On the Lagrangian formulation of gravity as a double copy of two … extended to loop-level...

University of Pisa

Department of Physics

Master’s Course in Theoretical Physics

Academic Year 2017/2018

Master’s Thesis

On the Lagrangian formulationof gravity as a double copy of

two Yang-Mills theories

Candidate: Supervisor:Pietro Ferrero Dott. Dario Francia

Ai miei genitori,

per il loro costante supporto e il loro amore incondizionato.

A Claudia,

per la leggerezza e l’amore che ha portato nella mia vita.

AbstractYang-Mills gauge theories and Einstein’s General Relativity represent the main pillars onwhich our understanding of the fundamental interactions is presently based. Althoughoriginally formulated on account of completely independent intuitions and procedures,many connections have been later uncovered between gauge theories and gravity, stem-ming in particular from the inspection of scattering amplitudes in the two theories. Inaddition to the long-known KLT relations new correspondences were recently discov-ered, the so-called double-copy relations, eventually leading to recognize that, with someprovisos, gravity can be understood as a suitably dened “square” of two Yang-Mills the-ories. While being essentially a theorem at tree level, this insight has been more recentlyextended to loop-level scattering amplitudes and to classical solutions, while still beingunclear both at the Lagrangian level and in its ultimate geometrical meaning.

In this Thesis we investigate the possibility of a Lagrangian formulation of this cor-respondence between gauge theories and gravity, focusing on the non-supersymmetriccase, in which the gravitational multiplet resulting from the product of two spin-oneelds also contains a two-form eld and a dilaton. To this end, we exploit the denitionof “double-copy eld” given by Du et al. in order to build a quadratic Lagrangian, whichhas a neat interpretation as the “square” of two Yang-Mills quadratic Lagrangians, whilealso discussing its geometrical meaning. The free theory is then extended to the inter-acting level by means of the Noether procedure, building its o-shell cubic vertices. Theinteractions allowed by the gauge symmetry at the cubic level contain a sector whichreproduces the results of the double copy for the three-point amplitudes, while also pos-sessing a number of notable features. In particular, the resulting Lagrangian is invariantunder a twofold Lorentz symmetry and is equivalent, up to suitable eld redenitions, tothe so-calledN = 0 Supergravity at the cubic level, which provides the natural outcomeof the double copy of two pure Yang-Mills theories. Last, we discuss the deformation tothe gauge transformation of the elds in the gravitational multiplet which is determinedfrom the Noether procedure. At this level we observe the need to modify the deni-tion of the scalar and of the graviton elds in terms of the double-copy eld by alsoallowing for quadratic terms amenable, as we also highlight, of a tantalizing geometricalinterpretation.

iv

AcknowledgementsNell’occasione di scrivere questa Tesi di Laurea Magistrale vorrei ringraziare tutte le per-sone che hanno contribuito, in diversi modi e in varia misura, ad aiutarmi a raggiungerequesto obiettivo, che no a qualche anno fa potevo solo sognare.

Prima di tutto vorrei ringraziare il mio relatore Dario Francia, che è stato il mioprincipale punto di riferimento accademico negli ultimi due anni, a livello scienticoe personale. Al di là di tutta la Fisica che mi ha insegnato, la sua disponibilità ad unconfronto aperto e sincero è stata cruciale per me, specialmente durante la scrittura diquesta Tesi. Spero di aver fatto mie, anche solo in minima parte, la sua precisione e lasua cura del dettaglio. Lo ringrazio per il tempo e l’attenzione che ha dedicato al miolavoro durante tutto l’anno.

Un ringraziamento speciale va ovviamente ai miei genitori e ai miei fratelli, che mihanno sempre sostenuto a distanza e sono stati un punto fermo nei momenti di dicoltà,nonché i primi a condividere con me le gioie di questo percorso accademico. Grazie peril vostro amore, per la ducia e per aver sempre creduto in me.

Non posso che dedicare un grande ringraziamento anche a Claudia, che è stata unacompagnia fondamentale durante tutto il lavoro di Tesi, dando un nale più dolce allegiornate di lavoro inconcludenti. Grazie per le risate che mi regali ogni giorno, per latua fame costante, per le dormite sul divano mentre io studio, per credere sempre in me.Quest’anno non sarebbe stato così bello senza di te.

Grazie a tutti i miei amici e compagni di corso, in particolare a Nicola e Mary, chein questi anni sono stati come fratelli per me. Grazie a Filippo, Paolo e Salvatore per glischerzi, le stanzate, i viaggi a mensa e i momenti di confronto durante questi due anni incollegio. Grazie a tutti gli altri per avermi fatto sentire come in una grande famiglia: Brig-gos, Bruno, Budrus, Chessa, Laura, Martino, Massimo, Matteo, Michele, Pierina, PierP,ilPresidente, Sofy e Stefano. Un grosso grazie va anche a Ivano, per aver ascoltato ognimio dubbio e per avermi aiutato durante la Tesi con discussioni sempre utili.

Grazie ai miei compagni di squadra di Pisa e a quelli di Bogliasco e grazie in partico-lare a Calzia, per le indimenticabili avventure vissute insieme. Grazie anche alle Dodsie,il miglior gruppo di amici che potesse capitarmi nel momento in cui più ne avevo bisogno.

Per concludere, vorrei ringraziare due persone molto diverse, ma altrettanto impor-tanti per la mia crescita personale: la prof. La Piana per i suoi insegnamenti e per la pas-sione che mi ha trasmesso, e il mio allenatore Pakito, che nella tana delle tigri mi ha in-segnato il valore del sacricio, una battaglia dopo l’altra e sempre col sangue agli occhi.

vi

Contents

Notation and conventions xi

1 Introduction 11.1 Goals and motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.3 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.4 Plan of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2 Scattering amplitudes and the double copy 132.1 Yang-Mills theory: Lagrangian and Feynman rules . . . . . . . . . . . . . 132.2 Tree-level Yang-Mills amplitudes and color-kinematics duality . . . . . . 16

2.2.1 Three gluons amplitude . . . . . . . . . . . . . . . . . . . . . . . 162.2.2 Four gluons amplitude . . . . . . . . . . . . . . . . . . . . . . . . 172.2.3 Color-kinematics duality . . . . . . . . . . . . . . . . . . . . . . . 212.2.4 Partial amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.5 Five gluons amplitude . . . . . . . . . . . . . . . . . . . . . . . . 262.2.6 A non-local Lagrangian for color-kinematics duality at ve points 332.2.7 Color-kinematics duality at loop level . . . . . . . . . . . . . . . . 37

2.3 General Relativity: Lagrangian and Feynman rules . . . . . . . . . . . . . 382.3.1 Three-graviton tree-level amplitude . . . . . . . . . . . . . . . . . 41

2.4 The double copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.4.1 Particle content of the double copy . . . . . . . . . . . . . . . . . 442.4.2 The double copy for three-particle amplitudes . . . . . . . . . . . 472.4.3 Three-particle amplitudes in N = 0 Supergravity . . . . . . . . . 502.4.4 The double copy at tree level . . . . . . . . . . . . . . . . . . . . . 522.4.5 The double copy at loop level . . . . . . . . . . . . . . . . . . . . 542.4.6 The zeroth copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3 Double copy for classical theories: a review 593.1 Relating Yang-Mills and gravitational symmetries . . . . . . . . . . . . . 60

3.1.1 Product of two Yang-Mills elds . . . . . . . . . . . . . . . . . . . 603.1.2 Field content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.1.3 Linearized symmetries . . . . . . . . . . . . . . . . . . . . . . . . 68

viii

3.1.4 Field strength and Riemann tensor . . . . . . . . . . . . . . . . . 703.1.5 Free equations of motion . . . . . . . . . . . . . . . . . . . . . . . 723.1.6 Sourced equations of motion . . . . . . . . . . . . . . . . . . . . . 75

3.2 BDHK Lagrangian and the double copy . . . . . . . . . . . . . . . . . . . 783.2.1 Color-kinematics duality in YM Lagrangian . . . . . . . . . . . . 793.2.2 Double copy for BDHK Lagrangian . . . . . . . . . . . . . . . . . 813.2.3 Comments on the BDHK approach . . . . . . . . . . . . . . . . . 90

3.3 The double copy of classical solutions . . . . . . . . . . . . . . . . . . . . 913.3.1 Kerr-Schild ansatz in General Relativity . . . . . . . . . . . . . . 913.3.2 Vacuum, stationary Kerr-Schild solutions and the double copy . . 953.3.3 The double copy for the Schwarzschild black-hole . . . . . . . . . 963.3.4 The double copy for (A)dS spacetime . . . . . . . . . . . . . . . . 98

4 Lagrangian formulation 1014.1 Quadratic Lagrangian for the double-copy . . . . . . . . . . . . . . . . . 101

4.1.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 1084.1.2 Gauge-xing and degrees of freedom . . . . . . . . . . . . . . . . 110

4.2 Troubles with the naive inclusion of double-copy interactions . . . . . . 1144.3 The Noether procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.3.1 Noether procedure for Yang-Mills theory . . . . . . . . . . . . . . 1184.3.2 Noether procedure for graviton cubic vertex . . . . . . . . . . . . 122

4.4 Cubic vertex for the double-copy Lagrangian . . . . . . . . . . . . . . . . 1274.4.1 Double-copy part of the cubic vertex . . . . . . . . . . . . . . . . 1324.4.2 Interpretation of δ1Hµν . . . . . . . . . . . . . . . . . . . . . . . . 1364.4.3 Full cubic vertex for Hµν . . . . . . . . . . . . . . . . . . . . . . . 143

5 Summary and outlook 147

Appendices 153

A Noether procedure for gravity 155A.1 Completion of the cubic vertex . . . . . . . . . . . . . . . . . . . . . . . . 155

B Noether procedure for the double-copy eld 159B.1 Completion of the double-copy cubic vertex . . . . . . . . . . . . . . . . 159B.2 Double-copy cubic vertex and N = 0 Supergravity . . . . . . . . . . . . 166

Bibliography 171

ix

Notation and conventionsWe adopt the “mostly plus” metric convention for the Minkowski metric:

ηµν = diag(−1,+1, ...,+1).

The symmetrization (antisymmetrization) of indices is performed with round (square)brackets, without normalization factors. For instance:

X(ij) := Xij +Xji,

X[ij] := Xij −Xji.

We use greek indices µ, ν, ..., as Lorentz indices in at Minkowski spacetime or as tensorindices in curved spacetime inD dimensions. We use latin indices i, j, ..., as little group,SO(D − 2) indices, thus with values 1, ..., D − 2.

We adopt the following sign convention for the Riemann tensor (without assuming atorsion-free connection):

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ.

For the gravitational coupling constant in D dimensions, we use κ2 = 8πG(D).

The Fourier transform is denoted with the symbol F , with normalization and sign de-ned as follows:

f(x) =∫ dDp

(2π)D e−ip·xFf(p),

with

Ff(p) =∫dDxeip·xf(x).

We denote withA scattering amplitudes in Yang-Mills theory, while withM we denote

x

amplitudes in gravitational theories (yet not necessarily pure graviton amplitudes).

When computing scattering amplitudes, we denote the scalar product between two Lorentzvectors vµ and wν as (vw) := vµw

µ.

When dealing with rank-two symmetric tensors hµν , we use the shorthands ∂ · hµ :=∂αhµα for the divergence and h := hαα for the trace.

We denote the convolution product of two functions with the symbol :

[f g](x) =∫dDyf(y)g(x− y).

We denote with 1 the inverse of the d’Alembert operator = ηµν∂µ∂ν , to be understood

as the Green’s function G of :

[ 1

(j)](x) :=∫dDyG(y)j(x− y) =

∫ dDp

(2π)DFj(p)−p2 e−ip·x.

We shall sometimes use the notation

đk = dDk

(2π)D .

xi

Chapter 1Introduction

The use of non-observable quantities in our description of physical phenomena repre-sents a cornerstone of modern Theoretical Physics. This is particularly evident for Quan-tum Field Theory (QFT), whose most fundamental objects are quantum elds and cor-relation functions, which are not necessarily in themselves physically observable quan-tities. The standard formulation of QFT is in terms of a Lagrangian density, dened as afunction of the elds and their derivatives, which is particularly convenient for the studyof symmetries. This formulation, however, typically brings about a lot of redundant in-formation, the main instance of which being incarnated in the local symmetry termed“gauge invariance”, unavoidable in particular to provide a covariant eld-theoretical de-scription of massless particles. Gauge invariance is ubiquitous in phenomenological andtheoretical models, including the Standard Model of particle physics and the Einstein-Hilbert description of the gravitational interaction, which provide our best established,yet incomplete, understanding of the laws of Nature.

From the Lagrangian formulation one can then compute scattering amplitudes, whichcan be viewed as an essential link between our theoretical description of the fundamentalinteractions and the experimental data, providing the dynamical information requiredto compute the cross sections. Scattering amplitudes are physically measurable quanti-ties, therefore their values cannot depend on arbitrary choices such as the gauge-xingor the eld basis, thus being insensitive to the redundancies of the Lagrangian formu-lation. Already in the sixties, inspired by ideas of Heisenberg, the so-called S-matrix

program [1] was initiated, with the aim of calculating S-matrix elements (i.e. scatteringamplitudes). The latter were to be calculated without relying on intermediate states, re-quiring the introduction of non observable quantities, but only building on fundamental

1

Chapter 1. Introduction 2

principles comprising unitarity, causality, analyticity, the superposition principle andLorentz-invariance. Originally motivated by the insuciency of theories like QED toaccount for the phenomenology of strong interactions, the S-matrix program was thenprogressively abandoned after the advent of QCD in the seventies. It was eventuallyrevived from a novel perspective in recent years, with the idea of disposing of the re-dundancies intrinsic of the eld-theoretical description of the interactions. Several tech-niques were developed in order to directly calculate scattering amplitudes, leading toa tremendous progress in our computational power as well as to a better understand-ing that, especially in gauge theories, the result of such calculations can be amazinglysimpler than what the direct evaluation of the Feynman diagrams would actually sug-gest. A remarkable example is provided by the Parke-Taylor formula [2] for the n-gluontree-level partial amplitude1 in four dimensions for the maximally helicity violating case,where all the gluons except for two have the same helicity2:

An[1+2+...i−...j−...n+] = 〈ij〉4

〈12〉〈23〉...〈n1〉 . (1.1)

The appealing feature of (1.1) is that it provides a compact, elegant formula for a class ofn-gluon amplitudes at tree level. By contrast, the number of Feynman diagrams requiredto compute the same amplitude with standard techniques displays a tremendous growthwith n, as shown in table 1.1 for the rst values of n [5].

n 4 5 6 7 8 9 10 ...# diagrams 4 25 220 2485 34300 559405 10525900 ...

Table 1.1: Number of Feynman diagrams contributing to the n-gluon tree-level ampli-tude.

One reason for the huge dierence between the Feynman diagrams calculation andthe simplicity of the nal expressions for amplitudes in gauge theories can be traced backto the fact that the individual Feynman diagrams are not gauge-invariant, together withthe fact that their evaluation involves o-shell intermediate states in the propagators. Onthe contrary, the amplitudes are gauge-invariant quantities which only involve on-shellphysical states: the passage from Feynman diagrams to scattering amplitudes eliminates

1We shall briey introduce the concept of partial amplitudes in Section 2.2.4. It provides the purelykinematic part of a gluon amplitude, without the color part.

2Formula (1.1) is written in the spinor-helicity formalism (see [3] or [4] for a review) where, in partic-ular, for real light-like momenta |〈ij〉|2 = 2pi · pj .

3 Chapter 1. Introduction

all the unphysical amount of information introduced together with the local symmetryof the theory. However, this observation in itself would not suce to account for thecompelling simplicity of scattering amplitudes. Indeed, it was recently realized that thelatter, at least in some theories, organize themselves according to hidden symmetrieswhich vastly constrain their expressions: these symmetry relations are additional withrespect to the ones that guided the very construction of the theory and are anyway notat all evident from the Lagrangian formulation [3].

Such observations, together with the necessity of precise computations to be com-pared with the experimental data, collected in particular at the Large Hadron Collider,led to the study of the so-called on-shell techniques (see [4] for a modern review), es-pecially employed in gauge theories, which helped to shed light on the hidden prop-erties of scattering amplitudes. The main tool employed include the aforementionedspinor-helicity formalism, the decomposition in partial amplitudes for (possibly super-symmetric) gauge theories and the unitarity methods [6–11], which allow to build loopamplitudes starting from tree-level ones. A crucial boost in the development of the eldwas the discovery of the so-called BCFW recursion relations [12, 13] in 2004. These re-late, in specic theories (including Yang-Mills theories and Einstein Gravity, togetherwith their supersymmetric extensions), n-point tree-level amplitudes to sums of prod-ucts of lower points amplitudes, with the result that, in such theories, all the tree-levelamplitudes can be derived from the three-point ones. This is an impressive result, whichimplements the idea that in theories with a local symmetry a large part of the Lagrangian(all the vertices with more than three elds, from this perspective), while needed to guar-antee gauge-invariance, may be regarded as being not strictly necessary to the goal ofbuilding physical quantities. The BCFW recursion relations actually are the main toolwhich is employed to prove, from the QFT perspective, two properties of certain scat-tering amplitudes which are of major importance for our work: the BCJ relations andthe double-copy relations (both conjectured in [14] and proven by means of the BCFWrelations in [15] and [16], respectively).

Among the hidden symmetries of scattering amplitudes that do not have a clearLagrangian counterpart, of central interest in this Thesis will be the so-called color-

kinematics (CK) duality [14, 17], a correspondence between the color factors and thekinematic factors of a particular set of cubic diagrams in Yang-Mills (YM) theories, whose


sum results in a given gluon amplitude. This symmetry, amazingly, points out a set ofrelations between YM and gravitational theories: when the color factors of the YM dia-grams are replaced by another copy of kinematic factors (possibly from a dierent YMtheory), the result is a gravitational amplitude. Correspondences of this type are collec-tively known as double-copy (DC) relations [14, 17], and provide the central theme ofour work.

Both the CK duality and the DC relations have been proven to hold at tree level (whiletheir realization at loop level is still a conjecture) and have been generalized to variousdierent theories, including the supersymmetric extensions of YM theory and of Ein-stein Gravity. However, in our work we shall focus on the simplest, yet fully non-trivialcase: the DC of pure YM theory which gives rise to a gravitational theory often referredto as N = 0 Supergravity, due to its eld content being the same as for the low-energylimit of the closed bosonic String Theory, i.e. a graviton coupled to a two-form eld andto a scalar.

It must be mentioned that the idea of relations between gravitational and gauge the-ory amplitudes was already present in the String-Theory literature since the discovery ofthe Kawai-Lewellen-Tye (KLT) relations [18]. These are relations between scattering am-plitudes in String Theory which, in their low-energy limit, reduce to relations betweentree-level gravitational amplitudes and products of tree-level gauge theory partial ampli-tudes. However, the DC relations are, from several perspectives, a considerable novelty.First, the KLT relations display rather cumbersome expressions for high number of par-ticles, while the DC relations, crucially relying on the validity of the CK duality, are farsimpler in their structure. Moreover, they provide a more drastic implementation of theidea that gravity is, in some sense to be claried, the “square” of two spin-one gaugetheories, in particular since they do not rely on the structure of the corresponding UVcompletion. Second, while the KLT relations only hold at tree level, the DC relations canbe straightforwardly generalized to loop level, consistently with a number of non-trivialchecks, although a complete proof is still lacking in the general case. Finally, the DC re-lations have also been extended to correspondences relating classical solutions in Yang-Mills theories and in General Relativity (possibly coupled to a two-form and a dilaton)[19–28]. For these reasons, one may regard the DC relations as being more fundamentalthan the KLT relations at the eld-theoretical level (despite the two being equivalent


for tree-level scattering amplitudes), at least in the sense that the DC “paradigm” can beextended beyond tree-level scattering amplitudes.

Besides the theoretical interest in trying to understand the origin of such compellingnew properties of quantum gravity, whose UV structure is one of the deepest and mostpuzzling open issues in Theoretical Physics, the DC relations own many aspects of phe-nomenological relevance. The rst, most tangible one, is that they signicantly sim-plify the calculations in perturbative quantum gravity, usually rather involved owing tothe complicated structure of the Einstein-Hilbert Lagrangian expressed in terms of thegraviton eld. Calculations in YM theories are instead much simpler, also by virtue ofthe aforementioned modern techniques for the computation of scattering amplitudes.Therefore, the possibility of trading a quantity in gravity for the product of two quanti-ties in YM theory may lead to relevant simplications. Another signicant advantage ofthe DC relations, still concerning the idea of “disentangling” the calculations in gravity,is that their extension to classical, radiative solutions [24–29] might be useful to the goalof computing quantities of interest for the study of gravitational radiation. This has be-come a central issue after the recent gravitational waves detections by LIGO and VIRGO([30–36], and will be more and more important as the sensitivity of the two observa-tories increases. Finally, it is worth mentioning an open problem, whose relevance isboth theoretical and phenomenological: the question about the UV-niteness of N = 8Supergravity in D = 4. Indeed, it was observed that this theory displays a UV behaviorsofter than expected, with the current consensus being that it is nite up to seven loops[37–43]. The analysis up to four loops, using modern on-shell techniques, shows thatthe critical dimension for the divergence of N = 8 Supergravity is compatible with thevalueDc = 6/L+4 [44–48], with L being the number of loops, pointing to its nitenessin D = 4 (L = ∞)3. The DC relations have recently been employed [53–55], togetherwith the unitarity method [6–11], in order to understand the UV properties of N = 8Supergravity to higher loops and it is hoped that they may help in leading to the nalanswer on this issue4.

3Indeed, the same value ofDc has been proven to hold forN = 4 Super-Yang-Mills (SYM) [44, 49–52].4It must be mentioned that the DC relations at loop level relate the loop integrands of scattering am-

plitudes, not the results of the integrals: this explains why, for instance, in D = 4 pure YM theory isrenormalizable, while the N = 0 Supergravity, as well as pure Einstein Gravity, is not. Therefore, theniteness of N = 4 SYM can only suggest a soft UV behaviour for N = 8 Supergravity, but it does notdirectly imply its niteness.


1.1 Goals and motivations

In spite of all the progress made on the side of understanding and implementing the DCrelations at various levels, very little is presently known about what could be regardedas a rather natural side of the problem: what is, if any, the Lagrangian counterpart of theDC structures derived at the level of amplitudes? Besides being a natural issue to raise,there are several motivations, in our opinion, to address such a question:

• As already mentioned, the existence of DC relations among amplitudes is a theo-rem at tree level but still a conjecture at loop level; we believe that a Lagrangianunderstanding of these properties might help to prove (or, possibly, disprove) theirloop-level validity. In the case of a positive answer the DC relations would be use-ful in several additional respects, e.g. in order to study the UV behavior of thegravitational theories, and especially of the aforementioned N = 8 Supergravityin D = 4, whose UV niteness has not yet been disproved.

• Basic pieces of insights into YM and gravitational theories derive from an under-standing of the underlying geometry[56]. We nd it hard to believe that the situ-ation may be radically dierent when it comes to assessing the actual meaning ofthe DC relations, if they are to be regarded as providing more than just a bunch ofuseful tricks helping to perform some computations. Understanding the geometry,however, requires relating the symmetries of the theories under scrutiny, and thisnaturally calls for trying to explore the DC structures at the Lagrangian level. Thegoal of the present work is to perform the rst steps of this exploration, buildingon the few results available to date.

• A Lagrangian formulation would allow to address the issue of asymptotic symme-tries, with the corresponding impact on soft amplitudes and memory eect (seee.g. [57]) from the perspective of the DC relations.

As of today, the two main contributions to the comprehension of the Lagrangian ori-gin of the DC relations came from Bern et al. [16] and from Du et al. [58]. In particular,a Lagrangian was proposed in [16] which reproduces the CK duality and the DC rela-tions only up to the ve-point amplitudes at tree level, though with the introduction ofnon-localities (which can be avoided only by introducing additional elds). A possibledrawback of the approach of [16], which we wish to address, is the partial gauge-xing


required by the proposed DC procedure. By contrast, in [58] a completely o-shell im-plementation of the DC relations on the elds was proposed, including a study of thecorresponding o-shell symmetries, although only at the linearized level and withoutarriving at a proposal for a DC Lagrangian. Let us also notice that in [59] the problem ofconnecting the YM equations and the (extended) linearized gravity equations of motionwas addressed, but again the introduction of non-localities proved necessary, this timewhen dealing with the coupling to sources.

In this Thesis we address the problem of the Lagrangian formulation of the DC rela-tions from a novel perspective, building on the results of [58].

1.2 Results

As a starting point for our work we consider the denition of the product betweentwo YM elds given in [58], which was shown to correctly reproduce the linearized (ex-tended) gravitational symmetries from the linearized YM ones. The “double-copy eld”resulting from this product, Hµν , carries the degrees of freedom of the N = 0 Super-gravity multiplet, both o shell and on shell. We added to this construction an o-shelldenition of the scalar eld in terms of Hµν , which was lacking in [58]. During thepreparation of the present work, however, the paper [59] appeared, where an analogousdenition of the scalar eld was also given. With an appropriate identication of theeld content of the N = 0 Supergravity in terms of YM elds we proceed to build aquadratic Lagrangian for the eld Hµν , to be interpreted as a “square”, in a concretesense that we specify, of two YM Lagrangians. The resulting Lagrangian correspondsto the quadratic part of the N = 0 Supergravity Lagrangian, being the sum of the freeLagrangians for a graviton, a two-form and a scalar. Moreover, when the eld Hµν isexpressed in terms of the YM elds, our quadratic Lagrangian can be viewed as the“product”, in some sense to be specied, of two YM free Lagrangians, thus conrmingour interpretation of a Lagrangian for the DC. Let us also mention that, when expressedin terms of a linearized eld strength for Hµν , it resembles the quadratic Lagrangiansbuilt in [60] for the Maxwell-like massless higher spin elds. The transverse part of ourquadratic Lagrangian matches the kinetic term obtained in [16] for the double-copy eldHµν , but we would like to stress that our construction is completely o shell and alsocontains pure gauge terms, which in [16] were discarded because of a preliminary gauge


xing performed in the YM theory.

Then, we extend such quadratic Lagrangian to the interacting level with the additionof cubic vertices. This is achieved by means of the Noether procedure: a perturbativealgorithm which allows to deform free gauge theories to interacting ones, while alsoconsistently building the deformation of the gauge transformation. Our cubic vertexcontains two sectors, identied by two arbitrary parameters. The selection of only oneof the two sectors allows to correctly reproduce the results obtained in the DC of twothree-gluon tree-level YM amplitudes and therefore we analyze this part in more detail.

We show that our Lagrangian extends the result of [16] at the cubic level, with theinclusion of longitudinal terms, and that it matches (modulo eld redenitions) the o-shell cubic vertices of theN = 0 Supergravity Lagrangian. This is a non-trivial result ifwe observe that, in the latter, the coupling of the scalar eld is not implied by the localsymmetry of the theory and thus is not an output of the Noether procedure. Therefore,since in our case a specic value of the coupling between the two-form and the scalar isselected, with a completely xed coecient that matches the same coupling of N = 0Supergravity, this must be seen as a result of the fact that we work in terms of Hµν asa fundamental variable. In other words, we provide a proof of the fact that the cubiccouplings of N = 0 Supergravity are indeed predicted from the assumption that thetheory is a square of two YM Lagrangians, being especially non-trivial that we also ob-tain those couplings that are not implied by the linearized gauge symmetry. Finally, wealso nd that the part of the cubic vertex which reproduces the DC relations has, withrespect to the remaining part of the vertex, an enhanced symmetry, which is also sharedby the quadratic Lagrangian. This corresponds to the possibility of performing two in-dependent Lorentz transformations on the left and right indices of Hµν , together witha discrete Z2 symmetry, dened as the exchange of left and right indices. This resultmatches similar properties observed at the level of DC relations, in which the gravityamplitudes are built from two copies of the YM ones, each with an independent Lorentzsymmetry. Moreover, we wish to stress that such symmetry is also shared by the termsin the Lagrangian which do not contribute to scattering amplitudes, which for the case ofthe three-point amplitudes comprise the pure gauge terms of the quadratic Lagrangianand the longitudinal parts of the cubic vertex. Such terms are precisely what we addedto the results of [16].


In addition, we analyze the other main output of the Noether procedure at the cu-bic level, computing the rst correction to the free gauge transformation of Hµν . Itssymmetric and antisymmetric parts correspond, as expected, to the Lie derivative of arank-two symmetric tensor and of a two-form, respectively. However, the correctionto the gauge transformation of the scalar gives rise to something dierent from the Liederivative of a scalar eld, at least as long as we stick to the denition of the scalar elditself given at the linear level, thus raising a possible issue on its validity. As a solutionto this puzzle we observe that our initial denition of the scalar was meant to guaranteeits gauge invariance at the free level, and there is no reason why such denition shouldnot receive corrections at the interacting level. Therefore, we propose to interpret thescalar eld as an unknown, non-linear function of Hµν , of which we dened only therst order in its perturbative expansion. From the requirement that the full non-linearscalar transforms with its Lie derivative in a putative complete theory, interpreted per-turbatively, we derive a system of equations which we solve at the second order in Hµν .The result has a clear interpretation from the geometrical viewpoint, since the correctionto the denition of the scalar is compatible with it being the convolution of the Green’sfunction of the Laplace-Beltrami operator with the Ricci scalar of a manifold with metricgµν = ηµν + 1

2H(µν) and Levi-Civita connection. It would be interesting to investigatethe physical meaning of this result. Similar considerations also apply to the denitionof the graviton in terms of Hµν , that we also modify accordingly.

1.3 Further remarks

Before summarizing our work, we would like to highlight part of the additionalprogress made in the study of the CK duality and of the DC relations, that is not ofprimary interest for our work and thus will not be discussed in this Thesis.

The rst remark is that, although the CK duality was originally discovered as a prop-erty of (possibly supersymmetric) YM theory, it was successively extended at least in twomain directions. The rst concerned the inclusion of gauge-coupled matter. Matter inthe fundamental representation was introduced in [61], with the aim of removing, bymeans of a particular ghost prescription, unwanted states in theories resulting from theDC, in order for example to obtain pure Einstein Gravity as a result. In [62] the CK


duality was generalized to the case of QCD amplitudes, where the matter content is pro-vided by the quarks. Furthermore, the CK duality was extended to other theories, such as(supersymmetric) Chern-Simons theories with matter [63, 64] and the non-linear sigmamodel (NLSM), where relations totally analogous to the ones of the CK duality undergothe name of avor-kinematics duality [65–67]. Interestingly, in the latter case the dualitybetween avor and kinematics was derived from the symmetries of an action [68]: al-though at the moment there is no understanding of an analogous situation in YM theory,this is surely encouraging for the program of the Lagrangian implementation of the CKduality.

The second remark we wish to make is that also the DC relations, initially conceivedas connecting YM and gravitational theories, have been observed several dierent con-texts, both in QFT and in String Theory. For instance, starting from the cubic diagramsin YM theory, it was noticed that one could replace kinematic factors by additional colorfactors, leading to amplitudes in a theory for a scalar eld transforming in the biadjointrepresentation of two gauge groups [69]. This construction was termed the zeroth copy

[70]. Furthermore, after the discovery of the avor-kinematics duality, it was realized[65, 71, 72] that the replacement of avor factors by new copies of kinematic factors inthe NLSM amplitudes leads to amplitudes in the special Galileon theory [73, 74]. In thiscase, as for the avor-kinematics duality, these relations have been understood from theLagrangian perspective [68]. Also, DC-like formulas for amplitudes in the Born-Infeld[75] and Dirac-Born-Infeld theories [76] were derived [67, 71, 72]. Such a universalityof the DC paradigm, which can be summarized in a “multiplication table” [77] (tab. 1.2),is not fully understood. As of today, the only hint of a unifying principle behind allthese theories can be traced back to the scattering equations of Cachazo, He and Yuan[70, 71, 78], who proposed a compact way of constructing the S-matrices for all the the-ories in tab. 1.2. Other generalizations of the DC paradigm comprise Einstein-Maxwelland Einstein-Yang-Mills theories [71, 72, 79–84], certain constructions in String Theory[85–89], conformal gravity [90] and gauge theories and gravity with the inclusion ofhigher derivative operators [91–93]. Another extension of the DC relations is their gen-eralization to curved backgrounds, both for scattering amplitudes [94] and for classicalsolutions [20, 21].

Moreover, as we mentioned several times, the CK duality and the DC relations also


BS NLSM YMBS BS NLSM YM

NLSM NLSM SG BIYM YM BI G

Table 1.2: Multiplication table of QFTs. BS=biadjoint scalar theory, NLSM=non-linearsigma modeal, YM=Yang-Mills theory, SG=special Galileon, BI=Born-Infeld theory,G=gravity.

hold for the supersymmetric extensions of YM theory and Einstein Gravity [14, 16]. Inparticular, N = 4 SYM provides the simplest theory in which to test the loop-level CKduality conjecture. Such theory is also of great interest since the DC of two N = 4SYM amplitudes is conjectured, also at loop level, to give rise to amplitudes in N = 8Supergravity, which is worth studying because of the issue on the UV niteness of thetheory. In the general case, the DC ofNL SYM withNR SYM gives rise toN = NL+NRSupergravity5.

We view all these developments as corroborations of the relevance of the CK dualityand of the DC relations, thus further motivating our interest in a deeper understandingof such universal features relating theories that appear otherwise independent in all re-spects. However, since this work is devoted to the quest for an o-shell implementationof these relations, which has hitherto proven to be rather dicult, we shall work in thesimplest possible setup, with the idea that what is understood in this case might be gen-eralized to more complicated scenarios in the future. Therefore, as mentioned, we shallonly consider pure YM theory, whose DC is the so-called N = 0 Supergravity.

1.4 Plan of the work

The material is organized as follows:

• In chapter 2 we review the CK duality and the DC relations for scattering ampli-tudes. We also present some explicit calculations, aiming both at clarifying somegeneral statement about these properties and at having a reference result, namelythe DC for the tree-level, three-point amplitudes, which will be useful throughout

5For the allowed values of NL and NR in the various dimensions where supersymmetry exists, theresulting Supergravity always have N ≤ Nmax,with Nmax the maximal number of supercharges.


the whole exposition.

• In chapter 3 we discuss the main results obtained so far in the attempt of under-standing the CK duality and the DC relations from the Lagrangian perspective.In particular, we focus on [58], in which the denition of the product of two YMelds is given, and on [16], also in order to make a comparison with our results atthe end of the work. Last, we briey review some results obtained in the study ofthe DC for classical solutions, focusing on [19].

• In chapter 4 we collect our original results. In the rst part, we introduce aquadratic Lagrangian for the DC and we discuss its interpretation, as well as theequations of motion which result from it together with their gauge-xing. Next,we move to consider interactions and we review the Noether procedure, present-ing as explicit examples of its application the cases of YM theory and of the cubicvertex of the Einstein-Hilbert action. We then apply the procedure to the cubicvertex for the self-interaction of the DC eld Hµν , comparing our solution withthe results of [16]. Last, we discuss the correction to the gauge transformation ofHµν , providing its decomposition into transformations for the elds in theN = 0Supergravity multiplet. We nd necessary to correct the denition of the scalareld and of the graviton with terms which are quadratic in Hµν . In particular,the correction to the scalar eld points to an amusing geometrical interpretation,in which the eld itself is interpreted as proportional to the Ricci scalar of thecorresponding manifold.

• In chapter 5 we summarize our main results and we provide an outlook on thepossible extensions and applications of our work.

• In appendix A we include, for pedagogical reasons, details on the evaluation ofthe graviton cubic vertices by means of the Noether procedure.

• In appendix B we include analogous details for the cubic vertices of the DC La-grangian, also proving its equivalence, up to eld redenitions, to the three-pointcouplings of the N = 0 Supergravity Lagrangian.

Chapter 2Scaering amplitudes and the

double copy

In this chapter we review some progress in our understanding of Yang-Mills (YM) theo-ries and of their connections to theories of gravity, recently achieved by looking at thecorresponding scattering amplitudes from a novel perspective. First, we will focus on theso-called color-kinematics (CK) duality [14, 17], a peculiar symmetry observed in puregluon amplitudes in YM theories. Secondly, we will introduce the main theme of thisThesis: the double-copy (DC) relations [14, 17], expressing gravitational amplitudes assome squares of YM amplitudes, in a sense to be claried, order by order in perturbationtheory. We will consider a rst, yet instructive, example, which will turn out to be use-ful throughout the whole exposition, before reviewing the DC relations in the generalcase. In order for this review to be self-contained, we shall also briey recall some basicsin the computation of scattering amplitudes, deriving the Feynman rules from YM andEinstein-Hilbert (EH) Lagrangian as well as the perturbative expansion of the latter.

2.1 Yang-Mills theory: Lagrangian and Feynman rules

A Yang-Mills theory is a gauge theory of several interacting massless spin-one eldsAaµ, with the index a labelling the various elds. As we will review in chapter 4, thepossible interactions are dictated by the gauge symmetry (up to higher derivative terms),resulting in the Lagrangian:

LYM = 12A

aµA

aµ + 12(∂ · Aa)2 − gfabc(∂µAaν)AbµAcν −

14g

2fabef cdeAaµAbνA

cµAdν ,

(2.1)

13

Chapter 2. Scattering amplitudes and the double copy 14

where g is the coupling constant of the theory, and where consistency demands thequantities fabc to be completely antisymmetric in their indices and to satisfy the Jacobiidentity:

fabef ecd + f bcef ead + f caef ebd = 0. (2.2)

Indeed, they provide the structure constants of a Lie algebra, with the gauge elds trans-forming in the adjoint representation:

δAaµ = ∂µεa + gfabcε

bAcµ. (2.3)

The geometrical interpretation of Aaµ is that of a connection on the principal bundle ofthe gauge group over spacetime and as such it allows to build a covariant derivative:

(Dµ)ac = δac∂µ − gfabcAbµ, (2.4)

as well as the curvature tensor of the principal bundle (or eld strength):

F aµν = g−1[Dµ, Dν ]a = ∂µA

aν − ∂νAaµ − gfabcAbµAcν . (2.5)

In terms of the curvature, the YM Lagrangian can be rewritten in a more geometricalfashion:

LYM = −14F

aµνF

µνa . (2.6)

The local symmetry of the theory can be viewed as a redundancy in the descrip-tion, that as such should not aect the value of physical quantities. Thus, one needs toimplement specic tools allowing to dispose of this redundancy in the computation ofscattering amplitudes in perturbation theory. To this end, we will exploit the Faddeev-Popov procedure (see e.g. [95]). In short, the procedure proposed by Faddeev and Popovbasically amounts to the insertion of an appropriate delta-function in the path integralof the theory, avoiding the redundant integration over all the equivalent eld congu-rations connected by gauge transformations. In practice, it corresponds to the additionof a term to the Lagrangian (known as the gauge-xing Lagrangian, LGF ) which ex-plicitly breaks the gauge symmetry and makes the derivative operator of the quadraticLagrangian invertible. Moreover, the procedure requires the introduction of ghost elds

15 Chapter 2. Scattering amplitudes and the double copy

(anticommuting bosons, which only enter as internal lines in loop diagrams). To ourpurposes however, since we will only compute tree-level diagrams, we will neglect theghost contribution and only keepLGF , both in the case of the YM theory and for gravity.

A common choice for a covariant gauge-xing in electrodynamics is the Lorenzgauge (∂ · A = 0). The corresponding gauge choice in the covariant quantizationof YM theories is the Lorenz ξ-gauge, where the gauge-xing Lagrangian has the form:

LGF = − 12ξ (∂ · Aa)2. (2.7)

For the sake of computational simplicity, we will further specify to the Feynmangauge (ξ = 1), where the Lagrangian becomes:

LYM + LGF (ξ = 1) = 12A

aµA

aµ − gfabc(∂µAaν)AbµAcν −14g

2fabef cdeAaµAbνA

cµAdν ,

(2.8)

while the propagator assumes a particularly simple form:

a, µ b, νp= −i δabηµν

p2 ,

where pµ is the momentum of the propagating gluon.

Both here and in the computation of scattering amplitudes, we will consider all theparticles as “entering” the diagram, so that momentum conservation reads:

n∑i=1

piµ = 0. (2.9)

With this convention the Feynman rules for YM theory are the following:

a1, µ1

a3, µ3a2, µ2

V µ1µ2µ33,a1a2a3 = gfa1a2a3ηµ1µ2(p1 − p2)µ3 + ηµ2µ3(p2 − p3)µ1

+ηµ3µ1(p3 − p1)µ2,


a1, µ1

a2, µ2 a3, µ3

a4, µ4

V µ1µ2µ3µ44,a1a2a3a4 = −ig2fa1a2efa3a4e(ηµ1µ3ηµ2µ4 − ηµ1µ4ηµ2µ3)

+fa1a4efa2a3e(ηµ1µ2ηµ3µ4 − ηµ1µ3ηµ2µ4)+fa1a3efa4a2e(ηµ1µ4ηµ2µ3 − ηµ1µ2ηµ3µ4).

In the following section we will often employ these rules in order to compute scat-tering amplitudes.

2.2 Tree-level Yang-Mills amplitudes and

color-kinematics duality

In this section we will compute some tree-level gluon scattering amplitudes aiming to il-lustrate both the color-kinematics (CK) duality and the double-copy (DC) relations in thesimplest cases, where analytical calculations are still manageable. We recall that scat-tering amplitudes are obtained dressing the connected and amputated Green’s functionswith the polarizations of the external particles. In the case of spin-one massless parti-cles inD spacetime dimensions, the physical polarizations are states in the fundamentalrepresentation of SO(D−2) (therefore, there areD−2 of them). In covariant notation,they can be embedded into D-dimensional vectors εµ(p) (with p the momentum of theparticle) subject to the constraint p · ε = 0 and, as we will detail in Section 2.4.1, tothe equivalence relation εµ ∼ εµ + cpµ. The latter condition is equivalent to a gaugetransformation on the spin-one eld in momentum space and identies an equivalenceclass of polarization vectors: Lorentz-invariance of the S-matrix requires the scatteringamplitudes to be unaected by the choice of the representative in the equivalence class.

2.2.1 Three gluons amplitude

The scattering amplitude among three massless particles, that we shall denote with A3,is obtained by dressing the Feynman rules for the cubic vertex with the external polar-izations, as well as imposing on-shell conditions (p2

i = 0) for each particle together withmomentum conservation (

∑i pi = 0). In the case of the three particles-kinematics these

conditions are so constraining that, in D = 4, they only allow for complex momenta.


However, to our purposes it is interesting to work out the expression of the amplitude inarbitraryD, as it will turn out to be our main explicit tool to investigate the double copy.

Exploiting transversality of the polarization vectors it is straightforward to derivethe following expression for A3:

A3 = ε1µ1ε

2µ2ε

3µ3V

µ1µ2µ33,a1a2a3 =

= gfa1a2a3(ε1ε2)[ε3 · (p1 − p2)] + (ε2ε3)[ε1 · (p2 − p3)] + (ε3ε1)[ε2 · (p3 − p1)]

= −2gfa1a2a3(ε1ε2)(ε3p2) + (ε2ε3)(ε1p3) + (ε3ε1)(ε2p1)

= −2gfa1a2a3εµ1εµ2εµ3ηµ1µ2pµ32 + ηµ2µ3pµ1

3 + ηµ3µ1pµ21 . (2.10)

Here and in what follows we use a notation in which scalar products between polariza-tions and momenta are written in parentheses and without the dot: for example insteadof ε1 ·ε2 we will write (ε1ε2). We will come back to this expression in Section 2.4.2, whenwe will employ it to build gravitational amplitudes.

