general — and hopefully also to learn from

such a proof something more about the nature of the geometrical object with this symmetry.

It should be clear how the diagrams generalize to all n. A chain of numerator factors

develops, corresponding precisely to the numerator factors in the BGK formula. A diagram

– 16 –

JHEP07(2013)075

for the case n = 8 is given here; the complete amplitude is the sum of the 120 permutations

of this over {34567}.

N = 7

8

7

6

5

4

3

2

1

(6.13)

Lastly we will examine the asymptotic behaviour in the shift parameter z. The shift we

are interested in is the shift that would be used with non-supersymmetric BCFW pivoted

on {6−1+}. This shift is

6]→ 6] + z 1]; 1〉 → 1〉 − z 6〉 .Note that this is the dual of the shift that is used in the N=7 BCFW recursion. Under

this shift the expression (6.12) for the 6-graviton MHV amplitude becomes:

∑

σ(345)

〈26〉7[23]{[56] + z[51]}{[4|5 + 6|1〉 − z[4|1 + 5|6〉}〈23〉〈34〉〈45〉〈56〉〈16〉{〈12〉 − z〈62〉}{〈13〉 − z〈63〉}{〈14〉 − z〈64〉}{〈15〉 − z〈65〉}

(6.14)

So every one of the six terms behaves as z →∞ like z2/z4 = z−2. For larger n the pattern

is the same, with one more numerator power and one more denominator power for each

increase in n. Thus for MHV amplitudes, an aspect of the N=7 formalism is that the z−2

behaviour of the complete amplitude is manifested in every term arising in the recursion.

6.1 Comparison of N=7 and N=8 formalism

Suppose we evaluate M(1−2+3+4−)by the N=8 formula (3.2), the external fields for these

helicities all having classical homogeneity (+2):

1

3 2

4

+

3 2

14

N = 8

(6.15)

If we wished we could perform the eight-fold fermionic integration and leave the sum

of two N=0 diagrams:

1

3 2

4

+

3 2

14

6 6

5

5

5

5 (6.16)

– 17 –

JHEP07(2013)075

These then give the correct amplitude as the sum:

[14]〈14〉6〈12〉〈34〉〈23〉3 −

[14]〈14〉6〈24〉〈13〉〈23〉3 =

[14]〈14〉7〈12〉〈34〉〈13〉〈24〉〈23〉2 . (6.17)

But instead of expanding the fermionic parts completely, we may integrate out only

the eighth of the eight super-components throughout. The effect is to leave an integral

with seven super-components still to be integrated:

1

3 2

4

+

3 2

14

N = 7(6.18)

in which the crucial change is that the lines from 3 to 1, 3 to 4, have become simple super-

boundary lines. This means that they can be combined through the triangular contour

identity, into a single diagram:

1

3 2

4

(6.19)

With an integration by parts to move the numerator factor, and the removal of a

double transform, this is simply:

1

2 4

3

N = 7 (6.20)

The agreement of the N=7 diagram recursion with the N=8 calculation is non-trivial.

The non-triviality is indicated by the use of the triangular contour identity 4.3, which

becomes far deeper for larger amplitudes. It has no analogue in reducing N=4 to N=3.

It is connected with the fact that the complete amplitude for N=8 gravity behaves better

than is required for the validity of BCFW recursion, decreasing asymptotically like z−2.

As is well known, this implies the existence of extra relations. As discussed in [2], one of

the most desirable features of a recursive procedure would be that it conforms to this z−2

behaviour at each stage, which can be taken as a sign that the procedure has made use of

these extra relations. It would be interesting to know if the N=7 recursion calculus is in

fact an effective way of ensuring this conformity. We leave this as an open question.6

6I am grateful for a communication [26] which indicates a demonstration that the N=7 formalism em-

bodies the bonus relations.

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JHEP07(2013)075

How have we contrived to squeeze the three inverse operators in DeWitt’s formula (3.5)

into a formalism which in gauge theory expresses only two? This question could have been

addressed in a more elementary way. We are given the amplitude with its (s12s13s14)−1

factors, and seek to encode it in a twistor calculus. If the external fields are represented

by functions of homogeneity degree (−6), analogous to the starting-point for gauge-field

theory with fields of degree (−4), then indeed we are driven into the splitting and double

pole solution. But twistor theory allows another representation of gravitons, by functions of

degree (+2). In this representation, the contraction of the spinor indices in ψ1ABCDψABCD2

translates into the operation (Iαβ ∂∂Zα1

∂

∂Zβ2)4. These derivatives in the numerator can cancel

the problematic inverse operator s−112 = (IγδZγ1Z

δ2)−1(Iαβ ∂

∂Zα1

∂

∂Zβ2)−1. This exploits the

subtle asymmetry in the DeWitt expression, in which the inverse operator s−112 is not on

the same footing as the others. The price paid for this choice of representation is the

eight-term summation of terms; but this is packaged into a single term through the magic

of N=7 super-symmetry.

