The Duality Between Color and Kinematics and its Applications
Zvi Bern, John Joseph Carrasco, Marco Chiodaroli, Henrik Johansson, Radu Roiban
UUCLA/TEP/2019/104 CERN-TH-2019-135 NUHEP-TH/19-11UUITP-35/19 NORDITA 2019-079
The Duality Between Color and Kinematicsand its Applications
Zvi Bern, ab John Joseph Carrasco, cd Marco Chiodaroli, e Henrik Johansson, ef Radu Roiban g a Mani L. Bhaumik Institute for Theoretical Physics,Department of Physics and Astronomy, UCLA, Los Angeles, CA 90095 b Theoretical Physics Department, CERN,1211 Geneva 23, Switzerland c Department of Physics and AstronomyNorthwestern University, Evanston, IL 60208, USA d Institute of Theoretical Physics (IPhT),CEA/CNRS-Saclay and University of Paris-SaclayF-91191 Gif-sur-Yvette cedex, France e Department of Physics and Astronomy,Uppsala University, 75108 Uppsala, Sweden f Nordita, Stockholm University and KTH Royal Institute of Technology,Roslagstullsbacken 23, 10691 Stockholm, Sweden g Institute for Gravitation and the Cosmos,Pennsylvania State University, University Park, PA 16802, USA
Abstract
This review describes the duality between color and kinematics and its applications, withthe aim of gaining a deeper understanding of the perturbative structure of gauge and gravitytheories. We emphasize, in particular, applications to loop-level calculations, the broad webof theories linked by the duality and the associated double-copy structure, and the issue ofextending the duality and double copy beyond scattering amplitudes. The review is aimedat doctoral students and junior researchers both inside and outside the field of amplitudesand is accompanied by various exercises. a r X i v : . [ h e p - t h ] S e p ONTENTS
Contents ⇒ supersymmetry . . . . . . . . . . . . . . . . . 362.7 General lessons from applying CK duality . . . . . . . . . . . . . . . . . . . 38 R symmetry . . . . . . . . . . . . . . . . . . . . 504.3 Local symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534.4 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.5 Soft theorems as tests of enhanced global symmetries . . . . . . . . . . . . . 57 N ≥ N = 2 supersymmetry . . . . . . . . 785.3.3 Homogeneous N = 2 Maxwell-Einstein supergravities . . . . . . . . . 815.3.4 Pure supergravities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 845.3.5 Theories with hypermultiplets and supergravities with N < ONTENTS N = 4 SYM theory . . . . . . . . . . . . 1086.2 One-loop examples of BCJ duality: SYM theories with reduced supersymmetry1166.3 Two-loop examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1196.4 Three-loop example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.5 Other examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 B.1 Basics of spinor helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170B.1.1 Massive spinor helicity . . . . . . . . . . . . . . . . . . . . . . . . . . 171B.2 On-shell superamplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Appendix C Generalized unitarity 176
C.1 One-loop example of unitarity cuts . . . . . . . . . . . . . . . . . . . . . . . 177C.2 Converting gauge-theory unitarity cuts to gravity ones . . . . . . . . . . . . 178C.3 Method of Maximal Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180C.3.1 Sewing superamplitudes . . . . . . . . . . . . . . . . . . . . . . . . . 1813
INTRODUCTION
Gauge and gravity theories play a crucial role in our understanding of physical phenom-ena. Yet, they appear to be distinct. The weak, strong and electromagnetic interactionsare manifestations of gauge theories, while gravity shapes the macroscopic evolution of theuniverse and spacetime itself. Finding a unified framework which seamlessly combines thesetwo classes of theories constitutes, arguably, the most important open problem in theoreticalphysics. It is by now clear that realizing this unification requires a departure from conven-tional approaches through new principles or novel symmetries. The double-copy perspectivereviewed here offers a radically different way to interpret gravity. Its relation to the otherforces through color/kinematics duality [1, 2] leads to remarkable new insights and powerfulcomputational tools.Despite their clear differences, gauge and gravity theories are already known to sharemany features, supporting the existence of an underlying unified framework, such as stringtheory. While many of these similarities are not apparent from a standard Lagrangian orHamiltonian standpoint, the study of objects closely related to observable quantities, such asscattering amplitudes, reveals deep and highly-nontrivial connections. This is most apparentin their perturbative expansions, which make it clear that the dynamics of these two classesof theories are governed by the same kinematical building blocks, even when their physicalproperties are strikingly different.The developments which exposed these features were systematized by the introductionof the duality between color and kinematics and of the double-copy construction. Thescattering amplitudes of many perturbative quantum field theories (QFTs) exhibit a double-copy structure. It is central to our ability to carry out calculations to very high loop ordersand a property of all supergravities whose amplitudes have been analyzed in detail. Thisleads to the natural question whether all (super)gravity theories are double copies of suitably-chosen matter-coupled gauge theories. Perhaps more importantly, the double copy realizesa unification of gauge and gravity theories in the sense of providing a framework wherecalculations in both theories can be carried out using an identical set of building blocks,yielding vast simplifications.The primary purpose of this review is to offer an introduction to the duality betweencolor and kinematics—also referred to as color/kinematics (CK) duality and Bern-Carrasco-Johansson (BCJ) duality—and the associated double-copy relation in the hope of stimulatingnew progress both inside and outside of the fairly well-understood setting of scatteringamplitudes. Beyond gauge and gravity theories, double-copy relations also provide a newperspective on QFT, generating a surprisingly wide web of theories through building blocksobeying the same algebraic relations.The duality essentially states that scattering amplitudes in gauge theories—and, moregenerally, in theories with some Lie-algebra symmetry—can be rearranged so that kinematicbuilding blocks obey the same generic algebraic relations as their color factors. Via theduality, we can not only constrain the kinematic dependence of each graph, but we can alsoconvert gauge-theory scattering amplitudes to gravity ones through the simple replacementcolor ⇒ kinematics . (1.1)Evidence provided by explicit calculations suggests that CK duality and the double-copy4 INTRODUCTION construction hold for a wide class of theories at loop level [2–23]. Formal proofs, using avariety of methods [24–29], have been constructed for only tree-level scattering amplitudes inthese theories. The duality also gave novel descriptions for tree-level amplitudes in bosonicand supersymmetric string theories, as well as in various effective field theories related tospontaneous symmetry breaking, and more. It has also been observed that, in the presenceof adjoint-representation fermions, the duality implies supersymmetry [30].The schematic rule (1.1) has served as a powerful guide for many studies in perturbativegravity and supergravity, especially on their loop-level ultraviolet (UV) properties (see e.g.Refs. [2, 6, 31–40]), showing a surprisingly tame behavior. For many supergravity theories,the physical degrees of freedom are obtained by the substitution (1.1). In others, such aspure Einstein gravity, the desired spectrum can only be obtained after a subset of the double-copy states are projected out. As we describe in some detail in Sec. 5, CK duality and theassociated double-copy properties hold for a remarkably large web of theories.Given the success at exploiting the double-copy structure for scattering amplitudes, it isnatural to wonder whether it also carries over to other areas of gravitational physics, espe-cially for understanding and simplifying generic classical solutions. Scattering amplitudeshave an important property that makes transparent the duality and double-copy structure:they are independent of the choice of gauge and field-variables. Generic classical solutions, onthe other hand, do depend on these choices, making the problem of relating gauge and gravityclassical solutions inherently more involved. Nevertheless, the prospect of solving problemsin gravity by recycling gauge-theory solutions is especially alluring. While the differenceswith scattering amplitudes are significant and make it a nontrivial challenge to implementthis program, there has been significant progress in unraveling both the underlying prin-ciples of CK duality [26, 41–49] and finding explicit examples of classical solutions relatedby the double-copy property [50–77]. One of the most promising applications of the doublecopy beyond scattering amplitudes relates to gravitational-wave physics, as highlighted byRefs. [57, 69, 78–82].The origins of the double copy can be traced back to the dawn of string theory, withthe observation of a curious connection between the Veneziano scattering amplitude [83], A ( s, t ), (later identified as an open-string scattering amplitude) and the Virasoro-Shapiroamplitude [84, 85], M ( s, t, u ), (later identified as a closed-string amplitude). With an ap-propriate normalization, these two amplitudes are related as [86] M ( s, t, u ) = sin( πα s ) πα A ( s, t ) A ( s, u ) , (1.2)where α is the inverse string tension. The arguments are the kinematic (Mandelstam)invariants of a four-point scattering process, s = ( p + p ) , t = ( p + p ) , u = ( p + p ) . (1.3)Equation (1.2) carries over to all string states, including the gluons of the open string andthe gravitons in the closed string. In the low-energy limit, when string theory reduces tofield theory, it yields a relation between scattering amplitudes in Einstein gravity and thoseof Yang-Mills (YM) theory [87], M tree4 (1 , , ,
4) = (cid:18) κ (cid:19) sA tree4 (1 , , , A tree4 (1 , , , , (1.4)5 INTRODUCTION b
12 3 a µ cν σ
Figure 1: Gauge theories have three- and four-point vertices in a Feynman diagrammatic descrip-tion. βα µ γν σ Figure 2: Gravity theories have an infinite number of higher-point contact interactions in a Feynmandiagrammatic description. where A tree4 (1 , , ,
4) is a color-ordered gauge-theory four-gluon partial scattering amplitude, M tree4 (1 , , ,
4) is a four-graviton tree amplitude and κ is the gravitational coupling to relatedto Newton’s constant via κ = 32 π G N and, for reasons that will become clear shortly, thepolarization vectors of gluons on the right-hand side of Eq. (1.4) are taken to be null. Wewill suppress the gravitational coupling by setting κ = 2 throughout this review. The color-ordered partial tree amplitudes are the coefficients of basis elements once the amplitude’scolor factors are expressed in the trace color basis, and the coupling g is set to unity. They aregauge invariant—see e.g. Refs. [88–92] for further details. Equation (1.4) is rather striking,asserting that tree-level four-graviton scattering is described completely by gauge-theoryfour-gluon scattering, bypassing the usual machinery of general relativity. Similar relationswere later derived for higher-point string-theory tree-level amplitudes [86], and generalized inthe field-theory limit to an arbitrary number of external particles [93]. Besides the remarkableimplication that the detailed dynamics of the gravitational field can be described in terms ofthe dynamics of gauge fields, Eq. (1.4) has other surprising features not visible in standardLagrangian formulations. For example, Eq. (1.4) implies that the four-graviton amplitudecan be re-arranged so that Lorentz indices factorize [94, 95] into “left” indices belonging toone gauge-theory amplitude and “right” indices belonging to another gauge theory. It is interesting to contrast the remarkable simplicity encoded in the relation (1.4) with themuch more complicated expressions that arise from standard Lagrangian methods. Scat-tering amplitudes for gauge and gravity theories can be obtained using the Feynman rulesderived from their respective Lagrangians L YM = − F aµν F a µν , L EH = 2 κ √− gR . (1.5)6 INTRODUCTION
Here F aµν is the usual YM field strength and R the Ricci scalar.Following standard Feynman-diagrammatic methods, we gauge-fix and then extract thepropagator(s) and the three- and higher-point vertices. For gravity we also expand aroundflat spacetime, taking the metric to be g µν = η µν + κh µν where η µν is the Minkowski metricand h µν is the graviton field. As illustrated in Figs. 1 and 2, with standard gauge choices,gauge theory has only three- and four-point vertices, while gravity has an infinite numberof vertices of arbitrary multiplicity. The complexity of each individual interaction term isperhaps more striking than their infinite number. Consider, for example, the three-gravitoninteraction. In the standard de Donder gauge, ∂ ν h νµ = ∂ µ h νν , the corresponding vertexis [96, 97], G µρ,νλ,στ ( p , p , p )= i Sym (cid:20) − P ( p · p η µρ η νλ η στ ) − P ( p ν p λ η µρ η στ ) + 12 P ( p · p η µν η ρλ η στ )+ P ( p · p η µρ η νσ η λτ ) + 2 P ( p ν p τ η µρ η λσ ) − P ( p λ p µ η ρν η στ )+ P ( p σ p τ η µν η ρλ ) + P ( p σ p τ η µν η ρλ ) + 2 P ( p ν p τ η λµ η ρσ )+ 2 P ( p ν p µ η λσ η τρ ) − P ( p · p η ρν η λσ η τµ ) (cid:21) , (1.6)where we set κ = 2, p i are the momenta of the three gravitons, η µν is the flat metric, “Sym”implies a symmetrization in each pair of graviton Lorentz indices µ ↔ ρ , ν ↔ λ and σ ↔ τ ,and P and P signify a symmetrization over the three graviton legs, generating three orsix terms respectively. The symmetrization over the three external legs ensures the Bosesymmetry of the vertex. In total, the vertex has of the order of 100 terms. This generallyundercounts the number of terms, because within a diagram each vertex momentum is alinear combination of the independent momenta of that diagram.We may contrast this to the three-gluon vertex in Feynman gauge, V abc µνσ ( p , p , p ) = gf abc (cid:20) ( p − p ) σ η µν + cyclic (cid:21) . (1.7)which does not appear to bear any obvious relation to the corresponding three-gravitonvertex (1.6). These considerations seemingly suggest that gravity is much more complicatedthan gauge theory. Moreover, the three-graviton vertex immediately appears to conflict withthe simple factorization of Lorentz indices into left and right sets visible in Eq. (1.4). Thefirst term in Eq. (1.6), for example, contains a factor η µρ which explicitly contracts a leftgraviton index with a right one.The reason why the three-graviton vertex is so complicated is that it is gauge-dependent. With special gauge choices and appropriate field redefinitions [94, 95, 98, 99], it is possibleto considerably simplifying the Feynman rules. Still, direct perturbative gravity calculationsin a Feynman diagram approach are rather nontrivial, especially beyond leading order, evenwith modern computers. To eliminate the gauge dependence we should instead focus onthe three-vertex with on-shell conditions imposed on external legs, by demanding that the While somewhat less complicated than the three-graviton vertex, the three-gluon vertex is also gauge-dependent. INTRODUCTION
21 34 32 14 13 24
Figure 3: The three Feynman diagrams corresponding to the s , t and u channels. vertex is contracted into physical states that satisfy, ε µρ = ε ρµ , p µ ε µρ = 0 , p ρ ε µρ = 0 , ε µµ ≡ η µν ε µν = 0 , (1.8)where p is a graviton momentum and ε µν the associated graviton polarization tensor. Thisremoves all trace and longitudinal terms, reducing the vertex to a simple form, G µρ,νλ,στ ( p , p , p ) = − i (cid:20) ( p − p ) σ η µν + cyclic (cid:21)(cid:20) ( p − p ) τ η ρλ + cyclic (cid:21) , (1.9)exposing its simple relation to the three-gluon vertex of gauge theory. This is a hint thatthere should be much better ways to organize the perturbative expansion of gravity. Wenow turn to four-graviton scattering amplitude, which is a better example as it correspondsdirectly to a physical process. Consider the full four-gluon tree amplitude in YM theory, which can be obtained, for exam-ple, by following textbook Feynman rules [100]. We write it as a sum over three channelscorresponding to the three diagrams in Fig. 3 i A tree4 = g (cid:18) n s c s s + n t c t t + n u c u u (cid:19) , (1.10)where the Mandelstam variables are defined in Eq. (1.3). The s -channel color factor, nor-malized to be compatible with the scattering amplitudes literature [88], is c s = − f a a b f ba a , (1.11)where the color-group structure constants f abc are the standard textbook ones [100]. Withthis normalization, the s -channel kinematic numerator, n s , is n s = − (cid:26)(cid:20) ( ε · ε ) p µ + 2( ε · p ) ε µ − (1 ↔ (cid:21)(cid:20) ( ε · ε ) p µ + 2( ε · p ) ε µ − (3 ↔ (cid:21) + s (cid:20) ( ε · ε )( ε · ε ) − ( ε · ε )( ε · ε ) (cid:21)(cid:27) , (1.12)where the momenta and polarization vectors satisfy on-shell conditions p i = ε i · p i = 0.The other color factors and numerators are obtained by cyclic permutations of the particlelabels (1 , , c t n t = c s n s (cid:12)(cid:12)(cid:12) → → → , c u n u = c s n s (cid:12)(cid:12)(cid:12) → → → . (1.13)8 INTRODUCTION
Feynman rules for gluons contain a four-gluon vertex, as in Fig. 1. Here we have absorbed itscontribution into the three diagrams in Fig. 3 according to the color factors, by multiplyingand dividing by an appropriate propagator. This is the origin of the term on the second lineof Eq. (1.12).A key property of the gauge-theory scattering amplitude (1.10) is its linearized gaugeinvariance. To check this, we need to verify that the amplitude vanishes with the replacement ε → p . Upon doing this replacement for the s -channel numerator we get, after somealgebra, the nonzero result n s (cid:12)(cid:12)(cid:12) ε → p = − s (cid:20) ( ε · ε ) (cid:16) ( ε · p ) − ( ε · p ) (cid:17) + cyclic(1 , , (cid:21) ≡ s α ( ε, p ) , (1.14)which is no surprise since individual diagrams are, in general, gauge dependent. The function α ( ε, p ) is clearly invariant under cyclic permutations of the labels (1 , , n s c s s + n t c t t + n u c u u (cid:12)(cid:12)(cid:12)(cid:12) ε → p = ( c s + c t + c u ) α ( ε, p ) , (1.15)where α ( ε, p ) is the expression in Eq. (1.14). Hence the amplitude is gauge invariant if c s + c t + c u vanish, i.e. c s + c t + c u = − f a a b f ba a + f a a b f ba a + f a a b f ba a ) = 0 . (1.16)This is the standard Jacobi identity, which indeed is satisfied by the group-theory structureconstants in a gauge theory.Consider the three-term sum over kinematic numerators in Eqs. (1.12) and (1.13), n s + n t + n u , analogous to the sum over color factors on the left-hand side of Eq. (1.16). Remarkably,this combination vanishes when the on-shell conditions are applied, n s + n t + n u = 0 . (1.17)We will refer to this relation as a kinematic Jacobi identity . This was originally noticedsome time ago for four-point amplitudes, as a curiosity related to radiation zeros in four-point amplitudes [101–103]. Generic representations of four-point amplitudes in terms ofdiagrams with only cubic vertices obey these identities, but at higher points nontrivial re-arrangements are needed. The significance of the identity Eq. (1.17) and its generality wasunderstood later [1, 2]. We refer to kinematic identities that are analogous to generic color-factor identities as a duality between color and kinematics. It turns out that they constitutean ubiquitous, yet hidden, structure not only of gauge theories, but also of an ever-increasingweb of theories, as described in Sec. 5.Exercise 1.1: Use Eqs. (1.12) and (1.13) to verify the numerator Jacobi identity (1.17).Redefine the numerators by eliminating c u in favor of c s and c t , defining new numerators n s and n t as the coefficient of c s /s and c t /t . The numerator n u vanishes by construction. Showthat the kinematic Jacobi identity still holds for these redefined numerators.The fact that the kinematic factors satisfy the same relations as the color factors suggeststhat they are mutually exchangeable. Indeed, we can swap color factors for kinematic factors9 INTRODUCTION in the YM four-point amplitude (1.10), which gives a new gauge-invariant object that, as wewill discuss momentarily, is a four-graviton amplitude, i A tree4 (cid:12)(cid:12)(cid:12)(cid:12) c i → ˜ n i g → κ/ ≡ i M tree4 = (cid:18) κ (cid:19) n s s + n t t + n u u ! . (1.18)The new amplitude M tree4 doubles up the kinematic numerators, and so we refer to it as adouble copy. (The i in front of the M tree4 is a phase convention.) The expression in Eq. (1.18)has the following properties: the external states are captured by symmetric polarizationtensors ε µν = ε µ ε ν , the interactions are of the two-derivative type, and the amplitude isinvariant under linearized diffeomorphism transformations. By choosing the polarizationvectors to be null ε = 0 (corresponding to circular polarization), implying that ε µν istraceless, this amplitude should describe the scattering of four gravitons in Einstein’s generalrelativity, up to an overall normalization. There are a number of ways to prove that thisis the case, including using on-shell recursion relations [41] and ordinary gravity Feynmanrules [94]; here we will show that Eq. (1.18) reproduces the Kawai-Lewellen-Tye (KLT) formof gravity amplitudes [86], derived using the low-energy limit of string theory.The diffeomorphism invariance of the amplitude requires some elaboration. Consider alinearized diffeomorphism of the asymptotic (weak) graviton field h µν . The diffeomorphismis parametrized by the function ξ µ and take the simple form δh µν = ∂ µ ξ ν + ∂ ν ξ µ . (1.19)Translating this to momentum space implies that a diffeomorphism-invariant amplitudeshould vanish upon replacing a polarization tensor as: ε µν → p µ ε ν + p ν ε µ . Applying this toleg 4 of the amplitude, we find n s s + n t t + n u u (cid:12)(cid:12)(cid:12)(cid:12) ε µν → p µ ε ν + p ν ε µ = 2( n s + n t + n u ) α ( ε, p ) = 0 . (1.20)Thus, we see that the kinematic Jacobi identity needs to be satisfied for the amplitude to beinvariant under linearized diffeomorphism transformations, in complete analogy to the colorJacobi identity in the gauge-theory amplitude.Returning to the YM amplitude, we note that the amplitude can be written in a mani-festly gauge-invariant form if we solve the Jacobi relation by choosing c t = − c u − c s , i A tree4 = g (cid:18) n s c s s + n t c t t + n u c u u (cid:19) = g (cid:18)(cid:18) n s s − n t t (cid:19) c s − (cid:18) n t t − n u u (cid:19) c u (cid:19) ≡ ig A tree4 (1 , , , c s − ig A tree4 (1 , , , c u . (1.21)The partial amplitudes A tree4 (1 , , ,
4) are gauge invariant because the color-dressed ampli-tude A tree4 is now decomposed in a basis of independent color factors, with elements c s and c u , and thus the gauge invariance of A tree4 implies the gauge invariance of the individualterms of this decomposition. 10 INTRODUCTION
It is not difficult to show that the partial amplitude can be written as A tree4 (1 , , ,
4) = − i t F st , (1.22)where t F ≡ h F F F F ) − Tr( F F )Tr( F F ) + cyclic(1 , , i (1.23)contains various Lorentz traces over four linearized Fourier transformed field strengths, F µνi ≡ p µi ε νi − ε µi p νi , (1.24)where the fields are replaced with polarization vectors. These are manifestly invariant underlinearized gauge transformations.We can also solve the kinematic Jacobi relation (1.17) by choosing n t = − n u − n s . Thepartial amplitudes then become iA tree4 (1 , , ,
4) = n s s − n t t = n s (cid:18) s + 1 t (cid:19) + n u t ,iA tree4 (1 , , ,
4) = n t t − n u u = − n u (cid:18) u + 1 t (cid:19) − n s t , (1.25)which may also be organized as a matrix relation i A tree4 (1 , , , A tree4 (1 , , , ! = s + t t − t − u − t ! n s n u ! . (1.26)It might seem that it is possible to solve for the numerators in terms of the partial amplitudesby inverting the two-by-two matrix of propagators. Existence of a solution would contradict,however, the fact that on the one hand numerators are gauge-dependent and on the otherpartial amplitudes are gauge-invariant. Indeed, the matrix of propagators has no inverseas its determinant is proportional to s + t + u = 0. At best, we can solve for one of thenumerators, say, n u , n u = itA tree4 (1 , , ,
4) + u n s s . (1.27)Replacing this into A tree4 (1 , , ,
4) in Eq. (1.25), the dependence on the undetermined kine-matic numerator n s cancels out, and we obtain the gauge-invariant relation A tree4 (1 , , ,
4) = su A tree4 (1 , , , . (1.28)Given the vanishing of the determinant of the above matrix of propagators, it is not surprisingto find that the two partial amplitudes are linearly dependent. In fact, one may phraseEq. (1.28) as the orthogonality condition of the left-hand side of Eq. (1.26) onto the nulleigenvector of the matrix of propagators.The existence of relations between partial amplitudes is a general feature. Such BCJamplitude relations exist whenever the duality between color and kinematics and gaugeinvariance conspire to prevent the relation between partial amplitudes and numerators to beinverted. These relations have been demonstrated in a variety of ways, including using bothstring theory [104–112] and field theory methods [113–118].11
INTRODUCTION
In string theory, one finds similar identities that follow from world-sheet monodromyrelations. For massless vector amplitudes of the open string, from world-sheet monodromyrelations [104, 105] one finds A tree4 (1 , , ,
4) = sin( πα s )sin( πα u ) A tree4 (1 , , , . (1.29)where α is the inverse string tension.We can also use the two relations, n t = − n u − n s and n u = tA tree4 (1 , , ,
4) + u n s /s , inEq. (1.18). The result is M tree4 (1 , , ,
4) = − i (cid:20) n s s + n t t + n u u (cid:21) = − i stu (cid:20) A tree4 (1 , , , (cid:21) , (1.30)where, as usual, we have suppressed the gravitational coupling setting κ = 2. As for thegauge-theory case, n s drops out; as in that case, this is to be expected as it would otherwiselead to a relation between gauge invariant and gauge-dependent quantities, M tree4 and n s respectively. We can put this equation into a more standard form using a relabeling identity(1.28), M tree4 (1 , , ,
4) = − isA tree4 (1 , , , A tree4 (1 , , , , (1.31)which is the simplest of the KLT relations between gravity and gauge-theory amplitudes.We derived it here as a consequence of CK duality and gauge-invariance constraints, but theoriginal derivation [86] comes from string theory. It is worth noting that these relations arenot unique given amplitude relations such as Eq. (1.28).Replacing the four-point YM amplitude in the from Eq. (1.22) into the KLT relation(1.31), we obtain an explicit form for the four-graviton amplitude M tree4 (1 , , ,
4) = − i t R stu , (1.32)where we define t R in terms of t F in Eq. (1.23) as t R ≡ (cid:16) t F (cid:17) . (1.33)As the notation suggest, t R can also be written as a contraction between a rank-16 tensor t and four linearized Riemann tensors, using the relationship to linearized gauge-theoryfield strengths in Eq. (1.24), R µνρσi = F µνi F ρσi = ( p µi ε νi − ε µi p νi )( p ρi ε σi − ε ρi p σi ) . (1.34) In this review, we will describe the duality between color and kinematics and the doublecopy, as proposed in the original work [1, 2], and later refined through various extensionsand applications. As indicated in Fig. 4, CK duality and double copy are intertwined withthe topics of several vigorous research fields. The areas that the review will mainly focuson include the web of theories, loop amplitudes and the classical double copy. The web of12
INTRODUCTION
Figure 4: Connections of CK duality to various topics. This review will discuss in some detail theconnection of CK duality to the topics in the upper right (with the main chapters indicated) andless so to the topics on the lower right. The various topics are intertwined with each other as well. theories allude to the large classes of known double-copy constructions and their underlyingsingle-copy theories, whose existence became clear after important theories, such as Chern-Simons [119], Yang-Mills-Einstein [120], Maxwell-Einstein [120, 121], spontaneously-brokentheories [122] and gauged supergravities [123] were observed to fit into the general framework.In addition, from the Cachazo, He and Yuan (CHY) formulation [124], it was observed [125]that also effective field theories such as the non-linear-sigma-model (NLSM) [126], (Dirac)-Born-Infeld (DBI) and special-Galileon theory played a central role.The usefulness of the duality and the double copy for loop amplitudes becames clearonce the framework was applied to obtain compact integrands for the three- [2] and four-loop [6] amplitudes in N = 4 SYM and in N = 8 supergravity. By now it is clear that loopamplitudes in many other theories can be obtained using the duality and double copy.When the double copy was shown to be applicable to problems of classical gravity, suchas the Schwarzschild and Kerr metrics [51] as well as other perturbatively constructablemetrics [58], it opened the door to further applications relevant to gravitational physics.With the discovery of gravitational waves from merging binary black holes and neutronstars [127, 128], it is becoming increasingly important to find better ways to accuratelycalculate classical observables in general relativity. The double-copy approach is still in itsinfancy, but it bears the promise of drastically changing the way we think of carrying outcomputations in gravity.In order to keep the discussion manageable, we will not discuss in much detail the chal-lenges of understanding gravitational radiation and potentials (see e.g. Refs. [129–132] for re-views and Refs. [78, 80, 82] for a state-of-the-art application of the double copy). Nor will webe thorough in describing the connections to string theory (see e.g. Refs. [104, 105, 133, 134]),the CHY construction [124, 135–137] and ambitwistor strings [138–148], all of which have13 THE DUALITY BETWEEN COLOR AND KINEMATICS interesting connections to CK duality and the double-copy construction.The outline of topics in each section is as follows: In Sec. 2, we describe the duality insome detail and give various examples, and show how the double copy implies diffeomorphisminvariance of gravity. In Sec. 3, we give a way to visualize how the duality can be thoughtof as specifying amplitudes in terms of boundary data on a graph of graphs and on makinguse of relabeling invariance. Then, in Sec. 4, we discuss the inheritance of symmetries in thedouble-copy theories from their component theories. Sec. 5 gives a detailed description of theweb of double-copy constructible theories, emphasizing the widespread applicability of theseideas. In Sec. 6, we give loop-level examples of the duality between color and kinematics. InSec. 7, we explain a generalized double-copy procedure that does not require loop integrandsto manifest the duality. Sec. 8 discusses the important issue of extending the double-copyprocedure to solutions of the classical equations of motion. Conclusions and prospects for thefuture are given in Sec. 9. In Appendix A, we collect acronyms and notation used throughoutthe review. Appendix B summarizes spinor helicity and on-shell supersymmetry, which willbe useful in various sections. Finally, Appendix C briefly describes generalized unitarity,used in Secs. 6 and 7.
