SOFT LIMITS AND COLOR-KINEMATIC DUALITIES IN A … · soft limits and color-kinematic dualities in perturbative quantum gravity a dissertation submitted to the department of physics

SOFT LIMITS AND COLOR-KINEMATIC DUALITIES IN

PERTURBATIVE QUANTUM GRAVITY

A DISSERTATION

SUBMITTED TO THE DEPARTMENT OF PHYSICS

AND THE COMMITTEE ON GRADUATE STUDIES

OF STANFORD UNIVERSITY

IN PARTIAL FULFILLMENT OF THE REQUIREMENTS

FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

Camille Boucher-Veronneau

May 2012

http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/xs037jq9002

© 2012 by Camille Boucher-Veronneau. All Rights Reserved.

Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution-Noncommercial 3.0 United States License.

ii



http://purl.stanford.edu/xs037jq9002

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein scope and quality as a dissertation for the degree of Doctor of Philosophy.

Lance Dixon, Primary Adviser


Renata Kallosh


Michael Peskin

Approved for the Stanford University Committee on Graduate Studies.

Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file inUniversity Archives.

iii

Abstract

This thesis is a study of the perturbative behavior of quantum gravity through explicit

tree- and loop-level supergravity calculations. First, I discuss how using universal soft

limits could allow us to reconstruct tree-level amplitudes. Second, I obtain loop-level

results in non-maximal (N < 8) supergravity using the double-copy prescription for

generating gravity amplitudes from gauge-theory amplitudes, which follows from the

recently discovered Bern-Carrasco-Johansson (BCJ) duality in gauge theory. I com-

bine N = 0, 1, 2 super-Yang-Mills (sYM) amplitudes with N = 4 sYM amplitudes to

obtain N = 4, 5, 6 supergravity amplitudes. I show how the double-copy prescription

reproduces known one-loop amplitudes, which correspond to the first explicit demon-

stration of the validity of the double-copy prescription in non-maximal (N < 8)

supergravity. I then present and discuss new amplitudes at two loops in N = 4, 5, 6

supergravity.

iv

Acknowledgements

I first and foremost would like to thank my advisor, Lance Dixon, for his mentorship

and guidance. I would like to acknowledge my collaborators: Zvi Bern, Henrik Jo-

hansson, and Andrew Larkoski. I also want to thank John-Joseph Carrasco for his

help and encouragement.

I would like to thank my committee members: JoAnne Hewett, Renata Kallosh,

Michael Peskin and Andras Vasy. I am grateful to my SLAC officemates, Marian-

gela, Yorgos, Michael, Martin, Tomas, Andrew, Jeff, Kassa, and Bart, for interesting

discussions and support. Finally, I also acknowledge all my friends and colleagues at

SLAC and Stanford. The figures of chapters 1, 3 and 4 were drawn using Jaxodraw

[1], based on Axodraw [2].

v

Contents

Abstract iv

Acknowledgements v

1 Introduction 1

1.1 Extended Supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 UV Behavior of Supergravity . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Color Structure in Gauge Theory . . . . . . . . . . . . . . . . . . . . 7

1.4 Spinor-Helicity Formalism . . . . . . . . . . . . . . . . . . . . . . . . 10

1.5 Universal Soft Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.6 Color-Kinematic Duality and Double Copies . . . . . . . . . . . . . . 14

1.6.1 Tree-Level BCJ Relations: String Theory Proof . . . . . . . . 18

1.6.2 BCJ at Loop Level . . . . . . . . . . . . . . . . . . . . . . . . 21

1.6.3 Gravity Amplitudes from Double Copies . . . . . . . . . . . . 23

1.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2 Amplitudes from Soft Limits 27

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2 Properties of Gauge Theory and Gravity . . . . . . . . . . . . . . . . 31

2.3 Inverse-Soft Construction of Gauge Theories and Gravity . . . . . . . 33

2.4 Reproducing the Soft Limits From BCFW Terms . . . . . . . . . . . 35

2.4.1 Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

2.4.2 Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.5 Reproducing NMHV Amplitudes . . . . . . . . . . . . . . . . . . . . 40

vi

2.5.1 Momentum Deformation in AR . . . . . . . . . . . . . . . . . 42

2.5.2 A Product of Soft Factors in Gauge Theory . . . . . . . . . . 43

2.5.3 A Product of Soft Factors in Gravity . . . . . . . . . . . . . . 46

2.5.4 Applicability of the procedure . . . . . . . . . . . . . . . . . . 47

2.5.5 Example: Gauge Theory NMHV . . . . . . . . . . . . . . . . 49

2.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3 One-Loop Amplitudes 53

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

3.2 Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.1 Duality between color and kinematics . . . . . . . . . . . . . . 56

3.2.2 Gravity as a double copy of gauge theory . . . . . . . . . . . . 58

3.2.3 Decomposing one-loop N ≥ 4 supergravity amplitudes. . . . . 60

3.3 Implications of the duality at one loop . . . . . . . . . . . . . . . . . 61

3.3.1 Implications for generic one-loop amplitudes . . . . . . . . . . 61

3.3.2 Four-point one-loop N ≥ 4 supergravity amplitudes . . . . . . 65

3.3.3 Five-point one-loop N ≥ 4 supergravity amplitudes . . . . . . 69

3.3.4 Comments on two loops . . . . . . . . . . . . . . . . . . . . . 75

3.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4 Two-Loop Amplitudes 79

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

4.2 Review of the BCJ duality and squaring relations . . . . . . . . . . . 81

4.3 Two-loop N = 8 supergravity . . . . . . . . . . . . . . . . . . . . . . 85

4.4 Two-Loop 4 ≤ N < 8 Supergravity . . . . . . . . . . . . . . . . . . . 87

4.5 Infrared poles and finite remainders . . . . . . . . . . . . . . . . . . . 90

4.6 Forward-scattering limit of the amplitudes . . . . . . . . . . . . . . . 104

4.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

A Six-point Gravity NMHV Amplitude 108

B One-loop five-point Yang-Mills amplitudes 110

vii

C Integrals 113

D One-loop expressions 116

Bibliography 120

viii

List of Tables

1.1 Supersymmetric multiplets . . . . . . . . . . . . . . . . . . . . . . . . 4

3.1 Particle content of relevant supergravity multiplets . . . . . . . . . . 60

4.1 State multiplicity as a function of helicity . . . . . . . . . . . . . . . . 87

ix

List of Figures

1.1 Feynman diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Color factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

1.3 Diagram contributing to the soft limit . . . . . . . . . . . . . . . . . 12

1.4 Four-point tree-level Jacobi identity . . . . . . . . . . . . . . . . . . . 15

1.5 Generic tree-level Jacobi identity . . . . . . . . . . . . . . . . . . . . 16

1.6 Contours . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

1.7 One-loop color Jacobi identity . . . . . . . . . . . . . . . . . . . . . . 22

2.1 Two-particle factorization BCFW diagram . . . . . . . . . . . . . . . 35

2.2 Diagrams entering the BCFW decomposition of NMHV amplitudes . 40

2.3 BCFW diagram that cannot be constructed with inverse soft . . . . . 48

3.1 One-loop m-gon master diagram . . . . . . . . . . . . . . . . . . . . . 62

3.2 Jacobi relation between three one-loop graphs . . . . . . . . . . . . . 63

3.3 Pentagon and box integrals . . . . . . . . . . . . . . . . . . . . . . . . 69

3.4 Two-loop cubic diagrams . . . . . . . . . . . . . . . . . . . . . . . . . 75

4.1 Planar and nonplanar cubic diagrams at two loops . . . . . . . . . . . 84

4.2 Two-loop diagrams related by a Jacobi identity . . . . . . . . . . . . 86

x

Chapter 1

Introduction

The Standard Model successfully describes the strong and electroweak forces. Cal-

culations and predictions are made using perturbation theory within the framework

of quantum field theory. Particles interact by exchanging force carriers which are

the gauge bosons of SU(3) × SU(2) × U(1). Namely, the gluon is the gauge boson

of SU(3), responsible for the strong force. SU(2) × U(1) governs the electroweak

interaction which is carried by the photon and the W and Z bosons.

In perturbation theory, processes are computed through a perturbative series in

powers of the coupling. This series can easily be visualized through Feynman diagrams

such as the ones depicted in figure 1.1. Additional loops correspond to higher orders

in quantum corrections.

Gravity is the only force not currently included in the Standard Model. Moreover,

when one tries to compute a graviton scattering amplitude through a perturbative

Figure 1.1: Two Feynman diagrams. The diagram on the left-hand side is classical or attree level. On the right-hand side, the diagram is at the one-loop level, or at the first orderin quantum corrections.

1

CHAPTER 1. INTRODUCTION 2

expansion, the series diverges at the two-loop level [3]. This led to the widespread

belief that one cannot create a ultraviolet (UV) finite theory of gravity with point-like

particles. However, supersymmetry is known to soften this UV behavior, delaying the

appearance of divergences at least when all particles are in the same multiplet as the

graviton. Whether maximal supergravity (N = 8) is finite is now an interesting open

question. The current status of symmetry constraints on possible counterterms in

supergravity is discussed in section 1.2.

Apart from the UV issues, perturbative gravity calculations are in general harder

to perform than analogous gauge-theory computations. The gravity Lagrangian is

complicated, with an infinite number of vertices. Recently, new techniques have

been developed to streamline calculations, including the use of recursion relations

or of universal limits. Another line of research finds ways to relate complicated

gravity amplitudes to gauge-theory building blocks. We will review these modern

calculational techniques in the latter part of this introduction.

1.1 Extended Supergravity

We will see in the next section that adding supersymmetries to Einstein’s theory of

general relativity can soften its UV behavior. In order to better understand how this

works, we now briefly review the representations of extended supergravity theories.

We will focus on how the particle states are organized in supermultiplets following

the treatment of [4].

The extended supersymmetry (SUSY) algebra with N supercharges is given by

adding N pairs of spinor generators, Qaα ,Q†αa, to the Poincare generators of Lorentz

transformations and translations. The generators Qaα, where a = 1, . . . ,N and

α = 1, 2, are complex, anticommuting spinors in the fundamental representation

of SL(2,C) while the Q†αa are in the conjugate representation. They enter the SUSY


algebra as

{Qaα,Q†αb} = 2σµααPµδab ,

{Qaα,Qbβ} = {Q†αa,Q†βb} = 0 ,

[Pµ,Qaα] = [Pµ,Q†α] = 0 , (1.1)

where σµαα = (1, σiαα), with σiαα the Pauli matrices. We now construct a massless

multiplet, choosing a frame such that pµ = (E, 0, 0, E). In this frame, we have,

{Qa1,Q†1b} = 4Eδba , {Qa2,Q†2b} = 0 , (1.2)

meaning that Qa2 plays no role. The generators Q†1a act as creation operators, each

increasing a state’s helicity by 1/2. Note that since Q†1a is an anticommuting spinor,

applying it twice to a given state will nullify it. Thus, starting with a state with

helicity λ, the highest helicity state in the multiplet will have helicity λ + N /2. In

gravity, we require |λ| ≤ 2 which forces N ≤ 8. Similarly, in gauge theory |λ| ≤ 1

forces N ≤ 4.

The number of states of a given helicity in various extended supergravity multiplets

is given in table 1.1. It can be understood by counting the number of ways a state of

a given helicity can be created using the Q†1a. Starting with a state of helicity λ, we

can create N states of helicity λ+ 1/2. To make a state with helicity λ+ 1, we need

to pick two different Q†1a, yielding N (N − 1)/2 states, etc. This procedure leads to a

total of 2N massless states in the supermultiplet. Note that by CPT invariance, the

number of positive helicity states needs to be equal to the number of negative helicity

states. This forces us to add to the supermultiplet the CPT -conjugates of the states

obtained through the procedure described above. For example, it is easy to see that

this forces N = 7 supergravity to be equivalent to N = 8 supergravity (at least in

particle content).


helicity 0 +1/2 +1 +3/2 +2

N = 8 supergravity 70 56 28 8 1N = 6 supergravity 30 26 16 6 1N = 5 supergravity 10 11 10 5 1N = 4 supergravity 2 4 6 4 1

N = 4 sYM 6 4 1N = 2 sYM 2 2 1N = 1 sYM 1 1N = 0 sYM 1

Table 1.1: Number of states with a given helicity occurring in the massless spectra forvarious extended supergravity and super-Yang-Mills (sYM) theories. The number of su-persymmetries (N ) corresponds to the number of gravitinos (spin 3/2 particles) in thegravity multiplet. Only positive helicity states are listed. By CPT invariance, the numberof negative helicity states is equal to the number of positive helicity states.

1.2 UV Behavior of Supergravity

One can see through power counting that gravity is nonrenormalizable. The cou-

pling constant κ is related to Newton’s constant, κ2 = 32πGN = 32π/M2PL, with

MPL the Planck mass. Remember that for a fixed number of external particles, the

scattering amplitude gains two additional vertices at each loop order (see for example

figure 1.1). Each of these vertices is proportional to κ, which has mass dimension

minus one. Since the amplitude as a whole has a fixed mass dimension, the rest of

the amplitude’s mass dimension must go up by two at each loop order to compen-

sate. Denoting the energy scale by E, the perturbative expansion of the four-graviton

scattering amplitude goes like M = E2GN + E4G2N + . . . . When the energy reaches

1/√GN , the Planck mass, the amplitude will become of order one and would thus

violate unitarity unless new physics or large loop corrections come into play. Thus,

perturbative gravity is a low-energy theory valid at most to the Planck mass unless

its behavior is strongly modified by loop corrections.

Another issue with perturbative quantum gravity comes from the possible prolif-

eration of UV divergences. In quantum field theory, UV divergences are associated


with local counterterms. In gravity, the coupling-stripped mass dimension of a coun-

terterm increases at each loop order, following the increase in mass dimension of the

coupling-stripped scattering amplitude. Thus, divergences appearing at each loop or-

der would need to be canceled by a different counterterm, meaning that we would need

an infinite number of counterterms to renormalize the theory, and losing predictivity.

However, in order for a counterterm to be allowed, it must satisfy all symmetries of

the theory. In pure gravity, counterterms associated with on-shell divergences must be

generally covariant; therefore, they take the form of a product of contracted Riemann

tensors. As we will see, supersymmetric theories have additional symmetries which

force additional constraints. If no local counterterm can be constructed below a cer-

tain loop order, the theory will be finite up to that order. However, having an allowed

candidate counterterm at a given loop order does not necessarily imply a divergence.

One needs to explicitly compute its coefficient to ensure that it is nonzero.

As we mentioned previously, pure Einstein gravity diverges at two loops. The

coefficient of the counterterm R3 ≡ RλρµνR

µνστR

στλρ , where Rµν

στ is the Riemann tensor,

was computed explicitly and found to be nonzero [3]. However, better UV behav-

ior can be obtained by adding supersymmetries to the theory. Namely, in all pure

supergravity theories, no two-loop counterterms are allowed [5–7]. This is because

the only candidate, R3, generates a four-graviton amplitude with helicity assignment

(±,+,+,+), which violates the supersymmetric Ward identities [8]. Note that this

argument only applies to pure supergravity theories, where all states are related by

supersymmetry to the graviton. This allows us to only study candidate counterterms

constructed of products of contracted Riemann tensors. Otherwise, counterterms of

the form TµνTµν , where T µν is the matter stress-energy tensor, would give a divergence

already at one loop [9].

At three loops, the counterterm R4, which stands for a particular contraction

of Riemann tensors, is allowed by pure N = 1 (minimal) supergravity [6,10]. How-

ever, explicit field-theory calculations showed that N = 8 supergravity is finite at

three loops [11,12]. It was later understood that some non-minimal theories contain

additional symmetries which prevent the appearance of the R4 counterterm. For in-

stance, N = 8 supergravity realizes a non-compact non-linear symmetry called E7(7).


The latter symmetry group is spontaneously broken to its maximal compact sub-

group, SU(8). The exceptional group E7(7) has 133 generators while SU(8) has 63

(82 − 1). We are then left with 70 scalars (133 - 63) acting as the Goldstone bosons.

It has been shown recently that this E7(7) symmetry prevents the appearance of a

three-loop counterterm. Stelle et al. showed that, while it is possible to write the

N = 8 supergravity theory in a manifestly E7(7)-invariant form, it is not possible

to write a E7(7)-invariant three-loop counterterm [13]. Moreover, as the Goldstone

bosons of the coset E7(7)/SU(8), the scalars satisfy relations analogous to Alder’s

soft pion theorems [14]. In contrast with the soft pion case, it was first noticed in

[15] that the N = 8 supergravity single-soft limits are always vanishing. However,

the six-point matrix element generated by R4 has a non vanishing single-soft limit,

which makes this three-loop counterterm inconsistent with the E(7)7 symmetry [16].

Moreover, the divergence appears to be delayed up to at least seven loops, where the

first E7(7)-invariant counterterm can be constructed [13,17]. The pure-gravity com-

ponent of this counterterm includes an operator of the form D8R4 with D a covariant

derivative.

Theories with N = 4, 5, 6 are invariant under smaller duality groups, which can

be used to extract constraints in analogy to the procedure carried out in N = 8 su-

pergravity. Namely, N = 6 supergravity is invariant under SO∗(12), the noncompact

version of SO(12). SO∗(12) is spontaneously broken to its maximal compact sub-

group, U(6), generating 30 Goldstones which are scalars of the theory (66 − 36).

The candidate counterterms R4 and D2R4 are not SO∗(12)-invariant, preventing the

presence of three- and four-loop divergences [13]. Similarly, N = 5 supergravity is in-

variant under SU(5, 1). The latter is broken to U(5), yielding 10 Goldstones (35−25).

R4 is also not invariant under SU(5, 1), delaying the first potential divergence to the

four-loop order.

However, N = 4 supergravity is invariant under SU(1, 1) (broken to U(1)) and

R4 is also invariant under this symmetry group. We would thus expect a divergence

already at the three-loop order. But, very recently the theory was shown to be finite

at this order through an explicit field-theory calculation [18]. Shortly after, Tourkine

and Vanhove speculated that this finiteness could be explained by the factorization


of a D2R4 term from the two-loop four-graviton string amplitude in the CHL model

[19].

It is interesting to note that while there has been a long-standing belief that su-

pergravity will diverge, not a single counterterm with a non-zero coefficient has been

found. It is thus important to perform more high-order calculations in a variety of

supergravity theories to test various counterterm predictions. However, as previously

mentioned, these computations are complicated and difficult to perform. In the fol-

lowing sections, we will explain recent techniques to simplify them. In addition, in

chapter 2, we will explore new methods based on kinematic limits and present ex-

plicit examples of their applicability to tree-level gravity-amplitude computations.

In chapters 3 and 4, we will use novel techniques to obtain loop-level supergravity

results.

1.3 Color Structure in Gauge Theory

Traditionally, amplitudes in quantum field theory have been computed using Feyn-

man rules derived from a Lagrangian. However, even though the method is well-

defined, it can soon become unwieldy by involving a very large number of diagrams,

of which a large number often ultimately cancels. Moreover, calculations involve off-

shell intermediate virtual states and individual terms are not gauge invariant. Thus,

various alternative calculational techniques have been introduced. In the following,

we will discuss some of these methods. We start with the idea of using universal kine-

matic limits to constrain or even reconstruct an amplitude. We will then study how

graviton scattering amplitudes can be related to simpler gauge-theory amplitudes at

tree and loop level. But before delving into these issues we will very briefly review

some general properties of amplitudes in order to set up notation. Namely, we will

discuss how to organize the color structure of gauge-theory amplitudes and review

the spinor-helicity formalism.

From now on, all external particles will be massless and will be defined as outgo-

ing. If desired, incoming particles can be obtained by crossing (an outgoing particle

with momentum p corresponds to an incoming particle of momentum −p). We now


consider a SU(Nc) gauge theory and focus on particles transforming in the adjoint

representation. For example, in quantum chromodynamics (QCD), we have Nc = 3

and the adjoint particles are the gluons in the octet representation. The generators

of SU(Nc) in the fundamental representation are the N2c − 1 traceless hermitian ma-

trices T ai , where ai = 1, . . . , N2c −1 is an adjoint index. We normalize them such that

Tr(T aT b) = δab. Their commutator defines the structure constants, fabc,

[T a, T b] = i√

2fabcT c , (1.3)

which we can rewrite as,

fabc =−i√

2

(Tr(T aT bT c)− Tr(T aT cT b)

). (1.4)

In a scattering amplitude with adjoint particles, Am, the SU(Nc) color factors

come in through a factor of fabc for each Feynman-diagram three-vertex and a factor

of contracted structure constants, fabcf cde, for each four-vertex. Our goal is to find

a color decomposition that will allow us to separate these color factors from the

kinematic factors in the amplitude. Namely, we wish to rewrite the amplitude as a

sum of products of color factors times color-stripped partial amplitudes. Using (1.4)

and Fierz rearrangement,

(T a)j1

i1(T a)

j2i2

= δj2

i1δ

j1i2− 1

Nc

δj1

i1δ

j2i2

, (1.5)

we can expand the structure constants and express a full-color tree-level gauge-theory

amplitude as [20–22]:

Am(a1, a2, . . . , am) = gm−2∑

σ∈Sm/Zm

Tr(T σ(1)T σ(2) . . . T σ(m))Am(σ(1), σ(2), . . . , σ(m)) ,

(1.6)

where the sum is over the (m − 1)!/2 cyclic permutations with reflections removed.

The color-ordered partial amplitudes on the right only contain diagrams that can be

drawn on the plane with one cyclic ordering of the external particles. Thus, they

can only depend on kinematic invariants constructed with adjacent particles such as


(ki + kj)2 = sij with j = i+ 1 (mod m) or, more generally, (ki + ki+1 + ki+2 + . . .)2 .

The color-ordered partial amplitudes are individually gauge invariant, making them

well-defined objects [21].

However, the color-ordered partial amplitudes of (1.6) are not independent. One

way to see this is to replace T a2 with a U(1) generator,

(T a2) ji → (T aU(1)) j

i =1√Nc

δ ji . (1.7)

Performing this substitution in (1.6) we obtain the amplitude for the scattering of a

U(1) “photon” with m−1 SU(Nc) gluons. This amplitude vanishes since all structure

constants involving the photon vanish (T aU(1) commutes with the SU(Nc) generators).

Thus, collecting all terms of (1.6) with the same trace structures after the substitution,

we obtain U(1)-decoupling identities of the form [20,21,23]

Am(1, 2, 3, . . . ,m) + Am(1, 3, 2, . . . ,m) + . . .+ Am(1, 3, . . . ,m, 2) = 0 . (1.8)

Moreover, it is possible to reduce the number of independent partial amplitudes fur-

ther, to (m− 2)! independent amplitudes, using the Kleiss-Kuijf (KK) relations [24],

Am(1, {α},m, {β}) = (−1)mβ∑

σ∈OP{α}{β}

Am(1, σ({α}, {βT}),m) . (1.9)

Above, {α} ∪ {β} = {2, . . . ,m − 1} and mβ is the number of elements in the set

{β}. The set {β}T contains the same elements as {β}, but in the reverse order. The

sum is over the ordered permutations OP which preserve the relative ordering of the

elements in {α} and {β}T . For example,

OP{2, 3}{4, 5}T =

{(2, 3, 5, 4), (2, 5, 3, 4), (2, 5, 4, 3), (5, 4, 2, 3), (5, 2, 4, 3), (5, 2, 3, 4)}.

Note that the U(1)-decoupling identity (1.8) is a special case of (1.9) in which {β}has a single element.


1

σ2σ2 σ3 σm−1

m

Figure 1.2: Graphical representation of a color factor from the color decomposition (1.10).Its value is obtained by replacing each vertex with a structure constant fabc and eachpropagator with a δab.

This redundancy in the color decomposition (1.6) leads to wonder if one could find

another decomposition in terms of fewer partial amplitudes. Indeed one can write

[25],

Am = (i√

2g)m−2∑

σ∈Sm−2

= fa1aσ2x1fx1aσ3x2 . . . fxm−3aσm−1amAm(1, σ2, . . . , σm−1,m) .

(1.10)

A color factor from this decomposition is represented graphically in figure 1.2. The

U(1)-decoupling identities are trivially satisfied since the color factors contain struc-

ture constants. We will use a generalization of this structure-constant based repre-

sentation to demonstrate the Bern-Carrasco-Johansson (BCJ) duality in section 1.6.

1.4 Spinor-Helicity Formalism

Amplitudes will be expressed in the spinor-helicity formalism. Our conventions in

this section follow from [26], which the reader is invited to consult for more details.

Other reviews of the spinor-helicity formalism can be found in [27]. Each particle i

(with momentum ki) is assigned a pair of spinors: one right-handed spinor, |i], and

one left-handed spinor, |i〉. In our conventions, they correspond to the definite-helicity

solutions of the Dirac equation for a massless particle, γµkµi u(ki) = /kiu(ki) = 0, where

γµ are the Dirac γ matrices. We have:

|i〉 = u+(ki) =1

2(1 + γ5)u(ki) , |i] = u−(ki) =

1

2(1− γ5)u(ki) ,

〈i| = u−(ki) = u(ki)1

2(1 + γ5) , [i| = u+(ki) = u(ki)

1

2(1− γ5) . (1.11)


These spinors can be contracted to form the antisymmetric inner products [ij] and

〈ij〉:〈ij〉 = u−(ki)u+(kj) , [ij] = u+(ki)u−(kj) . (1.12)

The particle’s momentum is given by

kµi =1

2[i|γµ|i〉 , (1.13)

Kinematic invariants are of the form

sij = (ki + kj)2 = 2(ki · kj) = 〈ij〉[ji] . (1.14)

We can see from (1.13) that in order to have real momentum, |i] and 〈i| need to

be complex conjugates of each other. Thus, the inner products of (1.12) correspond

to the square roots of the kinematic invariants sij up to a possible phase. In this

formalism, gauge-theory polarization vectors are expressed as

εµ+(i) =〈q|γµ|i]√

2〈qi〉, εµ−(i) = − [q|γµ|i〉√

2[qi], (1.15)

while gravity polarization tensors are

εµν+ (i) =〈q|γµ|i]〈r|γν |i]

2〈qi〉〈ri〉 , εµν− (i) =[q|γµ|i〉[r|γν |i〉

2[qi][ri]. (1.16)

In the equations above, q and r are reference spinors. It is easy to see that changing

their values correspond to a gauge transformation. The transverse conditions, kµi εµ =

0, and kµi εµν = 0, kνi εµν = 0, are also trivially satisfied.