This description still does not quite explain why the inverse derivative s−112 can somehow

be made innocuous, apparently not actually requiring any integration at all. The reason

why this is possible is that a twistor function of degree (+2) already embodies a great deal

of integration! The Penrose transform requires such a function to be differentiated four

times to get the physical Weyl spinors such as ψ1ABCD. They are in a sense potentials, but

of a kind that never introduces the spurious degrees of freedom that plague the conventional

potentials of gauge theory and gravity. Thus this geometric realization of DeWitt’s formula

actually goes to the the heart of the twistor representation.

On the other hand, the N=7 formalism completely fails to embody one of the main

ideas of the twistor representation, that the negative and positive helicity gravitons should

be represented in the same twistor space. Considerable effort has gone into finding solutions

to this ‘googly’ problem, but the formalism here seems to abandon any hope of so doing,

with gravitons of different helicities behaving like completely separate entities. We seem

also to have abandoned one of the main objectives described in [2], that of representing

the full Sn symmetry of gravitational amplitudes. Hopefully, since all the information of

N=8 theory is actually preserved in the N=7 formalism, some way will be found to bridge

from 7 to 8 and recover the full symmetry. Meanwhile we shall pursue the computational

advantages of the new formalism.

7 NMHV amplitudes

In this section we will evaluate the NMHV amplitude M(1+2+3+4−5−6−). We take {3+4−}as the pivots. The terms arising from BCFW are as follows.

First, one term from M(3+5−6−k) ◦M(4−1+2+k), with diagram

N = 7

6

1

2

435

(7.1)

We will refer to this as M356, characterising it by its factor of s356.

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Next, the term M(3+2+5−k) ◦M(4−6−1+k), with diagram

1

2

43

N = 7

5

6

(7.2)

and three similar terms obtained by 1 ↔ 2, 5 ↔ 6. We can refer to these as

M325,M315,M326,M316.

Then there is the term M(3+1+5−6−k) ◦M(4−2+k), with a diagram which has to be

written as a sum:

+5

2

4

1

6

3

5

2

4

1

6

3N = 7

(7.3)

There is a second term obtained by 1 ↔ 2, and the ‘flipped’ versions of these, i.e.

those obtained by the mapping (123456) ↔ (654321), 〈↔ [. We will refer to these as

M342,M341,M345,M346.

The amplitude is then the sum of these nine terms. (There are ten ways of dividing

the six momenta into two groups of three, but the tenth corresponds to M123 = 0.) Only

three types of term need to be considered.

Explicitly, evaluating the diagrams, we have

M356 =[12]7〈56〉7〈12〉[56]

s356[41][42]〈35〉〈36〉[1|2 + 4|3〉[2|1 + 4|3〉〈5|1 + 2|4]〈6|1 + 2|4](7.4)

M315 =[2|4 + 6|5〉7〈26〉[15]

s315[24][26][46]〈13〉〈15〉〈35〉〈1|3 + 5|4]〈3|1 + 5|6]〈3|1 + 5|4](7.5)

and M316,M325,M326 are similar, by 1↔ 2 and/or 5↔ 6. Lastly,

M342 =〈4|2 + 3|1]7[24]

s342[56]〈24〉〈34〉〈23〉[1|2 + 4|3〉[5|2 + 4|3〉[6|2 + 4|3〉

×{ 〈15〉〈36〉

[15]〈2|3 + 4|6]− 〈16〉〈35〉

[16]〈2|3 + 4|5]

}(7.6)

and then M341 is found by 1↔ 2; M345,M346 by (123456)↔ (654321), 〈↔ [.

The spurious pole structure is as follows. M412 has four spurious poles. M426,M416,

M425,M415 have only three, and M341,M342,M345,M346 have five. There are 18 different

spurious poles, and each of them occurs in just two terms. For instance, 〈2|3 + 4|5] occurs

– 20 –

JHEP07(2013)075

in the M342 and M345 terms, and demands that the combination

[24]〈35〉〈2|3 + 4|5]

