The double-copy method for supergravity amplitudes

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The double-copy method for supergravity

amplitudes

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Dissertation in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE WITH A MAJOR IN PHYSICS Department of Physics

By

Maor Ben-Shahar

Supervisor: Marco Chiodaroli

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Abstract

The double-copy construction enables the calculation of scattering amplitudes in theories of gravity by combining amplitudes from gauge theories. It relies on obtaining numer-ators that obey a duality between color and numerator factors, called color-kinematics duality. This construction is reviewed, along with the spinor-helicity formalism for on-shell states and supersymmetry in amplitudes. Using generalized unitarity, a one-loop amplitude is verified from literature for a N = 2 theory obeying color-kinematics dual-ity. This amplitude, along with a one-loop amplitude for a N = 0 theory are combined with the double copy in order to produce one-loop amplitudes from homogeneous super-gravities. The one-loop divergence is studied with the methods of counterterm analysis, that is, operators necessary to cancel the on-shell matrix element of the divergence are identified for the amplitudes studied. It is interesting to note that all vectors produced from the double copy behave in the same way, that is, have the same divergence, for the four special cases of the magical supergravities. Furthermore, one of the counterterms vanishes for these four special cases, which is likely related to the enhanced symmetry that these theories posses.

Svensk sammanfattning

Kvantfältteori kan användas för att räkna ut de resulterande partiklarna som uppst˚ar vid kollisioner av fundamentala partiklar. Dessa uträkningar är de s˚a kallade spridningsam-plituderna. En spridningsamplitud kan användas för att räkna ut sannolikheten för att ett visst event händer efter en kollision. I denna uppsats presenterar vi en uträkning av en spridningsamplitud som omfattar gravitoner, partiklarna som är ansvariga för den gravitionella kraften. Denna uträkning förlitar sig p˚a amplituder i gaugeteorier, och exempel p˚a s˚adana teorier är de som beskriver kvarkar, fotoner och elektroner. Am-plituderna fr˚an gaugeteorier kombineras med dubbelkopia metoden (“double-copy”) för att producera amplituder i gravitation. I kvantfältteorier är m˚anga amplituder faktiskt oändliga, och dessa oändligheter tas hand om att lägga till s˚a kallade “counterterms” i sin teori. Amplituderna i denna uppsats är ocks˚a oändliga, och vi presenterar de “counterterms” som behövs för att annulera oändligheterna.

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Table of Contents II

1 Introduction

1.1 Popular science introduction

Quantum field theory is a formalism that enables the combination of the principles of special relativity with those of quantum mechanics. It has had great success; the standard model of particle physics, which is the description of the strong, weak and electro-magnetic forces is a quantum field theory. However, no successful theory unifies this model with the fourth known fundamental force, that of gravity. One key difficulty in studying gravitation with the methods of quantum field theory is the sheer complex-ity of the calculations that arise. Some of these problems come about when one studies objects called “scattering amplitudes”. One generally would like to study some “incom-ing” states, then predict what other states would be observed a long way away. The probability of obtaining one outgoing state given another incoming state is the square of the calculated amplitude. Scattering amplitudes are therefore important observables. This thesis uses a modern technique, called the double copy, in order to produce am-plitudes for theories that contain gravity. The double copy takes, as input, amam-plitudes from gauge theories (like those describing quarks), that seemingly have nothing to do with gravitation, and by combining them produces an amplitude that would correspond to a theory that describes gravitation. Another issue that arises with many quantum fields theories is that when one calculates amplitudes, results are formally infinite. It is important to study precisely what kind of infinities are produced in various theories, as under some circumstances the theory can be tweaked in order to compensate for these. Compensating terms are generally called “counterterms”. Along with the double copy, another technique is used in the amplitude calculation called “generalized unitarity”. Perturbative calculations of amplitudes are usually expanded diagrammatically, starting with tree-level diagrams and obtaining corrections with increasing numbers of loops. Generalized unitarity enables one to calculate higher-loop corrections using results from lower order calculations, making the study of amplitudes iterative. The main calculation presented here is of a one-loop amplitude in a class of gravity theories called “homoge-neous supergravities”. Some of the divergences that arises can not be made zero for any choice of the parameters of the construction, not even in the particularly special cases called “magical supergravities”. This implies that the theories will most likely need an infinite number of corrections, making it an “effective field theory”.

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Introduction 2

1.2 Background

The duality between color and kinematics and the double copy construction [1, 2] has made higher-loop calculations in perturbative gravity feasible. This structure was first identified in string-theory perturbative amplitudes, [3] resulting in Kawai-Lewellen-Tye relations. These relations express tree-level supergravity amplitudes as the squares of gauge-theory amplitudes. In field theory, a key method in employing these relations has been unitarity [4,5]; that enables the calculation of higher-loop amplitudes in terms of tree-level diagrams. These methods have been successfully employed in the calculation of amplitudes in maximal supergravity up to five loops [6] which has enabled the explicit study of the ultraviolet properties of the theory [7].

In this thesis the double-copy method is employed in order to collect new informa-tion about the one-loop divergence of homogeneous supergravities with matter. The calculations presented in this thesis were used in a recently published paper [8] and are reproduced here with a focus on pedagogically obtaining the results. Some of the calcu-lations are motivated with background information, while others extended a little further than the original work. Motivation for using the double-copy approach, as opposed to direct computation, is presented in this chapter. In the second chapter we introduce the double-copy construction, and explain why color-kinematics duality is necessary for the double copy to be a valid construction. We study some examples of the double-copy constructions, aided by spinor-helicity formalism and little-group scaling properties of amplitudes. In the third chapter, supersymmetry is introduced from the perspective of amplitudes and we verify expressions for one-loop amplitudes presented in literature for a particular family of theories with N = 2 supersymmetry. These amplitudes obey the color-kinematics relations necessary for double-copy constructions. In the final chap-ter, we collect more one-loop results for non-supersymmetric gauge theories and use them, along with the supersymmetric results, in order to construct the homogeneous-supergravity amplitudes. Some of the one-loop four-point counterterms that appear for the theory are obtained there too.

1.3 Feynman rules for gravity - the direct approach

As motivation for studying amplitudes in gravity using the double copy, let us examine the direct approach to perturbative gravity amplitudes. Consider the Einstein-Hilbert action for gravity in D spacetime dimensions,

S= 2 κ2

Z

dDx√−gR , (1.1) with g = det(gµν) and R the Ricci scalar. Suppose that we are interested in fluctuations

in the metric about flat spacetime, written as

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Figure 1.1: Vertices in perturbative gravity. There are an infinite number of higher point vertices at higher orders of perturbation theory.

where κ is the gravitational coupling and can be assumed small. The Ricci scalar can be expanded from definition

R= gαβgγδRαγβδ , (1.3)

where the Riemann curvature tensor, which can be found in any General Relativity textbook [9], is Rαβγδ= 1 2[gλν,κµ− gµν,κλ− gλκ,νµ+ gµκ,νλ] + gαβ(Γ α νλΓβµκ− ΓακλΓβµν) . (1.4)

We have introduced the comma notation for derivatives, denoting ∂µφ by φ,µ. The

Christoffel symbols appearing in this definition are Γµ_νρ= 1

2g

µσ_(g

σν,ρ+ gσρ,ν− gνρ,σ) , (1.5)

and hence, since the inverse metric appears in these, they contain an infinite series of terms at higher and higher orders of κ.

The√−g appearing in the Lagrangian density can be expanded by using the identities log(det(g)) = Tr(log(g)) and det(η + h) = det(η)det(1 + η−1_{h). For simplicity, indices}

are removed and the metric is written as g = η + h, with this, the expression becomes p−det(g) = p−det(η)exp 1

2Tr[log(1 + η

−1_h)]

, (1.6)

whence one can see that series expansions of the logarithm and exponential produce an infinite number of terms in higher and higher orders of h. This expression will clearly lead to difficult Feynman rules. These have been worked out, famously in the lecture notes by Veltman [10], and in the articles by DeWitt [11, 12, 13]. For example, the three-graviton vertex has over 100 terms [14], and hence perturbation theory quickly becomes unmanageable.