1.5 Universal Soft Limits

Taking an external particle’s momentum to zero is usually referred as taking a

soft limit of the amplitude. In cases where the soft particle is a gauge boson which

couples to a conserved current, we obtain a universal behavior. Let us illustrate this in


kj

ki

Figure 1.3: A Feynman diagram contributing the the soft limit as the momentum of particlej goes to zero. The propagator connecting particle j to the rest of the diagram (the shadedblob) is singular.

the simplest case, a soft photon emitted from a tree-level quantum-electrodynamics

(QED) amplitude, AQEDm . Only diagrams where the soft photon is attached to an

external line will contribute as the propagator connecting the soft photon to the rest

of the amplitude diverges. We now sketch the contribution of the diagram illustrated

in figure 1.3. We have:

limkj→0DQEDm = DQED

m−1

ieQi

2kj · kiε(kj) · ki , (1.17)

where DQEDm−1 corresponds to the Feynman diagram of figure 1.3 with particle j re-

moved, e is the QED coupling constant, and Qi is the charge of particle i. The soft

limit is obtained by summing over all diagrams such that the photon can attach to

all possible charged external particles. After performing this sum, we will obtain

the factorization of the amplitude into a lower-point amplitude times a universal soft

factor:

limkj→0AQEDm =

m∑i=1

(i 6=j)

ieQi

2kj · kiε(kj) · ki

×AQEDm−1 . (1.18)

Analogous universal soft behavior is also present in Yang-Mills and in gravity

[28,29]. For a generic tree-level color-ordered Yang-Mills amplitude, we write,

limkj→0

Atreem (1, . . . , i, j, k, . . . , n) = S(j)× Atree

m−1(1, . . . , i, j, k, . . . , n) , (1.19)


where the bar over j on the right-hand side denotes the fact that gluon j has been

removed from the amplitude and the (m− 1)-particle amplitude is on-shell. Similar

expressions also exist at loop level. The universal soft factor S takes the following

form in gauge theory if j has positive helicity:

S(j+) =√

2

(−ki · ε+(j)

sij+kl · ε+(j)

sjl

)=〈il〉〈ij〉〈jl〉 . (1.20)

Note that it only depends on the momenta of the two particles, i and l, that are

adjacent to the particle taken soft. Because of the color ordering, the soft particle

only shares a color line with the two neighboring particles. Through parity, one

obtains the soft factor for negative helicity particle by conjugating (1.20). Namely,

S(j−) =−[il]

[ij][jl]. (1.21)

The gravity soft factor is more complicated than its gauge-theory analog. This is

in part due to the lack of color ordering, which requires its expression to be symmetric

under permutation of all external particles. To express the soft factor, we first define

the coupling-stripped amplitude Mm,

Mm(1, 2, . . . ,m) =(κ

2

)m−2

Mm(1, 2, . . . ,m). (1.22)

Taking particle 1 to be a soft graviton, the soft limit of the amplitude is

limk1→0

M treem (1, 2, . . . , n)

M treem−1(2, . . . , n)

=m∑i=2

kµi kνi εµν(1)

si1. (1.23)

Requiring an amplitude to satisfy soft limits of the form (1.19) or (1.23) for all glu-

ons and gravitons imposes strong constraints. One could also wonder if it is possible

to “invert” (1.19) and construct a higher-point amplitude by multiplying a lower-

point amplitude by an “inverse-soft factor”. This idea was first proposed in [30]. One

obvious issue is the fact that (1.19) is only valid in the limit where the soft parti-

cle’s momentum is null. One would then need to deform the momentum of the other


external particles in order to add a “soft” particle with nonzero momentum while

maintaining momentum conservation on both sides of (1.19). One may also wonder

whether a procedure based on soft limits alone could produce enough constraints to

reconstruct the higher-point amplitude. In chapter 2, we will study in detail in which

cases this construction is possible, both for gauge-theory and gravity amplitudes.

1.6 Color-Kinematic Duality and Double Copies

The color-kinematic duality of gauge theory can be observed when the amplitude

is expressed in a color decomposition similar to the structure-constant decomposition

of (1.10). Namely, we express any full-color gauge-theory amplitude with external

particles in the adjoint representation in terms of graphs with only cubic vertices as

follows,

Atreem = gm−2

∑j∈Γ3

njcj∏αjp2αj

, (1.24)

where g is the gauge coupling. The sum is over the set Γ3 of scalar diagrams j

with only three-point vertices, and the product in the denominator is over the Feyn-

man propagators associated with the internal lines αj of each of these cubic graphs.

The value of the color coefficient cj is obtained by dressing each three-vertex of the

associated diagram j with a structure constant defined as

fabc = i√

2fabc = Tr([T a, T b]T c) . (1.25)

Finally, the nj are kinematic numerators, which are a function of kinematic invariants

and spinors. Contact terms can easily be converted to cubic terms of the form (1.24)

using inverse propagators (multiplying the terms by 1 = p2j/p

2j) and assigned the

resulting expressions to various graphs according to the color structures. For example,

the four-gluon vertex has color factors of the form fabcf cde, facef bde, fadef bce, and its

contribution will be split among three cubic graphs.

The structure constants fabc satisfy the usual Jacobi identities,

fabe f cde = face f bde + fade f cbe , (1.26)


= +

1

2 3

4 1

2 3

4 41

2 3

4

Figure 1.4: Tree-Level Jacobi identity, cs = cu + ct.

which are represented graphically in figure 1.4. In this figure, we can see how Jacobi

identities relate the color factors of the three cubic graphs with four external particles.

The same identities can be used to relate diagrams with more external particles as

depicted in figure 1.5. There, we relate three graphs which are identical apart from a

region containing a four-point subgraph in which the Jacobi identity is applied. Note

that these relations among color factors, ci = cj+ck, allow us to shuffle terms around,

implying that the representation (1.24) is not unique.

A cubic representation (1.24) is said to satisfy the Bern-Carrasco-Johansson (BCJ)

duality if, for each triplet of diagrams with color factors cj related by a Jacobi identity,

the kinematic factors are also related by the same identity. Namely,

ci = cj + ck ⇒ ni = nj + nk . (1.27)

The numerators also need to obey the same antisymmetry as the structure constant

fabc under the interchange of two legs attached to a cubic vertex,

ci → −ci ⇒ ni → −ni . (1.28)

It is easy to find a BCJ-satisfying representation for tree-level amplitudes with

four external particles. We will follow the treatment in [31] and start by expanding


= +

Figure 1.5: Three color graphs related by a Jacobi identity. The identity is applied on thefour-point subgraphs inside the red, dashed circle. Everything outside of this circle is thesame in the three graphs.

the color-ordered partial amplitudes in terms of their poles:

Atree4 (1, 2, 3, 4) ≡ ns

s+ntt,

Atree4 (1, 3, 4, 2) ≡ −nu

u− ns

s,

Atree4 (1, 4, 2, 3) ≡ −nt

t+nuu, (1.29)

where s, t, u are the usual Mandelstam invariants (s = (p1 + p2)2, t = (p2 + p3)2,

u = (p1 + p3)2). Remember that color ordering prevents Atree4 (1, 2, 3, 4) from having

a u-channel pole since particle 1 is not adjacent to particle 3, explaining why each

amplitude in (1.29) is expanded as a sum of only two terms. The amplitudes (1.29)

are not independent. They are related by a U(1)-decoupling identity (1.8),

Atree4 (1, 2, 3, 4) + Atree

4 (1, 3, 4, 2) + Atree4 (1, 4, 2, 3) = 0 . (1.30)

Since this identity is valid for any helicity configuration of the external particles

and for any number of space-time dimensions, the cancellation cannot rely on four-

dimensional spinor identities and must rely solely on kinematic-invariant identities.

Namely, we must have

Atree4 (1, 2, 3, 4) + Atree

4 (1, 3, 4, 2) + Atree4 (1, 4, 2, 3) = (s+ t+ u)ξ = 0 , (1.31)

where ξ depends on the specific particle helicities and kinematics. Moreover, because

s and t are equivalent in A4(1, 2, 3, 4), the latter needs to be proportional to uξ =


−(s+ t)ξ. Similarly, we have:

Atree4 (1, 2, 3, 4) = uξ ,

Atree4 (1, 3, 4, 2) = tξ ,

Atree4 (1, 4, 2, 3) = sξ , (1.32)

which leads to the following relations amongst the partial amplitudes:

tAtree4 (1, 2, 3, 4)− uAtree

4 (1, 3, 4, 2) = 0 ,

sAtree4 (1, 2, 3, 4)− uAtree

4 (1, 4, 2, 3) = 0 ,

tAtree4 (1, 4, 2, 3)− sAtree

4 (1, 3, 4, 2) = 0 . (1.33)

Note that, as showed in (1.31), U(1)-decoupling identities reduce the number of in-

dependent four-point partial amplitudes to two. The BCJ relations (1.33) bring this

number down to one. Also, it is easy to see that only one of the relations (1.33) is

independent of the previously known U(1)-decoupling relations. One can check that

equations (1.33) are satisfied by the known four-gluon amplitude

Atree4 (1, 2, 3, 4) = i

〈jk〉4〈12〉〈23〉〈34〉〈41〉 , (1.34)

where j and k label the two negative-helicity gluons. For example, the first identity

becomes

i 〈jk〉4(

〈23〉[32]

〈12〉〈23〉〈34〉〈41〉 −〈13〉[31]

〈13〉〈34〉〈42〉〈21〉

)

=−i 〈jk〉4

〈12〉〈34〉〈41〉〈42〉(

[32]〈24〉+ [31]〈14〉)

= 0 , (1.35)

where the term in parenthesis on the second line vanishes by momentum conservation

(k1 + k2 + k3 + k4 = 0). Also note that the relations (1.33) can be obtained by other

methods such as by using the supersymmetric Ward identities.


By substituting (1.29) into (1.33), we see that the BCJ duality (1.27) is automat-

ically satisfied. Namely, the first line of (1.33) becomes

t(nss

+ntt

)− u(− nu

u− ns

s

)= 0

−ns + nt + nu = 0 . (1.36)

It is interesting to note that the BCJ duality was observed more than thirty years

ago for four external particles [32].

Similar relations follow from assuming the existence of a valid BCJ representation

for tree-level amplitudes with any number of legs [31]. For m legs, we have, in the

form of the relations given in [33],

0 = s12Am(1, 2, 3, . . . ,m) + (s12 + s23)Am(1, 3, 2, 4, . . . ,m)

+ (s12 + s23 + s24)Am(1, 3, 4, 2, 5, . . . ,m) + . . .

+ (s12 + s23 + s24 + . . .+ s2(m−1))Am(1, 3, 4, . . . ,m− 1, 2,m) . (1.37)

At four points, we can see that (1.37) reproduces equations (1.33),

s12Atree4 (1, 2, 3, 4) + (s12 + s23)Atree

4 (1, 3, 2, 4) = 0

sAtree4 (1, 2, 3, 4)− uAtree

4 (1, 4, 2, 3) = 0 , (1.38)

where we have used the reflection symmetry of color-ordered amplitudes in the second

line.

1.6.1 Tree-Level BCJ Relations: String Theory Proof

In this section, we will discuss the string-theory proof of the BCJ relations as first

presented in [34] (for a recent review, see [33]). We will show the derivation of the

five-point relation,

0 = s12A5(1, 2, 3, 4, 5) + (s12 + s23)A5(1, 3, 2, 4, 5) + (s12 + s23 + s24)A5(1, 3, 4, 2, 5) ,

(1.39)


where the A5 are color-ordered gauge-theory amplitudes. Generalization to higher-

point amplitudes is straightforward.

We start with the tree-level color-ordered open-string amplitude following the

treatment of [33],

Astr5 (2, 1, 3, 4, 5) ∼

∫C2

dx2(−x2)2α′k1·k2(1− x2)2α′k2·k4 ×∫C3

dx3(x3)2α′k1·k3(1− x3)2α′k3·k4(x3 − x2)2α′k2·k3 , (1.40)

where the inverse string tension is denoted by α′. The points x1 = 0, x4 = 1 and

x5 =∞ are fixed by the SL(2) symmetry and factors without branch cuts are omitted.

The contour of the x2 integral is illustrated with a dashed line in figure 1.6. It is useful

to write analogous expressions for other color orderings:

Astr5 (1, 2, 3, 4, 5) ∼

∫C2

dx2(x2)2α′k1·k2(1− x2)2α′k2·k4 ×∫C3

dx3(x3)2α′k1·k3(1− x3)2α′k3·k4(x3 − x2)2α′k2·k3

Astr5 (1, 3, 2, 4, 5) ∼

∫C2

dx2(x2)2α′k1·k2(1− x2)2α′k2·k4 ×∫C3

dx3(x3)2α′k1·k3(1− x3)2α′k3·k4(x2 − x3)2α′k2·k3

Astr5 (1, 3, 4, 2, 5) ∼

∫C2

dx2(x2)2α′k1·k2(x2 − 1)2α′k2·k4 ×∫C3

dx3(x3)2α′k1·k3(1− x3)2α′k3·k4(x2 − x3)2α′k2·k3 (1.41)

Note that since the amplitudes above are ordered, the x2 integration is from 0 to x3, x3

to 1 and 1 to∞ for Astr5 (1, 2, 3, 4, 5), Astr

5 (1, 3, 2, 4, 5) and Astr5 (1, 3, 4, 2, 5) respectively.

Thus, looking at figure 1.6, we see that we can deform the contour of (1.40) into the

contour of the three amplitudes of (1.41) by rotating in the upper-half plane.

However, we must introduce phase factors to not only match the contours, but

also to make the integrands the same. For example, the integrand of Astr5 (2, 1, 3, 4, 5)

is the same as the one of Astr5 (1, 2, 3, 4, 5) up to the following difference in the first


0x3 1

Figure 1.6: Contours of integration for various string amplitudes. The dashed line is thecontour of Astr

5 (2, 1, 3, 4, 5). It can be rotated into the solid line corresponding to the con-tours of the amplitudes of (1.41).

factor of the integrand: (−x2)2α′k1·k2 → (x2)2α′k1·k2 . To see how these factors can

be related, we follow [33] and consider the factor zc0 = (|z0|eiθ0)c = |z0|ceicθ0 with

Re(z0) < 0 and Im(z0) > 0. Consider (−z0)c on the branch −π < arg(z) < π,

(−z0)c = |z0|ceic(θ0−π) . (1.42)

We have θ0 − π ∈ [−π/2, 0) ⊂ (−π, π) and we are thus still on the correct branch.

Note that if we had taken the argument of (−z0) to be (θ + π) instead of (θ − π),

the argument would have ended up outside of the (−π, π) range after exponentiation.

Thus, we have from (1.42),

(z0)c = eiπc(−z0)c . (1.43)

Combining the contour deformation with the correct phase factors, we get the

following open-string relation,

Astr5 (2, 1, 3, 4, 5) = −

(ei2α

′πk2·k1Astr5 (1, 2, 3, 4, 5) + ei2α

′πk2·(k1+k3)Astr5 (1, 3, 2, 4, 5)

+ ei2α′πk2·(k1+k3+k4)Astr

5 (1, 3, 4, 2, 5)), (1.44)

where the minus sign comes from the change of orientation of the rotated contour

as can be seen in figure 1.6. Since the left-hand-side amplitude is real, we get the

following relation by taking the imaginary part of (1.44),

0 = sin(α′πs12)Astr5 (1, 2, 3, 4, 5) + sin(α′π(s12 + s23))Astr

5 (1, 3, 2, 4, 5)

+ sin(α′π(s12 + s23 + s24))Astr5 (1, 3, 4, 2, 5) . (1.45)


We then take the field-theory limit of (1.45) by expanding near α′ = 0. The open-

string amplitudes are simply mapped to the associated color-ordered gauge ampli-

tudes: Astr → A. We obtain

0 = s12A5(1, 2, 3, 4, 5) + (s12 + s23)A5(1, 3, 2, 4, 5) + (s12 + s23 + s24)A5(1, 3, 4, 2, 5) ,

(1.46)

which exactly matches the BCJ relation we were trying to prove (1.39).

Note that we can also take the real part of (1.44) to obtain a different relation:

−Astr5 (2, 1, 3, 4, 5) = cos(α′πs12)Astr

5 (1, 2, 3, 4, 5) + cos(α′π(s12 + s23))Astr5 (1, 3, 2, 4, 5)

+ cos(α′π(s12 + s23 + s24))Astr5 (1, 3, 4, 2, 5) . (1.47)

Taking the field-theory limit, we obtain

−A5(2, 1, 3, 4, 5) = A5(1, 2, 3, 4, 5) + A5(1, 3, 2, 4, 5) + A5(1, 3, 4, 2, 5) . (1.48)

which is a KK relation (1.9).1 This five-point argument can easily be generalized

to higher point amplitudes. The recently discovered BCJ relations are then directly

related to the known KK relations through string theory. Note that the KK rela-

tions come from group theory alone while the BCJ relations go beyond, containing

dynamical information. It is thus remarkable that they have a common origin!

1.6.2 BCJ at Loop Level

BCJ duality can be generalized to loop level. Loop-level amplitudes can also be

expressed in a cubic basis,

A(L)m = iL gm−2+2L

∑j∈Γ

(L)3

∫ L∏l=1

dDpl(2π)D

1

Sj

njcj∏αjp2αj

, (1.49)

1The field-theory relation (1.48) can be derived using only U(1)-decoupling (1.8). Indeed, forless than seven external particles, the KK relations are equivalent to the U(1)-decoupling identities.However, for a generic number of external particles, the real part of (1.44) can yield KK relationswhich are independent of the U(1)-decoupling identities.


= +

4

2 3

11

2 3

41

2 3

4

Figure 1.7: One-loop color Jacobi identity

where Sj gives the symmetry factor of the diagram and the sum is over the set Γ(L)3

of all L-loop cubic graphs. The rest of the notation is the same as in the analogous

tree-level formula (1.24). At loop level, color-Jacobi identities, ci = cj + ck, still

relate triplets of diagrams that are identical apart from a four-point tree subgraph, as

illustrated in figure (1.7). Again, a given representation will satisfy the BCJ duality

(1.27), (1.28) if the kinematic numerators satisfy the same relations as the color

factors.

For example, it is easy to see that the one-loop four-gluon N = 4 supersymmetric

Yang-Mills amplitude satisfies the duality. We have [35,36],

A1-loopN=4 (1, 2, 3, 4) = istg4Atree(1, 2, 3, 4)

(c1234I1234

4 + c1243I12434 + c1423I1423

4

), (1.50)

where I12344 is the scalar box integral,

I1234 =

∫dDp

(2π)D1

p2 (p− k1)2 (p− k1 − k2)2 (p+ k4)2, (1.51)

and the color factors are given by replacing the vertices with structure constants in

the associated boxes, for example,

c1234 = f ba1cf ca2dfda3ef ea4b . (1.52)

Looking at figure 1.7, we can see that since there is no triangle diagram in (1.50)

(or more generally in N = 4 sYM at one loop [37]), the duality forces the kinematic

numerators to be permutation symmetric. On the other hand, it is easy to verify that


(1.33) is equivalent to the permutation symmetry of stAtree:

n1234 = n1243 = n1423 = stAtree4 (1, 2, 3, 4) . (1.53)

That is, the four-gluon amplitude of equation (1.50) satisfies the BCJ duality. This

amplitude will be studied in more detail in section 3.3.2.

The BCJ relations can be used to simplify multiloop calculations by constraining

an ansatz for the numerator factors nj. Using BCJ-induced Jacobi relations between

numerators, the size of the ansatz can be greatly reduced, making it easier to compute

its coefficients using, for example, generalized unitarity. However, a perhaps more

striking consequence of BCJ can be found in the computation of gravity amplitudes

using the double-copy method, which we review in the next section.

1.6.3 Gravity Amplitudes from Double Copies

Two gauge-theory amplitudes in the form (1.49) can be combined to create a

gravity amplitude as long as one of them satisfies the BCJ duality. We get

M(L)m = iL+1

(κ2

)m−2+2L ∑j

∫ L∏l=1

dDpl(2π)D

1

Sj

njnj∏αjp2αj

, (1.54)

where either the nj or nj satisfy the duality (1.27), (1.28). One gets a N = 8 super-

gravity amplitude by squaring N = 4 sYM amplitudes:

N = 8 supergravity : (N = 4 sYM)× (N = 4 sYM) . (1.55)

However, the nj and nj do not need to come from the same theory. This allows

us to obtain non-maximal supergravity amplitudes by combining N = 4 sYM with

N = 0, 1, 2 sYM. Explicitly, we have

N = 6 supergravity : (N = 4 sYM)× (N = 2 sYM) ,


N = 4 supergravity : (N = 4 sYM)× (N = 0 sYM) . (1.56)


At the level of counting states, the constructions (1.55), (1.56) can be understood by

looking at the supermultiplet state counts listed in table 1.1. The helicity of two com-

ponent gauge-theory states are added to get the helicity of a tensor-product gravity

state. The total multiplicity of a supergravity with N supersymmetries obtained from

two gauge theories with NL and NR supersymmetries is given by 2N = 2NL × 2NR

Also, remember that as mentioned in section 1.1, N = 7 supergravity is equivalent to

N = 8 supergravity by CPT invariance. Similarly, N = 3 sYM is the same as N = 4

sYM.

At tree level, this double-copy prescription was proven recursively using the Britto-

Cachazo-Feng-Witten (BCFW) recursion relations in the cases of pure gravity and

N = 8 supergravity [38]. Tree-level N = 4, 5, 6 supergravity amplitudes are a subset

of theN = 8 supergravity ones. At loop level, the prescription’s validity can be argued

using unitarity. Many nontrivial checks were also performed. For N = 8 supergravity,

the double-copy construction has been checked explicitly at four points through four

loops [39,40] and at five points through two loops [41]. The first verifications in non-

maximal supergravity theories for one-loop four- and five-point amplitudes are the

objectives of chapter 3 of the present thesis. In chapter 4, we present the previously

unknown two-loop four-graviton amplitudes in N = 4, 5, 6 supergravity, which were

obtained through the double copy.

As a warm-up for the more involved double-copy constructions of chapters 3 and

4, we now show how to obtain the one-loop four-graviton amplitude in N = 8 su-

pergravity from two copies of the one-loop four-gluon amplitude in N = 4 sYM. We

studied this gauge-theory amplitude in section 1.6.2, showing that it satisfies the BCJ

duality. We reproduce it here for convenience,

A1-loopN=4 (1, 2, 3, 4) = ig4stAtree

4 (1, 2, 3, 4)(c1234I1234

4 + c1243I12434 + c1423I1423

4

). (1.57)

To obtain the N = 8 supergravity amplitude, we follow the prescription of equa-

tion (1.54). We first replace the gauge-theory coupling constant with the gravity one:

ig4 → −(κ/2)4. Then, we replace the color factors with their associated numerator


factors, which gives the prescription

(c1234, c1243, c1423)→ stAtree4 (1, 2, 3, 4) , (1.58)

where in this case all color factors are all replaced by the same kinematic numerator.

This invariance of the kinematic numerators under full crossing symmetry (see (1.53))

is unique to four-point N = 4 sYM. This explains why all color factors are replaced

by the same numerator here, but we will see numerous examples where this is not the

case in chapters 3 and 4 of the present thesis. After performing these replacements,

we get

M1-loopN=8 (1, 2, 3, 4) = −i

(κ2

)4(stAtree

4 (1, 2, 3, 4))2(I1234

4 + I12434 + I1423

4

), (1.59)

which agrees with the known amplitude [35,42].

Even through this fairly trivial four-point example, we were able to see the power

of the prescription. The double-copy construction allows us to obtain loop-level grav-

ity amplitudes directly from loop-level gauge-theory amplitudes. This method can

be contrasted with the traditional way [42] of relating gravity to Yang-Mills using

the Kawai-Lewellen-Tye (KLT) relations [43]. Since the latter relations are only valid

at tree level, they can only be used on the unitarity cuts of the loop-level ampli-

tudes. Reconstructing the amplitudes from the cuts, after the application of the KLT

relation, is a nontrivial task [11,44], which is avoided when using the BCJ relations.

1.7 Summary

In this introduction, we first described the known symmetry constraints on coun-

terterms. Currently, N = 8 supergravity has been shown to be finite up to at least

four loops through explicit field theory calculations [44]. Arguments based on the

non-linearly realized E7(7) symmetry suggest that the first allowed counterterm should

appear at the seven-loop order. Very recently, N = 4 supergravity was shown to be

finite at three loops. In any case, no non-zero coefficient for any potential counterterm

has yet been computed in any theory of supergravity.


We also reviewed novel techniques used in perturbative supergravity calculations.

We briefly discussed how using universal soft limits could allow us to reconstruct

tree-level amplitudes. This situation is analyzed in more detail in chapter 2. We then

focused on the novel BCJ duality, which leads to the double-copy prescription to

compute tree- and loop-level supergravity amplitudes. In chapter 3, we perform the

first checks of this prescription for non-maximal supergravity. Namely, we reproduce

the one-loop four- and five-point N = 4, 5, 6 supergravity amplitudes by combining

N = 0, 1, 2 sYM amplitudes with the BCJ-satisfying N = 4 sYM amplitude. Finally,

in chapter 4, we obtain the two-loop four-point N = 4, 5, 6 supergravity amplitudes.

We verify the expected infrared (IR) behavior and the forward-scattering limit, pro-

viding a highly non-trivial check of our results’ validity.

Chapter 2

Amplitudes from Soft Limits

C. Boucher-Veronneau and A. J. Larkoski, “Constructing Amplitudes from Their Soft

Limits,” JHEP 1109, 130 (2011).

2.1 Introduction

In the last twenty years, we have seen a resurgence of interest in the S-matrix

program of the 60’s [45] whose goal was to define a quantum field theory through the

analytic properties of its S-matrix. The unitarity [37,46] and generalized unitarity [47]

methods dramatically simplified loop-level calculations. At tree level, recent interest

was triggered in part by Witten’s remarkable description of gauge theory as a string

theory in twistor space [48]. Detailed studies of an amplitude’s analytic properties

have led, in particular, to the Britto-Cachazo-Feng-Witten (BCFW) recursion rela-

tions [49,50]. These relations exploit the analyticity of the amplitudes in a distinct

way from the old S-matrix program: external particles’ momenta are deformed by

a complex parameter and the factorization channels of the deformed amplitude are

studied. This procedure leads to very compact, on-shell, recursion formulas which

have been solved in generality for N = 4 sYM [51].

In the 1980’s, it was noticed that photon radiation amplitudes could be expressed

as products of soft factors times a lower-point amplitude [52]. A recent advancement

27

CHAPTER 2. AMPLITUDES FROM SOFT LIMITS 28

in this area due to Arkani-Hamed, et al., is a method called inverse soft which at-

tempts to construct amplitudes from their soft limits alone [30,53,54]. Inverse soft

was motivated by studying the residues of poles in the Grassmannian of [30]. The

Grassmannian has been conjectured to produce all leading singularities [55] in planar

N = 4 sYM from the residues of its poles. The relationship between leading singu-

larities at different loop orders, numbers of external legs and helicity configurations

was studied using inverse soft in [56]. In addition, it has been shown that inverse soft

reproduces tree-level maximally-helicity-violating (MHV) amplitudes and six-point

next-to-MHV (NMHV) amplitudes in N = 4 sYM [53].