2.2.2 Four gluons amplitude

The tree-level scattering amplitude with four gluons provides the simplest example ofcolor-kinematics duality. The amplitude is given by the sum of the Feynman diagramsin gure 2.1.

1

2 3

4

s

(a) s-channel.

1

2 3

4

t

(b) t-channel.

1

2 3

4

u

(c) u-channel.

1

2 3

4

(d) Contact term.

Figure 2.1: Feynman diagrams contributing to 4-gluon scattering

We will refer to the diagrams 2.1a, 2.1b and 2.1c as s−, t− and u-channel diagram respec-tively, from the names of the corresponding Mandelstam variables, dened as follows:

· s := q2s := (p1 + p2)2 = (p3 + p4)2 = 2p1 · p2 = 2p3 · p4, (2.11)


· t := q2t := (p1 + p4)2 = (p2 + p3)2 = 2p1 · p4 = 2p2 · p3, (2.12)

· u := q2u := (p1 + p3)2 = (p2 + p4)2 = 2p1 · p3 = 2p2 · p4, (2.13)

where we specialised to the case of massless particles. In this case the general identitys+ t+ u = −∑im

2i , reduces to s+ t+ u = 0.

The contributions of the three Mandelstam channels to the tree-level scattering am-plitude are:

2.1a = ε1µ1ε

2µ2ε

3µ3ε

4µ4g

2fa1a2bfa3a4cηµ1µ2(p1 − p2)ν + ηµ2ν(p2 + qs)µ1 − ηνµ1(qs + p1)µ2

× (−i)sδbcηνρηµ3µ4(p3 − p4)ρ + ηµ4ρ(p2 − qs)µ3 + ηρµ3(qs − p3)µ4

= −ig2

sfa1a2efa3a4e(ε1ε2)(p1 − p2)α + 2εα2 (ε1p2)− 2εα1 (ε2p1)

× (ε3ε4)(p3 − p4)α + 2ε4α(ε3p4)− 2ε3

α(ε4p3), (2.14)

2.1b = −ig2

tfa1a4efa2a3e(ε1ε4)(p1 − p4)α + 2εα4 (ε1p4)− 2εα1 (ε4p1)

× (ε2ε3)(p2 − p3)α + 2ε3α(ε2p3)− 2ε2

α(ε3p2), (2.15)

2.1c = −ig2

ufa1a3efa4a2e(ε1ε3)(p1 − p3)α + 2εα3 (ε1p3)− 2εα1 (ε3p1)

× (ε4ε2)(p4 − p2)α + 2ε2α(ε4p2)− 2ε4

α(ε2p4). (2.16)

Let us notice that the three channels share a similar structure: the same power ofthe coupling constant divided by a Mandelstam variable, a color factor composed of twostructure constants and a purely kinematic factor, containing Lorentz-invariant combi-nations of polarizations and momenta. It is possible to name the color factors accordingto the Mandelstam channel they appear in:

· cs := fa1a2efa3a4e, · ct := fa1a4efa2a3e, · cu := fa1a3efa4a2e. (2.17)

In terms of these variables, the Jacobi identity of the structure constants reads:

cs + ct + cu = 0. (2.18)

In the contribution stemming from the contact term (2.1d), on the other hand, allthe three color factors are present but there are no propagators. Upon multiplying and


dividing each color factor by the corresponding Mandelstam variable, one can write thecorresponding contribution to the amplitude as follows:

2.1d = ig2sscs[(ε1ε4)(ε2ε3)− (ε1ε3)(ε2ε4)] + t

tct[(ε1ε3)(ε2ε4)+

−(ε1ε2)(ε3ε4)] + u

ucu[(ε1ε2)(ε3ε4)− (ε1ε4)(ε2ε3)]

, (2.19)

thus leading to a structure similar to the contributions (2.14), (2.15) and (2.16), as it con-tains three terms of the type nc

p, where c is one of the three color factors dened in eq.

(2.17), p are denominators corresponding to Mandelstam variables (therefore they arescalar propagators) and n are purely kinematic factors at the numerator, containing mo-menta and polarizations in Lorentz invariant combinations. This allows us to performa splitting of the contact term contributions, assigning to each of the three Mandelstamchannels one of the addends in eq. (2.19), according to their color and propagator struc-ture (e.g. the term proportional to cs

sgets combined with the s-channel and similarly for

the other two channels).

This splitting procedure, namely the multiplication and division by appropriate prop-agators in order to reproduce the same color and propagator structures of the Feynmandiagrams with only trivalent vertices, can be implemented for amplitudes with morethan four particles as well. Therefore, it is possible to rearrange the amplitude in termsof a new set of diagrams (which are not Feynman diagrams) with only trivalent vertices.In the case of the four-particle amplitude they are represented in g. 2.2: where straightlines are used to represent the gluons, as opposed to the curly lines of the usual Feynmandiagrams. Each of these new diagrams subsumes the sum of the Mandelstam channeldiagram with a given color structure and the part of the contact term Feynman diagramwith the same color structure.

After the splitting of the contact term, properly multiplied and divided by propaga-tors, the tree-level four-gluon amplitude can be written in the simple form:

A4 = g2nscss

+ ntctt

+ nucuu

, (2.20)

where the kinematic factors at the numerator are given by:

ns = −i(ε1ε2)(ε3ε4)(u− t) + 2(ε1ε2)(ε4p3)ε3 · (p2 − p1) + (ε3p4)ε4 · (p1 − p2)


1

2 3

4

1

2 3

4

1

2 3

4

Figure 2.2: Trivalent diagrams contributing to 4-gluon scattering

+ 2(ε3ε4)(ε1p2)ε2 · (p3 − p4) + (ε2p1)ε1 · (p4 − p3) − 4(ε2ε3)(ε1p2)(ε4p3)

+ 4(ε1ε3)(ε2p1)(ε4p3)− 4(ε1ε4)(ε2p1)(ε3p4) + s(ε1ε3)(ε2ε4)− s(ε1ε4)(ε2ε3),(2.21)

nt = −i(ε1ε4)(ε3ε2)(s− u) + 2(ε1ε4)(ε3p2)ε2 · (p4 − p1) + (ε2p3)ε3 · (p1 − p4)

+ 2(ε2ε3)(ε1p4)ε4 · (p2 − p3) + (ε4p1)ε1 · (p3 − p2)+ 4(ε4ε3)(ε1p4)(ε2p3)

+ 4(ε1ε2)(ε4p1)(ε3p2)− 4(ε1ε3)(ε4p1)(ε2p3) + t(ε1ε2)(ε4ε3)− t(ε1ε3)(ε4ε2),(2.22)

nu = −i(ε1ε3)(ε2ε4)(t− s) + 2(ε1ε3)(ε2p4)ε4 · (p3 − p1) + (ε4p2)ε2 · (p1 − p3)

+ 2(ε2ε4)(ε1p3)ε3 · (p4 − p2) + (ε3p1)ε1 · (p2 − p4)+ 4(ε3ε2)(ε1p3)(ε4p2)

+ 4(ε1ε4)(ε3p1)(ε2p4)− 4(ε1ε2)(ε3p1)(ε4p2) + u(ε1ε4)(ε3ε2)− u(ε1ε2)(ε3ε4).(2.23)

Via direct calculation, exploiting momentum conservation and transversality of the po-larization vectors, it is possible to check that:

ns + nt + nu = 0, (2.24)

providing the kinematic counterpart of the Jacobi identity.

At the four particles level, this was actually known since [96] and [97]. Now it isunderstood to provide the simplest example of a more general property of YM scatter-ing amplitudes termed color-kinematics duality (CK), rst identied in [14]. Basically,it amounts to say that the numerator factors in the expression of the amplitude obeythe same algebraic relation as the color factors, in spite of the purely group-theoreticalnature of the latter. One can also observe that it is possible to perform a redenition of


the numerator factors ni, n′s = ns + sα(pi, εi),

n′t = nt + tα(pi, εi),

n′u = nu + uα(pi, εi),

(2.25)

such that the scattering amplitude is invariant:

A′4 = A4 + (cs + ct + cu)α = A4. (2.26)

In force of (2.18), the possibility of shifting of the kinematic factors ni as in (2.25)without aecting the physical amplitude is known as generalized gauge invariance [14].Both the color-kinematics duality and the generalized gauge are crucial ingredients tothe understanding of the double-copy structure of gravitational scattering amplitudes.Therefore, we will devote the next section to a more systematic illustration of thesecentral concepts.

2.2.3 Color-kinematics duality

The identity (2.24), whose existence we observed in the simple case of the tree-levelscattering amplitude among four gluons, turns out to be a general property of scatteringamplitudes in pure YM theory at tree level. This was rst understood by Bern, Carrascoand Johansson in [14] and is also known as BCJ duality. In this section we explore theduality in general terms, explaining its content for an arbitrary number of particles attree level, while we shall postpone to Section 2.2.7 a discussion of its validity at looplevel. We will not give any proofs since these are extremely technical, but we will makeclear what has been proven and what is still a conjecture.

For a general tree-level scattering amplitude in pure Yang-Mills theory it is alwayspossible to rearrange its expression as a sum over diagrams with only trivalent vertices,that we collectively denote as Γ3, in the form:

Atreen (1, ..., n) = gn−2 ∑i∈Γ3

nici∏αi sαi

, (2.27)

where ci is a color factor obtained by dressing each vertex of the diagram with a structureconstant, ni is an appropriate kinematic factor containing momenta and polarizations


while the denominator is the product of all the inverse propagators sαi corresponding tointernal lines. As showed in the previous section for a specic case, this representationcan be obtained by splitting (in the sense explained at the end of page 19) all the contri-butions from the contact terms.

As illustrated in [14], the representation of the kinematic factors is not unique. In-deed we can modify the latter with a shift of the form

ni → n′i = ni + ∆i(pi, εi), (2.28)

and still leave the amplitude invariant if the functions ∆i(pi, εi) are chosen such that:

∑i∈Γ3

∆ici∏αi sαi

= 0. (2.29)

This property was termed generalized gauge invariance in [14], since at the eld-theoreticallevel it corresponds to a gauge transformation plus a eld redenition or to the additionof higher derivative (possibly non-local) terms which leave the YM Lagrangian invariant.Bern, Carrasco and Johansson conjectured in [14] that, in the class of amplitudes equiv-alent under (2.28), there always exists a representation of the kinematic numerators nisuch that, to every Jacobi identity satised by the color factors of a triplet of diagrams inthe set Γ3, it corresponds an analogous identity satised by the kinematic numerators:

ci + cj + ck = 0 ⇒ ni + nj + nk = 0. (2.30)

Moreover, they conjectured the kinematic factors to share the same antisymmetry of thecolor factors:

ci = −cj ⇒ ni = −nj. (2.31)

For instance, if we consider two diagrams i and j which are identical, except for the ex-change of two external legs attached to the same vertex, owing to the antisymmetry ofthe structure constants the color factors of the two diagrams are related by ci = −cj . Theconjecture requires also the kinematic factors to share this antisymmetry, which, how-ever, is naturally encoded in the YM Feynman rules (Section 2.1) for the cubic vertex andin each of the three terms resulting from the splitting of the quartic vertex, as it can be


easily checked in the previous sections. Therefore, to satisfy the requirement (2.31), it issucient to wisely choose the generalized gauge transformation. This correspondencebetween the algebraic identities obeyed by the structure constants and analogous iden-tities relating the kinematical factors is known as the color-kinematics (CK) duality.

As already mentioned, the validity of the CK duality for an arbitrary number of ex-ternal particles at tree level was conjectured in [14]. The existence of duality-satisfyingnumerators was rst exemplied for any n in [98], while two independent solutionsto the problem of nding explicit expressions were proposed in [99] and [100]. Whatmakes this property quite remarkable is that, a priori, there is no apparent connectionbetween the algebra of the gauge group and the kinematic factors. To the present stateof knowledge, the reason for such a duality between color and kinematics is not yet un-derstood, with the only indication in this sense provided by [101], where the presenceof a kinematic algebra in the self-dual sector of YM theory in D = 4 was pointed out.

2.2.4 Partial amplitudes

As we will discuss in Section 2.4, the existence of the CK duality in YM theory is cru-cial for the validity of the so-called double-copy relations, connecting gravitational andgauge theory amplitudes, which are the main subject of this Thesis. However, the CKduality is also relevant from another viewpoint, as it highlights a new property of theso-called partial amplitudes, which we now introduce.

Let us consider an SU(N) YM theory and observe that all the pure gluon amplitudescontain products of structure constants, which one can express in terms of traces ofgenerators of the gauge group. The simplest case involves only one tensor fabc so that,using

[T a, T b] = ifabcT c, (2.32)

with T a the SU(N) generators in the fundamental representation1, one can extract

fabc = −2i

tr(T aT bT c)− tr(T aT cT b). (2.33)

1With normalization tr(T aT b) = 12δab. Let us mention that typically in the amplitude-related litera-

ture a dierent normalization is used, such that tr(T aT b) = δab.


From the Fierz identity:

(T a)ij(T b)kl = 12δilδjk −

1Nδijδkl

, (2.34)

one can derive an analogous expression for the product of two structure constants withone color index contracted:

fa1a2ef ea3a4 = 4

tr(T a1T a2T a3T a4)− tr(T a1T a2T a4T a3)

−tr(T a2T a1T a3T a4) + tr(T a2T a1T a4T a3). (2.35)

Similar results can be obtained for any products of structure constants, with dierentindex contractions, resulting in single traces or in products of traces of generators. Atn points there are only (n − 1)! independent single traces, due to the ciclicity of thetrace, but this set is not sucient to describe all the possible congurations of n gluons.However, it is possible to show [5] that, at tree level2:

Atreen = gn−2 ∑

σ∈Sn−1

An[1σ(2)σ(3)...σ(n)]tr(T a1T σ(a2)T σ(a3)...T σ(an)), (2.36)

where Sn−1 is the symmetric group of order n − 1 and An[1...n] are called partial orcolor-ordered amplitudes, owing to the fact that they have a xed ordering of externalparticles. The decomposition (2.36) is useful for two reasons: rst, partial amplitudesare easier to compute than full amplitudes, since the color-ordered Feynman rules (withdenite particle ordering) are simpler than the complete ones. Moreover, they can beproven to be gauge invariant and possess the following interesting properties [5]:

· Ciclicity: An[12...n] = An[2...n1], (2.37)

· Reection: An[12...n] = (−1)nAn[n...21], (2.38)

· U(1) decoupling identity: An[123...n] + An[213...n] + ...+ An[23...1n] = 0,(2.39)

together with satisfying the so-called Kleiss-Kuijf relations[102]:

An[1, α, n, β] = (−1)|β|∑

σ∈OP(α,βT )An[1, σ, n], (2.40)

2Loop-level amplitudes require the inclusion of products of traces.


where α and β denote two sets of external particle indices, “OP” the ordered per-mutations of the indices in the set α∪βT and βT a set of indices with the reverseordering with respect to β. Together with (2.37)-(2.39), (2.40) can be employed to re-duce the number of independent partial amplitudes: from (n− 1)! (corresponding withthe number of independent single traces of n generators) to (n− 2)!.

The discovery of the CK duality comes into play at this point, since it was realizedthat the new algebraic relations between the kinematic factors of the cubic diagramsintroduced in the previous sections can be exploited to express some of the partial am-plitudes in terms of other partial amplitudes and kinematic factors [14]. Then, one canperform a generalized gauge transformation of the type (2.28), setting to zero some ofthe kinematic factors, resulting in new relations between partial amplitudes. We willnot enter in the details of this construction, which are beyond the scope of this The-sis, limiting to state the result: the existence of the CK duality leads to the so-calledBCJ relations, which further constrain the partial amplitudes and leave only (n − 3)!independent ones. The original expression of the BCJ relations is the following:

An[1, 2, α, 3, β] =∑

σi∈POP(α,β)An(1, 2, 3, σi)Fi, (2.41)

with “POP” denoting partial-order-preserving permutations: permutations of set α ∪β preserving the relative ordering inside the set β, while Fi stands for complicatedkinematic factors containing Lorentz-invariant combinations of the particles momenta,whose explicit denition can be found in [14]. Apart from the exact expressions of suchfactors, the meaning of (2.41) is that all the partial amplitudes can be expressed in termsof the (n − 3)! partial amplitudes with particles 1, 2 and 3 in the rst positions, in thisorder. In addition, it was further shown [15] that (2.41) contains redundant information,and that it can be derived from the fundamental BCJ relation:

n∑b=3

( n−1∑c=1

p2 · pc)An[1, 3, ..., b− 1, 2, b, ..., n] = 0, (2.42)

where pi are the gluons momenta. Examples of BCJ relations in the four and ve gluonscase are the following:

s14A4[1234]− s13A4[1243] = 0, (2.43)


s12A5[21345]− s23A5[13245]− (s23 + s24)A[13425] = 0, (2.44)

with as usual sij = 2pi · pj .

There are many proofs of the BCJ relations, both from String Theory [103–109] andfrom Field Theory [15, 110, 111]. In [110] it was also shown that the existence of BCJrelations together with the Kleiss–Kuijf identities give rise to Jacobi-like identities forthe kinematic numerators.

2.2.5 Five gluons amplitude

Let us move to our last explicit computation in YM theory: the ve gluons amplitude.It is interesting to our purposes since it provides an example where the expression ofthe amplitude derived from Feynman diagrams (after splitting the contact terms) doesnot satisfy the CK duality. It is then necessary to perform a generalized gauge trans-formation to bring the amplitude in the form of eq. (2.27). We will also show explicitlythat, as observed in [16], this transformation is tantamount to the addition of a non-localve gluons contact term to the YM Lagrangian. The latter contributes non-trivially tothe trivalent diagrams Γ3, although it is identically zero due to the Jacobi identity andtherefore has no inuence on physical quantities.

The Feynman diagrams contributing to the amplitude contain vertices with eitherthree or four gluons; in the latter case each quartic vertex has to be split into threedistinct contributions to the trivalent vertices with xed color structure needed in or-der to enforce the color-kinematics duality. One can easily count the distinct tree-leveldiagrams with only trivalent vertices we can build at tree level: given the number ofdiagrams for n particles, call it Cn, in order to obtain a diagram for n + 1 particles wecan attach an additional leg either to one of the n external legs or to an internal line ofeach of the Cn diagrams for n particles. Denoting with E (I) the number of external (in-ternal) lines and with V the number of vertices, then the topology of tree-level diagramsimplies:

I = V − 1, (2.45)

which is a specic case of Euler’s formula for planar diagrams in the absence of loops.


In addition, for diagrams with only trivalent vertices, the following relation also holds:

3V = E + 2I, (2.46)

as can be understood from the fact that exactly three legs must meet in every vertex andexternal lines are attached to one vertex only, while internal lines connects two vertices.For E = n, (2.45) and (2.46) allow to derive the number of vertices and internal lines,which are given by:

V = n− 2,

I = n− 3.(2.47)

Since, as mentioned, tree-level diagrams with trivalent vertices and n + 1 external legsare built from analogous diagrams with n external legs, attaching an additional leg eitherto an external or to an internal line one nds:

Cn+1 = (E + I)Cn = (2n− 3)Cn ⇒ Cn = (2n− 5)!! (2.48)

Therefore, for n = 5, the set Γ3 contains (10−5)!! = 15 diagrams. Contributions to suchdiagrams come from Feynman diagrams with either only cubic vertices (g. 2.3a), whichare 15, or with one cubic and one quartic vertex (2.3b), which are

(52

)= 10. Since each

(a) (b)

Figure 2.3: Type of Feynman diagrams contributing to 5-gluon scattering

quartic term contains three dierent color structures, again we split their expressioninto three parts and multiply and divide by the appropriate inverse propagator, getting30 distinct contributions. Therefore, per each color structure we have 30/15=2 contribu-tions from diagrams with one quartic term (g. 2.3b).

To check the CK duality we choose three diagrams whose color factors sum up to


zero owing to the Jacobi identity, and compute the contributions to them from the ap-propriate Feynman diagrams. Without loss of generality, we can consider for instancethe diagrams with color structure

ci := fa1a2bf ba3cf ca4a5 , cj := fa1a2bf ba4cf ca5a3 , ck := fa1a2bf ba5cf ca3a4 , (2.49)

such that ci + cj + ck = 0, naming the corresponding diagrams in Γ3 with i, j and k.In gs. 2.4a, 2.4b and 2.4c we list the Feynman diagrams which contribute to diagramsi, j and k, respectively, where the vertical bar with cl means that only the part of thediagram proportional to cl, with l = i, j, k, has to be selected. As before, we consider

1

23

4

5

= + +

1

23

4

5 1

23

4

5 1

2

34

5

ci ci

(a) Contributions to the diagram with color factor ci.

1

24

5

3

= + +

1

24

5

3 1

24

5

3 1

2

45

3

cj cj

(b) Contributions to the diagram with color factor cj .

1

25

3

4

= + +

1

25

3

4 1

25

3

4 1

2

53

4

ck ck

(c) Contributions to the diagram with color factor ck .

Figure 2.4: Diagrams in Γ3 with color factor ci, cj , ck, respectively.

all the particles as ingoing and we label the inverse propagators of the internal lines assij := (pi + pj)2 = 2pi · pj . We wish rearrange the amplitude in the form:

An = gn−2 ∑i∈Γ3

nici∏αi sαi

, (2.50)


to then check whether ni + nj + nk = 0, given that ci + cj + ck = 0.

Let us focus, for instance, on the kinematic factor of diagram i (the others can beeasily obtained from cyclic permutations of the particle indices). Since each of the threeFeynman diagrams in g. 2.4a contributes to the kinematic numerator ni in a dierentway, we analyse them separately, starting from those in g. 2.5. As clear from g. 2.4,

1

2

3

4

5

Figure 2.5

the Feynman diagrams with only cubic vertices which contribute to the diagrams i, jand k in Γ3 share a common vertex, containing the external lines 1 and 2 and an internalline carrying a squared momentum (p1 + p2)2 = s12. We name V b,ν

12 this vertex (g.2.6a) contracted with the polarization vectors ε1 and ε2, while b (ν) is the color (Lorentz)index of the line which is not contracted with polarization vectors. The remaining partof the diagram consists in a gluon propagator, carrying squared momentum s12, whichis common to the three diagrams as well, and a part which is dierent in the three cubicdiagrams. This latter part, which we name V 345

b,ν in the case of diagram i, is representedin g. 2.6b (the corresponding terms for j and k are obtained from particle index per-mutations), where the polarization vectors ε3, ε4 and ε5 are contracted with three of theexternal lines, while as above b (ν) is the color (Lorentz) index of the line which is notcontracted with polarization vectors. With this factorization, the diagram of g. 2.5 canbe written as:

2.5 = V b,ν12 ·

−is12· V 345

b,ν , (2.51)

where V b,ν12 and V 345

b,ν are represented in g. 2.6a and 2.6b respectively. Their expressionis the following:

V b,ν12 = gfa1a2b(ε1ε2)(p1 − p2)ν + 2(ε1p2)εν2 − 2(ε2p1)εν1, (2.52)


1

2

b, ν

(a) V b,ν12 .

b, ν

3

4

5

(b) V b,ν345 .

Figure 2.6: Decomposition of the diagram 2.5

V b,ν345 = −i g

2

s45f ba3cf ca4a5εν3[4(ε5p4)(ε4p3)− (ε4ε5)s34 + 2(ε4ε5)s35 − 4(ε4p5)(ε5p3)]

− 2(ε3ε4)(ε5p4)(p3 − p4 − p5)ν + (ε4ε5)(ε3p4)(p3 − p4 − p5)ν

− (ε4ε5)(ε3p5)(p3 − p4 − p5)ν + 2(ε3ε5)(ε4p5)(p3 − p4 − p5)ν

+ 2(ε4ε5)(ε3p4)(p4 − p5)ν + 2(ε4ε5)(ε3p5)(p4 − p5)ν

− 4εν4[(ε5p4)(ε3p4) + (ε5p4)(ε3p5)] + 4εν5[(ε3p4)(ε4p5) + (ε3p5)(ε4p5)]. (2.53)

In order to obtain the sought for contribution to the kinematic factors ni in eq. (2.50),we must dispose of the coupling constant, of the color factors and of the propagators:we name V the objects V dened above, omitting these terms and without color indices(the free color index of the vertices 2.6a and 2.6b are simply contracted with each otherby the gluon propagator in Feynman gauge). Employing this prescription in eq. (2.52),we can read the denition of V ν

12:

V b,ν12 = gfa1a2b...ν ⇒ V ν

12 := ...ν . (2.54)

In a similar way, we dene V ν345, this time eliminating also the inverse propagator at the

denominator:

V b,ν345 = −i g

2

s45f ba3cf ca4a5...ν ⇒ V ν

345 := −i...ν . (2.55)

Moreover, we calculate explicitly only the part concerning diagram i (with particle order-ing 12345), while the contributions to diagrams j and k can be calculated easily perform-ing permutations of the particle indices. Therefore, diagrams of the type 2.5 contribute


to the sum of kinematic factors as follows:

[ni + nj + nk]2.5 = V b,ν12 (−i)V 345

b,ν + V 453b,ν + V 534

b,ν = −V 12b,ν(p3 + p4 + p5)ν

× [(ε4ε5)ε3 · (p4 − p5) + (ε5ε3)ε4 · (p5 − p3) + (ε3ε4)ε5 · (p3 − p4)]

+ εν3(ε4ε5)(s35 − s34) + εν4(ε5ε3)(s43 − s45) + εν5(ε3ε4)(s54 − s53).(2.56)

Using momentum conservation one has p3 + p4 + p5 = −p1− p2 and from the transver-sality of the polarization vectors and the mass-shell condition, also V 12

b,ν (p1 + p2)ν = 0,so that:

[ni + nj + nk]2.5 = −[(ε1ε2)(ε4ε5)(ε3p1)(s35 − s34)− (ε1ε2)(ε4ε5)(ε3p2)(s35 − s34)

+ 2(ε4ε5)[(ε2ε3)(ε1p2)− (ε1ε3)(ε2p1)](s35 − s34)]

+ [345→ 453] + [345→ 534], (2.57)

where with the notation +[abc → cab] we mean that the expression also contains thepermutations of the rst term with index a replaced with c, b with a and c with b. Wenotice that every term in ni contains either one or two (the integer part of n/2, in thegeneral case) factors of (εiεj), but in the sum of the kinematic numerators from thistype of diagrams only terms with two factors survive. This is essential in order for themto be cancelled by diagrams with a contact term, since the Feynman rules of the lattergenerate only terms with two factors. In order to nd the contribution to ni from thediagram in g. 2.7 we must select from the Feynman diagram the part proportional toci. Let us write down its full expression, with the appropriate insertions of sij

sij:

2.7 = ig2 (−i)s12

V 12b,νf

a1a2b × s45

s45f ba3cf ca4a5 [εν5(ε3ε4)− εν4(ε3ε5)]

+ s35

s35f ba4cf ca5a3 [εν3(ε4ε5)− εν5(ε4ε3)]

+ s34

s34f ba5cf ca3a4 [εν4(ε5ε3)− εν3(ε5ε4)]. (2.58)

Since in eq. (2.58) all the color factors of interest are present, we can easily extractfrom it the kinematic factors nl, with l = i, j, k, simply suppressing coupling constants,propagators and structure constants after having identied the color factors ci, cj, ck.


1

2

3

4

5

ci

Figure 2.7

Their sum is given by the following expression:

[ni + nj + nk]2.7 = s45(ε3ε4)[(ε1ε2)ε5 · (p1 − p2) + 2(ε2ε5)(ε1p2)− 2(ε1ε5)(ε2p1)]

− (ε3ε5)[(ε1ε2)ε4 · (p1 − p2) + 2(ε2ε4)(ε1p2)− 2(ε1ε4)(ε2p1)]

+ [345→ 453] + [345→ 534]. (2.59)

We can now collect the contribution to the sum of the kinematic factors coming fromdiagrams of the type 2.5 and 2.7, and observe that:

[ni + nj + nk]2.5+2.7 = 0. (2.60)

Therefore, any violation of the kinematic Jacobi identity must come from diagrams of thetype 2.8. Let us now compute the contribution of this diagram to ni, therefore extractingonly the part with color factor ci. Again, contributions tonj andnk can be easily obtainedupon permuting of the particle indices (3, 4, 5). We nd:

1

2

3

4

5

ci

Figure 2.8


2.8 = g3

s12s45cis122(ε1ε3)(ε2ε4)(ε5p4)− 2(ε1ε4)(ε2ε3)(ε5p4)

+ (ε2ε3)(ε4ε5)ε1 · (p4 − p5)− (ε1ε3)(ε4ε5)ε2 · (p4 − p5)

+ 2(ε1ε5)(ε2ε3)(ε4p5)− 2(ε1ε3)(ε2ε5)(ε4p5). (2.61)

We notice that with diagrams of the type 2.8 we have multiplied and divided by thesame factor s12 the contributions to diagrams i, j and k in order to render the correctpropagator structure. Therefore the kinematic factors nicely sum keeping s12 factorized,with the result:

[ni + nj + nk]2.8 = s122[(ε1ε4)(ε2ε5)− (ε1ε5)(ε2ε4)]ε3 · (p4 + p5)

+ (ε4ε5)[(ε2ε3)εα1 − (ε1ε3)εα2 ](p4 − p5)α+ [345→ 453] + [345→ 534] 6= 0. (2.62)

To summarize, though we were able to write the amplitude in the form indicated byeq. (2.50), still the CK duality is not yet satised and we need to look for a generalizedgauge transformation allowing for this property to hold. We shall not evaluate here thefactors ∆i appearing in eq. (2.29); rather, in the next section we will show that these canbe directly obtained from an appropriate modication of the YM Lagrangian.

2.2.6 A non-local Lagrangian for color-kinematics duality at ve

points

In the previous section we showed how to reorganize the Feynman diagrams expansionof the ve gluons tree-level amplitude, in such a way that it can be written as follows:

A5 =∑i∈Γ3

cini∏αi sαi

, (2.63)

We selected three diagrams out of the 15 present in the set Γ3, namely diagrams i, j andk, whose color factors obey a Jacobi identity: ci + cj + ck = 0. Then, we realized thatni + nj + nk 6= 0, thus seemingly resulting in a violation of the CK duality.

However, as explained, we still have the freedom to perform a generalized gaugetransformation, which does not aect the scattering amplitude but still sets ni + nj +


nk = 0. Instead of making the transformation directly on the numerators, we considerthe possibility of adding a 5-gluons contact term to the Yang-Mills Lagrangian, whichvanishes identically due to the Jacobi identity but contributes via Feynman diagrams toeach color structure in order to make the color-kinematics duality evident. This was rstproposed in [16], where the explicit expression of this Lagrangian term was also given:

L′5 = −g3

2 fa1a2bf ba3cf ca4a5

(∂[µA

a1ν]A

a2ρ A

a3µ + ∂[µAa2ν]A

a3ρ A

a1µ

+ ∂[µAa3ν]A

a1ρ A

a2µ) 1

(Aa4νAa5ρ). (2.64)

Even though in this form it is not clear that L′5 is identically zero, this can be easilyseen if we relabel the color indices in such a way that only one monomial is present,multiplied by a sum of color factors:

L′5 = −g3

2 (fa1a2bf ba3c + fa3a1bf ba2c + fa2a3bf ba1c)f ca4a5

×(∂[µA

a1ν]A

a2ρ A

a3µ) 1

(Aa4νAa5ρ), (2.65)

where the Jacobi identity is now apparent. It is interesting to notice that such a relevantsymmetry such the CK duality requires non-local contributions in order to be imple-mented at the Lagrangian level.

In order to extract the Feynman rules from this Lagrangian, we notice that, as usual,we must sum over the n! = 5! = 120 ways to contract the n = 5 elds in the Lagrangianwith the n = 5 external particles involved in the scattering. Exploiting the total anti-symmetry of the structure constants, we can organize these 120 contractions into the15 possible color structures (one per each diagram in Γ3), such that each color structurereceives contributions from 8 contractions. These contractions amount to the possi-ble associations of each external particle xi to each eld in L′5, or equivalently to eachcolor index ai. If we want to build, for instance, the color factor ci = fa1a2bf ba3cf ca4a5 ,we must contract x3 with a3 and x1, x2 (x4, x5) either with a1, a2 (a4, a5) or with a4, a5

(a1, a2) in each order. Therefore, with reference to the order of color indices present inthe Lagrangian of eq. (2.64), we have the freedom to switch a1 and a2 as well as a4 anda5, but also to exchange the pair (12) with the pair (45), overall amounting to a total of2! · 2! · 2 = 8 contractions. Let us notice that, when exchanging (12) with (45), the colorfactor stays the same (apart from signs), but 1/ changes value: indeed, when acting


on Aa4Aa5 (case “A”) its value is 1/s45 (to be multiplied by s12s12

), while when acting onAa1Aa2 (case “B”) it is 1/s12 (to be multiplied by s45

s45).

Here we compute separately the contributions to the kinematic factor of diagram i

coming from L′5, in the two possible cases A and B:

nAi = s12

2 (ε1ε4)(ε2ε5)ε3 · (p1 + p2)− (ε1ε3)(ε2ε5)ε4 · (p1 − p3)

+ (ε2ε4)(ε3ε5)ε1 · (p2 + p3) + (ε1ε5)(ε3ε4)ε2 · (p1 + p3)

+ (ε1ε5)(ε2ε3)ε4 · (p2 − p3)− (ε1ε5)(ε2ε4)ε3 · (p1 + p2)

− (ε1ε4)(ε3ε5)ε2 · (p1 + p3)− (ε2ε5)(ε3ε4)ε1 · (p2 + p3)

+ (ε1ε3)(ε2ε4)ε5 · (p1 − p3)− (ε2ε3)(ε1ε4)ε5 · (p2 − p3)

+ (ε1ε2)[(ε3ε5)εα4 − (ε3ε4)εα5 ](p1 − p2)α, (2.66)

nBi = −s45

2 (ε1ε4)(ε2ε5)ε3 · (p4 + p5) + (ε3ε4)(ε2ε5)ε1 · (p3 − p4)

+ (ε1ε5)(ε2ε3)ε4 · (p5 + p3) + (ε1ε3)(ε2ε4)ε5 · (p3 + p4)

+ (ε2ε4)(ε3ε5)ε1 · (p5 − p3)− (ε1ε5)(ε2ε4)ε3 · (p4 + p5)

− (ε1ε4)(ε2ε3)ε5 · (p3 + p4)− (ε2ε5)(ε1ε3)ε4 · (p3 + p5)

+ (ε1ε5)(ε3ε4)ε2 · (p4 − p3)− (ε3ε5)(ε1ε4)ε2 · (p5 − p3)

+ (ε4ε5)[(ε2ε3)εα1 − (ε1ε3)εα2 ](p4 − p5)α, (2.67)

where the nA,Bj,k are obtained from cyclic permutations of the particle indices (3,4,5), asusual. We notice that the coecients nBj,k contain exactly the same terms as nBi , exceptfor the sij terms: since via momentum conservation s34 + s25 + s45 = s12 we can sumthese terms to the A-type contributions, keeping s12 factorized. Therefore, if we denen′i = ni + nAi + nBi , we end up with:

n′i + n′j + n′k = (nAi + nBi + ni) + (nAj + nBj + nj) + (nAk + nBk + nk) = 0, (2.68)

which means that the addition of L′5 to the YM Lagrangian generates Feynman ruleswhich make the CK automatically satised up to ve points at tree level.

Furthermore, it is possible to consider the addition of another ve gluons contact


term to the YM Lagrangian, which again vanishes identically due to the Jacobi iden-tity and whose Feynman rules generate diagrams which automatically satisfy the color-kinematics duality at ve points:

L′′5 = −α12g

3fa1a2bf ba3cf ca4a5(∂(µA

a1ν)A

a2ρ A

a3µ + ∂(µAa2ν)A

a3ρ A

a1µ

+ ∂(µAa3ν)A

a1ρ A

a2µ) 1

(Aa4νAa5ρ). (2.69)

Per each value of α, the addition of such a term to the YM Lagrangian implements ageneralized gauge transformation which leaves the ve-gluons amplitude in the CK dualform, as can be seen via direct computation. The case in which 1

= 1s45

will be referred toas the “C” case, while we shall label with “D” the case in which 1

= 1s12

. Simply changingsome signs in the expressions of nAi and nBi (in particular, only terms proportional to thedierence of two momenta change sign), we nd (setting α = 1):

nCi = s12

2 (ε1ε4)(ε2ε5)ε3 · (p1 + p2) + (ε1ε3)(ε2ε5)ε4 · (p1 − p3)

+ (ε2ε4)(ε3ε5)ε1 · (p2 + p3) + (ε1ε5)(ε3ε4)ε2 · (p1 + p3)

− (ε1ε5)(ε2ε3)ε4 · (p2 − p3)− (ε1ε5)(ε2ε4)ε3 · (p1 + p2)

− (ε1ε4)(ε3ε5)ε2 · (p1 + p3)− (ε2ε5)(ε3ε4)ε1 · (p2 + p3)

− (ε1ε3)(ε2ε4)ε5 · (p1 − p3) + (ε2ε3)(ε1ε4)ε5 · (p2 − p3)

+ (ε1ε2)[(ε3ε4)εα5 − (ε3ε5)εα4 ](p1 − p2)α, (2.70)

nDi = −s45

2 (ε1ε4)(ε2ε5)ε3 · (p4 + p5)− (ε3ε4)(ε2ε5)ε1 · (p3 − p4)

+ (ε1ε5)(ε2ε3)ε4 · (p5 + p3) + (ε1ε3)(ε2ε4)ε5 · (p3 + p4)

− (ε2ε4)(ε3ε5)ε1 · (p5 − p3)− (ε1ε5)(ε2ε4)ε3 · (p4 + p5)

− (ε1ε4)(ε2ε3)ε5 · (p3 + p4)− (ε2ε5)(ε1ε3)ε4 · (p3 + p5)

− (ε1ε5)(ε3ε4)ε2 · (p4 − p3) + (ε3ε5)(ε1ε4)ε2 · (p5 − p3)

+ (ε4ε5)[(ε1ε3)εα2 − (ε2ε3)εα1 ](p4 − p5)α. (2.71)

Again, the terms contributing to the color structures j and k are obtained by cyclicpermutations of (3,4,5). Using s34 + s35 + s45 = s12 in the sum over D-type numerators


we get:

(nCi + nDi ) + (nCj + nDj ) + (nCk + nDk ) = 0, (2.72)

thus showing that L′′5 generates self-CK dual Feynman rules for ve gluon scattering.

2.2.7 Color-kinematics duality at loop level

We end our discussion on CK duality in YM amplitudes with an outline of how thisproperty is implemented when quantum corrections are concerned. The idea is that,as in the case of tree-level amplitudes, contributions from contact terms can always betreated is such a way that the amplitude gets rearranged as a sum over diagrams withonly trivalent vertices. The expression of a general L−loops amplitude is then writtenas:

AL-loopn = iLgn−2+2L ∑

i∈Γ3

∫ ( L∏k=1

dDlk(2π)D

) 1Si

cini∏αi sαi

, (2.73)

where the notation is the same of Section 2.2.3: Γ3 is the set of diagrams with trivalentvertices, c are color factors, n are kinematic numerators and 1/sαi are scalar-like prop-agators attached to the internal lines. Moreover, lk denote the loop momenta while Siis the symmetry factor of the corresponding diagram. The addition of diagrams withghosts as internal lines does not aect the possibility of deriving such representationof the amplitude: since the ghosts are states in the adjoint representation of the gaugegroup they behave similarly to the gluons. In particular, they only have trivalent cou-plings with the gauge eld while the corresponding vertex contains exactly one structureconstant, like the vertex with three gluons.

The content of the CK duality is totally analogous to the tree-level case: it was pro-posed in [17] that also at loop level a representation of the kinematic factors in (2.73)exists such that whenever the color factors of a triplet of diagrams i, j, k satisfy a Jacobiidentity (ci+cj+ck = 0), then the same identity is also satised by the kinematic factorsof the three diagrams (ni + nj + nk = 0). In contrast to the tree-level case, however,the validity of the CK duality at loop level is still a conjecture. Nonetheless, the conjec-ture has been tested working out explicit numerators in several non-trivial cases, bothin pure and Yang-Mills theory, as well as in its supersymmetric counterpart. Striking


examples are:

• 4 points, 4 loops in N = 4 SYM [17, 48],

• 5 points, 2 loops in N = 4 SYM [112],

• 4 points, 2 loops in pure YM (in D = 4) [17]

• 4 points, 1 loop in pure YM (in any D) [113],

• 4 points, 1 loop in pure YM with matter [114],

• 4 points, 1 loop with less than maximal supersymmetry [115].

It is worth emphasizing that, although most examples are carried out in simplied casessuch as maximal supersymmetry or xed-helicity amplitudes in D = 4, CK dualityseems to be a general property of every YM theory, in any dimension.

2.3 General Relativity: Lagrangian and Feynman rules

General Relativity is usually formulated as a eld theory for the eld gµν , playing therole of the dynamical metric of spacetime, which is a Lorentzian manifold endowed witha torsion-free Levi-Civita connection Γλµν(g). The Einstein-Hilbert (EH) action is ex-pressed in terms of the curvature as:

SEH = 12κ2

∫dDx√−gR, (2.74)

where g is the determinant of the metric, R the Ricci scalar and κ2 = 8πG(D) (in theusual c = 1 units), with G(D) the Newton constant in dimension D. We neglected thecosmological constant term, which will not be considered in this Thesis, while we choosethe following sign convention for the Riemann tensor:

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ. (2.75)

The components of the Levi-Civita connection Γλµν , when solved for the metric tensor,are expressed in terms of the Christoel symbols:

Γµαβ = 12g

µν∂αgνβ + ∂βgνα − ∂νgαβ

. (2.76)


Upon descarding a total derivative, the EH Lagrangian can be written in the equivalentform:

LΓ−Γ = − 12κ2√−ggµν

ΓβµνΓααβ − ΓβµαΓαβν

, (2.77)

with Γλµν given in (2.76), which turns out to simplify the perturbative expansion. Inorder to study scattering amplitudes on a at background, we dene the graviton as theuctuation over the at Minkowski metric ηµν :

gµν := ηµν + hµν := ηµν + 2κhµν , (2.78)

where hµν is the eld with the correct mass dimension, being:

[gµν ] = [ηµν ] = [hµν ] = 0, [κ] = −D − 22 , [hµν ] = D − 2

2 . (2.79)

In order to get a perturbative expansion of the EH Lagrangian in terms of the gravitoneld, we work out separately the rst terms of the expansion of its constituents:

•√−g. First, we use the identity log[det(A)] = tr[log(A)] to expand it as follows:

√−g =

√− det(η + h) = exp

12 log[− det(η) det(1 + η−1h)]

= exp

12 tr[log(1 + η−1h)]

. (2.80)

Then, we employ the expansions:

log(1 + x) =+∞∑k=1

(−1)k+1

kxk, ex =

+∞∑k=0

xk

k! , (2.81)

so that, up to cubic order, we nd:√−g = exp

12 tr[η−1h− 1

2(η−1h)2] + 13(η−1h)3 + o(h4)

= 1 + 1

2 tr(η−1h)− 14 tr[(η−1h)2] + 1

6 tr[(η−1h)3] + 18[tr(η−1h)]2

− 18 tr(η−1h)tr[(η−1h)2] + 1

48[tr(η−1h)]3 +O(h4)

= 1 + 12 h−

14 h

αβhαβ + 18 h

2 + 16 h

αβhγβhγα −18 hh

αβhαβ

+ 148 h

3 +O(h4). (2.82)


• gµν . We write the inverse metric as:

g−1 = (η + h)−1 = η−1η(η + h)−1 = η−1(1 + hη−1)−1. (2.83)

Then, we use:

(1 + x)−1 =+∞∑k=0

(−1)kxk, g−1 = η−1+∞∑k=0

(−1)k(1 + hη−1)k, (2.84)

with the result:

gµν = ηµν − hµν + hµαhνα − hµαhβαhνβ +O(h4). (2.85)

• Γβµα. Directly from eq. (2.76) and (2.78):

Γβµα = 12g

βλ(gαλ,µ + gµλ,α − gαµ,λ)

= 12(∂µhβα + ∂αh

βµ − ∂βhµα − hβλ∂µhλα − hβλ∂αhλµ + hβλ∂λhµα +O(h3)).