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The double-copy construction 4

2 The double-copy construction

Since directly calculating amplitudes in gravitational theories is a difficult task, it makes sense to seek out more efficient methods. The double-copy construction allows for build-ing amplitudes in theories that contain gravity, from two copies of gauge-theory ampli-tudes, which are often remarkably easier to calculate. The only relevant vertices in pure gauge theory have three-point and four-point interactions, as compared with perturbative gravity, which we saw has an infinite number of vertices. Furthermore, the expressions for off-shell vertices in gravity are far more complex than those in gauge theory, re-quiring much more computational power. In this chapter, the double-copy construction for gravity amplitudes is explained, and spinor-helicity formalism is briefly introduced. This will greatly simplify amplitude calculations in later chapters. Two introductory examples of double-copy constructions are discussed at the end of the chapter.

2.1 Little group

Here we discuss the little group scaling of on shell particles. Little-group scaling can be used to entirely fix three point functions in massless theories, up to a constant. However, this topic is presented here in order to motivate that the double copy gives the correct external states for gravitons. This section closely follows the discussion of Weinberg [15], Chapter 2.

Supposing a reference momentum kµ_{, a proper orthochronous Lorentz transformation}

leaves its square invariant, and does not affect the sign of k0_{. So any momentum p}µ_{can be}

related, by Lorentz transformations, to some reference momentum kµ _{with appropriate}

value of k2 _{and appropriate sign of k}0_{, that is,}

pµ= Lµ_ν(p; k)kν . (2.1) For any such Lorentz transformation, we assume the existence of a unitary operator U(L(p; k)) that will transform a reference one-particle state with momentum k to any other one-particle state with momentum p;

|p, σi = U (L(p; k))|k, σi . (2.2) The extra label σ on the states denotes any other particle labels which classify the state, these are taken to be discrete. Using the property that for two Lorentz transformations

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Λ1 and Λ2, U (Λ1)U (Λ2) = U (Λ1Λ2) we may expand the action of any arbitrary Lorentz

transformation Λ on a single state and obtain U(Λ)|p, σi = U (ΛL(p; k))|k, σi

= U (L(Λp; k))U (L−1(Λp; k)ΛL(p; k))|k, σi . (2.3) The operator L−1_{(Λp; k)ΛL(p; k) ˙}_{=G must belong to a group of operators that leave the}

momentum k invariant, which we call the little group. The little group is not a trivial group in general. The action of this operator G on a state must be of the form

U(G)|k, σi = Dσρ(G)|k, ρi , (2.4)

with an implicit sum over the label ρ, and D is a representation of the little group. Substituting this in the previous equation we get

U(Λ)|p, σi = U (L(Λp; k))Dσρ(G)|k, ρi . (2.5)

But of course the Dσρ are matrix elements and hence commute with the operator

U(L(Λp; k)), so this becomes

U(Λ)|p, σi = Dσρ(G)|Λp, ρi . (2.6)

This equation states that the Lorentz transformation of one-particle states is related to particle states of the transformed momentum via a representation of the little group. We consider massless momenta in this thesis, these can in general be transformed to (E, E, 0, 0) which is invariant under rotations in the y − z plane, described by the group SO(2). These are also invariant under translations, but if we consider non-trivial repre-sentations under translations along with rotations then states would have an additional continuous degree of freedom. Thus the label σ is related to an eigenvalue of the mo-mentum generator J3, which, without proof, is an integer or half integer.

Now we may consider the action of some little group element on our state. We may split any Lorentz transformation in to a composition of rotation, under which the state does not transform trivially, as well as other operators, denoted by O for which the state has always eigenvalue 0. These are other Lorentz transformations that leave the state invariant, as mentioned before. The action of the little group is

U(G)|p, σi = exp(O) exp(iJ3θ)|p, σi = exp(iθσ)|p, σi . (2.7)

From this we see that for massless particles the representation of the little group must be

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This implies that under Lorentz transformations, massless states acquire a phase pro-portional to their discrete helicity label σ in addition to having their momenta transform as usual. We shall return to little-group scaling in the next section, after introducing the spinor-helicity formalism. One of the reasons for introducing spinors is that their little group transformations are easy to work with.

2.2 The spinor-helicity formalism

The spinor-helicity formalism greatly simplifies the matter of calculating amplitudes, and will be used throughout this work. We follow the conventions of Elvang and Huang [16], differing only in the choice of a mostly-minus metric instead of a mostly-plus metric. We shall see how to express little group transformations for the case of spinors, and this will be helpful in determining the helicity of external states constructed as double copies.

We introduce the Feynman slash notation, defining /a = aµ_γ

µ. The conventions for

the gamma matrices are collected in the appendix, see (A.9). For massless fermions the Dirac equation for four component spinors is

/

pv±(p) = 0, u¯±/p= 0 . (2.9)

Spinors that satisfy the Dirac equation as above are often referred to as “on-shell”. The ± subscript enumerates the two independent solutions to the equation, denoting the positive or negative helicity of the wave function. Using the gamma matrix conventions in the appendix we may write the slashed momenta as

/

p= 0 pa˙b

p˙ab 0 !

. (2.10)

The 2 × 2 bispinors are defined

p_a˙b≡ p_µ(σµ)_a˙b ,

p˙ab≡ p_µ(¯σµ)˙ab _. _(2.11)

Note that the determinant of the matrix pa ˙a is equal to p2, which for massless momenta

is zero. Therefore the matrix can be written as an outer product of two two-component vectors

pa ˙a= |p]ahp|˙a , (2.12)

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For the momentum to have real components we must require |p]∗ _{= ±hp|. Pressing on,}

the independent solutions to the Dirac equation (2.9) are

v+(p) = |p]a 0 ! , v−(p) = 0 |pi˙a ! , (2.14) and ¯ u−(p) = (0, hp|˙a) , u¯+(p) = ([p|a,0) . (2.15)

The indices a, ˙a are raised with the antisymmetric symbols (A.2). These correspond to asymptotic states for spins ±1/2 that we encountered in the previous section. We can now define spinor products as hp|˙a|qi˙a=hpqi, and [p|˙ a|q]a=[pq]. Since the indices are˙

raised with antisymmetric Levi-Civita symbols, the products are antisymmetric hpqi = −hqpi. There can not be contractions between spinors with square and angle brackets, unless other objects are put between the spinors. For example, products of spinors with gamma matrices are defined by observing the usual action of the vectors u and v on the matrix: ¯ u−(p)γµv+(q) = (0, hp|˙a) 0 (σµ₎ a˙b (¯σµ)˙ab ₀ ! |q]b 0 ! ≡ hp|γµ|q] . (2.16)

Products of fermions with the same helicity and a gamma matrix are zero, hp|γµ_{|qi =}

0 = [p|γµ_|q].

Observe now that there is freedom in defining the spinors |pi and |p]. Equation (2.12) remains unchanged under the redefinition

|pi → t|pi , |p] → t−1|p] . (2.17) The introduced parameter is not totally free, since the angle and square brackets must satisfy |p]∗= hp| in order for the components of the momentum to be real, which is what will be done in this thesis. Hence t is any complex phase, i.e. there is U (1) = SO(2) freedom in defining the spinor components. This is precisely the little group freedom observed in the preceding section.

Massless vectors have polarizations transverse to momenta, µkµ= 0. We can find

a spinor expression for this; µ−(p) = − hp|γµ_|q] √ 2[qp] , µ +(p) = hq|γµ_|p] √ 2hqpi . (2.18) These satisfy the transversality condition. The momentum q has not been specified, it is an arbitrary reference momentum that can take any value, so long as it is not parallel to p. We see that the little-group scaling factor for these is t±2 under the transformation (2.17). In general, massless particles with helicity h scale with t−2h;

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This important formula can be used to fix amplitudes in three point interactions, but here it is mentioned as it will prove important in studying the external states of the amplitudes produced by the double copy. We shall see this in the next section.

2.3 The double copy

It is well known that higher-spin representations can be constructed by taking tensor products of lower spin representations. The double copy provides a way of constructing higher-spin amplitudes from lower-spin amplitudes, by effectively taking a tensor product of the lower spin amplitudes in a way that gives a reasonable result. That is, the result can be interpreted as an amplitude, meaning, for example, that it should be crossing symmetric, have the correct poles, and be invariant under gauge transformations. In this thesis we are confined to talking only about double-copy realizations that produce spins lower or equal to 2. It is not currently known if it is possible, in this framework, to consistently introduce higher massless spins. In QFT, higher-spin particles often pose a problem [15,17].