In [57], inverse soft was shown to reproduce MHV gravity amplitudes in the form

first given in [58]. Very recently, inspired in part by the inverse-soft procedure dis-

cussed in [57], Hodges presented new expressions for MHV amplitudes [59]. However,

other than this result, the application of inverse soft to gravity amplitudes has been

minimal. This is (possibly) due to several factors: gravity lacks a color expansion

(and hence color ordering) and the Planck mass is dimensionful. Moreover, the grav-

ity soft factor is a gauge-invariant sum of many terms and the inverse-soft procedure

exploits each term individually. Thus, there is not a unique form for the gravity soft

factor and so there is not a unique procedure for inverse soft in gravity. Here, we wish

to put inverse soft on a firmer ground for gravity amplitudes, as well as reviewing its

application to gauge theory amplitudes.

The main result of this chapter is to extend the applicability of inverse soft for

gauge theory and gravity amplitudes from MHV to NMHV. The gravity soft factor

in [57] cannot easily reproduce multiparticle factorization channels necessary to con-

struct NMHV amplitudes. We find that its most natural generalization that can do

so is a soft factor inspired by BCFW. We will use inverse soft to re-express terms in

the BCFW expansion of an n-point amplitude An as a sum of products of soft factors

times a lower point amplitude An−m.

For gauge theory we have

An =mmax∑m=1

2∑j=1

( m∏i=1

S ′(pi,j))A′n−m(pi,j) , (2.1)


where S is a gauge-theory soft factor. Here, m ranges over the possible BCFW terms;

m = 1 are the two-particle factorization terms, m = 2 the three-particle terms, etc.,

and pi,j represent appropriate particles added to the amplitude in the inverse-soft

construction. The lower-point amplitude An−m(pi,j) is the (n −m)-point amplitude

where particles pi,j were removed from An. The maximum value of m, mmax, depends

on the helicity configuration of the amplitude and corresponds to the highest-particle

factorization channel. For MHV, mmax = 1 since there is only two-particle factoriza-

tion; for NMHV, mmax is at most n/2− 1. The sum over j corresponds to summing

over the two distinct (m+ 1)-particle factorization diagrams. In general, to conserve

momentum, we will also need to deform the individual momenta of particles; this is

denoted by primes in Eq. 2.1. We will show that this formula holds for all MHV

amplitudes, for NMHV amplitudes with eight or fewer external legs (n ≤ 8) and for

classes of NMHV amplitudes with arbitrary number of legs.

For gravity, we can construct MHV amplitudes as follows:

MMHV(1, . . . , n+) =n−2∑i=1

G(n− 1, n+, i)MMHV(1, . . . , i′, . . . , (n− 1)′) , (2.2)

where G(n− 1, n+, i) is a “gravity soft factor” which is a term in the full gravity soft

factor arising when taking particle n soft. As mentioned before, the full gravity soft

factor is a sum of many terms and we use each of them individually in our inverse-soft

procedure. The sum over i corresponds to summing over the (n − 2) nonzero two-

particle BCFW diagrams or, equivalently, to adding the particle n next to all other

possible particles. The primes again denote the deformation of momentum required

to make room for the soft particle. At NMHV, we are able to construct amplitudes

with seven or fewer external particles as

Mn =

2(n−2)∑j=1

G(pj)A′n−1(pj) +

(n−2)(n−3)∑j=1

( 2∑i=1

G(pi,j)

)G ′(pj)A′n−2(pj) , (2.3)

where we sum over the distinct two- and three-particle diagrams in the first and

second term respectively. Using our method, four- or higher-particle factorization


channels cannot be constructed in gravity which limits us to at most seven-point

amplitudes.

We will begin in earnest in Sec. 2.4 by studying the two-particle factorization

terms in the BCFW expansion. The relationship between these terms and the soft

limits of the amplitude was first discussed in [60]. These terms will be shown to be

of the form S(i)A(i), where particle i has been removed from the amplitude factor.

We will use this result to uniquely define the form of the soft factor that we will

use in studying gravity amplitudes. It should be noted that the soft factor we find

is distinct from that presented in [28,57]. Since MHV amplitudes only contain two-

particle factorization terms in BCFW, we will present a compact inverse-soft recursion

relation for these amplitudes in gauge theory and gravity.

In Sec. 2.5, we continue by studying higher-point factorization terms in the BCFW

recursion. We show explicitly that arbitrary three-particle factorization terms can be

built-up from two-particle factorization terms using inverse soft. This leads to the

immediate result that any tree-level amplitude in gauge theory and gravity up to

seven points can be represented in the form of Eq. 2.1 or 2.3. There will be some

barriers to constructing arbitrary amplitudes in this manner which we will discuss in

detail. However, there exist classes of gauge-theory NMHV amplitudes that can be

straightforwardly constructed using inverse soft for any number of external legs1. We

also present explicit examples of the procedure for six-point NMHV amplitudes in

gauge theory and gravity.

This chapter is organized as follows. In Sec. 2.2, we review the soft limits in gauge

theory and gravity and BCFW recursion. In Sec. 2.3, we define inverse soft precisely

and discuss the philosophy of the procedure. As previously discussed, Secs. 2.4 and

2.5 are the meat of the chapter where we present the inverse-soft procedure and its

relationship to BCFW. In Sec. 2.6, we present our conclusions. In appendix A, we

show the inverse-soft construction of the NMHV six-point gravity amplitude.

1All NMHV gauge-theory amplitudes can be extracted from a single NMHV N = 4 sYM su-peramplitude. One might think that we could have used a supersymmetric version of inverse softto construct all NMHV amplitudes. However, such an approach is not expected to help constructgravity amplitudes as the problem with higher numbers of legs is due to the need to sum over manypermutations and there exists only one NMHV graviton amplitude for a given number of legs.


2.2 Properties of Gauge Theory and Gravity

In this chapter, we will consider tree-level gauge-theory and gravity amplitudes

with only gluons and gravitons on the external legs. In a Yang-Mills gauge theory, it

is well known that tree-level amplitudes can be expanded in a sum of color-ordered

partial amplitudes multiplied by single-trace color factors. For simplicity, we will

study these partial amplitudes in the following; the full amplitude can easily be

reconstructed from them. Due to the lack of color ordering, amplitudes in gravity

contain all possible orderings of external legs which will add some complications.

Because all external states are massless, it is very convenient to work in the spinor-

helicity formalism [26,27]. We will need expressions for the polarization vectors and

tensors of gauge theory and gravity in this formalism. In gauge theory, because

of gauge freedom, the polarization vector for an external particle i is defined with

reference to an arbitrary vector q. Gauge invariant amplitudes must be independent

of q, but particularly good choices can simplify computation greatly. Explicitly, the

polarization vectors are

εµ+(i) =〈qγµi]√

2〈qi〉, εµ−(i) = − [qγµi〉√

2[qi]. (2.4)

In gravity, symmetric tensor gauge freedom means that there are two arbitrary vectors

q and r which define the polarization tensor. The dependence on these reference

vectors must be symmetrized over and are

εµν+ (i) =〈qγ(µi]〈rγν)i]

2〈qi〉〈ri〉 , εµν− (i) =[qγ(µi〉[rγν)i〉

2[qi][ri]. (2.5)

With this setup, we can express the soft limits of gauge theory and gravity. In

general, taking an external particle’s momentum soft leads to a factorization of the

amplitude into a universal soft factor and a lower-point amplitude where the soft

particle has been removed [29]. The soft factor depends on the momenta of the

particles that were affected by taking that particle soft. In particular, in a color-

ordered amplitude in gauge theory, only particles adjacent to a soft particle appear


in the soft factor. This is because only adjacent external particles share a color line.

However, in gravity where there is no color, all external particles are affected by

the limit where one particle goes soft. All helicity information of the particle which

is taken soft is contained in the soft factor. The factorization of a color-ordered

amplitude in gauge theory is

limj→0

A(1, . . . , i, j, k, . . . , n) =√

2

(−i · ε(j)

sij+k · ε(j)sjk

)A(1, . . . , i, j, k, . . . , n) . (2.6)

Particles i and k are adjacent to the soft particle j and here j means that j has been

removed from the amplitude. The sum in parentheses is the soft factor, is gauge

invariant and has an especially simple form in spinor-helicity notation. If j has +

helicity, the soft factor is

√2

(−i · ε(j)

sij+k · ε(j)sjk

)=〈ik〉〈ij〉〈jk〉 . (2.7)

In contrast to gauge theory, there is no such compact form for the soft factor in

gravity. In general the soft factor when particle 1 is taken soft is [28,29]

lim1→0

M(1, 2, . . . , n)

M(2, . . . , n)=

n∑i=2

iµiνεµν(1)

si1. (2.8)

This sum is independent of q and r and using conservation of momentum and proper-

ties of the spinor products can be simplified slightly. Its independence on the choice

of the reference momenta q and r means that the individual terms in the sum can

be very different while keeping the sum fixed. In our analysis here, we will need a

particular form for each term and that form will be determined to satisfy some simple

requirements. The requirements will be discussed in later sections and will be related

to the BCFW on-shell recursion formula, which we now discuss.

The BCFW on-shell recursion relations are an efficient method for computing

amplitudes at tree level in gauge theories [49,50] and gravity [61,62]. Two external


particles, i and j, are singled out and their helicity spinors are deformed as:

i〉 → i〉 − z j〉 , i]→ i] ,

j〉 → j〉 , j]→ j] + z i] , (2.9)

where z is a complex variable. The BCFW recursion relation relates an amplitude to a

sum of products of lower-point, on-shell amplitudes with momenta of particles i and j

deformed as above. The amplitudes in the sum consist of all possible factorizations of

the amplitude with i and j on opposite sides of the cut. These products of amplitudes

are evaluated at the value of z determined by the location of the pole in the given

factorization channel. If the deformed amplitude A(z) vanishes as z → ∞, then the

recursion relation is schematically

A(1, . . . , n) =∑R,L

AL(i)1

P 2L

AR(j) . (2.10)

Here, the hats indicate to evaluate the amplitude at the shifted momenta and the

sum runs over all possible factorizations. A BCFW recursion exists in gauge theory

and gravity for specific helicity choices of the deformed particles i and j [61–63]. In

this chapter, we will explicitly develop a relationship between BCFW and the soft

limits of amplitudes.

In all expressions in this chapter, we will suppress the gauge and gravity couplings.

Namely a factor of ign−2 and i(κ/2)n−2, where n is the number of external particles,

is omitted for gauge theory and gravity respectively. However, the fact that the gauge

coupling is dimensionless and the Planck mass is dimensionful (in 3 + 1 dimensions)

leads to distinct behavior of the amplitudes in the soft limits.

2.3 Inverse-Soft Construction of Gauge Theories and Grav-

ity

The idea of inverse soft is to “undo” the soft limit of an amplitude. In particular,

in a gauge theory, motivated by Eq. 2.6, we can consider the following trial form for


an amplitude:

A(1, . . . , i, j, k, . . . , n) = S(i, j, k)A(1, . . . , i′, j, k′, . . . , n) . (2.11)

Here, S is a soft factor, for example, that given in Eq. 2.7. Unlike in Eq. 2.6, no soft

limit is taken here and so, to conserve momentum on the right side, the momenta of

adjacent particles i and k must be shifted to compensate for the removal of j; this

is indicated by the prime. The momentum shift depends on the helicity of particle j

and can be expressed as a deformation of the helicity spinors of particles i and k. If

j has + helicity, then the helicity spinors are deformed as

i〉 → i〉 , i]→ (i+ j)k〉〈ik〉 ,

k〉 → k〉 , k]→ (k + j)i〉〈ki〉 . (2.12)

This deformation conserves momentum: p′i + p′k = pi + pj + pk.

The expression in Eq. 2.11 only guarantees that the soft limit of j on the left side

is correct. To have the correct soft limits for all particles, and so to construct the

amplitude, a sum over a set of particles {j}must be taken on the right. This sum must

also produce all multiparticle factorization channels present in the true amplitude.

We will show in later sections in specific cases that, by combining several such inverse-

soft terms, the correct multiparticle factorization channels are generated. This will

be most easily seen in color-ordered gauge theory where only adjacent particles have

nonzero factorization channels.

Another important point is that the soft factor S is unambiguous in gauge theory;

it is a single, simple, gauge-invariant term. In gravity, the soft factor is a sum of terms,

each of which is not gauge invariant, but the sum is. Thus, when constructing an

object like Eq. 2.11 in gravity, to even reproduce the soft limit of a single particle, a

sum over particles i and k must be taken. This opens the possibility of having several

possible forms of inverse-soft construction in gravity. For each possible form of the

terms in the gravity soft factor, there exists another possible inverse-soft construction.


1+

2

3

n-

-P P

Figure 2.1: Two-particle factorization BCFW diagram. The hatted legs momenta are de-formed following Eq. 2.14.

2.4 Reproducing the Soft Limits From BCFW Terms

In this section, we will give a precise map from the inverse-soft construction to

the BCFW recursion relations. In particular, we will show that adding a single

particle with the inverse-soft procedure is identical to BCFW terms in which one of

the amplitude factors is a three-point amplitude. This will be straightforward and

unambiguous in gauge theory and will be used to define the form of the soft factor in

gravity, as discussed earlier.

First, for a BCFW recursion relation to exist in gauge theory and gravity, the

helicity of particles i and j as in Eq. 2.9 must be ++, −− or +−, respectively. The

−+ deformation does not lead to a BCFW recursion and it can be shown that in that

case all of the following analysis fails2. As we are only interested in reproducing the

amplitude, we will only need to show that inverse soft can construct all terms in the

BCFW recursion for a single shift. In this chapter we will only consider the +− shift.

Also, because only neighboring particles are affected by the soft limit in color-ordered

gauge-theory amplitudes, we will only consider adjacent BCFW deformations. This

is similar to the deformations in the original discovery of BCF(W) recursion [49].

The terms in the BCFW recursion we are considering are those whose form is

schematically illustrated in Fig. 2.1. We will refer to these as two-particle factorization

2This case can be studied supersymmetrically, however [53].


channels, similar to what was used in [60]. This diagram can be expressed as

D+ = AL(1+, 2,−P )1

s12

AR(P , 3, . . . , n−) , (2.13)

with the following momentum assignments:

1 = 1]〈1− 〈12〉〈n2〉1]〈n =

〈n1〉〈n2〉1]〈2 ,

n = n]〈n+〈12〉〈n2〉1]〈n =

(n+ 1)2〉〈n〈n2〉 ,

P = 1 + 2 =(2 + 1)n〉〈2〈2n〉 = 2′ . (2.14)

Note that with these assignments, AR corresponds to an n−1-particle amplitude with

the momenta of particles n and 2 shifted according to the inverse soft deformation,

Eq. 2.12. It now remains to be shown that

AL(1+, 2,−P )1

s12

(2.15)

is the soft factor for particle 1. We will show this explicitly for either helicity as-

signment of particle 2 in gauge theory and use the result in gravity to define the soft

factor.

2.4.1 Gauge Theory

From the two possible helicity assignments of particle 2, there are two nonzero

amplitudes, AL(1+, 2−,−P+) and AL(1+, 2+,−P−), which evaluate to the same result

when including the shifted momentum. In particular,

AL(1+, 2+,−P−) =[12]3

[2(−P )][(−P )1]= − [12]〈2n〉

〈1n〉 . (2.16)

Including the factor of 1/s12 gives exactly the soft factor for particle 1:

AL(1+, 2,−P )1

s12

= S(n, 1+, 2) =〈n2〉〈n1〉〈12〉 . (2.17)


Putting the pieces together, we have shown that two-particle factorized BCFW terms

can be computed with the inverse-soft procedure. Precisely, in gauge theory,

AL(1+, 2,−P )1

s12

AR(P , 3, . . . , n−) = S(n, 1+, 2)AR(2′, 3, . . . , n′) , (2.18)

where the hat and prime represent respectively the BCFW (Eq. 2.9) and inverse soft

momentum deformations (Eq. 2.12). With the P and C invariance of pure Yang-Mills

theories this result also holds for a negative helicity soft particle.

For MHV or MHV amplitudes in gauge theory, there only exist two-particle fac-

torization channels, so we can write down an explicit inverse-soft recursion relation

for these amplitudes. Specializing to MHV, these amplitudes are functions purely of

the angle-bracket spinors and so the momentum shifts of Eq. 2.12 do not explicitly

appear. That is,

AMHV(1, . . . , n+) = S(n− 1, n+, 1)AMHV(1, . . . , n− 1) ; (2.19)

to construct an MHV amplitude with one more particle we need only multiply by the

appropriate soft factor.

2.4.2 Gravity

Motivated by the relationship between inverse soft and BCFW in gauge theory, we

now turn to considering gravity. We begin as in gauge theory by constructing terms

in the BCFW recursion relation and then expressing them in inverse-soft language.

Much of the analysis from gauge theory carries over as three-point gravity amplitudes

are just the square of the corresponding gauge-theory amplitudes. In gravity, we have

ML(1+, 2+,−P−) =[12]6

[2(−P )]2[(−P )1]2=

[12]2〈2n〉2〈1n〉2 , (2.20)


where now the helicity labels mean helicity ±2. Including the propagator factor as

in the gauge theory case gives

ML(1+, 2,−P )1

s12

=〈n2〉2[21]

〈n1〉2〈12〉 ≡ G(n, 1+, 2) . (2.21)

We will take this as our definition of the gravity soft factor: an individual term in

the sum in Eq. 2.8. To connect with that equation, note that the soft factor here is

the i = 2 term from Eq. 2.8 with reference momenta q, r of graviton 1’s polarization

tensor set equal to the momentum of graviton n. Note that in contrast to the gauge-

theory soft factor of Eq. 2.17, Eq. 2.21 is not (anti-)symmetric under the exchange

of particles 2 and n. Particle n’s momentum is the reference momentum whereas

particle 2 defines the adjacent particle or a term in the sum of Eq. 2.8. To reproduce

the complete soft limit of particle 1 we must sum over terms of the form of Eq. 2.21,

replacing 2 successively by each particle in the amplitude. We will discuss in the next

section how multiparticle factorization channels are produced with this soft factor.

As in gauge theory, MHV gravity amplitudes are constructed by the BCFW recur-

sion from purely two-particle factorizations. Unlike their gauge-theory counterparts,

gravity MHV amplitudes have explicit dependence on both angle- and square-bracket

spinors. Unfortunately, this means that a simple inverse-soft recursion relation such

as the one that was written down for gauge-theory MHV amplitudes cannot be writ-

ten down for gravity since the inverse-soft deformations will appear explicitly in the

lower-point amplitude. In any case, the BCFW recursion relations imply the inverse-

soft recursion relation for MHV gravity amplitudes:

MMHV(1, . . . , n+) =n−2∑i=1

G(n− 1, n+, i)MMHV(1, . . . , i′, . . . , (n− 1)′) , (2.22)

where the primes on the right side indicate the inverse-soft momentum deformation.

Nguyen, et al., introduced in [57] a distinct inverse-soft construction of gravity


MHV amplitudes. In constructing MHV amplitudes, they used

n∑i=2

iµiνεµν(i)

si1=

n−2∑i=2

〈in〉〈i(n− 1)〉[i1]

〈1n〉〈1(n− 1)〉〈1i〉 (2.23)

as the form of the soft factor which first appeared in [28]. Using this soft factor has

the benefit of making a larger permutation symmetry manifest as well as leading to

a convenient “tree formula” for amplitudes. This tree formula has been shown to

reproduce a previously conjectured MHV-level result [58]. Motivated by this work,

Hodges presented a new formula for MHV gravity amplitudes in [59]. In our language,

Hodges’ formula for MHV amplitudes can be expressed as

MMHV(1, . . . , n+) =n−1∑i=3

[in]〈1i〉〈2i〉〈ni〉〈1n〉〈2n〉MMHV(1′, . . . , i′, . . . , (n− 1)) . (2.24)

Note that this equation has one fewer term than our corresponding formula, Eq. 2.22.

Indeed, these two expressions for MHV gravity amplitudes are related by momentum

conservation and a Schouten identity.

While the soft factor used in [57,59] leads to nice expressions for MHV level

amplitudes, difficulties arise when continuing to NMHV. Factorization channels with

particles 1 and 2 on the same side of the factorization have no simple way to be

constructed in this formalism. As we will see, this is another motivation for using the

BCFW-inspired soft factor

n∑i=2

iµiνεµν(i)

si1=

n−1∑i=2

〈in〉2[i1]

〈1n〉2〈1i〉 (2.25)

from which three-particle factorization channels can be constructed. Of course, the

two expressions of the soft factor are equal, but because we work term by term in the

sum, the BCFW-inspired soft factor leads more directly to multiparticle factorization.


(a)

MHVMHVMHV

(b)

Figure 2.2: The two types of diagrams that enter the BCFW decomposition of an NMHVamplitude.

2.5 Reproducing NMHV Amplitudes

We will now discuss how to construct NMHV amplitudes using the inverse-soft

procedure. Our goal is to express the amplitude as a sum of terms, each of which

is a a string of products of deformed soft factors times a lower-point amplitude.

Schematically we want

ANMHV ∼∑ (

m∏i=1

S ′(pi))A′n−m , (2.26)

where the pi have been removed from An−m. The prime indicates that the amplitudes

and soft factors are deformed to conserve momentum as particles are removed. We

will use the BCFW decomposition to determine which terms to include in the sum

so the problem is reduced to expressing all BCFW diagrams entering the amplitude

in the form of Eq. 2.26.

At NMHV level, there are two types of BCFW terms as illustrated in Fig. 2.2:

those with a three-particle amplitude in the factorization and those with only higher-

point MHV amplitudes. The former, illustrated in Fig. 2.2(a), was studied in the the

previous section where it was expressed as a single soft factor times a deformed am-

plitude with one less particle. However, the diagram of Fig. 2.2(b) is new and we will

now discuss how to express it in the form of Eq. 2.26. Note that AL(1+, 2, . . . , j,−P )

is an on-shell MHV amplitude which can be expressed as a product of soft factors

times a lower-point amplitude using Eqs. 2.19 or 2.22. One approach would then be


to express AL for gauge theory this way which would lead to the following expression

for the full diagram:

D = AL(1+, . . . ,−P )1

P 2AR(P , . . . , n−)

= S(i, j+, k)AL(1+, . . . , i′, j, k′, . . . ,−P )1

P 2AR(P , . . . , n−) , (2.27)

where particle j has been removed from the amplitude in the second line. While

Eq. 2.27 expresses the diagram as a product of a soft factor times lower-point ampli-

tudes, an extra explicit propagator is present and there is more than one lower-point

amplitude. Note also that P is the same in both lines of Eq. 2.27: it is the sum of

all left-hand-side particles’ momenta including particle j. It is not an original par-

ticle whose momentum was deformed using inverse soft (compare with two-particle

factorization in Eq. 2.14 where P is equal to 2′). Thus, Eq. 2.27 is not what we want

and it does not express the diagram in the desired form of Eq. 2.26.

We will then adopt a different approach. We will build up AL times the propagator

starting with a three-point amplitude times a propagator factor,

Dstart = AL(1+, j,−P )1

s1j

AR(P , j + 1, . . . , n−) = S(n, 1+, j)AR(j′, j + 1, . . . , n′−)

(2.28)

and will add particles until all that were contained in the original AL are included.

Namely, we will build the diagram of Fig. 2.2(b) starting with a two-particle diagram

where only two particles are in the left-hand-side MHV amplitude. We add particles

in turn to the left amplitude with soft factors and deforming the momenta appropri-

ately. Thus, a three-particle factorization will have two soft factors, a four-particle

factorization three and so on. We do not explicitly add particles to AR, but we must

make sure that the momentum shifts generated by the work on the left correspond

to the correct BCFW deformations. Thus, in order to achieve the form of Eq. 2.26,

we have two requirements: AL times the propagator must reduce to a product of soft

factors and all inverse-soft shifted momenta must reproduce the BCFW-deformed

momenta.

Unfortunately these requirements are not satisfied generically for any diagram of


the form illustrated in Fig 2.2(b). For instance, we will see that particles’ momenta

need to be deformed in a specific way and in a specific order to reproduce the BCFW

shift. On the other hand, we need to keep AL nonzero throughout the construction

which can restrict which particle can be added at a given step. For instance, the

nonzero three-point BCFW vertex with particle 1+ has helicity (+,+,−). Thus, the

first particle to be added on the left must have negative helicity. Complications will

also arise when trying to add a particle next to the unphysical momentum P . These

caveats will be discussed in detail in the next sections.

In section 2.5.1, we will discuss how the inverse-soft shift can reproduce the BCFW

deformation in AR. In sections 2.5.2 and 2.5.3, we will see how AL times the prop-

agator can be constructed as a string of deformed soft factors separately for gauge

theory and gravity. Finally, in section 2.5.4, we will summarize the above mentioned

caveats and address the applicability of the procedure.

2.5.1 Momentum Deformation in AR

In this section we will show how the inverse-soft shift can be equivalent to the

usual BCFW deformation in AR at each step of the construction. We begin with

a two-particle factorization diagram in Eq. 2.28 and the result from Eq. 2.14 that

the associated deformation is equivalent to the BCFW shift. Here, we will write the

deformed momentum differently from Eq. 2.14 to make the generalization to higher

points easier:

j′ = 1 + j +(1 + j)2

〈n(1 + j)1]1]〈n = P ,

n′ = n+(1 + j)2

〈n(1 + j)1]1]〈n = n . (2.29)

Now, let’s add particle 2 deforming particles 1 and j. Note that since the three-point

vertex with 1+ has helicity (+,+,−), particle 2 needs to have negative helicity. Thus,


the − helicity spinors are shifted and |1] is not touched. This leads to n′′ and j′′:

j′′ = 1′ + j′ +(1′ + j′)2

〈n(j′ + 1′)1]1]〈n = 1 + 2 + j +

(1 + 2 + j)2

〈n(1 + 2 + j)1]1]〈n = P ,

n′′ = n+(1′ + j′)2

〈n(1′ + j′)1]1]〈n = n+

(1 + 2 + j)2

〈n(1 + 2 + j)1]1]〈n = n . (2.30)

We can now add any number of positive helicity particles between particle 2 and

particle j. From Eq. 2.30, it is easy to see that the inverse-soft shifts on particles

2, . . . , j will continue to reproduce the correct BCFW deformation. Namely, at each

step of the construction, the momentum assignments in AR corresponds to the mo-

mentum assignments of a BCFW diagram with the particles currently present. It is

interesting to note that inverse soft gives us a definite procedure to extract the P ]

and 〈P components of P . Using the form of Eq. 2.14, we have:

j′′ =(j′ + 1′)n〉〈j′n〉〈j′ = [1j](1 + 2 + j)n〉

〈n(j + 2)1]× [1(j + 2)

[1j]= P ]〈P . (2.31)

Finally, note that the particles on the left had to be deformed in a specific order to

reproduce the BCFW shift. Namely, in going from the initial two-point diagram to a

three-point diagram the two particles on the left need to be deformed (particles 1 and

j in the example above). Afterward, the momentum of particle 1 can no longer be

touched. For instance, particles 2 and j would be shifted in Eq. 2.30 to add particle 3.