(2.86)

We can now expand the action perturbatively in powers of hµν or hµν , i.e. schematically:

L = 12κ2

+∞∑n=2L(n) = 1

2κ2

+∞∑n=2

∂2hn = 12

+∞∑n=2

κn−2∂2hn. (2.87)

The rst two terms of the expansion (quadratic Lagrangian and cubic interactions) turnout to be:

L(2) = −12∂µhαβ∂

µhαβ + ∂ · hµ∂ · hµ − ∂µh∂ · hµ + 12∂µh∂

µh. (2.88)

L(3) = κ

[(√g(1)gµν (0)) + (√g(0)gµν (1))][ΓΓ− ΓΓ](2)µν + [√g(0)gµν (0)][ΓΓ− ΓΓ](3)

µν

= κ

h[12∂µh∂

µh− ∂βh∂ · hβ −12∂µhαβ∂

µhαβ + ∂βhαµ∂αhβµ

]+ 2∂βhhαβ∂ · hα

− 2hµν∂βh∂βhµν + hµν∂µhαβ∂νhαβ − 2hµν∂βhαν∂αhβµ + 2hµν∂αhβν∂αhβµ

+ 2hµν∂βh∂µhβν − ∂βhhαβ∂αh+ 2hµν∂αhµν∂ · hα − 4hαλ∂αhβµ∂βhµλ. (2.89)

Let us stress once more that (2.88) and (2.89), whose calculation already took someeort, provide just the rst two terms of the expansion sketched in eq. (2.87), but thisactually contains an innite number of vertices. This makes computations in pertur-bative quantum gravity extremely dicult at higher points and represents a relevantcomplication when compared to the case of Yang-Mills theories.


2.3.1 Three-graviton tree-level amplitude

We will now compute the simplest scattering amplitude in General Relativity: the tree-level three-graviton amplitude, that will be useful in the next sections when we willintroduce the DC relations; for the moment we shall limit ourselves to state the result.We recall that in the case of a graviton in D spacetime dimensions the polarizationtensors are states in the rank-two, symmetric, traceless representation of SO(D − 2)(therefore, there are 1

2D(D − 3) of them). In covariant notation, they are transverse,symmetric, traceless tensors εµν(p) (p is the momentum of the particle) dened, as inthe spin-one case, up to an equivalence relation:

εµν ∼ εµν + cpµΛν + cpνΛµ, (2.90)

where Λµ is a Lorentz vector and c ∈ R.

The full expression of the Feynman rules is quite long already for the vertex withthree gravitons, but, luckily, we will not need it for our purposes. Indeed, since we areonly interested in the scattering amplitude between three gravitons, we can isolate thetransverse-traceless (TT) part of L(3): the part of the cubic vertex which does not containeither the trace (h) of hµν or its divergence ∂ · hµ). This is sucient in order to computethe three-graviton amplitude, since all the elds in the vertex are going to be contractedwhich the physical polarization tensors, which are themselves transverse and traceless.Setting h = 0 and ∂ · hµ = 0 in eq. (2.89), the expression of the TT vertex turns out tobe the following:

L(3)TT = κ

hµν∂µhαβ∂νh

αβ − 2hµν∂βhαµ∂αhβµ + 2hµν∂αhβν∂αhβµ − 4hαλ∂αhβµ∂βhµλ.

(2.91)

We can now integrate by parts and eliminate terms containing hµν , since they con-tribute to the amplitude with factors p2 which are zero for massless particles. Thereforsethe part of L3 actually contributing to the scattering among three gravitons is:

L(3)TT = κ

hµν∂µhαβ∂νh

αβ + 2hµν∂µhαβ∂βhαν. (2.92)


From the TT Lagrangian we can extract the part of the Feynman rules which allows tocompute the tree-level amplitude with three gravitons. Taking into account the symme-try of the graviton polarization tensors these are:

V µνστρλ3 = 2iκ

ηµσηντpλ2p

ρ2 + ηµληνρpσ1p

τ1 + ηλσηρτpµ3p

ν3

+ 2ηµσηρτpλ2pν3 + 2ηµτηνρpσ1pλ2 + 2ηλµηρτpλ1pν3. (2.93)

In order to get the actual amplitude, we contract eq. (2.93) with the physical polarizationtensors (εiµν = εiνµ, pi · εiµ = 0, εµ iµ = 0 ∀i) and exploit when necessary the mass-shellcondition (p2

i = 0 ∀i) together with momentum conservation (∑i pi = 0). The nal

result for the three-graviton tree-level scattering amplitude is:

M3 = ε1µνε

2στε

3ρλV

µνστρλ3

= 2iκε1αβε

αβ2 ε3

µνpµ2p

ν2 + ε2

αβεαβ3 ε1

µνpµ3p

ν3 + ε3

αβεαβ1 ε2

µνpµ1p

ν1

+ 2εµν1 ε2µσε

3νλp

σ1p

λ2 + 2εµν3 ε1

µσε2νλp

σ3p

λ1 + 2εµν2 ε3

µσε1νλp

σ2p

λ3

. (2.94)

This amplitude will be exploited in the last part of this chapter when dealing with thedouble copy, for the moment we only comment on the relative simplicity of the resultstemming from the fact that most of the terms in the Lagrangian only have the role toguarantee gauge invariance and do not contribute to physical quantities such as scatter-ing amplitudes.

2.4 The double copy

In this section we will present the main topic of interest for this Thesis: the double-copy(DC) relations [14, 17] connecting gravitational and YM amplitudes. The EH and YM La-grangians appear quite dierent in terms of the graviton and gluon elds, respectively.The most striking dierence is that the former contains an innite number of vertices,whereas the latter has only vertices with three and four elds.

The idea of a relation between the scattering amplitudes in the two theories rst ap-peared with the discovery of the Kawai-Lewellen-Tye (KLT) relations [18] in 1985. Theseare relations between tree-level scattering amplitudes in String Theory, which expressthe closed string amplitudes in terms of open string ones. In the low-energy limit, StringTheory reduces to Field Theory and the KLT relations express gravity tree-level ampli-


tudes in terms of a sum of products of “left” and “right” YM partial amplitudes (intro-duced in Section 2.2.4), possibly from two dierent gauge theories. The lower pointsKLT relations can be expressed as follows:

Mtree4 (1, 2, 3, 4) = −is12

(κ2)2Atree

4 (1, 2, 3, 4)Atree4 (1, 2, 3, 4), (2.95)

Mtree5 (1, 2, 3, 4, 5) = i

(κ2)3s12s34A

tree5 (1, 2, 3, 4, 5)Atree

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

+ i(κ

2)3s13s34A

tree5 (1, 3, 2, 4, 5)Atree

5 (3, 1, 4, 2, 5), (2.96)

where Mtreen are n-particle tree-level gravitational amplitudes, Atree

n and Atreen are the

aforementioned left and right YM partial amplitudes and sij = (pi+pj)2, as usual, whilethe arguments of the amplitudes label the external legs. Such relations hold for anyparticle state appearing in the closed String Theory and in particular, when the partialamplitudes are from pure YM theory, the gravitational amplitudes contain as externalstates not only the graviton, but also a two-form and a scalar, as we will discuss in Sec-tion 2.4.1. The KLT relations hold for any number n of external particles, although forhigh values of n the explicit expressions tend to be rather cumbersome, therefore we willnot write them. Moreover, the validity of the KLT relations is limited to tree-level am-plitudes, therefore connecting gravity and YM theories only at the semi-classical level.

The DC relations, equivalent to the KLT ones at tree level, provide an implementationof the more radical idea (still to be explored in all its aspects and meanings) that gravityis in a sense the “square” of a YM theory. These relations strongly rely on the existenceof CK duality in YM scattering amplitudes and when compared to the results obtainedby KLT they have the advantage that they can be extended also at loop level (with somecaveats), while also leading to less involved expressions.At tree level, given a YM amplitude in a CK-dual form,


nici∏αi sαi

, (2.97)

i.e. where ci + cj + ck = 0 ⇒ ni + nj + nk = 0, the basic tenet of the DC paradigm isthat it is possible to obtain a gravity amplitude simply substituting the color factors withanother copy of kinematic factors (ni), possibly from a dierent YM theory, appropriately


modifying the coupling constant [14]:

Mn = i(2κ)n−2 ∑i∈Γ3

nini∏αi sαi

. (2.98)

In what follows we will elaborate on the general meaning and implications of this ex-pression. In the next section we will focus on the explicit case of the product of twoYM tree-level scattering amplitudes with three gluons, interpreting the result as an am-plitude in an appropriate theory of gravity. Later, we will discuss the DC relations forgeneral amplitudes both at tree level and at loop level.

2.4.1 Particle content of the double copy

As we briey mentioned at the beginning of Section 2.4, the intuitive idea of the DC re-lations is that an appropriate “square” of scattering amplitudes in YM theory gives riseto amplitudes in a theory of gravity. As a rst step let us try to gure out the particlecontent of this putative theory.

When multiplying two YM amplitudes in the spirit of eq. (2.98), the product of kine-matic factors n gives rise, among other terms, to products of spin-onepolarizationvectors. Let us focus on a single particle and, since we are dealing with massless quanta,let us choose a frame where its momentum is

pµ = E(1, 0, ..., 0, 1), (2.99)

and its little group (LG), i.e. the subgroup of the Poincaré group which leaves this vectorunaltered, is ISO(D − 2). Particle states with nite spin are dened as reducible, yetundecomposable representations of the LG such that, in order to avoid continuous spinrepresentations, an equivalence relation between polarization tensors is assumed suchto trivialize the action of the commuting generators of ISO(D− 2). (See e.g. [116].) Forhelicity h = 1 two polarizations εµ and ε′µ belonging to the same equivalence class dierby an on-shell gauge transformation

ε′µ(p) = ε′µ(p) + α(p)pµ, (2.100)

with α(p) an arbitrary function. In the standard frame dened by (2.99) we can choose a


representative which satises ελ0 = ελD = 0 ∀λ (λ is the polarization), so that the (outer)product εµν := ελµε

λ′ν of two such vectors is a D ×D matrix with zeros on the rst and

on the last row and column:

εµν = ελµελ′

ν =

0 0 ... 0 00 ε11 ... ε1,D−2 0... ... ... ... ...

0 εD−2,1 ... εD−2,D−2 00 0 ... 0 0

=

0 0 ... 0 00 0... εij ...

0 00 0 ... 0 0

, (2.101)

where the latin indices are LG indices (i = 1, ..., D − 2).

The result of the outer product is a reducible SO(D−2) rank-two tensor; in order toextract its actual particle content we must decompose it into irreducible representations.In the language of Young tableaux this is easily done:

⊗ =∣∣∣∣T⊕ ⊕ •, (2.102)

where the three tableaux on the right hand side of (2.102) correspond to:

•∣∣∣∣T↔ symmetric, traceless tensor with D(D−3)

2 polarizations: graviton hµν ,

• ↔ antisymmetric tensor with (D−2)(D−3)2 physical polarizations: two-formBµν ,

• · ↔ trace part, corresponding to a scalar degree of freedom: scalar eld ϕ.

The sum of the physical degrees of freedom carried by the single representations, ofcourse, matches the number of degrees of freedom of the product of two spin-one po-larizations:

(D − 2)× (D − 2) = (D − 2)2 = D(D − 3)2 + (D − 2)(D − 3)

2 + 1. (2.103)

The explicit decomposition of the tensor εµν into irreducible representations requires atrace only on the LG indices. In order to make it covariant we need an auxiliary vector.Indeed, we can dene a matrix which is a (D− 2)× (D− 2) identity on the LG indices,


completed with zeros to a covariant D-dimensional matrix as follows:

1LGµν =

0 0 00 1(D−2)×(D−2)

ij 00 0 0

= ηµν + 12p(µqν), (2.104)

where qµ is an auxiliary vector which, in the frame where pµ = E(1, 0, .., 0, 1) takes theform qµ = E−1(1, 0, ..., 0,−1). (With this normalization p · q = 2.) With the LG identitydened by (2.104), we can work out the decomposition of the product of two spin-onepolarizations:


ν = 12ε

λ(µε

λ′

ν) −ελ · ελ′

D − 2

(ηµν + 1

2p(µqν)

)︸︷︷︸

graviton

+ 12ε

λ[µε

λ′

ν]︸︷︷︸two−form

+ ελ · ελ′

D − 2

(ηµν + 1

2p(µqν)

)︸︷︷︸

scalar

.

(2.105)

As we shall see, in order to implement this decomposition at the eld theoretical level, weshall be forced to perform a non-local projection as as to extract a gauge-invariant scalarmode. The same decomposition was also used in [11]. Neglecting the contributionscontaining the auxiliary vector q, amounting to an on-shell gauge transformation, weobtain alternative (but equivalent) decomposition:


ν = 12ε

λ(µε

λ′


D − 2 ηµν︸︷︷︸graviton

+ 12ε

λ[µε

λ′

ν]︸︷︷︸two-form

+ ελ · ελ′

D − 2 ηµν︸︷︷︸scalar

, (2.106)

in which the scalar “polarization” is the part proportional to ηµν in the last term, as wewould expect in a covariant theory. Moreover, we notice that now the graviton, denedas:

Gµν = 12ε

λ(µε

λ′


D − 2 ηµν , (2.107)

is in de Donder gauge: p · Gν = 12pνG. This conrms the fact that the dierence from

the decomposition in eq. (2.105), in which the graviton is transverse and traceless, andthe one in eq. (2.106), is simply an on-shell gauge transformation (i.e. with p2 = 0).This is also discussed in[117] from the vantage point of String Theory, in which case theantisymmetric component is absent.


Although this Thesis will not deal with supersymmetry, it is worth mentioning thatin the case of supersymmetric theories the particle content of the DC is richer and can bevaried accordingly to the choice of the two copies of super-YM theories which participateto the DC. In the case of no supersymmetry, the possibility of choosing two dierentYM theories reduces to the choice of two dierent gauge groups; however, since theonly information on the gauge group in pure YM amplitudes comes from the structureconstants, which are removed when performing the DC, the choice of the gauge groupdoes not aect the result3. On the contrary, in the case of supersymmetric theories, thereis the possibility to choose two super-Yang-Mills (SYM) theories with dierent amountof supersymmetry, giving rise to the various supergravities (SUGRA). In particular, therelation between the number of supersymmetry of the various theories (see e.g. [118])is:

[NL SYM ]⊗ [NR SYM ]→ [N = NL +NR SUGRA]. (2.108)

2.4.2 The double copy for three-particle amplitudes

We now want to make the content of the DC explicit in the simplest possible case: thetree-level amplitude with three particles. In this case, the prescription of eq. (2.98) issimply to eliminate the coupling constant and the structure constants from the two YMamplitudes and to multiply them, since at the three-particle level there are neither prop-agators nor any CK duality to satisfy. We label the particles of the rst gauge theorywith indices i = 1, 2, 3 and the particles of the second with barred indices i = 1, 2, 3.We stress that pi = pi ∀i, while in general εi and εi carry dierent polarizations, wherewe use a notation such that εi := ε.

Suppressing from the amplitude in eq. (2.10) the factors of g and fabc, we get:

A3(1, 2, 3)A3(1, 2, 3) ∼ 4(ε1ε2)(ε3p2) + (ε2ε3)(ε1p3) + (ε3ε1)(ε2p1)

× (ε1ε2)(ε3p2) + (ε2ε3)(ε1p3) + (ε3ε1)(ε2p1)

= 4(ε1ε2)(ε3p2)(ε1ε2)(ε3p2) + (ε1ε2)(ε3p2)(ε2ε3)(ε1p3) + (ε1ε2)(ε3p2)(ε3ε1)(ε2p1)

3In this sense, the DC relations display the somewhat unsatisfactory feature of “total color blindness”,as one can apparently build the same gravitational amplitudes from YM theories with dierent colorstructures.


+ (ε2ε3)(ε1p3)(ε1ε2)(ε3p2) + (ε2ε3)(ε1p3)(ε2ε3)(ε1p3) + (ε2ε3)(ε1p3)(ε3ε1)(ε2p1)

+ (ε3ε1)(ε2p1)(ε1ε2)(ε3p2) + (ε3ε1)(ε2p1)(ε2ε3)(ε1p3) + (ε3ε1)(ε2p1)(ε3ε1)(ε2p1)

= 4

[(ε1αβε

αβ2 ε3

µνpµ2p

ν2) + (123→ 231) + (123→ 312)]

+ [(ε1µ1ν1ε

µ1ν32 ε3

µ3ν3pν13 p

µ32 + ε1

µ1ν1εµ3ν12 ε3

µ3ν3pµ13 p

ν32 ) + (123→ 231) + (123→ 312)]

(2.109)

As in the previous section, we can dene for each particle i = 1, 2, 3 the product oftwo polarization vectors εµνi = εµi ε

νi to be split in a symmetric traceless part (graviton

Gµν), an antisymmetric part (two-form Bµν) and the trace part (scalar). This product ofYM amplitudes in principle decomposes in all the possible amplitudes that can consis-tently mix hµν , Bµν and ϕ and we wish to compute all these contributions. To this endwe dene:

· 1 123 = ε1αβε

αβ2 ε3

µνpµ2p

ν2, (2.110)

· 2 123 = ε1µ1ν1ε

µ1ν32 ε3

µ3ν3pν13 p

µ32 + ε1

µ1ν1εµ3ν12 ε3

µ3ν3pµ13 p

ν32 . (2.111)

Our strategy will be to split the polarizations in eq. (2.110) and (2.111), computing theirseparate contributions to all the possible amplitudes with gravitons (polarizations Gµν),two-forms (polarizations Aµν) and scalars (S), which arecollected in table 2.1. Then, asprescribed in eq. (2.109), we add analogous terms but with the two other cyclic per-mutations of particle indices. The list of amplitudes of table 2.1 distinguishes dierentparticle-orders: in the decomposition of εµν into Gµν , Aµν and S all the possible or-derings are met, and they produce dierent results in (2.110) and (2.111). Here we list

GGG GGA GGS GAG GSG GAA GAS GSA GSSAAA AAG AAS AGA ASA AGG AGS ASG ASSSSS SSG SSA SGS SAS SGG SGA SAG SAA

Table 2.1: Possible permutations of external particles in the DC amplitude A3A3

separately all the contributions of the type 1 and 2 to the amplitudes of table 2.1:

·AAA, single-A terms:

1 = 0 (G, S, pµ2pν2 are symm, A is antisymm)

2 = 0 ( 2 symm µ ↔ ν, A or AAA antisymm)


·GGG:

1 = G1

αβGαβ2 G3

µνpµ2p

ν2

2 = 2Gµν2 G3

µσG1νλp

σ2p

λ3·GGS:

1 = 0 (p2

2 = 0)

2 = 0 (G2 · p2 = 0)

·GSG:

1 = 0 (tr(G) = 0)

2 = 2D−2G

1µνG

µ3 λp

µ3p

λ2·GAA:

1 = 0 (G · A = 0)

2 = 2G1µνA

µλ2 A3

ρλpν3pρ2

·GSS:

1 = 0 (tr(G) = 0)

2 = 2(D−2)2G

1µνp

µ2p

ν3

·AAG:

1 = Aαβ1 A2

αβG3µνp

µ2p

ν3

2 = 2A1µνA

µλ2 G3

ρλpν3pρ2

·AAS:

1 = 0 (p2

2 = 0)

2 = 0 (p1 · A1 = 0) ·AGA:

1 = 0 (A ·G = 0)

2 = 2A1µνG

µλ2 A3

ρλpν3pρ2

·ASA:

1 = 0 (tr(A) = 0)

2 = 2(D−2)A

1µνA

3 µλ pµ3p

λ2· SSS:

1 = 0 (p2

2 = 0)

2 = 0 (p2 · p3 = 0)

· SSG:

1 = D

(D−2)2G3µνp

µ2p

ν2

2 = 0 (p2 ·G2 = 0) · SGS:

1 = 0 (tr(G) = 0)

2 = 0 (p2 ·G2 = 0)

· SGG:

1 = 0 (tr(G) = 0)

2 = 2D−2G

2µνG

3νλp

µ3p

λ2· SAA:

1 = 0 (tr(A) = 0)

2 = 2D−2A

2µνA

3 νλ pµ3p

λ2

Now, summing over particle-index even permutations, and taking into account thefactor of four in front of the curly bracket in eq. (2.109), we can compute all the non-vanishing amplitudes (taking into account momentum conservation and the propertiesof G and A):

·M3(GGG) = 4G1αβG

αβ2 G3

µνpµ2p

ν2 +G2

αβGαβ3 G1

µνpµ3p

ν3 +G3

αβGαβ1 G2

µνpµ1p

ν1

+ 2Gµν1 G2

µσG3νλp

σ1p

λ2 + 2Gµν

3 G1µσG

2νλp

σ3p

λ1 + 2Gµν

2 G3µσG

1νλp

σ2p

λ3. (2.112)

·M3(GGS) = (GGS)123 + (GSG)231 + (SGG)312

= 4ελ · ελ′ 2D − 2G

1µνG

2νλp

µ2p

λ1 + 2

D − 2G2µνG

µ1 λp

µ1p

λ3 = 0. (2.113)

·M3(GSS) = (GSS)123 + (SSG)231 + (SGS)312


= 4(ελ · ελ′)2 D

(D − 2)2G1µνp

µ3p

ν3 + 2

(D − 2)2Gµνpµ2p

ν3 = 4(ελ · ελ′)2

D − 2 G1µνp

µ3p

ν3.

(2.114)

·M3(GAA) = (GAA)123 + (AAG)231 + (AGA)312

= 4Aαβ2 A3

αβG1µνp

µ3p

ν1 + 2G1

µνAµλ2 A3

ρλpν3pρ2 + 2A2

µνAµλ3 G1

ρλpν1pρ3 + 2A3

µνGµλ1 A2

ρλpν2pρ1.

(2.115)

·M3(SAA) = (SAA)123 + (AAS)231 + (ASA)312

= 4ελ · ελ′ 2D − 2A

2µνA

3 νλ pµ3p

λ2 + 2

(D − 2)A3µνA

2 µλ pµ2p

λ1 = 16ελ · ελ′

D − 2 A2µνA

µ3 αp

ν3pα2 .

(2.116)

As a rst comment, let us observe that the scattering amplitude between three gravi-tons obtained from the product of two YM amplitudes between three gluons exactlymatches the result of eq. (2.94), when multiplied by κ/2. In the next section we aregoing to comment about all the other amplitudes.

2.4.3 Three-particle amplitudes in N = 0 Supergravity

Although in this Thesis we will not deal with it, a comment about String Theory is inorder. In fact, while the DC relations can be shown to hold from QFT arguments, at treelevel they are equivalent to the older KLT relations [18], which are derived from StringTheory. These are originally relations between scattering amplitudes of open strings(whose low-energy limit contains a spin-one gauge theory) and scattering amplitudes ofclosed strings (whose low-energy limit contains a graviton). Therefore, it is natural toexpect the theory described by the DC to bear some relations to the low-energy limit ofthe bosonic (in the case of no supersymmetry) closed sector of String Theory, which isexactly a theory of gravity coupled to a two-form eld, known as the Kalb-Ramond eld,and a scalar eld, known as the dilaton. In the DC literature this theory is sometimesreferred to as the N = 0 Supergravity (see e.g. [119] or [11]), and its action is:

SN=0 =∫dDx√−g 1

2κ2R−16e− 4κϕD−2HµνλH

µνλ − 12(D − 2)∂µϕ∂

µϕ, (2.117)

whereR is the Ricci scalar, ϕ is the dilaton andHµνλ is the eld strength of the two-formeld Bµν (H = dB). All the contractions of greek indices are performed by the dynam-ical metric eld gµν . We observe that the scalar normalization is clearly non-canonical:


we should rescale the dilaton by a factor of√D − 2 in order to achieve the usual 1/2

factor in front of the kinetic term. However, we will keep the normalization as it is inorder to simplify the comparison with the results of the DC: indeed, if we look at thedecomposition in eq. (2.105), we see that also in that case the normalization of the scalar“polarization” displays a factor of

√D − 2. Of course if we want to get the physical am-

plitude we must rescale it by an appropriate factor.

In order to compute the three-particles scattering amplitudes we can expand theelds about the IR vacuum expectation values 〈gµν〉 = ηµν , 〈Bµν〉 = 0, 〈ϕ〉 = 0. Weneglect the pure gravitational term, since it has already been computed in Section 2.3,while expanding gµν = ηµν + 2κhµν we extract the cubic interactions from the action ineq. (2.117), where now the greek indices are Lorentz indices, contracted with ηµν :

S(3)N=0 = S(3)

EH + κ∫dDx

− 1

2(D − 2)(hηµν − 2hµν)∂µϕ∂νϕ

+ 2D − 2ϕ(∂µBνλ∂

µBνλ + 2∂µBνλ∂νBλµ)

− 16(hηµαηνβηλγ − 2hµαηνβηλγ − 2ηµαhνβηλγ − 2ηµαηνβhλγ)

× (∂µBνλ + ∂λBµν + ∂νBλµ)(∂αBβγ + ∂γBαβ + ∂βBγα). (2.118)

From the cubic action we can compute all the non-vanishing tree-level scattering ampli-tudes (taking into account only the TT part of the vertices):

• S(3)TT [h] = κ

∫dDx

hµν∂µ∂νh

αβhαβ + 2hµν∂µhαβ∂βhαν

, from which:

M3(GGG) = 2iκG1αβG

αβ2 G3

µνpµ2p

ν2 +G2

αβGαβ3 G1

µνpµ3p

ν3 +G3

αβGαβ1 G2

µνpµ1p

ν1

+ 2Gµν1 G2

µσG3νλp

σ1p

λ2 + 2Gµν

3 G1µσG

2νλp

σ3p

λ1 + 2Gµν

2 G3µσG

1νλp

σ2p

λ3

.

(2.119)

• S(3)TT [ϕ, h] = κ

D−2∫dDxhµν∂

µϕ∂νϕ, from which:

M3(GSS) = 2iκD − 2G

1µνp

µ3p

ν3. (2.120)


• S(3)TT [ϕ,B] = 2κ

D−2∫dDxϕ(∂µBνλ∂

µBνλ + 2∂µBνλ∂νBλµ), from which:

M3(SAA) = 8iκD − 2A

2µνA

µ3 αp

ν3pα2 . (2.121)

• S(3)TT [h,B] = κ

∫dDx hµα(∂µBνλ∂

αBνλ + 2∂µBνλ∂λBαν + 2∂λBµν∂

αBνλ

+2∂λBµν∂λBαν + 2∂λBµν∂

νBλα), from which:

M3(GAA) = 2iκ(A2αβA

αβ3 G1

µνpµ2p

ν2 + 2G1

µνAµλ2 A3

ρλpρ2pν3

+ 2G1µνA

2ρλA

3νλp

µ2p

ρ3 + 2A2

µνGνλ1 A3

λρpµ2p

ρ3). (2.122)

When comparing these amplitudes with the ones calculated in Section 2.4.2 we no-tice that, when the latter are multiplied by κ/2 and ελ · ελ′ = 1, they are all equal,including the amplitudes which are identically zero. Let us stress that, while the cubicself-interacting gravitons vertex is xed by the gauge invariance of the free theory, it isnot at all obvious why the vertices involving the scalar elds should comply with thecouplings of the N = 0 Supergravity. In this sense it is a non-trivial result to observethat N = 0 Supergravity is, at least at the level of cubic vertices, the result of the DC.

2.4.4 The double copy at tree level

Having discussed an explicit example, which helps to clarify how is it possible to buildgravity amplitudes from gauge theory ones, we now want to clarify the content of theDC relations more in general. In this section, we deal with tree-level amplitudes, forwhich the DC relations were conjectured in [14]. As already mentioned, the idea is thatwhen a YM tree-level amplitude is written in a CK-dual form:


nici∏αi sαi

, (2.123)

with ci + cj + ck = 0 ⇒ ni + nj + nk = 0, the replacement of the color factors (c)with a new set ok kinematic factors coming from a dierent gauge theory (n) leads toan amplitude in a theory of gravity:

Mn = i(κ

2)n−2 ∑

i∈Γ3

nini∏αi sαi

. (2.124)


Initially it was believed that also the second YM amplitude, with kinematic factors n,should be in a CK-dual form. However, if we perform on one of the two amplitudes (saythe second, A, with kinematic factors n) a generalized gauge transformation (see Section2.2.3):

ni → n′i = ni + ∆i : A′n = An ⇔∑i∈Γ3

∆ici∏αi sαi

= 0. (2.125)

The last identity, which is the condition for the amplitude to be invariant, must hold forany gauge group, therefore its validity cannot depend on the specic values of the colorfactors (and hence of the structure constants), but only on their algebraic properties [16].Then, if the rst amplitude is in a CK-dual form, the kinematic factors n satisfy thesame algebraic identity of the color factors, therefore the following identity must hold:

∑i∈Γ3

∆ini∏αi sαi

= 0. (2.126)

This indicates that it is sucient that one of the two YM amplitudes is in a CK dual form,whereas the other does not have to:

∑i∈Γ3

nin′i∏

αi sαi=∑i∈Γ3

nini∏αi sαi

+∑i∈Γ3

ni∆i∏αi sαi︸︷︷︸

=0

. (2.127)

This allows a greater freedom in the representation of YM amplitudes, which is usefulwhen the two YM theories are dierent and CK-dual representations are not known forone of the two. Also, we observe that at tree level, due to the vertex structure of N = 0Supergravity, if the external states are all gravitons then the two-form and the dilatonare not allowed to propagate in internal lines and the result of the DC is pure Einsteingravity. It is therefore simple, at least at tree level, to select the pure gravity sectoramong the products of the DC. The situation is more complicated at loop level, as wewill comment in the following section.

At tree level, the validity of the DC relations was proven inductively in [16], withthe technique of complex shifts [13] and strongly relying upon the existence of the CKduality in YM amplitudes.


2.4.5 The double copy at loop level

at loop level, the existence of DC relations, as well as the CK duality, is still a conjecture(rst proposed in [17]). The idea is in many ways similar to what happens at tree level:again one exploits the possibility to “blow up” four-gluon vertices into three-gluon ones,so that the amplitude can be written as:

AL−loopn = iLgn−2+2L ∑i∈Γ3

∫ L∏j=1

(dLlj

(2π)D) 1Si

cini∏αi sαi

, (2.128)

where Si is the symmetry factor4 of the diagram and lj are the loop momenta. If it ispossible to nd a CK-dual representation for this amplitude, then the DC proceeds as attree level, i.e. the amplitude:

ML−loopn = iL+1

(κ2)n−2+2L ∑

i∈Γ3

∫ L∏j=1

(dLlj

(2π)D) 1Si

nini∏αi sαi

, (2.129)

is an amplitude in a theory of gravity, with n a second copy of kinematic factors. Atthe loop-level, however, even in the case of purely graviton amplitudes the two-form andthe scalar are allowed to propagate as internal states in the loops. Therefore, in this case,if one is interested in a scattering amplitude in pure gravity, it is necessary to employprojectors which select only gravitons in the internal lines [17]. Alternatively, in [61]ghosts states have been introduced in order to cancel the contributions of the dilatonand the two-form from loop amplitudes, obtaining pure gravity.

An important point to stress is that, as evident from eqs. (2.128) and (2.129), theDC relations at loop level relate the formal loop integrands, rather than the amplitudes(which are the result of the loop integral). This is crucial to account for the very dierentUV behavior of YM theory and gravity. For instance, in D = 4 the former is a renor-malizable theory while the latter is not. From the point of view of the DC, this dierentbehaviour in the UV is “explained” by the fact that the second copy of kinematic factorsn increases the powers of momenta in the numerator, thus enhancing the degree ofdivergence of the theory. This corresponds, from a Lagrangian perspective, to gravityhaving two-derivative interactions, as opposed to the one-derivative cubic vertex of YM

4Number of ways of interchanging internal or external lines without changing the topology of thediagram. The expression of a Feynman diagram is given by the product of the Feynman rules divided bythe symmetry factor, in order to prevent overcounting.


theory.

The non-existence of a proof for the DC relations at loop level is strictly related tothe lack of a proof for the validity of CK duality at loop level. Indeed, by means ofthe unitarity cuts [120] and assuming the existence of a CK-dual representation of YMamplitudes, formula (2.129) can be easily justied. The unitarity cuts are a powerfultool which allows to relate loop amplitudes to tree-level amplitudes, on account of theunitarity of the S−matrix of the theory. Indeed, if we write

S = 1 + iT, (2.130)

with T known as the transfer matrix (it contains the relevant information for the scat-tering), and impose unitarity we get:

SS† = 1 = 1 + iT − iT † + TT † ⇔ TT † = −i(T − T †) = 2Im(T ). (2.131)

Thus, upon expanding T in powers of the coupling constant, one realizes that beingSS† = 1 a quadratic constraint, it relates amplitudes at dierent order in perturtba-tion theory, such as for instance one-loop amplitudes and tree-level ones. The explicitrelations can be found inserting a complete set if intermediate states with the result:

2ImA(i→ f) = i∑X

∫dΠX(2π)DδD(pi − pX)A(i→ X)A∗(f → X), (2.132)

whereA(i→ f) is the amplitude of interest,X denotes the intermediate states, dΠX theLorentz-invariant measure for the phase space of the particle in state X andA(i→ X),A(f → X) the two amplitudes which are obtained by cutting the propagators ofA(i→f), i.e. putting on shell the momenta of the propagators without violating momentumconservation, with the insertion of X as intermediate states. In [7] and [6] the idea ofgeneralized unitarity was introduced: a technique which allows, relying on appropriateunitarity cuts, to entirely build loop amplitudes from tree-level ones, apart from ratio-nal terms. Rational terms are rational functions in the loop amplitudes which does notpossess branch cuts, being therefore undetectable using unitarity cuts (indeed, we seefrom (2.131), that the unitarity method reconstructs the imaginary part of the loop am-plitude). Such terms are known to be absent in certain theories (such as SYM), but theyare present for example in the case of pure YM theory.


In order to justify the DC formula at loop level [17], we assume the existence of a CKdual representation for loop amplitudes in YM theory and we apply the unitarity method.For the sake of concreteness, we consider a three-loops example [4], in which a cubicdiagram contributing to a gravity amplitude is built from tree-level diagrams by meansof generalized unitarity (g. 2.9). In the gure, the uncut propagators are named 1/p2

i

Figure 2.9: Unitarity cuts on loop amplitudes [4].

(i = 1, ..., 5), n is the numerator of the three-loop diagram and na, nb, nc are the nu-merators of the tree-level diagrams in terms of which the loop diagram is built with theunitarity method. If we consider the corresponding gauge theory diagrams, the assump-tion that the kinematic factors in n are in a CK-dual form implies that also na, nb, ncsatisfy the duality for the individual tree-level diagrams. Then, we can “square” the tree-level YM amplitudes individually, obtaining tree-level gravity amplitudes (since the DCrelations are proven to hold at tree level, as we mentioned): this guarantees that the“square” of the YM loop amplitude also result in a gravity loop amplitude. Despite thembeing not cut-constructible, formula (2.129) also correctly reproduces the rational termsin the amplitude: this is due to the fact that rational terms are obtained calculating theloop integral in higher dimensions, using the extra-dimensional momenta as regulators.However, the DC relations are valid in arbitrary spacetime dimensions: since formula(2.129) reproduces the correct cuts in anyD, it must also reproduce correctly the rationalterms.

More recently, the diculty in nding CK-dual representations of YM amplitudes atloop level has led to the development of generalized DC constructions [54, 55], whichcompensate the violation of BCJ relations in YM amplitudes when building gravity am-plitudes in cases where CK-dual kinematic factors are have not been found (though theCK duality is believed to hold).


2.4.6 The zeroth copy

It was also noticed in [121] that, starting from eq. (2.73) another replacement is possi-ble, which goes in opposite direction with respect to the double copy. Indeed, we caneliminate the kinematic factors n in favour of another copy of color factors c, com-ing from a YM theory with a possibly dierent gauge group. Inserting an appropriatecoupling constant y, the new amplitude reads:

T L−loopn = iLyn−2+2L ∑

i∈Γ3

∫ ( L∏k=1

dDlk(2π)D

) 1Si

cici∏αi sαi

, (2.133)

with the usual notation.

This is a scattering amplitude in a theory of scalar elds Φaa′ , transforming in theadjoint representation of two (possibly dierent) Lie algebras. The Lagrangian of thistheory, which correctly reproduces the scattering amplitudes of eq. (2.133), is:

LΦ = −12∂µΦaa′∂

µΦaa′ − y

3fabcfa

′b′c′Φaa′Φbb′Φcc′ , (2.134)

where unprimed indices belong to one of the two Lie algebras and primed indices to theother one.

Chapter 3Double copy for classical theories:

a review

Since the discovery of the DC relations, many endeavors have been made to extend theirscope beyond the realm of perturbative scattering amplitudes, pursuing the goal of betterunderstanding their origin and their breadth. In this chapter, we will review the main ap-proaches pursued in the attempt to generalize such relations, at the (semi-)classical level.

First, we will discuss the denition of the “product“ between two YM elds givenin [58], whose main advantage is to correctly reproduce the linearized dieomorphismsand two-form gauge, starting from the abelianized version of the YM gauge symmetry.With this denition, in [59] the issue of reproducing the linearized equations of motionfor the elds ofN = 0 Supergravity was also addressed, both in presence and in absenceof sources for the elds. We shall elaborate the results of [59] from a slightly dierentperspective, proposing in particular an alternative derivation of the equations of motion.

The second approach that we review was rst illustrated in [16]. It amounts to theconstruction of a YM Lagrangian which, order by order in the number of external gluons,produces Feynman rules which automatically encode the CK duality for tree-level scat-tering amplitudes, by means of the introduction of appropriate auxiliary elds. Fromsuch Lagrangian, at xed gauge, a “square” of the YM vertices is dened, building agravity Lagrangian which reproduces the correct scattering amplitudes at tree level, inagreement with the DC relations.

Last, for the sake of completeness, we will review some results obtained in the ap-

59

Chapter 3. Double copy for classical theories: a review 60

plication of the DC approach to classical solutions in YM theories and gravity. Sincethis latter lies a bit outside the scope of this Thesis we shall limit ourselves to illustratethe rst work on the topic [19], which already contains some of the key features of theclassical DC.

3.1 Relating Yang-Mills and gravitational symmetries

In this section we will review the o-shell denition of the product between YM eldsintroduced in [58], showing how it allows to reproduce the linearized local symmetriesof a theory containing a graviton, a two form and a scalar eld from the abelianized YMsymmetries, while also leading to a denition of the gravitational curvature tensors builtout of two spin-one abelianized eld strength.

Let us stress here that, while [58] deals with the supersymmetric versions of YM the-ory and gravity, in this Thesis we will be mainly concerned with the non-supersymmetriccase only. Barring this aspect, the results of [58] provide the basis on which our work isbuilt upon.

Last, following [59], we review the possibility of reproducing the linearized equationsof motion for the gravitational elds, both in presence and in absence of sources.

3.1.1 Product of two Yang-Mills elds

Inspired both by String Theory, where the massless states of the closed string arise fromtensor products of the massless states of the open string [122], and by the DC relations,it is natural to formulate the idea that gravity elds may be dened as “products” of YMelds. However, it was only in [58] that a precise denition of the product was given, inthe following schematic form:

Hµν(x) :=[Aaµ Φ−1

aa′ Aa′

ν

](x) :=

[Aµ ? Aν

](x), (3.1)

where Hµν contains, as we will explain, the gravity elds, is a convolution productand Aaµ, Aa′ν are two YM elds, which we will also denote as the left (L) and the right (R)factors respectively, transforming in the adjoint representation of two (possibly dier-ent) gauge groups GL, GR. Furthermore, the eld Φ−1

aa′ is a scalar eld with two colorindices belonging to the groups GL and GR, transforming in the adjoint representation

61 Chapter 3. Double copy for classical theories: a review

of both. Several comments are in order, to explain the rationale behind this denition.

First, we want to motivate the introduction of the convolution product and to re-call some of its basic properties. Both in the case of String Theory and of the DC con-structions, the result that gravity states are actual products of spin-one states arises inmomentum space (as in the decomposition of polarizations of Section 2.4.1), therefore itis somewhat natural to think of a convolution in coordinate space when attempting toimplement this product at the level of o-shell elds. We remind that the convolutionproduct is dened as:

[f g](x) :=∫dDyf(y)g(x− y), (3.2)

and satises the following properties:

· (f g) h = f (g h) (associativity), (3.3)

· f g = g f (commutativity), (3.4)

· (af + bg) h = a(f h) + b(f g), a, b ∈ R (linearity), (3.5)

· ∂µ(f g) = (∂µf) g = f (∂µg) (dierentiation rule), (3.6)

· Ff g = Ff · Fg (convolution theorem), (3.7)

where, as already mentioned, Ff denotes the Fourier transform of f .

Secondly, we want to explain and justify the presence of the scalar eld Φ−1aa′ . Its role

is to saturate the color indices of the YM elds, so as to dene a gravity eld Hµν whichdoes not contain color indices. This scalar eld will be always present in the productof two elds, or more generally functions, with color indices, in the sense of eq. (3.1).Therefore, apart from cases in which we will need its transformation properties to bemanifest, we will not write it explicitly and rather denote the operation “Φ−1

aa′” withthe symbol “?”, as in eq. (3.1). Moreover, following [59], the eld was introduced viaits inverse, since it enters as the convolution inverse of the biadjoint scalar eld Φaa′

introduced in Section 2.4.6, in the sense that:

[Φ−1aa′ Φbb′ ](x) = δabδa′b′δ

(D)(x), (3.8)

with δ(D)(x) aD-dimensional Dirac delta function. This has been motivated in two ways


in [59]. On the one hand, as explained in Section 2.4.6, the biadjoint scalar eld is in asense an output of the DC: the scattering amplitudes for a theory with cubic interactionsamong biadjoint scalars is indeed the zeroth copy of two YM theories [121]. Therefore,the actual biadjoint scalar should appear on the side of the “results” of the DC and thus,when we write it together with the YM elds its inverse should appear. On the otherhand, one has to account for the mass dimensions of the elds involved in eq. (3.2). In-deed, both the gravity eld Hµν (which, as we will see shortly, contains the graviton,the two-form and the scalar) and the YM elds should have mass dimension D−2

2 , com-patibly with the mass dimensions introduced by the convolutions. This is because themeasure dDx of the integrals performed for the convolution product brings a dimension[dDx] = −D, therefore, for instance, the convolution of two elds of dimension D−2

2 hasdimension 2D−2

2 −D = −2. Thus, if the biadjoint eld had dimension D−22 , as expected

from a scalar eld, it would produce a dimension for Hµν such that [Hµν ] = −D+62 ,

which is not correct. However, if we dene Φ−1aa′ as the convolution inverse of a scalar

eld with mass dimension one, we have [Φ−1aa′ ] = 3D+2

2 (since [δ(D)(x)] = D) and, cor-rectly, [Hµν ] = D−2

2 (from eq. (3.1)).

