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F A C U L T Y O F S C I E N C E
U N I V E R S I T Y O F C O P E N H A G E N
Master’s Thesis
Anders Øhrberg Schreiber
Scattering Amplitudes and the Power of Factorization
Supervisor: N. Emil J. Bjerrum-Bohr
July 20, 2016

Anders Ø. Schreiber: Scattering Amplitudes and the Power of Factorization, Master’s Thesis.

Abstract
Over the last three decades, we have seen fascinating advancements in the understanding
scattering amplitudes. A plethora of new techniques have been developed to not only study
the properties of scattering amplitudes but also to calculate scattering amplitudes with many
(greater than four) external particles and amplitudes with higher loop order corrections. These
techniques are extremely valuable to experimentalists when making high precision predictions
in scattering experiments. It is therefore of great importance that we study and develop new
and even better techniques in this field.
In this thesis we explore and develop some of these calculational techniques for scattering
amplitudes. We start out by giving a brief review of some standard methods and formalism
used throughout the literature. These include the spinor helicity formalism, which provides
compact notation, as well as the Britto-Cachazo-Feng-Witten (BCFW) recursion relations,
which allows one to construct higher point amplitudes from lower point amplitudes.
After the introductory review, we move on to supersymmetry and squaring of supermultiplets,
which hint at amplitude relations between chiral multiplets and super Yang-Mills multiplets.
We then explore the Kawai-Levellen-Tye (KLT) relations, described in the superfield and
superamplitude formalism, which provide an excellent foundation for finding new amplitude
relations. After studying supersymmetry reduction of supergravity multiplets, we develop a
reduction scheme to construct the desired amplitude relations between the chiral and super
Yang-Mills multiplets, which we call chiral KLT relations.
We then briefly move on to give a review of the Cachazo-Huan-Ye (CHY) formulas for
scattering amplitudes and the scattering equations. Here we make use of the KLT relations
to derive a compact form of the graviton amplitude in the CHY formalism.
Finally, we move away from tree-level amplitudes and move to loop amplitudes. We review
the unitarity method and Cutkosky’s cutting rules. Then, we move on to a very recently
developed method called the Q-cut method, which allows one to construct loop amplitude
integrands from on-shell tree-level amplitudes. Various examples of these integrands are
derived and compared to integrands obtained from the Feynman diagrams in scalar field
theory. As a conclusion we show how to integrate one of these integrands to obtain the full
loop amplitude using the maximally democratic contour prescription as well as Schwinger
parameterization of Feynman propagators.
iii

Acknowledgements
The work of this thesis has been a tremendously educational experience. The topics that have
been touched upon have taught how to work as a theoretical physicist and as an independent
researcher. I am very grateful for the environment that the Niels Bohr International Academy
has provided throughout my studies. I have had plenty a fruitful discussion with the amazing
people of the Academy in our lounge. I would especially like to thank the other M.Sc.
and Ph.D. students that have been at the Academy throughout my studies. These people
include Adam, Amel, Andreas (Jantzen), Asta, Bjarke, Carsten, Christian B., Christian B.-H.,
Christian B. J., Christine, Dennis, Emil A., Emil H., Gitte, Isak, Janet, Jeppe, Jules, Laure,
Meera, Melissa, Mikkel, Rasmus and Sebastian. Thank you all for being super awesome and
providing new and helpful perspective when I have run into walls.
I would like to thank Poul Henrik Damgaard and Changyong Liu for great discussions and
meetings throughout our KLT project in the fall of 2015. I would also like to thank Simon
Caron-Huot for helping me understand the i prescription for linear propagators in the Q-cut
method. Last but not least, I would like to thank my advisor Emil Bjerrum-Bohr, whom I
have known since he was the supervisor on my ”Projekt Forskerspirer” project, later as the
supervisor on my bachelor thesis and now as the advisor on my M.Sc. thesis. Thank you
for excellent supervision and guidance throughout this thesis. It has truly been a pleasure
working under your wings.
I would also like to give special thanks to Emil, Poul Henrik and Niels Obers for writing
letters of recommendation for my applications to US graduate schools. Without these letters,
my dream of doing a PhD in the US would have never come true.
Finally I would like to thank Jules and Lauren for helping me proof read parts of the thesis.
iv

Contents
List of Figures vii
List of Tables vii
Acronyms and initialisms viii
Notation and conventions viii
1 Introduction 1
1.1 Thesis outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2 Spinor helicty formalism 3
3 BCFW recursion relations and little group scaling 5
4 Supersymmetry and chiral squaring 7
4.1 Supersymmetry transformations in N = 4 Super Yang-Mills theory . . . . . . . 10
4.1.1 N = 4 super Yang-Mills theory from chiral multiplets: a dictionary . . . 10
4.2 N = 3 super Yang-Mills theory from chiral multiplets . . . . . . . . . . . . . . 11
4.5 Pure Yang-Mills theory from chiral multiplets . . . . . . . . . . . . . . . . . . . 13
5 Superamplitudes 14
6 Kawai-Levellen-Tye relations 17
7 Supersymmetric Kawaii-Levellen-Tye relations and diamond diagrams 19
7.1 KLT relations with less supersymmetry: the full map . . . . . . . . . . . . . . . 20
7.1.1 The equivalence of N = 3 and N = 4 super Yang-Mills theory . . . . . 22
7.2 Diamond diagrams and the NG < 8 KLT-relations . . . . . . . . . . . . . . . . 23
7.2.1 Diamond diagrams for the NG = 7 theory . . . . . . . . . . . . . . . . . 26
7.2.2 Diamond diagrams for the NG = 6 theories . . . . . . . . . . . . . . . . 26
8 Chiral Squaring and KLT relations 34
8.1 Chiral KLT relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
8.2 Some explicit checks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
v

9 Scattering equations and KLT orthogonality 40
9.1 The CHY formulas for amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 41
9.2 KLT orthogonality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
10 Loop amplitudes I: The Unitarity Method 44
10.1 The Unitarity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
10.2 Generalized unitarity cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
11 Loop amplitudes II: The Q-cut Method 51
11.1 Q-cut integrands in scalar ﬁeld theory . . . . . . . . . . . . . . . . . . . . . . . 53
11.1.1 Scalar integrands with bubble, triangle, and box topologies . . . . . . . 53
11.1.2 Color-ordered four-point amplitudes in φ4 theory . . . . . . . . . . . . . 56
11.1.3 Color-ordered four-point amplitudes in φ3 theory . . . . . . . . . . . . . 57
11.2 The Maximally Democratic Contour Prescription . . . . . . . . . . . . . . . . . 59
11.3 Integrating the Q-cut representation of the color-ordered four-point φ4-integrand 60
12 Conclusions 63
12.1 Outlook and future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
13 Appendix A: Color decomposition in Yang-Mills theory 65
13.1 MHV classiﬁcation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
14 Appendix B: The N = 4 SYM dictionary check 67
Bibliography 69
vi

List of Figures
4.1 Supersymmetry transformations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.1 Super Yang-Mills diamond diagrams . . . . . . . . . . . . . . . . . . . . . . . 21
7.2 Squaring of super Yang-Mills multiplets . . . . . . . . . . . . . . . . . . . . . 24
7.3 NG = 7 supergravity from a KLT product . . . . . . . . . . . . . . . . . . . . 26
7.4 Minimal NG = 6 supergravity from a KLT product . . . . . . . . . . . . . . . 27
7.5 NG = 6 supergravity equivalent to NG = 8 supergravity from a KLT product 28
7.7 NG = 5 supergravity equivalent to minimal NG = 6 supergravity from a KLT
product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7.9 NG = 4 supergravity equivalent to minimal NG = 5 supergravity from a KLT
product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
7.10 NG = 4 supergravity coupled to a vector multiplet from a KLT product . . . 30
7.12 NG = 3 supergravity coupled to a vector multiplet from a KLT product . . . 31
7.14 NG = 2 supergravity coupled to a hypermultiplet from a KLT product . . . . 32
8.1 Chiral superfields in the diamond diagram representation . . . . . . . . . . . 35
8.2 Vector superfields in the diamond diagram representation . . . . . . . . . . . 36
8.3 Chiral squaring with diamond diagrams . . . . . . . . . . . . . . . . . . . . . 37
10.1 Four-point one-loop amplitude after unitarity cut . . . . . . . . . . . . . . . . 45
10.2 Five-point box diagram with four cuts . . . . . . . . . . . . . . . . . . . . . . 49
11.1 Q-cut integrand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
11.2 Feynman diagrams for basis integals in φ3 theory . . . . . . . . . . . . . . . . 53
11.3 Four-point one-loop color-ordered φ4 theory Feynman diagrams . . . . . . . . 56
11.4 Four-point one-loop color-ordered φ3 Feynman diagrams . . . . . . . . . . . . 57
List of Tables
5.1 Picking out field operators from the N = 4 super Yang-Mills superfield . . . . 15
7.1 List of all supergravity theories based on Kawai-Levellen-Tye products . . . . 27
vii

Acronyms and initialisms
BCFW Britto-Cachazo-Feng-Witten.
SUSY Supersymmetry.
SYM Super Yang-Mills theory.
CPT Charge conjugation, parity, and time reversal.
MHV Maximal Helicty Violating.
KLT Kawai-Levellen-Tye.
SUGRA Supergravity.
CHY Cachazo-He-Yuan.
HV ’t Hooft-Veltman.
Notation and conventions
We will use the ”mostly positive” Lorentzian signature for metrics, i.e. Minkowski metrics will
have signature (−, +, +, +). Therefore, when we refer to any dimensionality, we will always
refer to D dimensions, meaning that D = d + 1, where d is the number of spatial dimensions.
We will work in D = 4 dimensions, unless otherwise stated.
For indices we will be using Greek letters from the beginning of the greek alphabet to denote
spinor indices (α, β, . . .), while greek letters from the middle of the alphabet will usually denote
Lorentz indices (µ, ν, . . .). Arabic letters as indices will denote gauge group indices or R-indices
in supersymmetry (e.g. SO(2) indices).
All amplitudes considered have all particles outgoing (i.e. a scattering process with ∅ →
A + B + C + . . .) unless otherwise stated. Amplitudes will not be expressed with coupling
constants.
All units are in natural units, = c = 1.

1 Introduction
The study of scattering amplitudes is an integral topic of particle physics as it lays the
foundation for predictions in scattering experiments. Making predictions from intricate quantum
field theories can be very complicated, and it is therefore the job of physicists to understand
the structure underlying the theories, such that predictions can more easily be made and
understood by experimentalists. This understanding of the mathematical structure of quantum
field theory has developed over that past eighty-five years with some incredible steps of
simplification and reduction in the computational demands for making predictions in scattering
experiments.
With the initial introduction of the Feynman diagrams [1], a revolution occured in the
business of calculating scattering amplitudes. Feynman rules and diagrams made the process
of writing down scattering amplitudes a great deal easier than before. This also initiated the
study of the S-matrix and its various axioms based on unitarity, causality and analyticity [2].
S-matrix theory however proved to be not quite useful, as it was too general and could not
be realized in any useful way for calculational purposes like the Feynman diagram method.
Feynman diagrams however also posed a huge problem in that the number of diagrams
for an amplitude would go approximately as n! for n external particles. This meant the
computational complexity would skyrocket with the number of external particles. The initial
steps for solving the problems of Feynman diagrams were seen in the extraordinary simplicity
of the Parke-Taylor formula [3] for the Maximally Helicity Violating (MHV) amplitude in pure
Yang-Mills theory (see Appendix A), which would initially be represented with a huge number
of Feynman diagrams, that would simplify down to one term. This incredible simplicity
hinted at a much simpler underlying structure for scattering amplitudes than what Feynman
diagrams could provide. Thus began the current era of amplitude research, where astonishing
mathematical structures have been found for calculating scattering amplitudes.
To name a few of the most important examples for tree-level amplitudes, we have the
Cachazo-Svrcek-Witten (CSW) expansion of amplitudes in terms of MHV vertices [4], the
Roiban-Spradlin-Volovich (RSV) formula which was the first step in the direction of what
is today the Cachazo-He-Yuan (CHY) formalism for scattering amplitudes [5,6], the Britto-
Cachazo-Feng-Witten (BCFW) recursion relations which simplifies calculating higher point
amplitudes through recursion of lower point amplitudes [7], the Bern-Carrasco-Johansson
(BCJ) relations allows one to reduce the basis of color-ordered tree-level amplitudes in Yang-
Mills theory [8]. The BCJ relations are also very important when considering the Kawai-
Levellen-Tye (KLT) relations in field theory [9], which has been an important tool for calculating
graviton amplitudes. A recent interesting development in the field is also the connection
between scattering amplitudes and the positive Grassmannian [10,11], which promises to find
an underlying structure of quantum field theory without the need of locality and unitarity.
A very recent development in calculations of loop amplitudes is the Q-cut method [12],
which makes it possible to calculate loop amplitude integrands from tree-level amplitudes.
An overarching theme in most of these new developments is that the field has moved away
from Feynman diagrams and that amplitudes are now calculable without reference to any
off-shell Lagrangians. Amplitudes can built recursively from a few on-shell building blocks.
1