Consider a general amplitude at some order in perturbation from a gauge theory, A(L)_m = iL−1gm−2+2LX i Z _dLD_l (2π)LD cini SiDi . (2.20)

Here L denotes the number of loops, and m is the number of external states. The factors ci are “color” factors, that is, the structure constants fabc or representation matrices

ta_R of the gauge group acting on fields in the theory. The sum runs over all graphs that can be constructed by contracting the color tensors such that the contraction is consistent with the choice of external particles. The denominators Di are propagators of

the relevant graph structure, and Si are the symmetry factors of the Feynman diagram.

The numerators ni are what is left; some functions of kinematical variables. These

contain polarization vectors, spinors, loop momenta, and so on.

Amplitudes are said to satisfy color-kinematics duality when identities satisfied by color factors are also satisfied by numerators, for example

ci+ cj+ ck= 0 ⇔ ni+ nj+ nk= 0 . (2.21)

If such relations hold for all amplitudes containing external spin-1 fields, then one can replace color factors with numerators in amplitudes, ci → ni, and so obtain an amplitude

from some theory containing massless spin-2 particles.

Wald shows [18] that under certain assumptions, requiring that the spin-2 fields couple to the stress-energy tensor of the matter in a theory forces the theory to have full diffeo-morphism invariance. For the double-copied amplitudes of spin-2 fields to be invariant under linearized diffeomorphisms we shall see that we are forced to use theories that satisfy the color-kinematics duality as building blocks. At loop level, color-kinematics

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relations are conjectured to exist [2], and if they do, the double-copy still provides a con-struction that is invariant under linearized diffeomorphisms and thus can be interpreted as a loop correction to a gravity amplitude.

We will now see that for a choice of external spin-1 particles in the numerator building blocks, the resultant double copy indeed produces an amplitude that is invariant under linearized diffeomorphisms. Recall that the Ward identity for spin-1 fields states that given some amplitude A = µAµ with an external spin one particle, it is invariant under

the transformation of the polarization vector → + k where k is the momentum of the particle in question. In other words, when a polarization vector is replaced by the momentum vector of the corresponding particle, the amplitude should vanish

A

µ_→kµ = Aµk

µ_{= 0 .} _(2.22)

Since the polarization vectors are absorbed in the numerator factors we may write the ni as nµiµ. In terms of these, we get the result

X i cini SiDi µ→kµ = X i ciniµkµ SiDi = 0 . (2.23)

Since the color factors do not need to be specified for this to hold, the cancellation relies purely on the algebraic relationships between color factors. For the cancellation to take place, the identities that the color factors obey must be employed in order to write the amplitude as a sum of independent color factors multiplied by new functions of momenta, that are each individually gauge independent, meaning are zero under the exchange → k.

Linearized diffeomorphisms are parametrized by the infinitesimal transformations δhµν = ∂µξν+ ∂νξµ . (2.24)

In momentum space, this becomes

δˆhµν = kµξˆν+ kνξˆµ . (2.25)

Amplitudes in any gravitational theory must be invariant under these transformations. When one is considering external states, this can be rephrased in terms of the transforma-tions on the polarization tensors of the graviton; when a polarization tensor is replaced by µν → kµqν+ kνqµ then the amplitude should vanish, where q is some reference

vec-tor. Gauge freedom can be used to impose that the polarization tensors are traceless and transverse to the momenta, requiring the infinitesimal gauge transformation parameter to be transverse to the momentum q·k = 0.

Let us suppose that an amplitude with numerators nithat satisfy the color-kinematics

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constructed by replacing the color factors in (2.20) with another copy of the numerators, and changing the coupling to be that of gravity, is

M(L)_m = iL−1κ 2 m−2+2L_X i Z dLDl (2π)LD nini SiDi . (2.26)

Since the numerators obey the same relations as the color factors, the summationP

i nini

Di

would still vanish if the polarization vector in one of the numerators is replaced with the momentum vector, since the same algebraic rearrangements are possible. Considering the double copied numerators, the new polarization tensor that captures the external state of what was a spin 1 field is now µν_{, and since we have two copies of the numerators}

that obey the color-kinematics duality then the entire summand vanishes under the replacement µν _{→ k}µν ₊µ_kν_; X i nini SiDi µν→kµν+kνµ = X i nini SiDi µν→kµν+ nini SiDi µν→kνµ = 0 . (2.27)

Moreover, this cancellation does not require both numerators to come from the same theory, they must only satisfy the same identities as the color factors for the opposite theory. Denoting the different numerators by ˜nand n, an example of this is

ci+ cj+ ck= 0 ⇔ ñi+ ñj+ ñk= 0 , (2.28)

˜

ci+ ˜cj+ ˜ck= 0 ⇔ ni+ nj+ nk= 0 . (2.29)

It is easy to see that by the above, the following holds; X i nin˜i SiDi µν→kµ˜ν+kνµ = X i nin˜i SiDi µν→kµ˜ν+ nin˜i SiDi µν→kνµ = 0 . (2.30)

The last term is zero since each of the sets of numerators obey the same linear relations as the color factors, by assumption.

More usefully, the double-copy construction only requires one of the two sets of numer-ators to satisfy the color-kinematics relations [2]. The only requirement on the second set of numerators is that they come from a theory that, in general, does have color-kinematics duality, however they may not be expressed in such a form directly. Let the duality satisfying numerators be ni. We assume that the other set, ˜n, is not duality

satisfying but there exist numerators ñ0_i that do satisfy the duality for the same theory. Then ñ = ñ0+ δi, where δi are some functions that depend on gauge choices and field

redefinitions. In order for the amplitudes written with ˜n and ˜n0 to be the same we require X i ˜ nici SiDi =X i ˜ n0_ici SiDi =X i ˜ nici SiDi + δici SiDi , (2.31)

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and therefore the gauge functions δ must satisfy X i δici SiDi = 0 . (2.32)

This cancellation must hold, once again, entirely due to the algebraic properties that the color factors satisfy, and so again we could replace them with some set of duality satisfying numerators and the cancellation will not be spoiled. Using a set of duality satisfying numerators and a set as above that does not satisfy the duality, the double-copy amplitude is M ∝X i nin˜i SiDi =X i ni˜n0i SiDi +X i niδi SiDi =X i nin˜0i SiDi . (2.33)

Hence, as before, this amplitude is invariant under linearized diffeomorphisms. Since the δicould be the outcomes of field redefinitions or gauge transformations in the Lagrangian

from which the numerators come, it may not be trivial to identify these and rewrite the amplitude with duality satisfying numerators. The fact that the double copy only requires one of the two sets of numerators to satisfy the duality makes it remarkably easier to construct amplitudes in gravitational theories, as we shall see in the main calculation of this thesis.

2.4 Tree-level examples

Pure Yang-Mills

This example is of a gauge theory with no other fields other than the gauge boson. The presentation here partly follows [14]. Only two diagrams, and their permutations, con-tribute to the four point, tree level pure Yang-Mills amplitude. The s-channel Feynman diagram in pure Yang-Mills with three-point interactions is given by

ig2 s f

a1a2b_fa3a4b₍₂

1· k2ν2+ 1· e2kν1− (1 ↔ 2))(2k4· 34ν+ 3· 4k3ν− (3 ↔ 4)) . (2.34)

The other channels can be obtained by permutation of the indices. The contribution from the four point contact term is

− ig2 fa1a2b_fa3a4b₍₍ 1· 4)(2· 3) − (1· 3)(2· 4)) + fa1a4b_fa2a3b₍₍ 1· 3)(4· 2) − (1· 2)(4· 3)) + fa1a3b_fa4a2b₍₍ 1· 2)(3· 4) − (1· 4)(3· 2)) . (2.35)

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The three color factors in the latter contribution match the color factors in the s,t and uchannels, and hence by multiplying and dividing by s,t or u the contribution from this diagram can be absorbed in to the other three. The new amplitude will have the form

Atree 4 = nscs s + ntct t + nucu u , (2.36)

with the s channel numerator

ns= (22· k1ν1+ 2· 1k2ν− (1 ↔ 2))(2k4· 34ν+ 3· 4k3ν− (3 ↔ 4))

+ s((4· 3)(1· 2) − (3· 1)(4· 2)) , (2.37)

and the corresponding color factor

cs= fa1ba2fa3a4b . (2.38)

The numerators and color factors for the other two channels are ctnt= csns 2→4→3→2 cunu= csns 2→3→4→2 . (2.39)

The gauge invariance of this amplitude implies that replacing 4 with k4 should make

the amplitude vanish. The individual diagrams however do not vanish under this replace-ment as the numerators ns,t,u are gauge dependent. Replacing 4 → k4 in ns produces

ns → s(1· 2(3· k1− 3· k2) + cyclic(1, 2, 3)) ≡ sf (i, ki) . (2.40)

Simplifications to this result are obtained once conservation of momentum is used, k1+ k2+ k3+ k4= 0, as well as the transversality of the polarization vectors 1· ki = 0.