3.1.2 Field content

We would now like to covariantly extract, from eq. (3.1), the actual physical elds. InSection 2.4.1 we achieved a similar goal, but for on-shell polarization vectors; in thissection we are going to perform a completely o-shell decomposition.

Being the graviton and the two-form respectively symmetric and antisymmetricrank-two tensors, the most natural idea might be to dene such elds as the symmetricand antisymmetric part of Hµν , respectively. This indeed reproduces the correct lin-earized gauge transformations of a graviton and of a two-form eld, as we will illustratein Section 3.1.3. However, this naive identication would miss two relevant aspects, aswe are now going to discuss.

Let us recall that the physical states of massless particles lie in irreducible represen-tations of SO(D − 2), whereas the corresponding covariant elds are representationsof GL(D,R). The representations of both groups can be described by means of Youngtableaux: in these terms, we can employ the same tableaux to describe a particle andthe corresponding covariant eld, provided that in the rst case (particle) the tableau is


interpreted as a tableau for SO(D − 2), while in the second as a tableau for GL(D,R).For instance, the physical states of a graviton are described by a symmetric, tracelesstensor hij (i, j = 1, ..., D − 2), while the covariant eld describing a graviton is a sym-metric tensor hµν (µ, ν = 0, ..., D − 1), with no constraints on the trace. Both casesare represented by the same Young tableau , but interpreted in dierent ways, asexplained. The relevant point for us is that in the covariant construction of the gravitoneld there is some degree of arbitrariness in the denition of its trace, and this leads topossible mixings with the scalar eld of the N = 0 Supergravity multiplet.

Indeed, while the denition of the graviton as the symmetric part of Hµν ,

hµν := HSµν := 1

2H(µν), (3.9)

correctly reproduces the linearized gauge transformation of a spin-two eld, as we willshow in Section 3.1.3 still, since we are now o shell, we can freely modify its traceprovided that we do not modify its transformation rule. Therefore, the graviton can beequivalently dened as

hµν := HSµν − γηµνϕ, (3.10)

where γ ∈ R is an arbitrary parameter, while ϕ is the (gauge invariant) scalar eld.For the above reasons, (3.10) is as good a denition of the covariant eld describing thegraviton for every real value of γ1.

Whatever choices one eventually makes, they will have an impact on the very de-nition of the scalar eld ϕ in terms ofHµν . In order to better frame the issue at stake, it isuseful to look back once again to Section 2.4.1, from which it is clear that, the scalar de-gree of freedom is a singlet of SO(D− 2), which arises from the trace of the product (inmomentum space) εiεj , with εi, εj physical spin-one polarizations and i = 1, ..., D − 2.On the other hand, the obvious candidate to play the role of the physical scalar, that isthe trace H := Hα

α of Hµν , However, is not o-shell gauge-invariant, thus leading tothe need to covariantly compensate its gauge transformation. In particular, since the

1In the following we will sometimes pause to comment on what appears to be the most suitable valuefor γ in dierent situations.


symmetric part of Hµν is subject to the gauge transformation (see Section 3.1.3):

δHSµν = ∂µξν + ∂νξµ, (3.11)

with ξµ a Lorentz vector, the trace of Hµν transforms as:

δH = 2∂ · ξ. (3.12)

Hence, in order to give a gauge-invariant denition of the scalar eld, we can subtractfrom H a term which compensates the variation (3.12). Such term must be unphysi-cal, since H already contains the physical scalar degree of freedom, and covariant, thusleading to the following denition:

ϕ := H − ∂ · ∂ ·H

, (3.13)

where a non-local projector enters, in the spirit of [123]. We will shortly comment moreabout the presence of 1

, but rst let us observe that (3.13) is a good, gauge-invarianto-shell denition of the scalar eld2. Indeed, we can rst x ∂ · ∂ ·H = 0 with a gaugeparameter ξµ such that ξµ 6= 0, being left with the possibility of performing residualgauge transformations with ξµ = 0 and ∂ · ξ = 0, which preserve the gauge-xing∂ · ∂ ·H = 0.

Let us also stress that the denition (3.13) of ϕ suggests a special value for the pa-rameter γ introduced in (3.10). Indeed, when the eld Hµν is gauge xed to be trans-verse with respect to both indices (∂αHαµ = ∂αHµα = 0), so that the scalar reducesto ϕ = Hα

α, then the graviton turns out to be in the harmonic (or de Donder) gauge ifγ = 1

D−2 . Indeed, from (3.10):

∂ · hµ = −γ∂µH,

h = H − γDH,⇒ ∂ · hµ = 1

2∂µh↔ −γ = 12(1− γD), (3.14)

which is solved precisely by γ = 1D−2 . Since the transverse gauge for Hµν is equiv-

alent to the Lorenz gauge condition for both the YM elds involved in the product2This is true at the linearized level; when interactions are included, as in Section 4.4, the denition of

the scalar has to be reconsidered.


(∂ · A = ∂ · A = 0), we can state that when γ = 1D−2 the Lorenz gauge on the YM

side implies the de Donder gauge for the graviton. Furthermore, this value of γ makesthe denition of the graviton similar to the one given in Section 2.4.1 for the on-shellgraviton polarization tensor.

Now, let us comment about the introduction of the inverse d’Alembertian operatorin eq. (3.13). First of all, we give a precise denition of what we mean by this inversion,listing some of its properties. We can introduce a Green’s function for the d’Alembertoperator = ηµν∂µ∂ν ,

xG(x− y) = δ(D)(x− y), (3.15)

which allows to nd the solution to equations of the type f = j in the following form

f(x) = [G j](x) + g(x), (3.16)

with g(x) satisfyingg = 0. The functionG denes the inverse d’Alembertian operator,that we shall interpret as

[ 1

(j)](x) := [G j](x) =∫ dDp

(2π)DFj(p)−p2 e−ip·x, (3.17)

where Fj(p) is the Fourier transform of j(x). In what follows, we will exploit mainlythe two following properties of 1

(see [59]):

· 1

(f) = ( 1f) = f when f 6= 0, (3.18)

· 1

(f g) = ( 1f) g = f ( 1

g). (3.19)

At the same time, the very denition of 1f relies on the fact that the function f is not

in the kernel of the operator, i.e. f 6= 0.Furthermore, let us stress that the presence of non-localities was expected, since the

scalar is dened projecting out covariantly the spin-zero part from a rank-two tensor,and such projections require the introduction of non-localities [123], which sometimesare “hidden” by the introduction of auxiliary elds. Let us now show explicitly how this


happens, starting from the spin-one case described by Maxwell’s equations

Aµ − ∂µ∂ · A = 0, (3.20)

which are invariant under the gauge transformation

δAµ = ∂µΛ, (3.21)

with Λ an arbitrary scalar function. If we covariantly split the eld into its transverseand longitudinal component, by putting:

Aµ = ATµ + ∂µχ, (3.22)

with ∂ ·AT = 0, and we plug it in eq. (3.20), we can observe that the Maxwell’s equationsonly rule the dynamics of the D − 1 transverse components ATµ , with the equation ofmotion:

ATµ = 0, (3.23)

while being

(∂µχ)− ∂µ∂α(∂αχ) = 0 (3.24)

identically, the longitudinal component χ is actually unconstrained by the equations ofmotion of Aµ. This is due to the fact that the longitudinal mode is non-physical, andcan be eliminated altogether by means of a gauge transformation δAµ = ∂µΛ, simplychoosing Λ = −χ. These are very well known facts, but we found useful to recall themin order to make the two following remarks. The rst is that, although it is true that theelds describing free massless particles satisfy wave equations of the type = 0, thisis not actually true for all their components: in the Maxwell case, it is not true that theequations imply χ = 0. The second remark is that, in the decomposition (3.22), weintroduced two auxiliary elds ATµ and χ in order to disentangle the transverse and thelongitudinal components of Aµ. However, if we insist on a description based solely onthe eld Aµ, it is possible to derive an expression for such auxiliary elds terms of Aµ,


but this necessarily requires the introduction of the inverse d’Alembertian operator:

∂ · A = χ ⇒ χ = ∂ · A

. (3.25)

If we now apply the same arguments toHSµν , we can show that eq. (3.13) is a perfectly

sensible covariant denition for the scalar eld. Indeed, we can decompose the eld asfollows:

HSµν = HS,T

µν + ∂µπTν + ∂νπ

Tµ + ∂µ∂νλ, (3.26)

with ∂ · HS,Tµ = 0 = ∂ · πT and λ a scalar. Now, the covariant and gauge invariant

equations of motion for HSµν (that we will derive in Section 4.1.1), cannot constrain the

components πTµ and λ, which are pure gauge just like the longitudinal modes in thespin-one case. Actually, the only components which must satisfy = 0 are the fullytransverse ones (HS,T

µν )3. Indeed, from (3.26), it is also possible to check that the trace ofHSµν and the trace of HS,T

µν are dierent, which is the reason why we could not denethe scalar simply as ϕ := H . The dierence can be expressed as follows (we omit theindex S since the contraction with ηµν when extracting the trace automatically selectsthe symmetric part of Hµν):

H = HT +λ. (3.27)

Then, we can work out an expression for λ in terms ofHµν , in a way which is analogousto eq. (3.25) (omitting S for reasons similar to above: ∂µ∂ν is symmetric):

∂ · ∂ ·H = 2λ. (3.28)

Therefore, we can conclude that a covariant expression for the trace of the transversepart of HS

µν , which comprises the physical modes, can be written entirely in terms ofHSµν itself, only upon introducing the inverse d’Alembertian operator:

HT = H − ∂ · ∂ ·H

. (3.29)

At any rate, let us anticipate that, as we shall discuss in Chapter 4, the theory which3Including the trace of HS,T

µν , which contains the propagating scalar.


we are going to build is completely local in terms of hµν , Bµν and ϕ.

To summarize, the physical eld content of our DC theory built upon the denition(3.1) is the following:

Aµ ? Aν = Hµν = hµν +Bµν + γηµνϕ, (3.30)

where hµν := HS

µν − γηµνϕ,

Bµν := HAµν ,

ϕ := H − ∂·∂·H .

(3.31)

Additional support to these identications will be given in the next section, where weshall discuss the gauge transformations of these elds.

3.1.3 Linearized symmetries

The aim of [58] was to reproduce the linearized symmetries of gravity, starting from thelinearized version of YM theory, i.e. the g → 0 limit of a YM theory, where g is thecoupling constant. In this limit the theory is equivalent to a set of free spin-one eldsAaµ, with local transformation rule (see eq. (2.3)):

δ0Aaµ = lim

g→0

(∂µε

a − gfabcεbAcµ)

= ∂µεa, (3.32)

with εa = εa(x) a set of local parameters and where the subscript 0 in δ0 refers tothe fact that this transformation holds at the linearized level. Also the eld strength is“abelianized” in the limit g → 0, indeed from eq. (2.5):

(F aµν)(0) = lim

g→0

(∂µA

aν − ∂νAaµ − gfabcAbµAcν

)= ∂µA

aν − ∂νAaµ. (3.33)

In the remainder of this chapter we will often employ eq. (3.32) and (3.33), but in orderto avoid cumbersome formulas we will omit the “0”, since we will only deal with thelinearized local transformation and eld strength.

We also take into account the global transformation rule of the YM elds under the


gauge group, which we write with a set of global (constant) parameters ϑa:

δϑAaµ = fabcAbµϑ

c, (3.34)

since the YM connection transforms in the adjoint representation, as already mentioned.Therefore, keeping into account both the local and the global transformation, under thegauge group:

δAaµ = ∂µε

a + fabcAbµϑc,

δAaµ = ∂µεa′ + fa

′b′c′Ab′µ ϑ

c′ .(3.35)

The eld Φ−1aa′ is chosen to transform only under global transformations, and as already

explained it transforms under the adjoint representations of the two groupsGL andGR:

δΦ−1aa′ = fabcΦ−1

ba′ϑc + fa

′b′c′Φ−1ab′ϑ

c′ . (3.36)

Given these transformations for the elds, exploiting the properties of the convolution(in particular eq. (3.6)) and the antisymmetry of the structure constants we nd:

δHµν = (∂µεa + fabcAbµϑc) Φ−1

aa′ Aa′

ν

+ Aaµ (fabcΦ−1ba′ϑ

c + fa′b′c′Φ−1

ab′ϑc′) Aa′ν + Aaµ Φ−1

aa′ (∂ν εa′ + fa

′b′c′Ab′

ν ϑc′)

= ∂µαν + ∂ναµ, (3.37)

where we have dened:αµ := εa Φ−1

aa′ Aa′µ = εa ? Aa

′µ ,

αµ := Aaµ Φ−1aa′ εa

′ = Aaµ ? εa′ ,

(3.38)

which will be the gauge parameters of the DC theory4. Splitting the result of eq. (3.37)into its symmetric (HS

µν := 12H(µν)) and antisymmetric (HA

µν := 12H[µν]) part:

δHS

µν = 12∂µ(αν + αν) + 1

2∂ν(αµ + αµ) := ∂µξν + ∂νξµ,

δHAµν = 1

2∂µ(αν − αν)− 12∂ν(αµ − αµ) := ∂µΛν − ∂νΛµ,

(3.39)


where we have introduced new gauge parameters, built from the ones of eq. (3.38):

ξµ := 1

2(αµ + αµ),

Λµ := 12(αµ − αµ).

(3.40)

The key observation of [58] is that the two equations in the system (3.39) correspondrespectively to the linearized version of the gauge transformation of a graviton and tothe gauge transformation of a two-form eld. However, as we discussed at length in theprevious section, the introduction of a gauge invariant scalar eld ϕ, together with thefact that an o-shell graviton has no constraint on its trace, introduces an ambiguityin the denition of the graviton. From eq. (3.39) we can check that, as anticipated, thedenition (3.13) of ϕ is gauge-invariant, therefore we can complete the denition of theelds in eq. (3.31) by including their gauge transformations:

hµν := HS

µν − γηµνϕ→ δhµν = ∂µξν + ∂νξµ,

Bµν := HAµν → δBµν = ∂µΛν − ∂νΛµ,

ϕ := H − ∂·∂·H → δϕ = 0.

(3.41)

3.1.4 Field strength and Riemann tensor

In [58], a linearized eld strength for Hµν was also dened, built from the “abelianized”eld strengths (dened in eq. (2.5)) of the two YM elds:

R∗µνρσ := −12F

aµν Φ−1

aa′ F a′

ρσ = −12Fµν ? Fρσ. (3.42)

R∗µνρσ is clearly gauge invariant, since it is built from two gauge invariant eld strengthtensors. If we expand (3.42) in terms of Hµν = HS

µν + HAµν = HS

µν + Bµν we get theexpression:

R∗µνρσ = 12∂ν∂ρH

Sµσ + ∂µ∂σH

Sνρ − ∂ν∂σHS

µρ − ∂µ∂ρHSνσ

+ 1

2∂ν∂ρBµσ + ∂µ∂σBνρ − ∂ν∂σBµρ − ∂µ∂ρBνσ

, (3.43)

In the rst curly bracket, we can recognize the linearized Riemann tensor of a manifoldwith Levi-Civita connection and a metric gµν = ηµν + HS

µν . The second curly bracket,instead, contains the eld srtength of the two-form, amenable to be identied as a con-


tribution to the Riemann tensor coming from the torsion part of the connection5. Indeeda Kalb-Ramond two-form minimally coupled to gravity contributes to the geometry ofthe spacetime manifold with an eective torsion, with totally antisymmetric torsion ten-sor given by the eld strength of the two-form eld. We consider the case in which thetorsion tensor is given by the eld strength Hµαβ of the Kalb-Ramond two-form Bµν :

−Tµαβ = Hµαβ = ∂µBαβ + ∂αBµβ + ∂βBµα. (3.45)

Even in the presence of torsion, the denition of the Riemann tensor in terms of theconnection is unaltered, with the only caveat that the connection is no longer symmetricin its lower indices:

Rρσµν = ∂µΓρνσ − ∂νΓρµσ + ΓρµλΓλνσ − ΓρνλΓλµσ, (3.46)

as in eq. (2.75). For our purposes, it is useful to split the connection into a torsion-freepart Γ, symmetric in its lower indices, and a pure torsion part:

Γµαβ = Γµαβ + 12T

µαβ. (3.47)

Now, Γµαβ is the usual Levi-Civita connection, determined by the metric as in Section 2.3,and exploiting the full antisymmetry of the torsion we can also split the Riemann tensorin a purely metric part R(Γ) and a part which contains the torsion:

Rρσµν = Rρ

σµν + 12D[µT

ρν]σ + 1

4Tρ[µ|λT

λ|ν]σ + 1

2TρλσT

λµν , (3.48)

whereDµ is the covariant derivative built with the Levi-Civita connection Γ. Expandinggµν = ηµν + hµν and linearizing eq. (3.48) on a at background, we obtain:

Rρσµν = 12∂µ∂σhρν + ∂ρ∂νhµσ − ∂µ∂ρhνσ − ∂ν∂σhρµ

+ 1

2∂µ∂σBρν + ∂ρ∂νBµσ − ∂µ∂ρBνσ − ∂ν∂σBρµ

, (3.49)

5We recall that the torsion tensor is dened as the antisymmetric part of the connection in its lowerindices (when expressed in a coordinate basis):

Tµαβ := Γµαβ − Γµβα = Γµ[αβ], (3.44)

where T is the torsion tensor and Γ is the connection.


which exactly matches eq. (3.43) if we identify HSµν = hµν . Therefore, we can interpret

the eld strength of Hµν given in eq. (3.42) as the linearized Riemann tensor of a man-ifold with metric gµν = ηµν + HS

µν and connection with torsion Tµνλ = −Hµνλ. Thisseems to indicate that the correct denition of the graviton is hµν = HS

µν (therefore,γ = 0 in eq.(3.31)) and (3.30), since the above reasoning shows that we can interpretHSµν as the metric uctuation over at spacetime.

Also, we observe that from eq. (3.48) we can calculate the Ricci scalar:

R = R− 34T

λµνTλµν , (3.50)

from which we can build a Einstein-Hilbert like action (recalling the identication (3.45):

S = 12κ2

∫dDx√−gR− 3

4HλµνHλµν

, (3.51)

which, apart from the normalization of the two-form (which can be made canonicalwith a eld redenition), is exactly the action of a two-form eld with eld strengthHµνλ minimally coupled to gravity.

3.1.5 Free equations of motion

In the present and in the following section we address the question of reproducing thelinearized equations of motion of the N = 0 Supergravity, given the linearized YMequations and the denition (3.1). Actually, we proceed as follows:

• We shall assume the free equations of motion for the elds in the N = 0 Super-gravity multiplet,

• We shall rewrite them in terms of the component gauge elds in order to checkwhether they are automatically satised when the gauge elds are on shell or, ifnot, what are the additional conditions to be imposed on Aaµ and Aa′µ .

This section deals with the free equations of motion (no sources), while in the fol-lowing we will study the sourced equations. The same equations have been explored in[59], although in a slightly dierent manner with respect to the one that we follow here.From now on we will not write Φ−1

aa′ explicitly anymore in the convolutions, rather we


shall systematically adopt the notation introduced in Section 3.1.1:

[X ? Y

]:=[Xa Φ−1

aa′ Y a′ .](x) (3.52)

We begin our analysis with the linearized Einstein-Hilbert equations, which in absenceof sources read:

Rµν −12ηµνR = 0, (3.53)

with Rµν and R being the linearized Ricci tensor and Ricci scalar, respectively. We ex-press the linearized Ricci tensor Rµν in terms of YM elds, exploiting the denition ofthe graviton given in eq. (3.31) and keeping γ arbitrary for the time being.

Since Rµν is gauge invariant at the linearized level, we try as much as possible toimplement its DC expression in terms of YM eld strengths, although, as we shall see,other types of contributions appear. Moreover, when ambiguities appear due to the factthat derivatives can act equivalently on both elds involved in the ?-product, we chooseto keep the expression symmetric in the left (A) and right (A) elds. Thus, exploitingthe denitions:

hµν = HSµν − γηµνϕ, (3.54)

HSµν = 1

2A(µ ? Aν), (3.55)

ϕ =(Aα ? Aα −

(∂ · A) ? (∂ · A)

), (3.56)

it is possible to derive the following form for the linearized Ricci tensor:

Rµν = 12∂µ∂ · hν + ∂ν∂ · hµ −hµν − ∂µ∂νh

= 1

4

(∂µAν + ∂νAµ) ? (∂ · A) + (∂ · A) ? (∂µAν + ∂νAν)

−(Aµ ? Aν + Aν ? Aµ)− 2∂µ∂νAα ? Aα

+ 12γ[(D − 2)∂µ∂ν + ηµν

]ϕ

= 14

(−Aµ + ∂µ∂ · A) ? Aν + Aν ? (−Aµ + ∂µ∂ · A)

+ Aα ? ∂ν(∂αAµ − ∂µAα) + ∂ν(∂αAµ − ∂µAα) ? Aα

+ 12γ[(D − 2)∂µ∂ν + ηµν

]ϕ

= 14− ∂αFαµ ? Aν − Aν ? ∂αFαµ + ∂νFαµ ? A

α + Aα ? ∂νFαµ


+ 12γ[(D − 2)∂µ∂ν + ηµν

]ϕ

= 14Fαµ ? F

αν + F α

ν ? Fαµ+ 12γ[(D − 2)∂µ∂ν + ηµν

]ϕ. (3.57)

Then, the Ricci scalar is:

R = −12Fαβ ? F

αβ + (D − 1)γϕ. (3.58)

Furthermore, we observe that:

Fαβ ? Fαβ = 2(Aα ? Aα)− 2(∂ · A) ? (∂ · A) = 2ϕ, (3.59)

hence the Ricci scalar can be equivalently written as:

R = γ(D − 1)− 1ϕ = 12γ(D − 1)− 1Fαβ ? Fαβ. (3.60)

As a next step we wish to explore to what extent assuming that the linearized YM equa-tions of motion hold6,

∂αF aαµ = Aaµ − ∂µ∂ · Aa ≈ 0, (3.61)

imply that the gravitational ones hold as well i.e. that the Ricci tensor vanishes.

First, let us observe that the free equations of motion for the scalar eld are easilysatised:

ϕ = Aα ? Aα − ∂ · A ? ∂ · A = (Aα − ∂α∂ · A) ? Aα = ∂µFµα ? Aα ≈ 0, (3.62)

thus implying that R ≈ 0, in force of (3.58). Therefore, the linearized version of (3.53)reduces to:

Rµν −12ηµνR ≈

14∂νFαµ ? A

α + Aα ? ∂νFαµ

+ 12γ(D − 2)∂µ∂νϕ

= 12[γ(D − 2)− 1]∂µ∂νAα ? Aα

+ 14∂ν

(Aµ −

∂µ∂ · A

)? ∂ · A+ 1

4∂ · A ?(Aµ −

∂µ∂ · A

). (3.63)

6With the symbol “≈” we denote an equality which only holds when the YM equations of motion aresatised.


If we want (3.63) to vansih, we can choose γ = 1D−2 to eliminate one term, while the

only option to eliminate the last line is to x the Lorenz gauge condition for both the YMelds: ∂ · A = ∂ · A = 0. Therefore, when the gauge elds are on shell and they satisfythe Lorenz gauge condition, the linearized Einstein equations of motion are satised too:

Rµν −12ηµνR ≈ 0. (3.64)

Similarly, we can recover the free e.o.m. for the two-form eld:

Bµν − ∂µ∂ρBρν + ∂ν∂ρBρµ

= 12(Aµ ? Aν − Aν ? Aµ)− ∂µ∂ρAρ ? Aν

+ ∂µ∂ρAν ? Aρ + ∂ν∂

ρAρ ? Aµ − ∂ν∂ρAµ ? Aρ

= 12

(Aµ − ∂µ∂ · A) ? Aν − Aν ? (Aµ − ∂µ∂ · A) + ∂ν∂ · A∂νAµ − Aµ ? ∂ν∂ · A.

(3.65)

Again, the result is that when the YM elds are on shell and in Lorenz gauge, the equa-tions of motion for Bµν are satised: Bµν − ∂µ∂ρBρν + ∂ν∂

ρBρµ ≈ 0.

To summarize, with the product of eq. (3.1) and the denition of the gravity eldsgiven in Section 3.1.2, the linearized equations of motion for the elds of N = 0 Super-gravity,

Rµν − 1

2ηµνR = 0,

Bµν − ∂µ∂ρBρν + ∂ν∂ρBρµ = 0,

ϕ = 0,

(3.66)

are indeed satised when the YM elds are on shell (∂µFµν = ∂µFµν = 0) and in Lorenzgauge (∂ · A = ∂ · A = 0), if, in addition, in the denition of the graviton given in eq.(3.31) one chooses γ = 1

D−2 .

3.1.6 Sourced equations of motion

In this section we consider the addition of sources for the YM elds and for the (inverseof the) biadjoint scalar eld. We refer to ideas presented in [59], but we proceed from a


slightly dierent perspective. Indeed, we introduce a new product (in the spirit of (3.1)),which we shall denote with “~” and we shall employ in order to build gravitationalsources from the YM ones7. The denition is as follows:

Jµν :=[jaµ [j(Φ)

aa′ ]−1 ja′ν](x) :=

[jµ ~ jν

], (3.67)

where Jµν is the “DC source”, jaµ, ja′ν are two YM currents and [j(Φ)aa′ ]−1 is the convolution

inverse of the source for the biadjoint scalar eld Φaa′ . In terms of these sources, thelinearized equations of motion for YM elds read:

∂µF a

µν = Aaµ − ∂µ∂ · Aa = jaν ,

∂µF a′µν = Aa′µ − ∂µ∂ · Aa

′ = ja′ν ,

(3.68)

while the equations of motion for Φ−1aa′ deserve a separate discussion. Indeed, we can start

from the biadjoint scalar, which in presence of a source satises the following equation:

Φaa′ = jΦaa′ , (3.69)

whose solution, focusing on the inhomogeneous part, is (in Fourier transform):

FΦaa′(k) = −FjΦaa′(k)k2 . (3.70)

Since Φ−1aa′ is dened to be the convolution inverse of the biadjoint scalar, their Fourier

transforms satisfy the following equation:

FΦ−1aa′(k) = − 1

FΦaa′(k) . (3.71)

Therefore, we can derive the equations of motion for Φ−1aa′ from (3.70) and (3.71):

FΦ−1aa′(k) = − k2

FjΦaa′(k) , (3.72)

or, in coordinate space:

Φ−1aa′ = [jΦ

aa′ ]−1. (3.73)

7We are grateful to Silvia Nagy for private communications in connection to the denition of thisproduct.


Let us comment more on the introduction of the ~-product in eq. (3.67). The un-derlying idea is that, in the DC spirit, as the gravity elds are built from YM elds, thesources for these elds should be built from the YM sources. Indeed, without introducing~, we would get convolutions of YM currents with YM elds, which should be identi-ed with the sources for the gravity elds. On the other hand, it is more natural to thinkthat the gravitational sources be determined only by the YM currents, Jgrav ∼ j ⊗ j,consistently with what expected from the DC of classical solutions, as we will discussin Section 3.3. Moreover, the use of the product ? also for sources, which would appearmaybe more natural, would also be problematic because of the mass dimension of thecurrents: [j ? j] = D+6

2 , since [j] = [j] = D+22 . The introduction of ~ solves this issue,

since [(jΦ)−1] = 3D−22 , therefore [j ~ j] = D+2

2 , which is the correct mass dimensionexpected from a source.

To summarize, we have dened two kinds of products: “?” and “~”. Their aim issimilar, in the sense that they are both used to perform the DC of two quantities in YMtheory. However, for the reasons which we detailed, we shall from now on employ “?”to multiply the YM elds and “~” to multiply the YM currents. More explicitly:

[X ? Y

]=[Xa ϕaa′ Y a′

](x) if X, Y are elds, (3.74)[

jX ~ jY]

=[jXa [j(Φ)

aa′ ]−1 jYa′](x) if jX , jY are currents, (3.75)

where [j(Φ)aa′ ]−1 is the convolution inverse of the source jΦ

aa′ of the biadjoint scalar. Fromthe last equation in the system (3.68), we can read the relation

X ? Y = X ~ Y, (3.76)

which we will use extensively in what follows.

We proceed employing the expressions for the equations of motion derived in eqs(3.57),(3.58),(3.65), then substituting the eqs. (3.68). First of all, we consider the scalareld, since this will turn out to be useful also in the case of gravity. As in eq. (3.62):

ϕ = ∂µFµα ? Aα = ∂µFµα ~A

α, (3.77)

therefore, again if the YM elds are in Lorenz gauge, and denoting with “≈” equalities


which hold on shell one has:

ϕ ≈ jα ~ jα. (3.78)

From now on we shall always assume the gauge elds to satisfy the Lorenz gauge condi-tion and γ = 1

D−2 , since in many occasions they both proved to be necessary conditionsto the goal of reproducing the equations of motion for the elds in the gravity theory.Then, from eqs. (3.57) and (3.58), we derive:

Rµν ≈ −14jµ ~ jν + jν ~ jµ+ 1

2(D − 2)ηµνjα ~ jα, (3.79)

R = −12∂αFαβ ? A

β + Aβ ? ∂αFαβ

+ D − 1D − 2ϕ ≈

1D − 2j

α ~ jα, (3.80)

from which we obtain the sourced EH equations in the form

Rµν −12ηµνR ≈ −

14j(µ ~ jν), (3.81)

where in particular we have a DC form, in the sense of the ~-product, for the energy-momentum tensor of matter. Last we consider the two-form for which, in the samesetup, one obtains:

Bµν − ∂µ∂ρBρν + ∂ν∂ρBρµ ≈

12j[µ ~ jν]. (3.82)

To summarize, with the choice γ = 1D−2 , when the YM elds are on shell and in the

Lorenz gauge, the sourced, linearized equations of N = 0Supergravity:

Rµν − 1

2ηµνR = −14j(µ ~ jν) := −Tµν ,

Bµν − ∂µ∂ρBρν + ∂ν∂ρBρµ = 1

2j[µ ~ jν] := j(B)µν ,

ϕ = jα ~ jα := j(ϕ).

(3.83)

3.2 BDHK Lagrangian and the double copy

In this section we review the proposal for a Lagrangian implementation of the DC rela-tions made in [16], trying to highlight its essential features. The program of [16] was tobuild a Lagrangian for pure YM theory with only cubic vertices, whose Feynman ruleswould automatically obey the CK duality, to then dene an appropriate “square” of such


a Lagrangian, whose tree-level scattering amplitudes would reproduce the results of theDC relations.

3.2.1 Color-kinematics duality in YM Lagrangian

As we explicitly showed in Section 2.2.2, the CK duality is a property of the tree-levelfour gluons scattering amplitude which can be derived directly from the Feynman rulesof YM theory, provided that the vertices are rearranged in a cubic (trivalent) form. In[16] Bern, Dennen, Huang and Kiermaier (BDHK) showed how to derive the CK dualityfor the four-point amplitude directly from a Lagrangian, suitably arranged to containonly cubic interactions by means of the introduction of an appropriate additional eld.

The BDHK Lagrangian is the following:

LBDHK = 12A

aµAaµ + 12(∂ · Aa)2 −BaρµνBa

ρµν − gfabc(∂µAaν + ∂ρBaρµν)AbµAcν ,

(3.84)

where Baρµν denotes new elds, which only propagate as internal lines in the Feynman

diagrams when computing gluon amplitudes. Given the form of its interaction in theLagrangian of eq. (3.84), it can be taken antisymmetric in its last two indices: Ba

ρµν =−Ba

ρνµ. Equivalence to the YM Lagrangian can be proven upon formally integrating outthe auxiliary eld, i.e. substituting in the Lagrangian the equations of motion for Ba

ρµν :

δLBDHK

δBaρµν

= 0⇒ Baρµν = g

2fabc∂ρ(AbµAcν), (3.85)

LBDHK(A,B(A)) = 12A

aµAaµ + 12(∂ · Aa)2

− gfabc∂µAaνAbµAcν −g2

4 fabcfadeAbµA

cνA

dµAeν , (3.86)

where in the intermediate steps algebraic manipulations of 1 where assumed to be well

dened.

The main advantage of (3.84) with respect to the conventional YM Lagrangian is thatit only contains cubic vertices. Since the amplitudes obtained from LYM automaticallysatisfy the CK duality up to four points, the same must true for the amplitudes obtainedfrom LBDHK when the external states are gluons, but with a simplication: in the YM


theory we were forced to split the four-gluons vertex into three-point contributions “byhand”, dividing and multiplying by appropriate inverse propagators, while starting from(3.84) this operation is automatically implemented when computing Feynman diagrams.As an example, let us compute the contribution to the s-channel in the tree-level four-gluon scattering. First, let us derive the Feynman rules from (3.84), representing the eldBaρµν with a straight continuous line and the gluon with a curly line, as usual:

a, µ b, νp= −i δabηµν

p2 ,a, µνρ b, αβγp = i

2p2

(δabηµαηνβηργ

),

a1, µ1

a3, µ3a2, µ2

= gfa1a2a3ηµ1µ2(p1 − p2)µ3 + ηµ2µ3(p2 − p3)µ1

+ηµ3µ1(p3 − p1)µ2,

a1, µ1

a2, µ2

a3, αβγ = gfa1a2a3pα3(ηβµ1ηγµ2 − ηβµ2ηγµ1

).

In Feynman gauge (eq. (2.8)), the contribution from the exchange of an auxiliary eld tothe s-channel in four gluons scattering is:

1

2 3

4 = gfa1a2b(−p1 − p2)α(ε1βε

2γ − ε2

βε1γ) i

(p1+p2)2 δbc

×ηραηλβησγgf ca3a4(p1 + p2)ρ(ε3λε

4σ − ε4

λε3σ)

= +ig2 ssfa1a2ef ea3a4

(ε1ε4)(ε2ε3)− (ε1ε3)(ε2ε4)

.

The result of this diagram is exactly the same as the part proportional to cs in eq. (2.19),i.e. the part of the contact term diagram with color structure cs. It is important to stressthe role of the kinetic term of the eld Ba

ρµν and of the derivative in the correspondinginteraction vertex: in the given example they provide an apparently irrelevant factorsss

= 1, which is actually instrumental in reproducing the “splitting” of the contact termperformed in Section 2.2.2. It is in this sense that Ba

ρµν is “auxiliary”, although its equa-


tions of motion are not algebraic and it actually propagates in the Feynman diagrams.Indeed, in order to recover the YM scattering amplitudes we must limit ourselves toconsider, among the possible amplitudes produced by (3.84), only the ones in which theexternal states are all gluons, and Ba

ρµν only propagates in internal lines.

Moreover, as we discussed in Section 2.2.6, in [16] it was shown how to introducea ve gluons (non-local) contact term in the YM Lagrangian, which on the one handis identically zero due to Jacobi identity, but on the other hand it contributes to theFeynman diagrams so as to render the CK duality apparent up to ve points. Similarlyto (3.84), it was also shown how to introduce additional “auxiliary”8 elds in order tomake all the interactions only cubic. While the procedure can be extended to in principlearbitrarily higher points, explicit results so far have been found only up to six points attree level [124].

3.2.2 Double copy for BDHK Lagrangian

In [16] a possible Lagrangian realization of the DC relations is also discussed. The ideais to perform, at the Lagrangian level, the same steps which guide the construction ofgravitational scattering amplitudes by means of the DC relations. We now review thisapproach, providing some comments about it in Section 3.2.3.

Let us review, step by step, the DC construction of scattering amplitudes, while dis-cussing its Lagrangian implementation as proposed in [16]. For the moment, we limitourselves to discuss the analysis up to the four-point scattering amplitudes.

• The rst step required to obtain gravitational amplitudes with the DC is to rear-range the expressions of YM amplitudes, writing them as a sum over diagrams withcubic vertices whose kinematic numerators respect the CK duality. At the four-point level, this is reproduced by the BDHK Lagrangian (3.84): as we discussed inthe previous section it contains only cubic vertices, whose Feynman rules auto-matically encode the CK duality.

• The DC of scattering amplitudes is naturally performed in momentum space, withthe color factors replaced by another copy of kinematic factors. Therefore, let us

8In the sense that, like Baµνρ in (3.84), they have a kinetic term but they contribute to the scatteringamplitudes only propagating in the internal lines of the diagrams, like ghosts in gauge theories quantizedà la Faddeev-Popov.


consider the BDHK Lagrangian written in momentum space, and introduce a pre-scription to remove the color indices and the structures constants. The idea of [16]is to consider a key feature of the CK duality the fact that the kinematic numera-tors of the cubic diagrams in YM theory, when in the CK-dual form, share the sameantisymmetry as the structure constants. Therefore, when suppressing the colorindices and the structure constants from the YM Lagrangian in momentum space,they proposed to make this antisymmetry explicit in the vertex. As an example ofthis procedure, let us consider the YM cubic vertex:

S1YM =

∫dDxfabc∂µA

aνA

µbA

νc , (3.87)

and let us write it in momentum space9

S1YM = −i

∫đk1đk2đk3δ

(D)(k1 + k2 + k3

)fabck1

µAaν(k1)Aµb (k2)Aνc (k3). (3.88)

Then, the prescription tells to remove color indices and structure constants, whilestill endowing the remaining part of the vertex with the same antisymmetry ofthe constants fabc. To this end, we explicitly write a sum over permutations withsign of the momentum indices 1, 2 and 3, since in (3.88) there is a one-to-onecorrespondence between the momentum indices and the color indices. The resultis the following10:

[S1YM ]no color = −i

∫đk1đk2đk3δ

(D)( 3∑i=1

ki) ∑σ∈S3

(−1)sgn(σ)kσ(1)µ Aσ(1)

ν Aµσ(2)Aνσ(3).

(3.89)

In the kinetic term, where no structure constants are present, the prescription isto simply suppress the color indices. The result, for the full BDHK Lagrangian in

Feynman gauge (this is crucial, as we will discuss), is:

SYMBDHK →12

∫đk1đk2δ

(D)(k1 + k2)k21

[A1µA

µ2 − 2Bρµν

1 B2ρµν

]9We introduce the notation: đk = dDk

(2π)D .10In order to avoid cumbersome formulas, from now on we adopt the notation Xi = X(ki), where X

is any eld.


−i∫

đk1đk2đk3δ(D)(

3∑i=1

ki)∑σ∈S3

(−1)sgn(σ)[(kσ(1)µ Aσ(1)

ν + kρσ(1)Bσ(1)ρµν

)Aµσ(2)A

νσ(3)

].

(3.90)

• Now that the color factors have been suppressed, with the vertices of (3.90) prop-erly antisymmetrized, let us insert a new copy of kinematic factors. To this end,it is necessary to adopt two dierent prescriptions for the kinetic term and forthe vertices. Indeed, in the DC of scattering amplitudes, while passing from YMto gravity the denominators of the YM diagrams, containing propagators, are leftunaltered, while their numerators are multiplied by another copy of kinematic nu-merators. Therefore, we insert a new copy of YM elds (with the “tilde”, since theycome from the second copy of YM theory), but leaving the momentum structureof the kinetic term unaltered, so as to obtain a two-derivative kinetic term also forthe gravitational theory:

12

∫đk1đk2δ

(D)(k1 + k2)k21

[A1µA

µ2 − 2Bρµν

1 B2ρµν

]→1

2

∫đk1đk2δ

(D)(k1 + k2)k21

[A1µA

µ2 − 2Bρµν

1 B2ρµν

][A1αA

α2 − 2Bγαβ

1 B2γαβ

].

(3.91)

On the contrary, in the interaction vertices, we insert a new copy of the elds andof the derivatives (or, equivalently, of the momenta) acting on them. For the timebeing, we work up to an overall constant. The result is the following:

∫đk1đk2đk3δ

(D)(3∑i=1

ki)∑σ∈S3

(−1)sgn(σ)[(kσ(1)µ Aσ(1)


)Aµσ(2)A

νσ(3)

]

→∫

đk1đk2đk3δ(D)(

3∑i=1

ki)∑σ∈S3

(−1)sgn(σ)[(kσ(1)µ Aσ(1)


)Aµσ(2)A

νσ(3)

]×∑τ∈S3

(−1)sgn(τ)[(kτ(1)α A

τ(1)β + kγτ(1)B

τ(1)γαβ

)Aατ(2)A

βτ(3)

]. (3.92)

• Finally, in the DC the states in the gravitational theory are built from productsof spin-one state, which do not carry color indices. The proposal of Bern andcollaborators is therefore to dene a physical “gravitational eld” Hµν with theidentication Aµ(k)Aν(k)→ Hµν(k), where Aµ(k) and Aν(k) are two YM elds,coming from the two YM theories which are involved in the DC, with their color


indices simply suppressed and carrying momentum kµ. As mentioned, the symbol“tilde” over the elds will be used to denote quantities in the second copy of YMtheory. The physical states of the eld Hµν arise from the product of spin-onestates, therefore they include a graviton, a two-form and a scalar, as discussed inSection 2.4.1. The other products of YM elds which appear in (3.91) and (3.92)contain at least one copy of the eld Ba

ρµν , and from these products new gravita-tional elds are dened, which will have, in the theory of gravity, a role similarto the one of Ba

ρµν in YM theory, as we shall discuss. Such elds are dened asAµBγαβ := gµγαβ , BρµνAα := gρµνα and BρµνBγαβ := fρµνγαβ .

Let us now study the kinetic part (3.91) and the vertices (3.92) separately, so as toinspect the DC Lagrangian which we just built following [16]. First, we consider (3.91),which can be easily written back in coordinate space, with the following result:

SDCkin = 12

∫dDx

[HµαHµα − 2gµγαβgµγαβ − 2gρµναgρµνα + 4fρµνγαβfρµνγαβ

].