〈4|2 + 3|1]7〈16〉s342[16][56]〈24〉〈34〉〈23〉[1|2 + 4|3〉[5|2 + 4|3〉[6|2 + 4|3〉

− 〈6|4 + 5|3]7[16]

s345〈16〉〈12〉[35][34][45][4|3 + 5|6〉[4|3 + 5|2〉[4|3 + 5|1〉

(7.7)

is non-singular when 〈2|3 + 4|5] = 0. It is straightforward, if tedious, to verify by spinor

algebra that this is the case.7

We may also check the asymptotic behaviour under momentum shifting. By an argu-

ment analogous to that used for the MHV amplitudes, the appropriate shift is

4]→ 4] + z 3]; 3〉 → 3〉 − z 4〉 . (7.8)

In the M356 term the numerator is invariant under the shift, whilst each factor in the

denominator is linear in z, so this term has behaviour like z−9. In M342, the numerator

factor 〈4|2 + 3|1] = 〈42〉[21] + 〈43〉[31] is again invariant under the shift. The numerator

[24] contributes linearly in z, and so do 〈36〉 and 〈35〉. In the denominator there are six

factors linear in z, so the overall behaviour is like z−4. In M315, the numerator [2|4 + 6|5〉7contributes a z7 power, but the denominator has z9 behaviour. Each of the nine terms is

separately consistent with the z−2 behaviour of the overall amplitude.

The nine-term expression is simpler, though not hugely simpler, than that given by [22].

It has fewer terms, no double poles, and fewer spurious poles; it follows directly from the

recursion rules without needing intermediate simplification of the sub-amplitude formulas.

As a nine-term expression it is not excessively long, considering that 60 × 3 terms are

needed for the complete gauge-theoretic NMHV amplitude, with all colour orders. The

structure also retains the feaure of the MHV diagrams, that they involve simple numerators

integrated over super-twistor regions. But of course we are far from seeing the S6 symmetry

of the full super-amplitude. To represent the full N=8 super-ampliitude we would need to

sum over 20 copies of the N=7 expression, one copy for each possible projections into N=7

multiplets. The N=8 symmetry would then be present but not manifest.

Furthermore, we have no geometrical insight into the spurious pole structure. In the

gauge-theoretic analogue, representation by momentum twistors, as introduced in [12], has

cast new light on the spurious poles. But in the gravitational case it is not clear how we

might use such a representation effectively. In any case, we have no hope of understanding

the geometry of the NMHV amplitudes unless the MHV amplitudes are first given a clearer

algebraic and geometrical form. We therefore return to question of MHV amplitudes.

8 A new form for the MHV amplitude

In this section we shall cast the MHV results into a more conventional momentum-space

form. We shall also step outside the pure N=7 theory, to borrow the insight from the N=8

7Nevertheless, the author is grateful to Jacob Bourjaily for applying his Mathematica application [27]

to check this algebra.

– 21 –

JHEP07(2013)075

theory that the MHV amplitudes are naturally written in terms of helicity-independent

reduced amplitudes Mn:

Mn(1−2−3+4+ . . . n+) = 〈12〉8Mn(1234 . . . n) (8.1)

so that Mn is totally symmetric in n arguments.

Thus M3(123) = (〈12〉〈23〉〈31〉)−2, M4 = [34](〈12〉2〈23〉〈34〉〈41〉〈13〉〈24〉)−1, and

M5(12345) =[34]〈24〉[12]〈31〉 − [24]〈34〉[31]〈12〉

〈12〉〈13〉〈14〉〈15〉〈23〉〈24〉〈25〉〈34〉〈35〉〈45〉 . (8.2)

8.1 A simple recursion for MHV amplitudes

Now we can write out in spinor terms the effect of using the N=7 BCFW formalism, as a

simple recursive formula. Our model is the gauge-theoretic analogue, where the recursive

formula

A(123 . . . n− 1, n) =〈n− 1, 1〉〈n− 1, n〉〈n1〉A(123 . . . n− 1) (8.3)

immediately produces the Parke-Taylor formula as solution. The N=7 recursion, when

written in terms of the Mn, is:

Mn(123 . . . n− 1, n) =

n−1∑

r=3

[rn]

〈rn〉〈1r〉〈2r〉〈1n〉〈2n〉Mn−1(1(r)23 . . . r . . . n− 1) , (8.4)

where

1(r)] =(1 + n)|r〉〈1r〉 , 1(r)〉 = 1〉, r] =

(r + n)|1〉〈r1〉 , r〉 = r〉 , (8.5)

so that 1(r) + r = 1 + r + n. The notation 1(r) is used to emphasise that the shifted

momentum 1 is different in each of the (n− 3) terms, depending on r.

This relation appears to differ from others in the literature which are similar in form;

thus the formula in [28] leads to sums over (n− 2)! terms; those from conventional BCFW

have double poles; the relation of [2] also gives rise to double poles, though it is of particular

interest for its derivation from physical ideas. But no claim is made here that this recursion

relation is new, since it is the natural relation satisfied by the BGK formula and may well

have been observed before. All that is new is its derivation from a principle based on N=7

recursion.