Since which momenta get shifted is related to the position of the added particle, this

will restrict which diagrams can be constructed. We will discuss this issue in more

detail in section 2.5.4.

2.5.2 A Product of Soft Factors in Gauge Theory

We now turn to showing that AL times the propagator can be constructed as a

string of deformed soft factors. In gauge theory this can be achieved by starting with

the three-point amplitude AL(1+, j,−P ) and adding particles between 1 and j. This

has the nice feature that no explicit P appears in the soft factors because particles


are never added next to P . However, note that

AL(1+, j+,−P+) = AL(1+, j−,−P−) = 0, (2.32)

where the second equality follows from the fact that 〈P | is proportional to 〈j| as can be

seen from Eq. 2.14. One consequence of Eq. 2.32 is that the initial nonzero three-point

amplitude will necessarily be MHV which is a function of square brackets. Momentum

shifts coming from adding + helicity particles will then be visible. Another more

direct consequence is that if we are in the situation of Eq. 2.32, we will have to start

with another three-point amplitude without particle j. The latter particle will need

to be added through a soft factor which contains P .

In the case where AL(1+, j, P ) is nonzero we begin with the single soft factor

A(1+, j,−P )1

s1j

= S(n, 1+, j), (2.33)

particles can be added to it. First add the negative helicity particle, particle 2 for

definiteness, between 1 and j as

A(1+, 2−, j,−P )1

s12j

= S(1, 2−, j)S(n, 1′+, j′) , (2.34)

where, in the second soft factor, we have to deform the momenta of particles 1 and j

according to inverse soft. Note also that s12j = s1′j′ . We can now add more positive

helicity particles in the exact same way by inserting them one by one between particles

2 and j.

Consider now the case where both j and P have negative helicity. We will start

with AL(1+, 2+,−P−) and add particle j− between 2 and P . We have

AL(1+, 2+, j−,−P−)1

s12j

= S(2, j−,−P )AL(1+, 2+,−P )1

s12j

, (2.35)

where we used the usual formula, Eq. 2.19, to decompose the four-point MHV ampli-

tude and both sides were multiplied by the propagator. Remember that because we

are adding a negative helicity particle to a MHV amplitude, the shifts on 2 and P are


not visible in the three-point amplitude. The BCFW shift on particle 1 also does not

appear. We can then instead imagine shifting the angle spinors of particles 1 and 2 to

make room for particle j. Then 1/s12j = 1/s1′2′ . Also note that P = 1+2+j = 1′+2′.

Thus, we can write

AL(1+, 2+, j−,−P−)1

s12j

= S(2, j−, P )AL(1′+, 2′+,−P−)1

s1′2′

= S(2, j−, P )S(n, 1′+, 2′)

= S(2, j−, 2′′)S(n, 1′+, 2′) , (2.36)

where we used Eq. 2.17 to combine the soft factor with the propagator. In going from

the second line to third line we have replaced P by 2′′ as in Eq. 2.31. Note also that

the gauge-theory soft factor is symmetric in P ↔ −P .

It is insightful to do an explicit example. Inserting the soft-factor expressions into

Eq. 2.36:

S(2, j−, 2′′)S(n, 1′+, 2′) =[22′′]

[2j][j2′′]

〈n2′〉〈n1′〉〈1′2′〉

=[2(1 + j)n〉

[2j][j(1 + 2)n〉〈n(2 + j)1][12]

〈n(1 + j)2](1 + 2 + j)2

=〈n(2 + j)1][12]

[j(1 + 2)n〉[2j](1 + 2 + j)2, (2.37)

where we used the value of 2′′] given in Eq. 2.31. We can now compare this with the

expression obtained directly through BCFW,

AL(1+, 2+, j−,−P−)1

s12j

=[12]3

[2j][jP ][P1]

1

s12j

=〈n(2 + j)1][12]

[j(1 + 2)n〉[2j](1 + 2 + j)2, (2.38)

and it agrees with the inverse-soft construction as expected.

In summary, it is always possible to go from a two-particle to a three-particle

diagram using inverse soft. The momenta of the two particles initially on the left

are always deformed irrespective of the location of the added third particle. One can


also see that it will not be possible to add particles next to P after this first step.

Each time a particle is added adjacent to P , the momentum of 1 must be deformed

to conserve momentum. In constructing a four-particle factorization channel we add

a + helicity particle which would induce a shift of |1], destroying the equivalence of

the BCFW and inverse soft shifts in AR; see Eq. 2.30. For the same reason, it is not

possible to add a + helicity particle next to particle 1.

2.5.3 A Product of Soft Factors in Gravity

As there is no color ordering in gravity, soft particles must be inserted at all

possible locations. This includes inserting the particle adjacent to the unphysical

particle P . As mentioned above, it is not possible to add next to P in a three-particle

diagram to create a four-particle diagram. Consequently, in gravity, at most three-

particle factorization diagrams can be created with inverse soft. Adding particle 2

next to particles j and P with particle 1’s momentum the reference momentum (see

Eq. 2.21), we find

M(1+, 2−, j,−P )1

s12j

=[G(1, 2−, j) + G(1, 2−,−P )

]G(n, 1′+, j′) (2.39)

=[G(1, 2−, j)− G(1, 2−, j′′)

]G(n, 1′+, j′) .

Note that the gravity soft factor is antisymmetric in j′′ ↔ −j′′ which explains the

minus sign in the second line of Eq. 2.39. To prove this, write the explicit values for

the soft factors in the equation above:

[G(1, 2−, j)− G(1, 2−, j′′)

]G(n, 1′+, j′)

=

[[j1]2〈2j〉[21]2[j2]

− [j′′1]2〈2j′′〉[21]2[j′′2]

]〈nj′〉2[1j]

〈n1′〉2〈j′1′〉

=[1j]〈j2〉〈n(1 + 2)j]

[21][j2]〈n(j + 1)2]× 〈n(j + 2)1]2[j1]2

〈n(1 + 2)j]2P 2

=[1j]3〈j2〉〈n(j + 2)1]2

[21][j2]〈n(j + 1)2]〈n(1 + 2)j]P 2, (2.40)


where we used the explicit value of j′′ given in Eq. 2.31. For completeness, we will

compute the left-hand side of Eq. 2.39 directly using the BCFW procedure:

M(1+, 2−, j,−P )1

s12j

= − 〈2j〉[P1]6

[2j][2P ][21][jP ][j1]× 1

P 2

=[1j]3〈j2〉〈n(j + 2)1]2

[21][j2]〈n(j + 1)2]〈n(1 + 2)j]P 2. (2.41)

Thus, the expression built using inverse soft agrees with the usual BCFW construction

as expected.

The requirement for a consistent inverse-soft construction, that such unphysical

soft factors be included in constructing multiparticle gravity factorization channels,

is very restrictive. The soft factor we have used to define the gravity inverse-soft

factor is exactly what is needed to reproduce individual BCFW terms. As mentioned

earlier, other soft factors, such as that from [57], could have significant difficulty in

reproducing the multiparticle factorization channels precisely because of this issue.

2.5.4 Applicability of the procedure

In this section, we will summarize which diagrams and which NMHV helicity am-

plitudes can be constructed with the inverse-soft procedure discussed above. First, for

gravity, we can construct all two- and three-particle factorization diagrams. However,

as previously mentioned, four-particle diagrams cannot be created as they require

adding a particle next to P in a three-particle diagram which destroys the BCFW de-

formation in AR. Thus, only NMHV gravity amplitudes with seven or fewer external

particles (n ≤ 7) can be constructed using inverse soft.

In gauge theory, we can also reproduce all two- and three-particle diagrams al-

lowing us to construct all n ≤ 7 amplitudes. However, we can do more as we can

add particles between particles 2 and j to create four- and higher-point diagrams.

Namely, we can create diagrams of the form

AL(1+, 2−, 3+, . . . , j±,−P∓)1

P 2AR(P±, . . . , n−) , (2.42)

AL(1+, 2+, 3+, . . . , j−,−P−)1

P 2AR(P+, . . . , n−) , (2.43)


MHV

1+ 9-

2+

3-

4+ 5+

6-

7+

8+-P- P+

Figure 2.3: A BCFW diagram for the amplitude A(1−, 2+, 3−, 4+, 5+, 6−, 7+, 8+, 9−) whichcannot be constructed with inverse soft.

where in the first line we start with AL(1+, j±,−P∓), add particle 2−, and fill in with

positive helicity particles between particles 2 and j. In the second line, we start with

AL(1+, 2+,−P−), add particle j− next to P , and fill in again between 2 and j.

Note that we can also take AL as our lower point amplitude and add particles to

AR. This will allow us to construct the additional diagrams

AL(1+, . . . ,−P+)1

P 2AR(P−, (j + 1)+, . . . , (n− 1)+, n−) , (2.44)

AL(1+, . . . ,−P−)1

P 2AR(P+, (j + 1)+, . . . , (n− 1)−, n−) , (2.45)

where in the first line we start with AR(P−, (n− 1)+, n−), add particle (j + 1)+, and

fill in with positive helicity par¡ticles between particles (j + 1) and (n − 1). In the

second line, we start with AR(P+, (n− 1)−, n−), add particle (j+ 1)+ next to P , and

fill in again between (j + 1) and (n− 1).

We now notice that any NMHV amplitude containing adjacent particles with

helicity (−,+,−) or (−,−,+) can be created using the inverse-soft procedure we

have developed here. In such a case, we factorize the amplitude in BCFW at the

location of the + helicity particle in this series: either (−, +|−) or (−, −|+). This

factorization results in the possible constructible terms discussed above. Such a fac-

torization is always possible in NMHV gauge-theory amplitudes up through eight

external legs. Problems begin at nine points as illustrated in Fig. 2.3. The amplitude

A(1+, 2+, 3−, 4+, 5+, 6−, 7+, 8+, 9−) does not contain either of the strings of adjacent

particles with helicity (−,+,−) or (−,−,+). The BCFW diagram in Fig. 2.3 would


require adding two particles adjacent to P or adding a + helicity particle next to

particle 1 and thus cannot be constructed using our inverse-soft procedure.

2.5.5 Example: Gauge Theory NMHV

To show the utility of inverse soft, we will give an explicit example of constructing

the six-point NMHV gauge-theory amplitude using inverse soft. Consider the gauge-

theory amplitude A(1+, 2−, 3+, 4−, 5+, 6−). We will deform the momenta of particles

3 and 4 for the BCFW construction. This deformation leads to three factorizations

which contribute to the amplitude:

D1 = A(2−, 3+,−P )1

s23

A(P , 4−, 5+, 6−, 1+) , (2.46)

D2 = A(1+, 2−, 3+,−P )1

s123

A(P , 4−, 5+, 6−) ,

D3 = A(6−, 1+, 2−, 3+,−P )1

s45

A(P , 4−, 5+) .

We will consider each of these terms in turn and show that they can be written simply

in the inverse-soft construction language. For compactness, we will only decompose

these diagrams down to soft factors times MHV or MHV amplitudes.

We first consider D1. As shown earlier, for the choice of helicity of P that gives a

nonzero value, A(2−, 3+,−P )(1/s23) = S(2, 3+, 4). Thus, we can equally express D1

as

D1 = S(2, 3+, 4)A(2′−, 1+, 6−, 5+, 4′−) . (2.47)

Note that the amplitude that remains is of MHV-type and so the inverse soft momen-

tum shift explicitly appears. The same arguments hold for D3; this time, re-expressing

the right side of the factorization. D3 can be written as

D3 = S(3, 4−, 5)A(6−, 1+, 2−, 3′+, 5′+) . (2.48)

D2 is slightly more complicated and requires some care. We will rewrite the left

side of the BCFW factorization as a product of two soft factors as in Eq. 2.34. To do

this, we will start with the soft factor S(1, 3+, 4) and add particle 2 between 1 and 3.


This gives

A(1+, 2−, 3+,−P )1

s123

=[13]

[12][23]

〈1′4〉〈1′3′〉〈3′4〉 ≡ S(1, 2−, 3)S(1′, 3′+, 4) , (2.49)

where the primes indicate the inverse soft momentum deformation with negative

helicity particle 2. The right side of the factorization is subtle but proceeds exactly as

previously. We first add particle 3 to the expression. This produces A(1′+, 6−, 5+, 4′−),

where the primes indicate the inverse soft momentum deformation with particle 3.

Next, we add particle 2 between particles 1 and 3. Thus, we now only deform the

momentum of particles 1 and 3 by the momentum of 2. To see how this affects the

momentum of the particles in A(1′+, 6−, 5+, 4′−), consider first 1′. We have

1′ =(1 + 3)4〉〈14〉〈1 . (2.50)

Deforming the momentum of 1 and 3 appropriately gives 1′′:

1′′ =(1 + 2 + 3)4〉[3(1 + 2)

〈4(1 + 2)3]. (2.51)

As it must, this equals the momentum flowing through the BCFW cut, P = 1+2+3.

We can now consider the deformation of 4′. Starting with

4′ =(4 + 3)1〉〈41〉〈4 , (2.52)

4′′ is then

4′′ = 4]〈4 +(1 + 2 + 3)2

〈4(1 + 2)3]3]〈4 = 4 . (2.53)

Finally, we can express D2 in inverse-soft language as

D2 = S(1, 2−, 3)S(1′, 3′+, 4)A(1′′+, 6−, 5+, 4′′−) , (2.54)

where the primes and double primes are as above.

Putting it all together, we can express the six-point, alternating helicity, NMHV


amplitude in gauge theory as

A(1+, 2−, 3+, 4−, 5+, 6−) = S(2, 3+, 4)A(2′−, 1+, 6−, 5+, 4′−)

+ S(3, 4−, 5)A(6−, 1+, 2−, 3′+, 5′+)

+ S(1, 2−, 3)S(1′, 3′+, 4)A(1′′+, 6−, 5+, 4′′−) . (2.55)

We present the inverse-soft expression of the NMHV six-point gravity amplitude in

appendix A.

2.6 Conclusions

We have shown that inverse soft, with BCFW as our guide, can be used to con-

struct gauge-theory and gravity amplitudes. In particular, for specific amplitudes,

each term in the BCFW expansion can be built up from soft factors multiplied by a

lower-point amplitude. This procedure works for all tree-level gauge-theory and grav-

ity amplitudes with seven or fewer external legs because, as was shown in Sec. 2.5,

inverse soft can produce three-particle factorization BCFW diagrams. Also, certain

classes of NMHV gauge-theory amplitudes can be constructed with any number of

legs which contain a set of consecutive particles with helicities (−,−,+) or (−,+,−).

As we have developed it, inverse soft does not explicitly need information from

the collinear limits of amplitudes to be able to reconstruct the amplitude. Presum-

ably, inverse soft could be used to compute amplitudes in any massless theory with

universal soft limits and a BCFW recursion. This property may seem surprising, es-

pecially because inverse soft can be used to construct NMHV amplitudes. However,

this property is similar to the way that BCFW exploits complex factorization to con-

struct amplitudes. In that case, only a subset of factorization channels are needed to

reproduce all factorization channels present in an amplitude. One can imagine con-

structing an inverse soft/collinear procedure which incorporates information about

both the soft and collinear limits to create amplitudes. Perhaps such a procedure is

necessary to extend the applicability to arbitrary helicity configurations and number

of legs.


Nevertheless, it is interesting to consider how far inverse soft can be extended.

In particular, it should be possible to use inverse soft to construct higher-point loop

amplitudes, perhaps along the lines of [56]. Using the supersymmetric BCFW formal-

ism of [64], supersymmetric inverse soft could be used to construct arbitrary NMHV

gauge-theory amplitudes. This is because we have an inverse-soft construction valid

for classes of NMHV amplitudes with any number of legs and the various helicity

orderings of NMHV Yang-Mills amplitudes can be extracted from a unique N = 4

sYM NMHV superamplitude. However, supersymmetry would likely not be useful in

finding a gravity all-point NMHV inverse soft construction because the problems we

encountered were caused by the need to sum over many permutations. To go further,

it is unclear if BCFW should continue to be our guide. Indeed, in gravity, it is not

clear what the “best” or most useful form of the gravity soft factor is for inverse soft.

Inverse soft has a particularly simple form in the twistor-space representation

[54]. Nguyen, et al., showed that their tree formula for gravity MHV amplitudes

also has a twistor-space representation. In [59], Hodges conjectured a new BCFW

recursion relation in N = 7 supertwistor space and used it to construct an inverse-

soft procedure for MHV amplitudes. It would then be interesting to see if N = 7

BCFW could be used as a guide to construct multiparticle factorization channels and

NMHV amplitudes with a possibly different soft factor. It would also be interesting

to study whether or not our inverse-soft procedure for gravity extended to N = 8

supergravity has a twistor-space representation. If so, what is the representation?

Are there symmetries of N = 8 amplitudes that become apparent, analogous to dual

superconformal symmetry [65] and the Yangian in N = 4 sYM [66]? How does the

E7(7) symmetry of the moduli space [67] manifest itself? More work in these directions

is necessary to fully elucidate the symmetries and simplicity of scattering amplitudes.

Chapter 3

One-Loop Amplitudes

Z. Bern, C. Boucher-Veronneau and H. Johansson, “N ≥ 4 Supergravity Amplitudes

from Gauge Theory at One Loop,” Phys. Rev. D 84, 105035 (2011).

3.1 Introduction

One of the remarkable theoretical ideas emerging in the last decade is the notion

that gravity theories are intimately tied with gauge theories. The most celebrated

connection is the AdS/CFT correspondence [68] which relates maximally supersym-

metric Yang-Mills gauge theory to string theory (and supergravity) in anti-de Sitter

space. Another surprising link between the two theories is the conjecture that to

all perturbative loop orders the kinematic numerators of diagrams describing grav-

ity scattering amplitudes are double copies of the gauge-theory ones [31,39]. This

double-copy relation relies on a novel conjectured duality between color and kine-

matic diagrammatic numerators of gauge-theory scattering amplitudes. At tree level,

the double-copy relation encodes the Kawai-Lewellen-Tye (KLT) relations between

gravity and gauge-theory amplitudes [43].

The duality between color and kinematics offers a powerful tool for construct-

ing both gauge and gravity loop-level scattering amplitudes, including nonplanar

contributions [39–41,69]. The double-copy property does not rely on supersymme-

try and is conjectured to hold just as well in a wide variety of supersymmetric and

53

CHAPTER 3. ONE-LOOP AMPLITUDES 54

non-supersymmetric theories. In recent years there has been enormous progress in

constructing planar N = 4 super-Yang-Mills amplitudes. For example, at four and

five points, expressions for amplitudes of this theory—believed to be valid to all loop

orders and nonperturbatively—have been constructed [70]. (For recent reviews, see

refs. [69,71].) Many of the new advances stem from identifying a new symmetry, called

dual conformal symmetry, in the planar sector of N = 4 super-Yang-Mills theory [72].

This symmetry greatly enhances the power of methods based on unitarity [37,46,73]

or on recursive constructions of integrands [74]. The nonplanar sector of the theory,

however, does not appear to possess an analogous symmetry. Nevertheless, the du-

ality between color and kinematics offers a promising means for carrying advances

in the planar sector of N = 4 super-Yang-Mills theory to the nonplanar sector and

then to N = 8 supergravity. In particular, the duality interlocks planar and nonpla-

nar contributions into a rigid structure. For example, as shown in ref. [39], for the

three-loop four-point amplitude, the maximal cut [75] of a single planar diagram is

sufficient to determine the complete amplitude, including nonplanar contributions.

Here we will explore one-loop consequences of the duality between color and kine-

matics for supergravity theories with 4 ≤ N ≤ 6 supersymmetries. These cases are

less well understood than the cases of maximal supersymmetry. (Some consequences

for finite one-loop amplitudes in non-supersymmetric pure Yang-Mills theory have

been studied recently [76].) Since the duality and its double-copy consequence re-

main a conjecture, it is an interesting question to see if the properties hold in the

simplest nontrivial loop examples with less than maximal supersymmetry. In par-

ticular, we will explicitly study the four- and five-point amplitudes of these theories.

These cases are especially straightforward to investigate because the required gauge

theory and gravity amplitudes are known. Our task is then to find rearrangements

that expose the desired properties. The necessary gauge-theory four-point amplitudes

were first given in dimensional regularization near four dimensions in ref. [77], and

later in a form valid to all orders in the dimensional regularization parameter [78].

At five points, the dimensionally regularized gauge-theory amplitudes near four di-

mensions were presented in ref. [79]. The four-graviton amplitudes in theories with

N ≤ 6 supersymmetries were first given in ref. [80]. More recently, the MHV one-loop


amplitudes of N = 6 and N = 4 supergravity were presented, up to rational terms in

the latter theory [81].1

Here we point out that the double-copy relations can be straightforwardly ex-

ploited, allowing us to obtain complete integrated four- and five-point amplitudes

of N ≥ 4 supergravity amplitudes as a simple linear combinations of corresponding

gauge-theory amplitudes. Because these relations are valid in any number of dimen-

sions, we can use previously obtained representations of QCD and N = 4 super-Yang-

Mills four-point amplitudes valid with D-dimensional momenta and states in the loop

to obtain such representations for N ≥ 4 supergravity. These D-dimensional results

are new, while our four-dimensional results reproduce ones found in refs. [80,81]. Rela-

tions between integrated N = 4 super-Yang-Mills and N = 8 supergravity four-point

one- and two-loop amplitudes had been described previously in refs. [82].

For cases with larger numbers of external legs, the loop momentum is expected

to become entangled with the relations making them more intricate. Nevertheless,

we expect that the duality should lead to simple structures at one loop for all mul-

tiplicity, and once understood these should lead to improved means for constructing

gravity loop amplitudes. Indeed, the duality has already been enormously helpful for

constructing four- and five-point multiloop amplitudes in N = 8 supergravity [39–

41,69].

This chapter is organized as follows. In section 3.2 we review some properties of

scattering amplitudes, including the conjectured duality between color and kinemat-

ics and the gravity double-copy property. Then in section 3.3, we give some one-loop

implications, before turning to supergravity. We also make a few comments in this

section on two-loop four-point amplitudes. We give our summary and outlook in sec-

tion 4.7. Two appendices, B and C, are included collecting gauge-theory amplitudes

and explicit forms of the integrals used in our construction.

1While completing the present chapter, version 2 of ref. [81] appeared, giving the missing rationalterms of the N = 4 supergravity five-point amplitudes.


3.2 Review

In this section we review some properties of gauge and gravity amplitudes perti-

nent to our construction of supergravity amplitudes. We first summarize the duality

between color and kinematics which allows us to express gravity amplitudes in terms

of gauge-theory ones. We then review decompositions of one-loop N = 4, 5, 6 su-

pergravity amplitudes in terms of contributions of matter multiplets, simplifying the

construction of the amplitudes.

3.2.1 Duality between color and kinematics

We can write any m-point L-loop-level gauge-theory amplitude where all particles

are in the adjoint representation as

(−i)Lgm−2+2L

Aloopm =

∑j

∫dDLp

(2π)DL1

Sj

njcj∏αjp2αj

. (3.1)

The sum runs over the set of distinct m-point L-loop graphs, labeled by j, with only

cubic vertices, corresponding to the diagrams of a φ3 theory. The product in the

denominator runs over all Feynman propagators of each cubic diagram. The inte-

grals are over L independent D-dimensional loop momenta, with measure dDLp =∏Ll=1 d

Dpl. The ci are the color factors obtained by dressing every three vertex with

an fabc = i√

2fabc = Tr{[T a, T b]T c} structure constant, and the ni are kinematic

numerator factors depending on momenta, polarizations and spinors. For supersym-

metric amplitudes expressed in superspace, there will also be Grassmann parameters

in the numerators. The Sj are the internal symmetry factors of each diagram. The

form in eq. (4.1) can be obtained in various ways; for example, starting from covariant

Feynman diagrams, where the contact terms are absorbed into kinematic numerators

using inverse propagators.

Any gauge-theory amplitude of the form (4.1) possesses an invariance under “gen-

eralized gauge transformations” [31,38,39,83,84] corresponding to all possible shifts,

ni → ni + ∆i, where the ∆i are arbitrary kinematic functions (independent of color)


constrained to satisfy ∑j

∫dDLp

(2π)DL1

Sj

∆jcj∏αjp2αj

= 0 . (3.2)

By construction this constraint ensures that the shifts by ∆i do not alter the amplitude

(4.1). The condition (3.2) can be satisfied either because of algebraic identities of the

integrand (including identities obtained after trivial relabeling of loop momenta in

diagrams) or because of nontrivial integration identities. Here we are interested in ∆i

that satisfy (3.2) because of the former reason, as the relations we will discuss below

operate at the integrand level. We will refer to these kind of numerator shifts valid at

the integrand level as point-by-point generalized gauge transformations. One way to

express this freedom is by taking any function of the momenta and polarizations and

multiplying by a sum of color factors that vanish by the color-group Jacobi identity,

and then repackaging the functions into ∆i’s over propagators according to the color

factor of each individual term. Some of the resulting freedom corresponds to gauge

transformations in the traditional sense, while most does not. These generalized gauge

transformations will play a key role, allowing us to choose different representations of

gauge-theory amplitudes, aiding our construction of gravity amplitudes from gauge-

theory ones.

The conjectured duality of refs. [31,39] states that to all loop orders there exists

a form of the amplitude where triplets of numerators satisfy equations in one-to-one

correspondence with the Jacobi identities of the color factors,

ci = cj − ck ⇒ ni = nj − nk , (3.3)

where the indices i, j, k schematically indicate the diagram to which the color factors

and numerators belong to. Moreover, we demand that the numerator factors have the

same antisymmetry property as color factors under interchange of two legs attaching

to a cubic vertex,

ci → −ci ⇒ ni → −ni . (3.4)

At tree level, explicit forms satisfying the duality have been given for an arbitrary


number of external legs and any helicity configuration [85]. An interesting conse-

quence of this duality is nontrivial relations between the color-ordered partial tree

amplitudes of gauge theory [31] which have been proven in gauge theory [86] and in

string theory [34]. Recently these relations played an important role in the impressive

construction of the complete solution to all open string tree-level amplitudes [87]. The

duality has also been studied from the vantage point of the heterotic string, which

offers a parallel treatment of color and kinematics [83]. A partial Lagrangian un-

derstanding of the duality has also been given [38]. The duality (3.3) has also been

expressed in terms of an alternative trace-based representation [88], emphasizing the

underlying group-theoretic structure of the duality. Indeed, at least for self-dual field

configurations and MHV amplitudes, the underlying infinite-dimensional Lie algebra

has been very recently been identified as area preserving diffeomorphisms [89].