(3.93)

The role of the new elds will be clear as soon as we study the interaction part of thisdouble-copy Lagrangian. Before doing this, let us write the various propagators of theelds, contracting independently the indices from the left and the right gauge theories:

〈Hµα(k1)Hνβ(k2)〉 = i

k21(ηµνηαβ)δ(D)(k1 + k2), (3.94)

〈gµαβγ(k1)gνδζη(k2)〉 = − i

2k21(ηµνηαδηβζηγη)δ(D)(k1 + k2), (3.95)

〈gµνρα(k1)gστλβ(k2)〉 = − i

2k21(ηµσηντηρτηαβ)δ(D)(k1 + k2), (3.96)

〈fµνραβγ(k1)fστλδζη(k2)〉 = i

4k21(ηµσηντηρληαδηβζηγη)δ(D)(k1 + k2). (3.97)

The interaction Lagrangian of eq. (3.92) contains 122 = 144 terms, that can be conve-niently grouped as follows: there are 36 vertices of the type HHH , 36 gHH vertices,36 gHH vertices, 24 ggH vertices and 12 fHH vertices. They can be derived writingexplicitly all the terms in the product of two sums over permutations in (3.92). In [16]the interaction vertices are not displayed because of their lengthy expressions; however,we realized that there is quite a bit of redundancy in the above counting, as it is possi-ble to collect groups of six terms which generate identical Feynman rules. Therefore, in


coordinate space the vertices can be rewritten as:

· LHHH = Hµν∂µ∂νHαβHαβ −Hµν∂µH

αβ∂νHαβ +Hµν∂µHαβ∂βHαν

−Hµν∂βHαν∂αHµβ +Hµν∂νHαβ∂αHµβ −HµνHαβ∂α∂νHµβ, (3.98)

· LgHH = ∂µ∂γgνγαβH

µαHνβ + ∂γgνγαβ∂µHναHµβ + ∂γgµγαβ∂

µHνβH αν

− ∂µ∂γgνγαβHναHµβ − ∂γgµγαβ∂µHναH βν − ∂γgνγαβ∂µHνβHµα, (3.99)

· LgHH = ∂α∂ρgρµνβH

µαHνβ + ∂ρgρµνα∂αHνβHµ

β + ∂ρgρµνβ∂αHµβHνα

− ∂α∂ρgρµνβHναHµβ − ∂ρgρµνα∂αHµβHνβ − ∂ρgρµνβ∂αHνβHµα, (3.100)

· LggH = ∂ρgρµνα∂γgνγαβHµ

β + ∂ρgρµνβ∂γgµγαβHν

α

− ∂ρgρµνα∂γgµγαβHνβ − ∂ρgρµνβ∂γgνγαβHµ

α, (3.101)

· LfHH = ∂ρ∂γfρµνγαβHµαHνβ − ∂ρ∂γfρµνγαβHµβHνα. (3.102)

Such a Lagrangian reproduces, by construction, the results of the DC in the case ofthree and four particles scattering amplitudes at tree level. Indeed, the interactions arebuilt exactly in such a way that, if the left and right YM elds contract independently,the vertices are products of two YM vertices which encode the CK duality, without thestructure constants but taking into account their antisymmetry. This is crucial: the YMFeynman rules always inherit from the structure constants the antisymmetry under theexchange of color indices (see Section 2.1), therefore when suppressing the structureconstants the antisymmetry must be enforced “manually”. Furthermore, since in the DCof the quadratic Lagrangian only the elds were squared, leaving the derivative part un-altered, the propagators are∼ 1

p2 as in the YM theory, as required by the DC prescriptionin order to leave the denominators of the amplitude unaltered. Finally, being all the ver-tices cubic, the “doubling” of the interactions in the Lagrangian exactly corresponds tothe “doubling” of the kinematic factors in the YM amplitudes, diagram by diagram.

We can represent schematically how the gravity vertices are built, drawing diagramswhere curly lines are gluons, straight lines represent the eldBa

ρµν , double wavy lines theeld Hµν , wavy+straight lines either gµγαβ or gρµνα (according to the order) and doublestraight lines the eld fρµνγαβ . With this notation, the DC of the BDHK Lagrangian isrepresented in g. 3.1. As an example, we can compute the three particles tree-levelscattering amplitude between gravitons, two-forms and scalars, and compare the resultwith what we obtained in Section 2.4.2. Integrating by parts and requiring transversality


2

=

Figure 3.1: Double copy for three particles vertex.

of the elds (inherited by the transversality of gluons polarization vectors), we can recast(3.98) into the form:

LTTHHH = 2Hµν∂µ∂νHαβHαβ + 2Hµν∂µH

αβ∂βHαν + 2Hµν∂νHαβ∂αHµβ, (3.103)

which, with the insertion of a factor κ2 (κ is the gravitational coupling constant), gives

exactly the amplitude of eq. (2.109).

If we adopt the same prescription for the coupling constant for all the cubic vertices,which is the natural thing to do if one thinks of the DC contruction of gravitationalscattering amplitudes, we can write the complete result for the Lagrangian proposed in[16] in order to reproduce the DC relations up to the four-point scattering amplitudes.Upon integration by parts of some terms in eqs. (3.98)-(3.102), it is possible to write theresult as follows:

LDCBDHK =1

2HµαHµα − gµγαβgµγαβ − gρµναgρµνα + 2fρµνγαβfρµνγαβ

+ κ

Hµν∂µ∂νH

αβHαβ +Hµν∂µHαβ∂βHαν +Hµν∂νH

αβ∂αHµβ

+ 12Hαβ∂νH

µν(∂µHαβ + ∂αH βµ ) + 1

2HµνHµβ∂

β∂αHαν

+ κ

2∂γgµγ[αβ]

(2∂νHµαHνβ +Hµα∂νH

νβ − ∂µHναH βν − ∂ρgρ[µν]αH β

ν

)+ ∂ρgρ[µν]α

(2∂βHµαHνβ +Hµα∂βH

νβ − ∂αHµβHνβ − ∂γgµγ[αβ]Hν

β

)− ∂ρgρ[µν]α∂γg

µγ[αβ]Hνβ + 1

2fρ[µν]γ[αβ]HµαHνβ

. (3.104)

As in the BDHK theory, integrating out the elds gµγαβ , gµνρα and fρµνγαβ mightin principle result in a non-local gravity Lagrangian in force of their non-trivial kineticoperators. However, it is possible to show that this does not happen in this case, owingto the particular form of the interaction vertices in eq. (3.104) and assuming that all in-termediate formal manipulations are well-dened.


Let us prove it, rst for the elds gµγαβ and gµνρα (the logic is the same for these twoelds) and then for fρµνγαβ . Let us write, schematically, the part of the action containinggµγαβ :

Sg =∫dDx

[− gµγαβgµγαβ + κ

2∂γgµγαβX

µαβg

], (3.105)

where with Xγαβg we denote the remaining part of the vertices which contain ∂γgµγαβ .

The key fact is that in all its interactions the eld gµγαβ appears with a derivative con-tracted with its second index, which allowed us to write (3.105) in this form. If we thenvary the action and we assume intermediate formal manipulations to be well dened,we can compute the equations of motion for the eld under inspection, which are thefollowing:

gµγαβ = − κ

4∂γXgµαβ. (3.106)

Last, replacing (3.106) into (3.105) we can “integrate out” gµγαβ , with a resulting action:

Sg =∫dDx

[− κ

4∂γXµαβ

g κ

4∂γX

gµαβ −

κ2

8∂γ∂γX

gµαβX

µαβg

]= −κ

2

16

∫dDxXµαβ

g Xgµαβ. (3.107)

From (3.107) it is evident that, as long as Xγαβg is a local quantity, integrating out gµγαβ

produces a local action even though its equations of motion (3.106) present a non-locality.The same argument applies to gµνρα, whose action can be written as:

Sg =∫dDx

[− gρµναgρµνα + κ

2∂ρgρµναX

µναg

]. (3.108)

The equations of motion for gρµνα are the following:

gρµνα = − κ2

4∂ρXgµνα, (3.109)

by means of the which one can integrate out gµνρα, with the result:

Sg = −κ2

16

∫dDxXµνα

g X gµνα. (3.110)


Similarly, we can write the action as for fρµνγαβ :

Sf =∫dDx

[+ 2fρµνγαβfρµνγαβ + κ

2∂ρ∂γfρµνγαβX

µναβf

], (3.111)

with the only dierence with respect to the previous cases being the fact that thereare two derivatives acting on fρµνγαβ in the interaction term. Indeed, from (3.111), theequations of motion for fρµνγαβ are the following:

fρµνγαβ = − κ

8∂ρ∂γXfµναβ. (3.112)

Integrating out fρµνγαβ from (3.111) by means of (3.112) yields the following result:

Sf =∫dDx

[2 κ

8∂ρ∂γXµναβ

f κ

8∂ρ∂γX

fµναβ −

κ2

16∂γ

∂ρ∂ρ∂γXfµναβX

µναβf

]= −κ

2

32

∫Xµναβf Xf

µναβ. (3.113)

Again, the action is local if Xµναβf itself is local.

In the above discussion we have introduced three quantities, namely Xγαβg , Xµνα

g

and Xµναβf , implicitly dened by eqs. (3.105), (3.108) and (3.111), respectively. They can

be straightforwardly derived from (3.104), with the following results:

·Xγαβg = 2∂νHµ[α|Hν|β] +Hµ[α|∂νH

ν|β] − ∂µHν[αH β]ν − ∂ρgρ[µν][αH β]

ν , (3.114)

·Xµναg = 2∂βH [µ|αH |ν]β +H [µ|α∂βH

|ν]β − ∂αH [µ|βH|ν]β − ∂γg[µ|γ[αβ]H

|ν]β , (3.115)

·Xµναβf = Hµ[α|Hν|β]. (3.116)

As a last step, we wish to display explicitly the action which results integrating outthe “auxiliary” elds, in order to obtain an action for the only physical eld Hµν . How-ever, comparing (3.106) and (3.109) with (3.114) and (3.115) it is evident that the equationsof motion of gµγαβ and gµνρα are coupled:

gµγαβ = − κ

4∂γ(2∂νHµ[α|Hν|β] +Hµ[α|∂νH

ν|β] − ∂µHν[αH β]ν − ∂ρgρ[µν][αH β]

ν

),

(3.117)

gρµνα = − κ

4∂ρ(2∂βH [µ|αH |ν]β +H [µ|α∂βH

|ν]β − ∂αH [µ|βH|ν]β − ∂γg[µ|γ[αβ]H

|ν]β

).

(3.118)


As a result, while Xµναβf is completely determined in terms of Hµν , Xγαβ

g also dependson gρµνα and Xµνα

g on gµγαβ . To settle this, we can solve the equations of motion (3.117)and (3.118) with a perturbative expansion in terms of powers ofHµν (or, equivalently, interms of powers of κ). The non-locality in the equations of motion of gµγαβ and gρµνα iscanceled as above: in (3.114) and (3.115) the two elds appear with one derivative con-tracted with the derivative factored out in the equations of motion (3.117) and (3.118),therefore formally eliminating 1

. The nal expression of the action is, therefore, com-pletely local, but can be determined only perturbatively.

Since this DC action allows, by construction, to compute gravitational scatteringamplitudes up to the four-point ones, we can compute the quartic vertices which re-sult integrating out gµγαβ , gρµνα and fµνργαβ . To this end, we can safely neglect theaforementioned mixing between gµγαβ and gρµνα: all the vertices produced by the abovemechanism contain two powers of the quantities X previously introduced, and all ofthem are at least quadratic in Hµν . Therefore, a non-zero value for gµγαβ and gρµνα in(3.114) and (3.115) would produce, from the respective equations of motion (3.117) and(3.118), terms at least cubic inHµν , which would result in vertices with at least ve elds.

Hence, the result of this discussion is that the quartic couplings of Hµν in the DCLagrangian proposed in [16] are the following:

LDC(4) = −κ2

16

12[Hµ[α|Hν|β]

][Hµ[α|Hν|β]

]+

+[2∂νHµ[α|Hν|β] +Hµ[α|∂νH

ν|β] − ∂µHν[αH β]ν

]×[2∂νHµ[α|Hν|β] +Hµ[α|∂

νHν|β] − ∂µHν[αHνβ]

]+[2∂βH [µ|αH |ν]β +H [µ|α∂βH

|ν]β − ∂αH [µ|βH|ν]β

]×[2∂βH[µ|αH|ν]β +H[µ|α∂

βH|ν]β − ∂αH[µ|βHβ|ν]

]. (3.119)

In [16], a generalization of this approach beyond four points is also suggested: theLagrangian we discussed in this section only reproduces the results of the DC up to fourexternal particles, since it is the DC of a YM Lagrangian with CK-dual Feynman diagramsup to the four gluons scattering amplitude. However, as discussed, in [16] and in [124],additional contributions to YM Lagrangians were built whose diagrams satisfy the CKduality up to six points. The proposal of [16] is to introduce further auxiliary elds in


order to bring these additional contributions (such as the one discussed in Section 2.2.6)into a local form with only cubic vertices. Once the YM Lagrangian is recast in this form,with manifest CK duality, it is possible to obtain a gravity Lagrangian with a procedureanalogous to the one discussed in this section for the BDHK Lagrangian.

3.2.3 Comments on the BDHK approach

The method proposed in [16] is, as of today, the only approach available in the litera-ture which actually implements the DC relations at the Lagrangian level, which makesit of extreme interest to our purposes. For this reason, we would like to add a few morecomments on its main features.

First, let us observe that two dierent prescriptions were employed in the DC proce-dure discussed in the previous section: as we already stressed, in the kinetic terms onlythe elds were “double-copied”, leaving the derivatives structure unaltered. On the con-trary, in the interaction part of the Lagrangian both the derivatives and the elds were“double-copied”, as expected from the DC procedure on scattering amplitudes. However,this procedure was successful only because in [16] the YM Lagrangian was chosen to bein Feynman gauge, where the quadratic term is simply∼ AA, with no derivatives con-tracted with the gauge elds. Had one chosen a dierent gauge or no gauge-xing at all,a prescription should have been specied on how to treat the derivatives contracted withthe elds in the quadratic part of the Lagrangian. Second, in this discussion the problemof symmetries and geometry, which is the main focus of this Thesis, was not addressed atall: it is not clear how the o-shell symmetries of gravity should emerge from YM theoryand indeed the gauge invariance of the resulting Lagrangian is not discussed at all, thereason being again that a gauge choice has been made before performing the “square”.Third, no prescription is given on how to treat the color indices of the YM elds in theDC: they are simply eliminated from the elds, together with the structure constants,and the identication Hµν(k) = Aµ(k)Aν(k) (and similarly for the auxiliary elds) isnot really mathematically claried, precisely because the color indices are simply “re-moved“, even though Aaµ in principle contains a multiplet of spin-one elds. Last, theauthors of [16] observed that they have found no indication that the procedure necessaryto build a Lagrangian with CK duality might eventually terminate, nor did they nd anypattern or symmetry principle behind their construction, which we ascribe once againto a lack of geometrical insight on the DC relations.


3.3 The double copy of classical solutions

Another interesting direction in which double-copy relations were studied and devel-oped is the realm of classical solutions. The rst work on the subject [19] explored thepossibility to relate the Kerr-Schild class of solutions to Einstein’s equations [125] toclassical solutions in YM theory. The program was then extended to consider dierentand possibly more complicated solutions, including the Taub-NUT spacetime [126, 127]as an example of double Kerr-Schild metric [22], Kerr-Schild metrics over curved back-grounds [20], perturbative solutions [119], accelerating particles and radiative solutions[24, 26–29, 128], GR and YM solutions over maximally symmetric spacetimes [21].

In the next few sections we shall review the basic, yet quite relevant case of the Kerr-Schild DC [19]. Indeed, while the DC relations only hold perturbatively, extending thediscussion to classical solutions opens up the possibility to relate exact, nonperturbativesolutions in gravity and YM theory, and this is exactly what was rst done in [19].

3.3.1 Kerr-Schild ansatz in General Relativity

The Einstein-Hilbert equations are a complicated set of non-linear partial dierentialequations, whose exact solutions are known only in a few cases. Among them, a par-ticularly interesting class was identied in [125]: the Kerr-Schild (KS) solutions. In thissection, we review the properties of KS solutions which turn out to be useful in the cor-responding double-copy construction.

The starting point is the KS ansatz for the metric:

gµν = ηµν + ϕkµkν , (3.120)where ηµν is the Minkowski metric, ϕ is a scalar eld and kµ a (co-)vector eld. Uponfurther requiring that kµ be null both w.r.t. Minkowski metric and to gµν

ηµνkµkν = gµνkµkν = 0, (3.121)

together with the condition

kµ∂µkν = 0, (3.122)

it is possible to prove that the metric (3.120) possesses a number of interesting properties:

1. gµν = ηµν − ϕkµkν .


Proof: let us try the ansatz gµν = ηµν + Xµν , with symmetric Xµν . Raising andlowering the indices of Xµν and kµ with ηµν we nd

gµαgαν = δµν = δµν +Xµν + ϕkµkν + ϕXµαkαkν , (3.123)

and therefore:

Xµν + ϕkµkν + ϕXµαkαkν = 0. (3.124)

Since ηµνkµkν = gµνkµkν = 0, then alsoXµνkµkν = 0; thus, by contracting (3.124)with kµ we nd:

kµXµν +

ϕkµkµkν +((((((

(ϕXµαkµkαkν = kµX

µν = 0. (3.125)

Substituting back into (3.124) yieldsXµν = −ϕkµν . Since gµνkν = ηµνkµ−ϕkµkµk

ν ,it turns out that the vector index of k can be raised/lowered equivalently with ηµνor with gµν .

2.√−g = 1.

Proof: let us write the determinant of the metric as in Section 2.3:

√−g = exp

12 tr[log(δµν + ηµαϕkαkν)]

, (3.126)

Given that

tr(ηµαϕkαkν) = ϕηµνkµkν = 0, (3.127)

and similarly all its powers vanish, one nds√−g = e0 = 1.

3. Γαµν = 12

∂µ(ϕkαkν) + ∂ν(ϕkαkµ)− ∂α(ϕkµkν) + ϕkαkµkνk · ∂ϕ

.

Proof: exploiting (3.121) and (3.122), we can write

Γαµν = 12g

αλ(∂µgνλ + ∂νgµλ − ∂λgµν)

= 12(ηαλ − ϕkαλ)[∂µ(ϕkλknu) + ∂ν(ϕkλkµ)− ∂λ(ϕkµkν)]

= 12∂µ(ϕkαkν) + ∂ν(ϕkαkµ)− ∂α(ϕkµkν) + ϕkαkµkνk · ∂ϕ

. (3.128)

If we then use

kλ∂µ(ϕkλkν) = ∂µϕk2kν + ϕ∂µkνk

2 + ϕkνkλ∂µkλ, (3.129)

kλ∂µkλ = 12∂µ(k2) = 0, (3.130)

the proof follows.


4. Γαµνkµkν = 0.Proof: one has to contract independently all the terms contained in the expressionfor Γλµν found at point 3.

∂µ(ϕkαkν)kµkν = ∂µϕkαkµk

2 + ϕk · ∂(kαkν)kν = 0, (3.131)

∂ν(ϕkαkµ)kµkν = ∂νϕkαk

2kν + ϕk · ∂(kαkµ)kµ = 0, (3.132)

∂α(ϕkµkν)kµkν = ∂αϕ(k2)2 + 2ϕkµ∂αkµk

2 = 0, (3.133)

ϕkαkµkνk · ∂ϕkµkν = ϕkα(k2)2k · ∂ϕ = 0. (3.134)

5. Γαµνkµkα = 0.Proof: as above, we simply evaluate all the contributions:

∂µ(ϕkαkν)kµkα = ∂µϕkνkµk

2 + ϕk · ∂(kαkν)kµ = 0, (3.135)

∂ν(ϕkαkµ)kµkα = ∂νϕ(k2)2kν + ϕ2kα∂νk

αk

2 = 0, (3.136)

∂α(ϕkµkν)kµkα = k · ∂ϕkνk2 + ϕkµk · (kµkν) = 0, (3.137)

ϕkαkµkνk · ∂ϕkµkα = ϕkν(k2)2k · ∂ϕ = 0. (3.138)

6. (Dµkν)kν = (∂µkν)kν = 0.Proof: (Dµkν)kν = (∂µkν)kν −Γαµνkαkν = 0. the proof then follows from therelation kλ∂µkλ = 1

2∂µ(k2) = 0.

7. (Dµkν)kµ = (∂µkν)kµ = 0.Proof: (Dµkν)kµ =

(∂µkν)kµ −Γαµνkαkν = 0. Thus, the null vector k is geodesicboth respect to gµν and to ηµν .

8. Rλν = 1

2

∂α∂

λ(ϕkαkν) + ∂α∂ν(ϕkαkλ)− ∂α∂α(ϕkλkν)

,where Rλ

ν = gλµRµν , but with the convention that ∂λ = ηλµ∂µ. The mixedconvention on how the indices are raised is useful since it allows to build a formof the Ricci tensor which is linear in the (non-perturbative) graviton eld denedas hµν = ϕkµkν .Proof: we simplify the Ricci tensor using the above properties.

Rλν = gλµ

∂µ∂ν ln(

√−g)− 1√

−g∂α(√−gΓαµν) + ΓβµαΓαβν

= (gλµ)

− ∂αΓαµν + ΓβµαΓαβν

, (3.139)


Then, we compute separately the two contributions, keeping in mind the conven-tion ∂λ = ηλµ∂µ:

gλµ∂αΓαµν = 12(ηλµ − ϕkµkλ)

∂α∂µ(ϕkαkν) + ∂α∂ν(ϕkαkµ)− ∂α∂α(ϕkµkν)

+ kµkν [(k · ∂ϕ)2 + ϕkαkβ∂α∂βϕ+ ϕk · ∂ϕ∂ · k]

= 12∂α∂

λ(ϕkαkν) + ∂α∂ν(ϕkαkλ − ∂α∂α(ϕkλkν) + kλkν [(k · ∂ϕ)2

+ ϕkαkβ∂α∂βϕ+ ϕk · ∂ϕ∂ · k]− 1

2ϕkλkµ[∂α∂µ(ϕkαkν)

+ ∂α∂ν(ϕkαkµ)− ∂α∂α(ϕkµkν)], (3.140)

ΓβµαΓαβν = 12Γαβν

∂µ(ϕkαkν) + ∂ν(ϕkαkµ)− ∂α(ϕkµkν) + ϕkβkαkµk · ∂ϕ︸︷︷︸

=0 contracted with Γ

= 14∂µ(ϕkβkα)∂ν(ϕkβkα) + ∂µ(ϕkβkα)∂β(ϕkνkα)

− ∂µ(ϕkβkα)∂α(ϕkβkν) + ∂α(ϕkβkµ)∂ν(ϕkβkα)

+ ∂α(ϕkβkµ)∂β(ϕkνkα)− ∂α(ϕkβkµ)∂α(ϕkβkν)

− ∂β(ϕkαkµ)∂ν(ϕkβkα)− ∂β(ϕkαkµ)∂β(ϕkνkα)

+ ∂β(ϕkαkµ)∂α(ϕkβkν) + ϕkαkβkνk · ∂ϕ[∂µ(ϕkαkβ) + ∂[α(ϕkβ]kµ)

= 12∂

α(ϕkµkβ)∂β(ϕkνkα) + 14kνk · ∂ϕϕk

αkβ∂µ(ϕkβkα)︸︷︷︸= 1

2∂µ[(ϕ2(k2)2)] = 0

. (3.141)

In the last equality we integrated by parts, exploiting the fact that total deriva-tive terms always contain k2 = 0. Then, contracting the last expression with theinverse metric tensor:

gλµΓβµαΓαβν = 12∂

α(ϕkλkβ)∂β(ϕkνkα)− 12ϕk

λ((((

(((kµ∂α(ϕkµkβ)︸︷︷︸k2 = kµ∂αkµ = 0

∂β(ϕkνkα). (3.142)

Therefore, from eq. (3.139):

Rλν = 1

2∂α∂

λ(ϕkαkν) + ∂α∂ν(ϕkαkλ)− ∂µ∂µ(ϕkλkν), (3.143)

since it is possible to show that other contributions cancel out integrating by partsand exploiting the above identities.

For what the KS DC is concerned, eq. (3.143) is extremely important: it shows that, forthe particular class of solutions to Einstein-Hilbert equations which admit a metric inthe KS form, the equations can be recast in a linear form, owing to the properties of the


vector eld kµ.

3.3.2 Vacuum, stationary Kerr-Schild solutions and the double

copy

The authors of [19] considered in particular stationary solutions, where the componentsof the metric (3.120) are time independent: ∂0ϕ = ∂0kµ = 0. Moreover, they observedthat redenitions of the type kµ → α(x)kµ, ϕ → ϕ/α(x)2 leave gµν unaltered, andexploited this freedom to set k0 = 1, absorbing all of its dynamics in the function ϕ.Under these conditions, the vacuum Einstein-Hilbert equations read, in components:

R00 = −1

2ϕ = −12∇

2ϕ = 0, (3.144)

Ri0 = 1

2∂i∂j(ϕkj)− ∂j∂j(ϕki)

= 1

2∂j[∂i(ϕkj)− ∂j(ϕki)

]= 0, (3.145)

Rij = 1

2∂i∂l(ϕkjkl) + ∂j∂

l(ϕklki)− ∂l∂l(ϕkikj)

= 0. (3.146)

The key prescription of [19] is to dene the single copy of this KS solution to be a YMeld of the form:

Aaµ = ϕakµ, (3.147)

where the ϕa’s are a set of scalars, which are assumed to solve the same equations as thescalar ϕ dened in eq. (3.120). The YM eld strength of this gauge eld turns out to belinear, since the dependence on the color and the Lorentz indices is factorized:

fabcAaµAbν = fabcϕaϕbkµkν = 0⇒ F a

µν = ∂µAaν − ∂νAaµ, (3.148)

due to the contraction of the symmetric productϕaϕb with the structure constants. Then,it is straightforward to check that the Einstein-Hilbert equations for the metric in theKS form imply the abelian Yang-Mills equations of motion for the gauge eld dened ineq. (3.147):

∂µF aµ0 = −∇2ϕa, (3.149)

∂µF aµi = −∂j

[∂i(ϕakj)− ∂j(ϕaki)

], (3.150)


since eq. (3.149) ((3.150)) is the same as eq. (3.144) ((3.145)). The fact that the YM equa-tions are in the abelian form (where the covariant derivative on the principal bundleis replaced with a partial derivative) reects the linear structure of the Einstein-Hilbertequations in KS coordinates.

As far as the general features of this approach to the DC are concerned, two remarksare in order. First, the KS DC relates YM solutions to GR solutions, but the two-formand the scalar eld typically present in the DC multiplet are absent. This is due to thefact that the graviton dened by this approach is hµν = ϕkµkν , which is automaticallysymmetric and traceless (k2 = 0). In this sense, the KS ansatz naturally selects only thepurely gravitational sector of the DC. Moreover, it is worth stressing that in this approachthe DC is dened in coordinate space, whereas in the case of scattering amplitudes it isproperly dened in momentum space.

3.3.3 The double copy for the Schwarzschild black-hole

As an application of the general construction presented in the previous section, let usreview a concrete example of the KS DC, namely the Schwarzschild black-hole. ByBirkho’s theorem it is the most general spherically symmetric solution to the Einsteinequations in vacuum. In Schwarzschild coordinates the line element reads:

ds2 = gµνdxµdxν = −F (r)dt2 + dr2

F (r) + r2dΩ2. (3.151)

For the sake of simplicity we consider the case of D = 4, in which F (r) = 1 −2GMr

is a scalar function and dΩ2 is the measure on the two-sphere. Even though theSchwarzschild solution is a vacuum solution, it is possible to source it with a static point-like mass M in the origin of the Schwarzschild coordinates:

Tµν = Mvµvνδ(3)(x), (3.152)

with vµ = (1, 0, 0, 0).

In order to illustrate the physical meaning of the KS DC in this case, rst of all weneed to perform a coordinate transformation so as to recast the metric in the KS form


(3.120). to this end, starting from eq. (3.151) we dene du := dt+ drS(r) :

ds2 = −F (r)du2+2drdu− dr2

F (r)+ dr2

F (r)+r2dΩ2 = −F (r)du2+2drdu+r2dΩ2. (3.153)

Then, introducing T := u− r:

ds2 = −F (r)(dT + dr)2 + 2drdT + 2dr2 + r2dΩ2

= dT 2 − dr2 − r2dΩ2(1− F (r))(dT + dr)2

= ηµνdxµdxν + (1− F (r))(dT 2 + 2xi

rdxidT + xixj

r2 dxidxj)

= ηµνdxµdxν + ϕkµkνdx

µdxν , (3.154)

where ϕ = 1− F (r) and kµ = (1, xir

), and where one can explicitly check that:

• k2 = 0: indeed, k2 = 1− xixir2 = 0.

• k ·∂(kµ) = 0: observe ∂j(1r) = −xj

r3 and ∂j(xir ) = −xixjr3 + δij

r. Clearly k ·∂(k0) = 0,

while k · ∂(ki) = xj

r(−xixj

r3 + δijr

) = 0.

This proves that a KS form is possible for every metric in the form (3.151).

In the specic case of the Schwarzschild black-hole 1 − F (r) = 2GMr

. Therefore, ifwe redene the scalar ϕ factorizing the gravitational coupling constant (κ2 = 8πG inour conventions) and the mass, i.e. ϕ = 1

4πr , while at the same time dening the gravitoneld hµν in gµν as gµν = ηµν + 2κhµν , one eventually nds:

hµν = κ

2Mϕkµkν = κ

2M

4πrkµkν . (3.155)

In order to build the corresponding gauge theory solution, we follow the prescriptionsof [19] to extract the single copy via the substitutions:

κ

2 → g, Mϕ→ ϕa = caϕ, kµkν → kµ. (3.156)

The resulting Yang-Mills eld is then:

Aaµ = gca

4πrkµ = gca

4πr(1, xir

). (3.157)

The goal of gaining a better physical interpretation of this gauge eld, we perform a


gauge transformation Aaµ → Aaµ + ∂µχa, with a time-independent scalar χ in order to

leave the zeroth component of Aaµ unaltered. It is then possible to completely eliminatethe spatial components of Aaµ choosing

∂iχa = −Aai ⇒ χa = −gc

a

4π

∫ xidxir2 = −gc

a

4π

∫ dr

r= −gc

a

4π ln(r

r0

). (3.158)

In this gauge, the single copy of a Schwarzschild black-hole turns out to be a Coulombpotential:

Aaµ =(gca

4πr ,~0). (3.159)

We can also compute the “single-copy” of the source exploiting the abelian Yang-Millsequations to extract jaν = ∂µF a

µν :

· ∂µF aµ0 = Aa0 − ∂0∂ · Aa = ∇2Aa0 = gca

4π ∇2 1r

= −gcaδ(3)(~x), (3.160)

· ∂µF aµi = Aai − ∂i∂ · Aa = 0. (3.161)

The resulting current is:

∂µF aµν = −gcavνδ(3)(~x) = jaν . (3.162)

Therefore, Tµν is a DC of jaµ with the same set of substitution rules given for the elds,except that kµkν → kµ is replaced by vµvν → vµ. The physical interpretation of jaµ isclearly that of a static color charge source located at the origin.

This result was obtained in D = 4 but it is easily generalizable to any spacetimedimensionD just by substituting 1/r → 1/rD−3 and accordingly modifying the angularmeasure.

3.3.4 The double copy for (A)dS spacetime

As an additional explicit example of the KS DC, we review the construction for the (A)dSspacetime [22]. Since the cosmological constant term can be considered as an eectiveenergy density, we can apply the KS DC construction in the presence of sources. (A)dSspacetime admits coordinates in which the line element reads as in eq. (3.151), although


with a dierent value of the function F (r) that takes the form:

F (r) =

1 + r2

l2AdS,

1− r2

l2dS,

(3.163)

with 1l2

= |Λ|3 in D = 4. With the same coordinate transformations employed in the

previous section (eqs. (3.153) and (3.154)), the (A)dS metric can be recast in the form:

ds2 = ηµνdxµdxν ∓ r2

l2(dT + dr)2 = ηµνdx

µdxν + ϕkµkνdxµdxν , (3.164)

with ϕ = r2/l2 and, as in the case of the Schwarzschild black-hole, kµ = (1, ki/r). Inthis case, the role of the gravitational charge is played by the cosmological constant, orequivalently by the factor 1/l2 = Λ

3 . The cosmological constant is related to the vacuumenergy density via Λ = 8πGρvac = κ2ρvac, therefore in this case to keep the correctmass dimensions we make the substitution ρvac → ρaYM, with ρaYM a uniform color chargedensity. All in all, with the graviton dened as before in gµν = ηµν + 2κhµν , the resultis:

hµν = 16κr

2ρvackµkν = κ

2ϕρvackµkν . (3.165)

Thus, performing the substitutions:

κ

2 → g, ρvacϕ→ ϕa = ρaYMϕ, kµkν → kµ, (3.166)

the single-copy of (A)dS spacetime is:

Aaµ = ∓gρaYMr2kµ = ∓gρaYMr

2(1, xir

). (3.167)

As in the case of the single copy of the Schwarzschild black-hole, the spatial part of thisgauge eld actually is pure gauge and as such it can be eliminated via a transformationAaµ → Aaµ + ∂µχ

a, with

∂iχa = −Aai ⇒ χa = ±gρaYM

∫rxidxi = ±gρaYM

∫r2dr = ±gρaYM

r3

3 . (3.168)


The resulting eld is an electrostatic eld with potential:

Aaµ = ∓gρaYMr2kµ = ∓gρaYMr

2(1,~0

). (3.169)

In this case we expect an extended source for the gauge eld conguration, since it is thesingle copy of a GR solution with non-vanishing energy-momentum tensor everywherein spacetime. The current can be extracted from the linearized YM equations,

· ∂µF aµ0 = Aa0 − ∂0∂ · Aa = ∇2Aa0 = ∓6gρaYM (∇2(r2) = 6), (3.170)

· ∂µF aµi = Aai − ∂i∂ · Aa = 0, (3.171)

and turns out to be:

∂µF aµν = ∓6gρaYMvν = jaν , (3.172)

with vµ = (1, 0, 0, 0). This current has the physical interpretation of a uniform distri-bution of (color) charge, with density ρa = ∓6gρaYM: it is the YM correspondent of acosmological constant term, which is a uniform energy density completely lling thespacetime.

Chapter 4Lagrangian formulation

In this chapter we collect the original results that we obtained so far in the attempt tounderstand the DC relations from the Lagrangian perspective. We shall exploit the prod-uct of two YM elds introduced in [58] and discussed in Section 3.1 in order to denethe graviton, the two-form and the scalar in terms of YM elds, to then build a suit-able quadratic Lagrangian for the corresponding DC system. Then we shall review theNoether procedure, which is a perturbative algorithm allowing to build interacting gaugetheories starting from a quadratic action with free gauge invariance, while also provid-ing some examples of concrete implementations of this method. We shall exploit theNoether procedure to build the cubic vertices for our DC quadratic Lagrangian, provingtheir consistency with the amplitude results of Section 2.4.2, 2.4.3 and comparing withthe o-shell formulation of [16], here discussed in Section 3.2.2. Last, we shall discuss thedeformation of the linear gauge transformation of Hµν which results from the Noetherprocedure, introducing the necessary non-linear deformations to the denitions of thegraviton and of the scalar eld.

4.1 Quadratic Lagrangian for the double-copy

In this section we build a quadratic Lagrangian which is the sum of the quadratic La-grangians of the gravity elds hµν , Bµν , ϕ, based on the denition of the eld Hµν interms of YM elds given in [58]. The nal result has a clear interpretation as the “square”,in a generalized sense, of two YM quadratic Lagrangians. In addition, we discuss theequations of motion and the gauge-xing which is needed to extract the physical de-grees of freedom propagated by our quadratic Lagrangian.

101

Chapter 4. Lagrangian formulation 102

Our starting point is the denition (3.1) of the DC eld Hµν , together with its gaugetransformation (3.37), that we reproduce here for the reader’s convenience:

Hµν : = Aµ ? Aν , (4.1)

δHµν = ∂µαν + ∂ναµ. (4.2)

In terms of this eld, we dened in (3.31) the graviton (hµν), the two-form (Bµν) and thescalar (ϕ):

hµν := HS

µν − γηµνϕ,

Bµν := HAµν ,

ϕ := H − ∂·∂·H ,

(4.3)

where we recall that

HSµν := 1

2H(µν) = 12(Hµν +Hνµ), (4.4)

HAµν := 1

2H[µν] = 12(Hµν −Hνµ). (4.5)

In order to build an action for Hµν we rst consider the most general quadratic La-grangian with two derivatives; taking into account that Hµν 6= Hνµ we have, up to totalderivatives:

L(H)0 = a1∂µHαβ∂

µHαβ + a2∂µHαβ∂µHβα + a3∂

αHαβ∂γHγβ + a4∂

αHαβ∂γHβγ

+ a5∂αHβα∂γH

βγ + a6∂αHαβ∂

βH + a7∂αH∂αH. (4.6)

Let us compute the gauge variation of each term separately to then impose gauge invari-ance of the full expression. Denoting with δi the variation of the term with coecientai in (4.6) and allowing for integrations by parts we obtain

· δ1 = 2∂µHαβ∂µ∂ααβ + 2∂µHαβ∂

µ∂βαα = 2∂αHαβαβ + 2∂βHαβα

α, (4.7)

· δ2 = 2∂µHαβ∂µ∂βαα + 2∂µHαβ∂

µ∂ααβ = 2∂βHαβαα + 2∂αHαβα

β, (4.8)

· δ3 = 2∂α(∂ααβ + ∂βαα)∂γHγβ = 2∂γHγβαβ + 2∂α∂β∂γHβγαα, (4.9)

· δ4 = αβ∂γHβγ + ∂α∂βαα∂γHβγ + ∂β∂γαγ∂αH

αβ +αβ∂αHαβ

= ∂βHαβ(α + α)α + ∂α∂β∂γHβγ(α + α)α, (4.10)

103 Chapter 4. Lagrangian formulation

· δ5 = 2∂α(∂βαα + ∂ααβ)∂γHβγ = 2∂α∂β∂γHβγαα + 2∂γHβγαβ, (4.11)

· δ6 = ∂α(∂ααβ + ∂βαα)∂βH + ∂αHαβ∂β(∂ · α + ∂ · α)

= ∂βH(α + α)β + ∂γ∂α∂βHαβ(α + α)γ, (4.12)

· δ7 = 2∂βH∂β(∂ · α + ∂ · α) = 2∂βH(α + α)β. (4.13)

Collecting separately the terms proportional to α and α and imposing δL(H)0 = 0 we

derive the conditions:

2a1 + 2a3 = 0,

2a1 + 2a5 = 0,

2a2 + a4 = 0,

2a3 + a4 + a6 = 0,

2a5 + a4 + a6 = 0,

a6 + 2a7 = 0.

(4.14)

The system admits a two-parameters family of solutions (a := a1, b := a2):

a3 = −a, a4 = −2b, a5 = −a, a6 = 2(a+ b), a7 = −(a+ b), (4.15)

leading to the Lagrangians:

L(H)0 (a, b) = a∂µHαβ∂

µHαβ + b∂µHαβ∂µHβα − a∂αHαβ∂γH

γβ − 2b∂αHαβ∂γHβγ

− a∂αHβα∂γHβγ + 2(a+ b)∂αHαβ∂βH − (a+ b)∂αH∂αH. (4.16)

In terms of the gravitational multiplet Hµν = hµν +Bµν + γηµνϕ one has:

L(H)0 (a, b) = (a+ b)

[∂µhαβ∂

µhαβ − 2∂ · hα∂ · hα + 2∂ · hα∂αh− ∂αh∂αh]+

+ γ(a+ b)∂µϕ[2(D − 2)∂ · hµ − 2(D − 2)∂µh− γ(D2 − 3D + 2)∂µϕ

]+

+ (a− b)[∂µBαβ∂

µBαβ − 2∂αBαβ∂γBγβ], (4.17)

where one has to keep in mind the relations connecting the eld in the scalar sector of


the theory:

h = H − γDϕ,∂·∂·h = ∂·∂·H

− γϕ,⇒ h− ∂ · ∂ · h

= [1− γ(D − 1)]ϕ, (4.18)

which we shall use many times in this chapter. In particular, integrating by parts theterms with a mixing between hµν and ϕ in (4.17) one nds

2(D − 2)∂µϕ∂ · hµ − 2(D − 2)∂µϕ∂µh = 2(D − 2)ϕ(h− ∂ · ∂ · h)

= 2(D − 2)[1− γ(D − 1)]ϕϕ, (4.19)

so that, upon substituting in (4.17), one eventually nds

L(H)0 = (a+ b)

[∂µhαβ∂

µhαβ − 2∂ · hα∂ · hα + 2∂ · hα∂αh− ∂αh∂αh]

− γ(a+ b)− 2(D − 2)[1− γ(D − 1)] + γ(D2 − 3D + 2)

∂µϕ∂

µϕ

+ (a− b)[∂µBαβ∂

µBαβ − 2∂αBαβ∂γBγβ]. (4.20)

Up to overall normalizations, (4.20) is the sum of the quadratic Lagrangians for a gravi-ton, a two-form and a scalar eld. We can exploit the parameters a and b in order to xthe canonical normalization for the graviton and the two-form:

a+ b = −1

2 ,

a− b = −12 ,

⇒ a = −12 , b = 0. (4.21)

The result is, denoting with LEH0 the quadratic part in the expansion of the Einstein-Hilbert Lagrangian and with LB0 the two-form quadratic Lagrangian (both with correctnormalizations):

L(H)0 = LEH0 + LB0 + 1

2γ(D − 2)

2− γ(D − 1)∂µϕ∂

µϕ, (4.22)

where the normalization of the scalar eld depends on the value of γ. In particular:• When γ = 0 the scalar is absent,

• When γ = 1D−2 , the normalization is D−3

2(D−2) , which has the wrong sign for D > 3and anyway does not have a clear interpretation,


• More generally, the normalization can be either positive or negative, according tothe value of γ.

Since at the level of gauge symmetries all the values of γ are equivalent, as we noticed inSection 3.1, it may be unexpected that for some values of γ the scalar is a ghost. We alsoobserved that, for some applications, two values of γ seem to be the most plausible ones:γ = 0 when dealing with the eld strength of Hµν in Section 3.1.4 and γ = 1

D−2 whendealing with the equations of motion in Section 3.1.5 and 3.1.6. In particular, γ = 1

D−2

seems to be promising also because it resonates with the decomposition of polarizationvectors of Section 2.4.1. However, the normalization of the scalar in eq. (4.22) apparentlyfavors other choices for γ.