The duality between color and kinematics is by now an extensive topic with a variety ofperspectives and applications. However, it is not always clear from the literature what rulesgovern this framework. In this section, the central aspects of CK duality will be described,with the aim of clarifying the reason for imposing various requirements as well as providingan understanding of when they can be relaxed.
CK duality in its original formulation states that it is possible to reorganize the perturbativeexpansion of tree-level amplitudes in D -dimensional pure YM theory with a general gaugegroup G in terms of cubic diagrams where the kinematic numerators obey the same Jacobirelations and symmetry properties as their color factors [1, 2]. While it is not a priori obvi-ous why such a reorganization is possible or even desirable, from a Lagrangian perspectivethis is a highly nontrivial statement about YM theory. The associated double-copy con-struction however, does make it clear that the duality is worth understanding because ofthe way it connects gravity to gauge theory. While there are tree-level proofs of the dualityfrom the amplitudes perspective [24, 25, 149], at present, only a partial Lagrangian-levelunderstanding has been achieved [41, 42, 150, 151].More generally, CK duality refers to the statement that in many gauge theories, extendingwell beyond YM theories with or without matter, it should be possible to reorganize theperturbative expansion so that there is a one-to-one map between the Lie-algebra identitiesof the color factors carried by certain diagrams (with cubic or higher-point vertices) andthe identities of the kinematic numerators of the same diagrams. In the broad class ofgeneral gauge theories, one can think of CK duality as a constraint that can be imposedon fields, gauge-group representations, interactions and operators, such that the theories14 THE DUALITY BETWEEN COLOR AND KINEMATICS give amplitudes that exhibit the duality structure. These constraints often result in theorieswith properties that are interesting for reasons not directly related to the duality [120–122, 152, 153].In generalizing beyond gauge theories, one can consider matter theories that are com-prised of spin < φ theory [44, 154]). The most remarkable aspect of CK dualityis that it naturally leads to scattering amplitudes in double-copy theories. Sec. 5 describesa remarkable web of theories that are connected by the duality and the double copy.Finally, for amplitudes that are not obtained from the standard QFT framework involvingFeynman diagrams, such as string-theory amplitudes, it is convenient to define CK dualityto mean that these amplitudes obey the same relations as if they were generated by aduality-satisfying diagrammatic expansion of the gauge-theory type. For example, the single-trace vector-amplitude sector of the heterotic string obeys the same relations as that of YMtheory [155]. Hence, we can write heterotic string amplitudes as a sum over cubic diagramswith duality-satisfying kinematic numerators, even if this might not seem completely naturalfrom a string-theory perspective. Consider scattering amplitudes in a nonabelian gauge theory with the following proper-ties: there is a gauge-group G under which all fields transform nontrivially; particles ofdifferent mass are assigned to various representations of the gauge group; the interac-tions are controlled by a gauge coupling constant g and a set of elementary color tensors C = n f abc , ( t a ) ji , . . . o . The set of elementary color tensors may include higher-rank tensorsas indicated by the ellipsis.An L -loop m -point scattering amplitude in this D -dimensional gauge theory can then beorganized as A ( L ) m = i L − g m − L X i Z d LD ‘ (2 π ) LD S i c i n i D i , (2.1)where the sum runs over the distinct L -loop m -point diagrams that can be constructed bycontracting the elements of C in various allowed ways (consistent with the choice of externalparticle representations, and where the valency of each vertex is determined by the tensorrank). We take each such diagram to correspond to a unique color factor c i . Each diagramhas an associated denominator factor D i which is constructed by taking a product of the Our conventions for the overall phase in the representations of gauge-theory and gravity amplitudesfollow the one in [156] rather than the original BCJ papers [1, 2]. THE DUALITY BETWEEN COLOR AND KINEMATICS (a) − =(b) − = Figure 5: Color-algebra relations in the adjoint (a) and fundamental representation (b). The curlylines represent adjoint representation states and the straight lines fundamental representation. Thevertices correspond to the color matrices in Eq. (2.4). denominators of the Feynman propagators ∼ / ( p − m j ) of each internal line of the dia-gram. For simplicity of notation, we assume that the color representation of the line uniquelyspecifies the mass m j of the propagator. Cases with differing masses, but the same color rep-resentation, are easily taken into account by setting appropriate masses and representationsequal at the end. The adjoint representation is by default massless and is associated to glu-ons (and, in some cases, additional fields). The remaining nontrivial kinematic dependenceis collected in the kinematic numerator n i associated with each diagram. The numerators n i are in general gauge-dependent functions that depend on external momenta p j , loop mo-menta ‘ l , polarizations ε j , spinors, flavor, etc., everything except for the color degrees offreedom. The integral measure is defined as d LD ‘ = Q Ll =1 d D ‘ l . Finally, S i are standardsymmetry factors that remove internal overcount of loop diagrams; they can be computedby counting the number of discrete symmetries of each diagram with fixed external legs.The color factors c i are in general not independent. They satisfy linear relations thatare inherited from the Lie algebra structure, such as the Jacobi identity and the definingcommutation relation, f dae f ebc − f dbe f eac = f abe f ecd , ( t a ) ki ( t b ) jk − ( t b ) ki ( t a ) jk = if abc ( t c ) ji , (2.2)and similar identities for other color tensors that might appear in the theory. In Eq. (2.2)we follow the standard textbook normalization of color generators [100],Tr( t a t b ) = δ ab . (2.3)Such Lie-algebra relations are directly tied to gauge invariance of amplitudes.In the amplitudes community, color generators differ from the textbook definition by a √ T a ≡ √ t a , ˜ f abc ≡ i √ f abc , (2.4)so that we have the identity Tr( T a T b ) = δ ab , (2.5)16 THE DUALITY BETWEEN COLOR AND KINEMATICS
With these changes in normalization the defining commutation relations are,˜ f dae ˜ f ebc − ˜ f dbe ˜ f eac = ˜ f abe ˜ f ecd , ( T a ) ki ( T b ) jk − ( T b ) ki ( T a ) jk = ˜ f abc ( T c ) ji , (2.6)as illustrated in Fig. 5. These identities imply that there exist relations between triplets ofcolor factors { c i , c j , c k } which take, for example, the form c i − c j = c k .The scattering amplitude (2.1) is said to obey CK duality if the kinematic numeratorfactors obey the same general algebraic relations as the color factors do, e.g. n i − n j = n k ⇔ c i − c j = c k , (2.7)which is a generalization of the kinematic Jacobi identity in Eq. (1.17). The relative signsbetween the terms depend on choices in defining the color factors for each diagram. Theessential point regarding the signs is that whatever choice is made for the color factors areinherited by the corresponding numerator factors. Another form of the duality in terms ofcolor traces has also been found [48, 157–161], but the most natural form is in terms of colorfactors of diagrams as described above.It is a nontrivial task to find duality-satisfying numerators since standard methods such asFeynman rules, on-shell recursion [162], or generalized unitarity [163–166], generally do notautomatically gives such numerators. A straightforward but somewhat tedious way to findsuch representations is to use an ansatz constrained to match the amplitude and manifest theduality [4, 6]. Constructive ways to obtain numerators have also been devised [24–29, 167–169]. Aside from amplitudes, the duality has also been found to hold for currents with oneoff-shell leg [9, 17, 21, 22, 114, 114, 170–172]. A natural way for making the duality validfor general off-shell quantities would be to find a Lagrangian that generates Feynman ruleswhose diagrams manifest the duality. At present, such Lagrangian is only known to a feworders in perturbation theory [41, 42, 150, 151]; an important problem is to find a closedform of such a Lagrangian valid to all orders.The color relations (2.6) have important implications for kinematic numerators of dia-grams. If we start with a set of numerators that satisfy the duality (2.7), and shift thenumerators, n i = n i − ∆ i . (2.8)subject to the constraint, X i Z d LD ‘ (2 π ) LD S i c i ∆ i D i = 0 , (2.9)the amplitude is unchanged. Because the color factors are not independent, nontrivial shiftsof the kinematic numerators can be carried out. In this way, without changing the amplitude,we can rewrite the amplitude in terms of a set of numerators n i not obeying the dualityrelations (2.7) starting from ones that do obey it. The ∆ i are pure gauge functions, i.e. theydrop out of the amplitude.When we have numerators n i that obey the same algebraic relations as the color factors c i , we can obtain sensible objects by formally replacing color factors by kinematic numeratorsas c i → n i , (2.10)17 THE DUALITY BETWEEN COLOR AND KINEMATICS in any given formula or amplitude. Given the algebraic properties are the same, this replace-ment is consistent with gauge-invariance properties inherited from the gauge theory. As wediscuss below, this color-to-kinematics replacement—or double-copy construction—gives usgravity amplitudes with remarkable ease.Consider two amplitudes A ( L ) m and e A ( L ) m , and organize them as in Eq. (2.1). Further-more, take the color factors to be the same in the two amplitudes, and label the two setsof numerators as n i and ˜ n i , respectively. If at least one of the amplitudes, say e A ( L ) m , man-ifests CK duality, we may now replace the color factors of the first amplitude with theduality-satisfying numerators ˜ n i of the second one. This gives the double-copy formula forgravitational scattering amplitudes [1, 2], M ( L ) m = A ( L ) m (cid:12)(cid:12)(cid:12)(cid:12) ci → ˜ nig → κ/ = i L − (cid:18) κ (cid:19) m − L X i Z d LD ‘ (2 π ) LD S i n i ˜ n i D i , (2.11)where the gravitational coupling κ/ κ/ κ = 2.For the replacement c i → ˜ n i to be valid under the integration symbol, it is important thatthe color factors are not explicitly evaluated by summing over the contracted indices. Atleast one contracted index per loop should not be explicitly summed over; this is required sothat the duality is not spoiled by treating color and kinematics differently. The numeratorsdepend on loop momenta n i = n i ( ‘ ) that is not yet integrated over, thus analogously thecolor factors should be thought of as depending on the unevaluated internal indices. If thissubtlety is ignored, it may happen that color factors explicitly vanish when combining thecolor sum with symmetries of particular color factors, and this vanishing behavior should notbe imposed on the un-integrated numerators. Stated differently, we do not wish to imposeany specific color-factor properties on the numerator factors, only generic ones.As the notation suggests, the two sets of numerators n i , ˜ n i can differ in several ways:(1) they can describe different gauge choices for the same scattering process, (2) they candescribe different external states in the same theory, and (3) they can originate from twodifferent gauge theories. The first case allows us to work with numerators where only oneset obeys the duality manifestly. The second case allows us to describe gravitational statesthat are not built out of a symmetric-tensor product(gravity state) = (gauge state) ⊗ ( (cid:94) gauge state) . (2.12)The third case allows us to describe gravitational theories that are not left-right symmetricdouble copies of gauge theories(gravity theory) = (gauge theory) ⊗ ( (cid:94) gauge theory) . (2.13)In Sec. 5, we will see that this latter case is crucial for probing the web of double-copy-constructible theories.When two different gauge theories are considered in the double-copy formula, it is im-portant that both, in principle, can be put into a form displaying CK duality, even if thisproperty needs only to be explicit in one of the amplitudes. This ensures that the generalized18 THE DUALITY BETWEEN COLOR AND KINEMATICS unitarity cuts of the loop-level double-copy formula will be unique and gauge invariant. Thelink between gauge invariance and BCJ amplitude relations has been explored in Refs. [173–177]. The amplitude relations can also be understood in terms of a symmetry that act asmomentum-dependent shifts on the color factors [178, 179]. Note that the precise form ofthe BCJ amplitude relations depends on the details of the gauge-group representations andelementary color tensors. The standard BCJ amplitude relations [1], for example, followfrom considering theories with only adjoint particles that interact via f abc color tensors.We will come back to the double-copy constructions of different theories in later sections,but for now we will focus on illustrating the details of CK duality on some familiar gaugetheories. Consider pure YM theory in D spacetime dimensions, consisting of gluons transforming inthe adjoint representation of a gauge group G , with Lagrangian L YM = −
14 ( F aµν ) , where F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . (2.14)Next consider m -point tree-level amplitudes. We know that the only color structure thatappears are contractions of f abc structure constants, thus the color factors must be in one-to-one correspondence with all possible cubic diagrams with m external legs.Cubic diagrams at multiplicity m = j + 1 can be built recursively by attaching a newleg to every possible edge of a multiplicity- j diagram. There are (2 j −
3) edges of a given j -point diagram, hence the recursion gives:number of cubic diagrams = 1 × × × × · · · × (cid:16) j − (cid:17) = (2 m − . (2.15)We organize the tree amplitude in terms of all such propagator-distinct diagrams withonly cubic vertices, A tree m ≡ A (0) m = − ig m − m − X i =1 c i n i D i , (2.16)where c i are the color factors that are straightforwardly obtained from the i -th diagram.Similarly, the D i denote the denominators of the propagators that correspond to the diagramlines. The n i are the corresponding kinematic numerators. Depending on the context we willalternate between using the diagram weights n i , c i , D i with subscripts indexed by a diagram-id number, as well as a functional maps from graph to their respective weights: n i ≡ n ( g i ), c i ≡ c ( g i ), and D i ≡ D ( g i ) where g i is the graph corresponding to the index i .It is useful to first clarify what we mean by independent diagrams. The least redundancyoccurs when we insist on only one instance of a diagram with the same propagator contribu-tion. This is distinct from the number of unique diagram topologies. Let us take a concreteexample at four-points. We have discussed in Sec. 1 that we need s , t , and u diagrams atfour point. They have same graphical topology, but different external labels, which resultsin different generic propagator contributions.As a trivial example, at four points for each distinct propagator structure we can relabelthe external legs without altering the propagators but flipping the signs of the color. For19 THE DUALITY BETWEEN COLOR AND KINEMATICS =12 3 mab cd ab cd
12 3 m − a db c m Figure 6: A Jacobi identity embedded in a generic diagram. The diagram can be either at treelevel or at loop level. The arrows indicate that the lines are oriented the same way. example, consider the s -channel diagram in Fig. 3 which we can label as g s :1 . Taking thegraph g s :2 to be g s :1 but with legs 1 and 2 swapped, we obtain the same propagator but thecolor factors are different: c ( g s :1 ) = ˜ f a a b ˜ f ba a ,c ( g s :2 ) = ˜ f a a b ˜ f ba a , (2.17)where we use the normalization in Eq. (2.4). The color factors, while distinct, are related bya negative sign inherited by the antisymmetry of the structure generators: c ( g s :2 ) = − c ( g s :1 ).For the purpose of describing scattering amplitudes in terms of functions of diagrams, wewill always take the kinematic weights of the diagrams to obey the same antisymmetry: n ( g s :2 ) = − n ( g s :1 ), whether or not we are discussing a CK-satisfying representation. Thismeans that for any multiplicity and loop order we will have in mind a canonical layout ofdistinct diagrams which determine the color factor and numerator signs. These signs cancelfrom color-dressed amplitudes because the numerator sign are correlated with the color signs.However, they will affect the signs appearing in the relation between color-ordered partialamplitudes and kinematic numerators, as well as the relative signs between terms in theJacobi identities.To be more explicit, as illustrated in Fig. 6, triplets of diagrams ( i, j, k ) satisfy Jacobirelations of the form c i − c j + c k = (cid:16) ˜ f dae ˜ f ebc − ˜ f abe ˜ f ecd + ˜ f dbe ˜ f eca (cid:17) C abcd = 0 . (2.18)where the last factor C abcd is a color tensor that is common to the diagrams in the triplet(external adjoint indices a , . . . , a m are suppressed). As noted above, the relative signs aresimply due to choices in the ordering of the color indices in the ˜ f abc s. While these relativesign choices are arbitrary, these signs are the same as for the corresponding kinematic Jacobiidentities.More generally, the (2 m − m there are ( m − m − m − − ( m − m − THE DUALITY BETWEEN COLOR AND KINEMATICS
Writing the adjoint generator matrices as ( ˜ f a ) bc ≡ ˜ f bac , defined in Eq. (2.4), we canwrite any color factor as products of ˜ f a i ’s, possibly involving commutators of the adjointgenerators. For example, pick a cubic tree diagram and find the unique path through thediagram that connect leg 1 and leg m . For each cubic vertex along this path, write down thecorresponding commutator of ˜ f a i ’s that describes the subdiagram that attaches this vertex.The product of these factors give c i for the full diagram. For example, consider the colorfactor of the following diagram c m − m = ( ˜ f a [ ˜ f a , ˜ f a ][ ˜ f a , [ ˜ f a , ˜ f a ]] · · · ˜ f a m − ) a a m , (2.19)where the adjoint indices of leg 1 and m correspond to the external matrix indices of theadjoint representation. The commutators arise from systematically eliminating subdiagramsinvolving ˜ f bac using the standard Lie-algebra identity ˜ f abc ˜ f c = [ ˜ f a , ˜ f b ]. Once only commu-tators of ˜ f ai ’s remain, they can of be written out as differences and sums of generators indifferent orders.In summary, any color factor can in general be written as c i = X σ ∈ S m − b iσ (cid:16) ˜ f a σ (2) ˜ f a σ (3) ˜ f a σ (4) · · · ˜ f a σ ( m − (cid:17) a a m , (2.20)where b iσ ∈ { , ± } are coefficients that depend on the permutation and on the specific colorfactor. They can be evaluated case by case, but their explicit values are not important herefor our purposes. The main result is that color factors c i in Eq. (2.16) can be eliminatedin favor of expressing the gauge-theory tree amplitude in terms of a sum over the possibleproducts of adjoint generators ˜ f a i , where the first and m -th leg is kept fixed. This givesa so-called Del Duca-Dixon-Maltoni (DDM) color decomposition [180] of the gauge-theorytree amplitude, A tree m = g m − X σ ∈ S m − A tree m (cid:16) , σ (2) , σ (3) , . . . , σ ( m − , m (cid:17)(cid:16) ˜ f a σ (2) ˜ f a σ (3) · · · ˜ f a σ ( m − (cid:17) a a m , (2.21)where the sum runs over ( m − m − A tree m (cid:16) , σ (2) , σ (3) , . . . , σ ( m − , m (cid:17) . This is usually called the Kleiss-Kuijf (KK) basis [181].The partial tree amplitudes in YM theory, A tree m (cid:16) , , . . . , m (cid:17) , have a number of usefulproperties [88]:• They are functions of kinematic variables only, ( ε i , p i ); the color dependence is onlyreflected by the ordering of the external particle labels.• They at most have poles in planar channels, i.e. when consecutive momenta add up toa null momentum (cid:16) P j ≤ i ≤ k p i (cid:17) = 0 (mod m ).21 THE DUALITY BETWEEN COLOR AND KINEMATICS • The amplitudes are invariant under cyclic permutations: A tree m (cid:16) , , . . . , m (cid:17) = A tree m (cid:16) , . . . , m, (cid:17) . (2.22)• Under reversal of the ordering, they at most change by a sign flip: A tree m (cid:16) m, . . . , , (cid:17) = ( − m A tree m (cid:16) , , . . . , m (cid:17) . (2.23)• They satisfy a photon-decoupling identity: X σ ∈ cyclic A tree m (cid:16) , σ (2) , . . . , σ ( m ) (cid:17) = 0 , (2.24)where cyclic permutations of all but one leg are summed over.• They satisfy KK relations [181]: A tree m (1 , α, m, β ) = ( − | β | X σ ∈ α (cid:1) β T A tree m (1 , σ, m ) , (2.25)where α and β are arbitrary-sized lists of the external legs, β T is used to representthe reverse ordering of the list β , and α (cid:1) β T is the shuffle product of these lists (i.e.permutations that separately maintain the order of the individual elements belongingto each list). | β | denotes the number of elements in the list β .• They obey BCJ relations, which in the simplest incarnation take the form [1]: m − X i =2 p · ( p + . . . + p i ) A tree m (2 , . . . , i, , i + 1 , . . . , m ) = 0 . (2.26)• After considering all permutations of the above BCJ relation, there are only ( m − A tree m (cid:16) , , σ (3) , . . . , σ ( m − , m (cid:17) can bechosen as the independent BCJ basis.The first property is obvious from our definition of the partial amplitudes; however, theremaining ones require some explanation.The fact that the partial tree amplitudes are invariant under cyclic permutations of theirarguments is most easily seen after a basis change of the color factors. We rewrite the colorfactors in terms of traces of generators, T a . From Eq. (2.6)˜ f abc ≡ i √ f abc = Tr([ T a , T b ] T c ) = Tr( T a T b T c ) − Tr( T b T a T c ) , (2.27)which follows from the identity (2.6) after multiplying both sides with T c , tracing over thefundamental indices, and using Eq. (2.5). This basis change implicitly assumes that wehave specialized to a gauge group were we can use ’t Hooft’s double-line notation [182], say G = U ( N c ). 22 THE DUALITY BETWEEN COLOR AND KINEMATICS