At loop level, less is known though some nontrivial tests have been performed. In

particular, the duality has been confirmed to hold for the one-, two- and three-loop

four-point amplitudes of N = 4 super-Yang-Mills theory [39]. It is also known to hold

for the one- and two-loop four-point identical helicity amplitudes of pure Yang-Mills

theory [39]. Very recently it has also been shown to hold for the four-loop four-point

amplitude of N = 4 super-Yang-Mills theory [40], and for the five-point one-, two-

and three-loop amplitudes of the same theory [41].

3.2.2 Gravity as a double copy of gauge theory

Perhaps more surprising than the gauge-theory aspects of the duality between

color and kinematics is a directly related conjecture for the detailed structure of

gravity amplitudes. Once the gauge-theory amplitudes are arranged into a form

satisfying the duality (3.3), corresponding gravity amplitudes can be obtained simply

by taking a double copy of gauge-theory numerator factors [31,39],

(−i)L+1

(κ/2)n−2+2LMloop

m =∑j

∫dDLp

(2π)DL1

Sj

njnj∏αjp2αj

, (3.5)


where Mloopm are m-point L-loop gravity amplitudes. The ni represent numerator

factors of a second gauge-theory amplitude, the sum runs over the same set of dia-

grams as in eq. (4.1). At least one family of numerators (nj or nj) for gravity must

be constrained to satisfy the duality (3.3) [38,39]. This is expected to hold in a large

class of gravity theories, including all theories that are low-energy limits of string

theories. We obtain different gravity theories by taking the ni and ni to be numer-

ators of amplitudes from different gauge theories. Here we are interested in N ≥ 4

supergravity amplitudes in D = 4. For example, we obtain the pure supergravity

theories as products of D = 4 Yang-Mills theories as,




N = 4 supergravity : (N = 4 sYM)× (N = 0 sYM) , (3.6)

where N = 0 super-Yang-Mills is ordinary non-supersymmetric Yang-Mills theory,

consisting purely of gluons. (N = 7 supergravity is equivalent to N = 8 supergravity,

so we do not list it.)

Since the duality requires the numerators and color factors to share the same

algebraic properties (3.3) and (4.4), eq. (3.2) implies that

∑j

∫dDLp

(2π)DL1

Sj

∆jnj∏αjp2αj

= 0 , (3.7)

so that the gravity amplitude (3.5) is invariant under the same point-by-point gener-

alized gauge transformation nj → nj + ∆j as in gauge theory.

At tree level, the double-copy property encodes the KLT [43] relations between

gravity and gauge theory [31]. The double-copy formula (3.5) has been proven at

tree level for pure gravity and for N = 8 supergravity, when the duality (3.3) holds

in the corresponding gauge theories [38]. At loop level a simple argument based on

the unitarity cuts strongly suggests that the double-copy property should hold if the


duality holds in gauge theory [38,39]. In any case, the nontrivial part of the loop-

level conjecture is the assumption of the existence of a gauge-theory loop amplitude

representation that satisfies the duality between color and kinematics. The double-

copy property (3.5) has been explicitly confirmed in N = 8 supergravity through

four loops for the four-point amplitudes [39,40] and through two loops for the five-

point amplitudes [41]. (The three- and four-loop N = 4 super-Yang-Mills and N = 8

supergravity four-point amplitudes had been given earlier, but in a form where the

duality and double copy are not manifest [11,12,44,90].)

3.2.3 Decomposing one-loop N ≥ 4 supergravity amplitudes.

scalars spin 1/2 spin 1 spin 3/2 spin 2

N = 8 70 56 28 8 1N = 6 gravity 30 26 16 6 1N = 5 gravity 10 11 10 5 1N = 4 gravity 2 4 6 4 1N = 6 matter 20 15 6 1N = 4 matter 6 4 1

Table 3.1: Particle content of relevant supergravity multiplets. The scalars are taken to bereal for counts in this table.

To simplify the analysis, we consider amplitudes with only gravitons on the exter-

nal legs. (One can, of course, use an on-shell superspace as described in ref. [91] to

include other cases as well.) At one loop it is well known that the graviton scattering

amplitudes of various supersymmetric theories satisfy simple linear relations dictated

by the counting of states in each theory. In table 3.1 we give the particle content

of relevant supergravity multiplets. (The N = 5 matter multiplet is the same as the

N = 6 matter one, hence, it is not explicitly listed. Similarly, the N = 8 supergrav-

ity multiplet is equivalent to the N = 7 one.) Looking at this table, we can easily

assemble some simple relations between the contributions from different multiplets


circulating in the loop,

M1-loopN=6 (1, 2, . . . ,m) = M1-loop

N=8 (1, 2, . . . ,m)− 2M1-loopN=6,mat.(1, 2, . . . ,m) ,

M1-loopN=5 (1, 2, . . . ,m) = M1-loop

N=8 (1, 2, . . . ,m)− 3M1-loopN=6,mat.(1, 2, . . . ,m) , (3.8)

M1-loopN=4 (1, 2, . . . ,m) = M1-loop

N=8 (1, 2, . . . ,m)− 4M1-loopN=6,mat.(1, 2, . . . ,m)

+ 2M1-loopN=4,mat.(1, 2, . . . ,m) ,

where the subscript “mat” denotes a matter multiplet contribution. Thus, in the

rest of the chapter, we will consider only one-loop amplitudes with the two types of

matter going around the loop in addition to the N = 8 amplitudes. The remaining

N ≥ 4 amplitudes (with generic amounts of N ≥ 4 matter) can be assembled by

linear combination of these three types.

3.3 Implications of the duality at one loop

In this section we first present a few general one-loop implications of the duality

between color and kinematics. Our initial considerations are general and apply as well

to non-supersymmetric theories. We will then specialize to N ≥ 4 supergravity four-

and five-point amplitudes, taking advantage of special properties of N = 4 super-

Yang-Mills theory.

3.3.1 Implications for generic one-loop amplitudes

As shown in ref. [25] all color factors appearing in a one-loop amplitude can be

obtained from the color factors of “ring diagrams”, that is the (m−1)!/2 one-particle-

irreducible (1PI) diagrams in the shape of a ring, as illustrated in fig. 3.1 for the cyclic

ordering 1, 2, . . . ,m. We will denote the color and kinematic numerator factors of such

a diagram with external leg ordering 1, 2, . . . ,m by c123···m and n123···m(p). Its color

factor is given by the adjoint trace,

c123...m = TrA[fa1 fa2 fa3 · · · fam ] , (3.9)


3

p1

2

m

Figure 3.1: The one-loop m-gon master diagram for the cyclic ordering 1, 2, . . . ,m.

where (fai)bc = f baic.

The color factors of the one-particle-reducible diagrams are simply given by anti-

symmetrizations of ring-diagram ones as dictated by the Jacobi relations (3.3). For

example, the color factor of the diagram with a single vertex external to the loop

shown in fig. 3.2 is

c[12]3···m ≡ c123···m − c213···m . (3.10)

If we have a form of the amplitude where the duality holds, then the numerator of

this diagram is

n[12]3···m(p) ≡ n123···m(p)− n213···m(p) . (3.11)

The color factors of other diagrams, with multiple vertices external to the loop, can

similarly be obtained with further antisymmetrizations such as c[[12]3]···m = c[12]3···m −c3[12]···m. In this way all color factors and numerators can be expressed in terms of

the ones of the ring diagram, so it serves as our “master” diagram.

It is also useful to consider representations where the dual Jacobi relations do not

hold. For any m-point one-loop amplitude, we can use the color-group Jacobi identity

to eliminate all color factors except those of the master diagram and its relabelings.

Indeed, this is how one arrives at the adjoint-representation color basis [25]. In this

color basis we express the one-loop amplitude in terms of a sum over permutations

of a planar integrand,

A1-loop(1, 2, . . . ,m) = gm∑

Sm/(Zm×Z2)

∫dDp

(2π)Dc123...m A (1, 2, . . . ,m; p) , (3.12)


3

p1

2

m

3

p1

2

m

3

p2

1

m

Figure 3.2: The basic Jacobi relation between three one-loop graphs that can be used toexpress any color factor or kinematic numerator factor for any one-loop graph in terms ofthe parent m-gons.

where A (1, 2, . . . ,m; p) is the complete integrand of the color-ordered amplitude,

A1-loop(1, 2, . . . ,m). The sum runs over all permutations of external legs (Sn), but

with the cyclic (Zm) and reflection (Z2) permutations modded out. In this repre-

sentation all numerator factors except for the m-gon ones are effectively set to zero,

since their color factors no longer appear in the amplitude. This is equivalent to a

generalized gauge transformation applied to the numerators2

n123···m(p) → n123···m(p) + ∆123···m(p) = A (1, 2, 3, . . . ,m; p)m∏α=1

p2α ,

ni → ni + ∆i = 0, for 1PR graphs i , (3.13)

where the product∏p2α runs over the inverse propagators of the m-gon master dia-

gram. In this representation the m-gon numerators are in general nonlocal to account

for propagators carrying external momenta present in the one-particle reducible (1PR)

diagrams but not in master diagrams. In general, the new numerators in eq. (3.13)

will not satisfy the duality relations (3.3).

Recall that generalized gauge invariance implies that only one of the two copies

of numerators needs to satisfy the duality in order for the double-copy property to

2Here we have absorbed a phase factor i into the numerator definition, i nj → nj , compared toeq. (4.1), as is convenient for one-loop amplitudes. For the remaining part of the chapter we willuse this convention.


work. For the first copy we use the duality-violating representation (3.13) where

all one-particle reducible numerator factors are eliminated in favor of nonlocal m-

gon master numerator factors. For the second copy we use the duality-satisfying

numerators, n12...m. Then according to the double-copy formula (3.5), by making the

substitution ci → ni in eq. (3.12), we obtain a valid gravity amplitude. We then have

M1-loop(1, 2, . . . ,m) =(κ

2

)m ∑Sm/(Zm×Z2)

∫dDp

(2π)Dn123...m(p) A (1, 2, . . . ,m; p) ,

(3.14)

where n12...m(p) is the m-gon master numerator with the indicated ordering of legs

and we have replaced the gauge-theory coupling constant with the gravity one.

At first sight, it may seem surprising that only the m-gon numerators are needed,

but as noted above, these master numerators contain all the nontrivial information in

the amplitudes. The nontrivial step in this construction is to find at least one copy

of m-gon numerators ni such that the duality relations (3.3) hold manifestly.

So far these considerations have been general. An important simplification occurs

if the numerators of one of the gauge-theory copies are independent of the loop mo-

menta, n123...m(p) = n123...m. We can then pull these numerators out of the integral

in eq. (3.14) giving relations between integrated gravity and gauge theory amplitudes.

Below we will identify two cases where this is indeed true: the four- and five- point

one-loop amplitudes of N = 4 super-Yang-Mills theory [35,41]. Taking one copy to

be the N = 4 super-Yang-Mills amplitude and the other to be a gauge-theory ampli-

tude with fewer supersymmetries, we then get a remarkably simple relation between

integrated one-loop (N + 4) supergravity and super-Yang-Mills amplitudes with Nsupersymmetries,

M1-loopN+4 susy(1, 2, . . . ,m) =

(κ2

)m ∑Sm/(Zm×Z2)

n123...mA1-loopN susy(1, 2, . . . ,m) , (3.15)

valid for m = 4, 5. This construction makes manifest the remarkably good power

counting noted in refs. [81,92]. We do not expect higher points to be quite this

simple, but we do anticipate strong constraints between generic one-loop amplitudes


of gravity theories and those of gauge theory.

3.3.2 Four-point one-loop N ≥ 4 supergravity amplitudes

We now specialize the above general considerations to four-point supergravity am-

plitude. There is only one independent four-graviton amplitude,M1-loopN susy(1−, 2−, 3+, 4+),

as the others either vanish or are trivially related by relabelings. As a warmup exer-

cise, we start with N = 8 supergravity and we reevaluate this supergravity amplitude

using the above considerations. Our starting point is the N = 4 super-Yang-Mills

one-loop four-point amplitude [35,36],

A1-loopN=4 (1, 2, 3, 4) = istg4Atree(1, 2, 3, 4)

(c1234I

12344 + c1243I

12434 + c1423I

14234

), (3.16)

where s = (k1 + k2)2 and t = (k2 + k3)2 are the usual Mandelstam invariants, and the

tree amplitude is

Atree(1−, 2−, 3+, 4+) =i 〈1 2〉4

〈1 2〉〈2 3〉〈3 4〉〈4 1〉 , (3.17)

where the angle brackets 〈i j〉 (also [i j] below) denotes spinor products. (See e.g.

ref. [26].) The function I12344 is the massless scalar box integral defined in eqs. (C.6)

and (C.7) of appendix C. The other box integrals are just relabelings of this one.

The expression in eq. (3.16) in terms of the box integral (C.6) is valid in dimensions

D < 10.

The first color factor in eq. (3.16) is given by

c1234 = f ba1cf ca2dfda3ef ea4b , (3.18)

and the others are just relabelings of this one. The kinematic numerator in each case

is

n1234 = n1243 = n1423 = istAtree(1, 2, 3, 4) . (3.19)

These numerators happen to have full crossing symmetry, but that is a special feature

of the four-point amplitude in N = 4 super-Yang-Mills theory. Because the triangle


and bubble diagrams vanish, eq. (3.19) is equivalent to the duality relations (3.3).

Thus, this representation of the amplitude trivially satisfies the duality.

Using eq. (3.14), by replacing color factors with numerators and compensating for

the coupling change, we then immediately have the four-point N = 8 supergravity

amplitude,

M1-loopN=8 (1, 2, 3, 4) = −

(κ2

)4

[stAtree(1, 2, 3, 4)]2(I1234

4 + I12434 + I1423

4

), (3.20)

which matches the known amplitude [35,42].

We now generalize to supergravity amplitudes with fewer supersymmetries. Specif-

ically, consider the one-loop four-graviton amplitudes with the N = 6 and N = 4

matter multiplets in the loop. These multiplets can be expressed as products of two

gauge-theory multiplets:

N = 6 matter : (N = 4 sYM)× (N = 1 sYM)mat. ,

N = 4 matter : (N = 4 sYM)× (scalar) , (3.21)

where the N = 1 Yang-Mills matter multiplet consists of a Weyl fermion with two

real scalars (this combination actually has two-fold supersymmetry so it can also be

thought of as a N = 2 matter multiplet), and on the second line “(scalar)” denotes a

single real scalar.

Following eq. (3.14), we get the gravity amplitude by taking the first copy of the

gauge-theory amplitude and replacing the color factors with the kinematic numerator

of the second copy, constrained to satisfy the duality (3.3), and switching the cou-

pling to the gravitational one. Because the duality satisfying N = 4 super-Yang-Mills

kinematic factors at four points (3.19) are independent of the loop momentum, they

simply come out of the integral as in eq. (3.15) and behave essentially the same way

as color factors. Thus, we have a remarkably simple general formula at four points,

M1-loopN+4 susy(1, 2, 3, 4) =

(κ2

)4

istAtree(1, 2, 3, 4)(A1-loopN susy(1, 2, 3, 4)

+ A1-loopN susy(1, 2, 4, 3) + A1-loop

N susy(1, 4, 2, 3)), (3.22)


where A1-loopN susy are one-loop color- and coupling-stripped gauge-theory amplitudes for

a theory with N (including zero) supersymmetries. We were able pull out an overall

stAtree(1, 2, 3, 4) because of the crossing symmetry apparent in eq. (3.19).

Using eq. (3.22) we can straightforwardly write down the four-graviton supergrav-

ity amplitudeM1-loopN=6,mat.(1

−, 2−, 3+, 4+) with the N = 6 matter multiplet in the loop.

We use the N = 1 one-loop amplitude representation3 from ref. [78] which is valid to

all order in the dimensional regularization parameter ε:

A1-loopN=1,mat.(1

−, 2−, 3+, 4+) = ig4Atree(1−, 2−, 3+, 4+)(tJ4(s, t)− I2(t)

),

A1-loopN=1,mat.(1

−, 2−, 4+, 3+) = ig4Atree(1−, 2−, 3+, 4+)(tJ4(s, u)− t

uI2(u)

),

A1-loopN=1,mat.(1

−, 4+, 2−, 3+) = ig4Atree(1−, 2−, 3+, 4+)(I2(t) +

t

uI2(u)

− tJ4(t, u)− tID=6−2ε4 (t, u)

), (3.23)

where the integrals I2, J4 and ID=6−2ε4 are defined in appendix C. Using eq. (3.22) we

can see that the bubble integrals cancel and we have the amplitude in a form valid

to all orders in ε. Also using the relation J4 = −εID=6−2ε4 , we get

M1-loopN=6,mat.(1

−, 2−, 3+, 4+) =(κ

2

)4 1

s[stAtree(1−, 2−, 3+, 4+)]2 (3.24)

×[ID=6−2ε

4 (t, u) + ε(−ID=6−2ε

4 (t, u) + ID=6−2ε4 (s, t) + ID=6−2ε

4 (s, u))].

Using the explicit value of ID=6−2ε4 given in eq. (C.14), we get the remarkably simple

result to order ε0,

M1-loopN=6,mat.(1

−, 2−, 3+, 4+)

=icΓ

2

(κ2

)4

[stAtree(1−, 2−, 3+, 4+)]21

s2

[ln2

(−t−u

)+ π2

]+O(ε)

= −icΓ

2

(κ2

)4 〈1 2〉4 [3 4]4

s2

[ln2

(−t−u

)+ π2

]+O(ε) , (3.25)

3Here we removed the factor of i(−1)m+1(4π)2−ε present in the integrals of ref. [78], where m is 2for the bubble, 3 for the triangle and 4 for the box. (Compare eq. (C.1) with eq. (A.13) of ref. [78].)


where the constant cΓ is defined in eq. (C.3). On the last line we plugged in the value

of the tree amplitude, stAtree(1−, 2−, 3+, 4+) = −i 〈1 2〉2 [3 4]2. Indeed, this reproduces

the known result from ref. [80].

Now consider the four-graviton amplitude with an N = 4 supergravity matter

multiplet going around the loop. We take the four-gluon amplitudes with a scalar in

the loop from ref. [78]. These are

A1-loopscalar (1−, 2−, 3+, 4+) = −ig4Atree(1−, 2−, 3+, 4+) (3.26)

×(1

tID=6−2ε

2 (t) +1

sJ2(t)− t

sK4(s, t)

),

A1-loopscalar (1−, 2−, 4+, 3+) = −ig4Atree(1−, 2−, 3+, 4+)

×( t

u2ID=6−2ε

2 (u) +t

suJ2(u)− t

sK4(s, u)

),

A1-loopscalar (1−, 4+, 2−, 3+) = −ig4Atree(1−, 2−, 3+, 4+)

×(−t(t− u)

s2J3(u)− t(u− t)

s2J3(t)− t2

s2I2(u)

− tu

s2I2(t)− t

u2ID=6−2ε

2 (u)− 1

tID=6−2ε

2 (t)− t

suJ2(u)− 1

sJ2(t)

+t

sID=6−2ε

3 (u) +t

sID=6−2ε

3 (t) +t2u

s2ID=6−2ε

4 (t, u)− t

sK4(t, u)

),

where the integral functions are given in appendix C. Using eq. (3.22), we immediately

have a form for the contributions of an N = 4 supergravity matter multiplet valid to

all orders in ε,

M1-loopN=4,mat.(1

−, 2−, 3+, 4+) =(κ

2

)4

[stAtree(1−, 2−, 3+, 4+)]2 (3.27)

×(−(t− u)

s3J3(u)− (u− t)

s3J3(t)

− t

s3I2(u)− u

s3I2(t) +

1

s2ID=6−2ε

3 (u) +1

s2ID=6−2ε

3 (t)

+tu

s3ID=6−2ε

4 (t, u)− 1

s2K4(t, u)− 1

s2K4(s, t)− 1

s2K4(s, u)

).

Expanding this through order ε0 and using integral identities from refs. [78,93] (see

also appendix C) to reexpress everything in terms of six-dimensional boxes, bubbles


1

2

3

5

4

1

2

3 4

5

Figure 3.3: Pentagon and box integrals appearing in the N = 4 super-Yang-Mills five-pointone-loop amplitudes. The complete set of such integrals is generated by permuting externallegs and removing overcounts.

and rational terms, we obtain

M1-loopN=4,mat.(1

−, 2−, 3+, 4+) =1

2

(κ2

)4 〈1 2〉2 [3 4]2

[1 2]2 〈3 4〉2[icΓs

2 + s(u− t)(I2(t)− I2(u)

)− 2ID=6−2ε

4 (t, u)stu]

+O(ε) , (3.28)

matching the result of ref. [80].

3.3.3 Five-point one-loop N ≥ 4 supergravity amplitudes

Our construction at five points is again directly based on eq. (3.14). We only

need to construct M1-loopN susy(1−, 2−, 3+, 4+, 5+); the other nonvanishing amplitudes are

related by parity and relabeling. Our starting point is the known one-loop five-

point amplitudes of N = 4 super-Yang-Mills theory. The original construction of the

amplitude [37,79] uses a basis of scalar box integrals. Rearranging these results into

the adjoint-representation color basis gives

A1-loop(1, 2, 3, 4, 5) = g5∑

S5/(Z5×Z2)

c12345A1-loop(1, 2, 3, 4, 5) . (3.29)

The sum runs over the distinct permutations of the external legs of the amplitude.

This is the set of all 5! permutations, S5, but with cyclic, Z5, and reflection symme-

tries, Z2, removed, leaving 12 distinct permutations. The color factor c12345 is the one

of the pentagon diagram shown in fig. 3.3, with legs following the cyclic ordering as in


eq. (3.9). The color-ordered one-loop amplitudes of N = 4 super-Yang-Mills theory

are

A1-loopN=4 (1, 2, 3, 4, 5) =

i

2Atree(1, 2, 3, 4, 5)

(s34s45I

(12)3454 + s45s15I

1(23)454 (3.30)

+ s12s15I12(34)54 + s12s23I

123(45)4 + s23s34I

234(51)4

)+O(ε) ,

where sij = (ki + kj)2 and the I

abc(de)4 are box integrals where the legs in parenthesis

connects to the same vertex, e.g. I(12)3454 is the box diagram in fig. 3.3. The explicit

value of I(12)3454 is given in eq. (C.9), and the values of the remaining box integrals are

obtained by relabeling. If we insert these explicit expressions in eq. (3.31) then the

polylogarithms cancel after using identities (see refs. [37,79]) leaving the expression for

A1-loopN=4 given in eq. (B.1) of appendix B. The representation (3.31) of the amplitude

does not manifestly satisfy the duality.

A duality satisfying representation of the amplitude was found in ref. [41]:

A1-loopN=4 (1−, 2−, 3+, 4+, 5+) = g5 〈1 2〉4

( ∑S5/(Z5×Z2)

c12345n12345I123455

+∑S5/Z2

2

c[12]345n[12]3451

s12

I(12)3454

), (3.31)

where I123455 is the scalar pentagon, and I

(12)3454 is the one-mass scalar box integral, as

shown in fig. 3.3. The explicit values of these integrals through O(ε0) are collected in

appendix C. Each of the two sums runs over the distinct permutations of the external

legs of the integrals. For I123455 , the set S5/(Z5 × Z2) denotes all permutations but

with cyclic and reflection symmetries removed, leaving 12 distinct permutations. For

I(12)3454 the set S5/Z

22 denotes all permutations but with the two symmetries of the

one-mass box removed, leaving 30 distinct permutations. Note that we pulled out

an overall factor 〈1 2〉4, which we do not include in the numerators. (If promoted to

its supersymmetric form it should then be included [41].) The numerators defined in


this way are then [41]

n12345 = − [1 2] [2 3] [3 4] [4 5] [5 1]

4iε(1, 2, 3, 4), (3.32)

and

n[12]345 =[1 2]2 [3 4] [4 5] [5 3]

4iε(1, 2, 3, 4), (3.33)

where 4iε(1, 2, 3, 4) = 4iεµνρσkµ1k

ν2k

ρ3k

σ4 = [1 2] 〈2 3〉 [3 4] 〈4 1〉 − 〈1 2〉 [2 3] 〈3 4〉 [4 1]. It

is not difficult to confirm that the duality holds for this representation, for example,

n12345 − n21345 = n[12]345 . (3.34)

A nice feature of this representation is that the numerator factors of both the pentagon

and box integrals do not depend on loop momentum, allowing us to use eq. (3.15).

This will greatly simplify the construction of the corresponding supergravity ampli-

tudes.

We first consider the one-loop five-point N = 8 amplitude. In this case we have

several useful representations. Proceeding as in section 3.3.2, using eq. (3.15), we can

obtain the five-point amplitude for N = 8 by replacing the color factors in eq. (3.29)

with the numerator factors of eq. (3.32), multiplying by the overall factor 〈1 2〉4, and

putting in the gravitational couplings. This yields

M1-loopN=8 (1−, 2−, 3+, 4+, 5+) = (3.35)

i

2

(κ2

)5

〈1 2〉4∑S5/Z2

n12345Atree(1−, 2−, 3+, 4+, 5+)s12s23I

123(45)4 +O(ε) ,

where the sum runs over all permutations of external legs, denoted by S5, but with

reflections Z2 removed. To obtain a second representation, we can instead replace

the color factors in eq. (3.31) with their corresponding numerator factors, yielding an


alternative expression for the amplitude,

M1-loopN=8 (1−, 2−, 3+, 4+, 5+) = (3.36)(κ

2

)5

〈1 2〉8( ∑S5/(Z5×Z2)

(n12345)2I123455 +

∑S5/Z2

2

(n[12]345)2 1

s12

I(12)3454

),

where the sums run over the same permutations as in eq. (3.31). We have checked

that in D = 4 both formulas (3.36) and (3.37) are equivalent to the known five-point

amplitude from ref. [58] (after reducing the scalar pentagon integrals to one-mass box

integrals),

M1-loopN=8 (1−, 2−, 3+, 4+, 5+) =

(κ2

)5

〈1 2〉8∑S5/Z2

2

d123(45)N=8 I

123(45)4 +O(ε) , (3.37)

where the box coefficient is given by

d123(45)N=8 ≡ −1

8h(1, {2}, 3)h(3, {4, 5}, 1) tr2[/k1/k2/k3(/k4 + /k5)] , (3.38)

and the “half-soft” functions are

h(a, {2}, b) ≡ 1

〈a 2〉2 〈2 b〉2, h(a, {4, 5}, b) ≡ [4 5]

〈4 5〉〈a 4〉〈4 b〉〈a 5〉〈5 b〉 . (3.39)

Indeed it is straightforward to check that

〈1 2〉4 d123(45)N=8 =

i

2s12s23

(n12345A

tree(1−, 2−, 3+, 4+, 5+)

+ n12354Atree(1−, 2−, 3+, 5+, 4+)

), (3.40)

where the pentagon numerator is given in eq. (3.32).