In order to clarify this issue let us notice that, given our denition of the scalar eld,there is another term which might be included in the Lagrangian of (4.6):

a8∂ · ∂ ·H

∂ · ∂ ·H

→ δ8 = −2a8∂µ∂ · ∂ ·Hαµ − 2a8∂

µ∂ · ∂ ·Hαµ, (4.23)

leading to the most general quadratic Lagrangian where one allows non-local terms inpure gauge contributions. Including (4.23) in (4.6), the system (4.14) changes to

2a1 + 2a3 = 0,

2a1 + 2a5 = 0,

2a2 + a4 = 0,

2a3 + a4 + a6 − 2a8 = 0,

2a5 + a4 + a6 − 2a8 = 0,

a6 + 2a7 = 0,

(4.24)

that admits a three parameters family of solutions (a := a1, b := a2, c := a8):

a3 = −a, a4 = −2b, a5 = −a, a6 = 2(a+ b+ c), a7 = −(a+ b+ c). (4.25)

The resulting Lagrangian is:

L(H)0 = a∂µHαβ∂

µHαβ + b∂µHαβ∂µHβα − a∂αHαβ∂γH

γβ − 2b∂αHαβ∂γHβγ

− a∂αHβα∂γHβγ + 2(a+ b)∂αHαβ∂βH − (a+ b)∂αH∂αH


+ c− 2H∂ · ∂ ·H +HH + (∂ · ∂ ·H)2

, (4.26)

and corresponds to the Lagrangian of eq. (4.16), plus a term

c− 2H∂ · ∂ ·H +HH + (∂ · ∂ ·H)2

= cϕϕ = −c∂µϕ∂µϕ, (4.27)

tantamount to the possibility of xing the normalization of the scalar independentlyof hµν and Bµν . However, the scalar kinetic term receives dierent contributions fordierent values of γ, as evident from (4.22). Thus, keeping c arbitrary but xing a =−1

2 , b = 0 (canonical normalization for hµν and Bµν), eq. (4.26) can be written as:

L(H)0 = −1

2∂µHαβ∂µHαβ + 1

2∂αHαβ∂γH

γβ + 12∂

αHβα∂γHβγ − (1− 2c)∂αHαβ∂βH

+ 12(1− 2c)∂αH∂αH − c∂µ

(∂ · ∂ ·H)

∂µ(∂ · ∂ ·H)

. (4.28)

An interesting possibility emerges from this Lagrangian: the choice c = 12 eliminates

two of its terms,

L(H)0 = 1

2HαβHαβ − 1

2Hαβ∂α∂γH

γβ − 12Hαβ∂

β∂γHαγ + 1

2(∂ · ∂ ·H

)(∂ · ∂ ·H

)= 1

2Hαβηαµηβν − ∂α∂µηβν − ∂β∂νηαµ + ∂α∂β∂µ∂ν

Hµν , (4.29)

with the appealing feature that the term in the curly brackets can be seen as the productof two YM kinetic operators, divided by :

LLYM = Aαaηαµ − ∂α∂µ

Aµa , LRYM = Aβa′

ηβν − ∂β∂ν

Aνa′ ,

ηαµηβν − ∂α∂µηβν − ∂β∂νηαµ + ∂α∂β∂µ∂ν

=ηαµ − ∂α∂µ

1

ηβν − ∂β∂ν .

(4.30)

Moreover, using the denition Hµν = Aµ ? Aν and integrating by parts

L(H)0 = 1

2Hαβ

Hαβ − ∂α∂γHγβ − ∂β∂γHαγ + ∂α∂β

∂ · ∂ ·H

= 1

2(∂µ∂νHαβ − ∂µ∂βHαν − ∂α∂νHµβ + ∂α∂βHµν)1∂µ∂νHαβ

= 18(∂µ∂νAα ? Aβ − ∂µ∂βAα ? Aν − ∂α∂νAµ ? Aβ + ∂α∂βAµ ? Aν)


× 1

(∂µ∂νAα ? Aβ − ∂µ∂βAα ? Aν − ∂α∂νAµ ? Aβ + ∂α∂βAµ ? Aν), (4.31)

therefore, even more suggestively in terms of the (abelian) eld strengths:

L(H)0 = 1

8(Fµα ? Fνβ) 1

(F µα ? F νβ). (4.32)

Amongst all our options, the Lagrangian (4.32) has the cleanest interpretation as the“square” of two Yang-Mills quadratic Lagrangians; while the non-local factor also ac-counts for the restoration of the correct dimension for a quadratic Lagrangian. Moreover,recalling from Section 3.1.4 the denition of the curvature tensor for Hµν :

R∗µνρσ := −12Fµν ? Fρσ, (4.33)

from eq. (4.32) we conclude that this Lagrangian can be written as:

L(H)0 = 1

2R∗µνρσ

1R∗µνρσ. (4.34)

This form of the quadratic Lagrangian is strongly reminiscent of the spin-two represen-tative in the class of quadratic Lagrangians found in [129] for Maxwell-like higher spinelds. In the case of [129] the gauge potential was chosen to be totally symmetric, andthe corresponding equations of motion where shown to propagate a massless gravitontogether with a massless scalar, but of course no two-form was present in the spectrum.Dierently, in (4.34) the generalized curvature R∗µνρσ also includes a torsion which, aswe explained in Section 3.1.4, is exactly due to the presence of the two-form eld1.

Therefore, we see that the choice c = 12 in eq. (4.28) leads to a Lagrangian with a clear

geometric interpretation (eq. (4.34)) which, when expressed in terms of gauge elds, canbe seen in a sense as the “square” of two linearized YM Lagrangians in the sense of eq.(4.32).

From now on, we shall stick to the choice c = 12 , which, in addition to the appealing

1Interestingly, the geometric Lagrangians of [129] have an interpretation as sectors of the tensionlessLagrangian of free Open String Field Theory, after integration over the auxiliary elds present in thecorresponding BRST construction [130, 131], while also relating to a Lagrangian description of masslesshigher spins allowing for the propagation of reducible spectra of particles, more akin to the spirit of theDC construction under scrutiny in this work [129, 132].


geometric interpretation, turns out to be promising also for other reasons. Let us goback to the expansion Hµν = hµν + Bµν + γηµνϕ in eq. (4.29), in order to study thenormalization of the scalar for dierent values of γ:

L(H)0 = LEH0 + LB0 −

12(D − 2)(D − 1)γ2 − 2γ(D − 2) + 1∂µϕ∂µϕ. (4.35)

Where the normalization N of the scalar kinetic term is given by:

N = −12(D − 2)(D − 1)γ2 − 2γ(D − 2) + 1. (4.36)

The discriminant of the polynomial is ∆ = −4(D−2), while the coecient of γ2 is alsopositive when D > 2, we conclude that the polynomial is always positive and thereforethe sign of N is always correct. Moreover, one can notice that

• γ = 0: N = −12 . In this case, the normalization of the scalar is canonical.

• γ = 1D−2 : N = − 1

2(D−2) . In this case the normalization is not canonical, butit corresponds to the normalization of the scalar eld in the action of N = 0Supergravity, given in eq. (2.117).

4.1.1 Equations of motion

From the Lagrangian (4.29), it is possible to derive the equations of motion for the eldHµν :

Eµν := δLHδHµν

= Hµν − ∂µ∂αHαν − ∂ν∂αHµα + ∂µ∂ν∂ · ∂ ·H

= 0, (4.37)

that we now scrutinize, in order to understand whether and how they contain the equa-tions of motion for all the elds in the gravitational multiplet. To this end we can studyseparately symmetric and antisymmetric parts of the equations of motion.

First, let us consider the antisymmetric part which, upon identifying Bµν = 12H[µν],

gives indeed:

12E[µν] = Bµν − ∂µ∂αBαν + ∂ν∂

αBµα = 0, (4.38)

i.e. the equations of motion for a two-form gauge eld. Recalling that, from the denition


(4.3) of the elds in the gravitational multiplet, we can use

∂ · ∂ ·H

= H − ϕ = h+ (γD − 1)ϕ, (4.39)

the symmetric part of (4.37) gives instead

12E(µν) = hµν − ∂µ∂ · hν − ∂ν∂ · hµ + ∂µ∂νh+ γηµνϕ+ [γ(D − 2)− 1]∂µ∂νϕ = 0.

(4.40)

We consider the two values of γ which proved to be the most relevant for our discussion,with the results:

• γ = 0: we can read the equations of motion in the two equivalent ways

12E(µν) = hµν − ∂µ∂ · hν − ∂ν∂ · hµ + ∂µ∂νh− ∂µ∂νϕ (4.41)

= hµν − ∂µ∂ · hν − ∂ν∂ · hµ + ∂µ∂ν∂ · ∂ · h

, (4.42)

where from (4.41) to (4.42) we used ϕ = h − ∂·∂·h , which hold when γ = 0. The

rst part of (4.41) is equivalent to Rlin.µν (h) (apart from an overall constant), i.e. the

linearized Ricci tensor in terms of a metric uctuation hµν . However, the presenceof the term ∂µ∂νϕ does not allow the interpretation of (4.40) with γ = 0 as thelinearized equations of motion for a graviton. We can conclude that, even thoughthe symmetric part of Eµν correctly propagates the degrees of freedom of a spin-two and of a spin-zero eld (as shown in [60] for equations in the form (4.42) andin Section 4.1.2), when γ = 0 the decomposition (4.3) into graviton and scalar doesnot reproduce the linearized Einstein-Hilbert equations of motion.

• γ = 1D−2 : one of the terms containing ϕ in eq. (4.40) cancels out, for the other we

can use eq. (4.18) with the given value of γ to write γϕ = h−∂ ·∂ ·h = Rlin.(h),i.e. the linearized Ricci scalar in terms of a metric uctuation hµν . Then, eq. (4.40)can be written as

hµν − ∂(µ∂ · hν) + ∂µ∂νh− ηµν(h− ∂ · ∂ · h) = 0, (4.43)

which is equivalent to

Rlin.µν (h)− 1

2ηµνRlin.(h) = 0. (4.44)


Last, we consider the trace of (4.37), which can be expressed as follows:

E = ηµνEµν = H − ∂ · ∂ ·H = ϕ. (4.45)

The last equation exactly corresponds to the equations of motion of a free scalar eld,therefore we can state that, when γ = 1

D−2 in the decomposition of Hµν , the equationEµν = 0 actually contains the free, linearized equations of motion for the elds hµν , Bµν

and ϕ.

4.1.2 Gauge-xing and degrees of freedom

The quadratic Lagrangian (4.29) possesses a large gauge invariance, which, as discussedin Section 3.1, originates from the spin-one gauge invariance and is given by (4.2):

δHµν = ∂µαν + ∂ναµ, (4.46)

In this section we would like to illustrate two possible gauge xings, aiming to highlightfrom a dierent angle the physical meaning ofHµν . The rst is to proceed from the pointof view of gravity, splitting:

δHµν = ∂µαν + ∂ναµ

= 12∂µ(αν + αν) + 1

2∂ν(αµ + αµ) + 12∂µ(αν − αν)−

12∂ν(αµ − αµ)

= ∂(µξν) + ∂[µΛν] = δHSµν + δHA

µν , (4.47)

where, ξµ and Λµ are the gauge parameters for HSµν and HA

µν , respectively. Since we aredescribing massless particles we want to reduce the equations of motion to Hµν = 0,and this can be achieved with the covariant gauge-xing

∂γHγµ = ∂γHµγ = 0. (4.48)

In order to see that (4.48) is indeed a possible gauge-xing, it is convenient to considerseparately the semi-sum and the semi-dierence of eq. (4.48):

Sµ := 1

2∂γ(Hγµ +Hµγ)→ δSµ = ξµ + ∂µ∂ · ξ,

Aµ := 12∂

γ(Hγµ −Hµγ)→ δAµ = Λµ − ∂µ∂ · Λ.(4.49)


In order to set Sµ = 0, we most solve the dierential equation

ξµ + ∂µ∂ · ξ = Sµ, (4.50)

which can be solved since the dierential operator+∂µ∂· has no zero modes and thus,in order to eliminate both longitudinal and transverse parts of Sµ, one has to solve forordinary wave equations with sources. This eliminates D degrees of freedom of Hµν ,corresponding to the D components of ξµ needed in order to x in order to set Sµ = 0.

For the two-form HAµν , on the other hand, the dierential operator entering δHA

µν

has zero modes, but, as usual for p-forms, the gauge transformation of a two-form eld,gauge transformation itself is reducible. Indeed, a transformation with parameter Λ′µ =Λµ + ∂µχ, is the same as one without the term in χ:

δHAµν = ∂µΛ′ν − ∂νΛ′µ = ∂µ(Λµ + ∂µχ)− ∂ν(Λµ + ∂µχ) = ∂µΛν − ∂νΛµ. (4.51)

One can then exploit this “gauge-for-gauge” symmetry2 and choose χ such that ∂ ·Λ = 0,so that in order to set Aµ = 0 one must solve once again the sourced wave equation:

Λµ = Aµ ⇒ Λµ = 1Aµ, (4.52)

while also taking into account that ∂ ·A ≡ 0. Altogether, at this stage we have eliminatedD+D− 1 = 2D− 1 degrees of freedom from Hµν . The gauge (4.48) corresponds to thede-Donder gauge for the graviton (∂ · hµ = 1

2∂µh) when the latter is dened as hµν =HSµν− 1

D−2ηµνϕ, and to the Lorenz gauge for the two-form eld (∂αBαβ = 0). Moreover,it reduces the denition of the scalar to ϕ = H , as expected from the discussion ofSection 3.1.2. The equations of motion (4.37) in their turn simplify to

Hµν = 0, (4.53)

thus conrming that the theory describes massless particles. If we choose a referenceframe where pµ = (E,~0i, E) (from now on latin indices denote little group indices:i = 1, .., D − 2), the gauge xing conditions ∂γHγµ = ∂γHµγ = 0 expressed in Fourier

2Such symmetry also implies that only D− 1 degrees of freedom of Λµ can be exploited to the aim ofperforming the gauge-xing.


space can be written as follows:

H00 +HD−1,0 = 0, (a)

H0i +HD−1,i = 0, (b)

H0,D−1 +HD−1,D−1 = 0, (c)

H00 +H0,D−1 = 0, (d)

Hi0 +Hi,D−1 = 0, (e)

HD−1,0 +HD−1,D−1 = 0. (f)

(4.54)

These 2(D − 2) + 4 = 2D equations are not all independent: indeed, the last one isrelated to the others via (f) = (a) + (c) − (d). The number of actually independentequations is therefore 2D − 1, matching the degrees of freedom counting performedbefore. In this reference frame, taking into account the gauge-xing conditions (4.54),we can write the independent (at this level) components of Hµν as:

Hµν =

a0 a1 ... aD−1 −a0

b1 −b1

... Hij ...

bD−1 −bD−1

−a0 −a1 ... −aD−2 a0

,

where Hij is a (D − 2) × (D − 2) matrix. As in the case of the Maxwell theory or oflinearized gravity, we can now exploit the equations of motion and the residual gaugeinvariance in order to eliminate further components ofHµν . To this end, we observe thateq. 4.50 is unaltered if we make a gauge transformation with parameter ξµ satisfyingξµ = 0 and ∂ · ξ = 0. With such a ξ we can remove D − 1 additional components ofHµν :

· δHS00 = 2ipξ0 → ξ0 = a0

2ip sets HS00 = H00 = 0. From the system (4.54):

H00 = 0 (a)⇒ HD−1,0 = 0 (f)⇒ HD−1,D−1 = 0 (c)⇒ H0,D−1 = 0. (4.55)

· δHS0i = ipξi → ξi = ai + bi

2ip sets HS0i = 0. (4.56)

· ξD−1 is xed by ∂ · ξ = 0.


Therefore, at this stage we are removing further D − 1 degrees of freedom from Hµν ,corresponding to the independent components of a vector ξµ satisfying ∂ · ξ = 0.

Then, we can exploit the residual gauge freedom with a parameter Λµ satisfyingΛµ = 0 and ∂ · Λ = 0, after xing its own residual gauge-for-gauge freedom, whichleaves it with only D − 2 independent components, just those that are needed in orderto satisfy the condition

δHS0i = ipΛi → ξi = ai − bi

2ip sets HA0i = 0, (4.57)

which completes the elimination of the components ofHµν with at least one index equalto 0 or D − 1, resulting in the fact that Hµν propagates the same number of degrees offreedom as the SO(D − 2)-reducible tensor Hij . Indeed, since Hµν is a non-symmetricmatrix with D2 components, we obtain the result that the number of physical degreesof freedom is given by:

#physical d.o.f. = D2 − (2D − 1)− (2D − 3) = D2 − 4D + 4 = (D − 2)2

= (D − 2)(D − 1)2︸︷︷︸

graviton

+ (D − 2)(D − 3)2︸︷︷︸

two-form

+ 1︸︷︷︸scalar

. (4.58)

At this stage, the actual decomposition of Hij into three irreducible representations canbe performed as in Section 2.4.1, with the result that the Lagrangian (4.29), together withthe gauge invariance of (4.46), actually describes the physical degrees of freedom of agraviton, a two-form and a scalar.

From an alternative perspective, one can focus on the fact that both the eld Hµν isdened in terms of Yang-Mills elds as in (3.1)

Hµν = Aµ ? Aν . (4.59)

Therefore, we can x the Lorenz gauge for the YM elds

∂ · Aa = 0 = ∂ · Aa′ , (4.60)


with the result that, as in (4.48)

∂γHγµ = ∂γHµγ = 0. (4.61)

Such a gauge-xing reduces the equations of motion to Aaµ = Aa′µ = 0 on the Yang-Mills side and Hµν = 0 on the gravity side. Still, it is possible to exploit the residualgauge invariance: as a result, the two Yang-Mills elds have only D − 2 physical com-ponents, reecting into (D− 2)2 physical degrees of freedom of Hµν . Then, the decom-position into graviton, two-form and scalar directly follows, exactly as in Section 2.4.1.

4.2 Troubles with the naive inclusion of double-copy

interactions

Our construction so far led to a free DC Lagrangian which can be suggestively ex-pressed in terms of YM elds as

L0 = F µν ? Fαβ 1Fµν ? Fαβ. (4.62)

When extending the theory to the interacting level, a natural idea might be to insertin (4.62) the full YM eld strength,

F aµν = ∂µA

aν − ∂νAaµ − gfabcAbνAcν , (4.63)

so as to obtain DC gravitational interactions in the most direct possible fashion. How-ever, this solution is problematic from various perspectives, and eventually does notseem to lead to a sensible result at least if one attempts its most straightforward imple-mentation.

First, the number of derivatives in the interaction vertices would not match the two-derivative interactions of gravity. Indeed, since the non-abelian part of (4.63) does notcontain derivatives, putative interactions built in this way would not match, by con-struction, the gravitational ones.

Second, the denition of the product (3.1) itself does not allow a clean interpretation


for products between more than two YM elds. Indeed, a “cubic vertex” derived from(4.62) with the inclusion of the non-abelian part of (4.63) would have the schematic form3

S1 =∫dDx[(A1A2) ? (A1A2)](x)[A3 ? A3](x). (4.64)

One can write (4.64) in momentum space, obtaining4

S1 =∫

đp1đp2δ(D)(p1 + p2)FA1A2(p1)FA1A2(p1)A3(p2)A3(p2), (4.65)

where we used the convolution theorem. As evident, the presence of more than one eldin the factors of the ?-product leads to a vertex containing the Fourier transform of theproduct between two elds, such as FA1A2(p1) in (4.65). This can be expressed asthe convolution (in momentum space) between the two elds

FA1A2(p1) =∫đqA1(q)A2(p1 − q). (4.66)

As a result, the vertex (4.65) can be written as

S1 =∫

đp1đp2đqđkδ(D)(p1 + p2)A1(q)A2(p1 − q)A1(k)A2(p1 − k)A3(p2)A3(p2),

(4.67)

from which it is clear that only the elds A3 and A3 come in pair with the same mo-mentum, allowing the identication A3(p2)A3(p2) → H(p2), with H a new eld. Onthe contrary, an actual vertex of the typeH3 is built from ?-products involving only oneeld on each side of the ?:

∫dDx(A1 ? A1)(A2 ? A2)(A3 ? A3). (4.68)

Finally, one can notice the most apparent of the diculties: how can it be possibleto obtain a non-polynomial theory like N = 0 Supergravity by squaring theories withat most quartic interactions?5

3For the sake of simplicity we neglect color indices, derivatives and structure constants. Also, we dropthe biadjoint scalar in the ?-product, since it does not play any role in this discussion, with the result thathere ? actually coincides with a convolution.

4Using the notation đk = dDk(2π)D already introduced in Section 3.2.2 and denoting once again with

Ff(p) the Fourier transform of a function f(x).5In principle, however, one may try to make contact with polynomial formulations of gravity like, e.g.,

the Palatini formulation. See also [133].


For all these reasons it appears that the extension of the double copy at the interactingLagrangian level, if at all possible, involves additional subtleties that defy an immediateintuitive understanding and require to go through a less straightforward route.

In the next sections we shall choose an alternative, systematic approach, leading to asuccessful construction of cubic interactions for the double-copy eld Hµν , in principleamenable to be extended to all orders.

4.3 The Noether procedure

In the previous section we built a quadratic Lagrangian for the eldHµν and highlightedits relation to the YM Lagrangian together with its particle content. One guiding prin-ciple has been to ensure invariance under the abelian gauge transformation of Hµν , in-herited from those of its YM constituents.

Now we want to add interactions among these massless particles. This is typicallyaccomplished by adding to the Lagrangian polynomials in the elds and their deriva-tives of degree higher than two. In general, a given deformation of the free theory bythe inclusion of higher-order polynomials would break its free gauge-invariance, so thedicult part of the construction of interactions is to select those vertices that do notbreak the free gauge symmetry, while possibly deforming it into a non-abelian one. Asystematic algorithm which allows the extension of a quadratic action with gauge invari-ance to the interacting level is the Noether procedure, which is dened perturbatively,order by order in the number of elds (or, equivalently, in powers of the coupling con-stant).

In order to outline how the Noether procedure works in general, let us consider theaction S[ϕ] of the putative complete theory which we would like to build and let usexpand it perturbatively in the number of elds, here collectively denoted with ϕ

S[ϕ] = S0[ϕ] + gS1[ϕ] + g2S2[ϕ] + ..., (4.69)

where g is the coupling constant of the theory while Sn[φ] includes all the vertices ofthe complete action with n − 2 elds, so that for instance S0 is the free theory. Cor-


respondingly, we expand the gauge variation δϕ of the eld ϕ in the complete theoryperturbatively in the number of elds, or equivalently in the power of the coupling con-stant g:

δϕ = δ0ϕ+ gδ1ϕ+ g2δ2ϕ+ ... . (4.70)

The requirement of gauge invariance of the action, interpreted order by order in pertur-bation theory, leads to the following system of equations:

δS = 0⇒

δ0S0 = 0,

δ0S1 + δ1S0 = 0,

δ0S2 + δ1S1 + δ2S0 = 0,

...

(4.71)

which is known as the Noether system. In order to nd a solution to the system, we canobserve that

δnS0 = δS0

δϕδnϕ ≈ 0, (4.72)

where, with “≈ 0”, we denote quantities that vanish when the free equations of motionare satised. This observation allows solve the system order by order, starting from thelowest non-trivial one:

δ0S1 + δ1S0 = 0⇒ δ0S1 ≈ 0. (4.73)

Therefore, in order to build the cubic vertex, one can make an ansatz for S1 with un-determined coecients, to be xed in order to obtain a quantity whose variation underthe free gauge symmetry is zero when the free equations of motion are satised. Oncesuch a S1 is obtained, we collect the terms proportional to the equations of motion inits variation and compute the rst correction to the gauge transformation of the eldsexploiting once more the Noether equation

δ1S0 = δS0

δϕδ1ϕ = −δ0S1. (4.74)

This procedure can be carried on at every order: at the n-th step one knows all the


vertices up to Sn−1, and the equation to be solved is

δ0Sn + ...+ δn−1S1 ≈ 0, (4.75)

which allows to determine Sn. Next, collecting all the terms proportional to the freeequations of motion in δ0Sn and solving

δnS0 = −δ0Sn − ...− δn−1S1, (4.76)

it is possible to nd n-th correction to the gauge variation δnϕ.

In this section, for pedagogical purposes, we provide two examples of applicationof the Noether procedure to YM theory and General Relativity. In the rst case, theprocedure terminates with the quartic vertices, and we shall indeed build the completetheory. In the case of gravity, the Einstein-Hilbert action expanded on at backgroundis non-polynomial in the uctuation hµν , while here we will only derive the expressionfor its cubic vertex, together with the rst correction to the free gauge transformationof the graviton eld.

4.3.1 Noether procedure for Yang-Mills theory

As a rst example of how the Noether procedure works in practice, we are going to buildthe full interacting Yang-Mills theory deforming the quadratic Lagrangian for a set ofN massless, spin-one elds Aaµ (a = 1, ..., N ), with free gauge invariance δ0A

aµ = ∂µε

a.The free Lagrangian is the sum of N Maxwell Lagrangians:

L0 = 12A

aµ

(Aaµ − ∂µ∂ · Aa

), (4.77)

where the sum over the internal indices is left implicit, so that the free equations ofmotion correspond to Maxwell’s equations for each eld:

Eaµ = Aaµ − ∂µ∂ · Aa ≈ 0. (4.78)

As a next step, we make an ansatz for a vertex with three elds trying to keep at aminimum the number of derivatives involved. Since we must contract all the Lorentzindices to obtain a scalar the number of derivatives must be odd, the simplest possibility


thus being

L1 = fabcAaµ(∂µAbν)Acν , (4.79)

with fabc a set of (at this stage) arbitrary real numbers. As a rst step, we compute thefree gauge variation of (4.79), isolating terms proportional to the equations of motionvia the identity Aaµ = Eaµ + ∂µ∂ · Aa. The variation is as follows:

δ0L1 = fabc∂µεa∂µAbνAcν + fabcA

aµ∂

µ∂νεbAcν + fabcAaµ∂

µAbν∂νεc

= fabc−εa[AcνAbν + ∂µAbν∂µAcν ] + εb[Acν∂ν∂ · Aa + Aaµ∂

µ∂ · Ac + ∂ · Aa∂ · Ac]

− εc[Aaµ∂µ∂ · Ab + ∂µAbν∂

νAaµ]

= εa−fabc[AcµEbµ + Acµ∂µ∂ · Ab + ∂µAbν∂µAcν ]− fcba[Acµ∂µ∂ · Ab + ∂µA

bν∂

νAcµ]

+ fbac[Acµ∂µ∂ · Ab + Abµ∂µ∂ · Ac + ∂ · Ab∂ · Ac + ∂µA

cν∂

νAbµ]

≈ εa

(−fabc − fcba + fbac + fcab)Acµ∂µ∂ · Ab + fbac∂ · Ab∂ · Ac

− fabc∂µAbν∂µAcν + (fbac − fcba)∂µAbν∂νAcµ, (4.80)

where we have integrated by parts in order to isolate the gauge parameter and, to obtainthe last expression, we have exploited the equations of motion (4.78). Moreover, we havecollected separately the dierent tensorial structures in the last two lines: all those termsmust vanish independently, upon imposing conditions on their coecients, since mustrequire that δ0L1 ≈ 0. To this end, we make the following observations:

• There is only one term of the form ∂ ·Ab∂ ·Ac: since it is symmetric in its indices,the only way it can drop out of the result is by imposing fbac = −fcab.

• Similarly, there is only one term of the form ∂µAbν∂

µAcν : in its turn, it is symmetricunder the exchange of its indices (b, c), therefore it can be eliminated only byrequiring its coecient to be antisymmetric under the same exchange: fabc =−facb.

• The relation fbac = −fcab obtained in the rst point also implies that anotherterms vanishes due to its symmetry:

(fbac − fcba)∂µAbν∂νAcµ = −fcba∂µAbν∂νAcµ. (4.81)

Being this the only term of this type, the only way to cancel it is, as above, to


require fcba = −fbca.

• The arguments of the previous points imply that the constants fabc are completelyantisymmetric in their indices. Moreover, since there is no other possible choiceto cancel the above terms, we can conclude that the solution is unique. As a laststep, we can check that the last term remaining in (4.80) also cancel due to theantisymmetry of fabc:

−fabc − fcba + fbac + fcab = 0. (4.82)

Therefore we can conclude that the complete antisymmetry of the fabc’s is a necessaryand sucient condition for δ0L1 to vanish on shell.

Having obtained δ0L1 ≈ 0, we collect the terms proportional to the equations ofmotion Eaµ in the expression (4.80) of δ0L1 in order to derive the correction to the gaugetransformation:

δ0S1 + δ1S0 =∫ δL0

δAaµδ1A

aµ + δL1

δAaµδ0A

aµ

=∫ Eaµδ1A

aµ + δL1

δAaµ∂µε

a

∫ Eaµδ1A

aµ − ∂µ

δL1

δAaµεa

=∫ Eaµδ1A

aµ + fabcε

aEbµAcµ, (4.83)

from which, exploiting the antisymmetry of the coecients fabc one nds

δ1Aaµ = fabcε

bAcµ. (4.84)

This is the correct result for a Yang-Mills theory, therefore we don’t expect further cor-rections to the gauge transformation, although we ought to be able to recover this in-formation from the procedure.

Still, at this stage δ1L1 6= 0 (as we will show in (4.88)), therefore we must add a newterm with four elds to compensate this piece of gauge variation. To this end, we mustsolve the following equation in the Noether system:

δ2L0 + δ1L1 + δ0L2 = 0. (4.85)

Since δ1L1 contains only one derivative, we choose L2 with no derivatives, so that δ0L2


has the same number of derivatives as δ1L1. Our ansatz for the quartic vertex is therefore:

L2 = KabcdAaµAbνAcµAdν , (4.86)

with Kabcd encoding arbitrary real coecients, at this stage symmetric under the ex-changes a↔ c, b↔ d and (ab)↔ (cd). Making use of these symmetries we obtain:

δ0L2 = 2Kabcd∂µεaAbνAcµAdν + 2KabcdAaµ∂νεbAcµAdν

= −2Kabcdεa∂µAbνAcµAdν + Abν∂ · AcAdν + AbνAcµ∂µA

dν

− 2Kbacdεa∂νAbµAcµAdν + Abµ∂νAcµAdν + AbµA

cµ∂ · Ad

= −4εaKabcd(Ab · Ad)(∂ · Ac) + 2KabcdAbνAc · ∂Adν. (4.87)

The other variation to be taken into account is:

δ1L1 = −fabcfadeεdAeµ∂µAbνAcν − fabcfbdeAaµ∂µ(εdAeν)Acν − fabcfcdeAaµ∂µAbνεdAeν= −εa−fcedfeabAb · Ad∂ · Ac + (fedbfeac + fcdefeab − fcedfeab)AbνAc · ∂Adν.

(4.88)

Since there are no terms proportional to the equations of motion, we can directly look fora solution with δ2A

aµ = 0, imposing δ1L1 + δ0L2 = 0. Requiring independent tensorial

structures to independently cancel and exploiting the symmetries of fabc and Kabcd, thisleads to the system:

4Kabcd = −feabfecd,

8Kabcd = 4Kabcd + 4Kcbad = −fedbfeac − feabfecd − feabfecd,(4.89)

which is solved by:

Kabcd = −1

4feabfecd,

feabfecd + feacfedb + feadfebc = 0.(4.90)

We observe that the solution obtained for Kabcd has less symmetry than originally ex-pected, but if we go bottom up and exploit only the symmetries of the solution, every-thing is consistent either way. The second expression in (4.89) is just the Jacobi identity,i.e. the condition under which the fabc’s are the structure constants of a Lie Group.


One can also check that these results solve the entire system: δ2Aaµ = 0, δ1S2 = 0,

therefore nothing else must be added. Thus, the solution of the Noether system is

L = 1

2Aaµ

(Aaµ − ∂µ∂ · Aa

)+ fabcA

aµ(∂µAbν)Acν − 1

4feabfecdAaµA

bνA

cµAdν ,

δAaµ = ∂µεa + fabcε

bAcµ.

(4.91)

If we dene F aµν := ∂[µA

aν] − fabcAbµAcν , the Lagrangian can be written as:

L = −14F

aµνF

µνa , (4.92)

with δF aµν = −fabcεbF c

µν , which is the usual form of a Yang-Mills theory.

4.3.2 Noether procedure for graviton cubic vertex

We now move to a more complicated case, which will also serve as a “warm-up” for thecomputation of the cubic vertex for the eld Hµν in the next section. We want to buildthe cubic self-interaction vertex for a massless spin-two eld, starting from the masslessFierz-Pauli Lagrangian by means of the Noether procedure. This approach reproducesthe perturbative expansion of the Einstein-Hilbert action, modulo boundary contribu-tions.

A massless spin-two particle can be described in a covariant fashion with a symmet-ric tensor hµν , together with an equivalence relation hµν ∼ hµν + ∂(µξν), where ξµ isan arbitrary vector which serves as a gauge parameter. An action principle can be for-mulated in terms of the massless Fierz-Pauli Lagrangian, which corresponds to the mostgeneral quadratic Lagrangian with two derivatives which is compatible with the gaugeinvariance of a massless spin-two eld, and coincides with the quadratic order of theexpansion of the Einstein-Hilbert Lagrangian in terms of hµν := gµν − ηµν , up to totalderivatives:

L0 = 12h

µνhµν − 2∂µ∂ · hν + ∂µ∂νh− ηµν(h− ∂ · ∂ · h)

, (4.93)


The free equations of motion for hµν are then:

hµν − ∂(µ∂ · hν) + ∂µ∂νh− ηµν(h− ∂ · ∂ · h) = 0. (4.94)

The trace of eq. (4.94) implies

h− ∂ · ∂ · h = 0. (4.95)

Therefore, the linearized equations of motion for hµν can be equivalently written as:

Rµν := hµν − ∂(µDν) = 0, (4.96)

Dµ := ∂ · hµ −12∂µh, (4.97)

where we wrote the linearized Ricci tensor Rµν in terms of the de-Donder tensorDµ. Interms of these variables, the quadratic Lagrangian can be rewritten as:

L0 = 12h

µνRµν −

12ηµνR

, (4.98)

whose gauge invariance relies on the Bianchi identity of the linearized Einstein tensor:

δ0S0 =∫ δS0

δhµνδ0hµν = 2

∫(Rµν −

12ηµνR)∂µξν = −2

∫ξν∂µ(Rµν −

12ηµνR) = 0

due to ∂ ·Rµ = 12∂µR. (4.99)

For later convenience, we also state two useful relations:

δ0Dµ = ξµ, δ0h = 2∂ · ξ. (4.100)

We can now move to building the cubic vertex S1 for the graviton self-interaction.To this end, we make a few observations which will simplify the calculation:

• As a rst step, we build the TT (transverse, traceless) part of the cubic Lagrangian,i.e. the part which doesn’t contain traces (h) and divergences (∂·hµ, or equivalentlyDµ) of the graviton. We make an ansatz and discard all terms containing Fµν , hand Dµ: these terms will enter the procedure at dierent stages.

• When two derivatives are contracted with each other we can make use of the


following identity (where we neglect total derivative contributions):

ϕ1∂µϕ2∂µϕ3 = 1

2(ϕ1ϕ2ϕ3 − ϕ1ϕ2ϕ3 − ϕ1ϕ2ϕ3). (4.101)

This generates two types of contributions: hµν = Fµν + ∂(µDν) (or its trace)andξµ = δDµ which, looking at the TT-level, can both be neglected. At the nextstep we include terms containingDµ or h. proportional to the equations of motionFµν .

• In order to avoid ambiguities related to the freedom of integrating by parts, weemploy the so-called cyclic ansatz: in the TT Lagrangian, every derivative actingon a eld must be contracted with an index of the previous one. Since our ansatzfor S1 contains two derivatives and three elds, every term which is not cyclic canbe brought in this form via integration by parts upon discarding non-TT termsand terms proportional to the equations of motion.

Given these prescriptions, the most general TT Lagrangian with three elds and twoderivatives has the form:

LTT1 = ahµν∂µ∂νhαβh

αβ + bhµν∂µhαβ∂αhβν , (4.102)

with a, b real coecients. We can compute the variation of LTT1 under a free gauge

transformation, exploiting the above prescriptions:

· δ0(hµν∂µ∂νhαβhαβ) = 2(∂µξν∂µ∂νhαβ + hµν∂µ∂ν∂αξβ)hαβ + 2hµν∂µ∂νhαβ∂αξβ

= hαβ∂νhαβξν −ξν∂νhαβhαβ − ∂νhαβhαβξν − hαβ∂αhµν∂µ∂ν ξβ− hµν∂µ∂νξβ∂ · hβ + hµν∂µ∂νh

αβ∂αξβ, (4.103)

· δ0(hµν∂µhαβ∂αhβν) = ∂(µξν)∂µhαβ∂αhβν + hµν∂µ∂

(αξβ) + hµν∂µhαβ∂α∂(βξν)

= 12∂αhβνh

αβξν − 12ξ

νhαβ∂αhβν −12h

αβ∂αhβνξν + ∂νξµ∂µh

αβ∂αhβν

+ 12h

µνhβν∂µξβ − 1

2hβνhµν∂µξ

β − 12∂µξ

βhµνhβν − ∂µξα∂αhβν∂βhµν

− hµν∂µξα∂α∂ · hν + hµν∂µhαβ∂α∂βξν − hµν∂µ∂νhαβ∂αξβ, (4.104)

where the underlined terms provide the TT part of the variation: requiring them to mu-tually cancel xes the coecients a and b to satisfy a = b/2. In what follows, we shallwork modulo an overall constant and set a = 1/2, b = 1 and only at the end we add


the appropriate normalization and coupling constant. We shall exploit ξµ = δDµ andwork withDµ instead of ∂ · hµ = Dµ + 1

2∂µh; every time we encounterhµν we use eq.(4.96), replacing it with Rµν + ∂(µDν).

For the full computations one can consult appendix A while here we only discuss theresults. Compensating the several non-TT terms which are generated by δ0LTT

1 , we canbuild the full vertex L1, which satises δ0L1 ≈ 0:

L1 = 12h

µν∂µ∂νhαβhαβ + hµν∂µh

αβ∂αhβν −14∂ · Dhαβh

αβ − 12h

αβ∂αhDβ. (4.105)

Up to boundary terms this result matches with the order h3 expansion of the EH ac-tion derived in Section 2.3. Then, we collect all the terms proportional to the equationsof motion in order to derive the correction to the gauge transformation δ1hµν . A keyobservation is in order at this point: terms in δ0LTT1 which contain the equations ofmotion

Eαβ := Rαβ −12ηαβR, (4.106)

and can be expressed as a Eαβδ0(F (h2)αβ), with F (h2) a function quadratic in hµν , canbe absorbed via a eld redenition. Indeed, since δ0Eαβ = 0, the corresponding term inL1 is Eαβ(F (h2)αβ), and if we truncate the Lagrangian at cubic order:

L0 + L1 = 12hαβE

αβ + EαβF (h2)αβ = Eαβhαβ + F (h2)αβ = Eαβh′αβ, (4.107)

with h′αβ = hαβ + F (h2)αβ the eld redenition which removes the vertex. This showsthat cubic vertices proportional to the free equations of motion are actually “fake” in-teractions, and would produce identically zero scattering amplitudes. Of course, sucheld redenitions will aect the higher order vertices Ln with n > 2, but since in thissection we only deal with the cubic interaction of the graviton, we can safely performthe eld redenition without worrying about its consequences on the Lagrangian of thecomplete theory. With this in mind, we move to the analysis of the part of δ0L1 whichvanishes on shell. Integrating by parts in order to isolate ξµ (without derivatives acting


on it) and replacing:

Rαβ → Eαβ + 1

2ηαβR,

R→ − 2D−2E ,

(4.108)

we obtain:

δ0L1 = Eαβ∂νh

αβξν + 12h

(αν(∂β)ξν − ∂νξβ))

+ 12Eαβh

αβ∂ · ξ − 14Eαβ∂

αhξβ

− 12Eαβ∂

α(hβνξν) + 14Eαβh∂

αξβ − 12(D − 2)Eh∂ · ξ

= Eαβ∂νh

αβξν + 12h

ν(α(∂β)ξν − ∂νξβ))

+ Eαβδ0(hαβh)− 1

8(D − 2)ηαβδ0(h2)

,

(4.109)

In addition we can observe that:

12h

(αν(∂β)ξν − ∂νξβ)) = hν(α∂β)ξν −12δ0(hναhβν ). (4.110)

Therefore, (4.109) can be rewritten as:

δ0L1 = Eαβ∂νh

αβξν + hν(α∂β)ξν

+ 14Eαβδ0

hαβh− 2hανhβν −

12(D − 2)η

αβh2.

(4.111)

The second part, as previously discussed, can be reabsorbed by the free Lagrangian withthe following eld redenition:

hµν → hµν + δhµν = hµν + 14hαβh− 2hανhβν −

12(D − 2)η

αβh2. (4.112)

Therefore, the rst part denes the actual correction to the gauge transformation,

δ1hαβ = ξν∂νhαβ + ∂(αξνhβ)ν , (4.113)

which corresponds to the Lie derivative of a rank-two symmetric tensor. Finally, bycomparison with the results of Section 2.3, we can x the overall constant of the vertexby comparison with the perturbative expansion of the Einstein-Hilbert action, discussed


in Section 2.3. The result is:

L1 = κhµν∂µ∂νhαβh

αβ + 2hµν∂µhαβ∂αhβν −12∂ · Dhαβh

αβ − hαβ∂αhDβ,

(4.114)

δhµν = 2κξν∂νhαβ + ∂(αξ

νhβ)ν, (4.115)

granting for the graviton and the gauge parameter to have the correct mass dimensions:

[hµν ] = D − 22 , [ξµ] = D − 4

2 , [κ] = −D − 22 . (4.116)

4.4 Cubic vertex for the double-copy Lagrangian

In this section we shall build the cubic vertices for the eld Hµν as a deformation ofthe DC quadratic Lagrangian (4.29), by means of the Noether procedure. Also, we shallderive and discuss the corresponding deformation of the gauge transformation. First, inorder to make this section self-consistent, let us summarize the key pieces of informationabout the free theory. The Lagrangian is

L0 = 12Hαβ

Hαβ − ∂α∂γHγβ − ∂β∂γHαγ + ∂α∂β

∂ · ∂ ·H

, (4.117)

invariant under the free gauge transformation:

δ0Hµν = ∂µαν + ∂ναµ. (4.118)

The free equations of motion are

Eαβ = Hαβ − ∂α∂γHγβ − ∂β∂γHαγ + ∂α∂β∂ · ∂ ·H

= Hαβ − ∂αDβ − ∂βDα ≈ 0, (4.119)

where we have dened: Dα := ∂γHγα − 1

2∂α∂·∂·H ,

Dα := ∂γHαγ − 12∂α

∂·∂·H .

(4.120)

As in the case of gravity, we shall rst build the TT part of the vertex, thus ignoring


terms which contain H , ∂αHαβ and ∂βHαβ , such as Dµ and Dµ. Moreover, also in thiscase we shall use the cyclic ansatz for the TT vertex. The most general local6 TT cubicLagrangian in the cyclic ansatz for the eld Hµν contains ten terms7:

LTT1 = a1H

µν∂µ∂νHαβHαβ + a2H

µν∂µ∂νHαβHβα

+ a3Hµν∂µH

αβ∂αHβν + a4Hµν∂νH

αβ∂αHβµ + a5Hµν∂µH

αβ∂βHαν

+ a6Hµν∂νH

αβ∂βHαµ + a7Hµν∂µH

αβ∂αHνβ + a8Hµν∂νH

αβ∂αHµβ

+ a9Hµν∂µH

αβ∂βHνα + a10Hµν∂νH

αβ∂βHµα. (4.121)

As a rst step, we require the gauge invariance at the TT order, ignoring for the momentin the variations all the non-TT contributions. Integrating by parts and neglecting suchterms, we can isolate the gauge parameters αµ and αµ, obtaining the ten variations

· δ1TT= −αβ∂αHµν∂µ∂νH

αβ − αα∂βHµν∂µ∂νHαβ − αβ∂αHµν∂µ∂νHαβ

− αα∂βHµν∂µ∂νHαβ, (4.122)

· δ2TT= −αβ∂αHµν∂µ∂νH

βα − αα∂βHµν∂µ∂νHβα − αα∂βHµν∂µ∂νHαβ

− αβ∂αHµν∂µ∂νHαβ, (4.123)

· δ3TT= −αµ∂µ∂νHαβ∂

αHβν + αµ∂αHνβ∂ν∂µHαβ + αν∂α∂βHµν∂µHαβ

+ αβ∂αHµν∂µ∂νHαβ, (4.124)

· δ4TT= −αν∂µ∂νHαβ∂

αHβµ + αµ∂αHβν∂ν∂µHαβ + αµ∂α∂βHµν∂νHαβ



βHαν + αν∂αHµβ∂ν∂µHαβ + αν∂α∂βHµν∂µHαβ

+ αα∂βHµν∂µ∂νHαβ, (4.126)


βHαµ + αν∂αHβµ∂ν∂µHαβ + αµ∂α∂βHµν∂νHαβ



αHνβ + αµ∂βHνα∂ν∂µHαβ + αν∂α∂βHµν∂µHαβ



αHµβ + αµ∂βHαν∂ν∂µHαβ + αν∂α∂βHµν∂µHαβ


6Even if the quadratic Lagrangian is non-local, we do not include non-localities in our TT ansatz, asthey only lie in pure-gauge, unphysical sectors of the theory and as such cannot have a direct inuenceon the physical degrees of freedom, contained in the TT components of Hµν .