The generators of U ( N c ) obey the completeness relation ( T a ) ji ( T a ) lk = δ li δ jk , implying thatproducts of several f abc can be expressed by merging several traces( ˜ f a ˜ f a · · · ˜ f a m − ) a a m = ˜ f a a b ˜ f b a b · · · ˜ f b m − a m − a m (2.28)= Tr( T a T a T a · · · T a m ) + ( − m Tr( T a m · · · T a T a T a ) + . . . where on the last line the suppressed terms corresponds to 2 m − distinct permutations of thetrace over m generators. Out of all the permutations that appear, only the two displayedterms have the property that the generators T a and T a m are adjacent (in the cyclic sense).This implies that, after replacing the DDM color factors with the trace-basis color factors inEq. (2.21), we can uniquely identify the location of, say, the Tr( T a T a T a · · · T a m ) factor.It appears only once in ( ˜ f a ˜ f a · · · ˜ f a m − ) a a m , which uniquely multiplies the partial treeamplitude A tree m (1 , , , . . . , m ). Hence, A tree m (1 , , , . . . , m ) must be the kinematic coefficientof Tr( T a T a T a · · · T a m ) in the trace-basis decomposition of the YM tree amplitude.By crossing symmetry in the trace basis, the decomposition into partial amplitudes hasthe form A tree m = g m − X σ ∈ S m − A tree m (cid:16) , σ (2) , σ (3) , . . . , σ ( m ) (cid:17) Tr( T a T a σ (2) T a σ (3) · · · T a σ ( m ) ) , (2.29)which can be straightforwardly verified starting from Eq. (2.21). Crossing symmetry requiresthe summation over ( m − m − m − − m Tr( T a m · · · T a T a T a ) in Eq. (2.28) always goes together withTr( T a T a T a · · · T a m ). The photon-decoupling identity follows from realizing that we canreplace one generator in Eq. (2.29) by the U (1) “photon” generator T U (1) = 1, which natu-rally belongs to the gauge group U ( N c ) = SU ( N c ) × U (1). However, gluons do not coupledirectly to photons since the latter have no charges. This can be seen directly by looking atthe structure constants ˜ f abU (1) = Tr([ T a , T b ] 1) = 0. Hence, the photon-decoupling identityfollows from the vanishing of the amplitude with one photon.The KK relations are explained by the fact that there are two different decompositions ofthe tree amplitude, the DDM (2.21) and the trace (2.29) decomposition, which use a differentnumber of partial amplitudes, ( m − m − m − m − f abc ’s, which obey the Jacobi relations, that we could findthe ( m − f abc will obey the KK relations.23 THE DUALITY BETWEEN COLOR AND KINEMATICS
The BCJ amplitude relations are a consequence of CK duality, specifically of its interplaywith gauge invariance. Consider the ( m − A tree m (1 , σ (2) , . . . , σ ( m − , m ) = − i X i ∈ planar b iσ n i D i , (2.30)where n i are the kinematic numerator weights of the diagrams canonical to some orderinglayout, D i are the propagators of the diagram, b iσ ∈ { , ± } are coefficients that depend onthe ordering σ , and the sum is only nonvanishing for planar diagrams with respect to theordering σ .We can impose kinematic Jacobi identities on the numerators, expressing all diagramnumerators in terms of ( m − m − m − m − m − m − A tree m (1 , , { α } , , { β } ) = X σ ∈ POP( { α } , { β } ) A tree m (1 , , , σ ) |{ α }| +3 Y k =4 F k (3 , σ, s ...k , (2.31)where |{ α }| is the length of the list { α } , and the sum runs over partially ordered permutations(POP) of the merged { α } and { β } sets. To be clear we are referring to leg labels, e.g. in s ...k ,with labels 4 through k as the first ( k −
3) entries of the ordered list { α, β } . Equation (2.31)gives all permutations of { α } S { β } consistent with the order of the { β } elements. Either α or β may be empty, trivially so for the α case. The function F k associated with leg k isgiven by, F k ( { ρ } ) = ( P m − l = t k S k,ρ l if t k − < t k − P t k l =1 S k,ρ l if t k − > t k ) + s ...k if t k − < t k < t k +1 − s ...k if t k − > t k > t k +1 , (2.32)where s ij...k ≡ ( p i + p j + · · · + p k ) , (2.33)and t k is the position of leg k in the set { ρ } , except for t and t |{ α }| +4 which are alwaysdefined to be, t ≡ t , t |{ α }| +4 ≡ . (2.34)For |{ α }| = 1 this means that t = t = t |{ α }| +4 = 0. The expression S i,j is given by, S i,j = ( s ij if i < j or j = 1 or j = 30 otherwise ) . (2.35)24 THE DUALITY BETWEEN COLOR AND KINEMATICS
14 3 25 g
14 2 35 g
12 4 35 g
12 5 34 g
23 1 45 g
13 5 24 g
34 1 25 g
23 5 14 g
24 1 35 g
34 2 15 g
23 4 15 g
24 3 15 g
12 3 45 g
13 2 45 g
13 4 25 g Figure 7: The color-dressed tree-level five-point amplitude an be organized using these fifteengraphs with only cubic vertices.
The so-called fundamental
BCJ relations (Eq. (2.26)) occur when the |{ α }| = 1. Theseamplitude relations were first identified in Ref. [1], and then proven, first as a low-energylimit of string-theory relations [104, 105], and then directly using the Britto-Cachazo-Feng-Witten (BCFW) recursion relations in field theory [113, 115].Consider the five-point amplitude (e.g. governing two-to-three scattering), which offersa first nontrivial example. In this case, 15 distinct cubic diagrams contribute, as illustratedin Fig. 7. Only five of these contribute to a given color-ordered partial amplitude. Let usconsider diagram nine from Fig. 7. To see which color-orderings (and with which signs) thisdiagram can contribute, we expand its canonical color-factor in the trace basis. The colorfactors follow from dressing with the structure functions ˜ f abc . Going to a trace basis we seethat the color weight associated with diagram nine is: c = Tr [ T a T a T a T a T a ] − Tr [ T a T a T a T a T a ] + Tr [ T a T a T a T a T a ]+ Tr [ T a T a T a T a T a ] − Tr [ T a T a T a T a T a ] − Tr [ T a T a T a T a T a ] − Tr [ T a T a T a T a T a ] + Tr [ T a T a T a T a T a ] . (2.36)This implies that diagram nine will contribute to multiple color-ordered partial amplitudes,defined as the coefficient of each color trace in the full amplitude, with a variety of signs. Thesigns associated with each diagram in a partial amplitude are easily determined for a givencolor ordering by reordering the legs of each diagram to match the color ordering withoutallowing lines to cross, and keeping track of the signs from permuting the ordering of legs ineach vertex.Taking the layout as depicted in Fig. 7, each diagram contributes to a color-orderedpartial amplitude according to whether we can flip the legs at each vertex (with a minussign for each flip) so that the cyclic ordering of legs matches the ordering of the argumentsof the partial amplitudes. For example, we have, iA tree5 (1 , , , ,
2) = n D + n D + n D − n D − n D , (2.37)25 THE DUALITY BETWEEN COLOR AND KINEMATICS as well as iA tree5 (1 , , , ,
4) = n D + n D + n D + n D + n D , (2.38)where the n i are the kinematic numerators and the 1 /D i are the products of Feynmanpropagators that can be read of from graph g i in Fig. 7.Jacobi relations imply that the n i of the diagrams in Fig. 7 are given as linear functionsof numerators { n , n , n , n , n , n } , which we take as the master numerators. In total thereare nine independent Jacobi relations, n = n − n , n = n − n , n = n − n , n = n − n , n = n − n ,n = n − n , n = n − n , n = n + n , n = n − n . (2.39)Solving this system in terms of the six master numerators gives n = − n + n , n = − n + n , n = − n + n , n = − n + n ,n = − n + n , n = n − n , n = n − n + n − n ,n = − n + n − n + n , n = n − n + n − n . (2.40)Remarkably, by using Eq. (2.40), we can show that the partial amplitudes (2.37) and(2.38) contain all information necessary to describe all other ordered amplitudes at fivepoint. For the sake of argument, solving Eqs. (2.37) and (2.38) for n and n gives: n = iD A tree5 (1 , , , , − n D D − n D D + ( n − n ) D D + ( n − n ) D D ,n = iD A tree5 (1 , , , , − n D D − n D D + ( n − n ) D D + ( n − n ) D D , (2.41)where we have replaced n and n with the master numerators, using Eq. (2.40). Using this,we express any other partial amplitude in terms of A tree5 (1 , , , ,
2) and A tree5 (1 , , , ,
4) byplugging in the solution (2.40) for these expressions for n and n . Consider, for example,the partial amplitude: iA tree5 (1 , , , ,
4) = − n D − n D − n D − n D + n D . (2.42)Jacobi relations constrain n = − n + n , and n = − n + n . Replacing all non-masternumerators with master numerators using Eq. (2.40), in conjunction with Eq. (2.41), wefind: iA tree5 (1 , , , ,
4) = iA tree5 (1 , , , , D D + iA tree5 (1 , , , , − − D D ! − D + 1 D + D D D + 1 D + D D D + D D D ! n + D + D D D + D D D + D D D ! n + D + 1 D + D D D + 1 D + D D D + D D D ! n − D D D + 1 D + D D D + D D D ! n . (2.43)26 THE DUALITY BETWEEN COLOR AND KINEMATICS
Using the explicit value of the propagators, a dramatic cancellation occurs under momentumconservation: A tree5 (1 , , , ,
4) = − A tree5 (1 , , , , (cid:18) s s (cid:19) + A tree5 (1 , , , , s s − n ( s + s + s + s ) s s s + n ( s + s + s + s ) s s s + n ( s ( s + s ) + s ( s + s ) + s ( s + s )) s s s s − n ( s ( s + s ) + s ( s + s ) + s ( s + s )) s s s s = − A tree5 (1 , , , , s s ! + A tree5 (1 , , , , s s . (2.44)All the coefficients in front of the remaining explicit numerators vanish, giving A tree5 (1 , , , , A tree5 (1 , , , ,
2) and A tree5 (1 , , , , n , n , n , n , once n and n are chosen as in Eq. (2.41), is that we can choose toset the former numerators to zero since they have no effect on any of the partial amplitudes.This forces n and n to be nonlocal since they must absorb the propagators of the diagramswhose numerators are set to zero. In this section we briefly review the Kawai-Lewellen-Tye (KLT) formulae for gravity tree-level amplitudes [86], first derived using string theory. We will show how they are intimatelytied to the BCJ double copy in Eq. (2.11), and how they can be used to directly constructduality-satisfying numerators in purely-adjoint gauge theories.Let us start by quoting the explicit KLT relations at three-, four-, five- and six-points, M tree3 (1 , ,
3) = iA tree3 (1 , , e A tree3 (1 , , , M tree4 (1 , , ,
4) = − is A tree4 (1 , , , e A tree4 (1 , , , , M tree5 (1 , , , ,
5) = i s s A tree5 (1 , , , , e A tree5 (1 , , , , i s s A tree5 (1 , , , , e A tree5 (1 , , , , , M tree6 (1 , , , , ,
6) = − is s A tree6 (1 , , , , , (cid:16) s e A tree6 (2 , , , , , s + s ) e A tree6 (2 , , , , , (cid:17) + P (2 , , , (2.45)where the M tree n are tree-level gravity amplitudes and the A tree n are color-ordered gauge-theory partial amplitudes, and P (2 , ,
4) represents a sum over all permutations of leg labels2 , , and 4.If A and ˜ A are the tree-level amplitudes of D -dimensional pure YM theory, then the mapbetween the two sets of on-shell gluon polarization vectors ε iµ , with SO ( D −
2) little-group27
THE DUALITY BETWEEN COLOR AND KINEMATICS indices i , and those of the double-copy fields can be made explicit,( ε h ) ijµν = ε (( iµ ε j )) ν (graviton) , ( ε B ) ijµν = ε [ iµ ε j ] ν ( B -field) , (2.46)( ε φ ) µν = ε iµ ε jν δ ij D − . On the first line the gluon polarizations are multiplied in symmetric-traceless combinationscorresponding to the ( D − D − − ( D − D −
3) states of an antisymmetric tensorfield. The completeness of the set of gluon polarization vectors implies that the right-handside of the third line of Eq. (2.46) is proportional to η µν up to momentum-dependent terms,so ( ε φ ) µν describes a single state. Adding them all up we find the ( D − states in thetensor product of two massless vectors, as we should.The double copy of D -dimensional pure YM theory gives gravity amplitudes M tree thatfollow from the Lagrangian [183, 184] S = Z d D x √− g " − R + 12( D − ∂ µ φ∂ µ φ + 16 e − φ/ ( D − H λµν H λµν , (2.47)where H λµν is the field strength of the two-index antisymmetric tensor B µν and the non-canonical normalization of the dilaton quadratic term is chosen to avoid non-rational depen-dence on the spacetime dimension D . The Z symmetry B µν → − B µν generates a consistenttruncation of this Lagrangian to Einstein gravity coupled to φ . The further Z symme-try of this truncation, φ → − φ , allows a further consistent truncation to Einstein gravity.The double copy analog of this truncation is realized by choosing gluon polarizations insymmetric-traceless combinations, as for the graviton polarizations in Eq. (2.46).To show the connection to the BCJ double copy, consider, for example, the five-pointtree amplitude. The double-copy amplitude in terms of Jacobi-satisfying numerators is, M tree5 (1 , , , ,
5) = − i X i =1 n i ˜ n i D i = − i ˜ n (cid:18) n D + n D + n D + n + n D + n + n D (cid:19) − i ˜ n (cid:18) n D + n D + n D + n + n D + n + n D (cid:19) , (2.48)where we used the solution (2.40) and the propagators 1 /D i can be read off from the diagramsin Fig. 7. As usual, where we suppress an overall factor of ( κ/ . Remarkably, after usingthe solution (2.40) for both the n i and ˜ n i , the result depends only on the numerators n , n and ˜ n , ˜ n . Using Eq. (2.41), we have, M tree5 (1 , , , ,
5) = − (cid:16) ˜ n A tree5 (1 , , , ,
5) + ˜ n A tree5 (1 , , , , (cid:17) = i s s A tree5 (1 , , , , e A tree5 (1 , , , , i s s A tree5 (1 , , , , e A tree5 (1 , , , , . (2.49)28 THE DUALITY BETWEEN COLOR AND KINEMATICS m m − Figure 8: An m -point half-ladder tree diagram. This implies that we can express the double copy in terms of partial tree amplitudes of thetwo gauge theories.This structures applies also at higher points, and is captured by the m -point formula [86]: M tree m = − i X σ,ρ ∈ S m − (2 ,...,m − A tree m (1 , σ, m − , m ) S [ σ | ρ ] e A tree m (1 , ρ, m, m − , (2.50)where we suppress an overall factor of ( κ/ m − . The formula makes use of a matrix S [ σ | ρ ]known as the field-theory KLT kernel. This is an ( m − × ( m − m − S [ σ | ρ ] = m − Y i =2 " p · p σ i + i X j =2 p σ i · p σ j θ ( σ j , σ i ) ρ , (2.51)where θ ( σ j , σ i ) ρ = 1 if σ j is before σ i in the permutation ρ , and zero otherwise. A compactdefinition, which reproduces Eq. (2.51) upon use of momentum conservation and on-shellconditions, can be given recursively as [169], S [ A, j | B, j, C ] = 2( p + p B ) · p j S [ A | B, C ] , S [2 | = s , (2.52)where multiparticle labels B = ( b , b , . . . , b p ) involve multiple external legs, and we use thenotation p B = p b + p b + . . . + p b p . Using the recursive formula, we can obtain four-, five-and six-point KLT relations as particular cases.In addition, the field-theory KLT kernel allows us to find explicit expressions for duality-satisfying tree-level numerators in the purely-adjoint case. The construction that we givehere was independently worked out in refs. [24, 149]. The idea is to define the numeratorsfor a subset of diagrams called half-ladder (or multi-peripheral) diagrams, whose structure isillustrated in Fig. 8. Corresponding to permutations of these half-ladder diagrams we specify( m − n (1 , σ (2 , . . . , m − , m − , m ) = − i X ρ ∈ S m − S [ σ | ρ ] e A tree m (1 , ρ, m, m − ,n (1 , τ (2 , . . . , m − , m ) (cid:12)(cid:12)(cid:12)(cid:12) τ ( m − = m − = 0 , (2.53)and take the remaining (2 m − − ( m − This recursive presentation of the KLT kernel has a string theory origin [24]. THE DUALITY BETWEEN COLOR AND KINEMATICS they obey the CK duality. However, to conclude that they define valid numerators we mustalso prove that they give correct amplitudes, both in gauge theory and in gravity.Consider the DDM decomposition introduced in Eq. (2.21) for gauge-theory amplitudeswith only adjoint particles. We use CK duality to replace the color factors in that formulawith the above defined numerators, M tree m = A tree m (cid:12)(cid:12)(cid:12)(cid:12) c i → n i = X τ ∈ S m − A tree m (cid:16) , τ (2 , . . . , m − , m ) n (1 , τ (2 , . . . , m − , m )= − i X σ,ρ ∈ S m − A tree m (cid:16) , σ, m − , m (cid:17) S [ σ | ρ ] e A tree m (1 , ρ, m, m − . (2.54)On the first line we have a DDM decomposition for gravity amplitudes, where the half-laddernumerators play the same role as the half-ladder color factors in Eq. (2.21). On the secondline we have plugged in the explicit numerators, and used the fact that only ( m − A tree m and e A tree m are gauge-theory amplitudes.If we take A tree m to be amplitudes in bi-adjoint φ -theory and e A tree m are YM amplitudes, thenthe above KLT formula gives back YM amplitudes. Hence the numerators in Eq. (2.53) givecorrect amplitudes.This completes the constructive proof showing that CK duality can be satisfied for gaugetheories with adjoint particles, given that all the BCJ amplitude relations (2.31) hold, whichimplies that the KLT formula hold. This argument relies on the availability of a DDMrepresentation of the amplitude and on the existence of the field-theory KLT kernel. PureYM theory, or N = 1 , , We now generalize the discussion in the previous subsection by introducing matter in thefundamental representation, as it appears in Quantum Chromodynamics (QCD) [188–190].To be specific, let us consider YM theory with gauge group G and with N f fundamentalfermions. For simplicity we call this theory QCD, given that it precisely matches QCDonce we specify the gauge group to be SU (3) and the number of quark flavors to be six; its A similar example of YM theory with scalars in matter representations will be discussed in Sec. 5.2. THE DUALITY BETWEEN COLOR AND KINEMATICS ˜ f abc = c ba c ( T b ) ji = c bi j ( T b ) ji ≡ c bi j = − ( T b ) ji Figure 9: Color vertices with planar ordering consistent with the color-ordered Feynman rules. k \ m Table 1: Number of cubic diagrams, ν ( m, k ), in the full m -point amplitude with k distinguishablequark-antiquark pairs and ( m − k ) gluons. Lagrangian is L QCD = −
14 ( F aµν ) + q α ( i (cid:0)(cid:0) D − M βα ) q β , where D µ = ∂ µ − igA aµ t a , (2.55)where α, β = 1 , . . . , N f are flavor indices, and spinor indices and fundamental gauge groupindices are suppressed. The mass matrix M βα is taken to be diagonal. The only color tensorsare in this case f abc and ( t a ) ji which both have three free indices. Thus, all color factorswill again correspond to cubic diagrams. A difference with the pure-adjoint case is that wenow need to decorate the lines of the diagrams with the appropriate representation: adjoint,fundamental or anti-fundamental. This is illustrated in Fig. 9. A general color decompositionof tree-level amplitudes with matter representations may be found in Ref. [189] (see alsoRefs. [191–193]).Without loss of generality we write the QCD m -point tree amplitude in terms of diagramswith cubic vertices, A tree m,k = − ig m − ν ( m,k ) X i ∈ cubic diag. c i n i D i , (2.56)where c i are color factors, n i are kinematic numerators, and D i are denominators encodingthe propagator structure of the cubic diagrams. The denominators (and numerators) may inprinciple contain masses, corresponding to massive quark propagators. For k quark-antiquarkpairs and ( m − k ) > ν ( m, k ) = (2 m − k − [189]. As exemplified in Table 1, the numbers grow modestly with thenumber of quarks.Amplitudes with multiple quarks of the same flavor and mass can be obtained fromdistinct-flavor amplitudes by setting masses to be equal and summing over permutationsof quarks with appropriate fermionic signs. Therefore, we do not lose generality by takingall k quark-antiquark pairs to have distinct flavor and mass. To be explicit, in Tab. 1 weprovide total counts ν ( m, k ) of cubic diagrams for different amplitudes up to eight particles31 THE DUALITY BETWEEN COLOR AND KINEMATICS n u
412 3 n t
12 34 n s Figure 10: The diagrams contributing to the two-quark two-gluon amplitude. and four quark pairs. It agrees with the usual counting of standard QCD Feynman diagramsrestricted to those diagrams that only have trivalent vertices.The color factors c i in Eq. (2.56) are constructed from the cubic diagrams using only twobuilding blocks: the structure constants ˜ f abc for three-gluon vertices and generators ( T a ) ji for quark-gluon vertices, as shown in Fig. 9. When separating color from kinematics, thediagrammatic crossing symmetry only holds up to signs dependent on the permutation oflegs. These signs are apparent in the total antisymmetry of ˜ f abc . For a uniform treatmentof the fundamental representation, it convenient to introduce a similar antisymmetry for thefundamental generators, ( T a ) ji ≡ − ( T a ) ji ⇔ ˜ f cab = − ˜ f bac . (2.57)This allows us to introduce a similar antisymmetry in color-ordered kinematic vertices, sothat they are effectively the same as for the adjoint representation. As noted in Eq. (2.6)the color factors obey Jacobi and commutation identities. They both imply color-algebraicrelations of the form given in Eq. (2.7), and differ only by the subdiagrams as drawn inFig. 6, but otherwise have common diagram structure. The interdependence among thecolor factors c i means that the corresponding kinematic coefficients n i /D i are in general notunique, as reflected by the underlying gauge dependence of the numerators.A first interesting example of an amplitude is the four-point amplitude for two gluonsand a quark-antiquark pair displayed in Fig. 10, A tree4 , (1¯ q, g, g, q ) = − i n s c s s − m q + n t c t t + n u c u u − m q ! , (2.58)where the numerators are n s = 12 ¯ u / ε (/ p δ α α − M α α )/ ε v , n u = 12 ¯ u / ε (/ p δ α α − M α α )/ ε v , n t = n u − n s , (2.59)and the color factors c s = ( T a T a ) i i , c u = ( T a T a ) i i , c t = c u − c s = ˜ f a a b ( T b ) i i . (2.60)Here the Greek indices α , α are global (flavor) indices carried by the fermions. When useful, we use the slightly-nonstandard notation A n (cid:0) , . . . , n Φ n (cid:1) to display explicitly the ex-ternal states in an amplitude. THE DUALITY BETWEEN COLOR AND KINEMATICS n s
21 34 n t
21 34
Figure 11: The diagrams contributing to the four-point pure-quark amplitude.