When the entire tree-level amplitude is considered under the replacement of 4 → k4 the

result is

Atree₄

4→k4 = (cs+ ct+ cu)f (ei, ki) , (2.41)

which is zero by the Jacobi identity for the structure constants of the group. This sug-gests that if the color factors were replaced with other objects that satisfy this identity, cs+ ct+ cu = 0 the amplitude would still satisfy the Ward identity. One set of objects

that satisfy this relation are the numerators of the theory themselves,

ns+ nt+ nu = 0 , (2.42)

which can be shown by direct computation. This is the previously mentioned color-kinematics duality, which happened to be produced almost directly from the Feynman rules. By replacing the color factors with an extra copy of numerators the new amplitude becomes Atree₄ ci→ni,g→κ₂ ≡ −iM tree 4 = κ 2 2 n2 s s + n2_t t + n2_u u , (2.43)

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Figure 2.1: Relevant diagrams for Majorana fermion scattering in a non-abelian gauge theory. Legs are labelled 1 − 4, clockwise from bottom left.

which has incoming polarization tensors µ_iν_i=˙ µν_i . These are still transverse to the momenta. The polarization vectors of the gauge theory can be chosen to satisfy 2

i = 0

which makes µν_i trace-less. We can study the spectrum of the possible helicities of this double-copy by observing how the possible external states transform under little group transformations. The set of possible external states’ helicities is {±1 ± 1} = {2, 0, −2}. Interestingly, the double-copy of pure Yang-Mills does not produce pure gravity but also has a comlpex scalar in the spectrum. In some situations, one may not want additional fields. There are methods of overcoming this, for example when constructing pure gravity, by the addition of ghosts [19].

Fundamental matter

This example is of a Yang-Mills Lagrangian with Majorana fermions in the adjoint representation. The relevant diagrams are shown in figure (2.1), where external states are labeled from 1 to 4 clockwise from bottom left. Prior to fixing a helicity, the amplitude for the scattering in four dimensions is given by

Atree

4 (1i,2j,3k,4l) =

ig2 (p1+ p2)2

ta_ijta_klv¯1γµv2v¯3γµv4+ cyclic(2, 3, 4) . (2.44)

The denominators are simplified using Mandelstam variables s, t and u for the respective channels. By taking out an overall factor of ig2_{, numerators can be extracted from the}

above:

ns= ¯v1γµv2¯v3γµv4 , nt= ¯v2γµv3v¯1γµv4 , nu = ¯v3γµv1v¯2γµv4 , (2.45)

and the color factors are

cs= taijtakl , ct= tajktail , cu = takitajl . (2.46)

Choosing the adjoint representation for the color factors, ta

ij = faij, they would obey

the Jaccobi identity, so we have

cs+ ct+ cu = 0 . (2.47)

Interestingly, the analogous identity for the numerator factors is required to hold in order for supersymmetry to exist at various dimensions, which points to interesting relations

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between color kinematics and supersymmetry [20]. However this is not discussed in this thesis.

We can specialize to four dimensions, and the numerators can be simplified by making a specific choice for the helicities of the incoming fermions. Each fermionic line must have one positive helicity fermion and one negative helicity fermion, for example one could consider Atree

4 (1+i ,2 − j,3 + k,4 −

l ). One of the numerators becomes zero regardless of

the choice of helicities. With this choice, and by using the Fierz identity (A.22), the numerators become

ns= [13]h24i , nt= [31]h24i , nu = 0 , (2.48)

whence it is clear that the numerators satisfy ns = −nt by using the antisymmetry of

the spinor product. Thus we have

ns+ nt+ nu = 0 . (2.49)

Once squaring the numerators, we obtain a spin one external state. This can be verified by examining the little group scaling of the squared amplitude. It is not immediate, however, that this should somehow come from a theory that includes gravity. Let us prove this. First define an internal momentum I = −p1−p2= p3+p4which we will set to

be on-shell later. Let us also only consider the s-channel numerator for the double-copied theory, and rearrange it as follows:

ngravity_s = n2₁₂= h42i2[31]2 = 1 s4 h12i[34]h42i[21]h43i[31] 2 = 1 s4h12i 2_[34]2_h4Ii4_[I1]4 = h4Ii 4_[I1]4 [12]2_h34i2 = h4Ii4 h34i2 × [I1]4 [12]2 . (2.50)

The final result is precisely the product of two three-point amplitudes, each with external spin-1 fields on legs 1, 2 and 3, 4 while leg I belongs to a spin-2 field. This can be observed of course from the little-group transformation properties of the spinors. This result is precisely the square of two fermion-fermion-gluon vertices, reflecting the fact that indeed we chose to simply square numerators of a theory of fermions and gluons. Note that the helicity of the graviton flips as it crosses from one diagram to the other, as it should as its helicity is flipped under parity inversion. It is positive on the side with particles 1 and 2. Another graviton contribution, with opposite helicities is also present, and can be produced by rearranging this result.

The numerator result can be thought of as the residue at the point where the prop-agator 1/s = 1/I2 _{, goes on-shell. This observation will be used in the next chapter to}

go in the opposite direction; that is, to construct loop-level amplitudes from products of tree level amplitudes by restoring propagators. One would think that a scalar exchange

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in the s-channel is possible too. On the left hand side of the cut, we would need to double copy two three point amplitudes, (ψ+_{, ψ}−_{, A}+_{) with (ψ}+_{, ψ}−_{, A}−_{). Those would}

be [12]−1[I1]2 _{and h12i}−1_h2Ii2_{. But two comments are due. Firstly, due to special}

three-point kinematics, either square or triangle brackets must all be zero, meaning that only one of these three point interactions is possible. Moreover, we must relax the condition that angle and square brackets are related by complex conjugation, and allow momenta to be complex in order for the three-point interactions to not vanish. For further details see [16]. Similar considerations also lead to the BCFW recursion relations [21], that allow for the construction of higher-point tree amplitudes from lower-point amplitudes by a construction much like the one seen here. Although we will not discuss this further, similar techniques related to unitarity (and the optical theorem for QFT) can be used to argue that internal gravitons are always present in such double copies.

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Supersymmetry in amplitudes 16

3 Supersymmetry in amplitudes

In this chapter we introduce supersymmetry, following references [22,23]. We will see how to calculate amplitudes in supersymmetric theories, and utilize cuts in order to ob-tain loop-level amplitudes. The amplitudes presented here have already been calculated in [20] and [8], however here the calculation is broken down in order to show all the steps taken in obtaining the results.

3.1 The supersymmetry algebra

The Coleman-Mandula theorem states that under reasonable assumptions the only possi-ble Lie group symmetry of a non-trivial theory is always a direct product of the Poincar´e group and an internal symmetry group [24]. Meaning the generators of the internal sym-metries must commute with generators of the Poincar´e group.1 _{However, there is the}

possibility of including a graded Lie algebra, an algebra with generators Qi

a for which

the anticommutators {Qi α, Q

j

β} can be related to other generators of the algebra. In a

graded Lie algebra, generators with anticommutation relations amongst themselves are called “odd” generators, while generators with commutation relations are called “even”. Anticommutators of odd generators give even generators, while commutators of even generators give even generators again. Commutators of even with odd generators, how-ever, give odd generators. Let Mµν be the generators of the Lorentz group, these obey

commutation relations and hence are even. Therefore the commutator [Qi

α, Mµν] is again

related to generators of the supersymmetry algebra, the only odd generators involved. This implies that Q must be in some representation of the Lorentz group. Since the Q generators have anticommutation relations amongst themselves it makes sense to take them in the spinor representation. A Fermionic field multiplying a bosonic field produces a Fermionic field, and hence the symmetry generated by the Q mixes between Fermionic and bosonic degrees of freedom, this is called a supersymmetry. Since the bosonic and Fermionic fields are related then the associated creation and annihilation operators must also be related to one another by the supersymmetry.

The simplest such algebra for the generators of the symmetry is {QA_α, ˜Q_βB˙ } = 2δBAσ

µ

α ˙βPµ , (3.1)

1

In theories of only massless particles, conformal symmetry is an exception [23]. The special conformal transformations and dilations do not commute with all other generators of the Poincar´e algebra. An example presented in this chapter, N = 4 SYM happens to be of a super conformal theory, but conformal symmetry is not utilized in this thesis.