Chapter 1. Introduction
1.1 Thesis outline
The structure of this thesis is divided into three parts, which are all connected through an
overarching theme of factorization properties of scattering amplitudes.
1. Supersymmetry, KLT relations and chiral squaring.
→ Chapters 4-8.
2. Scattering equations and the CHY formalism.
→ Chapter 9.
3. Loop amplitudes, unitarity and Q-cuts.
→ Chapters 10 and 11.
The content of part 1 can in part found in [13] and is mainly concerned with squaring of
supermultiplets and relations for amplitudes of supermultiplets from these squaring properties.
Part 2 surrounds the CHY formalism of scattering amplitudes, where amplitudes are expressed
as integrals localized on the scattering equations. Our goal in this part is to show how the
KLT relations can be used to construct a compact form for graviton amplitudes in the CHY
formalism. In part 3 we investigate properties of loop amplitudes. We introduce the unitarity
method for constructing amplitude integrands based on the discontinuities (branch cuts) of
the integrand. We then study the Q-cut method, which enables us to build loop integrands
from on-shell tree-level amplitudes.
Now we will introduce some basic tools used in general amplitude calculations, namely the
spinor helicity formalism in Chapter 2 and the BCFW recursion relations in Chapter 3.
2

2 Spinor helicty formalism
When calculating scattering amplitudes, expressions of the amplitudes can be very cumbersome
in regular relativistic notation [14], so here we will describe the spinor helicity formalism,
which is useful for writing down nice and compact expressions. The formalism is build
on solutions to the massless Dirac equation which are decomposed into two independent
commuting spinor solutions with opposite helicities. Helicity is the spin of a particle projected
onto its momentum, and for massless particles there will only two helicity configurations in four
dimensional spacetime as the spin can be either parallel or antiparallel with the momentum.
Note that helicity, together with momentum, are the only relevant quantum numbers for
massless particles. Let us however start by considering the massive Dirac equation [14]
(−i/∂ + m)Ψ(x) = 0 ⇒ Ψ(x) ∼ u(p)eipx
+ v(p)e−ipx
, (2.1)
where /∂ = ∂µγµ, px = pµxµ and the momentum is on-shell p2 = −m2. Solutions of the form
Ψ(x) = u(p)eipx + v(p)e−ipx, then satisfy
(/p + m)u(p) = 0, (−/p + m)v(p) = 0. (2.2)
Each of these equations have two independent solutions, which we label with a ± subscript,
giving us the full solution to the Dirac equation
Ψ(x) =
s=±
˜dp bs(p)us(p)eipx
+ d†
s(p)vs(p)e−ipx
, ˜dp =
d3p
(2π)32Ep
, (2.3)
and similarly for the conjugate field Ψ(x) ≡ Ψ†(x)γ0. We take the coefficients (b±(p), d±(p))
and conjugates to be anticommuting fermionic creation and annihilation operators when
canonically quantizing the theory. Thus u±(p) and v±(p) are commuting 4-component spinors.
u± and v± are eigenstates of the z-direction spin operators, so in the massless case the ±
subscript denotes the helicity of the particle. In terms of Feynman rules, we take v± to
describe an outgoing anti-fermion and u± to describe and outgoing fermion. When calculating
any scattering amplitude we will take all particles to be outgoing unless otherwise specified.
We can always use crossing symmetry (exchange outgoing particles with incoming, fermions
with antifermions and flip the helicity of particles) to construct amplitudes with incoming
particles. Crossing symmetry has the consequence that
u± = v , v± = u . (2.4)
To get to the spinor helicity formalism, we will write out the massless Dirac equation for
the u and v spinors (i.e. the Weyl equation)
/pv±(p) = 0, u±(p)/p = 0, /p =
0 pa˙b
p˙ab 0
, pα ˙β = pµ(σµ
)α ˙β. (2.5)
This means that u± and v± have the following solutions
v+(p) =
|p]α
0
, v−(p) =
0
|p ˙α ,
u−(p) = (0, p| ˙α), u+(p) = ([p|α
, 0),
(2.6)
3

Chapter 2. Spinor helicty formalism
where angle and square spinors are 2-component commuting spinors. To make the notation
simpler, we deﬁne the following shorthand notation (for several particles of momenta pi,
i = 1, 2, . . . , n) for spinor products
ij ≡ i| ˙α |j ˙α
= u−(pi)v−(pj), [ij] ≡ [i|α
|j]α = u+(pi)v+(pj). (2.7)
Note that spinor products where u and v spinors have opposite helicity vanish. We can use
the spinor completeness relation for massless spinors to write [14]
u−(p)u−(p) + u+(p)u+(p) =
0 |p]α p|˙b
0 0
+
0 0
|p ˙α
[p|β 0
= −/p ⇔
pα ˙β = −|p]α p| ˙β , p ˙αβ
= − |p ˙α
[p|β
.
(2.8)
Since the angle and square brakets have spinor indices, these can be raised and lowered
with Levi-Civita symbols. This results in spinor products being antisymmetric pq = − qp
and [pq] = −[qp]. Provided that momenta are real, the following identities will hold: [p|α =
(|p ˙α
) and |p]α = ( p| ˙α) , which results in [pq] = qp . An important identity for reducing
amplitudes to Mandelstam variables is [15]
pq [qp] = p| ˙β |q
˙β
[q|α
|p]α = Tr(q
˙βα
pα ˙β) = pµqν Tr[(σµ
)
˙βα
(σν
)α ˙β] = −2p · q = spq. (2.9)
The strength of the spinor helicity formalism is not only the possibility for compact notation,
but also the existance of a plethora of identities [15]
Charge conjugation of current : [i|γµ
|j = j| γµ
|i]
Fierz rearrangement : i| γµ
|j] k| γµ|l] = 2 ik [jl]
Gordon identity : i| γµ
|i] = 2pµ
i
Momentum conservation :
n
i=1
|i [i| = 0 ⇒
n
i=1,i=j,k
[ji] ik = 0
Schouten identity : ij kl = ik jl + il kj
(2.10)
An important part of the success of the spinor helicity formalism is that polarization vectors
for massless spin-1 particles can be written in terms of angle and square spinors
µ
+(pi; q) = −
q| γµ|i]
√
2 qi
,
/+(pi; q) =
√
2
qi
(|i] q| + |q [i|),
µ
−(pi; q) = −
i| γµ|q]
√
2[qi]
,
/−(pi; q) =
√
2
[qi]
(|i [q| + |q] i|),
(2.11)
with pµ
µ
±(p) = 0 due to the massless Weyl equation.
4

3 BCFW recursion relations and little group
scaling
In this chapter we give a brief review of the BCFW recursion and little group scaling (for a
more extensive review, see [15]). Little group scaling lets us determine on-shell three-point
scattering amplitudes up to coupling constants. One can then use the BCFW recursion
relations to build higher point tree-level amplitudes from the three-point amplitudes. With
these tools we avoid the cumbersome Feynman diagram method.
Little group scaling is realized through the following simultaneous transformations of the
angle and square spinors of an amplitude
|p → t |p and |p] → t−1
|p]. (3.1)
where t is the little group scaling parameter. Any tree level amplitude is always made
up of propagators, vertices and external states. Propagators for massless particles take the
form 1
p2
ij
∼ 1
ij [ij] but can also involve powers of momenta in the numerator. Propagators are
hence invariant under little group scaling. Vertices are also invariant under little group scaling.
External scalar states are just 1, so these are invariant. Fermions scale with a factor t−2h since
they have an angle or square spinor associated with their external state. Here h is the helicity
of the external state. Vectors also scale as t−2h because of the spinor helicity representation
of the polarization vectors (2.11). The same happens to gravitons as the polariazation tensor
factorizes into two polarization vectors. Thus for an n-point ampltude
An({|1 , |1], h1}, . . . , {ti |i , t−1
i |i], hi}, . . .) = t−2hi
i An({|1 , |1], h1}, . . . , {|i , |i], hi}, . . .).
(3.2)
Special kinematics lets us write three-point amplitudes purely in terms of either angle or
square spinors [15]. As a general ansatz for the three-point amplitude, we then have
A3(1h1
, 2h2
, 3h3
) = c 12 x12
13 x13
23 x23
, (3.3)
where c is a constant that depends on the specific theory (coupling constants etc.). Employing
(3.2), we get relationships between helicities of each particle and xij’s. Solving this leads to
the final form of the three-point amplitude
A3(1h1
, 2h2
, 3h3
) = c 12 h3−h1−h2
13 h2−h1−h3
23 h1−h2−h3
. (3.4)
Using this we can determine the form of any three-point amplitude. With this we will
now introduce the BCFW recursion relations, which allow us to build higher point tree-level
amplitudes with three-point amplitudes as building blocks.
The BCFW recursion relations comes from a more general set of recursion relations, utilizing
leg shifts (momentum shifts of external particles) with a complex parameter. The resulting
shifted amplitude (at tree level) is in general a rational function of this complex parameter.
One can thus easily find poles of the amplitude, appearing in the shifted propagators, and
5

Chapter 3. BCFW recursion relations and little group scaling
this captures the analytic structure of the amplitude. With this in mind, one can find the full
amplitude in the following way using the global residue theorem
An = Resz=0
Ân(z)
z
= −
zI
Ân(z)
z
+ Bn, (3.5)
where Ân(z) is the shifted amplitude as a function of the complex parameter z and zI are
poles appearing away from the origin. Bn is a possible pole at infinity, but usually Bn = 0,
as one can consider good or bad shifts in the sense that Bn = 0 if Ân(z) → 0 as |z| → ∞
(which is a good shift), while bad shifts do not have this feature. Due to relations between
shifted propagators and unshifted propagators (see [15]), the amplitude ends up factorizing
into a left and right part (with less than n legs), meaning that the n-point amplitude can be
build from ”less than n”-point amplitudes, thus giving us a recursive procedure for calculating
higher point amplitudes. Specifically
−Resz=zI
Ân(z)
z
= ÂL(zI)
1
P2
I
ÂR(zI) = L R
ˆPI
∧
∧
∧
∧
∧
∧
, (3.6)
where the hats indicate that legs have been shifted. When we sum over all the residues, we
end up with
An =
diagrams I
ÂL(zI)
1
P2
I
ÂR(zI) =
diagrams I
L R
ˆPI
∧
∧
∧
∧
∧
∧
. (3.7)
The BCFW recursion relations are a special case of the more general recursion relations
above in that only two legs are shifted. Legs i and j are taken as shifted and the shift of the
legs is deployed through the spinor helicity shifts
|î] = |i] + z|j], |ˆj] = |j], |î = |i , |ˆj = |j − z |i . (3.8)
Under such a shift, we end up with the following diagrammatic recursion relation for the
n-point amplitude
An =
diagrams I
ÂL(zI)
1
P2
I
ÂR(zI) =
diagrams I
L R
ˆPI
ˆjî
. (3.9)
The shifted legs have to appear on opposite sides of the diagram, because otherwise the
propagator between the two diagrams would not be shifted and there would be no residue. If
one wants to calculate a four-point amplitude, one would then get a diagram with both the
left and right amplitude have three legs and thus one calculates the four-point amplitude from
three-point amplitudes. The procedure continues in this way recursively.
6

4 Supersymmetry and chiral squaring
Supersymmetry (SUSY) is an important tool for simplifying the calculations of scattering
amplitudes. We will introduce the concept of SUSY transformations through Lagrangian
invariance under SUSY transformations. In this chapter we will focus on so-called chiral
squaring, which we will use to construct dictionaries between two chiral multiplets and super
Yang-Mills (SYM) multiplets.
We start by looking at SUSY transformations on an N = 1 chiral multiplet. We define this
as the following free Lagrangian with a lefthanded Weyl fermion ψ and a complex scalar field
φ [15]
Lchiral
0 = iψ†
σµ
∂µψ − ∂µφ∂µ
φ. (4.1)
This Lagrangian has obvious Poincaré symmetry but on top of that it also posseses a
symmetry that mixes the Weyl fermion and the scalar. Using an anti-commuting SUSY
parameter α, we define transformations
δφ = α
χα, δχα = −iσµ
α ˙α
† ˙α
∂µφ, (4.2)
and similarly for the conjugate fields φ and χ†
˙α. We can write the field content (in x-space)
in terms of canonically quantized creation and annihilation operators
φ(x) = ˜dp φ−(p)eipx
+ φ†
+(p)e−ipx
χα(x) =
s=±
˜dp χs(p)(PLus(p))αeipx
+ χ†
s(p)(PLvs(p))αe−ipx
(4.3)
= ˜dp χ−(p)|p]αeipx
+ χ†
+(p)|p]αe−ipx
,
where ˜dp = d3p
(2π)32ωp
and where φ±(p) satisfy the usual bosonic commutation relations and χ±
satisfy the usual fermionic anti-commutation relations. Also the ± notation indicate which
modes are related through SUSY. Combining (4.2) and (4.3), we get
δφ(p) = ˜dp χ−(p) α
|p]αeipx
+ χ†
+(p) α
|p]αe−ipx
. (4.4)
Taking eipx and e−ipx as independent directions in function space, we get the following
transformations for the annihilation operators of the scalar field
δφ−(p) = [ p]χ−(p), δφ+(p) = p χ+(p), (4.5)
where we have used α = | ]α. Similarly for the Weyl spinor, we get the transformation
δχα(p) = ˜dp pα ˙α
† ˙α
φ−(p)eipx
− pα ˙α
† ˙α
φ†
+(p)e−ipx
, (4.6)
7