Following the same steps as in pure YM theory one can show that the numerator relationtogether with the kinematics constraints at four points imposes massive BCJ relations forthe partial amplitudes,( s − m q ) A tree4 , (1¯ q, g, g, q ) = ( u − m q ) A tree4 , (1¯ q, g, g, q ) . (2.61)More generally, one can understand this relation as a consequence of gauge redundancy.We have two independent numerators, which are not invariant under gauge transformations.We can thus at most build one gauge-invariant quantity out of these, and hence all partialamplitudes must be related.At general multiplicity m , the BCJ amplitude relations in their simplest incarnation takethe form, m − X i =2 p · ( p + . . . + p i ) A tree m,k (2 , . . . , i, g, i + 1 , . . . , m ) = 0 , (2.62)where leg 1 must be a massless gluon in the adjoint. Unlike Eq. (2.26), here the particles2 , . . . , n may have any spin, mass, and gauge-group representation. The partial amplitudeis constructed as a sum over planar Feynman graphs in the same fashion as for the purelyadjoint case; however, the color decomposition for these mixed adjoint-generic-representationamplitudes is quite different. See Refs. [189, 191–193]) for details.As a further nontrivial example at four points, consider the fundamental representationfour-quark amplitude displayed in Fig. 11. This amplitude is given as a sum over twodisplayed diagrams, A tree4 , (1¯ q, q, q, q ) = − ig (cid:18) n s c s s + n t c t t (cid:19) , (2.63)where the color factors are c s = ( T a ) i i ( T a ) i i , c t = ( T a ) i i ( T a ) i i , (2.64)and the kinematic factors are n s = −
12 (¯ u γ µ v )(¯ u γ µ v ) δ α α δ α α , n t = −
12 (¯ u γ µ v )(¯ u γ µ v ) δ α α δ α α . (2.65)For this amplitude, neither the color nor the kinematic factors satisfy any relations amongthemselves, hence CK duality is trivially satisfied. Indeed, each kinematic numerator isgauge invariant by itself and thus the amplitude representation is necessarily unique. Wewill however see in Sec. 5 that in some cases it is possible or even necessary to imposeadditional numerator relations for matter amplitudes without external gluons.33 THE DUALITY BETWEEN COLOR AND KINEMATICS
We now look more in detail at the theory obtained from the double-copy formula with twosets of QCD numerators. It will consist of gravity coupled to a single massless complex scalar,as well as a set of massive photons and scalars. Massless and massive fields in this theoryoriginate from the double copy of adjoint and fundamental gauge-theory fields, respectively.As an example, we give the amplitude between four massive (complex) photons γ , M tree4 (1¯ γ, γ, γ, γ ) = − i n s ( n s | N f → ) s + n t ( n t | N f → ) t ! , (2.66)where we trivialize the number of flavors on one side in order to avoid a redundant descriptionwith a factorized flavor group in the gravitational theory.Writing out the expression we have M tree4 (1¯ γ, γ, γ, γ ) = − i h (¯ u γ µ v )(¯ u γ µ v ) i s δ α α δ α α + h (¯ u γ µ v )(¯ u γ µ v ) i t δ α α δ α α . (2.67)The square can be upgraded to a tensor product since the external spinors can be chosendifferently for the two numerator copies.In order to better understand the double-copy amplitude, we may write it in terms ofchiral spinors and explicitly write out the little group indices. For example, using the massivespinor-helicity variables reviewed in Appendix B, we simplify the above expression to obtain M tree4 (1¯ γ aa , γ bb , γ cc , γ dd ) = − i ((cid:16) h a c i [2 b d ] + [1 a c ] h b d i (2.68)+ h a d i [2 b c ] + [1 a d ] h b c i (cid:17) δ α α δ α α s + (cid:16) ↔ (cid:17)) . Why does the double copy of gauge-theory amplitudes yield amplitudes of some gravitytheory? A minimal criterion is that the expression obtained from the double-copy methodbe invariant under linearized diffeomorphisms. Here we show that invariance of the double-copy amplitudes under linearized diffeomorphisms is a direct consequence of color-kinematicsduality and gauge invariance of the two gauge-theory factors entering the construction.We start from a general linearized gauge transformation acting on a single external gluonwith momentum p . Its polarization vector transforms as: ε µ ( p ) → ε µ ( p ) + p µ . Gaugeinvariance of the amplitude implies that every diagram numerator should shift as n i → n i + δ i , δ i = n i (cid:12)(cid:12)(cid:12)(cid:12) ε → p . (2.69)Then, the entire amplitude is unaffected provided that the shifts δ i obey X i c i δ i D i = 0 , (2.70)which must hold since by assumption the gauge-theory amplitude is gauge invariant.34 THE DUALITY BETWEEN COLOR AND KINEMATICS
Aside from the explicit expressions for the numerator factors, the above equation mustrely exclusively on the generic algebraic properties of the color factors c i , namely antisym-metry and Jacobi identities. This means that, if we have CK-duality-satisfying numerators˜ n i in some gauge theory and consider their double copy with another set of gauge numera-tors n i , then any linearized gauge transform of the n i will leave the double-copy amplitudeinvariant: X i n i δ i D i = 0 . (2.71)We now analyze in more detail the significance of this transformation. A general coordinatetransformation can be used to impose both transversality and tracelessness on the on-shellasymptotic states that enter the definition of a scattering amplitude. This, in turn, resultsin imposing the conditions ε µν ( p ) p ν = 0 = ε µν ( p ) η µν on the graviton’s polarization tensor.After this choice of gauge, amplitudes will still be invariant under the subset of linearizeddiffeomorphisms that do not modify the above conditions. These will act as ε µν ( p ) → ε µν ( p ) + p ( µ q ν ) , (2.72)where q is a reference vector that obeys p · q = 0, but is otherwise generic. The parenthesisdenote symmetrization of spacetime indices.The first step of formulating a double-copy construction is to establish a map betweengravity asymptotic states and pairs of gauge-theory states. In general, the double-copygraviton will be obtained by taking the symmetric-traceless part of the product of the twogauge-theory gluons, i.e. its polarization tensor will be obtained from the gluon’s polariza-tions as ε µν = ε (( µ ˜ ε ν )) , where the double brackets indicate the symmetric-traceless part. We now study tree-level amplitudes obtained from the double-copy method. We take aset of duality-satisfying numerators n i only for one of the gauge-theory factors. The otherset of numerators is taken in the form˜ n i = ˜ n i + ˜∆ i , X i ˜∆ i c i D i = 0 . (2.73)While the numerators ˜ n i can violate CK duality, they can be obtained from a set of duality-satisfying numerators ˜ n i with a transformation of the form (2.8) with parameters ˜∆ i . Hence,we are assuming that there exists an amplitude presentation for which the duality is satisfiedalso for the second gauge theory. However, in the double-copy method, we use a set ofnumerators with different properties for one of the theories, a fact that will be advantageousin practical calculations.Starting from the double-copy gravity amplitude in Eq. (2.11) at tree level, a tree ampli-tude can then be expressed as M n = − i (X i n i ˜ n i D i + X i n i ˜∆ i D i ) = − i X i n i ˜ n i D i , (2.74) We also note that the antisymmetric and trace parts of the product of the two gauge-theory gluonpolarizations are identified with an antisymmetric tensor field and the dilaton. These two field are genericallypresent in amplitudes from the double copy unless additional steps are taken to ensure their removal, as wewill see in Sec. 5.3.4. THE DUALITY BETWEEN COLOR AND KINEMATICS where we have not included the overall ( κ/ n − . Because numerator factors n i obey thesame algebraic relations as the color factors c i , equation (2.73) implies the last equalityabove. Using Eq. (2.74), the variation of the double-copy amplitude under a linearizeddiffeomorphism of the form (2.72) becomes M n → M n − i X i δ i ˜ n i (cid:12)(cid:12)(cid:12) ˜ ε → q D i + X i n i (cid:12)(cid:12)(cid:12) ε → q ˜ δ i D i . (2.75)The two terms are of the form (2.71) and hence vanish because of CK duality. We thenconclude that invariance of the amplitude under linearized diffeomorphisms at tree levelfollows from gauge invariance of the gauge theories entering the double-copy constructionprovided that CK duality is obeyed. Diffeomorphism invariance of the amplitudes at looplevel can also be established through generalized unitarity [194]. We will see in the nextsubsection and in Secs. 4 and 5 that the double copy can also be used to engineer amplitudeswhich are invariant under other symmetries, including supersymmetry and gauge symmetry.In fact, one can think of the double copy as a clever procedure to write down amplitudesthat obey a prescribed set of on-shell Ward identities starting from gauge-theory data. Byconstruction, these amplitudes also obey standard factorization properties as well as crossingsymmetry. The basic intuition is that gauge invariance together with mild assumptions onthe singularity structure are sufficient to fix the form of amplitudes [173, 174]. + duality ⇒ supersymmetry In Sec. 2.4 we discussed CK duality in the context of YM theory with matter fermions. Wenow look at the case of adjoint fermions in arbitrary dimension. In this case, we will see thatthe duality is equivalent to the existence of supersymmetry, as argued in Ref. [30] (see also[117] for a related discussion). For concreteness, we specialize to D -dimensional YM theoryminimally coupled to a single adjoint Majorana fermion, described by the Lagrangian L = Tr (cid:20) − F µν F µν + i ψ (cid:0)(cid:0) Dψ (cid:21) . (2.76)In all dimensions, the four-gluon and two-gluon-two-fermion amplitudes respect the dual-ity between color and kinematics without any further constraint. However, four-fermionamplitudes leads to an interesting constraint. This amplitude is given by A tree4 (1 ψ, ψ, ψ, ψ ) = i (¯ u γ µ v )(¯ u γ µ v ) c s s + (¯ u γ µ v )(¯ u γ µ v ) c t t + (¯ u γ µ v )(¯ u γ µ v ) c u u ! , (2.77)where the ¯ u i and v i are spinor external states which obey ¯ u i γ µ v j = ¯ u j γ µ v i due to theMajorana condition, ¯ u i = v Ti C . In dimensions in which a Weyl representation can be chosenone of the terms above vanishes.The requirement that A tree4 (1 ψ, ψ, ψ, ψ ) obeys CK duality forces the gamma matricesto obey the relation(¯ u γ µ v )(¯ u γ µ v ) + (¯ u γ µ v )(¯ u γ µ v ) + (¯ u γ µ v )(¯ u γ µ v ) = 0 . (2.78)36 THE DUALITY BETWEEN COLOR AND KINEMATICS
Eq. (2.78) is the equivalent to the Fierz identity that appears in the supersymmetry trans-formation of the Lagrangian (2.76). This analysis can be repeated for pseudo-Majoranaspinors with analogous results. Overall, an identity of this form can be satisfied only for D = 3 , , ,
10, i.e. the dimensions for which the theory (2.76) is supersymmetric.The relation between CK duality with adjoint fermions and supersymmetry should notseem surprising in hindsight. In principle, adjoint fermions can be combined with gluonswith the double-copy procedure, resulting in a gravity theory which includes spin-3 / / u αµ ( p ) is u αµ ( p ) → u αµ ( p ) + p µ ξ α , (2.79)where ξ α is the transformation parameter which, in order to preserve the γ -tracelessnessof the gravitino wave function must obey the massless Dirac equation, p / ξ = 0. Then, thetransformation of a double-copy amplitude (after a discussion similar to the one that led toEq. (2.75)) is M tree m → M tree n + X i δ i ˜ n i (cid:12)(cid:12)(cid:12)(cid:12) ˜ u α → ξ α D i , (2.80)where ˜ u α is the spinor that, through the double copy, generates the gravitino under consider-ation. Since the parameter ξ of the supersymmetry transformation has the same propertiesas the original spinor ˜ u α it replaces, the factor ˜ n i (cid:12)(cid:12)(cid:12)(cid:12) ˜ u α → ξ α has the same properties as ˜ n i , inparticular it obeys Jacobi relations. Thus, the variation of the double-copy amplitude under(linearized) supersymmetry transformations vanishes, implying that the double-copy theoryexhibits local supersymmetry.More generally, we may expect that, under the right circumstances, the double copy ofa gauge theory with a theory that exhibits a global symmetry leads to a theory where theglobal symmetry is promoted to a local symmetry. We shall return to this point in Sec. 4.The emergence of supersymmetry from CK duality offers a novel perspective on themaximal number of gravitini that can consistently enter a supergravity theory. As we haveseen, a gauge theory coupled to fermions can exhibit CK duality in at most ten dimensionsdimensions. Thus, this is the highest dimension which a supergravity theory can be givena double-copy interpretation in the sense described here. Taking two such theories givestherefore the largest number of supersymmetries, which is two in ten dimensions or, upondimensional reduction, eight in four dimensions. This observation recovers the usual boundfollowing from the requirement that the exist multiplets of supersymmetry algebra containingfields of spin s ≤
2. 37
GEOMETRIC ORGANIZATION
In the previous subsections, we presented various concrete examples of theories which obeyCK duality. Statements about CK duality often depend on the details of the theories underconsideration and on what observables are being studied. Since the duality is often usedas a shortcut for computing gravitational amplitudes, one can restrict to gauge theoriessuitable for giving broad classes of consistent gravitational theories once the numerators areassembled via the double-copy method. Expanding on the examples discussed earlier, in therest of this review we will focus mostly on theories with the following general features:• There exists (at most) one massless gauge field, the gluon, that transform in the adjointof a gauge group G , and all fields of the gauge theory are charged under this group.We will see in Sec. 5 that this requirement translates to the equivalence principle ingravity.• The gauge group G is a completely general Lie group in the sense that no assumptionson its rank need be imposed on it. Note that throughout this section the only propertiesof the gauge group we have utilized are the Jacobi relations of its structure constantsand the commutation relations of its representation matrices, which do not require tospell out our choice of Lie group.• Amplitudes involving adjoint fields (gluons or adjoint matter) should admit perturba-tive expansions where the kinematic numerators obey the same Lie algebra relations(e.g. Jacobi identities) as the corresponding adjoint-valued color factors. We haveseen in Sec. 2.5 that this property is essential for obtaining a gravitational theoryafter the double copy. This condition also implies the universality of gravitationalself-interactions.• Amplitudes involving fields in generic representations of the gauge group should admitperturbative expansions where the kinematic numerators obey the same Lie algebrarelations as the generators of those representations. The simplest example of non-purely-adjoint theory has been discussed in Sec. 2.4.These general properties guarantee that every diagram in the perturbative expansion of anamplitude has a unique nontrivial color factor, which obeys the minimal constrains imposedby the Lie algebra of the gauge group, and furthermore that the coupling to the uniquegluon is universally controlled by the gauge-group representations. The kinematic factorscan then be constrained to obey the duality by enforcing the one-to-one map between colorand kinematic identities. Along these lines, in Sec. 5.1 we will articulate a more precise setof working rules which will define the properties of the gauge theories employed for obtaininga web of double-copy-constructible theories. The dual Jacobi identities give nontrivial relations between diagram numerators. Here wedescribe the systematics of these relations and how they can be used to express amplitudes’38
GEOMETRIC ORGANIZATION ���� � � ���� � � ���� � � Figure 12: Graph of graphs relevant to four-point tree-level scattering forms a triangle. Each vertexrepresents one of the three graphs in Fig. 3, and every edge represents a Jacobi or Whitehead move(c.f. Fig. 13). Every triangle in the graph of graphs represents a Jacobi relation that can be usedto constrain a dressing of one vertex graph in terms of dressings of the other two vertex graphs. integrands in terms of the contributions of a small set of master diagrams. This is generallyvery helpful at higher perturbative orders because it allows us to express an integrand interms of a (small) subset of all of its terms. To this end, we will first describe a usefulgeometric organization via a graph of graphs that offers insight into the information flow ofthe duality identities. We will then illustrate the general case through some examples. The duality between color and kinematics provides a set of relations between diagrams.Sec. 2 frames the discussion of the duality in terms of vector and matrix operations betweenlinear spaces of numerators of diagrams and linear spaces of scattering amplitudes. Here wegive an alternative perspective, using the language of graphs [196, 197]. This offers a usefulway to visualize how a small set of graphs is sufficient to describe the entire amplitude.We shall see that the minimal set of graphs whose numerators need to be specified can bethought of as boundary data on the graph of graphs describing the amplitude.As a simple example, consider the four-gluon tree amplitude. After absorbing any con-tact terms into graphs with only cubic vertices, this amplitude can be described by thethree graphs in Fig. 3, corresponding to the s -channel, t -channel, and u -channel. Both thekinematic and color numerators of these three graphs satisfy the (dual) Jacobi identities inEq. (1.16) and Eq. (1.17), c s + c t + c u = 0 and n s + n t + n u = 0. These equations areequivalent to the statement that any two “single-copy” numerator dressings determine the See Ref. [195] for a related application of such an approach towards identifying scattering forms ofamplitudes. GEOMETRIC ORGANIZATION ˆtˆu
12 3 mda bc
12 3 mda bc d ca b mda bc
12 3 m Figure 13: Operations that relate edges of one graph to edges of another. third one. We can draw this relationship as a graph of graphs, where each vertex representsa specific graph participating in the Jacobi relation and the edges connect to the other twovertices (i.e. graphs) that determine the first vertex.These relations can be summarized in the triangle shown in Fig. 12. Each vertex or nodein Fig. 12 is one of the three cubic graphs from Fig. 3 that would contribute to the four-pointamplitude. Every edge in this graph of graphs corresponds to a Jacobi move on the internalpropagator of one of the vertex graphs that transmutes it into another. In the mathematicsliterature these moves are known as Whitehead moves [198]. The basic moves for acting ongraphs represented by the edges in Fig. 12 are denoted ˆ t and ˆ u and are shown in Fig. 13.The first move, which we call ˆ t , takes the s -channel graph in Fig. 3 and converts it to the t -channel. Similarly, we call ˆ u the move that converts the s -channel graph to the u -channelone. The move that takes the t -channel graph to the u -channel graph can be understoodas the composition ˆ u ◦ ˆ t . Alternatively, we can view it as one of the same basic operationsas on the s -channel graph but acting on a graph with permuted labels. Strictly speaking,one should associate a direction with each move, but we will ignore this distinction because,up to relabelling, the reverse operation is identical to the forward one. It is not difficultto see that each edge in the triangle graph of graphs contains the graphs contributing to aparticular four-point color-ordered partial amplitude, and the entire triangle itself representsan occasion for Jacobi to be satisfied by a dressing of the graphs (whether color or kinematic).This basic structure generalizes straightforwardly to higher-point amplitudes. Considerit at five points: in total there are 15 distinct cubic graphs (constructible by starting withone five point cubic graph and applying ˆ t and ˆ u on each of it’s internal edges, and repeatingon new graphs until closure) contributing to the full color-dressed integrand as displayed in40 GEOMETRIC ORGANIZATION
Figure 14: Graph of graphs for the full color-dressed five-point partial amplitude, with vertexlabels corresponding to the graphs given in Fig. 7. Two color-ordered partial amplitude graphs arehighlighted, corresponding to A tree5 (1 , , , ,
2) and A tree5 (1 , , , , Fig. 14.Exercise 3.1: Draw individual five-point graphs for each vertex in Fig. 14. Label the edgeoperations to get from graph to graph.Let us focus on the subset of five graphs comprising the ordered partial amplitude A tree5 (1 , , , , t toits two internal edges to find and connect two other graphs with the same color order. It isinteresting to note that each edge in the graph represents a shared factorization channel forthe internal propagator not mutated between the two connected graphs. We keep applying ˆ t until closure. This procedure of repeated ˆ t application, building a graph of all the graphs ofa given color-order, carves out the skeletal graph of a polytope known as the associahedron .(cf. [200, 201]). Associahedra are often also called Stasheff polytopes. As ˆ t preserves colororder, each Stasheff skeletal graph is composed of all the graphs that contribute to a fixedcolor order [197]. Indeed every Stasheff subgraph of the full graph of graphs represents thecontributions of a particular ordered partial amplitude to the full amplitude. This is exem-plified by the two highlighted pentagonal subgraphs of Fig. 14 which represent the ordered Typically called a one-skeleton (cf. Ref. [199]). GEOMETRIC ORGANIZATION � �� �� � � � �� �� � � � �� �� � � � �� �� � �� � �� �� � � Figure 15: Graph of graphs relevant to the color-ordered five-point partial amplitude A tree5 (1 , , , , t on vertex-graph edges until closure. amplitudes A tree5 (1 , , , ,
4) and A tree5 (1 , , , , m − m − GEOMETRIC ORGANIZATION
Figure 16: Due to the nine Jacobi relations (six outer triangles, and three inner triangles), only thesix outer boundary graphs are needed to specify bulk data via Jacobi relations. color-ordered amplitudes do they correspond to? Is it a KK basis?The counting works differently for an amplitude where both the kinematic numeratorsand the color factors obey Jacobi identities. Consider the external boundary graphs ofFig. 16. As noted above, Jacobi identities are represented by triangles in the graph ofgraphs. By working inwards using Jacobi identities in Fig. 16, we see that each pair ofneighboring graphs in the six external boundary graphs completely specify all other graphs.So, for theories that can satisfy the dual Jacobi identities, we need only specify data on thisexternal boundary. More generally, for an m -point amplitude, it turns out that only ( m − m − u move on allinternal edges until no new graphs are generated. All graphs so generated will remain half-ladders with different labels and their corresponding graph of graphs forms the one-skeletonof a polytope known as a permutahedron [201].Exercise 3.4: Why is at least one half-ladder graph required at any multiplicity in the setof Jacobi master-graphs?As discussed in Sec. 2, the mismatch between the number of independent gauge-theoryamplitudes and the number of independent numerators leads to a gauge freedom that allows43 GEOMETRIC ORGANIZATION some numerators to take on arbitrary values, neither altering the amplitudes nor the BCJamplitude relations. At five points, four of the numerators can be arbitrarily chosen, i.e.even set to zero, making the remaining numerators nonlocal. In addition, the fact that thereare only ( m − m − m − m . While this is a reductionover the total (2 m − As a warm-up before turning to gauge theory, we consider the NLSM [202], as defined bythe Lagrangian in the Cayley parameterization [169, 203, 204], L NLSM = 12 Tr ( ∂ µ ϕ − λϕ ∂ µ ϕ − λϕ ) , (3.1)where ϕ is a Lie-algebra valued Goldstone-boson scalar field in the adjoint representation.Here the color symmetry is global. Although this theory has only even-point interactions,we can assign its data to graphs with only cubic vertices by multiplying and dividing byappropriate inverse propagators. In terms of diagrams with only cubic vertices, dimensionalanalysis dictates that each vertex effectively carries two powers of momentum. The firstnonvanishing amplitude is at four points. We will see that, by imposing Jacobi identitiesand relabeling symmetry on the kinematic numerators of the diagrams, we can obtain thefour-point scattering amplitude of this theory. At higher points, we will also either need toimpose additional conditions to uniquely fix the amplitudes to this theory. It is sufficientto impose the manifestation of only quartic poles in amplitudes [205], which in combinationwith color-kinematics encodes the necessary [206–208] vanishing soft-scalar limits, or Adlerzero conditions [209]. Higher-derivative deformations of the NLSM and their compatibilitywith the KK and the BCJ amplitudes relations have been discussed in [210]To start the construction of the four-point amplitude, consider the four-point half-laddergraph. Since we require the dimensions of the numerator to match that following fromthe NLSM Lagrangian and therefore carry four powers of momentum, we take a numeratoransatz: n ( a, b, c, d ) ≡ n a b c d = s ab ( αs ab + βs bc ) , (3.2)where s ab = ( p a + p b ) and α and β are to be constrained by symmetry and the kinematicJacobi identities. The other kinematic invariant is s ac = − s ab − s bc , and it is not independent.44 GEOMETRIC ORGANIZATION