8.2 Soft factor expressions

This section does not use the N=7 formalism directly, but is strongly influenced by what

it has brought to light. It is also influenced by remarks of Marcus Spradlin, Anastasia

Volovich, Nima Arkani-Hamed and others who have emphasised the significance of the

inverse soft factors which build up the n-graviton MHV amplitudes. In the diagrams we

see the addition of triangles in going from n to n + 1 gravitons, and these triangles are

equivalent to such factors. The diagrams can be thought of as the result of adding such

triangles in all possible ways to the original 3-amplitude.

– 22 –

JHEP07(2013)075

We are also mindful of the goal, emphasised by these authors, of representing the

complete Sn symmetry of the amplitudes. By finding a structure with Sn−2 symmetry

but requiring only (n − 3)! summands to represent it, we have taken a small step in this

direction. As this structure is strongly geometrical, residing in the homology of certain

many-dimensional integrals, we are led to look for emergent geometrical structure in the

amplitudes Mn.

Certainly the complete symmetry of M5, as given at (8.2), can be given a geometrical

characterization. The denominator is manifestly completely antisymmetric, and the nu-

merator has complete anti-symmetry since it is just εabcd pa1 p

b2 p

c3 p

d4, where

∑5i=1 pi = 0. For

higher n, the BGK formula does not seem to suggest any analogous geometrical pattern.

But a striking feature of the recursive formula we have distilled from the N=7 calculus

is that the [ij] spinors only enter in the combination [ij]/〈ij〉, which appears as a multi-

plicative factor. This is a momentum phase factor, since for real momenta, it is a complex

number with modulus 1. This suggests that there may be other solutions to the recursive

relation in which this phase factor plays a leading role, and which will point to new geo-

metrical pictures based on the physical concept of the inverse soft factor. In taking this

approach we are following the route already clearly identified by [2] and other authors.

It is now useful to define:

φij =[ij]

〈ij〉 (for i 6= j)

φii = −∑

j 6=i

[ij]〈jx〉〈jy〉〈ij〉〈ix〉〈iy〉 . (8.6)

This φii is the (negative of the) universal gravitational soft factor, as defined in [2], asso-

ciated with adding the ith field to the others. Momentum conservation ensures that the

definition is independent of the spinors x, y. Note that the spinorial weight of the φij is

(−2) on each index, this remaining true for φii. It is most important to note that φii is

only defined relative to a complete set of n momenta summing to zero; it has an implicit

dependence on the the other (n− 1) momenta.

This negative sign is chosen so that we have a convenient form for the vital linear

relation, from which everything flows:

n∑

j=1

φij πA′j π

B′j = 0 . (8.7)

This gives yet another way of writing the existing results:

M3(123) =1

〈12〉〈23〉〈31〉 〈12〉〈23〉〈31〉 , (8.8)

M4(1234) =φ14

〈12〉〈23〉〈31〉 〈23〉〈34〉〈42〉 =−φ44

〈12〉〈23〉〈31〉 〈12〉〈23〉〈31〉 , (8.9)

which is actually completely symmetric in {1234}, and

M5(12345) =φ1[4φ

25]

〈12〉〈23〉〈31〉 〈34〉〈45〉〈53〉 , (8.10)

– 23 –

JHEP07(2013)075

which is actually completely symmetric in {12345}. Note that the symmetry of M4(1234)

and M5(12345) depends upon the momentum conservation condition holding. Note also

that when anti-symmetrization is indicated by the use of square brackets round n indices,

it is not accompanied by a 1/n! factor; thus φ1[4φ25] means φ14φ

25 − φ15φ24.

There is a remarkable identity relevant to M6: given six spinors and six dual spinors,

denoted by |i〉, i] in the usual way, subject only to the condition that 〈ij〉 6= 0 when i 6= j,

φ1[4φ25φ

36]

〈12〉〈23〉〈31〉 〈45〉〈56〉〈64〉 (8.11)

is completely symmetric in {123456}. (Note that there is no momentum conservation

condition.) A proof of this fact was given in the 2011 preprint, together with a further

result relevant to M7. But the further advance reported in the 2012 preprint means that

it is now simpler to state and prove a general result for Mn.