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where ˜Q = Q† _{is in the conjugate spinor representation. Other commutators vanish}

{Q, Q} = 0 = { ˜Q, ˜Q}. Note that this is the algebra for massless representations, and this will be adhered to for the entirety of this thesis. A spinor index was introduced for the different generators. When the Latin index A, B 6= 1 this algebra is called an extended supersymmetry algebra without central charges, and the name of the supersymmetry is given based on the number of supersymmetry generators. For example, if A, B take values of 1, 2 we have N = 2 supersymmetry.

For motivation and derivation for the other commutation relations obeyed by the supersymmetry generators and generators of the Poincar´e algebra the reader is referred to reference [25], the results are:

[Qα, Pµ] = [ ˜Qαα˙, Pµ] = 0 ,

[Qα, Mµν] = σαµν βQβ , (3.2)

These are to be combined with the Poincar´e algebra to obtain the full algebra of the symmetries of the theory. The first two commutators in the equation above ensure that the generators obey the Jacobi identity with the momentum generators, while the second of these is the result of the fact that the generators transform as spinors.

We may suppose now another symmetry generator, R, that does not commute with the supersymmetry generators,

[QA

α, R] = VBAQBα 6= 0 . (3.3)

Considering the effects of these transformations on the generators, QA

α → VAJQJα and,

¯

Q_βB˙ → V_B†JQ˜_βJ˙ in the algebra of the supersymmetry generators (3.1) the non-trivial

commutator becomes VA_LV_B†K{Q_αL, ˜Q_βK˙ } = 2VALV †L B σ µ α ˙βPµ . (3.4)

In order to preserve the anticommutator we need VI LV

†L

J = δJI, that is, the matrix V

must be a member of the group U (N ). This symmetry is called R-symmetry. It does not need to be a symmetry of an action, moreover it could be spontaneously broken or violated by anomalies.

To find representations of this algebra, let pµ= (E, 0, 0, E), and then the commutator

(3.1) becomes {QI_α, ˜Q_βJ˙ } = 4E 1 0 0 0 ! α ˙β δI_J , (3.5) which implies that all Q2 are zero. Creation and annihilation operators can be defined

aA:= Q A 1 2√E , a †A _:= Q˜ A ˙1 2√E , (3.6)

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which have anticommutation relations

{aA, a†_B} = δA_B . (3.7) Recall that the angular momentum generators are defined as ji = ijkMjk, then using

(3.2) and the definition of the raising and lowering operators one finds that [a, J3] = 1₂a.

Therefore

J3a|pµ, λi= (λ −

1 2)a|p

µ_{, λi ,} _(3.8)

and so as their names suggest, each of the operators aA _{lowers the helicity of a state by} 1

2, and similarly, a †

A raises the helicity by 1

2. Starting from a vacuum state |λ0i which

is annihilated by all the aA_{, one can construct a set of states up to maximum helicity}

λ0+ N /2.

3.2 N

= 4 SYM

The spectrum of the N = 4 super Yang-Mills (SYM) theory contains massless states of helicities 1,1₂,0, −1₂, −1. They have the corresponding annihilation operators

a |{z} 1 gluon g+ aA |{z} 4 gluinos λA aAB |{z} 6 scalars SAB aABC | {z } 4 gluinos λABC a1234 | {z } 1 gluon g− . (3.9)

The Lagrangian for this theory can be found in reference [16], along with transformation properties for the fields under the SU (4) R-symmetry of the theory. The action of the supersymmetry generators is [QA, a(p)] = [p|aA(p) , [QA_{, a}B_{(p)] = [p|a}AB_{(p) ,} [QA, aBC(p)] = [p|aABC(p) , [QA, aBCD(p)] = [p|aABCD(p) , [QA, aABCD(p)] = 0 , [ ˜QA, a(p)] = 0 , [ ˜QA, aB(p)] = |piδABa(p) , [ ˜QA, aBC(p)] = |pi2!δ[Ba aC](p) , [ ˜QA, aBCD(p)] = |pi3!δ[Ba aCD](p) ,

[ ˜QA, aBCDE(p)] = |pi4!δ[Ba aCDE](p) .

The generator QA _{lowers the helicity by half, and adds an index. Similarly, the other}

generator has the opposite action. The mixing of fermionic and bosonic degrees of freedom is clear from the above, due to the half-integer change in the helicity after the action of either generator on the annihilation operators.

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3.2.1 On shell superspace

An off-shell representation of the supersymmetry (anti)commutation relations (3.1) for N = 1 is often introduced by defining

Q_a= −i ∂ ∂θa − (σ

µ₎

a ˙aθ¯˙a∂µ , Q¯˙a= i

∂ ∂ ¯θ˙a + iθ

a_(σµ₎

a ˙a∂µ , (3.10)

these generate the supersymmetry transformations of fields not restricted by any mass-shell conditions, relating tensor fields to spinor fields and vice versa [26]. The θa _are

anticommuting Grassmann variables. Following the discussions in [16], the anticommu-tation relations still hold with the replacement

Qa=

∂

∂θa , Q¯˙a= iθ a_(σµ₎

a ˙a∂µ . (3.11)

The Fourier-transformed anticommutator becomes ∂ ∂θa , −θ a_(σµ₎ a ˙apµ = −(σµ)a ˙apµ . (3.12)

However, we are interested in on-shell superspace for massless particles. This means that the momenta can be represented by spinors, and the relation above can be written

as ∂ ∂θa , θ a_|p] ahp|˙a = |p]ahp|˙a. (3.13)

This anticommutator does not look particularly symmetric, but this can be fixed by introducing a new Grassmann variable η = θa_|p]

a, which can be used to rewrite the θ

derivative as _∂θ∂a = |p]a_∂η∂ . The anticommutator above then becomes

|p]a ∂ ∂η , ηhp|˙a = |p]ahp|˙a. (3.14)

Of course the same relation holds for spinors with raised indices [p|a ∂ ∂η , η|pi ˙a = |pi˙a[p|a. (3.15) This supports the definition of “on shell supercharges”, now defined for any N

qAa ≡ [p|a ∂ ∂ηA

, q_A† ˙a≡ η_A|pi˙a . (3.16) We can now define the superfield

Ω = g++ ηAλA− 1 2ηAηBS AB₋1 6ηAηBηCλ ABC_{+ η} 1η2η3η4g− , (3.17)

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from which the individual fields can be picked out by the action of derivatives with respect to the ηI, followed by setting all remaining ηI to zero. The signs were chosen

here so that fields are picked out by the following derivatives: Field g+ λA SAB λABC g− Derivatives 1 ∂A _∂A_∂B _∂A_∂B_∂C _∂1_∂2_∂3_∂4

It is implicit that there is a momentum dependence Ωi _{= Ω(p}

i) in this definition.

The superamplitudes An(Ω1, ...,Ωn) are functions of the superfields associated to the

in-coming and out-going states. A specific field amplitude can be extracted from the superamplitude by taking appropriate derivatives with respect to the ηi

A. For example,

a four point superamplitude An(Ω1, ...,Ω4) is a polynomial in the Grassmann variables

η_Ai. To select a scalar on leg 1, one takes two derivatives ∂1

A∂B1 of the amplitude.

Returning to the supersymmetry generators, these are now the sum of all the on-shell supersymmetry generators for each particle

QA≡ n X i=1 qA_i = n X i=1 [i| ∂ ∂ηiA , Q˜B ≡ n X i=1 q†_B= n X i=1 |iiη_iB . (3.18)

These must annihilate the superamplitude, QA_A

n = 0 and ˜QAAn = 0. This condition

reproduces the Ward identities for superamplitudes, but it won’t be discussed here. We can begin by guessing that the superamplitude takes the form

An= δ(8)( ˜Q)P , (3.19)

where the Grassmann delta function δ(8)_{( ˜}_{Q) is there to ensure that the amplitude is}

annihilated by ˜Q, and P is some polynomial in the Grassmann variables. The following definition for the delta function is annihilated by multiplying on the left with ˜QB,

δ(8)( ˜Q) = 1 24 4 Y A=1 ˜ QA ˙αQ˜αA˙ = 1 24 4 Y A=1 n X i,j=1 hijiηi_Aη_Aj . (3.20)

To prove this, one must expand the definition of the delta function and observe that ηi B

would anti-commute with any of the η_Aj in the product of sums in the delta function. The interesting term is that which has B = A, it can be written asP

ijk|iihjkiηAi η j AηAk.