Let us now study amplitudes with fewer supersymmetries starting with the five-

graviton amplitude with the N = 6 matter multiplet running around the loop. We

pick the helicities (1−, 2−, 3+, 4+, 5+) for the gravitons; as noted above all other he-

licity or particle configurations can be obtained from this. For the N = 6 and N = 4


matter multiplets from eq. (3.15) we have

MN=6,mat.(1−, 2−, 3+, 4+, 5+) =(κ

2

)5

〈1 2〉4∑

S5/(Z5×Z2)

n12345A1-loopN=1,mat.(1

−, 2−, 3+, 4+, 5+) ,

MN=4,mat.(1−, 2−, 3+, 4+, 5+) =(κ

2

)5

〈1 2〉4∑

S5/(Z5×Z2)

n12345A1-loopscalar (1−, 2−, 3+, 4+, 5+) , (3.41)

where n12345 is given in eq. (3.32) and the sums run over all permutations, but with

cyclic ones and the reflection removed.

There are a number of simplifications that occur because of the permutation sum

in eq. (3.41) and because of the algebraic properties of the N = 4 sYM numerators

(n12345 and permutations). Because the matter multiplet contributions have neither

infrared nor ultraviolet divergences [94], all 1/ε2 and 1/ε divergences cancel. In N = 6

supergravity, this manifests itself by the cancellation of all bubble and triangle integral

contributions, as noted in ref. [81]. In the case of N = 4 supergravity, the cancellation

is not complete but the sum over bubble-integral coefficients vanishes to prevent the

appearance of a 1/ε singularity. A rational function remains which can be written

in a relatively simple form once the terms are combined and simplified. Our results

match those obtained in ref. [81].

The final form of the N = 6 results after simplifications are then [81]

M1-loopN=6,mat.(1

−, 2−, 3+, 4+, 5+)

= −(κ

2

)5

〈1 2〉8∑

Z3(345)

(〈1 3〉〈2 3〉〈1 4〉〈2 4〉〈3 4〉2 〈1 2〉2

)×(d

324(51)N=8 I

324(51)4,trunc + d

314(52)N=8 I

314(52)4,trunc

)+O(ε) , (3.42)

where the summation runs over the three cyclic permutations of legs 3, 4, 5 in the

box integrals and coefficients. The factor d123(45)N=8 is exactly the coefficient (3.38) of

the N = 8 theory and the integral I123(45)4,trunc given in eq. (C.9) of appendix C is the

one-mass box integral but with its infrared divergent terms subtracted out. Similarly,


the simplified N = 4 supergravity results are

M1-loopN=4,mat.(1

−, 2−, 3+, 4+, 5+) (3.43)

=(κ

2

)5[〈1 2〉8

∑Z3(345)

(〈1 3〉〈2 3〉〈1 4〉〈2 4〉〈3 4〉2 〈1 2〉2

)2

×(d

324(51)N=8 I

324(51)4,trunc + d

314(52)N=8 I

314(52)4,trunc

)+ icΓ

5∑i=3

(c1i ln(−s1i) + c2i ln(−s2i)) + icΓR5

]+O(ε) ,

where the coefficient of log(−s13) coming from the bubble integrals is

c13 =1

2

〈1 2〉4 [3 1] [5 2]

〈1 3〉〈2 5〉〈4 5〉

[− 〈2 4〉2 〈4|2 + 5|4] 〈1|3|4]2

〈3 4〉2 〈4 5〉〈4|1 + 3|4]2(3.44)

− 〈2 3〉〈3 4〉

(〈1 5〉〈2 5〉〈1|3|5] 〈5|2|4]

〈3 5〉2 〈4 5〉〈5|1 + 3|5]− 〈1 4〉〈2 4〉〈1|3|4] 〈4|2 + 5|4]

〈3 4〉2 〈4 5〉〈4|1 + 3|4]

)

+〈2 4〉〈3 4〉

(〈1 4〉〈2 3〉〈1|3|4] 〈3|2 + 5|4]

〈3 4〉2 〈3 5〉〈4|1 + 3|4]

+〈2 5〉〈5|2|4]

〈3 5〉〈4 5〉

(〈1 5〉〈1|3|5]

〈4 5〉〈5|1 + 3|5]− 〈1 4〉〈1|3|4]

〈4 5〉〈4|1 + 3|4]

))]+ (4↔ 5) ,

and the others are given by the natural label swaps, c1i = c13|3↔i and c2i = c1i|1↔2.

The rational terms follow the notation of ref. [81],

R5 = Rb5 +

∑Z2(12)×Z3(345)

Ra5 , (3.45)

where

Ra5 = −1

2〈1 2〉4 [3 4]2 [2 5] 〈2 3〉〈2 4〉

〈3 4〉2 〈2 5〉〈3 5〉〈4 5〉, Rb

5 = −〈1 2〉4 [3 4] [3 5] [4 5]

〈3 4〉〈3 5〉〈4 5〉 . (3.46)

The sum in eq. (3.45) corresponds to the composition of the two permutations of


1

2 3

41

2 3

4

Figure 3.4: The two-loop cubic diagrams appearing in the two-loop four-point N = 4 andN = 8 supergravity amplitudes.

negative-helicity legs 1 and 2 and the three cyclic permutations over the positive-

helicity legs 3, 4 and 5, giving six terms in total. (Results for general MHV amplitudes

may be found in ref. [81].)

Inserting the results from eq. (3.41) into eq. (3.8) immediately converts the results

we obtained for the matter multiplets into those for the N = 4, 5, 6 gravity multiplets

(the pure supergravities). For the N = 4 and N = 6 gravity multiplets these match

the results of ref. [81].

Thus we have succeeded in expressing the four- and five-point integrated ampli-

tudes ofN ≥ 4 supergravity amplitudes as simple linear combination of corresponding

gauge-theory ones. To generalize this construction to higher points, one would need

to find duality satisfying representations of m-point one-loop N = 4 super-Yang-Mills

amplitudes.

3.3.4 Comments on two loops

An interesting question is whether the same considerations hold at higher loops.

Consider the two-loop four-point amplitude ofN = 4 super-Yang-Mills theory [36,42]:

A2-loop4 (1, 2, 3, 4) = −g6stAtree

4 (1, 2, 3, 4)(cP

1234 s I2-loop,P4 (s, t) (3.47)

+ cP3421 s I

2-loop,P4 (s, u) + cNP

1234 s I2-loop,NP4 (s, t)

+ cNP3421 s I

2-loop,NP4 (s, u) + cyclic

),

where ‘+ cyclic’ instructs one to add the two cyclic permutations of (2,3,4) and

the integrals correspond to the scalar planar and nonplanar double-box diagrams

displayed in fig. 3.4. As at one loop, the color factor for each diagram is obtained by

dressing each cubic vertex with an fabc. It is then simple to check that all duality


relations (3.3) hold.

According to the double-copy prescription (3.5), we obtain the corresponding

N = 8 supergravity amplitude by replacing the color factor with a numerator fac-

tor,

cP1234 → is2tAtree(1, 2, 3, 4) , cNP

1234 → is2tAtree(1, 2, 3, 4) , (3.48)

including relabelings and then swapping the gauge coupling for the gravitational one.

Indeed, this gives the correct N = 8 supergravity amplitude, as already noted in

ref. [42].

As explained in section 3.2, generalized gauge invariance implies that we need

have only one of the two copies in a form manifestly satisfying the duality (3.3). The

color Jacobi identity allows us to express any four-point color factor of an adjoint

representation in terms of the ones in fig. 3.4 [25]. If the duality and double-copy

properties hold we should then be able to obtain integrated N ≥ 4 supergravity am-

plitudes starting from N ≤ 4 super-Yang-Mills theory and applying the replacement

rule (3.48). Indeed, in ref. [95], explicit expressions for the four-point two-loop N ≥ 4

supergravity amplitudes, including the finite terms, are obtained in this manner.

Two-loop supergravity amplitudes are UV finite and their IR behavior is given in

terms of the square of the one-loop amplitude [96,97]:

M(2-loop)4 (ε)/Mtree

4 =1

2

[M1-loop

4 (ε)/Mtree4

]2

+ finite . (3.49)

The amplitudes of ref. [95] satisfy this relation and the finite remainders are given in

a relatively simple form. These two-loop results then provide a rather nontrivial con-

firmation of the duality and double-copy properties for cases with less than maximal

supersymmetry.

3.4 Conclusions

The duality between color and kinematic numerators offers a powerful means

for obtaining loop-level gauge and gravity amplitudes and for understanding their


structure. A consequence of the duality conjecture is that complete amplitudes are

controlled by a set of master diagrams; once the numerators are known in a form that

makes the duality between color and kinematics manifest, all others are determined

from Jacobi-like relations. In this form we immediately obtain gravity integrands via

the double-copy relation.

In the present chapter, we used the duality to find examples where integrated

supergravity amplitudes are expressed directly as linear combinations of gauge-theory

amplitudes. In particular, we constructed the integrated four- and five-point one-loop

amplitudes ofN ≥ 4 supergravity directly from known gauge-theory amplitudes. This

construction was based on identifying representations ofN = 4 super-Yang-Mills four-

and five-point amplitudes that satisfy the duality. Because the relations are valid

in D dimensions, by using known D-dimensional forms of gauge-theory four-point

amplitudes we obtain corresponding ones for supergravity. The agreement of our

four- and five-point N ≥ 4 supergravity results with independent evaluations [80,81]

in D = 4 provides evidence in favor of these conjectures holding for less than maximal

supersymmetry. The two-loop results in ref. [95] provide further nontrivial evidence.

The examples we presented here are particularly simple because the numerator

factors of one copy of the gauge-theory amplitudes were independent of loop mo-

menta. In more general cases, we expect useful constraints to arise at the integrated

level. These constraints, for example, lead to KLT-like relations visible in box-integral

coefficients, such as those found in refs. [58,98]. It would be very interesting to further

explore relations between gravity and gauge theory after having carried out the loop

integration.

There are a number of other interesting related problems. It would of course be

important to unravel the underlying group-theoretic structure responsible for the du-

ality between color and kinematics. Some interesting progress has recently made for

self-dual field configurations and for MHV tree amplitudes, identifying an underlying

diffeomorphism Lie algebra [89]. Another key problem is to find better means for find-

ing representations that automatically satisfy the duality and double-copy properties.

Such general representations are known at tree level for any choice of helicities [85].

We would like to have similar constructions at loop level, instead of having to find


duality satisfying forms case by case. In particular, no examples have as yet been

constructed at loop level at six and higher points.

In summary, using the duality between color and kinematics we exposed a sur-

prising relation between integrated four- and five-point one-loop amplitudes of N ≥ 4

supergravity and those of gauge theory. We look forward to applying these ideas to

further unravel the structure of gauge and gravity loop amplitudes.

Chapter 4

Two-Loop Amplitudes

C. Boucher-Veronneau and L. J. Dixon, “N ≥ 4 Supergravity Amplitudes from Gauge

Theory at Two Loops,” JHEP 1112, 046 (2011).

4.1 Introduction

It is well known that pure Einstein gravity is ultraviolet (UV) divergent at two

loops [3]. This result, along with general power-counting arguments, has led to the

widespread belief that a UV finite pointlike theory of gravity cannot be constructed.

However, explicit calculations of scattering amplitudes in maximally supersymmet-

ric (N = 8) supergravity have displayed an ultraviolet behavior that is much better

than prior expectations, showing that the theory in four dimensions is finite up to

at least four loops. Furthermore, N = 8 supergravity exhibits the same UV be-

havior, when continued to higher spacetime dimensions, as does N = 4 super-Yang-

Mills (sYM) [40,44,90]. Surprising cancellations are also visible at lower loop or-

ders [11,12,41,42,58,64,92,99], and even at tree level where the amplitudes are nicely

behaved at large (complex) momenta [15,62–64,100].

In pure supergravity theories (where all states are related by supersymmetry to

the graviton) no counterterm can be constructed below three loops. This is because

the only possible two-loop counterterm, R3 ≡ RλρµνR

µνστR

στλρ , where Rµν

στ is the Rie-

mann tensor, generates non-zero four-graviton amplitudes with helicity assignment

79

CHAPTER 4. TWO-LOOP AMPLITUDES 80

(±,+,+,+) [5–7]. Such amplitudes are forbidden by the Ward identities for the

minimal N = 1 supersymmetry [8]. The counterterm denoted by R4 is allowed by su-

persymmetry and could appear at three loops [6,10]. However, as mentioned earlier,

N = 8 supergravity was found to be finite at this order [11,12]. It was recently under-

stood that the R4 counterterm is forbidden [16,101] by the nonlinear E7(7) symmetry

realized by the 70 scalars of the theory [67,102]. In fact, E7(7) should delay the di-

vergence in N = 8 supergravity to at least seven loops, where the first E7(7)-invariant

counterterm can be constructed [13,17,103]. Non-maximal (N < 8) supergravity does

not have this extra E7(7) symmetry, and may therefore diverge at only three loops in

four dimensions.

Recently, the constraints that the smaller duality symmetries of non-maximal

supergravities impose on potential counterterms have also been investigated [13,103].

In four dimensions, N = 6 supergravity is expected to be finite at three and four

loops, and N = 5 supergravity should be finite at three loops [13]. These results

still allow for a three-loop divergence in N ≤ 4 supergravities. In particular, for

N = 4 supergravity, although the volume of superspace vanishes on shell, it has been

argued that the usual three-loop R4 counterterm can appear [103]. The finiteness

results for N = 5, 6 could in principle be checked, and potential divergences for

N ≤ 4 investigated, via explicit three-loop amplitude calculations in non-maximal

supergravities. Because the same situation, in which the superspace volume vanishes

on shell, and yet a counterterm appears to be allowed, holds for N = 8 supergravity

at seven loops, as for N = 4 supergravity at three loops, this latter case may be of

particular interest.

On the other hand, relatively few loop amplitudes have been computed for any

non-maximal supergravities. At one loop, the four-point amplitudes with N ≤ 8

supersymmetries were presented in ref. [80,94], while the N = 6 supergravity all-

point maximally-helicity-violating (MHV) and six-point non-MHV amplitudes were

first obtained in ref. [81]. The N = 4 supergravity one-loop five-point amplitude

was also computed in refs. [81,104]. In the following, we present expressions for the

two-loop four-graviton amplitudes in N = 4, 5, 6 supergravity. The calculations were

performed using the gravity “squaring” relations [38,39], or double-copy property,


which follows from the color-kinematics, or Bern-Carrasco-Johansson (BCJ), duality

obeyed by gauge-theory amplitudes at the loop level [31].

The BCJ relations allow us to combine the N = 4 sYM amplitude [36] with the

N = 0, 1, 2 sYM amplitudes [105] in order to obtain the corresponding amplitudes

in supergravity. Although they have been tested now in several loop-level amplitude

computations [31,39–41], the underlying mechanism or symmetry behind the general

loop-level BCJ relations is still not well understood. (In the self-dual sector at tree

level, a diffeomorphism Lie algebra appears to play a key role. [89].) Therefore it is

important to validate results obtained using BCJ duality. We will verify the expected

infrared divergences and forward-scattering behavior for the two-loop amplitudes that

we compute.

This chapter is organized as follows. In section 4.2 we review BCJ duality and

the squaring relations for gravity. In section 4.3 we illustrate the method for N = 8

supergravity at two loops. In section 4.4 we present our main formula for the two-loop

amplitudes in N = 4, 5, 6 supergravity. In section 4.5 we expand the (dimensionally

regulated) amplitudes for D = 4−2ε around ε = 0. We discuss the infrared (IR) pole

structure, which agrees with general expectations, thus providing a cross check on

the construction. We present the finite remainders in the two independent kinematic

channels. In section 4.6 we examine the behavior of the amplitudes in the limit

of forward scattering. In section 4.7, we present our conclusions and suggestions

for future research directions. Appendix D provides some one-loop results that are

required for extracting the two-loop finite remainders.

4.2 Review of the BCJ duality and squaring relations

We now briefly review BCJ duality and the gravity squaring relations that follow

from it. For a more complete treatment see, for example, the recent reviews [33,69].

Here, we will focus solely on applications to loop amplitudes.

We can write any m-point L-loop-level gauge-theory amplitude, where all particles


are in the adjoint representation, as

A(L)m = iL gm−2+2L

∑j

∫ L∏l=1

dDpl(2π)D

1

Sj

njcj∏αjp2αj

, (4.1)

where g is the gauge coupling. The sum runs over the set of distinct m-point L-loop

graphs, labeled by j, with only cubic vertices, corresponding to the diagrams of a φ3

theory. The product in the denominator runs over all Feynman propagators of each

cubic diagram. The integrals are over {pµl }, a set of L independent D-dimensional

loop momenta. The ci are the color factors, obtained by dressing every three-vertex

with a structure constant, defined by fabc = i√

2fabc = Tr([T a, T b]T c

). The nj are

kinematic numerator factors depending on momenta, polarizations and spinors. The

Sj are the internal symmetry factors for each diagram. The form of the amplitude

presented in eq. (4.1) can be obtained in various ways. For example, one can start

from covariant Feynman diagrams in Feynman gauge, where the contact terms are

absorbed into kinematic numerators using inverse propagators, i.e. by inserting factors

of 1 = p2αj/p2

αj.

Triplets (i, j, k) of color factors are related to each other by ci = cj + ck if their

corresponding graphs are identical, except for a region containing (in turn for i, j, k)

the three cubic four-point graphs that exist at tree level. The relation holds because

the products of two fabc structure constants corresponding to the four-point tree

graphs satisfy the Jacobi identity

fabe f cde = face f bde + fade f cbe , (4.2)

and the remaining structure constant factors in the triplet of graphs are identical. The

relations ci = cj + ck mean that the representation (4.1) is not unique; terms can be

shuffled from one graph to others, in a kind of generalized gauge transformation [39].

A representation (4.1) is said to satisfy the BCJ duality if the three associated

kinematic numerators are also related via Jacobi identities. Namely, we must have:

ci = cj + ck ⇒ ni = nj + nk , (4.3)


where the left-hand side follows directly from group theory, while the right-hand side

is the highly non-trivial requirement of the duality. Moreover, we demand that the

numerator factors have the same antisymmetry property as the color factors under

the interchange of two legs attached to a cubic vertex,

ci → −ci ⇒ ni → −ni . (4.4)

The relations (4.3) were found long ago for the case of four-point tree amplitudes [32];

the idea that the relations should hold for arbitrary amplitudes is more recent [31,39].

As remarked earlier, the representation (4.1) is not unique. Work is often required

in order to find a BCJ-satisfying representation of a given amplitude in a particular

gauge theory. At loop level, such representations were found initially at four points

through three loops for N = 4 sYM, and through two loops for identical-helicity pure

Yang-Mills amplitudes [39]. A BCJ-satisfying representation was recently obtained

at five points through three loops in N = 4 sYM [41]. Very recently, a four-point

four-loop representation was found in the same theory [40].

As a remarkable consequence of the BCJ duality, one can combine two gauge-

theory amplitudes in the form (4.1), in order to obtain a gravity amplitude, as long

as one of the two gauge-theory representations manifestly satisfies the duality [38,39].

We have,

M(L)m = iL+1

(κ2

)m−2+2L ∑j

∫ L∏l=1

dDpl(2π)D

1

Sj

njnj∏αjp2αj

, (4.5)

where either the nj or the nj must satisfy eqs. (4.3) and (4.4). Here κ is the gravita-

tional coupling constant, which is related to Newton’s constant GN and the Planck

mass MPlanck by κ2 = 32πGN = 32π/M2Planck. The proof of eq. (4.5) at tree level is

inductive, and uses on-shell recursion relations [49,50] for the gauge and gravity the-

ories, which are based on the same complex momentum shift [38]. The extrapolation

to loop level is based on reconstructing loop amplitudes from tree amplitudes using

(generalized) unitarity.

The relations (4.5) are similar in spirit to the KLT relations [43]. Both types of

relations express gravity amplitudes as the “square” of gauge-theory amplitudes, or


1

2 3

41

2 3

4

a

b

Figure 4.1: The planar and nonplanar cubic diagrams at two loops. The marked (colored)propagators in the planar diagram are used in the text to describe different color andkinematic Jacobi identities.

more generally as the product of two different types of gauge-theory amplitudes, as

the ni and nj numerator factors may come from two different Yang-Mills theories.

However, the KLT relations only hold at tree level, which means that at loop level

they can only be used on the (generalized) unitarity cuts. Although the gravity cuts

can be completely determined by the KLT relations in terms of local Yang-Mills

integrands, the gravity integrand found in this way is not manifestly local. That

is, it does not manifestly have the form of numerator factors multiplied by scalar

propagators for some set of φ3 graphs. Reconstructing a local representation can be

a significant task [11,12,44].

In contrast, eq. (4.5) is a loop-level relation, and furnishes directly a local integrand

for gravity. Most of the applications of this formula to date have been to maximal

N = 8 supergravity, viewed as the tensor product of two copies of maximal N = 4

super-Yang-Mills theory. The squaring relations were shown to reproduce the N = 8

supergravity four-point amplitudes through four loops [39,40] and the five-point am-

plitudes through two loops [41]. Quite recently, in the first loop-level applications for

N < 8, the one-loop four- and five-point N ≤ 8 supergravity amplitudes were shown

to satisfy the double-copy property [104]. In this chapter, we would like to extend

this kind of analysis for N < 8 supergravity to two loops. First, however, we briefly

review the N = 8 case.


4.3 Two-loop N = 8 supergravity

In this section we review the construction of the two-loop four-graviton ampli-

tude in N = 8 supergravity based on squaring relations, as preparation for a similar

construction for N = 4, 5, 6 supergravity in the next section.

As mentioned previously, a manifestly BCJ-satisfying representation of the four-

gluon N = 4 sYM amplitude is known at two loops [36,39],

A(2)4 (1, 2, 3, 4) = −g6stAtree

4 (1, 2, 3, 4)(c

(P)1234 s I(P)

4 (s, t) + c(P)3421 s I(P)

4 (s, u) (4.6)

+ c(NP)1234 s I(NP)

4 (s, t) + c(NP)3421 s I(NP)

4 (s, u) + cyclic),

where s, t, u are the usual Mandelstam invariants (s = (k1 + k2)2, t = (k2 + k3)2,

u = (k1 + k3)2) and “+ cyclic” instructs one to add the two cyclic permutations of

(2,3,4). The tree-level partial amplitude is

Atree4 (1, 2, 3, 4) = i

〈j k〉4〈1 2〉〈2 3〉〈3 4〉〈4 1〉 , (4.7)

where j and k label the two negative-helicity gluons. The two-loop planar and non-

planar scalar double-box integrals are, respectively,

I(P)4 (s, t)=

∫dDp

(2π)DdDq

(2π)D1

p2 (p− k1)2 (p− k1 − k2)2 (p+ q)2q2 (q − k4)2 (q − k3 − k4)2,

I(NP)4 (s, t)=

∫dDp

(2π)DdDq

(2π)D1

p2 (p− k2)2 (p+ q)2 (p+ q + k1)2 q2 (q − k3)2 (q − k3 − k4)2,

and they are depicted in fig. 4.1. The color factors c(P,NP)ijkl are obtained by dressing each

vertex of the associated diagram with a factor of fabc, and each internal line with a δab.

All helicity information is encoded in the prefactor stAtree4 (1, 2, 3, 4), which is invariant

under all permutations, thanks to a Ward identity for N = 4 supersymmetry.


= +

1

2 3

4 1

2 3

4 1

2 3

4

a a

a

Figure 4.2: Two-loop diagrams related by a Jacobi identity. The Jacobi identity is applied tothe four-point tree-level subdiagram that contains the (light blue) intermediate line markeda. The rest of the diagram is unchanged.

Comparing eqs. (4.1) and (4.6) we can identify the numerators as

n(P)1234 = n

(P)3421 = n

(NP)1234 = n

(NP)3421 = s× stAtree

4 (1, 2, 3, 4) ,

n(P)1342 = n

(P)4231 = n

(NP)1342 = n

(NP)4231 = u× stAtree

4 (1, 2, 3, 4) ,

n(P)1423 = n

(P)2341 = n

(NP)1423 = n

(NP)2341 = t× stAtree

4 (1, 2, 3, 4) . (4.8)

It is easy to see that the two-loop expression (4.6) satisfies the duality [39]. For

instance, let’s look at the diagrams related by a Jacobi identity applied to a four-

point tree-level subdiagram of the planar double-box graph on the left-hand side

of fig. 4.1. The tree subdiagram is the one whose intermediate propagator is the light-

blue line marked a in the figure. We replace the “s-channel” tree subdiagram with

the corresponding t- and u-channel tree subdiagrams, by appropriately permuting

the attachments of line a to the rest of the graph. This Jacobi identity is illustrated

in fig. 4.2. Because the N = 4 sYM diagrams with triangle one-loop subdiagrams

all have vanishing coefficients in eq. (4.6), the duality (4.3) requires the equality of

the planar and nonplanar numerator factors, n(P)1234 = n

(NP)1234 . Similarly, applying a

Jacobi identity to the red propagator marked b in the planar double-box diagram

in fig. 4.1, we find two graphs, one of which again contains a vanishing triangle

subgraph. Therefore the numerator of the planar box graph should be symmetric

under the exchange of legs 1 and 2, or equivalently n(P)1234 = n

(P)3421. Looking at eq. (4.8),

we see that these two conditions are satisfied.

Having verified that eq. (4.6) satisfies the BCJ relations, we may combine two

copies of (4.6) following prescription (4.5) to obtain the two-loop four-graviton N = 8


amplitude. We obtain

M(2)4 (1, 2, 3, 4) = −i

(κ2

)6

[stAtree4 (1, 2, 3, 4)]2

(s2 I(P)

4 (s, t) + s2 I(P)4 (s, u)

+ s2 I(NP)4 (s, t) + s2 I(NP)

4 (s, u) + cyclic), (4.9)

which is precisely the known result [42]. We also recall that the four-graviton and

four-gluon tree-level partial amplitudes are related to each other by

stuM tree4 = −i [stAtree

4 (1, 2, 3, 4)]2 . (4.10)

4.4 Two-Loop 4 ≤ N < 8 Supergravity

helicity 0 +1/2 +1 +3/2 +2

N = 8 supergravity 70 56 28 8 1N = 6 supergravity 30 26 16 6 1N = 5 supergravity 10 11 10 5 1N = 4 supergravity 2 4 6 4 1

N = 4 sYM 6 4 1N = 2 sYM 2 2 1N = 1 sYM 1 1N = 0 sYM 1

Table 4.1: State multiplicity as a function of helicity for relevant supersymmetric multipletsin pure supergravities and super-Yang-Mills theories. By CPT invariance, the multiplicityfor helicity −h is the same as that shown for h.