8.3 The new formula for MHV amplitudes

It is convenient to define

cijk = cijk = {〈ij〉〈jk〉〈ki〉}−1 , (8.12)

so that the cijk are completely antisymmetric in their indices. Then the new result is that

the reduced gravitational MHV amplitude Mn is given simply by:

Mn(12 . . . n) = (−1)n+1 sgn(αβ) cα(1)α(2)α(3)cβ(1)β(2)β(3)φ

β(4)[α(4)φ

β(5)α(5) . . . φ

β(n)α(n)] , (8.13)

where α and β are any permutations of {123 . . . n}.The ±1 factors for the signature of the permutations are obviously necessary. Other-

wise, the overall sign is not of great importance, as the definition of the reduced amplitude

is conventional. But the (−1)n+1 ensures that M3(123) is simply c123c123 and that for n > 3

the formula is consistent with the definition of Mn given by the recursion relation (8.4), as

we shall soon show.

Equivalently, let Φ be the n×n symmetric matrix formed by the φij , and |Φ|rstijk be the

(n−3)× (n−3) minor determinant obtained by striking out rows i, j, k and columns r, s, t.

Then

Mn(12 . . . n) = (−1)n+1σ(ijk, rst) cijkcrst|Φ|rstijk , (8.14)

where σ(ijk, rst) = sgn((ijk12 . . . i/j/k/ . . . n) → (rst12 . . . r/s/t/ . . . n)). If we wish we can also

write the other factors as a determinant, since ca1a2a3cb1b2b3 = 2{det(〈aibj〉2)}−1.The new formula includes all the expressions given above and extends them. Thus we

now also have, for instance,

M5(12345) =φ4[4φ

55]

〈12〉〈23〉〈31〉 〈12〉〈23〉〈31〉 . (8.15)

– 24 –

JHEP07(2013)075

But the real force of the new expression begins with n = 6, where we obtain quite new

formulas, non-trivially related to the BGK expressions:

M6(123456) = −φ4[4φ

55φ

66]

〈12〉〈23〉〈31〉 〈12〉〈23〉〈31〉

=φ4[4φ

55φ

36]

〈12〉〈26〉〈61〉 〈12〉〈23〉〈31〉

= −φ4[4φ

25φ

36]

〈15〉〈56〉〈61〉 〈12〉〈23〉〈31〉

=φ1[4φ

25φ

36]

〈45〉〈56〉〈64〉 〈12〉〈23〉〈31〉 . (8.16)

8.4 Proof of the new formula

We first establish that formulas (8.13) and (8.14) are well-defined, i.e. that they are inde-

pendent of the permutations, and so enjoy Sn symmetry. To do this, it is useful to define

f ij = 〈i1〉〈i2〉φij . Then by (8.7),n∑

i=1

f ij = 0 . (8.17)

That is, the rows in the complete n× n matrix f ij all sum to zero. So

|f |123rst = −|f |124rst ,

by the elementary properties of determinants. Thus

c123|Φ|123rst = −c124|Φ|124rst (8.18)

where {r, s, t} may overlap with {1, 2, 3, 4}, without restriction. But now similarly

cijk|Φ|ijkrst = −cijm|Φ|ijmrst , and then any permutation can be composed from such transpo-

sitions. This completes the proof, and presents the Sn symmetry as a trivial consequence

of (8.7).

The expression (8.13) is thus well-defined, completely symmetric, and also of the right

spinorial weight. It remains to show that it satisfies the recursion relation (8.4).

We can exploit the freedom of representation offered by the new formula to choose a

helpful representation of the Mn−1 at each point of the recursion. The algebraic complexity

arises mainly from the ‘shifted’ momenta 1(r) and r, so we choose the triple {12r} for both

the excluded rows and the excluded columns. We have

[rn]

〈rn〉 = φnr ,〈1r〉〈2r〉〈1n〉〈2n〉c12r = c12n ,

so the consistency of the recursion relation (8.4) is equivalent to:

Mn = (−1)nn−1∑

r=3

φnr c12nc12r|Φ|12r12r . (8.19)

– 25 –

JHEP07(2013)075

Here the hatted Φ is an (n− 1)× (n− 1) matrix in which the objects φij are defined with

respect to the (n − 1) shifted momenta {1(r), 2, 3 . . . r − 1, r, r + 1, . . . n − 1} summing to

zero. Our choice of representation means that the only difference between φij and φij is

that within the rth term,

φkk = φkk + φnk〈1n〉〈rn〉〈1k〉〈rk〉 . (8.20)

Again it is useful to write:

f ij = 〈1i〉〈2i〉φij , f ij = 〈1i〉〈2i〉φij , fkk = fkk + fnk〈rn〉〈2k〉〈2n〉〈rk〉 . (8.21)

We define F as the (n− 3)× (n− 3) matrix with entries f ij for 3 ≤ i, j ≤ n− 1, with

its minor determinants indicated in the same way as for Φ, and F analogously. So in these

terms, it is required to prove that:

(−1)nn−1∏

k=3

〈1k〉〈2k〉 Mn = c12nc12n

n−1∑

r=3

fnr |F |rr . (8.22)

To do this, we expand the fnr |F |rr, ordering the sum by the number of shift-correction

factors used in each term of the expansion. The zeroth order term has no correction terms,

and gives fnr |F |rr. A typical first order term comes from taking just one correction, say for

f qq in one of the terms of the summation, say the pth, where p 6= q. This contributes

fnp fnq |F |pqpq

〈pn〉〈2q〉〈2n〉〈pq〉 .