This summand is zero by the Schouten identity (A.23).

The annihilation of the delta function by QA requires momentum conservation. The

multiplication of QA on the left of the delta function produces a result proportional to

P

kηkA

P

ihkii[i|, which is zero by momentum conservation (A.24). The SU (4)

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Grassmann variables.2 _{The delta function already is of degree 8, so the remaining factor}

must be some other function of momenta. We simply state that Atree(1234) =

δ(8)( ˜Q)

h12i...h41i (3.21) gives the Parke-Taylor formula when specialized to four external Gluons with helicities A4(1−,2−,3+,4+), and hence is the right result overall. We should make a brief note

about the The Parke-Taylor formula, and color-ordered amplitudes. Both topics are presented in Elvang and Huang [16] in chapters 2 and 3. The color ordering relies on the identity for the color factors

i ˜fabc= Tr(TaTbTc) − Tr(TbTaTc) , (3.22) which can be used to write the four-gluon s-channel diagram color factor as

˜

fa1a2b_f˜ba3a4 _{∝ Tr(T}a1_Ta2_Ta3_Ta4_{) − Tr(T}a1_Ta2_Ta3_Ta4_{) − Tr(T}a1_Ta2_Ta3_Ta4₎

+ Tr(Ta1_Ta2_Ta3_Ta4_{) .} _(3.23)

It seems redundant at first, however it allows to write the amplitude in a very compact form;

A₄ = g2_A

4(1, 2, 3, 4)Tr(Ta1Ta2Ta3Ta4) + perms(234) , (3.24)

where the A are referred to as partial amplitudes. We work with partial amplitudes for the remainder of this chapter.

3.2.2 Unitarity and one-loop amplitudes

The unitarity of the S matrix reads SS†= 1. If we expand the matrix as S = 1 + iT , this constraint becomes i(T†− T ) = T†T. Expanding T as a perturbation series, we see that this equation relates higher-order terms to products of lower-order terms. For example, one-loop diagrams can be related to tree level diagrams by i(T†(1l)− T(1l)_{) =}

T†(t)T(t). The product on the right hand side should be interpreted as one tree-level amplitude flowing into another, and one should sum over all possible tree amplitudes and intermediate momenta. We refer to the product of two tree amplitudes as a cut.3

The cut one-loop superamplitude for N = 4 SYM is decomposed into the product of two tree level amplitudes,

Cuts=

X

states

A4(−l1,1, 2, l2) × A4(−l2,3, 4, l1) . (3.25)

2_{This applies to the Maximally-helicity-violating (MHV) case, which is the only non-zero one for}

four-point amplitudes, other than anti-MHV. However, we restrict ourselves to MHV four-four-point amplitudes for the duration of the thesis.

3

The methods described here are usually referred to as “generalized unitarity” in literature [27], and are distinct from the Optical theorem and Cutkosky rules in that the latter two apply after integration,

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The sum over states is a notation that means that we must integrate over all possible exchanged states, as well as momenta. The sum over superstates amounts to integrating out the intermediate Grassmann variables [28]. The two loop momenta l1 and l2 are

related by conservation of momentum for the two diagrams −l1+ k1+ k2+ l2 = 0 and

l1+ k3+ k4− l2 = 0, they are taken to be on shell. We may plug in the color-ordered

superamplitude (3.21), and use the fact that δ(8)_(L)δ(8)_{(R) = δ}(8)_{(L + R)δ}(8)_{(R) =}

δ(8)( ˜Q)δ(8)_{(R) under the integration, the above equation becomes}

Cuts= δ(8)( ˜Q) h12ih34i Z d4ηl1d 4_η l2 δ(8)(R)

hl₁1ih2l2ihl2l1ihl23ih4l1ihl1l2i

. (3.26)

The d4_η

i is short hand for dηi1...dηi4. It should be understood that the color factors

associated with this component of the cut is simply the product of the color factors of the partial amplitudes. Employing the Grassmann product rules, the integration over d4ηl1d

4_η l2δ

(8)_{(R) gives a factor of hl}

2l1iN. The resultant cut superamplitude is

Cuts= −

δ(8)( ˜Q) h12ih34i

hl₁l2i2

hl₁1ih2l2ihl23ih4l1ihl1l2i

. (3.27)

This can be related back to the tree level four point superamplitude too, Cuts= −Atree4 (1, 2, 3, 4)

hl1l2i2h23ih41i

hl11ih2l2ihl23ih4l1ihl1l2i

. (3.28)

By employing the relation hpqi[pq] = (p + q)2_{, this can be written as}

Cuts= −Atree4 (1, 2, 3, 4)

hl₁l2i2h23ih41i[l11][2l2][l23][4l1]

(l1+ k1)4(l1+ k4)4

. (3.29) The simplification of the denominator was obtained by observing that since each sub diagram is on shell, then relations such as (l2+ k2)2 = (l1− k1)2 must hold. These are

the standard relations of Mandelstam variables for the sub-diagrams. The numerator poses a problem at this stage, but it too can be dramatically simplified:

Numerator = hl1l2i[l22]h23i[3l2]hl2l1i[l14]h41i[1l1]

= Tr+( /l1l/2k/2k/3l/2l/1k/4k/1) = −Tr+( /l1k/1k/2k/3k/4l/1k/4k/1) = −Tr+_{( /}_l 1k/1k/2k/3k/4k/1)2l1.k4 = −Tr+( /k1k/2k/3k/4)4l1.k4l1.k1 = −1 24(k1.k2k3.k4− k1.k3k2.k4+ k1.k4k2.k3)(l1+ k4) 2_(l 1+ k1)2 = su(l1+ k4)2(l1+ k1)2 . (3.30)

The first line is a simple rearrangement, keeping in mind that the spinor products are antisymmetric. To get the second line, 1

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realized by noting that −/p = |p]hp| + |pi[p|, and noting that the projection (1 + γ5)/2

selects the first of those. The notation Tr+_{(...) is short hand for Tr((1 + γ}

5)/2...).

The third line is obtained by using the masslessness of the momenta (/p/p = 0) and the conservation of momenta on each side of the cut. This implies /l1l/2k/2 = /l1( /l1 −

/

k1− /k2) /k2= − /l1k/1k/2, and similarly /k3l/2l/1 = /k3k/4l/1. The next two lines are obtained by

anticommutations, and once again the massless condition of momenta. The sixth equality is a standard gamma matrix relation, noting that the trace with γ5 becomes zero due

to the presence of a totally antisymmetric tensor Tr(γ5a//b/c/d) = −4iµνρσaµbνcρdσ, and

the fact that the four momenta are not independent. When plugged back into the cut amplitude, the obtained inverse propagators in the numerator cancel with those in the denominator, giving,

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

su

(l1+ k1)2(l1− k4)2

. (3.31)

We can now relate this result to the full amplitude. This cut must be related to a sum of one-loop Feynman diagrams, for which the propagators in the s channel go on shell. The cut has two remaining loop dependent propagators, and hence it must be just the cut box diagram. In this case this analysis was easy, and we can restore the cut propagators to obtain the result for the one-loop SYM amplitude

A1−loop₄ (1234) = suAtree₄ (1234)I4(p1, p2, p3) . (3.32)

The box integral is defined in the appendix (A.3), and is only IR divergent. This is just as should be expected for N = 4 SYM which is UV invariant.

3.3 Field-theory orbifolds

We can employ a method for truncating N = 4 SYM into field theories with less su-persymmetry. This is a useful method for obtaining amplitudes which obey the color-kinematics duality, as shown in [20]. We begin by defining a projector acting on a generic field of the theory. The fields carry R symmetry indices aias well as gauge group indices,

distinguished by hats ˆA. A generic field is therefore written ΦAˆ a1...anT

ˆ

A_{. In general such}

a field may also carry global symmetry indices, however we do not consider this case here. We take projections with some orbifold group Γ acting on both the gauge factors and the R symmetry indices

PΦA_aˆ₁_...a_nTAˆ = 1 |Γ| X (r,g)∈Γ rb1...bn a1...bnΦ ˆ A b1...bngT ˆ A_g†_. _(3.33) The tensor rb1...bn

a1...bn will be taken to be diagonal, meaning it has a purely multiplicative

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matri-Supersymmetry in amplitudes 24

ces. We can easily check that this operator is indeed a projection. To this end, we first consider the action of some element of Γ, denoted (q, h):

PΦA_aˆ₁_...a_nTAˆ = 1 |Γ| X (r,g)∈Γ (qr)b1...bn a1...bnΦ ˆ A b1...bnhgT ˆ A_g†_h† = 1 |Γ| X (qr,hg)∈Γ (qr)b1...bn a1...bnΦ ˆ A b1...bnhgT ˆ A_(hg)† _,

and since (rq, hg) ∈ Γ this is left invariant and we can infer that P2 _{= P. We can also}

define the action of the group elements g on gauge group indices by multiplying (3.33) by TBˆ _{on both sides and taking the trace. The new projection is}

PΦA_aˆ₁_...a_n = 1 |Γ| X (r,g)∈Γ rb1...bn a1...bnΦ ˆ B b1...bng ˆ A ˆB _, _(3.34) where we define gA ˆˆB_{= Tr(gT}Aˆ_g†_T_Bˆ ).