7Keep in mind that Hµν 6= Hνµ.



βHνα + αν∂βHµα∂ν∂µHαβ + αµ∂α∂βHµν∂νHαβ



βHµα + αν∂βHαµ∂ν∂µHαβ + αµ∂α∂βHµν∂νHαβ

+ αα∂βHµν∂µ∂νHαβ. (4.131)

Requiring that the TT part of the total variation is zero leads the following system:

−2a1 + a3 + a5 + a7 + a8 = 0,

−2a1 + a5 + a6 + a8 + a10 = 0,

−2a2 + a4 + a6 + a9 + a10 = 0,

−2a2 + a3 + a4 + a7 + a9 = 0,

a3 − a4 = 0,

a3 − a7 = 0,

a4 − a6 = 0,

a5 − a8 = 0,

a6 − a10 = 0,

a7 − a9 = 0,

a9 − a10 = 0,

(4.132)

which admits a two parameters family of solutions:

a5 = a8 = a, a3 = a4 = a6 = a7 = a9 = a10 = b, a1 = a+ b, a2 = 2b. (4.133)

Therefore, in contrast to the case of the graviton for which the same analysis at theTT level xes the coecients up to an overall constant, in this case we still have thefreedom to change a relative coecient arbitrarily, to the point of possibly “switch o”some interaction terms from the vertex. Indeed, using(4.133) we nd the following formfor the cubic TT Lagrangian

LTT1 (a, b) = a

[Hµν∂µ∂νHαβH

αβ +Hµν∂µHαβ∂βHαν +Hµν∂νH

αβ∂αHµβ

]+ b


αβ + 2Hµν∂µ∂νHαβHβα +Hµν∂µH

αβ∂αHβν

+Hµν∂νHαβ∂αHβµ +Hµν∂νH

αβ∂βHαµ +Hµν∂µHαβ∂αHνβ


+Hµν∂µHαβ∂βHνα +Hµν∂νH

αβ∂βHµα

]. (4.134)

Although, as mentioned, all possible values of a and b are possible at this level, thereis a unique choice of the parameters which reproduces the results of the DC for threeparticles amplitudes at tree level. Indeed, in the case a = 1, b = 0, the TT Lagrangian issimply:

LTT1 (a = 1, b = 0) = Hµν∂µ∂νHαβH


αβ∂αHµβ,

(4.135)

from which it is possible to compute the three particles scattering amplitudes, assigningto the eldHµν asymmetric transverse polarizations εµν , to be decomposed as in Section2.4.1. The result of the amplitude is the following:

A3 = ε1αβε

αβ2 ε3

µνpµ2p

ν2 + ε1

µ1ν1εµ1ν32 ε3

µ3ν3pν13 p

µ32 + ε1

µ1ν1εµ3ν12 ε3

µ3ν3pµ13 p

ν32 + perms. of 1,2,3,

(4.136)

which exactly matches the result of Section 2.4.2 up to the overall constant, that can bexed only moving to the quartic step of the Noether procedure.

Let us comment more on this choice of the parameters in (4.134). If we split Hµν

into its symmetric and antisymmetric parts, we can explicitly study the eect of theparameters a and b, so as to understand what are the minimal requirements which leadto the choice b = 0. To this end, we write Hµν = HS

µν + HAµν in (4.134) and we recall

that HSµν contains both the graviton and the scalar elds, while HA

µν coincides with thetwo-form eld. The result is the following

LTT1 (a, b) = (a+ 3b)

HµνS ∂µ∂νH

SαβH

αβS + 2Hµν

S ∂µHSαβ∂

αHβS ν

+ (a− b)

HµνS ∂µ∂νH

AαβH

αβA + 2Hµν

A ∂µHSαβ∂

αHβA ν

− 2HµνS ∂µH

Aαβ∂

αHβA ν − 2Hµν

A ∂µHAαβ∂

αHβS ν

. (4.137)

Since our goal is to build a DC Lagrangian, we wish to x the coecients a and b inorder for (4.137) to reproduce the results of Section 2.4.2. To this end, let us notice thatin (4.137) all the vertices containing the two-form eld (Bµν ≡ HA

µν) appear with a factor(a−b), while all the other vertices with a factor (a+3b). Therefore, in order to x a and


b, we can limit ourselves to consider the amplitudes with three gravitons (M3(GGG))and with one graviton and two two-forms (M3(GAA)). From the only requirement thatthese two amplitudes match the ones in Section 2.4.2, we derive the conditions

a+ 3b = κ,

a− b = κ,(4.138)

whose unique solution is a = κ, b = 0. Therefore, we have shown that the choice of thecoecients a and bwhich allows to reproduce the results of the DC does not involve theanalysis of amplitudes which contain the scalar eld. This is particularly important forwhat we will discuss in Section 4.4.1. For the remainder of this section let us consider,unless otherwise specied, a = 1, dropping the gravitational coupling constant for thesake of simplicity.

Let us also notice that the choice b = 0 selects of the only terms in the Lagrangianof eq. (4.121) which possess an enhanced Lorentz symmetry: indeed, the vertex of eq.(4.135) is invariant under two independent Lorentz transformations acting, respectively,on the left and on the right indices of Hµν . This is due to the fact that L-type (R-type)indices are always contracted only with other L-type (R-type) indices or with deriva-tives. Moreover, a closer inspection of the quadratic Lagrangian (4.117), shows that thesame factorization into L and R indices also holds for the free theory. This is the naturalconsistency feature of the DC construction, where the gravity amplitudes are obtainedfrom the product of two factors (the left and the right ones) which are separately Lorentzinvariant, each containing cubic vertices with exactly one derivative. The origin of sucha twofold Lorentz-invariance O(D− 1, 1)L⊗O(D− 1, 1)R can be traced back to StringTheory, and indeed a eld-theoretical linear realization of this symmetry in the low-energy eective action of the closed string was discussed already in [134–136], whilenow being studied in the context of Double Field Theory [137]. In [138], inspired by theKLT relations, a pure gravity action whose vertices respect this symmetry was built upto the quintic interactions and in [139] it was shown that it is possible to choose a eldbasis for the full EH action in which this symmetry is manifest. Also, as in the Dou-ble Field Theory and in [139], both the quadratic action (4.117) and the TT cubic vertex(4.135), are invariant under a Z2 symmetry which exchanges the L and the R indices ofthe eld Hαβ , together with the indices of the derivatives.


The emergence of this additional symmetry, together with the fact that this choicereproduces the results of the DC relations at the cubic level, gives a particular signi-cance to the case b = 0 (a = 1 still is an arbitrary choice at the cubic level), which we callthe double-copy part of the vertex. Therefore, we shall rst focus on this special case inthe following section, while in Section 4.4.3 we shall discuss the completion of the cubicvertex also including the case b 6= 0.

4.4.1 Double-copy part of the cubic vertex

As in the case of gravity, once we have xed (partially, in this case) the coecients ofthe ansatz for the TT Lagrangian with the TT part of its gauge variation, we proceedcollecting all the terms in δ0LTT

1 , and looking for possible counterterms so as to build thedouble-copy cubic vertex L1. For the details of the computation we refer to appendixB.1, while here we simply state the result:

L1 = Hµν∂µ∂νHαβHαβ +Hµν∂µH

αβ∂βHαν +Hµν∂νHαβ∂αHµβ

− 12∂ · DHαβH

αβ −Dβ∂α∂ · D

Hαβ − Dα∂β∂ · D

Hαβ. (4.139)

We remark that L1 shares the same O(D − 1, 1)L ⊗ O(D − 1, 1)R ⊗ Z2 symmetry asits TT part (eq. (4.135)). As a last step of the Noether procedure, we collect all the termsporportional to the equations of motion generated along the calculations of appendixB.1:

δ0L1 = 12Eαβ∂νHαβα

ν − ∂νEαβHαβαν + Eαβ∂µHαβαµ − ∂µEαβHαβα

µ

+ ∂βEανHαβαν − Eαβ∂βHαναν + EµνHαν∂µα

α − EανHµν∂µαα

+ ∂αEµβHαβαµ − Eαβ∂αHµβαµ + EµνHµβ∂να

β − EµβHµν∂ναβ

+ EανDααν + EανDναα − Eαβ∂α∂ · D

αβ − Eαβ∂β∂ · D

αα

= Eαβ[∂µHαβ(αµ + αµ) +Hαν(∂βαν − ∂ναβ) +Hµβ(∂ααµ − ∂µαα)

]+ Eαβδ0

(Hαβ

∂ · D

). (4.140)


Similarly to the case of the graviton vertex we notice that

Hαν(∂βαν − ∂ναβ) +Hµβ(∂ααµ − ∂µαα)

= Hαν∂β(αν + αν) +Hνβ∂α(αν + αν)− δ0(HανH βν ), (4.141)

therefore, with an appropriate eld redenition we eliminate the terms in the vertexwhich are proportional to the equations of motion:

Hµν → Hµν −HµαHαν + 1

2Hµν∂ · ∂ ·H

. (4.142)

As mentioned in Section 3.1.2, we should not worry about the presence of the possiblysingular term ∂·∂·H

, since it only aects a pure gauge sector of the theory and indeedit simply corresponds to the dierence H − ϕ, which has no reason to be singular. Bymeans of this redenition, we can recast the variation of L1 in the form:

δL1 = Eαβ∂µHαβ(αµ + αµ) +Hαν∂β(αν + αν) +Hνβ∂α(αν + αν)

, (4.143)

therefore from δ0S1 = −δ1S0 we can read the correction to the gauge transformation:

δ1Hαβ = 2ξν∂νHαβ +Hαν∂βξ

ν +Hνβ∂αξν, (4.144)

ξµ := 12(αµ + αµ). (4.145)

Finally, we can add the appropriate coupling constant, in order to match the three parti-cles scattering amplitudes of Section 2.4.2 one should multiply (4.139) and (4.144) by κ.

Having led the cubic step of the Noether procedure to completion, we can now com-pare this result with the expectation to reproduce the vertices of theN = 0 Supergravity.Then, in order to make full contact with the structure ofN = 0 Supergravity, in Section4.4.2, we shall discuss the correction to the gauge transformation of Hµν showing howto connect with the known gauge transformations of the elds in the gravitational mul-tiplet at this order.

We already observed that the double-copy vertex correctly reproduces the results ofthe DC of tree-level scattering amplitudes at the three particles level. The same ampli-tudes are also derived from theN = 0 Supergravity, as discussed in Section 2.4.3. Then,


it is natural to ask whether it is possible to show explicitly that the Lagrangian of eq.(4.139) matches the Lagrangian of the N = 0 Supergravity, if expanded perturbativelyat cubic order. In order to check this, we write Hµν = hµν + Bµν + γηµνϕ, and tryto make contact with the vertices derived in Section 2.4.3. The computations are, onceagain, rather convoluted, and require several eld redenitions which eliminate verticesproportional to the equations of motion of the various elds. Therefore, we collect themin appendix B.2, and we simply state the result here.

After some algebra, we nd that the Lagrangian of eq. (4.139) can be shown to cor-rectly match the perturbative expansion of the N = 0 Supergravity action (2.117) atthe cubic order, including the non-TT parts, but only in the case γ = 1

D−2 . This is animportant result, since it is an o-shell constructive proof (actually the rst proof, to ourknowledge) that N = 0 Supergravity actually matches the result of the DC at this or-der, with all the eld redenitions and the eld basis which lead to this result explicitlystated. The special role played by the value γ = 1

D−2 already emerged at the quadraticlevel, since it is only for this value of γ that the normalization of the scalar eld, givenin eq. (4.36), matches the normalization of the same eld in eq. (2.117).

Let us also mention an additional relevant feature of theN = 0 Supergravity action,that our Lagrangian (4.139) correctly reproduces. In the Lagrangian of N = 0 Super-gravity (2.117) the coupling between the scalar eld, the two-form and the graviton takesa very specic form involving the product

−16exp

[− 4κD − 2ϕ

]HµνλHαβγg

µαgνβgλγ. (4.146)

If we imagine to perform the Noether procedure separately for hµν ,Bµν and ϕ, the scalarcouplings, in general, would be very much unconstrained: given a scalar Lagrangiandensity, the multiplication by an arbitrary function F (ϕ) (not of ∂µϕ) would give rise toanother allowed Lagrangian density, to the extent that it would be perfectly consistentto have no cubic coupling at all with the two-form. However, implementing the Noetherprocedure on the DC eld Hµν , we reproduced at the cubic level exactly the couplingbetween ϕ and the two-form coming from the expansion of the exponential (4.146) torst order in ϕ. It would be interesting to check whether, extending the procedure tothe quartic interactions and beyond, also the following terms in the expansion of the


exponential are correctly reproduced. Let us also stress that, given the specic choice ofthe parameters made in the cubic TT Lagrangian of eq. (4.134) in order to reproduce theresults of the DC, it might seem that we have actually chosen the coupling of the scalarso as to reproduce (4.146). However, as we also stressed in Section 4.4, the value of thearbitrary parameters in (4.134) was actually xed without invoking amplitudes involv-ing the scalar, but only looking at the sector containing the graviton and the two-form.Therefore, the fact that the couplings of the scalar in our DC Lagrangian (4.139) can beviewed as a prediction of our construction.

Another relevant comparison is with the cubic vertex of the DC Lagrangian obtainedin [16], which was explicitly built to reproduce the results of the DC of scattering am-plitudes. Following [16], in Section 3.2.2 we dened a eld Hµν , which in momentumspace is the product of two YM elds, deprived of their color indices. This eld containsthe gravity elds, and we explicitly wrote its self-interacting cubic vertex, which we cancompare with (4.139). To this end, we display here the vertex for Hµν derived in Section3.2.2, and integrate it by parts in order to make contact as much as possible with eq.(4.139):

LHHH = Hµν∂µ∂νHαβHαβ −Hµν∂µH

αβ∂νHαβ +Hµν∂µHαβ∂βHαν

−Hµν∂βHαν∂αHµβ +Hµν∂νHαβ∂αHµβ −HµνHαβ∂α∂νHµβ

= Hµν(∂µ∂νHαβHαβ − ∂µHαβ∂νH

αβ) + 2Hµν∂µHαβ∂βHαν

+ 2Hµν∂νHαβ∂αHµβ −Hµν∂αHαν∂

βHµβ. (4.147)

Then, integrating by parts eq. (4.139) (with the appropriate coupling constant), the cubicvertex resulting from the Noether procedure can be written as

L1 = κ

2Hµν(∂µ∂νHαβHαβ − ∂µHαβ∂νHαβ) + 2Hµν(∂µHαβ∂βHαν

+ ∂νHαβ∂αHµβ)

+ κHαβ(DαDβ − Dα∂µHµβ −Dβ∂µHαµ). (4.148)

We conclude that our results match indeed the cubic vertices obtained in [16], when thelatter are multiplied by κ/2 (in agreement with the prescription of eq. (2.98) when n =3). However, our cubic Lagrangian also includes non-TT contributions which ensurethe gauge invariance of the theory at cubic order, whereas in [16] the issue of gaugeinvariance was not addressed at all.


4.4.2 Interpretation of δ1Hµν

We now turn to the study of the correction to the free gauge transformation δ1Hµν ,which was determined in eq. (4.144). We wish to understand how such a transforma-tion for the eld Hµν reects into the transformations of the gravitational multiplet. Inorder to accomplish this, we rst study the symmetric and antisymmetric parts of thetransformation separately, since in Section 3.1.2 this procedure directly led to the correct(linearized) gauge transformations of the graviton and the two-form (which, for the sakeof simplicity, we write here without a factor of 2κ). The result is:

δ1H

Sµν = ξ · ∂HS

µν + ∂µξαHS

αν + ∂νξαHS

µα,

δ1HAµν = ξ · ∂HA

µν + ∂µξαHA

αν − ∂νξαHAαµ.

(4.149)

As for what concerns the symmetric part, if we add it to the free gauge transformation,the total gauge variation is:

δHSµν = δ0H

Sµν + δ1H

Sµν = ∂µξν + ∂νξµ + ξ · ∂HS

µν + ∂µξαHS

αν + ∂νξαHS

µα, (4.150)

which corresponds to the Lie derivative of a metric tensor written as gµν = ηµν +HSµν

8.This matches with the analysis of Section 3.1, where we concluded that, at the linearizedlevel, everything is compatible with the fact that gµν = ηµν +HS

µν is a metric tensor.

Similarly, the antisymmetric part of eq. (4.149) corresponds to the Lie derivative of atwo-form9. This is compatible with the expectation, coming from N = 0 Supergravity,that we are building perturbatively a theory in which there is a two-form gauge eldcoupled to gravity.

8The Lie derivative of a metric tensor gµν = ηµν +HSµν is

Lξgµν = ξα∂αgµν + gµα∂νξα + gαν∂µξ

α

= ξα∂α(ηµν +HSµν) + (ηµα +HS

µα)∂νξα + (ηαν +HSαν)∂µξα

= ∂µξν + ∂νξµ + ξ · ∂HSµν + ∂µξ

αHSαν + ∂νξ

αHSµα = δHS

µν . (4.151)

9The Lie derivative of a two-form B = 12Bµνdx

µdxν is LξB = d(iξB) + iξ(dB). Therefore, the Liederivative of B reads:

LξBµν = ξ · ∂Bµν + ∂µξαBαν − ∂νξαBαµ. (4.152)


Last, we study the transformation of the scalar eld dictated by eq. (4.144). We recallthat our denition of the scalar is:

ϕ = H − ∂ · ∂ ·H

, (4.153)

which was designed in order to have a gauge-invariant scalar in the free theory. In orderto compute δ1ϕ, we rst compute separately the two variations:

δ1H = ξ · ∂H + (∂µξν + ∂νξµ)Hµν = ξ · ∂H + (∂µξν + ∂νξµ)HSµν , (4.154)

δ1∂ · ∂ ·H

= 1

(∂µ∂νξα∂αH

Sµν + 2(∂νξα + ∂αξν)∂α∂ ·HS

ν + ξα∂α∂ · ∂ ·HS

+ 2∂µ∂νξα∂µHSαν + 2ξα∂ ·HS

α +(∂µξν + ∂νξµ)HSµν

), (4.155)

Therefore, the correction to the transformation of ϕ reads

δ1ϕ = δ1H − δ1∂ · ∂ ·H

= ξ · ∂H + (∂µξν + ∂νξµ)HSµν −

1

(∂µ∂νξα∂αH

Sµν + 2(∂νξα + ∂αξν)∂α∂ ·HS

ν

+ ξα∂α∂ · ∂ ·HS + 2∂µ∂νξα∂µHSαν + 2ξα∂ ·HS

α +(∂µξν + ∂νξµ)HSµν

).

(4.156)

As evident from (4.156) δ1ϕ 6= Lξϕ, since the Lie derivative of a scalar isLξϕ = ξ ·∂ϕ.However, the transformation of a scalar eld under dieomorphisms in a theory of grav-ity must be its Lie derivative, in order to preserve the dieomorphism invariance of thefull theory. Thus, we face an apparent paradox.

As a solution, we propose to rethink the denition of the scalar ϕ. Indeed, the deni-tion (4.153) was designed somewhat “ad hoc” for the linearized theory, in the sense thatthe term ∂·∂·H

was introduced simply to compensate the free gauge variation of H , butthere is no reason to think that the same term should, in general, cancel the unwantedpart of the transformation of H when also the nonlinear part of the transformation isincluded. Since indeed this does not happen, we propose to think of the scalar in thecomplete theory as a non-linear function of Hµν , which one can reconstruct perturba-


tively. Therefore, we can expand:

ψ = ψ(1) + ψ(2) + ψ(3) + ..., (4.157)

with ψ(n) containing n powers of the eld Hµν and ψ(1) = ϕ as dened in (4.153). Then,we require the correct transformation rule under dieomorphisms for the scalar in thecomplete theory, and read this order by order:

δψ = ξ · ∂ψ ⇒

δ0ψ(1) = ξ · ∂ψ(0) = 0,

δ1ψ(1) + δ0ψ

(2) = ξ · ∂ψ(1),

δ2ψ(1) + δ1ψ

(2) + δ0ψ(3) = ξ · ∂ψ(2),

...

(4.158)

The rst line of the system is satised, by construction, by ϕ. We can now exploit thesecond line in order to determineψ(2) (then we stop, since at the cubic step of the Noetherprocedure we do not know δ2):

δ0ψ(2) = −HS

µνδ0(HµνS )− ξ · ∂∂ · ∂ ·H

+ δ1

(∂ · ∂ ·H

). (4.159)

In order to extract the value of ψ(2) from such expression, we formally integrate by partsthe inverse d’Alembertian using the relation

f(x)g(x) = 1(f(x)g(x)

)= 1

[(f

)g +

(g)f + 2∂µf∂µg

], (4.160)

so as to derive

−HSµνδ0(Hµν

S ) = −12δ0(HS

µνHµνS ) = − 1

δ0HSµνH

µνS + ∂αH

Sµν∂

αHµνS

, (4.161)

and

−ξ · ∂∂ · ∂ ·HS

= − 1

(ξα∂α∂ · ∂ ·HS +ξα∂α

∂ · ∂ ·HS

+ 2∂µ∂α∂ · ∂ ·H

S

∂µξα

).

(4.162)

This allows to sum the rst two terms on the r.h.s. of eq. (4.159) with the last one. After


some algebra, recalling that δ0HSµν = ∂(µξν), the result can be expressed as:

δ0ψ(2) = 1

δ0

(−Hµν

S HSµν + 1

2∂νHµα

S ∂αHSµν −

34∂

αHµνS ∂αH

Sµν + 2Hµν

S ∂µ∂ ·HSν

−HµνS ∂µ∂ν

∂ · ∂ ·HS

+ ∂ ·Hα

S∂ ·HSα − ∂α

∂ · ∂ ·HS

∂ ·HS

α

+ 14∂

α∂ · ∂ ·HS

∂α∂ · ∂ ·HS

). (4.163)

Interestingly, this expression is reminiscent of the second-order term in the perturbativeexpansion of the Ricci scalar of a manifold with Levi-Civita connection, in terms of ametric perturbation over at background HS

µν = gµν − ηµν . Indeed, if we expand theRicci scalar with respect to such perturbation, the rst two orders are:

R(1) = ∂ · ∂ ·HS −HS, (4.164)

R(2) = HµνS H

Sµν −

12∂

νHµαS ∂αH

Sµν + 3

4∂αHµν

S ∂αHSµν − 2Hµν

S ∂µ∂ ·HSν

+HµνS ∂µ∂νH

S − ∂ ·HαS∂ ·HS

α + ∂αHS∂ ·HSα −

14∂

αHS∂αHS. (4.165)

From the rst equation, it is tantalizing that our denition of the scalar eld can be readas:

ϕ = −R(1)

. (4.166)

Since we are looking for the full scalar ψ, which must be a scalar under dieomorphisms,we are naturally led to a geometrical guess for the general solution, in the form

χ := −R, (4.167)

where R is the Ricci scalar of the manifold, expressed in terms of the metric gµν =ηµν + HS

µν , while is the Laplace-Beltrami operator for a scalar eld, i.e. the covariantversion of the usual wave operator in at spacetime:

:= gµνDµ∂ν = gµν(∂µ∂ν − Γαµν∂α), (4.168)

with Dµ the covariant derivative on the spacetime manifold. We can expand also in


powers of HSµν , with the result:

(0) = ηµν∂µ∂ν = , (4.169)

(1) = −HµνS ∂µ∂ν − ∂ ·Hα

S∂α + 12∂

αHS∂α. (4.170)

Then, we can compare the result of eq. (4.163) with the perturbative expansion of eq.(4.167):

χ = −R

= −R(1) +R(2) + ...

+ (1) + ...

= −( 1(1 + (1) · 1

+ ...)R(1) + R(2)

+ ...

)= − 1

(R(1) −(1)R

(1)

+R(2) + ...

).

(4.171)

Therefore, from eqs. (4.165) and (4.170), the order two term in this expansion is:

χ(2) = − 1

(HµνS H

Sµν −

12∂

νHµαS ∂αH

Sµν + 3

4∂αHµν

S ∂αHSµν − 2Hµν

S ∂µ∂ ·HSν

+HµνS ∂µ∂νH

S − ∂ ·HαS∂ ·HS

α + ∂αHS∂ ·HSα −

14∂

αHS∂αHS

−HµνS ∂µ∂νH

S +HµνS ∂µ∂ν

∂ · ∂ ·HS

− ∂ ·Hα

S∂αHS

+ ∂ ·HαS∂α

∂ · ∂ ·HS

+ 1

2∂αHS∂αH

S − 12∂

αHS∂α∂ · ∂ ·HS

). (4.172)

Since δ0HS = δ0

∂·∂·HS

, one can easily check that δ0χ(2) = δ0ψ

(2), therefore at this orderwe can safely state that the non-linear correction to the dilaton is compatible with thefact that the scalar eld in the complete theory is:

ψ = R

. (4.173)

Now that we have dened a “full scalar” with the correct transformation propertiesunder dieomorphisms, we can wonder whether the denition of the graviton as:

hµν := HSµν − γηµνϕ, (4.174)

is still a good denition for every value of γ ∈ R, i.e. if it displays the expected trans-


formation rules. At the order δ1, the expected transformation for a graviton is:

δ1hµν = Lξhµν = ξ · ∂hµν + hµα∂νξα + hαν∂µξ

α. (4.175)

However, while this is indeed the transformation of HSµν , the cumbersome expression

(4.156) of δ1ϕ spoils our expectations. To solve this issue the natural attitude, followingthe interpretation of the scalar, is to assume that also the denition of the graviton interms of Hµν gets corrections in the complete theory. Thus, we are led to propose thefollowing ansatz

hµν = HSµν − γXµνψ, (4.176)

where ψ is the scalar of the complete theory, as discussed above, andXµν is a symmetricrank-two tensor, whose expression is to be determined. As in the case of ψ, we deneXµν perturbatively in powers ofHS

µν , withX(0)µν = ηµν (then, δnX(0)

µν = 0 ∀n). We requirefrom eq. (4.176) to reproduce the correct transformation properties of the graviton, butsince we only worked out the cubic order of the Noether procedure, all we can ask fromXµν is that:

δ1hµν = ξ · ∂h(1)µν + h(1)

µα∂νξα + h(1)

αν∂µξα

= δ1HSµν − γδ0X

(1)µν ψ

(0) − γX(0)µν δ0ψ

(1) − γX(0)µν δ1ψ

(0), (4.177)

where h(1)µν is dened as the order-one contribution in the expansion of eq. (4.176):

h(1)µν := HS

µν − γηµνϕ. (4.178)

We can exploit the second equation in the system (4.158), rearranging eq. (4.177) in theform:

δ1hµν = ξ · ∂(HSµν − γηµνψ(0)

)+(HSµα − γηµαψ(0)

)∂νξ

α +(HSαν − γηανψ(0)

)∂µξ

α+

+γ(∂µξν + ∂νξµ − δ0X(1)µν )ψ(0) = ξ · ∂h(1)

µν + h(1)µα∂νξ

α + h(1)αν∂µξ

α, (4.179)

which is satised if:

δ0X(1)µν = ∂µξν + ∂νξµ. (4.180)


Hence, we can conclude that:

Xµν = X(0)µν +X(1)

µν +O(H2) = ηµν +HSµν + ∆µν +O(H2), (4.181)

with δ0∆µν = 0. At this order we cannot exclude the presence of a non-vanishing ∆µν ,but we can argue that, in order for ∆µν to be gauge invariant, it must be proportional toηµνϕ. Therefore, at this order, the denition of the graviton as

hµν = HSµν − γ

[(ηµν +HS

µν)ϕ+ ηµνψ(1)], (4.182)

possesses the correct transformation property under dieomorphisms for every value ofγ. However, such a non-linear redenition of the elds might generate new cubic ver-tices (we do not care about the quartic ones at this level) when employed in the quadraticaction. This aspect may not be a problem in general, but since in the previous section weshowed that the double-copy vertex L1 for the self-interaction of Hµν contains exactlythe same (o-shell) vertices of the N = 0 Supergravity, we do not want this nice resultto be spoiled by the modied denition of the elds in the gravitational multiplet.

If we include the quadratic corrections, the decomposition (4.182) of Hµν can bewritten as:

Hµν = h(1)µν +Bµν + γηµνϕ+ γδSµν , (4.183)

with h(1)µν = HS

µν−γηµνϕ and δSµν =(X(1)µν ϕ+ηµνψ(2)

)symmetric. If we replace (4.183)

in the quadratic Lagrangian of eq. (4.117), we get a new cubic vertex:

δL(0)H = γδSαβ(ηαµηβν − ∂α∂µηβν − ∂β∂νηαµ + ∂α∂β∂µ∂ν

)(h(1)

µν +Bµν + γηµνϕ)

= γδSµν(h(1)µν − ∂µ∂ · h(1)

ν − ∂ν∂ · h(1)µ + ∂µ∂νh

(1)) + γδSϕ

− γδSµν∂µ∂ν[(h(1) − ∂ · ∂ · h(1)

)+ γϕ

]. (4.184)

The second line of eq. (4.184) contains vertices proportional to the linearized equationsof motion of the graviton (Rµν = 0) and of the scalar eld (ϕ = 0), respectively,therefore they can be trivially absorbed in the quadratic Lagrangian. The last line, how-ever, contains a term which is in general non-vanishing. At the linearized level, with


h(1)µν = HS

µν − γηµνϕ, as we already mentioned the following identity holds:

h(1) − ∂ · ∂ · h(1)

= [1− γ(D − 1)]ϕ. (4.185)

Therefore, from eq. (4.184) we can read that the quadratic correction to the denitionof the gravity elds (in particular hµν and ϕ, in this case) manifestly does not generatenew vertices if γ = 0, or:

1− γ(D − 1) + γ = 0 ⇒ γ = 1D − 2 , (4.186)

thus conrming the fact that among all possible values for γ the two choices γ = 0 andγ = 1

D−2 are somewhat special, as they allow to display in a more explicit fashion anumber of relevant features of the construction.

4.4.3 Full cubic vertex for Hµν

In the course of our construction of the cubic vertex for the DC Lagrangian in Section4.4, we found an arbitrariness in the coecients of the cubic TT vertex. However, owingto the particular features of one of the solutions, in Section 4.4.1 we discussed only whatwe called the double-copy part of the vertex, hence neglecting some terms of LTT whichare not otherwise forbidden by gauge invariance at the cubic level. Indeed, the full TTcubic Lagrangian can be written as in (4.134) that we report here for additional clarity:

LTT1 = a



αβ∂αHµβ

]+ b



αβ∂αHβν




αβ∂βHµα

], (4.187)

with arbitrary a and b. In Section 4.4.1 we only discussed the case b = 0 (a = 1 is notactually a limitation, since the cubic level does not x the overall constant, as alreadystressed). Now, we explore the completion of the full TT vertex (4.187). However, sincecalculations are very close to the ones of appendix B.1 (only the position of some indiceschanges), we shall limit ourselves to state the result.


The full o-shell cubic vertex L1, satisfying δ0L1 ≈ 0, is:

L1 = aHµν∂µ∂νHαβH


αβ∂αHµβ

− 12∂ · DHαβH



Hαβ

+ b

Hµν∂µ∂νHαβH


αβ∂αHβν




αβ∂βHµα

− 12∂ · DHαβH

αβ − ∂ · DHαβHβα −Dβ∂α∂ · D

Hαβ

− Dα∂β ∂ · D

Hαβ − 2Dβ∂α∂ · D

Hβα − 2Dα∂β ∂ · D

Hβα

. (4.188)

Then, collecting terms proportional to the free equations of motion, the complete gaugevariation of the vertex can be written as follows

δ0L1 = Eαβ

(a+ b)[∂µHαβ(αµ + αµ) +Hαν(∂βαν − ∂ναβ) +Hµβ(∂ααµ − ∂µαα)

+δ0(Hαβ∂ · D

)]

+ 2b[∂µHβα(αµ + αµ) +Hνα(∂βαν − ∂ναβ)

+Hβµ(∂ααµ − ∂µαα) + δ0(Hβα∂ · D

)]. (4.189)

As in Section 4.4.1, it is now possible to perform a eld redenition:

Hµν →Hµν + (a+ b)−HµαH

αν + 1

2Hµν∂ · ∂ ·H

(4.190)

+ 2b−HναH

αµ + 1

2Hνµ∂ · ∂ ·H

,

in order to eliminated vertices proportional to the equations of motion. Implementingthis redenition, the correction to the gauge transformation reads:

δ1Hαβ = 2(a+ b)[ξν∂νHαβ +Hαν∂βξ

ν +Hνβ∂αξν]

+ 4b[ξν∂νHβα + 2Hνα∂βξ

ν + 2Hβν∂αξν], (4.191)

again with ξµ = 12(α + α)µ. It is possible to observe that the choice a = b in eq.

(4.188) decouples the two-form because of the symmetries under exchange of indices,whereas for general b 6= a this Lagrangian produces the same type of amplitudes of thedouble-copy vertex, but with dierent relative coecients between them. We expect


the extension of the Noether procedure to the level of quartic vertices to teach us moreabout the allowed values of a and b; this happens indeed both in the case of gravityand in the case of YM theory, where consistency with the quartic vertices require thestructure constants (i.e. the couplings of the cubic vertex) to satisfy the Jacobi identity.

Chapter 5Summary and outlook

Let us summarize the results presented in this Thesis, so as to look at them from a broaderperspective.

We analyzed the product between two YM elds given in [58], nding useful to re-visit the denitions of the graviton and of the scalar eld also at the linearized level.Our conclusion, given the discussion of Section 3.1.2 on the scalar, the results on thelinearized equations of motion of Section 4.1.1 and the idea of reproducing the N = 0Supergravity o-shell cubic vertices, is that the more appropriate decomposition of Hµν

at the linearized level is:

Hµν := Aµ ? Aν := hµν +Bµν + 1D − 2ηµνϕ, (5.1)

with hµν := HS

µν − 1D−2ηµνϕ,

Bµν := HAµν ,

ϕ := H − ∂·∂·H .

(5.2)

In particular, the scalar-graviton mixing is chosen so as to reproduce the N = 0 Super-gravity action in our “canonical” basis. Given these denitions, we were able to build aquadratic Lagrangian for the eld Hµν that can be written in several forms:

L(H)0 = 1

2Hαβηαµηβν − ∂α∂µηβν − ∂β∂νηαµ + ∂α∂β∂µ∂ν

Hµν (5.3)

= 18(Fµα ? Fνβ) 1

(F µα ? F νβ) (5.4)

147

Chapter 5. Summary and outlook 148

= 12R∗µνρσ

1R∗µνρσ, (5.5)

where

Fµν = ∂µAν − ∂νAµ, (5.6)

R∗µνρσ = −12Fµν ? Fρσ = 1

2∂ν∂ρHµσ + ∂µ∂σHνρ − ∂ν∂σHµρ − ∂µ∂ρHνσ

. (5.7)

The Lagrangian (5.3) can be split as a sum of quadratic Lagrangians for a graviton, a two-form eld and a scalar, thus making its particle content explicit. In addition, it possessestwo notable features. First, the form (5.4) makes it explicit that it can be interpreted asthe “square”, in a sense which was specied, of two YM Lagrangians. Second, when ex-pressed in terms of the linearized eld strength R∗µνρσ for Hµν (5.5), our Lagrangian isanalogous to the “geometric” Maxwell-like Lagrangian derived in [60] from tensionlessstrings. In this direction, a suggestive outlook for the future is to extend R∗µνρσ at thenon-linear level, looking for a “geometric” Lagrangian for the N = 0 Supergravity.

The presence of non-localities in our Lagrangian formulation of the DC might be asource of perplexity. However, as we commented extensively in our analysis, all thesenon localities arise purely from the need to covariantly project the scalar degree of free-dom out of the double-copy eld Hµν , while also ensuring its gauge invariance. Indeed,in the other covariant approaches that we reviewed in this Thesis, aiming at an o-shellinterpretation of the double copy, non localities are always present, either as a tool toensure the CK duality to be manifest in YM amplitudes [16], or, when sources are in-cluded, in the linear equations of motion of [59].

Then, we extended our quadratic Lagrangian to the interacting level, with the in-sertion of cubic vertices for Hµν . Among the vertices allowed by the Noether proce-dure it is possible to select a sector which reproduces the amplitude results of the DCat the cubic level, while also displaying an enhanced Lorentz symmetry of the formO(D − 1, 1)L ⊗ O(D − 1, 1)R ⊗ Z2. We showed that these vertices are equivalent, upto eld redenitions, to the cubic terms in the perturbative expansion of the N = 0Supergravity Lagrangian. Furthermore, we compared our vertices with the ones foundin [16] nding agreement between the two proposals, up to longitudinal terms at thecubic level. Therefore, following the discussion of Section 4.4.1, one can say that ourresults provide a completion of what found in [16]. In this respect, the obvious exten-

149 Chapter 5. Summary and outlook

sion of our work would be to carry on the Noether procedure (at least) to the followingorder, with the inclusion of quartic vertices. Also upon comparing with [16], quartic ver-tices might tell us something about the role of the CK duality in the DC relations, sincethe rst non-trivial example of the former manifests itself for the four-point amplitude.Moreover, our work suggests the possibility of rewriting the full action of the N = 0Supergravity in terms of Hµν , which contains the complete multiplet, thus in particularmaking the O(D − 1, 1)L ⊗O(D − 1, 1)R ⊗ Z2 symmetry explicit at the action level.

The other main output of the Noether procedure at the cubic level, namely the rstcorrection to the gauge transformation of Hµν , was also analyzed. The non-linear partof the YM transformation does not contain derivatives, dierently from the gravitationalone, therefore one could expect their connection to be harder to spot (or even to denein principle) than the one between cubic vertices, where there is a natural match be-tween the number of derivatives on the two sides. Nevertheless, if the Lagrangian westarted to build is to be thought of as the “square” of two YM Lagrangians, a link be-tween the two gauge transformation is somewhat expected also at the non-linear level,and we maintain that an investigation in this direction could provide deeper insights onthe geometric origin of the DC relations.

The main lesson that one gets from the rst non-linear corrections to the gaugetransformation is the need to consistently deform the denitions of the scalar and ofthe graviton, by means of the insertion of terms which are quadratic in Hµν . Amus-ingly, we found glimpses of geometry behind this observation, since the redenition ofthe scalar eld is compatible, to this order, with the convolution of the Green’s func-tion of the Laplace-Beltrami operator with the Ricci scalar of a manifold with metricgµν = ηµν + HS

µν and Levi-Civita connection. The insertion of quartic vertices wouldprovide a validation or a refutation of this result. A hidden geometrical structure inthe denition of the scalar eld in terms of Hµν might also justify the structure of itscouplings to the other elds in the N = 0 Supergravity multiplet, which is (somewhatunexpectedly) also reproduced by the Noether procedure.

Apart from these direct extensions of this Thesis, further and more far-reaching di-rections can be envisaged as applications or continuations of our work:

Chapter 5. Summary and outlook 150

• As we stressed several times, although we focused on the DC of pure YM theory,the CK duality and the DC relations also hold for the supersymmetric extensionsof YM theory and Einstein Gravity. Our work can be viewed as an extension of[58], since we exploited the product between YM elds and the analysis of thelinearized gauge symmetries given in that work. However, since in [58] also thesupersymmetric case is considered, a natural extension of our results is to applythe same machinery to supersymmetric theories.

• As we discussed, our quadratic Lagrangian, together with the appropriate sectorof the cubic vertices, is physically equivalent to the N = 0 Supergravity one, upto the cubic interactions. This theory has been considered, although with dierentgoals, also from the Double Field Theory (DFT) perspective [135–137]. The latterdescription and ours share the idea of using a eld basis such that the three eldsof the theory are viewed as components of a unique tensor, and the comparisonof our results with the DFT ones might teach something more on the geometry ofthe theory, and possibly on the DC relations1.

• In recent years, renewed attention was paid to a rather old subject: the study ofasymptotic symmetries. It was recently shown how these symmetries can, bothin spin-one gauge theories and in General Relativity, be linked to Weinberg’s softtheorems and to the memory eect (see [57] for a review and [140] for the exten-sion to spin greater than two). The recent derivation of a unied soft theorem validfor amplitudes with gravitons, two-form elds and scalars [141], in which the DCconstruction for scattering amplitudes was also employed, calls for the study ofits asymptotic symmetry origin. Our work should be relevant in this sense sincethe study of asymptotic symmetries requires a Lagrangian formulation, or equiv-alently the knowledge of the o-shell gauge symmetry.

• Among the various extensions of the DC paradigm, we briey reviewed the clas-sical DC in Section 3.3. We consider it a particularly promising direction to the

1Fore instance, one can easily write the correction to the free gauge transformation of Hµν (4.144) ina dierent basis as follows

δ1Hαβ = (αµ + αµ)∂µHαβ +Hαν(∂βαν − ∂ναβ) +Hµβ(∂ααµ − ∂µαα), (5.8)

where in particular the dierential structure is same as the generalized Lie derivative encountered in thecontext of the DFT. Let us also observe, however, that a similar possibility exists in the case of pure gravityas well, at least to cubic order in the Noether procedure.

151 Chapter 5. Summary and outlook

goal of getting a better understanding of the DC, in view of the fact that it relatesexact solutions in YM theory and gravity. Since classical solutions are naturallyderived from a Lagrangian (via the Euler-Lagrange equations) and, at least whenthe Kerr-Schild DC is concerned, the considered solutions are in a sense “linear”(as detailed in Section 3.3), we hope that with our Lagrangian it can be possibleto discern the origin of the Kerr-Schild DC. In particular, one of its most puzzlingproperties is that in this case the DC arises in coordinate space, while the DC ofscattering amplitudes is naturally identied in momentum space.

• One of the natural motivations for the study of the DC relations, at least fromthe phenomenological perspective, is that they provide tremendous simplicationsin gravity calculations. In particular, an ongoing problem is the study of binaryblack-hole systems, for future applications to LIGO and VIRGO. In this sense, theDC was shown to work for radiative solutions in the post-Minkowskian regime[142], both at leading order [26, 27, 128] and, recently, at next-to-leading order[29], non-trivially generalizing the CK duality to the perturbative expansion ofclassical solutions. The analysis of the post-Newtonian regime [143, 144] was ad-dressed only at leading order [25], but one may hope that DC constructions, inparticular with the support of a Lagrangian formulation, could help in the futureto increase the precision of analytic calculations of interest for the gravitationalwaves observatories.

• In view of a recent work [94] generalizing the DC construction for scattering am-plitudes to the three-point amplitudes on a curved background and of the exten-sion of the classical DC to non-at backgrounds [20, 21], it is conceivable that theDC relations might be applied, in the future, to the AdS/CFT correspondence [145].