Now we move to the main subject of this chapter, the construction of the two-

loop four-graviton amplitudes for N = 4, 5, 6 supergravity. As we mentioned earlier,

only one of the two gauge-theory amplitudes entering the double-copy formula (4.5)

needs to satisfy the BCJ duality. We will combine the duality-satisfying N = 4 sYM

amplitude (4.6) with four-gluon amplitudes for N ≡ NYM = 0, 1, 2 sYM, in order

to obtain the corresponding two-loop four-graviton amplitudes in supergravities with


N = 4 + NYM = 4, 5, 6. Looking at the multiplicities of states for various super-

gravities and super-Yang-Mills theories in table 4.1, we can see that at the level of

counting states,



N = 4 supergravity : (N = 4 sYM)× (N = 0 sYM) , (4.11)

where N = 0 sYM refers to pure Yang-Mills theory with only gluons. Because the

gauge theories with N < 4 supersymmetry are consistent truncations of maximal

N = 4 sYM, and similarly on the gravity side, these equivalences also hold at the

level of amplitudes, through either the KLT relations (at tree level) or the double-

copy relations (4.5).

In ref. [25], it was shown that one could write a color decomposition of any one-

loop full-color all-adjoint gauge-theory amplitude in terms of color factors called “ring

diagrams”. The diagrammatic representation of these color factors have all the ex-

ternal legs connected directly to the loop. Other conceivable color factors, in which

nontrivial trees are attached to the loop, can be removed systematically by using

Jacobi identities, in favor of ring graphs with different cyclic orderings of the external

legs. This decomposition is independent of the (adjoint) particle content in the loop.

In the same way, we can use the Jacobi identities at two loops to rewrite any full-color

four-gluon amplitude in a theory with only adjoint particles, in terms of only the color

factors c(P)1234 and c

(NP)1234 of the diagrams of fig. 4.1 (plus permutations).

For super-Yang-Mills theory with N = NYM supersymmetries, we write

A(2)NYM

(1, 2, 3, 4) (4.12)

= −g6(c

(P)1234A

(P)1234,NYM

+ c(P)3421A

(P)3421,NYM

+ c(NP)1234 A

(NP)1234,NYM

+ c(NP)3421 A

(NP)3421,NYM

+ c(P)1342A

(P)1342,NYM

+ c(P)4231A

(P)4231,NYM

+ c(NP)1342 A

(NP)1342,NYM

+ c(NP)4231 A

(NP)4231,NYM

+ c(P)1423A

(P)1423,NYM

+ c(P)2341A

(P)2341,NYM

+ c(NP)1423 A

(NP)1423,NYM

+ c(NP)2341 A

(NP)2341,NYM

),

where A(P)1234 is the integrated color-ordered subamplitude associated with the color


factor c(P)1234. For example, for the N = 4 sYM representation (4.6), we read off

A(P)1234,NYM=4 = stAtree

4 (1, 2, 3, 4)× s I(P)4 (s, t) .

Normally, to implement the double-copy formula (4.5), we would need to have a

representation for the integrand of the gauge-theory amplitudes, in particular for the

N = 0, 1, 2 sYM amplitudes we are combining with those for N = 4 sYM. However,

at two loops the numerator factors for N = 4 sYM have no dependence on the loop

momenta. The same feature holds for the one-loop four- and five-point amplitudes

studied in ref. [104]. Therefore, just as in those cases, we can remove the N = 4 sYM

numerator factors from the loop integrals in eq. (4.5). Using eq. (4.8) for the N = 4

sYM numerator factors, we obtain the remarkably simple general formula,

M(2)NYM+4(1, 2, 3, 4) = −i

(κ2

)6

stAtree4 (1, 2, 3, 4)

×(sA

(P)1234,NYM

+ sA(P)3421,NYM

+ sA(NP)1234,NYM

+ sA(NP)3421,NYM

+uA(P)1342,NYM

+ uA(P)4231,NYM

+ uA(NP)1342,NYM

+ uA(NP)4231,NYM

+ t A(P)1423,NYM

+ t A(P)2341,NYM

+ t A(NP)1423,NYM

+ t A(NP)2341,NYM

). (4.13)

In summary, we obtain the N = 4, 5, 6 supergravity amplitudes by first expressing the

N = 0, 1, 2 sYM helicity amplitudes from ref. [105] in terms of the color basis (4.12)1.

We then replace g6 → i(κ/2)6 and perform the following additional replacements

(plus their relabelings):

c(P)1234 → stAtree

4 (1, 2, 3, 4)× s, c(NP)1234 → stAtree

4 (1, 2, 3, 4)× s. (4.14)

Because stAtree4 (1, 2, 3, 4) is permutation-invariant, only the single factors of s, t, u

persist inside the parentheses in eq. (4.13).

In order to preserve supersymmetry, we use the four-dimensional helicity variant

of dimensional regularization [77] for both copies of the gauge-theory amplitudes.

The results (4.13) can be expressed in terms of master integrals for the two-loop

1We thank Zvi Bern for providing us with the expressions in this format.


planar and nonplanar double-box topologies, plus various other integrals with fewer

propagators present. However, in this form the results are rather lengthy. Instead of

presenting them here, we expand the dimensionally-regulated results, for D = 4− 2ε,

around ε = 0, as discussed in the next section.

4.5 Infrared poles and finite remainders

At two loops, all pure supergravity amplitudes are ultraviolet finite [5–7]. There-

fore all of their divergences are infrared in nature, either soft or possibly collinear. As

two massless external particles become collinear, gravitational tree amplitudes have

singularities only in phase, not in magnitude. The same universal “splitting ampli-

tude” that controls the phase behavior governs loop amplitudes as well as tree am-

plitudes [58]. Correspondingly, there are no virtual divergences from purely collinear

regions of integration [106]. Soft divergences were studied long ago and found to

exponentiate [107]. More recent, explicit analyses can be found in refs. [106,108,109].

At one loop, the IR pole behavior is [80,94,96,107,110],

M(1)4 =

( κ8π

)2 2

ε

(s ln(−s) + t ln(−t) + u ln(−u)

)Mtree

4 + O(ε0). (4.15)

At L loops, the leading divergence is at order 1/εL. We first checked that the leading

divergence of our two-loopN = 4, 5, 6 supergravity amplitudes is indeed at order 1/ε2.

Moreover, the exponentiation of soft divergences implies that the full two-loop IR

behavior can be expressed in terms of the one-loop amplitude as follows:

M(2)4 (ε)

Mtree4

=1

2

[M(1)4 (ε)

Mtree4

]2

+( κ

8π

)4

F(2)4 + O(ε) , (4.16)

where F(2)4 is the finite remainder in the limit ε → 0. This infrared behavior was

checked explicitly for the four-point N = 8 supergravity amplitude [96,110], and was

conjectured to hold for all supersymmetric gravity amplitudes [109]. We have checked

that our expressions indeed satisfy eq. (4.16). We remark that the lack of any addi-

tional (ultraviolet) poles in ε confirms the absence of UV divergences for N = 4, 5, 6


supergravity in four dimensions at two loops [5–7].

In order to verify eq. (4.16) and extract F(2)4 , we need the O(ε0) and O(ε1) co-

efficients in the expansion of the corresponding one-loop amplitude M(1)4 . That is

because M(1)4 appears squared in eq. (4.16), and the 1/ε pole in eq. (4.15) can mul-

tiply the O(ε1) coefficient to generate a finite term. We give the required one-loop

expansions in appendix D.

Next we present the finite remainders F(2)4 for the different theories under consid-

eration. It is convenient to express the remainders for N < 8 supergravity in terms

of the N = 8 remainder plus an additional term. The result for N = 8 supergravity

was first presented in refs. [96,110]. We always consider the helicity configuration

(1−, 2−, 3+, 4+). There are three separate physical kinematic regions: the s channel,

with s > 0 and t, u < 0; the t channel (t > 0 and s, u < 0); and the u channel (u > 0

and s, t < 0). The s channel is singled out by the fact that it has identical-helicity

incoming gravitons. For all the supergravity theories, the (1−, 2−, 3+, 4+) helicity

configuration chosen is symmetric under 3↔ 4. Therefore we do not have to present

results separately for the u channel; they can be obtained from the t-channel results

by relabeling t ↔ u. In the case of N = 8 supergravity, an N = 8 supersymmetric

Ward identity implies that the results in the t channel (normalized by the tree am-

plitude) can be obtained simply by relabeling s ↔ t. For N < 8, this property no

longer holds, and we will have to quote the s- and t-channel results separately.

The N = 8 finite remainder was expressed in refs. [96,110] partly in terms of

Nielsen polylogarithms Sn,p(x). Here we give a representation similar to ref. [96],

and a second representation entirely in terms of classical polylogarithms Lin, for

consistency with the forms we present below for N < 8. The finite remainder is

F(2),N=84

∣∣∣s−channel

= 8

{t u[f1

(−ts

)+ f1

(−us

)]+ s u

[f2

(−ts

)+ f3

(−ts

)]+s t

[f2

(−us

)+ f3

(−us

)]}, (4.17)


where

f1(x) = S1,3(1− x) + ζ4 +1

24ln4 x+ iπ

[−S1,2(1− x) + ζ3 +

1

6ln3 x

]= − Li4(x) + ln xLi3(x)− 1

2ln2 xLi2(x) +

1

24ln4 x− 1

6ln3 x ln(1− x)

+ 2 ζ4 + iπ

[Li3(x)− lnxLi2(x) +

1

6ln3 x− 1

2ln2 x ln(1− x)

], (4.18)

f2(x) = S1,3

(1− 1

x

)+ ζ4 +

1

24ln4 x+ iπ

[S1,2

(1− 1

x

)− ζ3 +

1

6ln3 x

]= Li4(x)− lnxLi3(x) +

1

2ln2 xLi2(x) +

1

6ln3 x ln(1− x)

− iπ[Li3(x)− lnxLi2(x)− 1

2ln2 x ln(1− x)

], (4.19)

and

f3(x) = Li4(y)− ln(−y) Li3(y) +1

2

[ln2(−y) + π2

]Li2(y)

+1

6

[ln3(−y) + 3 π2 ln(−y)− 2 i π3

]ln(1− y) , (4.20)

with y = −x/(1 − x). The N = 8 supergravity remainder in the t channel is given

simply by relabeling the s-channel result, exchanging s and t:

F(2),N=84 (s, t, u)

∣∣∣t−channel

= F(2),N=84 (t, s, u)


. (4.21)

It was noted previously [96,110] that F(2),N=84 has a uniform maximal transcen-

dentality. That is, all functions appearing are degree-four combinations of polyloga-

rithms, logarithms, and transcendental constants. A pure function is a function with

a uniform degree of transcendentality, having only constants (rational numbers) mul-

tiplying the combinations of polylogarithms, etc. A pure function f has a well-defined

symbol, S(f), which can be obtained by an iterated differentiation procedure [111–

114]. In the representation (4.17), the functions f1, f2 and f3 are pure functions with


very simple, one-term symbols:

S(f1) = x⊗ x⊗ x⊗ x

1− x , (4.22)

S(f2) = x⊗ x⊗ x⊗ (1− x) , (4.23)

S(f3) = − x

1− x ⊗x

1− x ⊗x

1− x ⊗ (1− x) . (4.24)

We have shuffled terms slightly with respect to refs. [96,110] in order to make this

property manifest. For example, our function f1(x) is very similar to the function

h(t, s, u) given in eq. (2.26) of ref. [96], after multiplying it by 1/8 and setting −s/t→x. However, eq. (4.18) contains a term 1

24ln4 x in place of the term 1

24ln4(1 − x) in

h/8. Because only the sum f1(x) + f1(1−x) appears in eq. (4.17), this swap of terms

does not affect the total, but it does ensure that the branch cut origins are in the

same place for all terms in f1, and correspondingly it simplifies the symbol S(f1). The

functions f2 and f3 are related to f1 by crossing: f2 by the map x → 1/x (s ↔ t),

and f3 by the map x→ −(1− x)/x (s→ t→ u→ s).

Curiously, the symbol of f1 obeys a certain “final entry” condition recently ob-

served to appear in the context of the remainder function for planar N = 4 sYM

amplitudes or Wilson loops [115,116]. Furthermore, f1(x) obeys the generalization of

this condition to functions, namely

df1

dx=

p(x)

x(1− x), (4.25)

where p(x) is also a pure function, in this case

p(x) =1

6ln3 x+

iπ

2ln2 x . (4.26)

When the finite remainder of the four-graviton amplitude in N = 8 supergravity

becomes available at three loops (for example by computing the integrals for one

of the three available expressions for it [11,12,39]), it will be very interesting to see

whether it can also be expressed in terms of pure functions of degree six with simple

symbols. Perhaps the functions will even obey a relation like eq. (4.25).


We return now to two loops and N < 8 supergravity. We present the finite

remainder for N = 6 supergravity, first in the s channel:

F(2),N=64


= F(2),N=84


+ t u

[f6,s

(−ts

)+ f6,s

(−us

)], (4.27)

where

f6,s(x) = f6,s;4(x) + f6,s;3(x) (4.28)

gives the decomposition into a degree-four function,

f6,s;4(x) = 20 Li4(x)− 4 (1− x) Li4

( −x1− x

)− 12 lnxLi3(x) + 4 ln2 xLi2(x)

− 4 (1− x) ln

(x

1− x

)Li3(x)− 1

4x (1− x)

[ln4

(x

1− x

)+ π4

]+π2

2

[x lnx+ (1− x) ln(1− x)

]2

+2

3x ln4 x

− 2

3lnx ln(1− x)

[(1 + x) ln2 x− 9

4lnx ln(1− x)

]− 4 ζ2

[xLi2(x) + 2 lnx ln(1− x)

]− 41

2ζ4

+ i π

[−12 Li3(x) + 8 lnx (Li2(x) + ζ2)− 2

3(1− 2x) ln x (ln2 x+ π2)

+ 4 (1− x) ln2 x ln(1− x)

], (4.29)

and a degree-three one,

f6,s;3(x) = −4

3x lnx

[ln2 x+ 3 ln2(1− x) + π2

]+ 8

[Li3(x)− lnxLi2(x)− ζ3

2+ iπ ζ2

]. (4.30)

It has been observed [110] that at one loop the four-graviton amplitude in N = 6

supergravity has maximal transcendentality (degree two). This result extends to

one-loop amplitudes with more gravitons, thanks to the absence of bubble inte-

grals [81,104]. However, the degree-three nature of eq. (4.30) shows that this property

is broken at two loops. The breaking comes from both the two-loop amplitudeM(2)4 ,


but also from the square of the one-loop amplitudeM(1)4 , which has to be subtracted

in eq. (4.16). As can be seen from eqs. (D.5) and (D.8), the one-loop N = 6 ampli-

tude has degree-two terms as well as degree-three terms at O(ε); the former terms

multiply the 1/ε degree-one terms from the IR pole shown in eq. (4.15) to generate

degree-three contributions to eq. (4.30). On the other hand, these contributions are

purely logarithmic; the polylogarithmic terms in eq. (4.30) can be traced to M(2)4 .

The complexity of the expressions (4.29) and (4.30), in terms of their power-law de-

pendence on x, makes it unprofitable to try to separate the N < 8 finite remainders

into pure functions and to compute their symbols.

Because of the helicity assignment (1−, 2−, 3+, 4+), the s-channel remainder is

always symmetric under t↔ u. However, in the t channel there is no such symmetry.

The N = 6 remainder in this channel is,

F(2),N=64


= F(2),N=84


+ t u

[f6,t;4

(−ut

)+ f6,t;3

(−ut

)], (4.31)

where the degree-four part is

f6,t;4(x) = −20 Li4(1− x)− 20 Li4

(1− x−x

)− 4

1 + x

1− x(

Li4(x)− ζ4

)+ 16 ln xLi3(1− x)− 12 ln(1− x)

(Li3(x)− ζ3

)+ 4

4− 3x

1− x lnx[Li3(x)− ζ3 +

1

2ln(1− x) ln2 x

]+ 4 ln x

(lnx− 2 ln(1− x)

)Li2(1− x) + 4 ζ2

7− 5x

1− x Li2(1− x)

− 1

6

5− 8x

(1− x)2ln4 x− 6 ln2(1− x) ln2 x− 2 ζ2

13− 19x+ 12x2

(1− x)2ln2 x

+ 16 ζ21− 2x

1− x lnx ln(1− x) + i π

[16 Li3(1− x)

+4

1− x(

Li3(x)− ζ3

)− 8 ln(1− x) Li2(1− x)

+2

3

1− 2x+ 4x2

(1− x)2ln3 x+ 2

2 + x

1− x ln2 x ln(1− x)

− 2 lnx ln2(1− x)− 4 ζ24− x1− x lnx

], (4.32)


and the degree-three part is

f6,t;3(x) =4

3

x

1− x lnx(

ln2 x− 2 π2)− 8

(Li3(x)− lnxLi2(x)

)+ 4 ln(1− x)

(ln2 x− 4 ζ2

)+ 4 i π

[x

1− x ln2 x− 2(

Li2(1− x) + ζ2

)].

(4.33)

In the s channel, the finite remainder for N = 5 supergravity at two loops is given

by,

F(2),N=54


= F(2),N=84


+ t u

[f5,s

(−ts

)+ f5,s

(−us

)], (4.34)

where

f5,s(x) = f5,s;4(x) + f5,s;3(x) + f5,s;2(x) (4.35)


f5,s;4(x) = −12

{(1− x)

[Li4

( −x1− x

)− ζ2 Li2(x)

]− 2

(1 + x (1− x)

)Li4(x)

+[(2− x2) ln x− (1− x)2 ln(1− x)

]Li3(x)− 1

2ln2 xLi2(x)

}− 1

16x (1− x)

[5 ln4

(x

1− x

)+ 34 π2 ln2

(x

1− x

)]+

1

2x ln4 x

− (1− x) ln3 x ln(1− x) +3

4

(3− 4x (1− x)

)ln2 x ln2(1− x)

+π2

2

[−(1− x) (3− 2x) ln2 x+

3

2ln2

(x

1− x

)]− 3

8ζ4

(72 + 323 x (1− x)

)+ i π

{−12

[(1 + 2 x (1− x)

)Li3(x)− lnxLi2(x)

]− (1− 2x) (1− x) lnx

(ln2 x+ π2

)+ 3

(2 (1− x)2 + x

)ln2 x ln(1− x) + 2 π2 lnx

}, (4.36)


a degree-three function,

f5,s;3(x) = 12

{(1 + x2)

[Li3(x)− lnxLi2(x)

]− 1

2

(1− x (1− x)

)ln2 x ln(1− x)

}− 2x (1− x) ln3 x− 4π2 x lnx− 12 ζ3 + 12π i

[(1− x) Li2(x)

+1

4

(x lnx+ (1− x) ln(1− x)

)2

+ζ2

2

(2− 3x (1− x)

)], (4.37)

and a degree-two function,

f5,s;2(x)=−3

[(x lnx+ (1− x) ln(1− x)

)2

− π2 x (1− x) + 4 π i x lnx

]. (4.38)

The N = 5 remainder function in the t channel is,

F(2),N=54


= F(2),N=84


+ t u

[f5,t;4

(−ut

)+ f5,t;3

(−ut

)+ f5,t;2

(−ut

)],

(4.39)



f5,t;4(x) = 12

{−1 + x

1− x(

Li4(x)− ζ4

)− 2

(1− x

(1− x)2

)[Li4(1− x) + Li4

(1− x−x

)]−[(

1− 2x

(1− x)2

)ln(1− x)−

(2− x2

(1− x)2

)lnx

](Li3(x)− ζ3

)+ 2 ln xLi3(1− x)− 1

2lnx

(lnx− 2 ln(1− x)

)Li2(x)

+ 2 ζ22− x1− x Li2(1− x)− 8− 21x

96 (1− x)2ln4 x

+(1− 2x)(5− x)

12 (1− x)2ln3 x ln(1− x) +

1

8

(3 +

4x

(1− x)2

)ln2 x ln2(1− x)

− ζ2

4

10− 12x+ 11x2

(1− x)2ln2 x+ ζ2

1− 5x+ 2x2

(1− x)2lnx ln(1− x)

+ i π

[Li3(x)− ζ3

(1− x)2+ 2 Li3(1− x)− ln(1− x) Li2(1− x)

+3

8

x2

(1− x)2ln3 x+

1

24

2 + x

1− x ln2 x(

lnx+ 6 ln(1− x))

− 1

4lnx ln2(1− x)− ζ2

2

4− x1− x lnx

]}, (4.40)


the degree-three part is

f5,t;3(x) = 12

{1 + x

1− x[Li3(1− x)− ln(1− x) Li2(1− x)− 1

2lnx ln2(1− x)

]−(

1 +x2

(1− x)2

)(Li3(x)− lnxLi2(x)

)+

x ln3 x

6 (1− x)2+

1

2ln2 x ln(1− x)

− ζ2

[x (1− 4x)

(1− x)2lnx+ ln(1− x)

]− ζ3

1− 2x

(1− x)2

+ i π

[−1 + (1− x)2

(1− x)2Li2(1− x)− 1

(1− x)2lnx ln(1− x)

+1

2

x

1− x(

ln2 x+ 2 ζ2

)]}, (4.41)

and the degree-two part is

f5,t;2(x) = −6

[(ln(1− x) +

x

1− x lnx)2

− π2 x

1− x

+2 i π

1− x(

ln(1− x) +x

1− x lnx)]

. (4.42)

The results for N = 4 supergravity are the lengthiest of all. In the s channel, the

finite remainder for N = 4 supergravity at two loops is given by,

F(2),N=44


= F(2),N=84


+ t u

[f4,s

(−ts

)+ f4,s

(−us

)], (4.43)

where

f4,s(x) = f4,s;4(x) + f4,s;3(x) + f4,s;2(x) + f4,s;1(x) + f4,s;0(x) (4.44)



f4,s;4(x) = 4(

9− 4x (1− x))

Li4(x)− 4 (8− 2x− 9x2 + 8x3) ln xLi3(x)

− 4 (1− x) (3 + 3x− 8x2)

[Li4

( −x1− x

)+ ζ2 Li2

( −x1− x

)− ln(1− x) Li3(x)

]+ 4

(2− x (1− x)

)lnx (lnx+ 2 i π) Li2(x)

− 4 i π(

5− 2x (1− x))

Li3(x)

+1

6x (4− 8x− 5x2 + 21x3 − 9x4 + 3x5)

×(

ln4 x− 4 ln3 x ln(1− x) + 2 π2 ln2 x+π4

2

)− 2

3(2− x (1− x)) (1− 3x)

(ln2 x (lnx− 6iπ) ln(1− x)

+ iπ lnx (ln2 x− π2))

+2

3i π x lnx

[(2− 13x+ 8x2) ln2 x

+ 3 (4 + 10x− 5x2) ln x ln(1− x)

+ (14 (1 + x2)− 19x)π2]

+1

2(2− x (1− x)) (1− x (1− x))2 lnx ln(1− x)

×(

3 ln x ln(1− x)− 2 π2)

− 2 ζ2 x (8− 16x+ 11x2) ln2 x− 3

2ζ4 (44− 17x (1− x)) , (4.45)


a degree-three function,

f4,s;3(x) = −(

53

6+ x2

)[Li3

( −x1− x

)− ln

(x

1− x

)Li2

( −x1− x

)]− 1

18(59− 12x2 + 8x3 + 54x4 + 36x (1− x)4)

× lnx[lnx

(lnx− 3 ln(1− x)

)+ π2

]−(

31

3− 12x+ 10x2

)ln2 x ln(1− x)

− i π[(1− 2x) ln2 x− 9x (1− x)

(lnx ln

(x

1− x

)+π2

2

)]+ζ2

3(59− 156x+ 132x2) (lnx+ i π)− 33 ζ3 x (1− x) , (4.46)

a degree-two function,

f4,s;2(x) = −6− 7x+ 4x2

2 (1− x)lnx (lnx+ 2 i π)

+

[3(

1 + x2 (1− x)2)− 13

3x (1− x)

] [lnx ln

(x

1− x

)+π2

2

]+

1

3ζ2 x

2(

6x2 − (1− x) (23− 24x)), (4.47)

a degree-one function,

f4,s;1(x) = −1

3x(

4 (1− x)2 − x (1− 2x))

(lnx+ i π) , (4.48)

and a rational part,

f4,s;0(x) = −1

4

(2 + x (1− x)

). (4.49)

The N = 4 remainder function in the t channel is,

F(2),N=44


= F(2),N=84


+ t u

[f4,t;4

(−ut

)+ f4,t;3

(−ut

)+ f4,t;2

(−ut

)+ f4,t;1

(−ut

)+ f4,t;0

(−ut

)], (4.50)



f4,t;4(x) = −4

(9 +

4x

(1− x)2

)[Li4(1− x) + Li4

(1− x−x

)+(

ln(1− x) +iπ

2

)(Li3(x)− ζ3

)+ ζ2 Li2(1− x)

]− 4

1 + x

1− x

(3− 8x

(1− x)2

)[Li4(x)− ζ4 −

iπ

2

(Li3(x)− ζ3

)]+ 4

8− 22x+ 11x2 − 5x3

(1− x)3

×[lnx

(Li3(x)− ζ3

)+ ζ2

(2 Li2(1− x) + ln x ln(1− x)

)]+ 4

(2 +

x

(1− x)2

){(2 ln(1− x) + 3 i π

)(Li3(x)− ζ3)

+(

ln2 x+ 4 ζ2

)Li2(1− x)

+ 2 (lnx+ i π)(

2 Li3(1− x)− ln(1− x) Li2(1− x))

+ i π

[(1

6+x (1− x2 + x3)

2 (1− x)4

)ln3 x

+2 + x

2 (1− x)ln2 x ln(1− x)− 1

2lnx ln2(1− x)

− ζ24− x1− x lnx

]}− 9− 48x+ 104x2 − 129x3 + 87x4 − 26x5

6 (1− x)6ln2 x (ln2 x− 4π2)

+2

3

23− 52x+ 49x2 − 17x3

(1− x)3ln3 x ln(1− x)

−(

11 +5x

(1− x)2

)ln2 x ln2(1− x)

− 4 ζ2x (14− 9x (1− x))

(1− x)3lnx ln(1− x)

− 2 ζ243− 71x+ 100x2 − 25x3

(1− x)3ln2 x ,

(4.51)


the degree-three part is

f4,t;3(x) = −(

56

3+

2x

(1− x)2

)[Li3(x)− (lnx+ i π) Li2(x)

− 2

3i π3 − 5

3π2 ln(1− x)

]+x (24− 15x+ 13x2 − 32x3 + 28x4)

9 (1− x)5lnx (lnx+ i π) (lnx+ 2 i π)

+

(1

3+ 2

5− 4x (1− x)

(1− x)2

)[ln(1− x)

((lnx+ i π)2 − 3 π2

)− 2 i π3

]− 2 + 10 x− x2

(1− x)2

[π2 (lnx− 2 ln(1− x))− i π (ln2 x+ π2)

]− 1 + x

1− x

[(lnx+ i π) ln(1− x) (ln(1− x) + 2 i π) +

iπ

1− x ln2 x

]+ 66 ζ3

x

(1− x)2, (4.52)

the degree-two part is

f4,t;2(x) = −6− 5x+ 3x2

2 (1− x)

(ln2 x− 2 ln(1− x) (lnx+ i π) + π2

)+

(3 +

x (13 (1− x)2 + 9x)

3 (1− x)4

)((lnx+ i π)2 + 2 ζ2

)+

3 (1 + x2)− 8x

2xln(1− x) (ln(1− x) + 2 i π)

+ ζ214 (1 + x2)− 3x

(1− x)2, (4.53)

the degree-one part is

f4,t;1(x) = −1

3

[ln

(x

1− x

)− 1 + x+ 4x2

(1− x)3(lnx+ i π)

], (4.54)

and the rational part is

f4,t;0(x) = −(2− x)(1− 2x)

2 (1− x)2. (4.55)


4.6 Forward-scattering limit of the amplitudes

We now inspect the behavior of the two-loop supergravity amplitudes in the limit

of small-angle, forward scattering, i.e. small momentum transfer at fixed center-of-

mass energy. In particular, we want to verify the contributions from matter exchange,

versus graviton exchange, in the forward-scattering limit. The results are sensitive

to the helicity configuration, or for fixed helicity configuration, to which invariant is

time-like and which of the two space-like invariants is becoming small.