The orbifold employed in this thesis is a Z3 orbifold. Assuming an SU (3N ) gauge

group, the action of the orbifold group is governed by the following

rn=       1 0 0 0 0 1 0 0 0 0 einθ ₀ 0 0 0 e−inθ       , gn=    IN 0 0 0 einθIN 0 0 0 e2inθIN    , θ= 2π 3 . (3.35) Under this projection the gauge group is broken down to SU (3N ) → SU (N )3_{× U (1)}2_.

We break the index Â into three components, Â = (â,α, ˆˆ α), where each runs over the¯ representations ˆ a: (N2− 1, 1, 1) ⊕ (1, N2− 1, 1) ⊕ (1, 1, N2− 1) ⊕ 2(1, 1, 1) , ˆ α: (N, ¯N,1) ⊕ (1, N, ¯N) ⊕ ( ¯N,1, N) , ˆ ¯ α: ( ¯N, N,1) ⊕ (1, ¯N, N) ⊕ (N, 1, ¯N) . (3.36) The first of these carries the the vector multiplet. This group is invariant under the action of gn, and hence in order to be invariant under the action of P, fields in equation

(3.17) must either have neither of the R symmetry indices 3, 4 or both a 3 and a 4. The fields in this multiplet are

V = g+_{+ η}

1λ1+ η2λ2− η1η2S12+ η3η4 − S34+ η1λ341+ η2λ342+ η1η2g− . (3.37)

These transform in the adjoint of the gauge group SU (N ), produced by the projection (PGΦ)aˆ =

X

Γ

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This equation tells us how to relate the multi-index ˆAto the indices of fields in the adjoint representation. We may therefore write this projector as carriying two indices, (PG)a ˆˆA,

and having the action (PG)ˆa ˆAΦAˆ= Φaˆ on the fields. The remainder of the field content

are two half-hypermultiplets, transforming in conjugate bifundamental representations [29,20] with projection operators

(PRΦ)αˆ = X Γ r₃3g_α_ˆBˆΦBˆ = Φαˆ , (3.39) (P_RΦ)αˆ¯ = X Γ r₄4gα ˆˆ¯BΦBˆ = Φαˆ¯ . (3.40)

The field content for these is the set of fields that have either of the R symmetry indices 3 or 4 in equation (3.17). The corresponding superfields are

Q = χ++ ηrφr+ η1η2χ˜− , r = 1, 2 , (3.41)

and its CPT conjugate, ¯

Q = ˜χ++ ηrφr+ η1η2χ˜+ , r= 1, 2 . (3.42)

In a pseudo-real representation, the two multiplets are the same, ¯Q = Q and ˜χ±= χ±.

The representation matrices are given by ˜ Tˆa ˆ_α_ˆβ = −(PR) ˆ A ˆ α (PR) ˆ β ˆB_(P G)a ˆˆCf˜ ˆ A ˆB ˆC _. _(3.43)

We have once again introduced indices to the projection operators, for both the R and ¯

R representations.

3.4 One-loop results obeying color-kinematics duality

In this section we will see how to obtain the one-loop results from the truncation de-scribed above, as presented in [20] and [8]. The full double-copy construction will be done in the next chapter. The Lagrangian under consideration has a N = 2 supersymmetry, consisting of a vector multiplet along with a half hypermultiplet,

L_L= −1 4F ˆ a µνFâµν+ i ¯ψΓµDµψ+ DµϕD¯ µϕ+ i 2χΓ¯ µ_D µχ+ √ 2 ¯ψâϕT¯ _Râχ+√2 ¯χT_Râϕψâ . (3.44) The Lagrangian given is in six spacetime dimensions, meaning that each six-dimensional fermion is made up of two four-dimensional fermions put together. Four-dimensional

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scalars in the adjoint vector multiplet are obtained from the six dimensional vectors by dimensional reduction. The covariant derivatives for this theory are

(Dµψ)â = ∂µψâ+ gfâˆbˆcA ˆ_b

µψˆc , (3.45)

Dµχ = ∂µχ − igTRˆaAaµˆχ , (3.46)

Dµϕ = ∂µϕ − igTRaˆAˆaµϕ . (3.47)

The fermion ψˆa _{and gluon form the vector multiplet, both transform in the adjoint of}

the group. The fermion obeys a chirality condition

Γ7ψ= ψ , Γ7 = Γ0· · · Γ5 . (3.48)

The remaining scalar and six-dimensional fermion, ϕ and χ, form the half hypermulti-plet, and transform in a pseudo-real representation of the gauge group. They obey the conditions,

¯

χ= χt_{CV ,} _Γ

7χ= χ . (3.49)

Note that the half-hypermultiplet is CPT self-conjugate, however the vector multiplet is not, hence the CPT conjugate fields are to be considered too. Writing this theory in six dimensions makes it simple to apply Feynman rules in order to produce the amplitudes. Despite this we employ the projection technique described before in order to quickly obtain the tree level amplitudes, which will then be used with generalised unitarity to obtain loop level results.

Figure 3.1: Basic amplitude building blocks, solid lines represent half-hypermultiplets and curly lines represent vector multiplets. Labeling of external lines begins from bottom left and goes clockwise.

The building blocks for the cuts are shown in figure (3.1). Labeling of the incoming momenta starts at the bottom left and goes around clockwise.

We must establish a convention regarding the charge flow of the half hypermultiplet, displayed with arrows, and the undisplayed helicity of the vector multiplet. For this, recall that the fermionic delta function for the N = 4 theory takes the form δ8_{( ˜}_Q

1,2) =

Q4

A=1

P4

i,j=1hijiηAi ηAj . We are only concerned with the components for which A =

3, 4. These are degree four polynomials in Grassmann variables, generically written η_i3η3_jη4_kη_l4hijihkli. If, for instance, i, j, k, l 6= 1, by observing equation (3.17) we see that this fixes leg 1 to be of helicity 1,1₂ or 0, meaning it must be in the positive helicity vector multiplet. Similarly if say i, k = 1 then leg 1 has helicity 0, −1

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i= 1 or k = 1, leg 1 belongs to the half hypermultiplet. Let us define now that outgoing charge arrows will be denoted with a bar over the leg number, ¯1 for both diagrams in figure 3.1, and will correspond to the Grassmann variable with R index 4, so we have η4 1

in the superamplitude. We also denote vector multiplets with negative helicities with a bar over the leg number.

For the left diagram we must consider A(¯123¯4) which has a Grassmann coefficient η₂3η3₄η4₁η₄4h14ih24i. We must also consider A(¯12¯34) which has coefficient η3

2η33η41η43h13ih23i.

Grassmann derivatives applied to this amplitude should come in the order they appear, so for A(¯abc ¯d) the derivatives will be ∂4

a∂b3∂c3∂d4. Thus we obtain the amplitudes

A(1¯23¯4) = −h24iδ

4_{( ˜}_Q)

h12ih23ih34i , A(1¯2¯34) =

−h13iδ4_{( ˜}_Q)

h12ih34ih41i . (3.50) With the fermionic delta function δ4_{( ˜}_{Q) =}Q

A=1,2

P4

i,j=1hijiηiAηjA. For the amplitude

with only external hypermultiplets we similarly obtain A(¯12¯34) = −δ

4_{( ˜}_Q)h23i

h12ih23ih34i (3.51) We must also associate each cut with a color factor. Prior to projecting, each diagram has color factors built from the rescaled4 _{structure constants ˜}_fA ˆˆB ˆC_{. For example, the}

first diagram in figure (3.1), prior to projecting, has color structure ˜fA ˆˆS ˆDf˜B ˆˆC ˆS, we work in the so called Del Duca-Dixon-Maltoni (DDM) basis [30,31]. Following the projection, the color factor becomes ˜fαˆˆ¯σ ˆdf˜βˆˆcˆσ¯. Latin indices refer to the adjoint indices, and Greek indices with a bar or non bar refer to the outgoing and incoming half-hypermultiplet states respectively. Indices that are summed over are labeled with σ or s. We can use equation (3.43) in order to rewrite this in terms of the representation matrices, obtaining ˜Tdˆˆσ ˆα¯T˜ˆg ˆβ ˆσ¯. for convenience we may drop the usage of the bar on conjugate half-hypermultiplets by lowering all non-conjugate indices, finally producing ( ˜TdT˜g) α

β .