Chapter 4. Supersymmetry and chiral squaring
where we have defined pµσµ
α ˙α ≡ pα ˙α. Again taking independent directions, we find the
transformations of the Weyl spinor annihilation operators
δχ−(p) = p φ−(p), δχ+(p) = [ p]φ+(p). (4.7)
Let us now introduce a Majorana spinor for the SUSY generator: QM =
|Q]α
|Q† ˙α . We
want to figure out, how this acts on our annihilation operators. In general a symmetry
transformation of an operator, with a symmetry generator, can be written as
δO = [λG, O], (4.8)
where G is the symmetry generator and λ is a parameter for the symmetry transformation. If
we arrange the SUSY parameter as a Majorana spinor M =
| ]α
| ˙α , then we can write the
SUSY transformation as
δO = [ M QM , O] = [ Q] + Q†
, O , (4.9)
where [. , .] is a graded bracket, meaning that it is an anti-commutator if O is fermionic and
a commutator if O is bosonic. With this tool we can explicitly construct how the SUSY
generator acts on the annihilation operators
δφ+(p) = [[ Q], φ+(p)] + [ Q†
, φ+(p)] = p χ+(p),
δχ+(p) = [[ Q], χ+(p)] + [ Q†
, χ+(p)] = [ p]φ+(p),
δχ−(p) = [[ Q], χ−(p)] + [ Q†
, χ−(p)] = p φ−(p),
δφ−(p) = [[ Q], φ−(p)] + [ Q†
, φ−(p)] = [ p]χ−(p).
(4.10)
From the righthand side of these equations, we see that we can impose the following
condition on the lefthand side
[[ Q], φ+(p)] = 0, [ Q†
, χ+(p)] = 0,
[[ Q], χ−(p)] = 0, [ Q†
, φ−(p)] = 0.
(4.11)
For (4.10) to be satisfied, we must have |Q]α ∝ |p]α and |Q†
α ∝ |p α. Additionally
we have commutation relations for the creation and annihilation operators [φ±(p), φ†
±(p )] =
(2π)32ωpδ3(p−p ) and {χ±(p), χ†
±(p )} = (2π)32ωpδ3(p−p ). We can use these to write down
the SUSY generators as
|Q]α = ˜dp|p]α[φ+χ†
+ − χ−φ†
−],
|Q† ˙α
= ˜dp |p ˙α
[φ−χ†
− − χ+φ†
+],
(4.12)
such that equations (4.10) are satisfied. With this SUSY generator we can only relate plus
operators to plus and minus operators to minus. However, if we want to relate a chiral
righthanded multiplet to a chiral lefthanded multiplet, then we need to relate plus operators
to minus operators. To achieve this we want to construct an extra SUSY generator, effectively
giving us an N = 2 chiral multiplet. Constructing this new SUSY generator is redundant,
when we consider only a single chiral multiplet, but it will be crucial later when we consider
chiral squaring.
8

χ−0
φ+
χ+
φ−
0
Q†
Q†Q
Q
Q′
Q′
Figure 4.1 – A diagram over how fields in the chiral multiplet transform under SUSY after
adding an SO(2) rotated supercharge.
To construct this new SUSY generator, consider first the action of an SO(2) transformation
on the complex scalar fields
φ → φ =
φ−
φ+
=
0 −1
1 0
φ−
φ+
=
−φ+
φ−
, (4.13)
and equivalently for hermitian conjugate fields. Continuing this transformation to the SUSY
generator we can obtain a new primed SUSY generator
|Q ]α = ˜dp|p]α[φ−χ†
+ + χ−φ†
+],
|Q † ˙α
= ˜dp |p ˙α
[−φ+χ†
− − χ+φ†
−].
(4.14)
It is now easy to see that under this SO(2) rotated SUSY generator the field operators
transform in the following way
[Q , χ+(p)] = |p]φ−(p),
[Q , χ−(p)] = 0,
[Q †
, χ+(p)] = 0,
[Q †
, χ−(p)] = − |p φ+(p),
[Q , φ+(p)] = −|p]χ−(p),
[Q , φ−(p)] = 0,
[Q †
, φ+(p)] = 0,
[Q †
, φ−(p)] = |p χ+(p).
(4.15)
We can also combine the SUSY generators and fields like (4.13): φi = (φ+, φ−) and Qi =
(Q, Q ). This makes the transformations more compact (see also Figure 4.1)
[Qi
, χ+(p)] = |p]φi
(p),
[Qi
, φj
(p)] = ij
|p]χ−(p),
[Qi
, χ−(p)] = 0,
[Q†
i , χ+(p)] = 0,
[Q†
i , φj
(p)] = δj
i |p χ+(p),
[Q†
i , χ−(p)] = ij |p φj
(p),
(4.16)
where ij is the Levi-Civita symbol with 12 = 1. We note that we can raise and lower the these
indices with ijφj = φi and jiφj = φi. Equation (4.16) are now the SUSY transformation
relations of an N = 2 chiral multiplet. This multiplet has the same field content as the N = 1
chiral multiplet, so it is simply an enhanced chiral multiplet.
9

4.1 Supersymmetry transformations in N = 4 Super Yang-Mills
theory
We are interested in exploring dictionaries between the enhanced N = 1 chiral multiplet
with SUSY transformation relations (4.16) and SYM multiplets. Dictionaries here refers to
taking a tensor product between the field content of two chiral multiplets. In order to figure
out what the resulting field content is, we want check the resulting SUSY transformation
relations consistent with any known theory (in this case SYM theory). This is the first step in
the direction of finding possible scattering amplitude relations between amplitudes with spin-1
particles (with supersymmetry) and chiral amplitudes. We will explore amplitude relations
later, but the existance of dictionaries will be a strong indication of how one can relate different
types of scattering amplitudes through factorization.
We start out with the maximally supersymmetric Yang-Mills multiplet, namely the N = 4
SYM multiplet. This multiplet is the maximally supersymmetric, as it is a spin-1 multiplet,
and four supercharges is the maximal number of supercharges that can lower the helicity of a
spin-1 particle with positive helicity to itself with negative helicity. This involves four steps
of 1/2 helicity lowerings. The field content of the N = 4 SYM multiplet consists of one gluon
with helicity h = +1 (g+), four gluinos with h = 1
2 (fa
+), six scalars with h = 0 (sab), four
gluinos with h = −1
2 (fabc
− ), and one gluon with h = −1 (gabcd
− ) where a, b, c, d = 1, . . . , 4 are
SU(4)R indices [15]. It is however convenient to rewrite the fields as
g+, fa
+, sab
=
1
2!
abcd
scd, f−
a =
1
3!
abcdfbcd
− , g− = −
1
4!
abcdgabcd
− . (4.17)
Since we working in the N = 4 SYM multiplet we have four supercharges so we add an
index (SU(4)R index) to the supercharge Q → Qa and Q† → Q†
a. Analogous to the N = 1
chiral calculation, we now have a SUSY hierachi going from right to left in (4.17). This means
that we can go through the hierachi of SUSY transformations and construct these with the
antisymmetry of index switching in mind
[Qa
, g+(p)] = |p]fa
+(p),
[Qa
, fb
+(p)] = |p]sab
(p),
[Qa
, sbc
(p)] = |p] abcd
f−
d (p),
[Qa
, f−
b (p)] = −|p]δa
b g−(p),
[Qa
, g−(p)] = 0,
[Q†
a, g+(p)] = 0,
[Q†
a, fb
+(p)] = |p δb
ag+(p),
[Q†
a, sbc
(p)] = |p 2!δ[b
a f
c]
+(p),
[Q†
a, f−
b (p)] = |p sab(p),
[Q†
a, g−(p)] = − |p f−
a (p).
(4.18)
These are the SUSY transformation relations we need, in order to show that the squaring
of the enhanced chiral multiplet (4.16) results in an N = 4 SYM multiplet.
4.1.1 N = 4 super Yang-Mills theory from chiral multiplets: a dictionary
Now that we have the SUSY transformation relations between the N = 4 SYM fields and the
N = 1 enhanced chiral multiplet fields, we can construct a dictionary between two copies of
enhanced chiral multiplets and a SYM multiplet and show how the dictionary gives rise to
the correct SUSY transformations. We have SU(4)R indices in the SYM multiplet a, b, . . . =
1, . . . , 4 and we split these into a set of left SU(2)R indices i, j, . . . = 1, 2 and a set of right
SU(2)R indices r, s, . . . = 1, 2. With these sets, we can construct operators with SU(4)R
indices in terms of tensor products of SU(2)R operators OL ⊗ ÕR, where SUSY generators are
10

split as well such that they act on the left or right sector only. We now propose a dictionary
g+(p) = χ+(p) ⊗ ˜χ+(p),
fa
+ =
fi
+ = φi(p) ⊗ ˜χ+(p)
fr
+ = χ+(p) ⊗ ˜φr(p)
,
sab
(p) =



sij(p) = ijχ−(p) ⊗ ˜χ+(p)
sir(p) = φi(p) ⊗ ˜φr(p)
srs(p) = rsχ+(p) ⊗ ˜χ−(p)
,
f−
a (p) =
f−
i = φi(p) ⊗ ˜χ−(p)
f−
r = χ−(p) ⊗ ˜φr(p)
,
g−(p) = χ−(p) ⊗ ˜χ−(p).
(4.19)
With this dictionary, we can do a consistency check between our SUSY transformation
relations (4.16) and (4.18). Consider the SUSY transformation relation [Qa, g+(p)] = |p]fa
+(p).
We can plug in the dictionary prescription for g+(p) (4.19) and check what happens when we
let the SUSY generator act on the left or right sector respectively
[Qa
, g+(p)] = [Qa
, χ+(p) ⊗ ˜χ+(p)] =
|p]φi(p) ⊗ ˜χ+(p) = |p]fi
+(p)
χ+(p) ⊗ |p]˜φr(p) = |p]fr
+(p)
= |p]fa
+(p), (4.20)
where in the upper half of the bracket we have let the SUSY generator act on the left sector
while in the lower half we have let it act on the right sector and then used the transformations
(4.16) in both cases. We see that the resulting transformation matches the N = 4 SYM SUSY
transformation (4.18). We go through the rest of the transformations in Appendix B.
Based on the dictionary (4.19), we conclude that we have a complete map between two
enhanced chiral multiplets and the N = 4 SYM multiplet
[N = 4 SYM] = [NL = 2 chiral] ⊗ [NR = 2 chiral]. (4.21)
This is exactly what we have refered to earlier as chiral squaring, since we are multiplying
two chiral multiplets to form a SYM multiplet.
4.2 N = 3 super Yang-Mills theory from chiral multiplets
Let us now construct a dictionary between the N = 3 SYM multiplet and two chiral multiplets
(one enhanced and one regular N = 1 chiral multiplet). The N = 3 SYM multiplet consists
of one gluon with helicity h = +1 (g+), four gluinos with h = 1
2 (fa
+ and f
123(4)
+ ), six scalars
with h = 0 (sab and sab(4)), four gluinos with h = −1
2 (f
a(4)
− and f123
− ), and one gluon with
h = −1 (g
ab(34)
− ), where a, b, c = 1, 2, 3 are SU(3) indices. The ﬁelds can be written as
g+, fa
+ f
123(4)
+ , sab
sab(4)
, f−
a(4) = abf
b(4)
− f−
123 =
1
4!
abcfabc
− , g− = −
1
4!
abcg
abc(4)
−
(4.22)
For the N = 3 SYM multiplet we have two multiplets, one more positively-valued helicity
multiplet and its CPT conjugate. For the positive helicity multiplet we have the following
SUSY transformation relations
[Qa
, g+(p)] = |p]fa
+(p),
[Qa
, fb
+(p)] = |p]sab
(p),
[Qa
, sbc
(p)] = |p] abc
f−
123(p),
[Qa
, f−
123(p)] = 0,
[Q†
a, g+(p)] = 0,
[Q†
a, fb
+(p)] = |p δb
ag+(p),
[Q†
a, sbc
(p)] = |p 2!δ[b
a f
c]
+(p),
[Q†
a, f−
123(p)] = |p abcsbc
(p),
(4.23)
11