Imposing the Jacobi constraints, n ( a, b, c, d ) = n ( c, a, d, b ) + n ( d, a, b, c ) , (3.3)relates α and β according to0 = αs ab − αs bc − βs bc s ac − αs ac = − αs ab s bc + βs ab s bc − αs bc + βs bc , (3.4)where we used s ac = − s ab − s bc . Given that the Mandelstam invariants s ab and s bc areindependent, we find β = 2 α ; thus, the numerator is uniquely fixed up to its overall scale,which may be identified as the coupling of the model: n ( a, b, c, d ) ∝ s ab ( s ab + 2 s bc ) . (3.5)One can verify that this expression satisfies all necessary antisymmetry constraints, e.g. itchanges sign with a ↔ b or c ↔ d . The full amplitude, up to overall normalization is then A treeNLSM ∝ c (1 , , , n (1 , , , s + c (3 , , , n (3 , , , s + c (4 , , , n (4 , , , s , (3.6)where the color factors are obtained by dressing each vertex with a structure constant f abc .The key lesson is that, by taking the numerators to be functions of the graph labels, only asingle numerator needs to be specified.Exercise 3.5: Repeat the above analysis assuming only degree-one monomials in the Man-delstam invariants. What theory could this construction correspond to?Next, consider the case of YM theory at four points. In this case the numerator ansatzis constructed out of external momenta and polarization vectors { ε , . . . , ε } , subject tothe requirements that every ε i appears once in each term and that every term has exactlytwo momenta. These constraints guarantee consistency with the structure of Feynman rules.The possible third-degree monomials are constructed from the following independent Lorentzinvariants: (cid:26) s , s , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( p · ε ) , ( ε · ε ) , ( ε · ε ) , ( ε · ε ) , ( ε · ε ) , ( ε · ε ) , ( ε · ε ) (cid:27) . (3.7)There are 30 possible combinations, leading to an ansatz with an equal number of param-eters. Besides constraining it with the kinematic Jacobi identity (3.3), we also impose theantisymmetry constraints at the two vertices, n ( a, b, c, d ) = − n ( a, b, d, c ) = − n ( b, a, c, d ) , (3.8)matching the antisymmetry of the color factors. See Sec. 2.1, Eq. (2.17) and discussion below it. GEOMETRIC ORGANIZATION
Applying these constraints on the ansatz built from the monomials in Eq. (3.7) fixes allbut five of the ansatz’ coefficients. Further imposing gauge invariance on one external legthen fixes the form of the numerator. In fact, it is sufficient to impose gauge invariance onone leg when the amplitude is factorized on the pole of a given channel, i.e. for s ab → n ( a, b, c, d ) | s ab → , and ε a → p a → . (3.9)This gives n ( a, b, c, d ) ∝ (cid:26)(cid:20) ( ε a · ε b ) p µa + 2( ε a · p b ) ε µb − ( a ↔ b ) (cid:21)(cid:20) ( ε c · ε d ) p cµ + 2( ε c · p d ) ε dµ − ( c ↔ d ) (cid:21) + s ab (cid:20) ( ε a · ε c )( ε b · ε d ) − ( ε a · ε d )( ε b · ε c ) (cid:21)(cid:27) , (3.10)in agreement with Eq. (1.12). We note that, as explained in Refs. [173, 174], one can alsodetermine the amplitude using other constraints, in particular from gauge invariance andmild assumptions on the singularity structure.Exercise 3.6: Verify explicitly that the four-point YM numerator given above satisfies theJacobi constraint.Emboldened by the success to obtain four-point amplitude by imposing dual-Jacobi re-lations, we continue to the next multiplicity for the NLSM. As for four points, the dualityinvolves not only imposing kinematic Jacobi relations, but also the same antisymmetry car-ried by color factors. For the half-ladder numerators, n ( a, b, c, d, e ) ≡ n a b c ed , (3.11)these antisymmetry constraints read: n ( a, b, c, d, e ) = − n ( a, b, c, e, d ) = − n ( b, a, c, d, e ) = − n ( d, e, c, a, b ) . (3.12)The Jacobi identities corresponding to the two independent propagators of the five-pointhalf-ladder graphs are: n ( a, b, c, d, e ) = n ( a, c, b, d, e ) + n ( c, b, a, d, e ) ,n ( a, b, c, d, e ) = n ( a, b, d, c, e ) + n ( b, a, e, c, d ) . (3.13)One can immediately see the need to impose two Jacobi relations from the two triangles thattouch every vertex in the graph of graphs for the five-point tree as drawn in Fig. 16.At five points, we have a 35-parameter ansatz comprised of all degree-three monomialswith factors from { s ab , s ac , s ad , s bc , s bd } . (3.14)The constraints in Eqs. (3.12) and (3.13) fix 34 parameters, leaving us with a unique expres-sion, up to an overall coefficient, n ( a, b, c, d, e ) ∝ ( s ac + s bc ) ( s ad ( s bd + s be ) − { a ↔ b } ) . (3.15)46 GEOMETRIC ORGANIZATION
Remarkably, the five-point amplitudes obtained from these numerators actually vanish, inline with the fact that odd-point amplitudes vanish in the NLSM. Moreover, this amplitudevanishes without having to impose the requirement that the underlying theory has no three-point vertex (i.e. that there is no two-particle factorization channel). This in turn suggeststhat there do not exist CK-satisfying scalar two-derivative theories with only fields in theadjoint representation that are not the NLSM.Exercise 3.7: Verify that the above numerator satisfies the two independent Jacobi relationsat five points.Exercise 3.8: Verify explicitly that a color-ordered amplitude, say A tree5 (1 , , , , s ... k → s ... k A tree n (1 , . . . , n ) = 0 , (3.16)or, equivalently, the vanishing of the residue of the simple pole in s ... k : X states A tree2 k +1 (1 , . . . k, p ) A tree n − k +1 ( − p, k + 1 , n ) (cid:12)(cid:12)(cid:12) p =0 = 0 . (3.17)These conditions guarantee recursively consistency with the vanishing of the odd-point treeamplitudes with multiplicity smaller than n . Let us illustrate this for the six-point amplitude.For our scalar theory, we have nine independent external momentum invariants; from themwe can construct 495 degree-four monomials thus obtaining a 495-parameter ansatz for thehalf-ladder graph. CK duality alone constrains all but 23. Imposing the vanishing of thecolor-ordered factorization X states A tree3 (1 , , p ) A tree5 ( − p, , , ,
6) = 0 , (3.18)leaves six unconstrained parameters. Although individual diagrams depend on them, theseparameters always appear in the same linear combination in front every color-ordered par-tial amplitude, which indeed reproduce the six-point partial NLSM amplitudes, up to theoverall normalization. Other factorization limits, e.g. 0 = Res s =0 ( A tree6 (1 , , , , , P states A tree3 (2 , , p ) A tree5 ( − p, , , , GRAVITY SYMMETRIES AND THEIR CONSEQUENCES for building tree-level and low-loop numerators that satisfy the kinematic Jacobi identi-ties [14, 24–26, 28, 29, 149, 156, 211–216].In general, higher-loop integrands is a more involved problem, perhaps more in gauge the-ories than for the NLSM. Direct approaches based on constraining ansätze have proven aneffective means of generating gauge-theory loop integrands [6, 217]. We will see explicit exam-ples in Sec. 6. It turns out, however, that it can be difficult to find gauge-theory numeratorsthat manifest the duality between color and kinematics thus complicating the constructionof corresponding gravity integrands. Nevertheless, a generalized double-copy procedure out-lined in Sec. 7 can be used to convert gauge-theory integrands in generic representations tointegrands in gravity theories; for example, this procedure was used to obtain the five-loopfour-point integrand of N = 8 supergravity and determine its UV behavior [38, 218]. Symmetries are essential for understanding the properties of gauge and gravity theories. Inthe context of the framework provided by CK duality and the double copy, which relatesQFTs order-by-order in perturbation theory, it is hence interesting to explore how sym-metries originate and transfer. Not all symmetries of double-copy theories are currentlywell-understood from this perspective; likewise, the consequences of certain symmetries ofsingle-copy parent theories have yet to be properly understood. In this section we reviewthe current status of the relation between the symmetries of the single- and double-copy the-ories. We begin by outlining which (part) of the symmetries of a Lagrangian can be identifiedand analyzed through scattering-amplitude techniques emphasizing that, while the linearizedpart of symmetries can be manifest, nonlinear symmetries affect only special momentum con-figurations of scattering amplitudes. We then proceed to discuss the linearly-realized globalsymmetries and to extend the diffeomorphism and local-supersymmetry discussion in Sec. 2to also include nonabelian gauge symmetry. All these symmetries are inherited from thesymmetries of their single-copy parents. We then discuss certain enhanced symmetries, i.e.symmetries which, while unrelated to any of the single-copy symmetries, act linearly on thedouble-copy asymptotic states. The ability to efficiently compute amplitudes and analyzethem for special momentum configurations is essential to explore the emergence of nonlinearsymmetries in the double copy.
Lagrangians exhibiting nonlinear symmetries—such as supersymmetry in a formulation with-out auxiliary fields, or nonabelian gauge symmetries—are usually constructed through aniterative Noether procedure. One starts with the free-field theory with the desired spectrum,which is invariant under the linearized form of the desired symmetries and simultaneouslydeforms the action and the transformation rules such that the resulting action is invariantoff-shell under the deformed transformations. The resulting symmetry algebra closes up to Moreover, symmetries of sectors of a single-copy parent theories that relate to the gauge group—such assymmetries of the planar sector–seem difficult to capture because of “contamination" from other sectors. GRAVITY SYMMETRIES AND THEIR CONSEQUENCES the equations of motion. Thus, this approach leads to actions and transformation rules ofthe form S = S + S + S + ... ,δ = δ (0) + δ (1) + δ (2) + ... , (4.1)where the n -field term S n in the action determines the ( n − δ ( n − in thetransformation rules. For example, δ (0) S + δ (1) S = 0 ,δ (0) S + δ (1) S + δ (2) S = 0 . (4.2)The first relation implies that the cubic term is invariant under the undeformed transforma-tions up to terms proportional to the free equations of motion.Quantum mechanically, symmetries are realized through Ward identities, which relatetime-ordered correlation functions of the fundamental fields of the theory. Nonlinear trans-formation rules imply that the relevant Ward identities contain correlation functions of dif-ferent multiplicities. For example, for a transformation δφ ∝ φ k , an n -point Green’s functionis related to ( n + k − k ≥
2, these k fields are at the same spacetime point(s).Upon Lehmann-Symanzik-Zimmermann (LSZ) reduction, Ward identities simplify con-siderably. For asymptotic states with momenta p , . . . , p n , the amputation leading to the n -point amplitude selects the most singular term, proportional to Q ni =1 ( p i − m i ) − . For an( n + k − k = 2 , , ... ) Green’s function resulting from a nonlinear term in a (symme-try) transformation, momentum conservation requires that it has a different pole structure.Thus, all such terms are amputated away and all effects of nonlinear terms in the symmetrytransformations that underlie the structure of off-shell Ward identities are projected out bythe LSZ reduction. The resulting on-shell Ward identities imply that, for generic momenta,S-matrix elements are invariant only under the linearized symmetry transformations. Thisargument fails when the additional fields appearing in nonlinear symmetry transformationsall carry vanishing momenta; indeed, in this case, the off-shell ( n + k )-point Green’s func-tion develops a pole Q ni =1 ( p i − m i ) − and gives a nonvanishing contribution after the LSZ n -point amputation. We shall return in Sec. 4.5 to this special momentum configuration andinterpret it as the soft limit of a higher-point scattering amplitude.It is not difficult to identify these features in nonabelian YM theory: the amplitudesvanish if the polarization vector of a gluon ε µ ( p ) is replaced by the momentum δ (0) A µ = ∂ µ Λ −→ δε µ ( p ) = p µ Λ( p ) . (4.3)They are also invariant under the global part of the gauge group, which is the only remnantof the nonlinearity of the gauge transformation. Here we assume that external states have generic masses m i . Assuming from the outset that externalstates are massless does not alter the conclusion. It is worth mentioning that, from the perspective of the gauge-fixed theory, the transformation ε µ → ε µ + Λ( p ) p µ can also be interpreted as ε µ not being a proper Lorentz vector [173]. Indeed, the polarizationvector is constrained to obey p · ε = 0 so, on shell, any transformation of ε can include a shift by p µ . GRAVITY SYMMETRIES AND THEIR CONSEQUENCES
Not all symmetry transformations have a linearized approximation. An outstandingclass of examples are the U-duality symmetries of extended supergravity theories, such asthe E duality group of N = 8 supergravity. It turns out that only their maximal compactsubgroup, which is isomorphic to the on-shell R -symmetry group, has such an approxima-tion. It is therefore an interesting question whether on-shell methods can probe symmetrytransformations which are inherently nonlinear.A possible approach, put forward in Ref. [219] and further explored in Refs. [220, 221],effectively amounts to constructing the quantum one-particle irreducible (1PI) effective ac-tion and studying its symmetries. Indeed, the quantum 1PI effective action is determined by the S-matrix of the theory up to terms proportional to the free equations of motion.Consequently, up to the corresponding contact terms and assuming absence of anomalies,the off-shell Ward identities of all symmetries—in particular of the nonlinear ones—shouldhold. In this formalism, anomalies appear as violations of the Ward identities of the corre-sponding symmetries which cannot be removed by the addition of finite local countertermsto the (effective) action. These counterterms may be simultaneously interpreted both aspart of the definition of the theory and as an ambiguity in the construction of the effectiveaction from the S-matrix.Another approach geared towards the exploration of nonlinearly-realized symmetries wasfirst described in Ref. [222] for the E symmetry of N = 8 supergravity in four dimensions.It amounts to (1) the vanishing of scattering amplitudes in the limit in which momenta ofone scalar field vanish and (2) the identification/extraction of the structure constants of thenonlinearly-realized part of the symmetry group from the limit in which two scalar fields havevanishing momenta. In Sec. 4.5 we will outline this approach and summarize some of itsmany generalizations to nonlinearly-realized (Volkov-Akulov) supersymmetry [220, 223, 224],Bondi-Metzner-Sachs (BMS) symmetry [225–227], anomalous symmetries [228], effective the-ories [229], string theory [230–232] and theories with spontaneously-broken conformal invari-ance [233]. For discussions of soft theorems at the quantum level see Refs. [234–238]. Whileneither of these two approaches is specifically tied to the double-copy construction, they mayprovide strategies to understanding aspects of symmetries of the double-copy theories andtheir relation to their single-copy parents. R symmetry In the absence of anomalies, the scattering amplitudes of a theory exhibit its off-shell sym-metries to all orders in perturbation theory. Below, we shall review how the double copyexpresses this property.As we reviewed at length in previous chapters, at tree level the KLT relations buildgravity scattering amplitudes from gauge-theory amplitudes. More generally, for all double-copy theories (including the non-gravitational ones), there exist analogous relations thatbuild their scattering amplitudes in terms of data the single-copy parent theories. It istherefore clear that, at tree level, the global symmetry group G of a double-copy theory is One may use other methods, such as those outlined in Ref. [219], to construct the effective action. This assumes the existence of a regulator that preserves these symmetries. GRAVITY SYMMETRIES AND THEIR CONSEQUENCES at least as large as the product of the global symmetry groups G , of the parent theories: G ⊃ G ⊗ G . (4.4)In a Feynman-diagram approach to the construction of scattering amplitudes, one canarrange that each diagram exhibits all off-shell global symmetries of the classical Lagrangian.The construction of tree-level CK-satisfying numerators in terms of tree-level amplitudes [24,149] implies that, at tree level, the same is true for each single-copy parent theory if one alsodemands that the amplitude obey CK duality. Thus, Eq. (4.4) also holds in this approach.In the presence of a symmetry-preserving regulator, generalized unitarity then guaranteesthat the regularized cuts of higher-loop amplitudes of the double-copy theory also inherit allthe global symmetries of the single-copy parent theories.Exercise 4.1: Explore if there is a general statement that can be made about anomalousglobal symmetries, i.e. whether all anomalous global symmetries of the single-copy parenttheories remain anomalous in the double-copy theory. To this end, consider the example ofa four-dimensional gauge theory with chiral fermions and construct examples of the double-copy amplitudes that involve scattering amplitudes of this theory that are sensitive to thechiral anomaly.Not all global symmetries of a double-copy theory are inherited; in fact, inheritance ofsome symmetries demands that others be enhanced. Consider, for example, the case of thedouble copy of two theories with N and N -extended supersymmetry, respectively. Theirsupersymmetry algebras have SU ( N ) and SU ( N ) R symmetry (perhaps with additionaldecoupled U (1) factors) and, according to the previous discussion, the double-copy theorywill be invariant under at least SU ( N ) × SU ( N ) transformations. However, the ( N + N )-extended supersymmetry algebra that is expected based on the number of supercharges hasa larger R symmetry, SU ( N + N ). Thus, to extend R ⊗ R to the complete R symmetrygroup, it is necessary to identify further 2 N N + 1 generators. The first 2 N N generatorswere constructed in Refs. [239, 240] in terms of the supersymmetry generators of the twosingle-copy parent theories. In four dimensions, they are G I ˜ J = Q + I ˜ Q − ˜ J and G I ˜ J = Q − I ˜ Q + ˜ J , (4.5)where Q and ˜ Q are the supersymmetry generators of the two single-copy parent theories,respectively, and the ± indices represent their helicity. These generators have vanishing totalhelicity. From their structure it is clear that they change the helicities of the two single-copycomponents in opposite ways, such that the helicity of the double-copy state is unchanged.For the case of N = 8 supergravity, their action on states is given in fig. 2.The remaining (Cartan) generator which is necessary to recover the complete (and ex-pected) R -symmetry group may in principle be obtained from the closure of the off-diagonal G I ˜ J and G I ˜ J . In Sec. 4.4, we shall review another way of identifying it, as well as its physicalinterpretation.An interesting feature which has been observed in explicit examples, some of which aredescribed in Sec. 5, is that certain gravity theories have two distinct double-copy realizations.In these cases, each version of the construction exhibits different manifest symmetries and,while following the pattern above, the details of the symmetry enhancement are different.51 GRAVITY SYMMETRIES AND THEIR CONSEQUENCES g + f + I φ IJ f − IJK g − IJKLQ − Q − Q − Q − h + ψ + I A + IJ φ IJKL χ + IJK ˜ g + ψ +˜ I A + I ˜ I χ + IJ ˜ I χ − IJKL ˜ I φ + IJKL ˜ I ˜ f +˜ I A +˜ I ˜ J χ + I ˜ I ˜ J φ + IJ ˜ I ˜ J A − IJKL ˜ I ˜ J χ + IJK ˜ I ˜ J ˜ φ ˜ I ˜ J χ +˜ I ˜ J ˜ K φ + I ˜ I ˜ J ˜ K χ − IJ ˜ I ˜ J ˜ K ψ − IJKL ˜ I ˜ J ˜ K A − IJK ˜ I ˜ J ˜ K ˜ f − ˜ I ˜ J ˜ K φ ˜ I ˜ J ˜ K ˜ L χ − I ˜ I ˜ J ˜ K ˜ L A − IJ ˜ I ˜ J ˜ K ˜ L h − IJKL ˜ I ˜ J ˜ K ˜ L ψ − IJK ˜ I ˜ J ˜ K ˜ L ˜ g − ˜ I ˜ J ˜ K ˜ L ˜ Q − ˜ Q − ˜ Q − ˜ Q − Table 2: Action of the G I ˜ J generator defined in Eq. (4.5) on the states of N = 8 supergravity. Theaction of the generators G I ˜ J is obtained by reversing the direction of the arrows. An example discussed in Ref. [241] from a double-copy perspective and in [242] from a string-theory point of view, is N = 4 supergravity with two vector multiplets, which can be realizedboth as ( N = 4 SYM) × (YM+2 scalars) and ( N = 2 SYM) × ( N = 2 SYM). While in theformer construction the complete SU (4) R symmetry of supergravity is manifest, the latteronly has a manifest SU (2) × SU (2) symmetry. A more dramatic example is provided bythree-dimensional N ≥ H µν = h µν + B µν + φη µν = A aµ ? Φ − aa ? ˜ A a ν , (4.6)where A aµ and ˜ A a µ are the fields in the left and right gauge theory, respectively. The resultingdouble-copy field H µν (sometimes referred to as the “fat graviton” [58], see also Sec. 8) needsto be decomposed in irreducible representations of the Lorentz group, giving the gravitonfield, the dilaton and an antisymmetric tensor B µν . This version of the construction isformulated in position space and hence relies on the convolution among linearized superfields,which is defined as [ f ? g ]( x ) = Z d yf ( y ) g ( x − y ) . (4.7)Crucially, Φ aa is a bi-adjoint scalar field which is employed to contract the gauge indices ofthe left and right fields. The action of a symmetry transformation on left and right fields isthen written as δA aµ = ∂ µ Λ a + f abc A bµ θ c + ˆ δA aµ ,δ ˜ A aµ = ∂ µ ˜Λ a + f abc ˜ A bµ ˜ θ c + ˆ δ ˜ A aµ , (4.8)52 GRAVITY SYMMETRIES AND THEIR CONSEQUENCES where Λ a , ˜Λ a , θ a and ˜ θ a are the parameters of local abelian and global nonabelian gaugetransformations, while ˆ δ indicates a global transformation under the (super)Poincaré group.The bi-adjoint scalars are designed to offset the left and right gauge transformations, andtransform as [253] δ Φ − aa = − f bac Φ − ba θ c − f b a c Φ − ab ˜ θ c + ˆ δ Φ − aa . (4.9)While this approach treats the action of the gauge-theory symmetries in an elegant way,the full dictionary is known only at the linearized level. As it was shown in Ref. [256], thelinearized gravitational equations of motion can be obtained from the linearized gauge-theoryones. It remains an open question how to include interactions in this formalism (which arenaturally incorporated from the perspective of scattering amplitudes).Exercise 4.2: Use Eq. (4.9) to show that the fat graviton