We first consider configurations, or channels, for which the associated tree-level

amplitudes have a pole at small momentum transfer. These configurations are dom-

inated by the exchange of soft gravitons, and require helicity conservation along

the forward-going graviton line. (They also require helicity conservation along the

backward-going line, but this second condition follows automatically from the first

one, for the MHV amplitudes that we study.) To see the helicity conservation explic-

itly, we rewrite the tree amplitude as,

M tree4 (1−, 2−, 3+, 4+) = −is2

(1

t+

1

u

)[〈1 2〉[12]

[34]

〈3 4〉

]2

, (4.56)

where the quantity in brackets is a pure phase. Expanding eq. (4.56) for small t at

fixed s in the physical s channel (s > 0 kinematics), one gets a leading term of O(s2/t)

as t→ 0. Because the s-channel amplitude is symmetric under t↔ u, one could also

have taken the small u limit and gotten a pole-dominated behavior. However, in

the physical t channel, one has to take u small in order to conserve helicity at both

vertices. Then the leading tree-level behavior is O(t2/u) as u → 0. In contrast, the

limit of small s in the t channel violates helicity conservation, and the tree amplitude

is heavily power-law suppressed with respect to the dominant pole behavior, having

a leading term of O(s3/t2) as s→ 0.

Interestingly, in the helicity-conserving channels described above, the two-loop

remainders, F(2)4 , for N = 4, 5, 6, 8 supergravity amplitudes are all power-law sup-

pressed. The forward-scattering leading behavior is thus fully determined by the

square of the one-loop amplitude. Moreover, the dominant one-loop behavior is the


same for all 4 ≤ N ≤ 8 supergravity amplitudes. Namely, at one loop as t → 0 in

the s channel, we have

M(1)4 (ε)

Mtree4

=( κ

8π

)2

(−2πi) s

[1

ε+ ln

(s

−t

)+ε

2ln2

(s

−t

)]+ O(ε2, t) , (4.57)

and at two loops we have

M(2)4 (ε)

Mtree4

=1

2

[M(2)

4 (ε)

Mtree4

]2

+ O(ε, t) . (4.58)

Both equations hold for any number of supersymmetries. We also verified the analo-

gous equations in the limit u→ 0 in the physical t channel (t > 0 kinematics).

As discussed in refs. [117], in the physical s channel only the s-channel ladder

and crossed-ladder diagrams (shown in fig. 4.1 with s flowing horizontally) contribute

to the eikonal limit t → 0. The limit is dominated by graviton exchanges because

the coupling of a particle of spin J exchanged in the channel with small momentum

transfer is proportional to EJ , where E is the center-of-mass energy. The s-channel

ladder and crossed-ladder diagrams allow for the maximum number of attachments

of gravitons to a hard line (one with energy of order E). This property explains

why eqs. (4.57) and (4.58) are independent of the number of supersymmetries at high

energy. The possible Reggeization of gravity, discussed in ref. [118], remains an open

question. However, this issue cannot be resolved by studying forward-scattering or

eikonal limits. The t-channel ladder diagrams (obtained from fig. 4.1 by rotating by

90◦ or permuting 1→ 2→ 3→ 4→ 1), which should contribute to Reggeization, are

subleading by powers of t/s because they have fewer attachments to the high-energy

lines.

It is also interesting to consider the helicity-violating limit in which s → 0 for

t > 0 kinematics (u ' −t). As mentioned before, the associated tree-level amplitude is

power-suppressed in this limit with respect to the dominant pole behavior; its leading

behavior is O(s3/t2). In this limit, many of the finite-remainder expressions naively

appear to blow up (see for instance eq. (4.32) as x→ 1). However, one can check in

all cases that these spurious singularities cancel, and the leading behavior of the ratio


of the one- and two-loop amplitudes to the tree amplitude is of O(tL), L = 1, 2. Thus

the one- and two-loop amplitudes never have a power (1/s) enhancement over the

tree amplitude in the helicity-violating limit, but are of the same order in s. (There

is a ln(s) enhancement, but only in the pure N = 8 supergravity terms, not in any of

the matter contributions.)

4.7 Conclusions

In this chapter, we have computed the full four-graviton two-loop amplitudes

in N = 4, 5, 6 supergravity. As expected, their IR divergences can be expressed in

terms of the square of the corresponding one-loop amplitudes. The finite remainders

were presented in a simple form. We also noted that the finite remainder in N = 8

supergravity can be expressed in terms of permutations of a pure function f1(x)

possessing a simple, one-term symbol.

The N = 4, 5, 6 supergravity results were obtained using the double-copy property

of gravity, which is a consequence of the recently-conjectured BCJ duality. The former

property allowed us to combine the BCJ-satisfying N = 4 sYM representation with

knownN = 0, 1, 2 sYM gauge-theory amplitudes, in order to obtain the corresponding

supergravity amplitudes, including all loop integrations.

Our task was vastly simplified by the fact that both sets of Yang-Mills ampli-

tudes entering the double-copy formula were known, as well as by the lack of loop-

momentum dependence for the N = 4 sYM amplitudes in this case. As mentioned

in the introduction, generic N < 8 supergravity theories are expected to diverge at

three loops (but not N = 5 or 6 [13,103]), because the counterterm R4 is allowed

by supersymmetry. It would thus be very interesting to compute explicit three-loop

non-maximal supergravity amplitudes. If one computes in N ≥ 4 supergravity, then

one can use the double-copy formula, because a BCJ-satisfying form exists for one of

the two copies, namely the three-loop N = 4 sYM amplitude [39].

However, for the other gauge-theory copy, N < 4 sYM, the three-loop amplitudes

are not known. Full-color amplitudes (including nonplanar terms) are required, and

they should be known at the level of the integrand, because the BCJ form for the


three-loop N = 4 sYM amplitude contains loop-momentum dependence in its numer-

ator factors. BCJ duality for N < 4 sYM could help simplify these gauge-theory

calculations. For instance, for the three-loop four-point N = 4 sYM amplitude, the

duality reduced the computation of the full amplitude to the evaluation of the max-

imal cut [75] of a single diagram [39]. Non-maximal amplitude calculations are not

expected to be as simple, however. More powers of loop momentum will appear in

the numerator factors, and graphs containing triangle and bubble subgraphs will also

arise. It would be interesting nonetheless to investigate the simplifications that may

be provided by BCJ duality in these cases.

Appendix A

Six-point Gravity NMHV

Amplitude

In this appendix, we present the six-point NMHV gravity amplitude as expressed

using inverse soft. We will consider the amplitude M(1+, 2+, 3+, 4−, 5−, 6−) and its

BCFW representation by deforming the momenta of particles 3 and 4. There are 13

108

APPENDIX A. SIX-POINT GRAVITY NMHV AMPLITUDE 109

BCFW diagrams and using the techniques of Sec. 2.5 can be expressed as:

M(1+, 2+, 3+, 4−, 5−, 6−)

= G(4, 3+, 1)M(1′+, 2+, 4′−, 5−, 6−)

+ G(4, 3+, 2)M(1+, 2′+, 4′−, 5−, 6−)

+ G(4, 3+, 5)M(1+, 2+, 4′−, 5′−, 6−)

+ G(4, 3+, 6)M(1+, 2+, 4′−, 5−, 6′−)

+ G(3, 4−, 1)M(1′+, 2+, 3′+, 5−, 6−)

+ G(3, 4−, 2)M(1+, 2′+, 3′+, 5−, 6−)

+ G(3, 4−, 5)M(1+, 2+, 3′+, 5′−, 6−)

+ G(3, 4−, 6)M(1+, 2+, 3′+, 5−, 6′−)

+[G(3, 5−, 1)− G(3, 5−, 1′′)

]G(4, 3′+, 1′)M(1′′+, 2+, 4′′−, 6−)

+[G(3, 5−, 2)− G(3, 5−, 2′′)

]G(4, 3′+, 2′)M(1+, 2′′+, 4′′−, 6−)

+[G(3, 6−, 1)− G(3, 6−, 1′′)

]G(4, 3′+, 1′)M(1′′+, 2+, 4′′−, 5−)

+[G(3, 6−, 2)− G(3, 6−, 2′′)

]G(4, 3′+, 2′)M(1+, 2′′+, 4′′−, 5−)

+[G(3, 6−, 5)− G(3, 6−, 5′′)

]G(4, 3′+, 5′)M(1+, 2+, 4′′−, 5′′−) .

(A.1)

As in Sec. 2.5.1, primes indicate inverse soft momentum deformations with respect to

the last added particle (corresponding to the leftmost soft factor) and double primes

indicate inverse soft momentum deformations with respect to the first added particle

and then the second added particle.

Appendix B

One-loop five-point Yang-Mills

amplitudes

This appendix collects the five-point one-loop Yang-Mills amplitudes used to con-

struct the five-point supergravity amplitudes. The external states are gluons and all

amplitudes can be obtained from two configurations, (1−, 2−, 3+, 4+, 5+) and

(1−, 2+, 3−, 4+, 5+) using relabeling and parity. These results are from ref. [79] which

the reader is invited to consult for further details. The results are presented in the

four-dimension helicity (FDH) regularization scheme [77], which is known to preserve

supersymmetry at one loop.

The five-gluon color-ordered and coupling-stripped amplitudes with the N = 4,

N = 1 matter multiplet and a real scalar going around the loop can be expressed as:

A1-loopN=4 (1, 2, 3, 4, 5) = cΓV

gAtree5 ,

A1-loopN=1,mat.(1, 2, 3, 4, 5) = −cΓ(V fAtree

5 + iF f ) ,

A1-loopscalar (1, 2, 3, 4, 5) =

1

2cΓ(V sAtree

5 + iF s) , (B.1)

110

APPENDIX B. ONE-LOOP FIVE-POINT YANG-MILLS AMPLITUDES 111

where the tree amplitudes are

Atree5 (1−, 2−, 3+, 4+, 5+) =

i〈1 2〉4〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 1〉 ,

Atree5 (1−, 2+, 3−, 4+, 5+) =

i〈1 3〉4〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 1〉 . (B.2)

The function,

Vg = − 1

ε2

5∑j=1

(−sj,j+1)−ε +5∑j=1

ln

( −sj,j+1

−sj+1,j+2

)ln

(−sj+2,j−2

−sj−2,j−1

)+

5

6π2 . (B.3)

is independent of the helicity configuration. In contrast to ref. [79], we have set the

dimensional-regularization scale parameter, µ, to unity. For the (1−, 2−, 3+, 4+, 5+)

helicity configuration we have,

V f = −1

ε+

1

2[ln (−s23) + ln (−s51)]− 2 , V s = −1

3V f +

2

9,

F f = −1

2

〈1 2〉2 (〈2 3〉 [3 4] 〈4 1〉+ 〈2 4〉 [4 5] 〈5 1〉)〈2 3〉〈3 4〉〈4 5〉〈5 1〉

L0

(−s23−s51

)s51

,

F s = −1

3

[3 4] 〈4 1〉〈2 4〉 [4 5] (〈2 3〉 [3 4] 〈4 1〉+ 〈2 4〉 [4 5] 〈5 1〉)〈3 4〉〈4 5〉

L2

(−s23−s51

)s3

51

− 1

3F f

− 1

3

〈3 5〉 [3 5]3

[1 2] [2 3] 〈3 4〉〈4 5〉 [5 1]+

1

3

〈1 2〉 [3 5]2

[2 3] 〈3 4〉〈4 5〉 [5 1](B.4)

+1

6

〈1 2〉 [3 4] 〈4 1〉〈2 4〉 [4 5]

s23 〈3 4〉〈4 5〉 s51

,

and the corresponding functions for the (1−, 2+, 3−, 4+, 5+) helicity configuration,

V f = −1

ε+

1

2[ln (−s34) + ln (−s51)]− 2 , V s = −1

3V f +

2

9,

F f = −〈1 3〉2〈4 1〉[2 4]2

〈4 5〉〈5 1〉Ls1

(−s23−s51 ,

−s34−s51

)s2

51

+〈1 3〉2〈5 3〉[2 5]2

〈3 4〉〈4 5〉Ls1

(−s12−s34 ,

−s51−s34

)s2

34

− 1

2

〈1 3〉3(〈1 5〉 [5 2] 〈2 3〉 − 〈3 4〉 [4 2] 〈2 1〉)〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 1〉

L0

(−s34−s51

)s51

,

APPENDIX B. ONE-LOOP FIVE-POINT YANG-MILLS AMPLITUDES 112

F s = −〈1 2〉〈2 3〉〈3 4〉〈4 1〉2[2 4]2

〈4 5〉〈5 1〉〈2 4〉22 Ls1

(−s23−s51 ,

−s34−s51

)+ L1

(−s23−s51

)+ L1

(−s34−s51

)s2

51

+〈3 2〉〈2 1〉〈1 5〉〈5 3〉2[2 5]2

〈5 4〉〈4 3〉〈2 5〉22 Ls1

(−s12−s34 ,

−s51−s34

)+ L1

(−s12−s34

)+ L1

(−s51−s34

)s2

34

+2

3

〈2 3〉2〈4 1〉3[2 4]3

〈4 5〉〈5 1〉〈2 4〉L2

(−s23−s51

)s3

51

− 2

3

〈2 1〉2〈5 3〉3[2 5]3

〈5 4〉〈4 3〉〈2 5〉L2

(−s12−s34

)s3

34

(B.5)

+L2

(−s34−s51

)s3

51

(1

3

〈1 3〉 [2 4] [2 5] (〈1 5〉 [5 2] 〈2 3〉 − 〈3 4〉 [4 2] 〈2 1〉)〈4 5〉

+2

3

〈1 2〉2〈3 4〉2 〈4 1〉 [2 4]3

〈4 5〉〈5 1〉〈2 4〉 − 2

3

〈3 2〉2〈1 5〉2 〈5 3〉 [2 5]3

〈5 4〉〈4 3〉〈2 5〉

)+

1

6

〈1 3〉3 (〈1 5〉 [5 2] 〈2 3〉 − 〈3 4〉 [4 2] 〈2 1〉)〈1 2〉〈2 3〉〈3 4〉〈4 5〉〈5 1〉

L0

(−s34−s51

)s51

+1

3

[2 4]2[2 5]2

[1 2][2 3][3 4]〈4 5〉[5 1]− 1

3

〈1 2〉〈4 1〉2[2 4]3

〈4 5〉〈5 1〉〈2 4〉[2 3][3 4]s51

+1

3

〈3 2〉〈5 3〉2[2 5]3

〈5 4〉〈4 3〉〈2 5〉[2 1][1 5]s34

+1

6

〈1 3〉2 [2 4] [2 5]

s34 〈4 5〉 s51

.

In contrast to ref. [79], in eqs. (B.4) and (B.5) we use unrenormalized amplitudes; this

distinction actually has no effect on the corresponding gravity amplitudes since the

difference drops out in eq. (3.41). The functions appearing in the above expressions

are

L0(r) =ln(r)

1− r , L1(r) =ln(r) + 1− r

(1− r)2, L2(r) =

ln(r)− (r − 1/r)/2

(1− r)3,

Ls1(r1, r2) =1

(1− r1 − r2)2

[Li2(1− r1) + Li2(1− r2) + ln r1 ln r2 −

π2

6(B.6)

+ (1− r1 − r2)(L0(r1) + L0(r2))].

As discussed in section 3.3.3, these gauge-theory amplitudes serve as building blocks

for the corresponding N ≥ 4 supergravity amplitudes.

Appendix C

Integrals

In this appendix we collect the integrals used in our expressions from various sources

and adjust normalization to match our conventions. The m-point scalar integrals in

D dimensions are defined as:

Im =

∫dDp

(2π)D1

p2(p−K1)2(p−K1 −K2)2 . . . (p−K1 −K2 − . . .−Km−1)2, (C.1)

where the Ki’s are the external momenta which can be on- or off-shell.

The D = 4− 2ε bubble with momentum K is

I2(K2) =icΓ

ε(1− 2ε)(−K2)−ε , (C.2)

where

cΓ =1

(4π)2−εΓ(1 + ε)Γ2(1− ε)

Γ(1− 2ε). (C.3)

The D = 4− 2ε one-mass triangle is

I3(K21) =

−icΓ

ε2(−K2

1)−1−ε , (C.4)

where K1 is the massive leg momentum and the two-mass triangle is

I3(K21 , K

22) =

−icΓ

ε2(−K2

1)−ε − (−K22)−ε

(−K21)− (−K2

2), (C.5)

113

APPENDIX C. INTEGRALS 114

where K1 and K1 are the two massive leg momenta.

For amplitudes with four massless external particles we have the zero-mass box

I12344 ≡ I4(s, t) where s = (k1 + k2)2, t = (k2 + k3)2 and the ki are massless momenta.

An all-order in ε expansion in terms of hypergeometric functions is [119]:

I4(s, t) =2icΓ

ε2st

[t−ε 2F1

(−ε,−ε; 1− ε; 1 +

t

s

)+ s−ε 2F1

(−ε,−ε; 1− ε; 1 +

s

t

)],

(C.6)

which through order ε0 is

I4(s, t) =icΓ

st

[2

ε2

((−s)−ε + (−t)−ε

)− ln2

(−s−t

)− π2

]+O(ε) . (C.7)

Similarly, the one-mass box through ε0 is [119],

I(12)3454 = − 2icΓ

s34s45

{− 1

ε2

[(−s34)−ε + (−s45)−ε − (−s2

12)−ε]

(C.8)

+ Li2

(1− s12

s34

)+ Li2

(1− s12

s45

)+

1

2ln2

(s34

s45

)+π2

6

}+O(ε) ,

where legs 1 and 2 are at the massive corner. An all orders in ε form in terms of

hypergeometric functions may be found in ref. [119]. The integral I(12)3454,trunc is given by

dropping the term multiplied by 1/ε2,

I(12)3454,trunc = − 2icΓ

s34s45

{Li2

(1− s12

s34

)+ Li2

(1− s12

s45

)+

1

2ln2

(s34

s45

)+π2

6

}+O(ε) .

(C.9)

Finally, we use the pentagon integral whose expansion to order ε0 is [119]

I123455 =

∑Z5

−icΓ(−s51)ε(−s12)ε

(−s23)1+ε(−s34)1+ε(−s45)1+ε

[1

ε2+ 2 Li2

(1− s23

s51

)]+ 2 Li2

(1− s45

s12

)− π2

6+O(ε) , (C.10)

where the sum is over the five cyclic permutations of external legs.

APPENDIX C. INTEGRALS 115

We also need integrals in higher dimensions. The triangle and bubble integrals

are obtained by direct integration and the box integrals by dimension-shifting rela-

tions [119]. Explicitly, the D = 6− 2ε bubble is

ID=6−2ε2 (K2) =

−icΓ

2ε(1− 2ε)(3− 2ε)(−K2)1−ε , (C.11)

whereas the D = 6− 2ε one-mass triangle is

ID=6−2ε3 (K2

1) =−icΓ

2ε(1− ε)(1− 2ε)(−K2

1)−ε . (C.12)

The zero-mass D = 6 − 2ε box can be expressed as a linear combination of the

four-dimensional one-mass boxes and one-mass triangles:

ID=6−2ε4 (s, t) =

1

s+ t

(st

2I4(s, t)− icΓ

ε2

((−s)−ε + (−t)−ε

)). (C.13)

Note that it is finite and equal to

ID=6−2ε4 (s, t) = −i cΓ

2(s+ t)

[ln2

(−s−t

)+ π2

]+O(ε) . (C.14)

We also make use of the integral combination from ref. [78],

Jm = −εID=6−2εm , Km = −ε(1− ε)ID=8−2ε

m . (C.15)

Through order ε0, these become

J4 = 0 +O(ε) , K4 = − i

6(4π)2+O(ε) , J3 =

i

2(4π)2+O(ε) . (C.16)

Appendix D

One-loop expressions

In this appendix we give the O(ε0) and O(ε1) coefficients in the expansion of the

one-loop four-graviton amplitude M(1)4 in the various supergravity theories, because

they enter the extraction of the two-loop finite remainder F(2)4 according to eq. (4.16).

These amplitudes were first computed through O(ε0) in ref. [80,94] for N = 4 and

N = 6 supergravity (and the N = 5 case is trivially related to N = 6 at one loop).

Expressions valid to all orders in ε, in terms of box, triangle and bubble integrals,

can be found in ref. [104].

We write

M(1)4 =

( κ8π

)2(

4π e−γ µ2

|s|

)εMtree

4

[2

ε

(s ln(−s) + t ln(−t) + u ln(−u)

)+ F

(1)4

],

(D.1)

where ln(−s) → ln |s| − iπ in the s channel, ln(−t) → ln |t| − iπ in the t channel.

We will give the O(ε0) and O(ε1) coefficients for F(1)4 for each theory in these two

channels.

For N = 8 supergravity in the s channel we have,

F(1),N=84


= s[gs

(−ts

)+ gs

(−us

)], (D.2)

116

APPENDIX D. ONE-LOOP EXPRESSIONS 117

where

gs(x) = 2x (lnx+ iπ) ln(1− x) (D.3)

+ ε

{−2 (2− x)

[Li3(x)− ζ3

3+ (lnx+ iπ) Li2(1− x)

+1

2ln(1− x) (ln2 x− 4 ζ2)

]+

1

3x ln3 x− iπ (1− x) (ln2 x− 4 ζ2)− lnx (lnx+ iπ) ln(1− x)

}.

The t-channel result for the N = 8 supergravity amplitude, divided by the tree, is

obtained by exchanging s and t in the corresponding s-channel result. (This is not

quite the case for F(1),N=84 , due to the explicit factor of |s|−ε extracted in eq. (D.1).)

We express the finite remainders for N < 8 supergravities in terms of the one for

N = 8 supergravity. For N = 6 supergravity we find, in the s channel,

F(1),N=64


= F(1),N=84


+ s

[g6,s

(−ts

)+ g6,s

(−us

)], (D.4)

where

g6,s(x) =1

2x (1− x)

[ln2

(x

1− x

)+ π2

](D.5)

+ ε

{2x (1− x)

[Li3(x)− lnxLi2(x)− 1

3ln3 x− π2

2lnx

]− 1

2

[x (lnx+ iπ) + (1− x) (ln(1− x) + iπ)

]2

− π2

2(1− x (1− x))

}.

The t-channel result is

F(1),N=64


= F(1),N=84


+ s g6,t

(−ut

), (D.6)


where

g6,t(x) = − x

(1− x)2lnx (lnx+ 2 iπ) (D.7)

+ ε

{2x

(1− x)2

[Li3(x)− ζ3 − (lnx+ iπ) (Li2(x)− ζ2)

+1

3ln3 x+ 2 ζ2 lnx+

iπ

2ln2 x− lnx (lnx+ 2 iπ) ln(1− x)

]−[(ln(1− x) + iπ) +

x

1− x (lnx+ iπ)

]2

− 1− x (1− x)

(1− x)2π2

}.

The corresponding one-loop results for N = 5 supergravity are trivially related

to those for N = 6, because the difference in field content from N = 8 is due to the

same matter multiplet, just three copies instead of two. Therefore we have,

F(1),N=54


= F(1),N=84


+3

2s

[g6,s

(−ts

)+ g6,s

(−us

)], (D.8)

F(1),N=54


= F(1),N=84


+3

2s g6,t

(−ut

). (D.9)

The s-channel one-loop finite remainder for N = 4 supergravity is given by

F(1),N=44


= F(1),N=84


+ s

[g4,s

(−ts

)+ g4,s

(−us

)], (D.10)

where

g4,s(x) =[2− x (1− x)

]g6,s(x) + x (1− x)

[(1− 2x) ln x+

1

2

]− ε

6

{x(3− x2 (12− 15x+ 5x2)

)1− x ln2 x

− 5x2 (1− x)2[lnx ln(1− x)− π2

2

]+ iπ

[2

x2

1− x (7− 12x+ 6x2) lnx+ 1

]− 2x (6− 24x+ 17x2) ln x− 10x (1− x)

}. (D.11)


The t-channel expression is

F(1),N=44


= F(1),N=84


+ s g4,t

(−ut

), (D.12)

where

g4,t(x) =

[2 +

x

(1− x)2

]g6,t(x)− x

(1− x)2

[1 + x

1− x (lnx+ iπ) + 1

]+ε

6

x

(1− x)2

{(3− x (1− x))

[ln2

(x

1− x

)+ π2

]− 5x

(1− x)2lnx (lnx+ 2iπ)

+1− x+ 3x2

x2ln(1− x) (ln(1− x) + 2iπ)

+ 21− 12x− 6x2

x (1− x)(lnx+ iπ)

− 21− 5x+ x2

xln

(x

1− x

)− 20

}. (D.13)

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SOFT LIMITS AND COLOR-KINEMATIC DUALITIES IN A … · soft limits and color-kinematic dualities in perturbative quantum gravity a dissertation submitted to the department of physics

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