and similarly for the CPT conjugate multiplet. We propose a dictionary for between the
N = 3 SYM field content and two chiral multiplets (one enhanced chiral multiplet and one
regular N = 1 chiral muliplet)
g+(p) = χ+(p) ⊗ ˜χ+(p),
fa
+ =
fi
+ = φi(p) ⊗ ˜χ+(p)
f3
+ = χ+(p) ⊗ ˜φ+(p)
,
sab
(p) =
s12(p) = χ−(p) ⊗ ˜χ+(p)
si3(p) = φi(p) ⊗ ˜φ+(p)
,
f−
123 = χ−(p) ⊗ ˜φ+(p),
(4.24)
where a = 1, 2, 3 and i = 1, 2. There is a similar dictionary for the CPT conjugated multiplet.
We can go through the same procedure as for the N = 4 SYM multiplet and check that this
dictionary constructs a consistent map using (4.16). We conclude that
We can construct dictionaries between the N = 2 SYM multiplet and chiral multiplets in two
ways:
[N = 2 SYM] = [NL = 2 chiral] ⊗ [NR = 0 chiral],
[N = 2 SYM] = [NL = 1 chiral] ⊗ [NR = 1 chiral].
(4.26)
However only the former will work. The latter is equivalent to [NL = 2 chiral] ⊗ [NR =
2 chiral] (the enhanced and regular chiral multiplets have the same field content) and this will
be equivalent to N = 4 SYM multiplet. In reality though, we will end up with an N = 2 SYM
multiplet coupled to a matter hypermultiplet, but we will not explore this possbility here.
The N = 2 SYM multiplet consists of one gluon with helicity h = +1 (g+), two gluinos
with h = 1
2 (fa
+), two scalars with h = 0 (s12 and s(34)), two gluinos with h = −1
2 (f
a(34)
− ),
and one gluon with h = −1 (g
ab(34)
− ) where a, b, c, d = 1, 2 are SU(2) indices. The fields can
be rewritten as
g+, fa
+, s12
s(34)
, f−
a(34) = abf
b(34)
− , g− = −
1
2!
abg
ab(34)
− . (4.27)
We again have two multiplets, where the more positive helicity multiplet has the following
SUSY transformation relations
[Qa
, g+(p)] = |p]fa
+(p),
[Qa
, fb
+(p)] = |p] ab
s12
(p),
[Qa
, s12
(p)] = 0,
[Q†
a, g+(p)] = 0,
[Q†
a, fb
+(p)] = |p δb
ag+(p),
[Q†
a, s12
(p)] = |p abfb
+(p),
(4.28)
and similarly for the CPT-conjugate multiplet. The dictionary for the N = 2 SYM multiplet
will be (NL = 2) ⊗ (NR = 0)
g+(p) = χ+(p) ⊗ ˜χ+(p),
fa
+ = φa
(p) ⊗ ˜χ+(p),
s12
(p) = χ−(p) ⊗ ˜χ+(p),
(4.29)
12

where a = 1, 2. It is now easy to check, using (4.16), that this dictionary constructs a
consistent map
The N = 1 SYM multiplet can be constructed from chiral multiplets with [N = 1 SYM] =
[NL = 1 chiral] ⊗ [NR = 0 chiral].
The N = 1 SYM multiplet consists of one gluon with helicity h = +1 (g+), one gluino with
h = 1
2 (f+), one gluino with h = −1
2 (f−), and one gluon with h = −1 (g−). The positive
helicity multiplet has SUSY transformation relations
[Q, g+(p)] = |p]f+(p),
[Q, f+(p)] = 0,
[Q†
a, g+(p)] = 0,
[Q†
, f+(p)] = |p g+(p),
(4.31)
and similarly for the CPT conjugate multiplet. We can now construct a dictionary between
N = 1 SYM and the two chiral multiplets
g+(p) = χ+(p) ⊗ ˜χ+(p),
f+ = φ+(p) ⊗ ˜χ+(p).
(4.32)
It is now easy to check that this construction is consistent
4.5 Pure Yang-Mills theory from chiral multiplets
Finally for pure Yang-Mills theory, which consists of one gluon with helicity h = +1 (g+) and
one gluon with h = −1 (g−), there is no interesting chiral squaring relations as the chiral side
of the dictionary consists only of N = 0 chiral multiplets. These only contain a Weyl fermion,
so they cannot have any interactions. We are interested in scattering amplitude relations
based on these dictionaries, so in the case of pure Yang-Mills there will be no amplitude
relations.
As a statement, we can say that from tensor multiplying two N = 0 chiral multiplets, we
get a theory with a gluon and two complex scalar ﬁelds, which is not pure Yang-Mills theory,
but rather pure Yang-Mills coupled to two scalars.
13

5 Superamplitudes
With the chiral squaring relations (dictionaries) from Chapter 4, we now want to extend
these squaring relations to amplitude relations. Since we are relating supermultiplets through
squaring of field operators then, when we want to formulate amplitude relations, it will
be convenient to go to the superamplitude formalism where amplitudes have superfields -
collections of fields in a supermultiplet acompanied with on-shell superspace variables - as
external states rather than specific particle states. This should make the chiral squaring more
manifest, as we expect chiral squaring relations manifest themselves through multiplying states
of the same two legs of two amplitudes. Schematically this means that for two amplitudes A1
and A2, we look at leg i in both amplitudes and expect relations like
A1(. . . , X, . . .) × A2(. . . , Y, . . .) ∼ A3(. . . , X ⊗ Y, . . .), (5.1)
such that the resulting amplitude A3 has the ith leg resulting from multiplying the ith legs
from A1 and A2. We shall make this much more concrete later on in the thesis.
To make the superfield formalism concrete, we first need to talk about the superspace
formalism [16]. This is a convenient way to describe the field content of a supersymmetric field
theory. The superspace formalism consists of an extension to the usual spacetime coordinates
xµ in the form of Grassmann valued coordinates. Specifically superspace coordinates will
be included with spacetime coordinates as (xµ, θα, ˜θ ˙α) where θ and ˜θ are Grassmann valued
variables and α and ˙α are spinor indices. If we only have one SUSY generator (supercharge)
in our theory, meaning that we are in an N = 1 supermultiplet, we then want the supercharge
and its conjugate to anticommute with itself
{Q†
˙α, Q†
˙β
} = 0, {Qα, Qβ} = 0, (5.2)
since applying the supercharge twice to a field operator should annihilate it. The anticommutator
of the supercharge with its conjugate is constructable. In fact that there is no representation
of the Lorentz group, which is symmetric in two spinor indices. However we can use the only
available object, the four-momentum pµ = −i∂µ, to construct a non-zero righthand side
{Qα, Q†
˙α} = i(σµ
)α ˙α∂µ. (5.3)
We can realize the above anticommutation relations by introducing the following specific
expressions for the supercharge and its conjugate
Qα =
∂
∂θα
, Q†
˙α = iθα
(σµ
)α ˙α∂µ → θα
pα ˙α. (5.4)
Now consider what happens if we let pµ become lightlike. In this case we can use the
spinor-helicity formalism and introduce the on-shell Grassmann variables η = θα|p]α with
∂
∂θα = |p]α
∂
∂η and then write supercharges as
Q† ˙α
=
n
i=1
|i ˙α
η, Qα =
n
i=1
|i]α
∂
∂η
. (5.5)
14

Chapter 5. Superamplitudes
ΦN=4(pi): g+(pi) fa
+(pi) sab(pi) fabc
− (pi) g1234
− (pi)
Operator: 1 ∂a
i ∂a
i ∂b
i ∂a
i ∂b
i ∂c
i ∂1
i ∂2
i ∂3
i ∂4
i
Table 5.1 – How to pick out specific fields from the superfield. Here
∂a
i = ∂
∂ηa
p=pi
. After applying the differential operator to
pick out the field, set all remaining η’s to zero.
Using this, we can now write down all of the field content of N = 4 SYM theory in terms
of a superfield, as a polynomial expansion of fields in the multiplet with respect to on-shell
Grassmann variables η’s [17]
ΦN=4
= g+ + ηafa
+ +
1
2!
ηaηbsab
+
1
3!
ηaηbηcfabc
− + η1η2η3η4g1234
− , (5.6)
where the Grassmann variables ηa are labeled with the SU(4)R indices, with a = 1, 2, 3, 4.
Supercharges now have SU(4)R index as well
Q† ˙α
a =
n
i=1
|i ˙α
ηa, Qa
α =
n
i=1
|i]α
∂
∂ηa
. (5.7)
Note that we can reduce the amount of supercharges by setting η’s to zero or integrating
them out.
With the on-shell superspace in the baggage, we will now show how to extract specific fields
from the superfield. We can single out a specific field by applying differential operators with
respect to the superspace variables and then set the remaining superspace variables to zero.
See Table 5.1.
We will now introduce the superamplitude - an amplitude expressed with superfields as
external states
AN=4
n (Φ1, Φ2, . . . , Φn). (5.8)
With superamplitudes we do not have specify helicity or external state configurations
of an amplitude. To get a specific amplitude (with a specific helicity and external state
configuration) from the superamplitude, we simply follow the prescription in Table 5.1 for the
desired external state configuration for each leg.
Superamplitudes can be expanded in terms of MHV superamplitudes, NMHV superamplitudes
and so on (see Appendix A). An MHV amplitude (for pure gluons) has two minus helicity
gluons while the rest of the gluons have positive helicity. In parallel, we can define the
MHV part of the superamplitude as the part of the amplitude proportional to (η)8 (since
this corresponds to a pure gluon MHV amplitude). For a pure gluon NMHV amplitude, we
have three minus helicity gluons, so we define the NMHV part of the superamplitude to be
proportional to (η)12. Continuing this logic, we can schematically write [15]
AN=4
n = AMHV
n (η)8
+ ANMHV
n (η)12
+ · · · + AMHV
n (η)4n−8
, (5.9)
where AMHV
n , ANMHV
n etc. are basis amplitudes in the superamplitude expansion. These can
be reduced to a minimal basis through superward identities [18]. The reason that (5.9) works,
is a consequence of the prescription in Table 5.1. Since we can single out specific states of the
superfield, we can apply this to all legs of the superamplitude to single out specific amplitudes.
15

Chapter 5. Superamplitudes
Thus there will exist an expansion like (5.9) where independent component amplitudes (say
the ANkMHV
n amplitudes) can be singled out using the prescription in Table 5.1. We note
that each SU(4)R index of the ηa’s appear in equal numbers in each monomial. Otherwise
we would break SU(4)R symmetry of the superfields and the superamplitude. The SU(4)R
symmetry of the superamplitude corresponds to super Ward identities, i.e. we can transform
different indices into one another and get different amplitudes that are equal.
16

6 Kawai-Levellen-Tye relations
To realize the chiral squaring relations for amplitudes, we will now turn our attention to a
very powerful tool: The KLT relations [19]. The KLT relations originally came from String
Theory, but were later shown to have field theory applications [9,20] from a limiting case of
the stringy KLT relations. The full stringy KLT relations factorize a closed string amplitude
into a product of two open string amplitudes. Specifically it has the form [20]
Mclosed
n =
γ,β
Ãopen
n (n − 1, n, γ, 1)Sα [γ|β]p1 Aopen
n (1, β, n − 1, n), (6.1)
where we sum over two sets of (n − 3) permutations β and γ. The momentum kernel, Sα ,
’glues’ the two open string amplitudes, Ãopen
n and Aopen
n , together to form the closed string
amplitude Mclosed
n . Sα has the explicit form from [20]:
Sα [i1, . . . , im|j1, . . . , jm]p ≡ (πα /2)−m
m
t=1
sin πα p · kit +
m
q>t
θ(it, iq)kit · kiq , (6.2)
which involves a Heaviside stepfunction
θ(ia, ib) ≡
1 if ia appears after ib in the sequence {j1, . . . , jm}
0 if ia appears before ib in the sequence {j1, . . . , jm}
. (6.3)
To get to the field theory limit of the KLT relations (6.1), we take the limit where the
string tension is taken to infinity, which means taking universal Regge slope, α , to zero. The
field theory KLT relations express graviton amplitudes (from the closed string graviton) in
terms of gauge theory amplitudes1. An important note about these gauge theory amplitudes
is, that they are color-stripped color-ordered amplitudes of the gauge theory (as opposed to
full amplitudes of the gauge theory). See Appendix A for more details about color-ordering
and color decomposition. The field theory KLT relations have an explicit expression, proven
in [9], with a manifest (n − 3)! permutation symmetry
Mn =
γ,β∈Sn−3
Ãn(n − 1, n, γ, 1)S[γ|β]p1 An(1, β, n − 1, n), (6.4)
where γ and β are permutations over the legs 2, . . . , n − 2 and the momentum kernel in this
1
The fact that open strings give rise to gauge theory can be seen by considering two D-branes with an open
string end attached to each D-brane. In the limit of the D-branes being coinciding, one finds that the
massless modes on the open string will correspond to gauge fields.
17