Appendices

153

Appendix ANoether procedure for gravity

A.1 Completion of the cubic vertex

We write here the full computations which led to the derivation of the graviton cubicvertex L1 in Section 4.3.2. We recall that we started from a TT Lagrangian in the cyclicansatz and we derived a constraint on its coecient, with the result:

LTT1 = 1

2hµν∂µ∂νhαβh

αβ + hµν∂µhαβ∂αhβν , (A.1)

up to an overall coecient (which we x only at the end, by comparison with the per-turbative expansion of the EH Lagrangian). After a free gauge transformation δ0hµν =∂(µξν), the variation of this Lagrangian is:

δ0LTT1 = 12Fαβ∂νh

αβξν+∂αDβ∂νhαβξν−12δ0Dν∂νhαβhαβ

:::::::::::::::::

−12∂νFαβh

αβξν

− ∂ν∂αDβhαβξν − hµν∂µ∂νξβDβ −12h

µν∂µ∂νξβ∂µh+12∂αFβνh

αβξν

+ 12∂α∂βDνh

αβξν+12∂α∂νDβh

αβξν−12δ0Dνhαβ∂αhβν−

12F

αβ∂αhβνξν

−12∂

αDβ∂αhβνξν. . . . . . . . . . . . . . . . . . .

− 12∂

βDα∂αhβνξν+12F

µνhβν∂µξβ+1

2∂µDνhβν∂µξβ

. . . . . . . . . . . . . . . . . . .

+ 12∂

νDµhβν∂µξβ−12Fβνh

µν∂µξβ − 1

2∂βDνhµν∂µξ

β − 12∂νDβh

µν∂µξβ

−12∂µδ0Dβhµνhβν − hµν∂µξα∂αDν −

12h

µν∂µξα∂α∂νh−Dµ∂µhαβ∂αξβ

:::::::::::::::

155

Appendix A. Noether procedure for gravity 156

− 12∂

µh∂µhαβ∂αξβ. (A.2)

In the previous expression, some terms were underlined in dierent ways: we collectterms with equal underlining and treat separately these terms.

· : terms proportional to the equations of motion. They will have the same underlin-

ing also in what follows since they have to be collected all together at the end of

this step ofthe procedure, for the moment we neglect them since we rst want to

build L1 such that δ0L1 ≈ 0.

· :: :− 12δ0Dν∂νhαβhαβ −Dµ∂µhαβ∂αξβ = −1

4δ0Dν∂νhαβhαβ −12D

µ∂µhαβδ0hαβ

= 14δ0(∂ · D)hαβhαβ + 1

2∂ · Dhαβδ0hαβ + 1

2hαβDµ∂µδ0hαβ

= δ0(14∂ · Dh

αβhαβ) + hαβDµ∂µ∂αξβ. (A.3)

· . . . :− 12∂

αDβ∂αhβνξν + 12∂

µDνhβν∂µξβ = −14δ0DνDβhβν +

14D

βhβνξν

+ 14hβνD

βξν + 14hβνξ

βDν −1

4Dνhβνξ

β − 14δ0DβDνhβν

= 12FβνD

βξν + 12∂βDνD

βξν + 12∂νDβD

βξν − 12δ0DνDβhβν . (A.4)

· :− 12δ0Dνhαβ∂αhβν −

12∂µδ0Dβhµνhβν = −1

2∂α(δ0Dνhβν)hαβ

= 12δ0DνhβνDβ + 1

4δ0Dνhβν∂βh. (A.5)

· : + ∂αDβ∂νhαβξν + 12∂α∂νDβh

αβξν = −∂αDβhαβ∂ · ξ − ξν∂ν∂αDβhαβ

+ 12ξ

ν∂α∂νDβhαβ = −12h

αβ∂αDβδ0h−12ξ

ν∂α∂νDβhαβ

= 12h

αβDβδ0(∂αh) + 12D

βDβδ0h+ 14∂

βhDβδ0h−12ξ

ν∂α∂νDβhαβ. (A.6)

Next, we integrate by parts all the terms in δ0LTT1 and in the expressions above whichcontain derivatives of ξ, in order to isolate the gauge parameter. The list below containsall the necessary integration by parts, together with their results (terms proportional tothe equations of motion are underlined as above):

· − hµν∂µ∂νξβDβ = −∂ · ∂ · hξβDβ − hµν∂µ∂νDβξβ − 2∂ · hµ∂µDβξβ

= −12FξβD

β − 2∂ · DξβDβ − hµν∂µ∂νDβξβ − 2Dµ∂µDβξβ − ∂µh∂µDβξβ, (A.7)

157 Appendix A. Noether procedure for gravity

· − 12h

µν∂µ∂νξβ∂βh = −1

4FξβDβ − ∂ · Dξβ∂βh

− 12h

µν∂µ∂ν∂βhξβ −Dµ∂µ∂βhξβ −

12∂

µh∂µ∂βhξβ, (A.8)

· + 12∂

νDµhβν∂µξβ = −12∂

ν∂ · Dhβνξβ −12∂

νDµ∂µhβν∂µξβ, (A.9)

· − 12∂βDνh

µν∂µξβ = 1

2hµν∂µ∂βDνξβ + 1

2∂βDνDνξβ + 1

4∂βDν∂νhξβ, (A.10)

· − 12∂νDβh

µν∂µξβ = 1

2hµν∂µ∂νDβξβ + 1

2∂νDβDνξβ + 1

4∂νDβ∂νhξβ, (A.11)

· − 12h

µν∂µξα∂αDν = hµν∂µ∂αDνξα + ∂αDνDνξα + 1

2∂αDν∂νhξα, (A.12)

· − 12h

µν∂µξα∂α∂νh = 1

2hµν∂α∂µ∂νhξ

α + 12∂α∂νhD

νξα + 14∂α∂νh∂

νhξα, (A.13)

· − 12∂

µh∂µhαβ∂αξβ = −1

4∂αδ0Dβhhαβ + 14hh

αβ∂αξβ + 14h

αβh∂αξβ

= 14δ0Dβ∂αhhαβ + 1

4δ0DβhDβ + 18δ0Dβh∂βh+ 1

4Fhαβ∂αξβ

+ 12∂ · Dh

αβ∂αξβ+14F

αβh∂αξβ + 14∂

αDβh∂αξβ + 14∂

βDαh∂αξβ

= 14δ0Dβ∂αhhαβ + 1

4hDβξβ + 1

4ξβhDβ + 1

2ξβ∂αh∂αDβ + 1

8h∂βhξβ

+ 18ξβh∂

βh+ 14ξβ∂αh∂

α∂βh+14Fh

αβ∂αξβ −12∂α∂ · Dh

αβξβ

− 12∂ · DD

βξβ −14∂ · D∂

βhξβ+14F

αβh∂αξβ −14D

βhξβ −14∂

αDβ∂αhξβ

− 14∂

β∂ · Dhξβ −14∂

βDα∂αhξβ, (A.14)

· + hαβDµ∂µ∂αξβ = ∂α∂ · Dhαβξβ + ∂µDβDµξβ + 12∂µ∂

βhDµξβ + ∂ · DDβξβ

+ 12∂ · D∂

βhξβ + ∂µhαβ∂αDµξβ, (A.15)

· + 12D

βDβδ0h = −ξα∂α(DβDβ) = −2ξαDβ∂αDβ, (A.16)

· + 14∂

βhDβδ0h = −12∂α∂βhD

βξα −12∂

βh∂αDβξα. (A.17)

Finally, we substitute these results in the expression of δ0LTT1 :

δ0LTT1 =1

2Fαβ∂νhαβξ

ν − 12∂νFαβh

αβξν + 12∂αFβνh

αβξν − 12F

αβ∂αhβνξν

+ 12F

µνhβν∂µξβ − 1

2Fβνhµν∂µξ

β − 12FD

βξβ −14F∂

βhξβ + 12FβνD

βξν

Appendix A. Noether procedure for gravity 158

+ 14FD

βξβ + 18F∂

βhξβ + 18∂βFhξ

β + 14Fh

αβ∂αξβ + 14F

αβh∂αξβ

+[12h

αβDβδ0(∂αh) + δ0(14∂ · Dhαβh

αβ + 14δ0Dνhβν∂βh+ 1

4δ0Dβ∂αhhαβ)]

+(− ∂ · DDβξβ −

12∂αh∂

αDβξβ −12∂α∂βhD

βξα − 12∂αh∂ · Dξ

α). (A.18)

The terms in the curly brackets are proportional to the equations of motions, and theywill be considered in the last step. The terms in the squared brackets can be summedand expressed as:

δ0(14∂ · Dhαβh

αβ + 12h

αβ∂αhDβ)− 12δ0h

αβ∂αhDβ

= δ0(14∂ · Dhαβh

αβ + 12h

αβ∂αhDβ)

+1

2FDβξβ + ∂ · DDβξβ + 1

2∂αh∂αDβξβ + 1

2∂α∂βhDβξα + 1

2∂αh∂ · Dξα. (A.19)

We observe see the terms in the curly brackets of eq. (A.19) exactly cancel the ones inthe round brackets of eq. (A.18). Therefore, the remaining part is:

δ0LTT1 ≈ δ0(14∂ · Dhαβh

αβ + 12h

αβ∂αhDβ). (A.20)

Thus we have identied the terms which must be added to the TT Lagrangian in or-der to have on-shell gauge invariance for L1 at the linearised level. The correspondingLagrangian is:

L1 = 12h

µν∂µ∂νhαβhαβ + hµν∂µh

αβ∂αhβν −14∂ · Dhαβh

αβ − 12h

αβ∂αhDβ. (A.21)

Appendix BNoether procedure for the

double-copy eld

B.1 Completion of the double-copy cubic vertex

In this appendix, we write the computation which leads to the completion of the double-copy cubic vertex L1, such that δ0L1 ≈ 0, starting from the TT Lagrangian:

LTT1 = Hµν∂µ∂νHαβH


αβ∂αHµβ. (B.1)

Before starting the procedure, we collect some useful identities and gauge variationswhich will be employed extensively during the computation:

· ∂ · ∂ ·H = 2∂ · D = 2∂ · D, (B.2)

· ∂αEαβ = ∂αEβα = 0, (B.3)

· H = E + 2∂ · D, (B.4)

· ∂γHγµ = Dµ + ∂µ∂ · D

, (B.5)

· ∂γHµγ = Dµ + ∂µ∂ · D

, (B.6)

· Hµν = Eµν + ∂µDν + ∂νDν , (B.7)

· αµ = δDµ + ∂µ∂ · Λ, (B.8)

· αµ = δDµ − ∂µ∂ · Λ, (B.9)

· Λµ = 12(αµ − αµ), (B.10)

159

Appendix B. Noether procedure for the double-copy eld 160

· δ0H = δ0(∂ · ∂ ·H

)= 2δ

(∂ · D

)= ∂ · α + ∂ · α. (B.11)

The variations of the terms in eq. (B.1) after a free gauge transformation are:

· δ0(Hµν∂µ∂νHαβHαβ) = (∂µαν + ∂ναµ)∂µ∂νHαβH

αβ +Hµν∂µ∂ν(∂ααβ

+ ∂βαα)Hαβ +Hµν∂µ∂νHαβ(∂ααβ + ∂βαα) = 12H

αβ∂νHαβαν

− 12δ0Dν∂νHαβH

αβ − 12∂

ν∂ · Λ∂νHαβHαβ − 1

2∂νHαβHαβαν

+ 12H

αβ∂νHαβαν − 1

2δ0Dν∂νHαβHαβ + 1

2∂ν∂ · Λ∂νHαβH

αβ

− 12∂νHαβH

αβαν −Hαβ∂αHµν∂µ∂ναβ −Hµν∂µ∂ναβ∂αH

αβ

−Hαβ∂βHµν∂µ∂ναα −Hµν∂µ∂ναα∂βH

αβ +Hµν∂µ∂νHαβ(∂ααβ + ∂βαα). (B.12)

· δ0(Hµν∂µHαβ∂βHαν) = (∂µαν + ∂ναµ)∂µHαβ∂βHαν +Hµν∂µ(∂ααβ

+ ∂βαα)∂βHαν + +Hµν∂µHαβ∂β(∂ααν + ∂ναα) = 1

2∂βHανHαβαν

− 12δ0DνHαβ∂βHαν −

12∂

ν∂ · ΛHαβ∂βHαν −12H

αβ∂βHαναν

+ ∂ναµ∂µH

αβ∂βHαν − ∂αHµν∂µαβ∂βHαν +−Hµν∂µα

β∂β∂αHαν+

+ 12H

µνHαν∂µαα − 1

2HµνHαν∂µα

α − 12H

µνHαν∂µδ0Dα

+ 12H

µνHαν∂µ∂α∂ · Λ +Hµν∂µH

αβ∂α∂βαν

− (Hµν∂µ∂νHαβ + ∂νH

µν∂µHαβ)∂βαα. (B.13)

· δ0(Hµν∂νHαβ∂αHµβ) = (∂µαν + ∂ναµ)∂νHαβ∂αHµβ +Hµν∂ν(∂ααβ

+ ∂βαα)∂αHµβ +Hµν∂νHαβ∂α(∂µαβ + ∂βαµ) = ∂µαν∂νH

αβ∂αHµβ

+ 12∂αHµβH

αβαµ − 12δ0DµHαβ∂αHµβ + 1

2∂µ∂ · ΛHαβ∂αHµβ

− 12H

αβ∂αHµβαµ + 1

2Hµν∂να

βHµβ −12H

µν∂νδ0DβHµβ

− 12H

µν∂ν∂β∂ · ΛHµβ −

12HµβH

µν∂ναβ − ∂βHµν∂να

α∂αHµβ

−Hµν∂ναα∂α∂

βHµβ +Hµν∂νHαβ∂α∂βαµ

− (Hµν∂µ∂νHαβ + ∂µH

µν∂νHαβ)∂ααβ. (B.14)

161 Appendix B. Noether procedure for the double-copy eld

Exploiting the substitutions listed in eqs. (B.2-B.11), we can derive:

δ0LTT1 = 12E

αβ∂νHαβαν+1

2∂αDβ∂νHαβα

ν + 12∂

βDα∂νHαβαν−1

2δ0Dν∂νHαβHαβ

:::::::::::::::::::

− 12∂

ν∂ · Λ∂νHαβHαβ−1

2∂νEαβHαβαν − 1

2∂ν∂αDβHαβαν − 1

2∂ν∂βDαHαβαν

+12E

αβ∂µHαβαµ+1

2∂αDβ∂µHαβα

µ + 12∂

βDα∂µHαβαµ−1

2δ0Dµ∂µHαβHαβ

:::::::::::::::::::

+ 12∂

µ∂ · Λ∂µHαβHαβ−1

2∂µEαβHαβαµ − 1

2∂µ∂αDβHαβαµ − 1

2∂µ∂βDαHαβαµ

−Hµν∂µ∂ναβ(Dβ + ∂β

∂ · D

)−Hµν∂µ∂ναα

(Dα + ∂α

∂ · D

)+1

2∂βEανHαβαν + 1

2∂α∂βDνHαβαν+1

2∂ν∂βDαHαβαν−1

2δ0DνHαβ∂βHαν

− 12∂

ν∂ · ΛHαβ∂βHαν−12E

αβ∂βHαναν − 1

2∂αDβ∂βHανα

ν−12∂

βDα∂βHαναν

. . . . . . . . . . . . . . . . . . . . .

−Hµν∂βDν∂µαβ −Hµν∂β∂ν∂ · D

∂µαβ+1

2EµνHαν∂µα

α+12∂

µDνHαν∂µαα

. . . . . . . . . . . . . . . . . . . .

+ 12∂

νDµHαν∂µαα−1

2EανHµν∂µα

α − 12∂αDνH

µν∂µαα + 1

2HµνHαν∂µ∂

α∂ · Λ

− 12∂νDαH

µν∂µαα−1

2HµνHαν∂µδ0Dα−Dµ∂µHαβ∂βαα

::::::::::::::::− ∂µ∂ · D

∂µH

αβ∂βαα

+12∂αEµβH

αβαµ+12∂µ∂αDβH

αβαµ + 12∂α∂βDµH

αβαµ−12δ0DµHαβ∂αHµβ

+ 12∂

µ∂ · ΛHαβ∂αHµβ−12E

αβ∂αHµβαµ−1

2∂αDβ∂αHµβα

µ

. . . . . . . . . . . . . . . . . . . . .− 1

2∂βDα∂αHµβα

µ

+12E

µν∂ναβHµβ + 1

2∂µDν∂ναβHµβ+1

2∂νDµ∂ναβHµβ

. . . . . . . . . . . . . . . . . . . .−1

2Hµν∂νδ0DβHµβ

− 12H

µν∂ν∂β∂ · ΛHµβ−

12EµβH

µν∂ναβ − 1

2∂µDβHµν∂να

β − 12∂βDµH

µν∂ναβ

−Hµν∂αDµ∂ναα −Hµν∂α∂µ∂ · D

∂ναα−Dν∂νHαβ∂ααβ

::::::::::::::::− ∂ν ∂ · D

∂νH

αβ∂ααβ.

(B.15)

In the previous expression, some terms were underlined in dierent ways. We collectterms with equal underlining and treat them separately:

· : terms proportional to the equations of motion. They will have the same underlin-


ing also in what follows since they have to be collected all together at the end of

this step ofthe procedure, for the moment we neglect them since we rst want to

build L1 such that δ0L1 ≈ 0.

· :: :− 12δ0Dν∂νHαβH

αβ − 12δ0Dµ∂µHαβH

αβ − Dµ∂µHαβ∂βαα −Dν∂νHαβ∂ααβ

= −14(δ0Dµ + δ0Dµ)∂µ(HαβHαβ) + ∂ · DHαβ(∂ααβ + ∂βαα) +DµHαβ∂ν∂ααβ

+ DµHαβ∂µ∂βαα = δ0(1

2∂ · DHαβHαβ

)+DµHαβ∂ν∂ααβ + DµHαβ∂µ∂βαα.

(B.16)

· . . . :− 12∂

βDα∂βHαναν + 1

2∂µDνHαν∂µα

α − 12∂

αDβ∂αHµβαµ + 1

2∂νDµ∂ναβHµβ

= −14δ0DνDαHαν −

14∂

ν∂ · ΛDαHαν

+14D

αHαναν + 1

4HανDααν

+ 14HανDναα −

14δ0DαDνHαν + 1

4∂α∂ · ΛDνHαν −

14D

νHαναα

− 14δ0DµDβHµβ + 1

4∂µ∂ · ΛDβHµβ + +1

4HµβDβαµ +1

4DβHµβα

µ

+ 14HµβDµαβ −

14D

µHµβαβ − 1

4δ0DβHµβDµ −14∂

β∂ · ΛHµβDµ

= 12EανD

ααν + 12∂αDνD

ααν + 12∂νDαD

ααν+12EανD

ναα + 12∂αDνD

ναα

+ 12∂νDαD

ναα − 12δ0DαDβHαβ −

12δ0DβDαHαβ −

12D

αHαβ∂β∂ · Λ

+ 12∂

α∂ · ΛHαβDβ. (B.17)

· :− 12δ0DνHαβ∂βHαν −

12H

µνHαν∂µδ0Dα −12H

µν∂νδ0DβHµβ

− 12δ0DµHαβ∂αHµβ = −1

2Hαβ∂β(δ0DνHαν)−

12H

αβ∂α(δ0DµHµβ)

= 12δ0DαDβHαβ + 1

2δ0DβDαHαβ + 12∂

α∂ · D

Hαβδ0Dβ

+ 12δ0DαHαβ∂

β ∂ · D

. (B.18)

· : + 12∂

αDβ∂νHαβαν + 1

2∂βDα∂νHαβα

ν + 12∂

αDβ∂µHαβαµ + 1

2∂βDα∂µHαβα

µ

+ 12∂ν∂βDαH

αβαν + 12∂µ∂αDβH

αβαµ

= −12(∂αDβ + ∂βDα)Hαβ(∂ · α + ∂ · α)− 1

2∂µ(∂αDβ + ∂βDα)Hαβ(αµ + αµ)

+ 12∂

µ∂αDβHαβαµ + 12∂

µ∂βDαHαβαµ = −(∂αDβ + ∂βDα)Hαβδ0(∂ · D

)

− 12∂

µ∂βDαHαβαµ −12∂

µ∂αDβHαβαµ = DβHαβδ0(∂α∂ · D

)


+DαHαβδ0(∂β ∂ · D

) +DβDβδ0(∂ · D

) + DαDαδ0(∂ · D

)

+ (Dα + Dα)∂α∂ · D

δ0(∂ · D

)− 12∂

µ∂βDαHαβαµ −12∂

µ∂αDβHαβαµ.

(B.19)

Terms with ∂ · Λ and terms proportional to the equations of motion will be consideredseparately. We integrate by parts all the other terms in δ0LTT1 and in the expressionsabove which contain derivatives of αµ, αµ, in order to isolate the gauge parameters andsum the variations. The list below contains the results of such integrations:

· −HµνDβ∂µ∂ναβ = −2∂ · DDβαβ −Hµν∂µ∂νDβαβ −Dν∂νDβαβ

− ∂ν ∂ · D

∂νDβαβ − Dµ∂µDβαβ − ∂µ∂ · D

∂µDβαβ, (B.20)

· −Hµν∂µ∂ναβ∂β ∂ · D

= −2∂ · D∂β ∂ · D

αβ −Hµν∂µ∂ν∂β ∂ · D

αβ

−Dν∂ν∂β∂ · D

αβ − ∂ν∂ · D

∂ν∂β∂ · D

αβ − Dµ∂µ∂β∂ · D

αβ

− ∂µ∂ · D

∂µ∂β∂ · D

αβ, (B.21)

· −HµνDα∂µ∂ναα = −2∂ · DDβαβ −Hµν∂µ∂νDβαβ −Dν∂νDβαβ

− ∂ν ∂ · D

∂νDβαβ − Dµ∂µDβαβ − ∂µ∂ · D

∂µDβαβ, (B.22)

· −Hµν∂µ∂ναβ∂β ∂ · D

= −2∂ · D∂β ∂ · D

αβ −Hµν∂µ∂ν∂β ∂ · D

αβ

−Dν∂ν∂β∂ · D

αβ − ∂ν∂ · D

∂ν∂β∂ · D

αβ − Dµ∂µ∂β∂ · D

αβ

− ∂µ∂ · D

∂µ∂β∂ · D

αβ, (B.23)

· −Hµν∂βDν∂µαβ = Dν∂βDναβ + ∂ν∂ · D

∂βDναβ +Hµν∂µ∂βDναβ, (B.24)

· −Hµν∂β∂ν∂ · D

αβ = Dν∂ν∂β∂ · D

αβ + ∂ν∂ · D

∂ν∂β∂ · D

αβ

+Hµν∂µ∂ν∂β∂ · D

αβ, (B.25)

· + 12∂

νDµHαν∂µαα = −1

2∂ν∂ · DHανα

α − 12∂

νDµ∂µHαναα, (B.26)

· − 12∂αDνH

µν∂µαα = 1

2Hµν∂µ∂αDναα + 1

2∂αDνDναα + 1

2∂αDν∂ν ∂ · D

αα,

(B.27)


· − 12∂νDαH

µν∂µαα = 1

2Hµν∂µ∂νDααα + 1

2∂νDαDναα + 1

2∂νDα∂ν ∂ · D

αα,

(B.28)

· − ∂µ∂ · D ∂µHαβ∂βαα = αα∂

µHαβ∂µ∂β∂ · D

+ αα∂µDα∂µ

∂ · D

+ αα∂µ∂ · D

∂µ∂α∂ · D

= 12δ0DαHαβ∂

β ∂ · D− 1

2∂α∂ · ΛHαβ∂

β ∂ · D

−12 ααE

αβ∂β∂ · D− 1

2 αα∂αDβ∂β

∂ · D− 1

2 αα∂βDα∂β

∂ · D

− 12 ααH

αβ∂β∂ · D + αα∂µDα∂µ

∂ · D

+ αα∂µ∂ · D

∂µ∂α∂ · D

, (B.29)

· + 12∂

µDν∂ναβHµβ = −12∂

µ∂ · DHµβαβ − 1

2∂µDν∂νHµβα

β, (B.30)

· − 12∂µDβH

µν∂ναβ = 1

2Hµν∂µ∂νDβαβ + 1

2Dµ∂µDβαβ + 1

2∂µ∂ · D

∂µDβαβ,

(B.31)

· − 12∂βDµH

µν∂ναβ = 1

2Hµν∂ν∂βDµαβ + 1

2Dµ∂βDµαβ + 1

2∂µ∂ · D

∂βDµαβ,

(B.32)

· −Hµν∂αDµ∂ναα = Dµ∂αDµαα + ∂µ∂ · D

∂αDµαα +Hµν∂α∂νDµαα, (B.33)

· −Hµν∂α∂µ∂ · D

∂ναα = Dµ∂µ∂α

∂ · D

αα + ∂µ∂ · D

∂µ∂α∂ · D

αα

+Hµν∂µ∂ν∂α∂ · D

αα, (B.34)

· − ∂ν ∂ · D ∂νHαβ∂ααβ = αβ∂

µHαβ∂µ∂α∂ · D

+ αβ∂µDβ∂µ

∂ · D

+ αβ∂µ∂ · D

∂µ∂β∂ · D

= 12δ0DβHαβ∂

α∂ · D

+ 12∂

β∂ · ΛHαβ∂α∂ · D

−12αβE

αβ∂α∂ · D− 1

2αβ∂αDβ∂β

∂ · D− 1

2αβ∂βDα∂β

∂ · D

− 12ααH

αβ∂β∂ · D + αβ∂µDβ∂µ

∂ · D

+ αβ∂µ∂ · D

∂µ∂β∂ · D

, (B.35)

· DνHαβ∂ν∂ααβ = ∂α∂ · DHαβαβ +Dν∂νDβαβ +Dν∂ν∂β∂ · D

+ ∂ · DDβαβ + ∂ · D∂β∂ · D

αβ + ∂αDν∂νHαβαβ, (B.36)

· DµHαβ∂µ∂βαα = ∂β∂ · DHαβαα + Dµ∂µDααα + Dµ∂µ∂α∂ · D

αα

+ ∂ · DDααα + ∂ · D∂α∂ · D

αα + ∂βDµ∂µHαβαα, (B.37)

· DβDβδ0(∂ · D

) = −αα∂αDβDβ − αα∂αDβDβ, (B.38)


· DαDαδ0(∂ · D

) = −αα∂αDβDβ − αα∂αDβDβ, (B.39)

· Dα∂α∂ · D δ0(∂ · D

) = −12(αα + αα)∂αDβ∂β

∂ · D− 1

2(αα + αα)Dβ∂α∂β∂ · D

,

(B.40)

· Dα∂α∂ · D δ0(∂ · D

) = −12(αα + αα)∂αDβ∂β

∂ · D− 1

2(αα + αα)Dβ∂α∂β∂ · D

.

(B.41)

Exploiting these in the expression of δ0LTT1 yields the following result (apart from termsproportional to the equations of motion):

δ0LTT1 ≈− 1

2∂ν∂ · ΛHαβ∂βHαν + 1

2∂µ∂ · ΛHαβ∂αHµβ + 1

2HµνHαν∂µ∂

α∂ · Λ

− 12H

µνHµβ∂ν∂β∂ · Λ− 1

2∂α∂ · ΛHαβ∂β

∂ · D

+ 12∂β∂ · ΛH

αβ∂α∂ · D

− 12∂

β∂ · ΛHαβDα + 12∂

α∂ · ΛHαβDβ + Λα∂α(Dβ∂β

∂ · D

)

− Λα∂α(Dβ∂β

∂ · D

)

+[δ0(1

2∂ · DHαβHαβ) +Dβδ0∂

α(∂ · D

)Hαβ

+ δ0Dβ∂α∂ · D

Hαβ + Dαδ0(∂β ∂ · D

)Hαβ + δ0Dα∂β∂ · D

Hαβ

]−(αβ∂

αDβ∂α∂ · D

+Dβ∂ · Dαβ + ∂ · D∂α∂ · D

αα −Dβ∂β∂α∂ · D

αα

+ ∂ · D∂β ∂ · D

αβ + Dα∂α∂β∂ · D

αβ + ∂βDα∂β∂ · D

αα + Dα∂ · Dαα).

(B.42)

The curly brackets contain terms with ∂ · Λ, and integrating by parts it can be shownthat they cancel exactly. The terms in the square brackets can be summed and expressedas:

δ0(1

2∂ · DHαβHαβ +Dβ∂α∂ · D

Hαβ + Dα∂β ∂ · D

Hαβ

)−Dβ∂α∂ · D

δ0Hαβ − Dα∂β

∂ · D

δ0Hαβ

= δ0(1



Hαβ

)+αβ∂

αDβ∂α∂ · D

+Dβ∂ · Dαβ + +∂ · D∂α∂ · D

αα +Dβ∂β∂α∂ · D

αα

+ ∂ · D∂β ∂ · D

αβ + Dα∂α∂β∂ · D

αβ + ∂βDα∂β∂ · D

αα + Dα∂ · Dαα. (B.43)


There is an exact cancellation between the terms in the curly brackets of eq. (B.43) andthe ones in the round brackets of eq. (B.42), hence the remaining part is:

δ0LTT1 ≈ δ0(1



Hαβ

). (B.44)

Thus, we have identied the counterterms which must be added toLTT1 and the resultingcubic vertex is:

L1 = Hµν∂µ∂νHαβHαβ +Hµν∂µH

αβ∂βHαν +Hµν∂νHαβ∂αHµβ

− 12∂ · DHαβH



Hαβ. (B.45)

B.2 Double-copy cubic vertex and N = 0 Supergravity

We now show that the double-copy cubic vertex of eq. (B.45) is equivalent, in an appro-priate basis, to the cubic vertex of theN = 0 Supergravity, including the non-TT terms.As a rst step, we decompose as usual Hµν = hµν + Bµν + γηµνϕ and consequentlyexpand L1 in terms of the gravity elds.

· Hµν∂µ∂νHαβHαβ = hµν∂µ∂νh

αβhαβ + hµν∂µ∂νBαβBαβ + γhµν∂µ∂νhϕ

+ γhµν∂µ∂νϕh+ γ2Dhµν∂µ∂νϕϕ+ γϕhαβhαβ + γϕBαβBαβ

+ γ2ϕϕh+ γ2ϕ2h+ γ3Dϕ2ϕ. (B.46)

· Hµν∂µHαβ∂βHαν +Hµν∂νH

αβ∂αHµβ = 2hµν∂µh

αβ∂βhαν + hµν∂µBαβ∂βBαν

+ γhµα∂µhαβ∂βϕ+ γhµν∂µϕ∂ · hν + γ2hµν∂µϕ∂νϕ+Bµν∂µB

αβ∂βhαν

+Bµν∂µhαβ∂βBαν + γBµα∂µBαβ∂

βϕ+ γBµν∂µϕ∂αBαν + γϕ∂µhαβ∂βhαµ

+ γϕ∂µBαβ∂βBαµ + 2γ2ϕ∂ · hβ∂βϕ+ γ3ϕ∂µϕ∂µϕ. (B.47)

All the terms in which the two-form appears an odd number of times (BBB, Bhh,

Bhϕ, Bϕϕ) cancel out in the sum for symmetry reasons.

· − 12∂ · DHαβH

αβ = −14∂ · ∂ ·HHαβH

αβ = −14(∂ · ∂ · h+ γϕ

)HαβH

αβ

=(− 1

2∂ · ∂ · h+ 14h+ 1

4(∂ · ∂ · h−h)− 14ϕ

)HαβH

αβ

= −12∂ · ∂ · hh

αβhαβ −12∂ · ∂ · hB

αβBαβ −12Dγ

2∂ · ∂ · hϕ2 − γ∂ · ∂ · hϕh

+ 14hh

αβhαβ + 14hB

αβBαβ + 14Dγ

2hϕ2 + 12γhϕh


+ 14[γ(D − 2)− 1]ϕHαβHαβ. (B.48)

We employed the identity h− ∂ · ∂ · h = [1− γ(D − 1)]ϕ.

· −Dβ∂α∂ · D Hαβ − Dα∂β∂ · D

Hαβ

= −12Hαβ

∂µH

µβ∂α∂ · ∂ ·H

+ ∂µHαµ∂β

∂ · ∂ ·H

− ∂α∂ · ∂ ·H

∂β∂ · ∂ ·H

= −hαβ∂ · hβ∂αh− (γD − 1)hαβ∂ · hβ∂αϕ−Bαβ∂µB

µβ∂αh

− (γD − 1)Bαβ∂µBµβ∂αϕ− γϕ∂ · hα∂αh− γ(γD − 1)ϕ∂ · hα∂αϕ

− γhαβ∂αϕ∂βh− γ(γD − 1)hαβ∂αϕ∂βϕ− γ2ϕ∂αϕ∂αh+

− γ2(γD − 1)ϕ∂αϕ∂αϕ+ 12hαβ∂

αh∂βh+ (γD − 1)hαβ∂αh∂βϕ

+ 12(γD − 1)2hαβ∂

αϕ∂βϕ+ 12γϕ∂

αh∂αh+ γ(γD − 1)ϕ∂αh∂αϕ

+ 12γ(γD − 1)2ϕ∂αϕ∂αϕ. (B.49)

Again, all the terms in which the two-form appears an odd number of times cancel

out in the sum for symmetry reasons. Moreover, we employed the identity (already

introduced in Section 4.1):∂ · ∂ ·H

= h+ (γD − 1)ϕ. (B.50)

We recall, once again, that interaction vertices proportional to the equations of motionare actually fake interactions, which can be eliminated by means of appropriate eldredenitions. We will exploit this fact several times in what follows, therefore we remindthat the linearized equations of motion for the gravity elds are:

hµν − ∂µ∂ · hν − ∂ν∂ · hµ + ∂µ∂νh = 0,

Bµν − ∂µ∂αBαν + ∂ν∂αBαµ,

ϕ = 0.

(B.51)

Thus, for instance, all the terms proportional to ϕ are fake interactions, and we willneglect them from now on. In order to compare the vertices we obtained with the ones ofN = 0 Supergravity, we collect separately interactions of dierent type and exploitingthe freedom to integrate by parts we rearrange their expressions in order to make themas similar as possible to the vertices derived in Section 2.4.3.


• ϕϕϕ: all the terms of this kind are either ∼ ϕ2ϕ, which is proportional to theequations of motion of ϕ, or ∼ ϕ∂µϕ∂µϕ, which is again in the form ∼ ϕ2ϕ ifintegrated by parts. Therefore, there are no interactions among three scalars ϕ.

• hhh: the Lagrangian vertices are:

Lhhh1 = hµν∂µ∂νhαβhαβ + 2hµν∂µhαβ∂αhβν − hαβ∂αh∂ · hβ

− 12hαβ∂

αh∂βh− 14∂ · ∂ · hhαβh

αβ. (B.52)

The last term can be rewritten as:

−14∂ · ∂ · hhαβh

αβ =[− ∂ · ∂ · h+ 1

2h−12(h− ∂ · ∂ · h)

]hαβhαβ. (B.53)

Sinceh−∂ ·∂ ·h ∝ ϕ, we can neglect it, and what remains is exactly the cubicvertex of the EH Lagrangian derived in appendix A.

• BBϕ: the Lagrangian vertices are:

LBBϕ1 = γϕBαβBαβ + 2γBµα∂µBαβ∂βϕ+ 2γBµν∂µϕ∂

αBαν

+ 2γϕ∂µBαβ∂βBαµ − (γD − 1)Bαβ∂µBµβ∂αϕ = ϕ−4γ∂µBαβ∂βBµα

+Bαβ[γBαβ + (γD − 1)∂α∂µBµβ] + [(γD − 1)− 2γ]∂αBαβ∂µBµβ.(B.54)

By comparison with the expansion ofN = 0 Supergravity action in Section 2.4.3,we can observe that the last term should cancel, and this is possible if

γD − 1− 2γ = 0 ⇒ γ = 1D − 2 . (B.55)

Moreover, making use of the equations of motion for Bµν (we neglect verticesproportional to the free equations of motion):

ϕBαβ∂α∂µBµβ = BαβBαβ, (B.56)

and

ϕBαβBαβ = −∂µϕ∂µBαβBαβ − ϕ∂µBαβ∂µBαβ, (B.57)


but since

∂µϕ∂µBαβB

αβ = −12ϕB

αβBαβ, (B.58)

we can ignore this term (it contains ϕ). Summarizing, with γ = 1D−2 , we end up

with:

LBBϕ1 = − 2D − 2ϕ∂

µBαβ∂µBαβ + 2∂µBαβ∂βBµα. (B.59)

From now on, we will always use γ = 1D−2 , since also in the next cases it is the

value which reproduces the correct vertices.

• hhϕ: the Lagrangian vertices are:

Lhhϕ1 = γhµν∂µ∂νhϕ+ γhµν∂µ∂νϕh+ γϕhαβhαβ + 2γhµα∂µhαβ∂βϕ

+ 2γhµν∂µϕ∂ · hν + 2γϕ∂µhαβ∂βhαµ − γ∂ · ∂ · hϕh+ 12γhϕh

− (γD − 1)hαβ∂ · hβ∂αϕ− γϕ∂ · hα∂αh− γhαβ∂αϕ∂βh

+ (γD − 1)hαβ∂αh∂βϕ+ 12γϕ∂

αh∂αh = 1D − 2ϕ

hµν [hµν

− ∂(µ∂ · hν) + ∂µ∂νh] + 12hh+ 1

2∂αh∂αh

. (B.60)

The rst term in the curly brackets contains the linearized equations of motion ofhµν , therefore we neglect it. For the remaining part, we observe that

12ϕ∂

αh∂αh = 14ϕh

2 − 12ϕhh. (B.61)

Therefore, neglecting also the term proportional to ϕ, the vertex reduces toL1(hhϕ) = 0.

• ϕϕh: the Lagrangian vertices are:

Lϕϕh1 = γ2Dhµν∂µ∂νϕϕ+ γ2ϕϕh+ γ2ϕ2h+ 4γ2ϕ∂ · hβ∂βϕ

+ 2γ2hµν∂µϕ∂νϕ−12Dγ

2∂ · ∂ · hϕ2 + 14Dγ

2hϕ2 − γ(γD − 1)ϕ∂ · hα∂αϕ

− γ(γD − 1)hαβ∂αϕ∂βϕ− γ2ϕ∂αϕ∂αh+ 1

2(γD − 1)2hαβ∂αϕ∂βϕ

+ γ(γD − 1)ϕ∂αh∂αϕ = 1(D − 2)2−(D − 2)hµν∂µϕ∂νϕ


− 12(D + 2)h∂αϕ∂αϕ− ϕ2h = 1

D − 2(− hµν + 1

2ηµνh

)∂µϕ∂νϕ.

(B.62)

• BBh: the Lagrangian vertices are:

LBBh1 = hµν∂µ∂νBαβBαβ + 2hµν∂µBαβ∂βBαν + 2Bµν∂µB

αβ∂βhαν

+ 2Bµν∂µhαβ∂βBαν −

12∂ · ∂ · hB

αβBαβ + 14hB

αβBαβ

−Bαβ∂µBµβ∂αh = −2hµν

[Bαν∂

α∂βBµβ +Bαβ∂α∂νBµβ

]+ hµν

[− 2∂µBαβ∂βBνα − 2∂βBαν∂

αB βµ − 2∂αBαβ∂νBµβ − ∂µBαβ∂νBαβ

]+ h

[12B

αβ(Bαβ + 2∂α∂µBµβ) + 12∂µBαβ∂

µBαβ + ∂αBαβ∂µBµβ].

(B.63)

In the second square bracket, we integrate one term by parts:

−2hµν∂αBαβ∂νBµβ = 2∂αhµνBαβ∂νBµβ + 2hµνBαβ∂α∂νBµβ. (B.64)

With this, the vertex can be written as:

LBBh1 = h[12∂µBαβ∂

µBαβ + ∂αBµβ∂µBαβ]

+ hµν[− 4∂µBαβ∂βBνα

− 2∂βBαν∂αB β

µ − 2∂αBµβ∂αBβν − ∂µBαβ∂νBαβ

]+

2∂αhµνBαβ∂νBµβ − 2hµνBαν∂α∂βBµβ + 2hµν∂µBαβ∂βBνα

+ 2hµν∂αBµβ∂αBβν − h∂µBαβ∂

αBβµ + 12hB

αβBαβ

+ hBαβ∂α∂µBµβ + h∂αBαβ∂µB

µβ. (B.65)

The terms in the square brackets match with the expected vertex ofN = 0 Super-gravity, therefore we must show that all the terms in the curly brackets cancel. tothis end, we perform the following integrations by parts:

?− h∂µBαβ∂αBβµ = ∂αh∂

µBαβBµβ + h∂µ∂αBαβB

βµ

= −∂α∂µhBαβBβµ − ∂αhBαβ∂µBβµ + h∂µ∂αBαβB

βµ.

? h∂αBαβ∂µBµβ = −∂αhBαβ∂µB

µβ − hBαβ∂α∂µBµβ

= ∂µ∂αhBαβBµβ + ∂αh∂

µBαβBµβ − hBαβ∂α∂µBµβ. (B.66)


? 2∂αhµνBαβ∂νBµβ = −2∂α∂ · hµBαβBµβ − 2∂αhµν∂νBαβBµβ. (B.67)

? 2hµν∂αBµβ∂αBβν = hµνBµβB

βν − 2hµνBµβB

βν . (B.68)

With these replacements, the terms which should cancel in eq. (B.65) become:

... = BµβBνβ[hµν − ∂(µ∂ · hν) + ∂µ∂νh] + 1

2hBαβ[Bαβ − ∂[α|∂

µBµ|β]]

+∂µ∂νhB

µβBνβ + ∂νh∂µ(BµβBν

β)− 2hµνB β

ν

(Bµβ + ∂β∂

αBαµ

)− 2∂αhµν∂νBαβBµβ + 2hµν∂µBαβ∂

βB αν . (B.69)

The terms in the square brackets of the rst line contain respectively the equationsof motion of hµν andBµν , therefore we neglect them, moreover the curly bracketsis identically zero if the second term inside it is integrated by parts. With anotherintegration by parts:

−2∂αhµν∂νBαβBµβ = −2hµν∂µBαβ∂βB α

ν + 2hµνB βν ∂µ∂

αBαβ. (B.70)

With this, the last term cancels and the only leftover is:

−2hµνB βν (Bµβ + ∂β∂

αBαµ − ∂µ∂αBαβ), (B.71)

which contains the equations of motion of Bµν and is therefore another fake in-teraction. This completes the proof that this vertex is equivalent, modulo eldredenitions, to:

L1BBh = h

[12∂µBαβ∂

µBαβ + ∂αBµβ∂µBαβ]

+ hµν[− 4∂µBαβ∂βBνα − 2∂βBαν∂

αB βµ − 2∂αBµβ∂αB

βν − ∂µBαβ∂νBαβ

].

(B.72)

Summarizing, the double-copy cubic vertex (B.45) is equivalent to the cubic verticesof the N = 0 Supergravity when Hµν is expanded as:

Hµν = hµν +Bµν + 1D − 2ηµνϕ, (B.73)

modulo eld redenitions and the inclusion of the gravitational coupling constant κ infront of the cubic interactions.

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