defined in Eq. (4.6) is inert undernonabelian gauge transformations. Moreover, show that its local transformation rules are alinear combination of linearized diffeomorphisms and gauge transformations of a two-indexantisymmetric tensor field. In Sec. 2, we discussed in detail the emergence of diffeomorphism invariance and local super-symmetry in gravity scattering amplitudes obtained from the double-copy construction. Theformer is a direct consequence of the gauge invariance of the two single-copy gauge theoriesand manifest CK-satisfying form for at least one of the two gauge theories [156, 173, 174].The latter is a consequence of the gauge invariance of one of the single-copy gauge theories,supersymmetry of the second, and manifest CK-satisfying form for at least one of them. Theon-shell supersymmetry Ward identities of the double-copy theory follow from those of thesingle-copy parents. In this section we review how similar mechanisms lead to other localsymmetries in double-copy theories.As discussed in the beginning of this section, scattering amplitudes in theories with localsymmetries that act nonlinearly on fields are invariant under the global part of the symmetrygroup (if it acts linearly) as well as under its linearized local transformations. The converse,however, does not necessarily hold: scattering amplitudes that are invariant under someglobal symmetry group G and under abelian local transformations do not necessarily describea QFT with a local G symmetry. For example, they may correspond to a field theory withdim( G ) abelian vector fields. It is of course not difficult to distinguish between these twopossibilities by inspecting the scaling dimension of certain scattering amplitudes, which isdifferent according to whether the theory involves abelian and nonabelian vector fields.From the discussion in Sec. 2, it is clear that, in any double-copy theory, each vector fieldwhose asymptotic states are realized as a product of a scalar- and a vector-field asymptoticstates exhibits a Maxwell gauge symmetry, which is a consequence of the corresponding gaugesymmetry of the vector field in the single-copy parent theory. In order to associate thesevector fields to a local nonabelian symmetry, the corresponding amplitudes must exhibitseveral properties: (1) be invariant under the adjoint action of some a global nonabelian We note that the single-copy origin of Maxwell gauge symmetry is not obvious if the vectors are realizedas in terms of two fermions. GRAVITY SYMMETRIES AND THEIR CONSEQUENCES symmetry group on the asymptotic states of vector fields and (2) have the correct dimensionto be consistent with minimal coupling. The second property demands that a three-vectoramplitude have unit dimension, [ A (0)3 ] = 1 , (4.10)as in a standard nonabelian gauge theory. To obtain such amplitudes through double copy, atleast one of the single-copy parent theories must have amplitudes of dimension zero. Lorentzinvariance and locality imply then that the corresponding single-copy fields labeling suchamplitudes must be scalars. To satisfy property (1), the corresponding amplitude must bemomentum-independent and coming from a Lagrangian of the type L = · · · + f abc F ABC φ aA φ bB φ cC + . . . , (4.11)where the ellipsis stand for other interactions. This reasoning led to the double-copy real-ization of Yang-Mills-Einstein (YME) theories, as described in Ref. [120]. It was also usedin Ref. [257] to obtain amplitudes in the same theory through a KLT-like construction. Thisprocedure can be extended to give a double-copy realization of spontaneous breaking of YMgauge symmetry of supergravity theory [122]; spontaneous breaking of this symmetry is re-lated to explicit breaking of a global symmetry of one of the single-copy parent theories. Weshall review its applications more thoroughly in Sec. 5.3.7.The same analysis implies that the double-copy fields that are realized as products ofsingle-copy fields with nonzero spin cannot couple directly to the nonabelian vector potentialand can couple only to its field strength. Indeed, in a conventional gauge theory, any three-point amplitude with at least one field with nonzero spin has unit dimension. Thus, thecorresponding double-copy three-point amplitude has dimension 2 and cannot be given by aminimal-coupling term.The analysis above can be extended to interactions of gravitini with abelian or nonabeliangauge potentials. Such interactions are the tell-tale of gauged supergravities—that is, su-pergravities in which part of the R symmetry is gauged. The gravitino minimal couplingaround Minkowski space is L ∼ ¯ ψ µ γ µνρ D ν ψ ρ , (4.12)and, thus, as for the case of lower-spin fields, the two-gravitini-vector amplitudes have againunit dimension. Since all three-point amplitudes in conventional gauge theories that couldhave this spin content in their product have at least unit dimension, it follows that theirdouble copy can only describe the coupling of gravitini and field strengths and thus, a morerefined argument is needed to accommodate minimal couplings of gravitini.The observation that sidesteps the difficulty exposed above [123] is that, assuming thatthe theory has a Minkowski ground state, the three-point amplitude following from theminimal coupling of a gravitino with a vector field spontaneously breaks supersymmetry.Consequently, some of the gravitini must be massive and therefore their double-copy real-ization must involve a single-copy theory with massive vector fields and another one withmassive fermions. The former must therefore be a spontaneously-broken gauge parent theorywhile the latter turns out to exhibit explicit supersymmetry breaking (this construction willbe illustrated in detail in Sec. 5.3.8). The general pattern is that, through the double copy,explicit breaking of a global symmetry can be promoted to spontaneous breaking of the localversion of the same symmetry. 54 GRAVITY SYMMETRIES AND THEIR CONSEQUENCES
We have seen in Sec. 4.2 that double-copy theories inherit all the global symmetries of theirsingle-copy parents and that some of the single-copy symmetries combine in nontrivial ways(e.g. two supersymmetry generators combine to become a bosonic R -symmetry generator)to enhance the inherited symmetries. Lagrangian-based supergravity considerations suggestthe existence of much larger symmetries — the U-duality symmetries — which are typicallynoncompact. In pure supergravities in various dimensions, these symmetries were originallydiscussed in Refs. [258, 259]; their maximal compact subgroup is isomorphic to the on-shell R -symmetry group of the theory. Depending on the dimension, they are either symmetriesof the equations of motion (in four dimensions) or symmetries of the Lagrangian (e.g. infive dimensions). In four dimensions, the field strengths and their duals form an irreduciblerepresentation of the U-duality group, which therefore contains electric/magnetic dualityas one of its generators. While the general understanding U-duality symmetries from thedouble-copy perspective is currently an open problem, their dimension-dependent propertiessuggest that their realization should involve transformations that are not off-shell symmetriesin the single-copy parent theories.Ref. [260] showed that a universal generator of the U-duality groups of four-dimensionalsupergravities can be realized as the difference of the little-group generators (helicity) of thetwo single-copy parent theories; the charges of the double-copy fields under this generatorare Q = q ( h − ˜ h ) . (4.13)Note that this transformation acts on the positive and negative helicity vector fields in thedouble-copy theory with opposite phases. Because of this property, the above transformationcan be identified as an electric/magnetic duality transformation acting on vector fields,combined with additional transformations of other fields [261, 262]. It is not a priori clearwhy this transformations should be a symmetry of the double-copy theory at tree level. Onecan nonetheless check that for N ≥ R symmetry of the theory andthus part of the maximal compact subgroup of the U-duality. For example, decomposing thepositive-helicity (denoted with the index “+” below) and scalar states of N = 8 supergravityin representations of the ( SU (4) , (cid:94) SU (4)) U (1) subgroup of the SU (8) R symmetry (of whichonly SU (4) × SU (4) is manifest in the double copy) one finds + = ( , ) , + = ( , ) q ⊕ ( , ) − q , + = ( , ) q ⊕ ( , ) − q ⊕ ( , ) , (4.14) + = (¯ , ) q ⊕ ( , ¯ ) − q ⊕ ( , ) q ⊕ ( , ) − q , = ( , ) q ⊕ ( , ) − q ⊕ (¯ , ) q ⊕ ( , ¯ ) − q ⊕ ( , ) . As pointed out in Ref. [260], the U (1) charges resulting from this decomposition are exactlygiven by eq. (4.13). The decomposition of the negative helicity states is obtained by conju-gating the first four lines of Eq. (4.14); the U (1) charge changes sign under conjugation andmay be also identified as being proportional to the net number of indices of the states in Ta-ble 2. From this table, we also see that supersymmetry generators change the U (1) charge by55 GRAVITY SYMMETRIES AND THEIR CONSEQUENCES / R -symmetry generators enhancing SU (4) × ˜ SU (4) → SU (8)change the U (1) charge by one unit. While we illustrated here its relevance for N = 8 su-pergravity, the U (1) symmetry described by eq. (4.13) is required for obtaining the completeon-shell R -symmetry for all N ≥ ≤ N ≤
4, this symmetry is present at tree level but it is anomalous [260, 263].This anomaly sources certain loop-level amplitudes which vanish at tree level. In the real-ization of these theories as double copies with one non-supersymmetric gauge-theory factor,these anomalous amplitudes [260] can be traced to a self-duality anomaly of YM theory[264]. When realized as double copies of supersymmetric gauge theories, the identification ofanomalous amplitudes is more subtle: they arise from µ -terms which, in each gauge theory,give only O ( (cid:15) ) terms but give finite terms only after the double copy [20]. It turns out [265]that, at least for N = 4 supergravity, the anomalous amplitudes can be canceled at one loopby the addition of a finite local counterterm to the classical action; this counterterm restoresthe U (1) symmetry at the expense of breaking other symmetries that do not appear toimpose any obvious selection rules on amplitudes. The same counterterms also cancels thetwo-loop anomalous amplitudes [266]. The full consequences of these cancellations remainto be explored. It is instructive to consider the U (1) transformation with charges (4.13) vis à vis theobservation discussed in Sec. 4.2 that the same supergravity theory may have (two or perhapsmore) different double-copy realizations. While a general analysis is yet to be carried out,it is not difficult to see on a case-by-case basis that this symmetry may play different roles.To this end, consider N = 4 supergravity realized as ( N = 4 SYM) × (YM+2 scalars)and ( N = 2 SYM) × ( N = 2 SYM), both of which can be obtained as different (orbifold)truncations of the double-copy constructions of N = 8 supergravity. In the former, the SU (4) R symmetry in manifest and the U (1) symmetry is part of the SU (1 ,
1) dualitygroup of N = 4 supergravity. In the latter, only SU (2) × SU (2) ⊂ SU (4) is manifest andthe U (1) symmetry is required to enhance it to the complete SU (4) R symmetry. In thisformulation the origin of U (1) ⊂ SU (1 ,
1) is not clear. Similarly, the further enhancement tothe SO (6 , × SU (1 ,
1) complete U-duality group (see Sec. 5) is currently an open problem,on the same footing as SU (8) → E in N = 8 supergravity. The definition of this universal U (1) symmetry in (4.13), does not single out super-gravities as the only double-copy theories that exhibit this symmetry. There exist many µ -terms are numerator terms proportional to the extra-dimensional parts of loop momenta. Such termsvanish identically if the integrand is evaluated in four dimensions. They are the two generators that, together with U (1), form the SU (1 ,
1) classical U-duality group of thetheory. For N = 0 supergravity, defined as the double copy of two pure YM theories, this symmetry is alsopresent; it represents the U (1) rephasing of the dilaton-axion. It survives at the quantum level because thereare no fields that can contribute to its anomaly. See Sec. 5.2.2 for details on field-theory orbifolds in this and related contexts. For supergravity theories for which the scalar fields parametrize the locally-homogeneous space
G/H with H being the maximal compact subgroup of G , the noncompact part of G can be identified once H andits representations carried by scalars are determined [267]; see also Refs. [239, 240, 268] for further detailsfrom the double-copy perspective at the noninteracting level. It is nevertheless not clear how to constructthe noncompact G -generators in terms of operators in the single-copy theories. GRAVITY SYMMETRIES AND THEIR CONSEQUENCES non-gravitational four-dimensional field theories exhibiting electric/magnetic duality; for allthose that have a double-copy realization, the transformations of the asymptotic states underduality have the same form 4.13. An example is the Born-Infeld theory; in this case dualityimplies that only split-helicity amplitudes (i.e. amplitudes with an equal number of positiveand negative vector fields) are nonvanishing. While this can be proven to all multiplicitiesthrough various techniques [261, 262], it would be interesting to understand this propertyfrom the perspective of the double-copy construction. Quite generally, it remains an inter-esting open question to understand the consequences of duality from the perspective of thesingle-copy parent theories.The one-loop all-multiplicity all-plus and single-minus amplitudes of the Born-Infeld the-ory were constructed using D -dimensional unitarity and supersymmetric decomposition in[269] and integrated using dimension-shifting relations in [270]. The amplitudes in bothclasses turn out to be nonvanishing, implying that, similarly to N = 4 supergravity, dualityappears to be anomalous in the non-supersymmetric Born-Infeld theory in this regularizationscheme. It remains an open question [269] whether the anomaly is physical or whether it canbe removed by a finite local counterterm at the expense of other symmetries. Argumentspresented in [262] suggest that it should be possible to restore duality with a countertermthat breaks Lorentz invariance. As described at length in Sec. 4.1, supergravity considerations suggest the existence of a muchlarger symmetry group then the one manifestly realized on on-shell scattering amplitudes.Part of this (U-duality) group acts nonlinearly and thus does not impose standard selectionrules on scattering amplitudes; consequently, the corresponding generators cannot be realizedmanifestly (i.e. linearly) on scattering amplitudes simultaneously with supersymmetry andLorentz invariance. Because of these features, it is not currently known how to identify thesingle-copy origin of the noncompact U-duality transformations. The discussion in Sec. 4.1and the ability to compute scattering amplitudes efficiently (both at tree and loop level)gives us an alternative route to probe the existence of these symmetries, borrowing from thesupergravity knowledge that scalar fields of the theory parametrize coset space of the form
G/H , where G is the U-duality group and H its maximal compact subgroup. In this section,departing from the philosophy in the rest of this review, we shall assume that supergravityamplitudes are available (through the double-copy or by some other means) and describehow to identify the hidden existence of the noncompact U-duality symmetries.This problem was first discussed in detail in Ref. [222], where it was shown that the exis-tence of nonlinearly realized symmetries of this type can be identified through the vanishingof single-soft-scalar limit of scattering amplitudes, while the precise group structure can beinferred from the limit in which the momenta of two scalar fields become simultaneouslysoft. We review this construction, which was also extended to other nonlinearly-realized or We note that a Lagrangian formulation of N = 8 that has manifest E symmetry was constructed inRef. [271] and further explored in Ref. [272]. This formulation, however, breaks manifest Lorentz invariance.Moreover, diffeomorphism transformations on vector fields are realized in a nonstandard way. It would beinteresting to explore the scattering amplitudes of N = 8 supergravity in this formulation and compare themto the standard form. GRAVITY SYMMETRIES AND THEIR CONSEQUENCES spontaneously-broken symmetries, as well as to fields with nontrivial Lorentz-transformationproperties, in [220, 224, 228, 231, 236]. A thorough analysis of the soft limits in effectivefield theories was carried out in Refs. [229, 273].Consider, following Ref. [222], a symmetry group G with generators falling into two sets, T and X , broken to the subgroup H generated by T . Schematically, the commutationrelations are [ T, T ] ∼ T , [ T, X ] ∼ X , [ X, X ] ∼ T . (4.15)From a Lagrangian point of view (if one is available), there exists a (Nambu-Goldstone)scalar for each of the broken generators X . In general, this Lagrangian has many degeneratevacua; moving from one to another amounts to giving nonzero vacuum expectation values(VEVs) to the Nambu-Goldstone scalars. From the perspective of scattering amplitudes,a vacuum-expectation value of a field corresponds to a condensate of the zero-momentummode. Thus, exploring the change in vacuum state is equivalent to exploring the propertiesof scattering amplitudes in the zero-momentum limit for some of the scalars.A similar conclusion may be reached by revisiting the argument in Sec. 4.1 showingthat, for generic momentum configurations, LSZ reduction renders scattering amplitudesinsensitive to nonlinear field transformations. Assuming a generic transformation rule δφ ∼ φ k ≥ , the same argument implies that, if all but one of the fields on the right-hand side of thetransformation carry vanishing momentum, then the transformed Green’s function has thesame poles as the original one and therefore survives the LSZ reduction. Thus, the nonlinearparts of a symmetry transformation should have a reflection on higher-multiplicity scatteringamplitudes in which the additional asymptotic states have vanishing momenta.Starting with some vacuum state | i , a neighboring one is obtained through a G trans-formation with parameters given by the VEVs of the old scalars in the new vacuum: | i θ = e iX α θ α | i . (4.16)Since the G -symmetry requires that amplitudes around the two vacua be the same, theconclusion is therefore that the scattering amplitudes with at least one zero-momentumscalar field vanish identically. For a single soft scalar, this reproduces the celebrated Adlerzero [209]. One may turn the single-soft-scalar limit argument around and infer [121] that,in a theory that has vanishing single-soft-scalar limits, the scalar fields belong to a locally-homogeneous space (i.e. a space that has a transitive local group action).A more involved argument [222] extracts the structure constants of the broken symmetrygroup from the double-soft-scalar limit of scattering amplitudes: M n +2 (1 , , , . . . , n + 2) p ,p → −−−−−−−−→ n +2 X i =3 p i · ( p − p ) p i · ( p + p ) T M n (3 , . . . , n + 2) , (4.17)where T is the G -generator given by the commutator of the X generators corresponding tothe two soft scalars. The momenta of the two scalars should be taken soft at the same rate.Exercise 4.3: As we discussed in Sec. 4.1 we have seen that scattering amplitudes withgeneric momenta are insensitive to nonlinear terms in symmetry transformations becausethe LSZ reduction projects out their contribution. An interesting unexplored problem is the58 GRAVITY SYMMETRIES AND THEIR CONSEQUENCES contribution of terms with special momentum configurations. Consider a nonlinear symme-try transformation whose nonlinear parts contains bilinears and cubic terms. Assuming thatonly one of the fields in the nonlinear terms carries nonzero momentum, explore the featuresof the single- and double-soft-scalar limits of amplitudes by applying LSZ construction toGreen’s functions acted upon by such special transformations.The construction reviewed above does not refer to any specific order in perturbationtheory and thus relies on absence of U-duality anomalies. Its conclusions have been usedto constrain and characterize possible counterterms of N = 8 supergravity, which should besuch that their contributions to scattering amplitudes have soft limits following the samepattern. Through this reasoning it was shown that a suggested three-loop R countertermis inconsistent with the E symmetry of N = 8 supergravity [273]. Along the same lines,Ref. [274] argued that the first deformation of N = 8 supergravity that is consistent with thesoft-scalar behavior required by E symmetry can appear at seven loops and correspondsto a supersymmetric completion of a D R operator.Generic diffeomorphism transformations are nonlinear. As we discussed in the beginningof this section and in Sec. 4.4, infinitesimal/linearized diffeomorphisms are symmetries ofscattering amplitudes: shifting the graviton polarization tensor ε ( p ) µν ε ( p ) µν + p ( µ Λ ν ) withΛ µ being the parameter of the transformation, leaves amplitudes invariant. By definition,large diffeomorphisms do not have a linearized approximation; the BMS transformations(named after Bondi, van der Burg, Metzner and Sachs [275–277]) arise, in a certain gauge, asresidual diffeomorphism symmetries of asymptotically-flat spacetimes which do not fall off atinfinity. It was argued in Refs. [278–280] that the Ward identities of these symmetries implythe tree-level single-soft-graviton behavior of scattering amplitudes. Quantum correctionshave been discussed in Refs. [235, 281], with the conclusion that they affect the linear orderin the small momentum if all other momenta are generic. The identification of the BMSalgebra in the double-soft-graviton limit of scattering amplitudes was discussed in Ref. [227].Other symmetries can also be probed through double-soft limits. For example, by explic-itly inspecting the tree-level amplitudes of a certain Akulov-Volkov theory [223], Ref. [224]showed that the double-soft-goldstino limit yields the supersymmetry algebra. Moreover, for4 ≤ N ≤ N = 16 supergravity in three dimen-sions, tree-level scattering amplitudes have a universal behavior in the double-soft-fermionlimit which is analogous to the scalar one. The photon and graviton soft theorems werediscussed from an effective-field-theory standpoint in Ref. [229], where a complete classifica-tion of local operators responsible for modifications of soft theorems at subleading order forphotons and subsubleading order for gravitons was derived.The original discussion [222] of the U-duality symmetries in supergravity and its subse-quent generalizations assumed absence of anomalies of the spontaneously-broken symmetry.Possible anomalies have been included in this framework in Ref. [228], from the perspectiveof the effective action; the conclusion of the analysis is that, while the single-soft limits re-ceive corrections signaling the anomalous breaking of the symmetry, double-soft limits areunaffected. This is probably a reflection of the anomaly (defined as the nonvanishing of thedivergence of the symmetry current) being invariant under the classical symmetry.59 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Figure 17: Schematic rendition of the web of theories. Nodes represent the main double-copy-constructible theories discussed in this section, which include gravitational theories (rectangularnodes), string theories (oval nodes) and non-gravitational theories (octagonal nodes). Undirectedlinks are drawn between theories that have a common gauge-theory factor in their construction(different gauge-theory factors correspond to different colors). Directed links connect theoriesobtained by modifying/deforming both gauge-theory factors (e.g. adding matter, assigning VEVs).Details are given throughout Sec. 5.3.