Chapter 6. Kawai-Levellen-Tye relations
limit simply takes the form
S[i1, . . . , im|j1, . . . , jm]p1 ≡ lim
α →0
(πα /2)−m
m
t=1
sin πα p1 · kit +
m
q>t
θ(it, iq)kit · kiq
= 2m
m
t=1
p1 · kit +
m
q>t
θ(it, iq)kit · kiq
=
m
t=1
sit,1 +
m
q>t
θ(it, iq)sit,iq ,
(6.5)
where θ(ia, ib) is deﬁned in (6.3). This version of the KLT relations, (6.4), only works for
pure gravity and pure Yang-Mills theory. It is however possible to extend these relations to
supersymmetric versions of gravity and Yang-Mills theory, which take the exact same form as
(6.4) when written in the superamplitude formalism.
18

7 Supersymmetric Kawaii-Levellen-Tye
relations and diamond diagrams
With the KLT relations written down in a field theory form (6.4), we will now expand
these to supersymmetric versions of gravity and gauge theory amplitudes. We introduced
the superamplitude formalism with (5.8). We can write down the exact same KLT relations
as (6.4) for the supersymmetric version of the KLT relations [21].
As a starting point for our discussion of the super KLT relations, we first consider the
maximally supersymmetric case of the relations, i.e. for NG = 8 supergravity (SUGRA)
and N = 4 SYM multiplets - the G subscript in NG indicates that this number counts the
supercharges in the SUGRA multiplet. We already have the N = 4 SYM superfield in (5.6),
so we will now write down the NG = 8 SUGRA superfield to have a complete description of
the super KLT relations.
The NG = 8 SUGRA multiplet is the maximally supersymmetric spin-2 multiplet, as the
eight supercharges is the maximal number of supercharges that can lower the helicity of
a spin-2 particle with positive helicity to itself with negative helicity. This involves eight
steps of 1/2 helicity lowerings. The NG = 8 SUGRA multiplet consists of one graviton h±,
eight gravitinos ψ±, twenty-eight graviphotons ν±, fifty-six graviphotinos χ± and seventy real
scalars ψ. This can all be represented in the superfield formalism with on-shell superspace
variables [21]
ΦNG=8
= h+ + ηAψA
+ +
1
2!
ηAηBνAB
+ +
1
3!
ηAηBηCχABC
+ +
1
4!
ηAηBηCηDφABCD
+
1
5!
ηAηBηCηDηEχABCDE
− +
1
6!
ηAηBηCηDηEηF νABCDEF
−
+
1
7!
ηAηBηCηDηEηF ηGψABCDEFG
− + η1η2η3η4η5η6η7η8h12345678
− ,
(7.1)
and with the corresponding superamplitude for NG = 8 SUGRA
MNG=8
n (Φ1, Φ2, . . . , Φn). (7.2)
This can also be expanded in terms of component amplitudes MNkMHV
n dressed with strings
of ηi,A’s where capital letters A, B, . . . = 1, . . . , 8 are the SU(8)R indices (like in (5.9)). Like
for SYM superamplitudes, each SU(8)R index appears an equal number of times in each
monomial such that the superamplitude is SU(8)R invariant. This results in super Ward
identities for the different component amplitudes.
With the superamplitudes for the N = 4 SYM multiplet (5.8) and the NG = 8 SUGRA
multiplet (7.2), we can write down the n-point super KLT relations [21]
MNG=8
n =
γ,β∈Sn−3
Ã
Ñ=4
n (n − 1, n, γ, 1)S[γ|β]p1 AN=4
n (1, β, n − 1, n). (7.3)
where γ, β are permutations over legs 2, . . . , n − 2 and the momentum kernel, S[γ|β]p1 , is
defined in (6.5). The above super KLT-relation has been proven in [22]. The superamplitude
19

Chapter 7. Supersymmetric Kawaii-Levellen-Tye relations and diamond diagrams
expansions correctly yield all the correct component relations when the η’s on the supergravity
side are identified with the unions of η’s of the two SYM multiplets.
The relation (7.3) contains all the relevant information, also for reduced supersymmetry
(reduced meaning less than eight supercharges). Since SU(8)R ⊃ SU(4)R ⊗SU(4)R there is a
perfect matching between SU(4)R indices 1,2,3,4 of Ñ = 4 and the SU(4)R indices 5,6,7,8 of
N = 4 for the amplitudes Ã
Ñ=4
n and AN=4
n . The product of these amplitudes match exactly
with the NG = 8 SU(8)R indices 1,2,. . .,8. On the righthand side of (7.3) we get strings of
ηi,a’s and ηi,b’s where i = 1, . . . , n, a = 1, 2, 3, 4 and b = 5, 6, 7, 8. These are matched with the
lefthand side with strings of ηi,A’s with A = 1, . . . , 8. We can thus pick out the appropriate
coefficients of η-strings on the left- and righthand side of (7.3) and then get KLT relations for
the component amplitudes.
We can also derive vanishing identities from the super KLT relations. Consider a violation
of SU(8)R symmetry on the SUGRA side, while each of SU(4)R symmetries of the two SYM
superamplitudes are kept intact. Each SYM superamplitude should be SU(4)R invariant. For
the tilded side we can have the same power k of each ηa, a = 1, 2, 3, 4 while on the non-tilded
side we have another power k of each ηb, b = 5, 6, 7, 8. Now for k = k we break the SU(8)R
symmetry on the SUGRA side and hence it vanishes
0 =
γ,β∈Sn−3
ÃNkMHV
n (n − 1, n, γ, 1)S[γ|β]p1 ANk MHV
n (1, β, n − 1, n). (7.4)
We will not make anymore comments on these vanishing relations, rather we just note
their existance. We will now move on to explore super KLT relations for non-maximal
supersymmetry.
7.1 KLT relations with less supersymmetry: the full map
To get super KLT relations for NG < 8 SUGRA multiplets, there exists a procedure for
removing supercharges through the on-shell superspace formalism. Since each ηa corresponds
to a supercharge, we can remove supercharges by removing ηa’s from the superfield. There two
ways to remove η’s from the superfields: either one sets the desired η to zero or one integrates
it out of the superfield. Both possibilities should be included in the SUSY reduction procedure,
as each method results in a different type of superfield. The two resulting superfields ends
up being each others CPT conjugate, so incorporating both SUSY reduction procedures, we
actually get the full CPT invariant multiplet (both CPT conjugated multiplets). Consider
now the full NG = 8 superfield (7.1) and let us set η8 = 0 in one case and integrate η8 out in
the other case. This will result in the two superfields for NG = 7 SUGRA multiplet.
Consider now the same procedure on the N = 4 SYM superfield (5.6). We introduce what
we call the Φ − Ψ formalism, which is a two-superfield (one superfield Φ and another Ψ)
formalism for reduced SUSY multiplets. The Φ superfield is one where we reduce the number
of supercharges by setting η’s to zero:
ΦN<4
= ΦN=4
ηN +1,...,η4→0
. (7.5)
The N = 3 Φ superfield is obtained by setting η4 → 0 in (5.6)
ΦN=3
= g+ + ηafa
+ +
1
2!
ηaηbsab
+ η1η2η3f123
− , (7.6)
20

−1(1234)
+1
N = 0
−1
2
(234)
−11(234)
+1
+1
2
1
N = 1
+1
012
+1
2
a
0(34)
−1
2
a(34)
−112(34)
N = 2
(3)
(3)
(3)
(3)
+1
2
(4)
0a(4)
−1
2
ab(4)
−1123(4)
−1
2
123
0ab
+1
2
a
+1
N = 3
(6)
(4)
(4)
+1
+1
2
a
0ab
−1
2
abc
−11234
N = 4
Figure 7.1 – Diamond diagrams for superfields of SYM multiplet with different numbers of
supercharges. The SU(N)R indices a, b, c are labeled with a < b < c and they
range from a, b, c = 1, 2, . . . , N. Indices in parenthesis have been integrated
out and these diamonds thus represent the field content of the Ψ superfields.
The numbers inside diamonds show the number of states with a given helicity.
The helicty of the state is noted outside of the diamond.
where a, b = 1, 2, 3 now. To get the CPT conjugated superfield, the Ψ superfield, we start
with the full superfield (5.6) and integrate out η’s
ΨN<4
=
4
a=N+1
dηaΦN=4
. (7.7)
The N = 3 Ψ superfield is (see [14] for how to integrate Grassmann variables)
ΨN=3
= dη4 g+ + ηafa
+ +
1
2!
ηaηbsab
+
1
3!
ηaηbηcfabc
− + η1η2η3η4g1234
−
= f4
+ − ηasa(4)
+
1
2!
ηaηbf
ab(4)
− − η1η2η3g
123(4)
− ,
(7.8)
where we have put the index 4 in a parenthesis to indicate that is has been integrated out.
Our indices are now a, b = 1, 2, 3 so we are left with SU(3)R symmetry. The Φ − Ψ formalism
is sufficient to describe all states in the N = 3 SYM multiplet.
Consider now an N < 4 superamplitude with n legs and have legs i1 < i2 < · · · < im in the
Ψ superfield representation while the remaining legs j1 < j2 < · · · < jl are in the Φ superfield
represention (so m + l = n). We can write such a superamplitude as
AN<4
n,i1...im
=


4
a1=N+1
dηi1,a1 · · ·
4
am=N+1
dηim,am AN=4
n (Φ1, . . . , Φn)


ηN +1,...,η4→0
. (7.9)
With this tool in hand, we are now ready to tackle super KLT relations with less than
maximal supersymmetry. However before moving on we will introduce a tool for keeping track
of the states in supermultiplets and how these states are related by supercharges. This tool is
called a diamond diagram [21] and it is used to describe the field content of a superfield. One
single diamond diagram will correspond to one superfield. We have for example listed all SYM
diamond diagrams in Figure 7.1. For the N = 4 SYM multiplet there is only one superfield
and therefore only one diamond diagram. The points in the diagrams correspond to states of
the supermultiplet, while the lines indicate how the action of a supercharge can transform one
state into another. One cannot go from one diamond to another using supercharges (as we can
only do supersymmetry transformations within a single superfield). Diamonds with indices
in parenthesis represent the content of the Ψ superfield while diamonds with no indices in
parenthesis represents the Φ superfield. We can easily obtain all states of SUGRA multiplets
21

by (tensor) multiplying diamonds two SYM diamonds together, much like we did in Chapter
4 for chiral squaring. For the remainder of this chapter however we will not write out the
SU(N)R indices on the states of the diamond diagrams (like we did in Figure 7.1) to simplify
notation.
7.1.1 The equivalence of N = 3 and N = 4 super Yang-Mills theory
Before working out the super KLT relations with less than maximal supersymmetry, we will
first establish the equivalence between N = 3 and N = 4 SYM multiplets. By equivalence
we mean that the two multiplets have the same field content and hence give rise to the same
physics. This can be seen by using diamond diagrams. Looking at the N = 3 diamond
diagram in Figure 7.1, we see that in combined states of the two diamonds contribute one
gluon, four fermions and six scalars, which is exactly the same field content as the N = 4
SYM multiplet. In fact we can recover the N = 4 superfield exactly from the Φ−Ψ formalism
of the N = 3 multiplet by comparing (5.6) to (8.1) and (8.2)
ΦN=3
+ η4ΨN=3
= g+ + ηafa
+ +
1
2!
ηaηbsab
+ η1η2η3f123
−
+ η4f4
+ − η4ηasa(4)
+
1
2!
η4ηaηbf
ab(4)
− − η4η1η2η3g
123(4)
−
= g+ + ηifi
+ +
1
2!
ηiηjsij
+
1
3!
ηiηjηkfijk
− + η1η2η3η4g1234
−
= ΦN=4
,
(7.10)
where indices a, b = 1, 2, 3 and i, j, k = 1, 2, 3, 4. The same fact is true in SUGRA where the
NG = 7 superfields (in the Φ−Ψ formalism) can be combined into the NG = 8 superfield.The
NG = 8 superfield is given in (7.1), so correspondingly we can write down the NG = 7 Φ − Ψ
superfields. The Φ superfield is given by
ΦNG=7
= h+ + ηAψA
+ +
1
2!
ηAηBνAB
+ +
1
3!
ηAηBηCχABC
+ +
1
4!
ηAηBηCηDφABCD
+
1
5!
ηAηBηCηDηEχABCDE
− +
1
6!
ηAηBηCηDηEηF νABCDEF
−
+ η1η2η3η4η5η6η7ψ1234567
− ,
(7.11)
where indices A, B, . . . , F = 1, 2, . . . , 7. The Ψ superfield is given by
ΨNG=7
= ψ
(8)
+ − ηAν
A(8)
+ +
1
2!
ηAηBχ
AB(8)
+ −
1
3!
ηAηBηCφABC(8)
+
1
4!
ηAηBηCηDχ
ABCD(8)
− −
1
5!
ηAηBηCηDηEν
ABCDE(8)
−
+
1
6!
ηAηBηCηDηEηF ψ
ABCDEF(8)
− − η1η2η3η4η5η6η7h
1234567(8)
− ,
(7.12)
where indices A, B, . . . , F = 1, 2, . . . , 7. It is now straightforward to check that
ΦNG=7
+ η8ΨNG=7
= ΦNG=8
. (7.13)
Hence we conclude that there is a similar equivalence between NG = 7 SUGRA and NG = 8
SUGRA, in that the two multiplets contain the exact same field content and hence the same
physics.
22