Summing over all p and all q, and so symmetrising over p and q, the Schouten identity

gives this simple expression for the sum of all first-order corrections:∑

3≤p<q≤n−1fnp f

nq |F |pqpq .

A typical second order term comes from taking two corrections in the pth term of the

summation, say from f qq and f rr , thus contributing

fnp fnq f

nr |F |pqrpqr

〈pn〉〈2q〉〈2n〉〈pq〉

〈pn〉〈2r〉〈2n〉〈pr〉 .

Adding in the contribution from fpp and f rr in the qth term of the summation, and fpp and

f qq in the rth term of the summation, it is clear that every disjoint set {p, q, r} contributes

fnp fnq f

nr |F |pqrpqr

〈pn〉〈2q〉〈2n〉〈pq〉

〈pn〉〈2r〉〈2n〉〈pr〉 + (p↔ q) + (p↔ r) ,

which is readily seen to be

fnp fnq f

nr |F |pqrpqr ,

and so contributing a total∑

3≤p<q<r≤n−1fnp f

nq f

nr |F |pqrpqr .

– 26 –

JHEP07(2013)075

The same simplification occurs at every order, by repeated use of the Schouten identity.8

So it remains only to show that:

(−1)nn−1∏

k=3

〈1k〉〈2k〉 Mn = c12nc12n

n−3∑

i=1

∑

3≤p1<p2...pi≤n−1fnp1 . . . f

npi |F |p1p2...pip1p2...pi . (8.23)

But this too is simple. Consider the (n− 3)× (n− 3) matrix H with entries

hij = f ij + δijfnj for 3 ≤ i, j ≤ n− 1 .

Thus H has the same entries as F , but with the addition of fni to elements down the main

diagonal. Each row of H sums to zero, by (8.17), so its determinant |H| vanishes. On the

other hand, we can also expand |H| in terms of the number of fni factors. The zeroth order

part is just |F |. The ith order part is

∑

3≤p1<p2...pi≤n−1fnp1 . . . f

npi |F |p1p2...pip1p2...pi .

It follows that

0 = |H| = |F |+n−3∑

i=1

∑

3≤p1<p2...pi≤n−1fnp1 . . . f

npi |F |p1p2...pip1p2...pi .

Thus the claim we are checking reduces to

(−1)nn−1∏

k=3

〈1k〉〈2k〉 Mn = −c12nc12n|F | ,

which indeed is true, being equivalent to

Mn = (−1)n+1c12nc12n|Φ|12n12n .

This observation concludes the recursive proof of the new formula for Mn.

It is striking that whilst determinants are naturally thought of as generating anti-

symmetry, the minor determinants of the symmetric Φ matrix naturally yield a Sn-

symmetry — exactly as needed for a gravitational amplitude. This suggests scope for

generalization beyond MHV tree amplitudes.9

The new formula is much simpler than the BGK-Mason-Skinner expression, in a very

concrete sense. Given numerical data for the spinors, it requires only O(n2) operations

to find the φij , after which the determinant of a symmetric matrix of order (n − 3) must

be computed. This is easily achieved in (better than) O(n3) time. In contrast, the BGK-

Mason-Skinner formula requires summation over (n−3)! terms and so grows exponentially.

8This is equivalent to the identity (35) noted by Conde and Rajabi [29].9Such generalizations have subsequently been located, notably in [9–11]. It has also been subsequently

noted that the earlier work of [2] came close to identifying this determinant structure. The matrix-tree

theorem provides a link between their summation over trees and the new formula. The work of Clifford

Cheung [31] has drawn particular attention to earlier work on this connection.

– 27 –

JHEP07(2013)075

The new gravitational formula even compares well with gauge theory, where the simplicity

of the Parke-Taylor formula emerges only after the separation into (n− 1)!/2 colour-order

sectors, all of which must be considered. For a general gluon interaction, therefore, the

complexity is exponential in n. This formula might be seen as an indication of the emergent

simplicity of gravitational scattering, notably advanced in [30].