The results that we will reproduce are collected in Table (3.1), with both color and numerator factors included.

i G Ci, ni 1 R R R R k1 k2 k3 k4 l C1 = ( ˜TaˆT˜ ˆ_b )_α_ˆˆδ( ˜TaˆT˜ˆb)_ˆγˆ β n1 = − s2 h12ih34iδ 4 _Q (3.52) √ √

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Supersymmetry in amplitudes 28 2a R R R R k1 k4 k2 k3 l C2a = ( ˜TâT˜ ˆ b₎ γˆ ˆ α( ˜T ˆ b_T_ãˆ₎ ˆδ ˆ β n2a = st h12ih34iδ 4 _Q (3.53) 2b R R R R k1 k4 k2 k3 l C2b = ( ˜T ˆ a_T_˜ˆb₎ ˆδ ˆ α( ˜T ˆ_b_˜ Tâ)_ˆˆγ β n2b = su h12ih34iδ 4 _Q (3.54) 3 R R R R k1 k3 k4 k2 l C3 = ( ˜TaˆT˜ ˆ_b )_α_ˆˆγ( ˜TâT˜ˆb)δˆ ˆ β n3 = − s2 h12ih34iδ 4 _Q (3.55) 5a k4 k1 k2 k3 R R R R l C5a = − ˜Tâ ˆ_β_ˆγ( ˜T ˆ b_T_˜â_T_˜ˆb₎δˆ ˆ α = T(G) − 2C(R) ˜Tâ ˆ_α_ˆδT˜â ˆ_ˆγ β n5a = − st h12ih34iδ 4 _Q (3.56) 5b k4 k1 k2 k3 R R R R l C5b = T˜â ˆ_β_ˆγf˜âˆbˆc( ˜T ˆ b_T_˜cˆ₎δˆ ˆ α = −T (G) ˜Tâ ˆαˆδT˜ ˆ a ˆγ ˆ β n5b = st h12ih34iδ 4 _Q (3.57) 7a k2 k3 k4 k1 R R R R l C7a = − ˜Tâ ˆαˆδ( ˜T ˆ_b_˜ TâT˜ˆb)_ˆγˆ β = T(G) − 2C(R) ˜Tâ ˆ_α_ˆδT˜â ˆ_ˆγ β n7a = − st h12ih34iδ 4 _Q (3.58) 7b k2 k3 k4 k1 R R R R l C7b = T˜â ˆαˆδfãˆˆbˆc( ˜T ˆ b_T_˜ˆc₎ γˆ ˆ β = −T (G) ˜T ˆ a ˆδ ˆ α T˜ ˆ a ˆγ ˆ β n7b = st h12ih34iδ 4 _Q (3.59) 8a k4 k2 k1 k3 R R R R l C8a = − ˜Tâ ˆ_α_ˆγ( ˜T ˆ b_T_˜â_T_˜ˆb₎δˆ ˆ β = T(G) − 2C(R) ˜Tâ ˆ_α_ˆγT˜â ˆ_ˆδ β n8a = − su h12ih34iδ 4 _Q (3.60)

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8b k4 k2 k1 k3 R R R R l C8b = − ˜Tâ ˆαˆγf˜ ˆ aˆbˆc_{( ˜}_Tˆb_T_˜ˆc₎ ˆδ ˆ β = −T (G) ˜T ˆ a ˆγ ˆ α T˜ ˆ a ˆδ ˆ β n8b = − su h12ih34iδ 4 _Q (3.61) 9a k1 k3 k2 k4 R R R R l C9a = T˜â ˆ_βˆδ( ˜T ˆ b_T_˜â_T_˜ˆb₎γˆ ˆ α = − T(G) − 2C(R) ˜Tâ ˆ_α_ˆγT˜â ˆ_ˆδ β n9a = su h12ih34iδ 4 _Q (3.62) 9b k1 k3 k2 k4 R R R R l C9b = − ˜Ta ˆˆ_βˆδf˜âˆbˆc( ˜T ˆ_b_˜ Tˆc)_α_ˆˆγ= T (G) ˜Tâ ˆ_α_ˆγT˜â ˆ_ˆδ β n9b = − su h12ih34iδ 4 _Q (3.63) 11a k1 R R k3 l k2 R R k4 C11a = T˜â ˆ_α_ˆγf˜âˆbˆcf˜ ˆ_bˆ_{c ˆ}_d_˜ Tâ ˆ_ˆδ β = −2T (G) ˜T ˆ a ˆγ ˆ α T˜ ˆ a ˆδ ˆ β n11a = 2 su h12ih34iδ 4 _Q (3.64) 11b k1 R R k3 l k2 R R k4 C11b = − ˜Tâ ˆ_α_ˆγTr( ˜TâT˜ ˆ b_{) ˜}_Ta ˆˆ δ ˆ β = −2T (R) ˜T ˆ a ˆγ ˆ α T˜ ˆ a ˆδ ˆ β n11b = − su h12ih34iδ 4 _Q (3.65) 12a k1 R R k4 l k2 R R k3 C12a = T˜â ˆαˆδfãˆˆbˆcf˜ ˆ_bˆ_{c ˆ}_d_˜ Tâ ˆ_ˆγ β = −2T (G) ˜T ˆ a ˆδ ˆ α T˜ ˆ a ˆγ ˆ β n12a = 2 st h12ih34iδ 4 _Q (3.66) 12b k1 R R k4 l k2 R R k3 C12b = − ˜Tâ ˆαˆδTr( ˜TâT˜ ˆ_b ) ˜Tâ ˆ_ˆγ β = −2T (R) ˜T ˆ a ˆδ ˆ α T˜ ˆ a ˆγ ˆ β n12b = − st h12ih34iδ 4 _Q (3.67)

Table 3.1: One-loop superamplitudes with four external hypermultiplets in a complex representation. The reader should note an overall minus sign from the results of [8], in which equation (3.4) defines superamplitudes with an overall minus sign to the definition here.

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Relations amongst color factors can be obtained from the commutation relations, [ ˜Taˆ, ˜Tˆb] = ˜fˆaˆbˆcT˜ˆc:

C1− C2b = C5b= C7b ,

C3− C2a = C8b= −C9b ,

C11a = 2C8b= −2C9b ,

C12a = 2C5b= 2C7b . (3.68)

Additional relations can be obtained from the Jacobi identity, which after projecting reads,

˜

fâ_α_ˆ ˆγf˜â_β_ˆ δˆ+ ˜fâ_α_ˆ ˆδfãˆ_γ_ˆ βˆ+ ˜fâ_{α ˆ}_ˆ_βf˜âˆδˆγ= 0 . (3.69) The last term must be zero, and this can be rewritten with the representation matrices Tâ ˆ_α_ˆγTâ ˆ_ˆδ β = T ˆ a ˆδ ˆ α T ˆ a ˆγ ˆ

β . The relations obtained are

C1 = C3 ,

C2b = −C8a= C9a ,

C2a = −C5a= −C7a ,

C11b = C8a = −C9a ,

C12b = C7a = C5a . (3.70)

The color factors of all diagrams will be written in terms of a set of independent box diagrams. These will soon be produced by the cuts, which we will use to verify the amplitudes in Table (3.1). All of these relations must be taken into account when making the ansatz for the cuts, as they stem from the Jacobi identities of the parent theory and these were assumed when the amplitudes were written in the DDM basis prior to projecting.

3.4.1 Vector multiplet cut

1

4 2

3 l1

Figure 3.2: Diagram for the vector multiplet cut, the free loop momentum l1 flows to

the left as can be inferred from the amplitudes.

Let us consider a loop diagram in which we cut two vector multiplet lines. The left diagram in the cut has external (incoming) momenta 1, 4, −l2, l1 and correspondingly