7.2 Diamond diagrams and the NG < 8 KLT-relations
As we have seen, we can obtain N < 4 SYM fields by integrating η’s out or setting η’s to zero
(see (8.1) and (8.2)). We have seen that we can do the same for NG < 8 SUGRA. In this
section we will write out the SUGRA field content as a tensor product of two SYM multiplets
like chiral squaring in Chapter 4.
We can identify MNG≤8
n superamplitudes, which has SU(NG)R symmetry, with two SYM
superamplitudes Ã
Ñ
n and AN
n through the super KLT relations (7.3) by splitting the SU(NG)R
indices into two subsets 1, 2, . . . , Ñ and 5, 6, . . . , N, which will be associated with SU( Ñ)R or
SU(N)R indices of the tilded and untilded SYM superamplitudes seperately.
When we have reduced the supersymmetry of the SYM multiplets, we can then correspondingly
translate this to the external states of the SUGRA superamplitudes being in one of four
representations (˜Φ, Φ), (˜Φ, Ψ), (˜Ψ, Φ), and (˜Ψ, Ψ) for each leg. This means that we can write
down super KLT relations, where we have used notation like in (7.9)
γ,β∈Sn−3
Ã
Ñ≤4
n,˜i1,...,˜i ˜m
(n − 1, n, γ, 1)S[γ|β]p1 AN≤4
n,i1,...,im
(1, β, n − 1, n).
=
γ,β∈Sn−3


4
ã1= Ñ+1
dη˜i1,ã1
· · ·
4
ã ˜m= Ñ+1
dη˜i ˜m,ã ˜m
Ã
Ñ=4
n (n − 1, n, γ, 1)


η Ñ +1,...,η4→0
× S[γ|β]p1 ×


8
a1=N+5
dηi1,a1 · · ·
8
am=N+5
dηim,am AN=4
n (1, β, n − 1, n)


ηN +5,...,η8→0
=


4
ã1= Ñ+1
dη˜i1,ã1
· · ·
4
ã ˜m= Ñ+1
dη˜i ˜m,ã ˜m
8
a1=N+5
dηi1,a1 · · ·
8
am=N+5
dηim,am
×
γ,β∈Sn−3
Ã
Ñ=4
n (n − 1, n, γ, 1)S[γ|β]p1 AN=4
n (1, β, n − 1, n)


η Ñ +1,...,η4→0
ηN +5,...,η8→0
=


4
ã1= Ñ+1
dη˜i1,ã1
· · ·
4
ã ˜m= Ñ+1
dη˜i ˜m,ã ˜m
8
a1=N+5
dηi1,a1 · · ·
8
am=N+5
dηim,am MNG=8
n


η Ñ +1,...,η4→0
ηN +5,...,η8→0
≡ MNG≤8
n,(˜i1,...,˜i ˜m);(i1,...,im)
,
(7.14)
where subscripts (˜i1, . . . ,˜i˜m) and (i1, . . . , im) label the external legs in the ˜Ψ and Ψ representations
respectively on the SYM side of the super KLT relations, with ˜m ≤ n and m ≤ n. The
superfields on the SUGRA side will be in one of the four representations mentioned earlier
ˆ (˜Φ, Φ) representation: If the set of legs k are not in the ˜Ψ or Ψ representation on the
SYM side, i.e. k /∈ (˜i1, . . . ,˜i˜m) and k /∈ (i1, . . . , im), then we set all ηk, Ñ+1, . . . , ηk,4 and
ηk,N+5, . . . , ηk,8 to zero and the resulting superfield is
ΦNG= Ñ+N
k = ΦNG=8
k
ηk, Ñ +1,...,ηk,4;ηk,N +5,...,ηk,8→0
(7.15)
ˆ (˜Ψ, Ψ) representation: If the set of legs k are in the both the ˜Ψ and Ψ representations
on the SYM side, i.e. k ∈ (˜i1, . . . ,˜i˜m) and k ∈ (i1, . . . , im), then we integrate out all
23

(6)
(4)
(4)
+1
0
−1
0 ⊗ (6)
(4)
(4)
+1
0
−1
0 =
(8)
(28)
(56)
(70)
(56)
(28)
(8)
+2
00
−2
Figure 7.2 – The tensor multiplication of two N = 4 SYM multiplets into the NG = 8
SUGRA multiplet.
ηk, Ñ+1, . . . , ηk,4 and ηk,N+5, . . . , ηk,8. The resulting superfield is
ΨNG= Ñ+N
k =
4
a= Ñ+1
dηk,a
8
b=N+5
dηk,bΦNG=8
k (7.16)
ˆ (˜Ψ, Φ) representation: If the set of legs k are in the ˜Ψ representation and not in the
Ψ representation on the SYM side, i.e. k ∈ (˜i1, . . . ,˜i˜m) and k /∈ (i1, . . . , im), then we
integrate out all ηk, Ñ+1, . . . , ηk,4 and set all ηk,N+5, . . . , ηk,8 to zero
ΘNG= Ñ+N
k =
4
a= Ñ+1
dηk,a ΦNG=8
k
ηk,N +5,...,ηk,8→0
(7.17)
ˆ (˜Φ, Ψ) representation: If the set of legs k are in not in the ˜Ψ representation and are the
Ψ representation on the SYM side, i.e. k /∈ (˜i1, . . . ,˜i˜m) and k ∈ (i1, . . . , im), then we set
all ηk, Ñ+1, . . . , ηk,4 to zero and integrate out ηk,N+5, . . . , ηk,8
ΓNG= Ñ+N
k =
8
b=N+5
dηk,b ΦNG=8
k
ηk, Ñ +1,...,ηk,4→0
(7.18)
As a note, it happens that the last two superfields, ΘNG and ΓNG combine to form an
SU(NG) matter multiplet when Ñ < 3 and N < 3, as they will not contain a graviton
in this case.
These are the superfields of the superamplitudes of the SUGRA side of (7.14).
We will now show how it is possible to write the SUGRA multiplet states in terms of SYM
multiplet states and how this relates to the diamond diagrams introduced in Figure 7.1. With
the diamond diagrams we can illustrate the super KLT relations squaring of states just like we
did for chiral squaring in Chapter 4. Consider the NG = 8 SUGRA multiplet. We can write
down the states of this multiplet as tensor products of two N = 4 SYM multiplets. First off,
we have the graviton state, h±,
(+1) ⊗ (+1) and (−1) ⊗ (−1).
24

The numbers in the parentheses denote the helicity of the states multiplied together (in this
case two gluons with the same helicity from each multiplet). The 8 × 2 gravitino states ψ±
are given by
(+1/2)4
⊗ (+1), (+1) ⊗ (+1/2)4
and
(−1/2)4
⊗ (−1), (−1) ⊗ (−1/2)4
.
Here superscripts denote the degeneracies of the states. The 28 × 2 vector states v± are
given by
(+1/2)4
⊗ (+1/2)4
, (+1) ⊗ (0)6
, (0)6
⊗ (+1) and
(−1/2)4
⊗ (−1/2)4
, (−1) ⊗ (0)6
, (0)6
⊗ (−1).
The 56 × 2 spin-1/2 fermions χ± come from
(+1/2)4
⊗ (0)6
, (0)6
⊗ (+1/2)4
, (+1) ⊗ (−1/2)4
, (−1/2)4
⊗ (+1) and
(−1/2)4
⊗ (0)6
, (0)6
⊗ (−1/2)4
, (−1) ⊗ (+1/2)4
, (+1/2)4
⊗ (−1).
Finally the seventy scalar fields φ are given by
(+1) ⊗ (−1), (+1/2)4
⊗ (−1/2)4
, (0)6
⊗ (0)6
, (−1/2)4
⊗ (+1/2)4
, (−1) ⊗ (+1).
We can sum up this whole procedure with the tensor multiplication of diamond diagrams of
the N = 4 SYM multiplet, as we have done in Figure 7.2. We can make similar representations
of the NG < 8 SUGRA multiplets through tensoring diamond diagrams of SYM multiplets
with less than maximal SUSY. We will call these KLT products of supermultiplets. The KLT
products of diamond diagrams are equivalent to taking the tensor product of two supefields of
SYM multiplets to form SUGRA superfields like (7.15)-(7.18). This is also the reason why we
have the different representations (˜Φ, Φ), (˜Ψ, Ψ), etc., as these denote that the superfield of a
specific representation is constructible by taking the KLT product of the two SYM superfields
of the representation. For example ΦNG= Ñ+N = ˜Φ
Ñ ⊗ ΦN . All possible SUGRA multiplets
can be categorized into three categories in terms of these KLT products:
ˆ Category I: This category consists of maximal NG = 8 SUGRA, its equivalent NG = 7
SUGRA and also the special case of NG = 6 SUGRA coming from the KLT product
[ Ñ = 3 SYM] ⊗ [N = 3 SYM]. These all have the same field content, so they can be
considered as equivalent multiplets and will give rise to the same physics. The super
KLT relations for these multiplets are (7.3).
ˆ Category II: This category consists of all minimal SUGRA multiplets for 4 ≤ NG < 8.
A SUGRA multiplet is minimal if it only contains a SUGRA multiplet (and no matter
multiplets), meaning that these multiplets only have two diamonds with ΦNG and ΨNG
superfields. These multiplets arise from KLT products of the form [ Ñ = 4] ⊗ [N ≤ 2].
For this category, we get super KLT relations
MNG=4+N
n (ΦNG
i1,...,im1
, ΨNG
j1,...,jm2
) =
γ,β∈Sn−3
Ã
Ñ=4
n (Φ
Ñ=4
1,...,n)S[γ|β]p1 AN≤2
n (ΦN≤2
i1,...,im1
, ΨN≤2
j1,...,jm2
), (7.19)
where (i1, . . . , im1 ) and (j1, . . . , jm2 ) denote legs in the Φ and Ψ representation of the
untilded sector respectively (with m1 + m2 = n).
25

(6)
(4)
(4)
+1
0
−1
0 ⊗ (3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
= 0 00
−3/2
(7)
(21)
(35)
(35)
(21)
(7)
(7)
(21)
(35)
(35)
(21)
(7)
+2
−2
+3/2
Figure 7.3 – The tensor multiplication, in the diamond diagram representation, of the
N = 4 SYM multiplet with the N = 3 SYM multiplet resulting in the NG = 7
SUGRA multiplet.
ˆ Category III: This category consists of the remaining multiplets, namely mininmal
SUGRA multiplets coupled to a variety of matter multiplets. These multiplets have
four diamonds (two for the minimal SUGRA part and two for the matter part) to
describe the complete CPT-complete state space. The super KLT relations for these
multiplets are
MNG= Ñ+N
n (ΦNG
i1,...,im1
, ΨNG
j1,...,jm2
, ΘNG
k1,...,km3
, ΓNG
l1,...,lm3
) =
γ,β∈Sn−3
ÃN≤2
n (Φ
Ñ≤2
i1,...,im1 ;l1,...,lm3
, Ψ
Ñ≤2
j1,...,jm2 ;k1,...,km3
)
× S[γ|β]p1 × AN≤2
n (ΦN≤2
i1,...,im1 ;k1,...,km3
, ΨN≤2
j1,...,jm2 ;l1,...,lm3
),
(7.20)
where (i1, . . . , im1 ), (j1, . . . , jm2 ), (k1, . . . , km3 ), and (l1, . . . , lm3 ) are all labels of external
legs in the Φ, Ψ, Θ, and Γ representation on the SUGRA side of the super KLT relations
(with m1 + m2 + 2m3 = n). If the number of Θ legs are not equal to the number of Γ
legs the SU(NG)R symmetry is violated.
All SUGRA multiplets, their KLT products and their descriptions (according to the above
categorization) are summarized in Table 7.1.
In the following sections, briefly discuss the diamond diagram multiplication associated with
the KLT products in Table 7.1.
7.2.1 Diamond diagrams for the NG = 7 theory
The equivalence between the N = 3 and N = 4 SYM multiplets, shown in Section 7.1.1,
ensures that we have an equivalence between NG = 7 and NG = 8 SUGRA (as we also showed
in Section 7.1.1). The construction of the NG = 7 SUGRA multiplet can be done in terms of
diamond diagrams. This is shown in Figure 7.3. This is the only way to construct the NG = 7
SUGRA multiplet.
7.2.2 Diamond diagrams for the NG = 6 theories
When constructing NG = 6 SUGRA multiplets, we have two KLT products available: [ Ñ =
4] ⊗ [N = 2] and [ Ñ = 3] ⊗ [N = 3]. The former case is minimal NG = 6 SUGRA (see Table
7.1). Minimal NG = 6 SUGRA contains one graviton h±, 6 gravitinos ψ±, 16 vectors v±, 26
26