As we have seen in the previous sections, the duality between color and kinematics andthe double-copy construction express amplitudes of gravitational theories in terms of sim-pler building blocks from gauge theory. It has become clear that this property is not anaccident of few very special theories, but extends to large classes of gravitational and non-gravitational theories. Seemingly unrelated theories have been shown to share—and thusbe connected by—the same set of building blocks, yielding a “web of theories” which canbe analyzed with double-copy methods (see Fig. 17). In this section, we aim to probe thisweb more in detail. Particularly prominent results will be the classification of homogeneous N = 2 Maxwell-Einstein supergravities [282], which can be reproduced and streamlinedby double-copy methods, the double-copy construction for YME [120, 125, 257, 283] andgauged supergravities [123, 284], and the construction for Dirac-Born-Infeld (DBI) theories60 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Supergravities Free Parameters Scalar geometry N > N = 4 supergravity number of vector multiplets symmetric space N = 3 supergravity number of vector multiplets symmetric space N = 2, vector multiplets,5D uplift C IJK -tensor very-special Kähler geometry N = 2, vector multiplets,4D only free degree-two holomorphicfunction (prepotential) special Kähler geometry N = 2, hypermultiplets,from c -map C IJK -tensor or prepotential special/very specialquaternionic Kähler geometry N = 2, hypermultiplets,general See text quaternionic Kähler geometryTable 3: Freedom in specifying the two-derivative action in extended (ungauged) supergravitieswith 2 ≤ N ≤ [125, 285]. We will also see that some of the building blocks which appear, for example, inthe double-copy construction for conformal supergravities play a role in a family of “stringy”double-copy constructions. Similar webs of theories have appeared, for example, in the con-texts of the scattering equations formalism [125], amplitude transmutation [285], and softlimits [286].The simplest examples of double-copy-constructible theories we have discussed so farinclude N ≥ N < N ≥ N = 4, it becomes possible to have various matter contents. While N = 3 , N = 2 supersymmetry.Supergravity theories generically involve scalar fields, which can be regarded as the coordi-nates of a manifold. While extended N > N ≤ ≤ N ≤
8, together with the corresponding geome-tries. It should be noted that theories with N = 2 have different geometrical propertiesdepending on whether or not they have a five-dimensional uplift. Theories with vectors mul-61 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES tiplets which can be lifted to five dimensions are uniquely specified by a symmetric constanttensor C IJK whose indices run over the total number of vector fields. Since this tensor canbe obtained from inspecting specific three-point interactions, supergravities of this sort havethe pleasant property of being entirely constructible from their three-point amplitudes, aproperty that we will utilize extensively later in this section. Intrinsically-four-dimensionaltheories are significantly less constrained. They are fully specified by a homogeneous degree-two holomorphic function—the prepotential—which is otherwise arbitrary. Theories withhypermultiplets possess even more freedom: N = 2 supersymmetry only require the hyper-multiplet scalar manifold be quaternionic-Kähler (that is, to admit an hermitian metric andthree complex structures which satisfy the quaternionic algebra). At the same time, a subsetof these theories can be regarded as the image of supergravities with vector multiplets underan operation known as c -map; specifying these theories requires the same information astheir vector counterparts. When studying supergravities with reduced supersymmetry it isimportant to keep in mind how this freedom is reflected in the gauge-theory data enteringthe double-copy construction.Supergravities studied in the double-copy context have thus far been mostly theories ofthe Maxwell-Einstein class, i.e. theories in which all vector fields are abelian and there areno charged matter fields. From a Lagrangian perspective, supergravities with nonabeliangauge interactions have also been studied, see [287, 288] for reviews. They can be fur-ther divided into YME theories and proper gauged supergravities. In the former class, anonabelian subgroup of the isometry group of the scalar manifold is promoted to a gaugesymmetry. In the latter case, part of the R symmetry is promoted to gauge symmetry. Thisprocedure, customarily referred to as gauging, does not introduce additional vector fields.It minimally couples some of the existing vector fields while also giving them nonabelianself-interactions and extending the resulting theory so that it is invariant under the requirednumber of supercharges. In an amplitude context, YME theories have been studied from avariety of perspectives, including scattering equations [125, 283, 289], collinear limits of gaugetheory amplitudes [290], BCFW recursion [214, 216], string theory [133, 134], ambitwistorstrings [140] and, of course, the double-copy construction [120]. Through this work, it hasbecome clear that amplitudes in such theories may be written as linear combinations of(color-ordered) amplitudes or ordinary YM theory [133, 214, 216, 289]. We will see later inthis section that the above property has a very straightforward double-copy interpretation.Gauged supergravities display a considerably more involved structure. Once a subset ofthe R symmetry is gauged (i.e. some of the R -symmetry generators appear in the covariantderivatives), supersymmetry requires a scalar potential to appear in the theory. Accordingto whether the potential vanishes or not at a critical point, the theory admits Minkowski,Anti-de Sitter or de Sitter vacua. Minkowski vacua break supersymmetry spontaneously(partly or completely), resulting in massive gravitini. The study of gauged supergravities inthe double-copy framework is still in the early stages, but encouraging results are availablewhich will be reviewed later in this section.A growing body of work seems to suggest that the existence of a double-copy structure isnot merely an accidental feature of highly-supersymmetric theories, but a generic propertyof very large classes of gravities. To determine whether the double-copy property is a hiddenstructure of gravitational interactions it is necessary to identify the gauge-theory counterpartsof all data required to specify a generic gravity theory, whether it be ungauged, YME or62 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Gravity Gauge theories Refs. Variants and notes N > • N = 4 SYM theory • SYM theory ( N = 1 , ,
4) [1, 2, 31, 291,292] N = 4supergravity withvector multiplets • N = 4 SYM theory • YM-scalar theory from dim.reduction [1, 2, 31, 293] • N = 2 × N = 2 constructionis also possiblepure N < • (S)YM theory with matter • (S)YM theory with ghosts [188] • ghost fields in fundamental repEinstein gravity • YM theory with matter • YM theory with ghosts [188] • ghost/matter fields infundamental rep N = 2Maxwell-Einsteinsupergravities(generic family) • N = 2 SYM theory • YM-scalar theory from dim.reduction [120] • truncations to N = 1 , • only adjoint fields N = 2Maxwell-Einsteinsupergravities(homogeneoustheories) • N = 2 SYM theory with halfhypermultiplet • YM-scalar theory from dim.reduction with matter fermions [121, 294] • fields in pseudo-real reps • include Magical Supergravities N = 2supergravities withhypermultiplets • N = 2 SYM theory with halfhypermultiplet • YM-scalar theory from dim.red. with extra matter scalars [121, 240] • fields in matter representations • construction known inparticular cases N = 2supergravitieswith vector/hypermultiplets • N = 1 SYM theory with chiralmultiplets • N = 1 SYM theory with chiralmultiplets [239, 241, 295] • construction known inparticular cases N = 1supergravities withvector multiplets • N = 1 SYM theory with chiralmultiplets • YM-scalar theory with fermions [188, 239, 241,295] • fields in matter reps • construction known inparticular cases N = 1supergravities withchiral multiplets • N = 1 SYM theory with chiralmultiplets • YM-scalar with extra matterscalars [188, 239, 241,295] • fields in matter reps • construction known inparticular casesEinstein gravitywith matter • YM theory with matter • YM theory with matter [1, 188] • construction known inparticular cases A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES R + φR + R gravity • YM theory + F + F + . . . • YM theory + F + F + . . . [296] • extension to N ≤ • DF theory • (S)YM theory [152, 153] • N ≤ • involves specific gauge theorywith dimension-six operators3 D maximalsupergravity • BLG theory • BLG theory [119, 243, 297] • D only Table 4: Non-inclusive list of ungauged gravities and supergravities for which a double-copyconstruction is presently known. Theories are given in four dimensions unless otherwise stated. gauged. While this program has not yet been completed, important progress has beenmade in formulating double-copy constructions for theories which include, among others,pure supergravities, homogeneous N = 2 Maxwell-Einstein supergravities, homogeneous N = 2 theories with hypermultiplets, large classes of YME or gauged theories, and conformalsupergravities. A list of ungauged and gauged theories for which a double-copy constructionis currently known can be found in Table 4 and Table 5, respectively. Gauge theorieswith fields in various matter (non-adjoint) representations of the gauge group are a rathercommon building block for this class of extended constructions. Useful tools for treatingmatter representations in a way that makes manifest color and numerator relations will beintroduced in Sec. 5.2. We will then discuss systematics of the process of identifying thegravity theory given, through double copy, by a pair of gauge theories and study severalexamples in Sec. 5.3.Double-copy constructibility is a property that goes beyond gravitational theories. Vari-ous theories without a graviton, most prominently some variants of the DBI theory have alsobeen shown to possess this property (see Table 6). We shall briefly review their constructionin Sec. 5.3.11. To capture as many gravities as possible, we need to consider gauge theories which are moregeneral than the ones discussed at length in previous sections. At the same time, havingin mind a double-copy construction which leads to a sensible gravity theory with desirablebasic properties, it makes sense to impose some requirements on the gauge theories underconsideration. Some additional requirements will also be imposed for simplicity reasons; inboth cases, one can contemplate generalizations in which some of the stated rules of thegame bent or broken.First of all, for simplicity, we choose to focus on theories for which amplitudes can beorganized exclusively in terms of cubic graphs. This is a natural generalization of the gaugetheories from the previous sections, which possess this property, and is a natural choice for64
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Gravity Gauge theories Refs. Notes
YMEsupergravities • SYM theory • YM + φ theory [120, 125, 133,134, 140, 214,216, 257, 283,285, 289] • trilinear scalar couplings • N = 0 , , , • SYM theory (Coulomb branch) • YM + φ theory with extramassive scalars [122] • N = 0 , , , • massive fields in supergravity U (1) R gaugedsupergravities • SYM theory (Coulomb branch) • YM theory with SUSY brokenby fermion masses [123] • ≤ N ≤ • SUSY is spontaneously broken • only theories with Minkowskivacuagaugedsupergravities(nonabelian) • SYM theory (Coulomb branch) • YM + φ theory with massivefermions [284] • SUSY is spontaneously broken • only theories with Minkowskivacua Table 5: Gauged/YME gravities and supergravities for which a double-copy construction ispresently known. describing gravities that are entirely specified by their three-point interactions. Hence, werestrict the space of gauge theories under consideration according to the following rule:
Working Rule 1:
Consider gauge theories with only cubic invariant tensors or,alternatively, theories for which amplitudes can be organized in terms of cubicgraphs.Allowed invariant tensors will include, for example, structure constants, representation ma-trices and cubic Clebsch-Gordan coefficients. It should be emphasized that the gauge the-ories under consideration can and will possess quartic vertices. Our requirement constrainshigher-point interaction vertices to be made of color building blocks which are cubic. If thisproperty is satisfied, amplitudes can be expressed in terms of cubic graphs by including asuitable number of inverse propagators in the numerator factors. While this rule is quitedesirable for the sake of simplicity, it can in principle be broken. A notable violation are thethe Bagger-Lambert-Gustavsson (BLG) and Aharony-Bergman-Jefferis-Maldacena (ABJM)theories, which are most naturally organized in terms of quartic graphs [119, 243, 297].Within the class of cubic theories, however, we need to consider cases which are as generalas possible. This motivates the second rule:
Working Rule 2:
The gauge theories will include matter fields transforming ingeneral (not necessarily irreducible) representations of the gauge group (which isnot necessarily semisimple). Only one adjoint representation will be allowed.Considering general gauge groups and representations will allow us to capture very largefamilies of (super)gravities which would not otherwise be accessible through double-copymethods. The main observation is that there is nothing in the double-copy construction65
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Double copy Starting theories Refs. Variants and notes
DBItheory • NLSM • (S)YM theory [125, 126, 285,298–301] • N ≤ • also obtained as α → • NLSM • SYM theory (external fermions) [125, 302–308] • restriction to external fermionsfrom supersymmetric DBISpecial Galileontheory • NLSM • NLSM [125, 285, 301,306, 309] • theory is also characterized byits soft limitsDBI + (S)YMtheory • NLSM + φ • (S)YM theory [125, 126, 156,285, 298–300,306, 310] • N ≤ • also obtained as α → • NLSM • YM + φ theory [125, 126, 156,285, 298–300] Table 6: List of non-gravitational theories constructed as double copies. that requires that representations be divided into irreducible blocks. At the same time, wewant to obtain theories with a single graviton. This forces us to combine all gauge-theorygluons in a single adjoint representation, even when the gauge group is the product of severalfactors each possessing its own adjoint representation. In case of more than one semi-simplefactor in the gauge group, we need to take all gauge coupling constants to be the same. Sinceall fields in the gauge theory have canonical couplings with gluons, our second rule can alsobe regarded as the double-copy incarnation of the Equivalence Principle.Additionally, massive fields are typically assigned to non-adjoint representations suchthat all the fields in a given representation have the same mass. This will be accompaniedby mass-matching conditions of the spectrum of the two sides of the double copy.Combining the first two rules, we obtain a generic amplitude structure that involves cu-bic graphs in which internal and external legs carry definite representations of the gaugegroup. Cubic vertices between three representations are allowed only when it is possibleto extract a gauge singlet in their tensor product (or, alternatively, there exist a nonvan-ishing invariant tensor with the three corresponding indices). Whenever a vertex involvestwo lines carrying the same representation, its symmetry or antisymmetry will be dictatedby the representations under consideration (real representations will imply antisymmetry,pseudo-real representation will imply symmetry). Additionally, color factors will obey three-term identities following from the Jacobi relations, the generators’ commutation relationsand additional algebraic relations which may also involve the Clebsch-Gordan coefficients.Consequently, the duality between color and kinematics must to be imposed in the followingway: 66
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Working Rule 3:
Numerator factors in a duality-satisfying presentation of anamplitude need to have the same algebraic properties as the color factors. Thisincludes symmetry properties as well as obeying two- and three-term identities.As discussed in Sec. 2, this rule ensures that the gravity theory obtained through doublecopy is invariant under linearized diffeomorphisms. If there exists (massive) vector fieldsthat transform in non-adjoint representations, additional gauge-group Lie algebra relationsare needed to guarantee that gauge invariance aside from Jacobi and commutation relations.The same relations should be imposed on the kinematic numerators for all fields that trans-form in the same representations as the vectors. For some classes of constructions, it willbe convenient to consider a slight variant of Working Rule 3 which instructs to impose thealgebraic properties of the color factors of one theory on the numerators of the other theoryentering the double-copy construction (and vice versa).Finally, we need a procedure for consistently pairing representations in the two gaugetheories when we substitute color factors with numerator factors following the double-copyprescription. A priori, several choices are possible. However, the following criterion is con-venient, elegant and easy to implement:
Working Rule 4:
Each state in the double-copy (gravitational) theory correspondsto a gauge-invariant bilinear of gauge-theory states. For this to be possible, we willidentify the gauge groups of the two theories entering the construction.A concrete consequence of this rule is that gauge-theory states in the adjoint representationwill double copy among themselves, but not with states in matter non-adjoint represen-tations. Similarly, states in two matter representations will be combined only when thetensoring of the representations includes a singlet. Considering general graphs, two numer-ators will be combined only when Working Rule 4 is satisfied by each internal and externalline. We will see that this requirement is essential for preventing the gravity from the doublecopy from having too many gravitini.The space of all possible gauge theories is quite vast (though perhaps not quite as vast asthat of gravitational theories). The purpose of the working rules we laid out is to restrict thisspace to a subset which is sufficiently large to capture a considerable number of theories andyet sufficiently small to allow a thorough analysis. It is not difficult to enlarge it by relaxingsome of the rules. We emphasize that many gauge theories which might be naively rejectedas unphysical, such as theories with ghost fields, may be admissible—even in some sensenecessary—from a double-copy perspective. This is because, through double copy, gauge-theory data is deconstructed and reassembled in a highly-nontrivial way and undesirablefeatures of gauge theories can be rendered harmless by this process.The rules stated in this section should be slightly modified when constructing not grav-itational. In this case, the gauge group should be replaced by a global symmetry group inthe theories entering the construction.
Having established the general rules of the game, we will now analyze particular examples.We start by considering a YM-scalar theory with only adjoint fields and trilinear cubic67
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES couplings [120]. Its Lagrangian can be written as L YM+ φ = − F ˆ aµν F µν ˆ a + 12 ( D µ φ A ) ˆ a ( D µ φ A ) ˆ a − g f ˆ a ˆ b ˆ e f ˆ e ˆ c ˆ d φ A ˆ a φ B ˆ b φ A ˆ c φ B ˆ d + 13! λgF ABC f ˆ a ˆ b ˆ c φ A ˆ a φ B ˆ b φ C ˆ c . (5.1)The indices ˆ a, ˆ b, ˆ c are gauge-group adjoint indices. A, B, C = 1 , . . . , n are global indicescarried by the scalars. The theory has a SO ( n ) global symmetry which is broken bythe trilinear couplings to the subgroup preserved by the F ABC tensor. Field strengths andcovariant derivatives are F ˆ aµν = ∂ µ A ˆ aν − ∂ ν A ˆ aµ + gf ˆ a ˆ b ˆ c A ˆ bµ A ˆ cν , ( D µ φ A ) ˆ a = ∂ µ φ A ˆ a + gf ˆ a ˆ b ˆ c A ˆ bµ φ A ˆ c . (5.2)To understand the constraints imposed by CK duality on the parameters of this theory wefirst analyze the four-scalar amplitudes. There is a clean separation between the contributionfrom the trilinear scalar coupling and the one from gluon exchange (the latter including thecontact term). After a short calculation, the s -channel numerator can be written as n s = δ AB δ CD ( t − u ) − ( δ AC δ BD − δ AD δ BC ) s − λ F ABE F ECD , (5.3)while the other numerators can be obtained by relabeling the external lines. The threecorresponding color factors obey standard Jacobi relations; imposing the duality betweencolor and kinematics then results in the condition λ ( F ABE F ECD + F BCE F EAD + F CAE F EBD ) = 0 . (5.4)The λ part of the numerator factors satisfies the duality automatically. This follows fromthe YM-scalar theory with λ = 0 being the dimensional reduction of a pure YM theory inhigher dimension, which is known to satisfy the duality at arbitrary multiplicity. At order λ , the kinematic Jacobi relations imply that F ABC -tensors must themselves obey Jacobirelations. This implies that they can be regarded as the structure constants of some globalgroup which is unrelated to the gauge group. What remains to be done is to consideramplitudes involving vectors and amplitudes at higher points. It turns out that no furtherconstraint on the theory appears. It has been explicitly checked that the Lagrangian obeysCK duality up to at least six points [120].A second example which we review in detail is YM theory with complex scalars in amatter representation [122]; an analogous example involving matter fermions was discussedin Sec. 2. For such a field content, trilinear couplings are forbidden by gauge symmetry; thefirst possible scalar self-interaction is quartic, so the Lagrangian is L scalar = D µ ϕD µ ϕ − a g ϕt ˆ a ϕ )( ϕt ˆ a ϕ ) , (5.5) In this section, we frequently use hatted indices for gauge-group indices of the gauge theories that enterthe double copy, to help distinguish them from global indices and gauge indices that appears in gravitationaltheories. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES where ϕ is a complex scalar, t ˆ a are representation matrices and a is a constant. Products areunderstood in the sense of matrix multiplication, as representation indices are not displayedexplicitly to avoid cluttering the expression. The two-scalar two-gluon amplitude in thistheory is A (cid:16) ϕ ˆ ı , ϕ ˆ , A ˆ a , A ˆ b (cid:17) = ig ( ε · k )( ε · k ) + t ( ε · ε ) t ( t ˆ a t ˆ b ) ˆ ı ˆ + (3 ↔ ! + i ε · k )( ε · k ) − ε · k )( ε · k ) + ( u − t )( ε · ε ) s f ˆ a ˆ b ˆ c ( t ˆ c ) ˆ ı ˆ ) , (5.6)where we have displayed explicitly the gauge representation indices ˆ ı, ˆ . As a consequenceof the commutation relation for the group generators, the color factors obey a three-termidentity, [ t ˆ a , t ˆ b ] = if ˆ a ˆ b ˆ c t ˆ c → c t − c u = c s . (5.7)It is easy to verify that the same identity is automatically satisfied by the numerators in theabove amplitude, n t − n u = n s . (5.8)This is possibly the simplest nontrivial example of the duality between color and kinematicsfor theories with non-adjoint fields. While CK duality for the two-scalar two-gluon amplitudeis a rather straightforward generalization of the case of amplitudes with fields in the adjointrepresentation, the four-scalar amplitude exposes new subtleties. This amplitude is A (cid:16) ϕ ˆ ı , ϕ ˆ , ϕ ˆ k , ϕ ˆ l (cid:17) = ig (cid:26) s − u − att ( t ˆ a ) ˆ ı ˆ l ( t ˆ a ) ˆ ˆ k + (3 ↔ (cid:27) . (5.9)Due to the scalar being complex, the amplitude involves only two terms; in principle, wemay consider imposing the extra identity( t ˆ a ) ˆ ı ˆ l ( t ˆ a ) ˆ ˆ k − ( t ˆ a ) ˆ ˆ l ( t ˆ a ) ˆ ı ˆ k = 0 . (5.10)However, this identity is not satisfied except for special gauge groups and representations,so it would seem that our Working Rule 3 does not compel us to impose (5.10) in thegeneral case. At the same time, the numerator factors can easily obey the correspondingtwo-term kinematic identity if we fix a = 1. Whether of not this choice should be madedepends on the situation in which the kinematic numerators are used. For example, we mightchoose to use this theory in a double-copy construction that involves massive W fields in amatter representation (we will see that this is required, for example, for constructing Higgsedsupergravities). In these cases, the spontaneously-broken gauge symmetry results in Wardidentities that can be satisfied only if the massive vectors belong to specific representationsfor which color factors obey additional relations which are of the form (5.10). Hence, itwill be appropriate to impose two-term identities on the numerators of the second gaugetheory entering the double copy. In contrast, whenever (5.10) is not necessary for derivingsome Ward identity in the gravity theory, there is no particular reason for imposing thecorresponding numerator identity. We use the slightly-nonstandard notation, e.g. A n (cid:0) , . . . , n Φ n (cid:1) , which displays explicitly the externalstates. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
So far, we have presented two examples of theories which obey CK duality. In some cases,it is sufficient to write down simple gauge theories, verify that they obey the duality up toat least a certain multiplicity, and feed the corresponding numerators in the double-copyapparatus. However, this approach quickly becomes inconvenient as the number of matterrepresentations increases. Hence, we would like to have systematic tools for obtaining moregeneral theories which obey the duality from simpler ones. These tools will be reviewed inthe next three subsections.