9 The momentum-twistor numerator for MHV amplitudes

In this final section we investigate the representation in momentum-twistor space of the

gravitational MHV amplitudes. This development is logically independent of the preceding

material, since we are not concerned now with how the amplitude is computed. However, it

is partly motivated by the structure elucidated by the N=7 diagrams, in which an integrand

numerator appears to express the conformal-symmetry-breaking feature of gravity. Can

momentum-twistor space provide a formulation in which a simple numerator appears in

the statement of the amplitude? It is also motivated by the simplicity of the new formula,

in which momentum conservation is encoded into a simple matrix structure.10

The essence of the gravitational setting is that we do not have any ring-ordering of the

external fields, whilst the definition of momentum-twistors requires an ordering. We will

adopt the point of view that the resulting structure should be order-independent, and that

this independence will encode the Sn symmetry. We can define order-independence in a

way that does not depend on translating back into the original spinors but is intrinsic to

momentum-twistor space.

Let σ be a permutation on (123 . . . n), and let (1′2′3′ . . . n′) denote the momentum

twistors as defined relative to the ordering (σ(1), σ(2), σ(3), . . . σ(n)). Then Fn(123 . . . n)

will be said to be order-independent if for all σ,

Fn(σ(1)′σ(2)′σ(3)′ . . . σ(n)′)∏i〈σ(i)σ(i+ 1)〉 =

Fn(123 . . . n)∏i〈i, i+ 1〉 . (9.1)

Rather than check all σ, it is necessary and sufficient (i) that F is cyclic, and (ii) that F

transforms correctly under a transposition of just one pair of neighbouring elements in the

chosen ordering. Under such a transposition, we may take all but two of the momentum

twistors to remain unchanged, and then there is a simple formula for the other two. Ex-

plicitly, suppose the momentum twistors Zαi are defined by the order {1234 . . . n}, and Z ′αiby the order {1324 . . . n}. Then Z ′αi = Zαi except for:

Z ′α2 = (Zα3 〈42〉 − Zα4 〈32〉)〈34〉−1, Z ′α3 = (Zα2 〈31〉 − Zα1 〈32〉)〈12〉−1 . (9.2)

Thus it is sufficient to have cyclicity plus:

〈12〉〈34〉Fn(13′2′4 . . . n) = −〈13〉〈24〉Fn(1234 . . . n) . (9.3)

As a precursor, we consider the gravitational soft factor, as introduced at 8.6. As

the invariance of this expression depends upon momentum conservation, this suggests that

10It should be noted that earlier interesting and profitable use has been made of momentum-twistors in

a super-gravity context in [32].

– 28 –

JHEP07(2013)075

momentum-twistor variables, in which this conservation is automatically encoded, may

give it a simple form. In fact, if we take momentum twistors with respect to the order

{1234 . . . n}, we find that for n ≥ 4, the soft factor is economically expressed by:

φnn =

n−2∑

j=2

〈n− 1, n1j〉〈j − 1, j + 1〉〈1n〉〈n− 1, n〉〈j − 1, n〉〈jn〉〈j + 1, n〉 . (9.4)

This is encouraging, and leads us to investigate the form taken by the full gravitational

MHV amplitude in such co-ordinates.

As usual we begin with the n = 3 amplitude, and note that it may be written as:

1∏i〈i, i+ 1〉 ∏i<j〈ij〉

Considering next the n = 4 amplitude, we may take momentum-twistors defined relative

to the order (1234), and find:

M4(1234) =〈1234〉∏

i〈i, i+ 1〉 ∏i<j〈ij〉(9.5)

The notation M has been used because this is not the same as the reduced amplitude M ;

the arguments are momentum twistors rather than pairs of spinors, and it is no longer the

co-efficient of the delta-function in momenta.

The denominator factor∏i〈i, i+1〉 is essentially a Jacobian, giving the effect of trans-

lating into momentum-twistor space. The other denominator, the product of all the pos-

sible physical poles, can also be written as a (Vandermonde) determinant. The numerator

〈1234〉 captures the essential content of the 4-graviton amplitude. It is easy to see that

〈1234〉, combined with∏i<j〈ij〉, satisfies the criterion (9.3) for order-independence.

For n = 5 we have the expression (8.2) in which the denominator is already given as∏i<j〈ij〉. Encoding the numerator with momentum twistors defined relative to (12345) we

find:

M5(12345) =〈1234〉〈2345〉〈51〉 − 〈1234〉〈3451〉〈25〉 − 〈5123〉〈2345〉〈14〉∏

i〈i, i+ 1〉 ∏i<j〈ij〉. (9.6)

The remarkable thing about this expression is that it is indeed a pure numerator, i.e. a

polynomial N5(12345) of degree 2 in each of the five momentum twistors. In fact it can

be shown that the n-field amplitude will have this form. The emergence of this numerator

is closely connected with the new formula.