NG
˜N ⊗ N Description
8 4 ⊗ 4 Maximal NG = 8 supergravity (Category I)
6 4 ⊗ 2 Minimal NG = 6 supergravity with an SU(6) R-symmetry (Category II)
4 2 ⊗ 2 NG = 4 supergravity coupled to a vector multiplet (Category III)
2 1 ⊗ 1 NG = 2 supergravity coupled to a hypermultiplet (Category III)
1 1 ⊗ 0 NG = 1 supergravity coupled to a chiral multiplet (Category III)
0 0 ⊗ 0 Einstein gravity coupled to two scalars (Category III)
Table 7.1 – All possible SUGRA multiplets constructed through KLT products of SYM
multiplets. The ﬁrst column denotes the number of supercharges in the
SUGRA multiplet. The second column denotes the KLT product, i.e. which
SYM multiplets are tensor multiplied to construct the SUGRA multiplet. The
third column is a description of the SUGRA multiplet according to the
categorization given in this section.
(6)
(4)
(4)
+1
0
−1
0 ⊗
+1
0
−1
0
= 0 00
+2
−1
+1
−2
(6)
(15)
(20)
(15)
(6)
(6)
(15)
(20)
(15)
(6)
N = 4 SYM multiplet with the N = 2 SYM multiplet resulting in the
minimal NG = 6 SUGRA multiplet.
spin-1/2 fermions χ± and 30 scalars (see Figure 7.4). The latter KLT product is equivalent
to NG = 8.
27

(3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
⊗ (3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
=
+2
−1
+3/2
−3/2
+3/2
−3/2
+1
−2
0 0 0 00
(6)
(15)
(20)
(15)
(6)
(6)
(15)
(20)
(15)
(6)
(6)
(15)
(20)
(15)
(6)
(6)
(15)
(20)
(15)
(6)
N = 3 SYM multiplet with the N = 3 SYM multiplet resulting in an NG = 6
SUGRA multiplet coupled to matter multiplets. This SUGRA multiplet is
equivalent to the NG = 8 SUGRA multiplet
To make the equivalence to the NG = 8 SUGRA multiplet clear, we consider the diamond
diagram representation of the KLT product in Figure 7.5. At a superfield level, we can write
the ΦNG=8 superfield (see (7.1)) in terms of Φ (setting η4, η8 → 0), Ψ (integrating out η4 and
η8), Θ (integrate out η4 and set η8 → 0) and Γ (setting η4 → 0 and integrate out η8). It can
then be shown that
ΦNG=8
= ΦNG=6
+ η4η8ΨNG=6
+ η4ΘNG=6
+ η8ΓNG=6
. (7.21)
To get the general minimal SUGRA superamplitudes (with no matter multiplets), we start
with the super KLT relation (7.14) and we set m = ˜m and ij = ˜ij, j = 1, . . . , m, such that
the relations simplify to the form
MNG≤8
n,(i1,...,im);(i1,...,im) =
4,8
a1= Ñ+1,N+5
dηi1,a1 · · ·
4,8
am= Ñ+1,N+5
dηim,am
×
γ,β∈Sn−3
Ã
Ñ=4
n (n − 1, n, γ, 1)S[γ|β]p1 AN=4
n (1, β, n − 1, n)
η Ñ +1,...,η4→0
ηN +5,...,η8→0
,
(7.22)
since we only have ΦNG and ΨNG on the SUGRA side, which is exactly what we have in
minimal SUGRA multiplets.
For five supercharges in the SUGRA sector, we have two possible KLT products:
[ Ñ = 4] ⊗ [N = 1] and [ Ñ = 3] ⊗ [N = 2]. The former KLT product is represented in terms
of diamond diagrams in Figure 7.6. Here we have one graviton h±, five gravitinos ψ±, ten
vectors v±, eleven spin-1/2 fermions ψ± and ten scalars. This constitutes the external states
for minmal NG = 5 SUGRA. For the latter KLT product, we end up with a diamond diagram
representation with four diamonds, which is shown in Figure 7.7.
Due to the correspondance between Ñ = 3 and Ñ = 4 SYM multiplets, the SUGRA
multiplet, from [ Ñ = 3]⊗[N = 2], is equivalent to minimal NG = 6 SUGRA. This equivalence
can be shown by considering
ΦNG=6
= ΦNG=5
+ η6ΘNG=5
, ΨNG=6
= ΨNG=5
+ η6ΓNG=5
. (7.23)
28

(6)
(4)
(4)
+1
0
−1
0 ⊗ −1
2
−1
+1
+1
2 =
−1/2
+2
0 0 00
(5)
(10)
(10)
(5) (5)
(10)
(10)
(5)
−2
+1/2
(3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
⊗
+1
0
−1
0
= 0 0 0 00
+2
−1/2
+3/2
−1
+1
−3/2
+1/2
−2
(5)
(10)
(10)
(5)
(5)
(10)
(10)
(5)
(5)
(10)
(10)
(5)
(5)
(10)
(10)
(5)
Figure 7.7 – The tensor multiplication, in the diamond diagram representation, of an
N = 3 SYM multiplet with an N = 2 SYM multiplet resulting in the NG = 5
SUGRA multiplet coupled to matter multiplets. This SUGRA multiplet is
equivalent to the NG = 6 SUGRA multiplet.
For four supercharges, there exists three possible KLT products resulting in SUGRA multiplets:
[ Ñ = 4] ⊗ [N = 0], [ Ñ = 3] ⊗ [N = 1], and [ Ñ = 2] ⊗ [N = 2]. The first case corresponds
to minimal NG = 4 SUGRA, which is illustrated in Figure 7.8. This minimal case has one
graviton h±, four gravitinos ψ±, six vectors v±, four spin-1/2 fermions χ± and two scalars. The
second KLT product is equivalent to the minimal NG = 5 SUGRA, which forms a diamond
diagram representation shown in Figure 7.9. The equivalence can be seen by considering
ΦNG=5
= ΦNG=4
+ η5ΘNG=4
, ΨNG=5
= ΨNG=4
+ η5ΓNG=4
. (7.24)
The third KLT product is shown in the diamond diagram representation in Figure 7.10.
We end up with minimal NG = 4 SUGRA multiplet coupled to two vector multiplets. The
vector multiplets combined consist of two vectors v±, 8 spin-1/2 fermions χ± and 12 scalars
(the prime notation means that these fields are not from the minimal SUGRA multiplet).
29

(6)
(4)
(4)
+1
0
−1
0 ⊗
−1
+1
= 0
0
+2
−2
(4)
(6)
(4)
(4)
(6)
(4)
(3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
⊗ −1
2
−1
+1
+1
2 =
+2
0
+3/2
−1/2
+1/2
−3/2
0
−2
(4)
(6)
(4)
(4)
(6)
(4) (4)
(6)
(4)
(4)
(6)
(4)
minimal NG = 4 SUGRA multiplet coupled to matter multiplets. This
multiplet is equivalent to the NG = 5 SUGRA multiplet.
+1
0
−1
0
⊗
+1
0
−1
0
=
+2
0
+1
−1
+1
−1
0
−2
(4)
(6)
(4) (4)
(6)
(4)
0
(4)
(6)
(4) (4)
(6)
(4)
minimal NG = 4 SUGRA multiplet coupled to a two vector matter
multiplets.
30

(3)
(3)
(3)
(3)
+1
2
0
−1
−1
2
0
+1
⊗
−1
+1
=
+2
+1/2
+3/2
0
0
−3/2
−1/2
−2
(3)
(3) (3)
(3)
(3)
(3)(3)
(3)
N = 3 SYM multiplet with the N = 0 SYM multiplet resulting in an NG = 3
SUGRA multiplet equivalent to the minimal NG = 4 SUGRA multiplet.
+1
0
−1
0
⊗ −1
2
−1
+1
+1
2 =
+2
+1/2
+1
−1/2
+1/2
−1
−1/2
−2
(3)
(3)
(3)
(3) (3)0 0 0
(3)
(3)
(3)
N = 2 SYM multiplet with the N = 1 SYM multiplet resulting in an
NG = 3 SUGRA multiplet coupled to a vector matter multiplet.
For three supercharges, we can make two KLT products that result in SUGRA multiplets:
[ Ñ = 3] ⊗ [N = 0] and[ Ñ = 2] ⊗ [N = 1]. In the former case we get an NG = 3
SUGRA multiplet which is equivalent to minimal NG = 4 SUGRA with a diamond diagram
representation in Figure 7.11. This equivalence can be seen from the relations
ΦNG=4
= ΦNG=3
+ η4ΘNG=3
, ΨNG=3
= ΨNG=3
+ η4ΓNG=3
. (7.25)
For the latter KLT product we get an NG = 3 SUGRA coupled to a vector multiplet
(consisting of one vector, four spin-1/2 fermions and six scalars). It has a diamond diagram
representation found in Figure 7.12.
For two supercharges, we can construct two KLT products: [ Ñ = 2] ⊗ [N = 0] and
31

+1
0
−1
0
⊗
−1
+1
=
+2
+1
+1
0
0
−1
−1
−2
N = 2 SYM multiplet with an N = 0 SYM multiplet resulting in an NG = 2
SUGRA multiplet coupled to a vector matter multiplet.
−1
2
−1
+1
+1
2 ⊗ −1
2
−1
+1
+1
2 = 0 00
+1
+2
+1/2 +1/2
−1/2−1/2
−1
−2
NG = 2 SUGRA multiplet coupled to a matter hypermultiplet.
[ Ñ = 1] ⊗ [N = 1]. The former KLT product results in an NG = 2 SUGRA multiplet coupled
to a vector multiplet (consisting of one vector, two spin-1/2 fermions and two scalars). It has
a diamond diagram representation shown in Figure 7.13. The latter KLT product consists of
an NG = 2 SUGRA coupled to a hypermultiplet (consisting of two spin-1/2 fermions and four
scalars). This has a diamond diagram representation found in Figure 7.14.
For one supercharge there is only KLT product: [ Ñ = 1] ⊗ [N = 0]. This gives an NG = 1
SUGRA multiplet coupled to a chiral multiplet (consisting of one spin-1/2 fermion and two
scalars). It has a diamond diagram represenation found in Figure 7.15.
For the NG = 0 case, we can only write the KLT product [ Ñ = 0] ⊗ [N = 0]. The
SUGRA states are one graviton h± and two scalars (see Figure 7.16 for the diamond diagram
32

−1
2
−1
+1
+1
2 ⊗
−1
+1
=
+2
+3/2
+1/2
0
0
−1/2
−3/2
−2
NG = 1 SUGRA multiplet coupled to a chiral matter multiplet.
−1
+1
⊗
−1
+1
=
+2
0
0
−2
NG = 0 SUGRA multiplet coupled to two scalar matter ﬁelds.
representation). This is exactly Einstein gravity coupled to two scalar ﬁelds.
33

8 Chiral Squaring and KLT relations
With the full KLT map between SUGRA and SYM multiplets, we now move on to explore
other ways of deriving KLT maps from super KLT relatios (7.3). As we stated in Chapter 4,
our goal is to find amplitude relations between Yang-Mills type supermultiplets and chiral
multiplets. We derived evidence for such relations based on dictionaries between chiral
multiplets and SYM multiplets. With the super KLT relations in our toolbox, we are ready
to derive new amplitude relations. However, as we are going to see, these relations will
not be between SYM amplitudes and chiral amplitudes, but rather between vector multiplet
amplitudes and chiral amplitudes. The most notable difference is that there is no color-
ordering for the vector multiplet amplitudes, like there is for Yang-Mills amplitudes.
In the previous chapter, we introduced a reduction method, for removing supercharges from
a supermultiplet. This involved integrating out on-shell superspace variables as well as setting
some of these variables to zero. We saw how this gave us the Φ − Ψ formalism for reduced
supermultiplets. However for SUGRA superfields, we also saw the Θ and Γ matter multiplet
fields in (7.17) and (7.18). These superfields came from a mixture of SUSY reductions (i.e.
a mix of integrating out and setting superspace variables to zero) as opposed to the Φ − Ψ
superfields which only contained one reduction type or the other. We will utilize this mixture
reduction on the SYM superfield (5.6) and show how we can reduce the N = 4 SYM multiplet
to an enhanced chiral multiplet.
To start out let us see how the regular SUSY reduction methods work on the N = 4 SYM
superfield (5.6). The regular SUSY reduction will simply lead to N < 4 SYM superfields in
the Φ − Ψ formalism. To get the N = 3 SYM superfields, we start by removing η4. First of
the Φ superfield is given by
ΦN=3
= ΦN=4
η4→0
= g+ + η1f1
+ + η2f2
+ + η3f3
+
+ η1η2s12
+ η1η3s13
+ +η2η3s23
+ η1η2η3f123
− .
(8.1)
The N = 3 Ψ superfield is given by
ΨN=3
= η4ΦN=4
= f
(4)
+ − η1s1(4)
− η2s2(4)
− η3s3(4)
+ η1η2f
12(4)
− + η1η3f
13(4)
− + η2η3f
23(4)
− − η1η2η3g
123(4)
− ,
(8.2)
where we again use the notation of indices in parenthesis, which are indices that have been
integrated out. The two N = 2 SYM superfields are found by continuing this procedure
ΦN=2
= ΦN=4
η3,η4→0
= g+ + η1f1
+ + η2f2
+ + η1η2s12
, (8.3)
ΨN=2
= dη3dη4ΦN=4
= −s(34)
− η1f
1(34)
− − η2f
2(34)
− − η1η2g
12(34)
− . (8.4)
Finally the N = 1 SYM superfields are given by
ΦN=1
= ΦN=4
η2,...,η4→0
= g+ + η1f1
+, (8.5)
ΨN=1
=
4
a=2
dηaΦN=4
= −f
(234)
− + η1g
1(234)
− . (8.6)
34

Chapter 8. Chiral Squaring and KLT relations
(2) (2)0
+1/2
−1/2
Figure 8.1 – The diamond diagram representation of the two superfields (8.7). The helicity
of states are written outside of diamonds, while the number of states are
written inside the diamond.
We now move on to a new procedure for reducing the number of supercharges in a superfield.
The Φ − Ψ formalism removes the fields of either from the bottom or the top of the helicity
hierarchy from the original N = 4 SYM superfield (5.6). If we instead truncate the N = 4
SYM superfield from both the top and the bottom at the same time, we can construct an
N = 2 chiral superfield. The way to do this is to integrate out one of the superspace variables
in the set (η1, η4) and set the other to zero. This will yield two chiral superfields, as we
have two choices for which superspace variable we integrate out (and we always set the other
variable to zero)
χN=2
1 = dη1ΦN=4
η4→0
= f
(1)
+ + η2s(1)2
+ η3s(1)3
+ +η2η3f
(1)23
− ,
χN=2
2 = dη4ΦN=4
η1→0
= f
(4)
+ − η2s2(4)
− η3s3(4)
+ η2η3f
23(4)
− .
(8.7)
The superfields of (8.7) can also be represented in terms of diamond diagrams, which has
been done in Figure 8.1.
When deploying the SUSY reduction procedure of (8.7) on both sets of superspace variables
on the SYM side of the super KLT relations, we do the exact same thing on the SUGRA side
of the relations. This means that we get four different ways of reducing the N = 8 SUGRA
superfield. We get the following four superfields on the SUGRA side of the super KLT relations
νN=4
1 = −ν
(15)
+ − ηAχ
A(15)
+ −
1
2!
ηAηBφAB(15)
−
1
3!
ηAηBηCχ
ABC(15)
− − η2η3η6η7ν
2367(15)
− ,
νN=4
2 = −ν
(45)
+ − ηAχ
A(45)
+ −
1
2!
ηAηBφAB(45)
−
1
3!
ηAηBηCχ
ABC(45)
− − η2η3η6η7ν
2367(45)
− ,
νN=4
3 = −ν
(18)
+ − ηAχ
A(18)
+ −
1
2!
ηAηBφAB(18)
−
1
3!
ηAηBηCχ
ABC(18)
− − η2η3η6η7ν
2367(18)
− ,
νN=4
4 = −ν
(48)
+ − ηAχ
A(48)
+ −
1
2!
ηAηBφAB(48)
−
1
3!
ηAηBηCχ
ABC(48)
− − η2η3η6η7ν
2367(48)
− ,
(8.8)
where we now have SU(4)R indices on the SUGRA side: A, B, C = 2, 3, 6, 7. We can also
represent the superfields of (8.8) in terms of diamond diagrams, as in Figure 8.2.
Next up, we will investigate how this new way of SUSY reduction will affect the super KLT
relations (7.3).
8.1 Chiral KLT relations
With the derivation superfields for N = 2 chiral multiplets from the N = 4 SYM multiplet,
(8.7), and the corresponding vector superfields from the SUGRA multiplet (8.8), we can now
35

(4)
(6)
(4)
(4)
(6)
(4)
(4)
(6)
(4)
(4)
(6)
(4)
000
+1/2
+1
−1/2
−1
−1/2
−1
+1/2
+1 +1
+1/2
−1/2
−1
Figure 8.2 – The diamond diagram representation of the fields (8.8). The helicity of states
are written outside of diamonds, while the number of states are written inside
the diamond.
construct explicit KLT relations for chiral squaring. In [23] the existence of such relations
was suggested based on squaring relations between chiral multiplets and the N = 4 SYM
multiplet. However we now arrive at different chiral squaring amplitude relations, as our
chiral KLT relations relate vector amplitudes to chiral amplitudes and take the explicit form
MN=4
n (ν1)N=4
i1,...,im1
, (ν2)N=4
j1,...,jm2
, (ν3)N=4
k1,...,km3
, (ν4)N=4
l1,...,lm4
=
γ,β∈Sn−3
Ã
Ñ=2
n (χ1)
Ñ=2
i1,...,im1 ;k1,...,km3
, (χ2)
Ñ=2
j1,...,jm2 ;l1,...,lm4
× S[γ|β]p1 × AN=2
n (χ1)N=2
i1,...,im1 ;j1,...,jm2
, (χ2)N=2
k1,...,km3 ;l1,...,lm4
,
(8.9)
where m1 +m2 +m3 +m4 = n and legs i1, . . . , im1 are in the ν1 representation, legs j1, . . . , jm2
are in the ν2 representation, and so on. The ordering on the chiral side of (8.9) is the same as
on the SYM side of (7.3) (i.e. color-ordered amplitudes, see Appendix A) and the momentum
kernel is given in (6.5). The field content of (8.9) is given in (8.7) for the chiral superfields and
(8.8) for the vector superfields. We note that there is no restriction on how many times each
field can to appear, as each superfield on both the chiral side and the vector side contains the
same Grassmann variables.
In Figure 8.3 we have shown how the squaring of states can be represented in terms of the
diamond diagrams presented in Figures 8.1 and 8.2.
With these new chiral KLT relations we also need an interpretation of the amplitudes. On
the chiral side of (8.9) we have amplitudes that looks like Yukawa theory amplitudes. They
are however color-ordered, which means that they get some additional vanishing relations for
these amplitudes. On the vector side of (8.9) we have amplitudes involving vector multiplets.
One would perhaps think that these amplitudes are SYM amplitudes, but this is not the case.
The amplitudes are fully symmetric with respect to momentum permutations. There exists no
three-point MHV/anti-MHV amplitude, since the three-vector amplitude would factorize into
two three-fermion amplitudes, which both vanish since observables have to be Lorentz scalars.
For abelian gauge theory, there are no three-point MHV/anti-MHV amplitudes either. Indeed
from the SUGRA multiplet, the vector particle of the multiplet is the graviphoton which is
actually the gravitational equivalent of a photon with a U(1) gauge group [24, 25]. The
graviphoton couples to the energy-momentum tensor, but has a repulsive force for graviphoton
exchange between two (anti)matter particles. The interpretation is that the vector side of (8.9)
describes interactions of matter with an anti-gravitational force.
We will now go into some explicit checks of (8.9) to ensure that the relation actually works
in practice.
36


 (2) (2)0
+1/2
−1/2

 ⊗

 (2) (2)0
+1/2
−1/2


=




(4)
(6)
(4)
(4)
(6)
(4)
(4)
(6)
(4)
(4)
(6)
(4)
000
+1/2
+1
−1/2
−1
−1/2
−1
+1/2
+1 +1
+1/2
−1/2
−1




Figure 8.3 – In this figure we see how the two chiral supermultiplets squares to four vector
multiplets. Each pair of the chiral supermultiplets come from a reduction of
the N = 4 SYM multiplet as in (8.7), whereas the four vector multiplets come
from a reduction of the N = 8 SUGRA multiplet as in (8.8). See also Figures
8.1 and 8.2.
8.2 Some explicit checks
To simplify matters of making these explicit calculations with (??), we will only consider
examples where all legs are in the ν1 representation on the vector side (see (8.8)). The legs
on the chiral side will therefore be in the χ1 representation for both sectors (see (??)). The
resulting simplified chiral KLT relation is then
MN=4
n [(ν1)i] =
γ,β∈Sn−3
Ã
Ñ=2
n [(χ1)i] S[γ|β]p1 AN=2
n [(χ1)i] . (8.10)
We will do explicit checks of three- and four-point relations. The three-point chiral KLT
relations are
MN=4
3 [1, 2, 3] = Ã
Ñ=2
3 [1, 2, 3] AN=2
3 [1, 2, 3] , (8.11)
with the definition S[∅|∅]p1 = 1. The four-point relations take the form
MN=4
4 [1, 2, 3, 4] = s12
Ã
Ñ=2
4 [3, 4, 2, 1] AN=2
4 [1, 2, 3, 4] . (8.12)
We pick out any desired particles for the external states by applying derivatives, in a
similar way to what we did in Chapter 5 in Table 5.1. Let us first consider specific three-point
amplitude could realistically come from local interactions. On the vector side we can consider
a three-point amplitude with two fermions and a vector particle [26]
MN=4
3 [1, 2, 3] → M3 f2+
1 f2+
2 3−
. (8.13)
The notation 3− means that we have a vector particle on leg 3 with negative helicity and f2+
1
means a fermion on leg 1 with positive helicity and R-index 2. On the chiral side of (8.11),
we get
A3[φ2
1φ2
2f−
3 ]A3[f+
1 f+
2 f−
3 ] = 0. (8.14)
37

This product is zero since both amplitudes are vanishing. However we can consider another
chiral KLT relation, which is non-vanishing
M3 f2+
1 f6+
2 3+
= A3[φ2
1f+
2 f+
3 ]A3[f+
1 φ6
2f+
3 ]. (8.15)
There are four different versions of this relation (depending on the choice of SU(4)R indices
on the fermions on the vector side). We can calculate any of these three-point amplitudes
using the spinor helicity notation introduced in Chapter 2 and little group scaling introduced
in Chapter 3. For the vector amplitudes we get M3 f+
1 f+
2 3+ = [13][23]. For the chiral
amplitudes we get A3[φ1f+
2 f+
3 ] = [23] and A3[f+
1 φ2f+
3 ] = [13]. Thus we see that the the
relations (8.15) work out fine. For the all minus helicity version of equation (8.15), we switch
out square brackets with angle brackets. We can also consider a non-minimal coupling [26]
between two graviphotons and a scalar
M3 1−
2−
φ26
3 = A3[f−
1 f−
2 φ6
3]A3[f−
1 f−
2 φ6
3]. (8.16)
Again using little group scaling, we find the vector amplitude
M3 1−
2−
φ26
3 = 12 2
, (8.17)
and the chiral amplitude
A3[f−
1 f−
2 φ6
3] = 12 = A3[f−
1 f−
2 φ6
3], (8.18)
which means that the relation works out.
Having determined some of the three-point relations we move on to do some more nontrivial
checks at four-point (8.12). Consider the exchange of a graviphoton between two fermions
M4 f2+
1 f6+
2 f267−
3 f236−
4 = s12A4 φ2
1f+
2 φ2
3f−
4 A4 f+
1 φ6
2f−
3 φ6
4 . (8.19)
Both sides are constructible, using the BCFW recursion relations from Chapter 3, from the
amplitudes in (8.15). The resulting amplitude on the vector side is quite simple:
M4 f2+
1 f6+
2 f267−
3 f236−
4 = s12
34 2
12 2 . On the chiral side we get amplitudes A4 φ2
1f+
2 φ2
3f−
4 =
A4 f+
1 φ6
2f−
3 φ6
4 = 34
12 . This means that (8.19) works out perfectly.
Consider now a case where the external particles on the vector side are only graviphotons.
We take the four-point chiral KLT relation (8.12) with two negative helicity graviphotons on
legs 1 and 2 and two positive helicity graviphotons on legs 3 and 4
M4 1−
2−
3+
4+
= s12A4[f−
1 f−
2 f+
3 f+
4 ]A4[f−
1 f−
2 f+
4 f+
3 ]. (8.20)
We construct amplitudes from the three-point amplitudes in equations (8.17) and (8.18).
In the BCFW recursion relations there are now internal scalars running. For the amplitudes
(8.17), we have six choices of scalars from the vector multiplet. However only four of these
choices yield a nonzero amplitude. For example
M3 1−
2−
φ23
3 = A3[f−
1 f−
2 f+
3 ]A3[f−
1 f−
2 f−
3 ] = 0. (8.21)
Similar vanishing happens when picking the scalar φ67. There is a similar non-trivial
counting for constructing the chiral amplitudes in equation (8.20), since there are two possible
BCFW-internal scalars in each chiral multiplet. Taking this into account, we get a counting
factor for each amplitude. The vector amplitude ends up taking the form
M4 1−
2−
3+
4+
= 4
12 2
[34]
34
, (8.22)
38

where the factor of 4 count the four scalars that can run internally in the BCFW recursion
relations. Similar counting for the chiral multiplet results in
A4[f−
1 f−
2 f+
3 f+
4 ] = 2
12
34
. (8.23)
It can now easily be checked that the relation (8.20) holds.
39

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