Theorem:

Mn(123 . . . n) =Nn(123 . . . n)∏

i〈i, i+ 1〉 ∏i<j〈ij〉(9.7)

where Nn is a polynomial of degree (n − 3) in each of the n momentum twistors, and of

degree (n− 3)(n− 4)/2 in I.

– 29 –

JHEP07(2013)075

Proof. We have

Mn(12 . . . n) = (−1)nφ1[2φ

34φ

56φ

77φ

88φ

99 . . . φ

nn]

〈13〉〈35〉〈51〉 〈24〉〈46〉〈62〉 . (9.8)

Translating this into momentum twistors as defined by the ordering (123 . . . n), it is obvious

that just two denominator factors of 〈12〉 will occur in each term of the expansion. These

are safely absorbed in∏i〈i, i + 1〉 ∏i<j〈ij〉, and the numerator function is therefore not

singular in 〈12〉. By cyclicity, it cannot be singular in any other 〈i, i + 1〉. A similar

argument applies to all the other 〈ij〉 factors, and so it must be a polynomial.11

In the case n = 6 we have:

N6(123456) = 〈123I456〉〈234I561〉〈345I612〉+〈123I456〉〈5612〉〈2345〉〈14〉〈36〉+〈234I561〉〈6123〉〈3456〉〈25〉〈14〉−〈345I612〉〈1234〉〈4561〉〈36〉〈25〉+〈1234〉〈3456〉〈5612〉〈14〉〈25〉〈36〉+〈2345〉〈4561〉〈6123〉〈25〉〈36〉〈14〉 . (9.9)

Here 〈123I456〉 represents εαβγκZα1 Z

β2Z

γ3 I

κλZµ4Zν5Z

π6 ελµνπ. It is easy to see that

N6(123456) = −N6(234561), as required for cyclicity.

The existence of these polynomials opens a new field of geometrical description. The

general polynomial Nn is of degree (n− 3) in the twistors and degree (n− 3)(n− 4)/2 in

I. So it is a (n − 3)-degree polynomial in the n!/(n − 4)!4! objects like 〈1234〉πA′5 . . . πN′

n .

It remains to be seen whether this can be given a useful geometric characterization (e.g.

a volume in the (n − 3)-dimensional space of symmetrised spinors.) In the case of the

five-graviton numerator N5, there is such a geometric description. For i = 1, 2, 3, 4, 5, we

may define zA′

i = 〈i+ 1, i+ 2, i+ 3, i+ 4〉πA′i , where Zαi = (ωA′

i , πiA′). Then by the 5-term

Schouten identity on twistors, zA′

1 + zA′

2 + zA′

3 + zA′

4 + zA′

5 = 0 in C2. These five vectors

therefore form a closed pentagon on C2. The quantity N5 = z1.z5 + z2.z5 + z1.z4 then

has a geometrical meaning as (twice) the (complexified) area of the pentagon with vertices

(12), (34), (51), (23), (45) in order. It is thus manifestly cyclic. This pentagonal area is

reminiscent of similar expressions in the geometric description of amplitudes [33].

However, recent developments in the theory of amplitudes, based on developments of

Arkani-Hamed et al. [8], suggest regarding the zeroes of the numerator as the key to its

geometric description. The numerator N5 is in fact IαβGαβ where Gαβ is orthogonal in CP5

to all of (12), (23), . . . (51). This gives another global and manifestly cyclic characterization

of it. Similar features can be noted in the case of N6. The extension of these observations

is a matter of active investigation.

11A similar proof can be given that the gravitational MHV amplitude, as given by the new formula, has

1/z2 behaviour, and that it has just one simple pole in every 〈ij〉.

– 30 –

JHEP07(2013)075

Acknowledgments

The N=7 recursion was first formulated during my May-June 2010 visit to the Institute for

Advanced Study, thanks to an invitation from Nima Arkani-Hamed, and I am most grateful

for generous support from that institution. A second visit in May 2011 allowed further

highly stimulating discussions of this material and specific suggestions from Nima Arkani-

Hamed. The interest and encouragement of Marcus Spradlin and Anastasia Volovich has

also been notable, and I am grateful for the invitation to join them in April 2011 at the

Kavli Institute for Theoretical Physics, UC Santa Barbara, for the Harmony of Scattering

Amplitudes workshop. At Oxford, I am particularly grateful to Lionel Mason and Tsou

Sheung Tsun for organizing the special conference in July 2011 marking the eightieth

birthday of Sir Roger Penrose, FRS, OM. The influence of Roger Penrose’s ideas, over 40

years, remains paramount.

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