A first step for generating theories with fields transforming in matter representations in away that preserves the duality is to start from the adjoint representation of a larger gaugegroup and decompose it into representations of a subgroup. This amounts to splitting theadjoint index of the larger groups ˆ A asˆ A → (ˆ a, ˆ α , . . . , ˆ α p ) , (5.11)where ˆ a is the adjoint index of the smaller subgroup and ˆ α , . . . , ˆ α p are indices of otherrepresentations. While it is always possible to choose them to correspond to irreduciblerepresentations, we will not do so here. The structure constants of the original gauge groupare broken down as follows: { f ˆ A ˆ B ˆ C } → { f ˆ a ˆ b ˆ c , f ˆ a ˆ α i ˆ β i , f ˆ α i ˆ β j ˆ γ k } . (5.12)Here f ˆ a ˆ b ˆ c are the structure constants of the unbroken subgroup, f ˆ a ˆ α i ˆ β i give the representationmatrices of the i -th matter representation and f ˆ α i ˆ β j ˆ γ k give Clebsch-Gordan coefficients forrepresentations i, j , and k . We note that f ˆ a ˆ b ˆ α = 0 from closure of the algebra of the unbrokengauge group. The notation above suggests that we have assumed the matter representationsabove to be real; the complex case can be treated analogously by introducing a pairingbetween some representations i, j, k and their conjugate, denoted as ¯ ı, ¯ , ¯ k . The breaking ofthe adjoint representation acts in the following way on color and numerator factors:• Color factors are split into different pieces according to the representations carried byinternal and external lines (see Fig. 18); color identities are preserved by this opera-tion, but one needs to take into account that some color factor may vanish upon directevaluation.• Numerator factors are unchanged. Graphs with the same topology but different rep-resentation labels will inherit the same numerator factors as the original graphs of theunbroken theory. Whenever numerators obey a three-term identity in the unbrokentheory, the identity will be inherited by the broken theory.Two-term identities of the form (5.10) deserve a more detailed discussion. Before decompo-sition into representations of a subgroup, color factors obey the standard Jacobi relations,70 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES c
12 34 → c
12 34 , c
12 34 ,c
12 34 , c
12 34 , · · · Figure 18: Breaking of an adjoint representation into representations of a smaller subgroup. Curlylines denote the adjoint representation of the smaller group; double lines denote matter represen-tations. which at four points can be written as c
12 34 − c
12 34 = c
12 34 . (5.13)If we consider the case in which the initial adjoint representation is broken into three pieces(adjoint, a single complex matter representation, its conjugate), the corresponding identityfor external matter is c
12 34 = c
12 34 , (5.14)because the structure constant does not contain a component with two indices in the samecomplex representation. Hence, the original three-term color identity has collapsed into atwo-term identity. However, the decomposition of color factors with respect to a subgroupdoes not affect the kinematic numerators so, if the original theory obeys CK duality, thenumerators still obey three-term kinematic identities after the decomposition. In otherwords, a nonvanishing numerator is associated to a vanishing color factor. We will see thatthis will require extra care, e.g. in the construction of Higgsed supergravities in Sec. 5.3.7.Note that a nonzero kinematic numerator can be associated with a vanishing color factoralso in theories with only adjoint fields, as we shall see in Sec. 6. If we consider a gauge theory which possesses a certain number of global/flavor symmetries(which may include the R symmetry), it is always possible to truncate it to its sector which71 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES is invariant under the combined action of some elements of the global and gauge groups.Specifically, a generic adjoint field Φ of the original theory will transform asΦ → RF g Φ g † , (5.15)where g is the gauge-group element and R, F are the corresponding elements of the R -symmetry and global-flavor group ( R -symmetry and global indices are not explicitly dis-played). It is convenient to consider elements ( g, R, F ) which belong to a discrete subgroupΓ of the symmetry group of the theory we are considering, that is we have elements g, R, F such that g k = I, F k = I, R k = I for some k . Theories obtained with this constructionare referred to as field-theory orbifolds in the literature [311]. Given ( g, R, F ) above, we canimmediately write a projector P Γ Φ = 1 | Γ | X ( g,R,F ) ∈ Γ RF g Φ g † , (5.16)where | Γ | denotes the rank of Γ. It is easy to verify that this projector sets to zero allcomponents of Φ which are not invariant under Γ.To give a simple example, we start from N = 4 SYM theory with SU (2 N ) gauge groupand consider an Γ = Z orbifold with generators r = diag (cid:16) , , − , − (cid:17) , g = I N − I N ! . (5.17)The matrix r gives the action of the unique nontrivial generator of Z on the fundamental R -symmetry indices; the action of Z on other representation of the R symmetry can beobtained by taking tensor products of r . In this case, it is convenient to represent the actionof the projector on the components of a on-shell N = 4 superfield V ˆ A N =4 which is written as(B.12). The part of this superfield which survives the orbifold projection (5.16) is V ˆ A N =4 → A ˆ a + + η i λ ˆ ai + + η r λ ˆ αr + + η η φ ˆ a + η i η r φ ˆ αir + η η φ ˆ a + η η η r λ ˆ αr − + η i η η λ ˆ ai − + η η η η A ˆ a − , (5.18)where i, j = 1 , r, s = 3 ,
4. The gauge-group indices ˆ a, ˆ b and ˆ α, ˆ β run over the (reducible) SU ( N ) × SU ( N ) × U (1) adjoint representation and the bi-fundamental representation, re-spectively. This result can be organized in N = 2 on-shell superfields as V ˆ A N =4 → V ˆ a N =2 + η r Φ ˆ αr N =2 + η η V ˆ a N =2 , r = 3 , , (5.19)where Φ N =2 is the on-shell hypermultiplet superfield. Hence, we see that the theory resultingfrom the orbifold projection (5.16) with Γ = Z acting as (5.17) is an N = 2 SYM theory withgauge group SU ( N ) × SU ( N ) × U (1) and one matter hypermultiplet in the bi-fundamentalrepresentation.Exercise 5.1: Work out spectrum and on-shell superfield organization for the Z orbifoldprojection of N = 4 SYM theory with generators r = diag (cid:16) − , − , − , − (cid:17) , g = I N − I N ! . (5.20) Other subgroups can also be considered. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
What is the residual supersymmetry?Exercise 5.2: Formulate an orbifold projection of N = 4 SYM preserving N = 1 supersym-metry. Work out the multiplet structure of the on-shell superfields.Field-theory-orbifold amplitudes are constructed through a set of Feynman rules whichare obtained directly by taking the Feynman rules of the parent theory and dressing bothinternal and external lines with projectors of the form (5.16). For a tree-level amplitude, onecan use invariance of the propagators and vertices under global and gauge symmetries tomove all projectors from internal to external lines. The result is that all tree-level amplitudesof a theory constructed as an orbifold can be obtained from the amplitudes of the parenttheory by inserting projectors on the external legs or, alternatively, by ensuring that theasymptotic states are invariant under the orbifold group. Particularly relevant to us, thisproperty has the consequence that all numerator relations of the parent theory are preservedby the orbifold construction [30].Exercise 5.3: Consider N = 4 SYM theory with SU (2 N + 1) gauge group. Show that theprojection with orbifold group generators r = diag (cid:16) , , − , − (cid:17) , g = − I N
00 1 ! , (5.21)yields a N = 2 theory with a hypermultiplet in the fundamental representation.The reader may wonder whether there is a straightforward way to extend this result toloop level. At one loop, using the symmetries of propagators and vertices, projectors can beremoved from all but one internal line (which can be chosen freely). Additionally, particularclasses of loop-level amplitudes (for example, planar amplitudes in the large- N limit) areinherited from the parent theory (for a subset of so-called regular orbifolds) [311, 312]. Loop-level amplitudes can of course be constructed from tree-level ones with unitarity methods.For general amplitudes and choices of orbifold groups, the properties at loop-level will not bedirectly related to the ones of the parent theory and the orbifold construction will be used toobtain tree-level building blocks to be employed with unitarity methods. This constructionhas been instrumental, for example, in the study of one-loop amplitudes for supergravitiesthat can be embedded in the N = 8 maximal theory [30, 241]. Once representations of a larger gauge group are broken into smaller pieces, it is in principlepossible to introduce nonzero masses for some of the fields. At the same time, if we intendto consider more general theories of gravity coming from the double copy, we need someprocedure for generating massive states (for which the most natural choice is the Higgsmechanism). If we consider gauge theories that can be written in higher dimension, astraightforward way to create mass terms from an amplitude perspective consists of assigningto some of the fields momenta in one of the extra (compact) dimensions.73
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Our starting point is to consider adjoint fields in a higher-dimensional theory which arewritten as A µ ˆ A ( ~x, x D +1 ) (cid:12)(cid:12)(cid:12)(cid:12) D +1 = (cid:16) e ix D +1 m (cid:17) ˆ A ˆ B A µ ˆ B ( ~x ) ,φ a ˆ A ( ~x, x D +1 ) (cid:12)(cid:12)(cid:12)(cid:12) D +1 = (cid:16) e ix D +1 m (cid:17) ˆ A ˆ B φ a ˆ B ( ~x ) , a = 1 , . . . , n , (5.22)where m ˆ A ˆ B is a mass matrix with adjoint indices ˆ A, ˆ B , ~p is the D -dimensional momentumand p D +1 is the momentum in the D +1 internal direction. If the mass matrix vanishes, this isequivalent to ordinary dimensional reduction. The condition above can also be implementedin position space through the differential equation ∂ D +1 A µ ˆ A φ a ˆ A ! (cid:12)(cid:12)(cid:12)(cid:12) D +1 = i m ˆ A ˆ B A µ ˆ B φ a ˆ B ! . (5.23)Introducing this mass term has the effect of breaking the adjoint representation of the gaugegroup into various representations with respect to which m ˆ A ˆ B is block-diagonal. We choose m ˆ A ˆ B to be given by m ˆ A ˆ B = igV f ˆ0 ˆ A ˆ B . (5.24)Fields that commute with the gauge-group generator t ˆ0 will not have a mass since thatimplies that f A ˆ B vanish. We can now explicitly show that the kinetic term of the scalars in( D + 1) dimensions is identical to a kinetic term in D dimensions plus a φ -term in which ascalar acquires a VEV:12 (cid:16) D µ φ a ˆ A (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) D +1 → (cid:16) D µ φ a ˆ A (cid:17) − (cid:16) i m ˆ A ˆ B φ a ˆ B + gf ˆ A ˆ B ˆ C φ B φ a ˆ C (cid:17) = 12 (cid:16) D µ φ a ˆ A (cid:17) + g (cid:16) [ V t + φ , φ a ] (cid:17) , (5.25)where we have renamed the gauge field in the internal direction, A ˆ AD +1 → φ A (the globalindex a does not include a = 0). We then inspect the ( D + 1)-dimensional vector-field kineticterm, − (cid:16) F ˆ Aµν (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) D +1 → − (cid:16) F ˆ Aµν (cid:17) + 12 (cid:16) ∂ µ φ A − i m ˆ A ˆ B A ˆ Bµ + gf ˆ A ˆ B ˆ C A ˆ Bµ φ C (cid:17) = − (cid:16) F ˆ Aµν (cid:17) + 12 (cid:16) ( D µ φ ) ˆ A − i m ˆ A ˆ B A ˆ Bµ (cid:17) = − (cid:16) F ˆ Aµν (cid:17) + 12 (cid:16) ( D µ φ + D µ h φ i ) ˆ A (cid:17) . (5.26)This term is identical to the D -dimensional vector-field kinetic term plus the kinetic termfor φ in the presence of a VEV h φ i = V t ˆ0 . (5.27)Adding the quartic potential terms for the scalars, one sees that the ( D + 1)-dimensionalmassless (S)YM Lagrangian in the presence of a compact momentum of the form (5.22) is in-deed equivalent to a spontaneously-broken D -dimensional SYM Lagrangian with VEV given74 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES by (5.27). Strictly speaking, we have shown that this procedure works only in the presence ofa quartic potential generated by dimensional reduction of a higher-dimensional pure (S)YMtheory. The case of more general scalar potentials need to be considered separately. Theargument in this subsection gives a prescription for finding amplitudes of theories with fieldsbecoming though Higgs mechanism in terms of higher-dimensional massless amplitudes. Ifthe higher-dimensional theory obeys CK duality, the massive amplitudes will inherit thesame algebraic properties [122]. BCJ amplitude relations with massive particles were alsoderived in [116] using the CHY formalism.
At this point, we have developed some basic techniques to start from the amplitudes of anarbitrary theory which is known or can be shown to obey the duality between color andkinematics and generate the amplitudes of more involved theories, which may include fieldsin non-adjoint representations and mass terms from the Higgs mechanism, in a way thatpreserves numerator relations. In principle, we can use the numerators from various theoriesobtained with this procedure for producing amplitudes through the double-copy technique.As discussed in Secs. 2 and 4, these amplitudes will obey the Ward identities related toinvariance under linearized diffeomorphisms and hence should be the amplitudes from somegravitational theory. Identifying the precise theory, however, is not always straightforward.In principle, one could consider a generic Lagrangian involving the Einstein-Hilbert term(or, in case of conformal gravity, some Weyl gravitational action) and arbitrary matterinteractions. Up to terms which vanish due to the equations of motion, this Lagrangiancan be fixed order by order by comparing its amplitudes with the ones from the double-copy method. In practice, the implementation of this program is limited by one’s desireto evaluate higher-point tree amplitudes. For many theories however, minimal informationabout symmetries and lower-point interactions can be sufficient for identifying the theorycompletely and, in principle, for writing down its Lagrangian (with some help from therelevant supergravity literature). More specifically:• Symmetry considerations are sufficient for identifying supergravities with extended N ≥ N = 2 supersymmetrywhich can be lifted up to at least five spacetime dimensions can be uniquely specified bytheir three-point interactions (specifically, three-vector amplitudes in five-dimensions).In a sense, these theories constitute a natural testing ground for double-copy construc-tions with reduced supersymmetry [120, 121].• More generally, there exist amplitudes which capture physical features of the desiredsupergravity theory. For example, YME theories or gauged supergravities with non-abelian gauge group will possess non vanishing three-point amplitudes between three75 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES gluons [120]. Gauged supergravities will have nonvanishing amplitudes between twogravitini and one vector [123]. Knowledge of these amplitudes can either allow identi-fication of the theory or point to the general class to which the theory belongs.• Some theories are characterized in terms of their soft limits. These include e.g. theo-ries with homogeneous target spaces [121, 222], the NLSM and some of its extensions[125, 285, 306], and the special Galileon theory [309].
We now proceed to discussing some examples. A list of the main double-copy constructibletheories at the time of this writing can be found in Tables 4, 5, and 6.
N ≥ supersymmetry Pure supergravities with N = 4 and 8 have been originally formulated from a Lagrangianperspective in [313, 314] and [258, 315], respectively, while the N = 5 and N = 6 Lagrangianswere obtained by truncation [316] from that of the N = 8 supergravity. Amplitudes oftheories with extended N ≥ N = 4 SYM theory together with a YM or SYM theory. The possibilities are the following[31]: N = 8 supergravity : ( N = 4 SYM) ⊗ ( N = 4 SYM) , N = 6 supergravity : ( N = 4 SYM) ⊗ ( N = 2 SYM) , N = 5 supergravity : ( N = 4 SYM) ⊗ ( N = 1 SYM) , N = 4 supergravity : ( N = 4 SYM) ⊗ ( N = 0 YM) . (5.28)All double copies above involve gauge theories with only adjoint fields. These are cases inwhich the symmetries of the desired supergravity single out the correct construction, withoutany free parameters. Supergravities with N > N = 4 supergravities onecan add matter in the form of N = 4 vector multiplets, which correspond to adding adjointscalars in the non-supersymmetric gauge theory.Perturbative mass spectra and on-shell superfield structure of the theories listed abovecan be straightforwardly obtained from the on-shell superfields of the gauge theories. Al-ternatively, all theories with N > N = 4 theories can be seen astruncations of N = 8 supergravity using a field-theory orbifold construction. N = 4 supergravity has an alternative double-copy construction, in terms of two N = 2SYM theories coupled to hypermultiplets in matter representations. Apart from the massspectra, it has been verified that tree-level and four-point one-loop amplitudes in the tworealizations are the same, including anomalous amplitudes [20, 241].We now look more in detail at the pure N = 4 supergravity in four dimensions. Thetheory involves one complex scalar, whose asymptotic states are obtained by taking thedouble copy of gauge-theory gluons with opposite polarizations. Geometrically, the scalar76 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES can be regarded as the complex coordinate of the coset space M D = SU (1 , U (1) . (5.29)As we will discuss more in detail later, the fact that the scalar lives in an homogeneous spacecan be confirmed by checking the vanishing of the scalar soft limits at tree level. Moreexplicitly, the bosonic part of the Lagrangian for pure N = 4 supergravity has a relativelysimple form, e − L = − R ∂ µ τ ∂ µ ¯ τ (Im τ ) −
14 Im τ F
Iµν F Iµν −
18 Re τ e − (cid:15) µνρσ F Iµν F Iρσ = − R ∂ µ τ ∂ µ ¯ τ (Im τ ) + iτ ( F + Iµν ) − i ¯ τ ( F − Iµν ) ! , (5.30)where τ = ie − φ + χ is the dilaton-axion scalar, ˜ F Iµν = ( i/ e(cid:15) µνρσ F Iρσ , F ± Iµν = ( F Iµν ± ˜ F Iµν ) / I = 1 , . . . , N = 4 su-pergravities can be easily modified by adding extra adjoint scalars in the non-supersymmetricgauge theory. This can be done by considering a YM theory coupled to N scalars, which isthe reduction to four dimensions of D = ( N + 4) pure YM theory. This theory is invari-ant under an SO ( N ) symmetry, which is the subgroup of the D -dimensional Lorentz grouptransverse to four dimensions, SO (1 , N ) → SO (1 , × SO ( N ). Under this symmetry,the vector fields are inert while the scalars transform in the vector (fundamental) represen-tation. Since these scalars transform in the adjoint representation of the gauge group, theycan be double-copied with the N = 4 vector multiplet to yield N vector multiplets in thesupergravity theory. Their scalars transform in the ( N , ) representation of SO ( N ) × SO (6),where latter factor is the R -symmetry group. In N = 4 supergravity scalars fields outside the graviton multiplet parametrize a homo-geneous space of the form M D = GH , (5.31)where the stabilizer group H is the symmetry which is linearly realized and thus visiblein amplitudes involving scalars. Thus, the symmetry which is manifest in the double-copyconstruction cannot be larger than H . In our case, we have SO ( N ) × SO (6) ⊆ H . Thissuggests that the 6 N vector multiplet scalars parametrize SO (6 , N ) / ( SO (6) × SO ( N )) and, In N = 4 supergravity the single-soft-scalar limit no longer vanishes at one loop [260] due to an anomalyof the U (1) symmetry in Eq. (5.29). Finite local counterterms can be used to restore this symmetry (at theexpense of the other SU (1 ,
1) generators) [265]. This counterterm also restores the vanishing single-soft-scalarlimit. Recall that dimensional reduction is an operation which is known to preserve CK duality. These theorieswill sometimes be denoted as YM DR . Vector fields in this theory are of two types: graviphotons, which are part of the gravition multiplet andtransform in ( , ) and vectors which are part of the additional vector multiplets, which transform as ( N , ). A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES together with Eq. (5.29), that the (6 N + 2) real scalars in the four-dimensional N = 4supergravity theory parametrize the symmetric space M D = SO (6 , N ) SO (6) × SO ( N ) × SU (1 , U (1) . (5.32)The double-copy construction for N ≥
N ≥ N = 4 supergravity have been shown to be UV-finiteat three loops and UV-divergent at four loops in four dimensions [33, 36, 37, 293]. Thefour-loop UV-divergence appears to be related to a U (1) anomaly [260, 263, 265, 266].Full one- and two-loop four-point amplitudes in N = 4 supergravity are given inRefs. [31, 32].• Four-point amplitudes for N = 5 supergravity are finite at least through four loops infour dimensions [292]. Despite various attempts, there is no standard symmetry expla-nation for the “enhanced cancellations” that lead to this improved UV behavior [317].See, however, Refs. [318, 319] for arguments suggesting that U-duality invariance maybe ultimately responsible. It is of considerable interest to settle the origin of thesecancellations, and to know whether they continue to higher orders.• The complete two-loop four-point amplitude of N = 6 supergravity may be foundin Ref. [32]. As yet there have not been any direct studies of the critical dimensionof this theory at high loop orders, altough it follow from the calculations in N = 5supergravity that divergences cannot appear before five loops. Standard symmetryconsiderations imply that divergences are delayed until at least five loops [317, 320].• UV properties of four-point amplitudes in N = 8 supergravity have been analyzedin detail through five loops [38]. In contrast to the case of N = 5 supergravity, N = 8 supergravity at five loops does not appear have enhanced cancellations, butit is possible that this is an artifact of the fact that the analysis is carried out inthe fractional critical dimension D = 24 /
5, where from various considerations [321,322] divergences are first expected to appear. A proper study of this issue in themost interesting dimension D = 4 requires a seven-loop computation, as suggestedby symmetry considerations [274, 321–326]. The complete three-loop four-point andtwo-loop five-point amplitudes of N = 8 supergravity have been obtained [327, 328],starting from integrands constructed via the double copy [2, 4]. The construction of N = 8 one-, two- and three-loop integrands via the double copy is described in Sec. 6. N = 2 supersymmetry In this section we discuss amplitudes in theories with N = 2 supersymmetry in four di-mensions (eight supercharges). Theories of this type are no longer specified solely by their78 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES matter content. Hence, we need a strategy to conveniently classify the interactions consis-tent with N = 2 supersymmetry. An efficient approach is to focus on theories that canbe uplifted to five dimensions. The Lagrangians for these theories have long been knownexplicitly [329–332]. Here we will write only the bosonic part of the Lagrangian: e − L = − R − ◦ a IJ F Iµν F Jµν + 12 g xy ∂ µ φ x ∂ µ φ y + e − √ C IJK ε µνρσλ F Iµν F Jρσ A Kλ . (5.33)All vectors in the Lagrangian are taken to be abelian (YME theories will be discussed inSec. 5.3.6). The index I = 0 , , . . . , n runs over the number of vectors in the theory with I = 0corresponding to the graviphoton. F Iµν are the field strengths, while ◦ a IJ and g xy are functionsof the physical scalars φ x ( x = 1 , . . . , n ). The key insight is that the symmetric constanttensor C IJK is sufficient to specify the theory completely, i.e. to fix all functions appearingin the two-derivative Lagrangian. The formalism manifesting this feature introduces anauxiliary ambient space with coordinates ξ I and dimension equal to the number of vectorsin the theory which, together with the C IJK -tensor, are used to define a cubic polynomial V ( ξ ) ≡ C IJK ξ I ξ J ξ K . (5.34)In turn, this is used to define a metric on the ambient space: a IJ ( ξ ) ≡ − ∂∂ξ I ∂∂ξ J ln V ( ξ ) . (5.35)The scalar manifold M D is defined as the hypersurface obeying the equation V ( h ) = C IJK h I h J h K = 1 , h I = s ξ I . (5.36)The functions ◦ a IJ ( φ ) and g xy ( φ ) which appear in the Lagrangian are given by the restrictionof the ambient-space metric to M D and the pullback to that surface of the ambient spacemetric, respectively: ◦ a IJ ( φ ) = a IJ (cid:12)(cid:12)(cid:12) V ( h )=1 ; g xy ( φ ) = 32 ∂ξ I ∂φ x ∂ξ J ∂φ y a IJ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) V ( h )=1 . (5.37)The functions appearing in the fermionic part of the Lagrangian can also be expressed interms of the C IJK -tensor. Since the C IJK -tensor can be obtained by inspecting three-pointamplitudes, N = 2 Maxwell-Einstein theories in five dimensions are uniquely specified bytheir three-point interactions. This is in contrast to Maxwell-Einstein theories that onlyexist in four dimensions as well as theories with hypermultiplets. It is in principle possibleto compute amplitudes from the Lagrangian (5.33) using Feynman rules. To this end oneshould first expand around a scalar-base point (i.e. some background values for the scalarfields) at which the scalar and vector kinetic terms are positive-definite. The quadratic termsshould then be diagonalized in order to find the spectrum, the propagators, and the vertices. For consistency with the rest of this review, this Lagrangian is written with a metric of mostly-minussignature, in contrast to most of the supergravity literature. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
For practical calculations, it is often convenient to reduce the theory to four dimensions anduse the spinor-helicity formalism.To identify the simplest supergravity theories we will utilize symmetry considerationstogether with minimal information on the trilinear interaction terms. A natural startingpoint is to consider a double copy of the form N = 2 supergravity : ( N = 2 SYM) ⊗ ( N = 0 YM) , in which the non-supersymmetric theory is a pure (4 + n )-dimensional YM theory reduced tofour dimensions. Bosonic asymptotic states from the double copy are identified with thosefrom the supergravity Lagrangian as follows [120]: A − − = ¯ φ ⊗ A − , h − = A − ⊗ A − , A − = φ ⊗ A + , h + = A + ⊗ A + ,A − = φ ⊗ A − , i ¯ z = A + ⊗ A − , A = ¯ φ ⊗ A + , − iz = A − ⊗ A + ,A A − = A − ⊗ φ A , i ¯ z A = ¯ φ ⊗ φ A , A A + = A + ⊗ φ A , − iz A = φ ⊗ φ A . (5.38)The index A above has range A = 1 , , . . . , n . Note that, in four dimensions, an extra vectorfield ( A − µ ) is present. φ denotes the single complex scalar in the N = 2 SYM theory. Sinceall gauge-theory fields are in the adjoint representations, each field bilinear is associated toa supergravity state. Overall, the construction produces a supergravity with the followingproperties:1. has N = 2 supersymmetry in four dimensions;2. has ( n + 1) vector multiplets in four dimensions. Scalars obtained as φ ⊗ φ A transformunder a U (1) × SO ( n ) symmetry;3. uplifts to five dimensions whenever n > N = 4 supergravity with the samenumber of vector multiplets.Putting together all available information, the scalar manifold of the resulting theory turnsout to be M D = SO ( n, SO ( n ) × SO (2) × SU (1 , U (1) . (5.39)This infinite family of theories is known in the supergravity literature as the generic Jordanfamily of N = 2 Maxwell-Einstein supergravities. The corresponding cubic polynomial inthe natural basis is [329]: V ( ξ ) = √ (cid:16) ξ ( ξ ) − ξ ( ξ i ) (cid:17) , i = 2 , , . . . , n . (5.40) The phase in the map between the asymptotic states from the supergravity Lagrangian and the onesfrom the double copy was chosen to match the phase conventions in the supergravity literature, see e.g.[121, 329, 332]. Note that z is the same scalar as τ from the previous subsection. The detailed form of the C -tensor may be changed by field redefinitions without changing the scatteringamplitudes. See Ref. [329] for a discussion of the canonical and natural basis. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Exercise 5.4: Calculate explicitly ◦ a IJ and g xy corresponding to the cubic polynomial above.Exercise 5.5: Calculate the three point amplitude M tree3 (cid:16) A − , A A − , z B (cid:17) using the double-copy prescription and the map (5.38). N = 2 Maxwell-Einstein supergravities
We now want to consider more general theories with N = 2 supersymmetry and homogeneousscalar manifolds. A scalar manifold is said to be homogeneous if it admits a transitive groupof isometries. From an amplitude perspective, not all these isometries will linearly realized,i.e. some of them correspond go constant shifts of the scalars which modify the vacuum of thetheory. Hence, in the homogeneous case, all coordinates of the manifold are Goldstone bosonsand, consequently all single-soft limits of scalar amplitudes vanish (see for example [222]).In short, drawing from the discussion in Sec. 4.5, we have the following criterion:A necessary condition for a theory to possess a (locally) homogeneous scalar man-ifold is that all single-soft limits of scalar amplitudes vanish.We also note that double-soft limits can be used to identify the particular homogeneousspace under consideration (i.e. G in G/H ) [222]. More generally, each independent vanishingsingle-soft scalar limit will correspond to an isometry of the scalar manifold.We now return to the double-copy construction outlined in Sec. 5.3.2. A natural extensionconsists of adding some matter fields in both gauge theories. Hypermultiplets are the onlyavailable matter that can be coupled to N = 2 SYM theory. One hypermultiplet consistsof four real scalars and two Majorana fermions and is an irreducible representation of the N = 2 supersymmetry algebra. For the construction described below, we will need toassign matter representations of the gauge group to hypermultiplets. If the gauge-grouprepresentation is pseudo-real, an additional option becomes available: we may consider a half-hypermultiplet instead of a full one. A single half-hypermultiplet is by itself a representationof the supersymmetry algebra, but one is forced to include the Charge-Parity-Time reversal(CPT)-conjugate states unless its gauge-group representation is pseudo-real. This leadsto a full hypermultiplet. Taking a single half-hypermultiplet, i.e. choosing the smallestrepresentations of supersymmetry algebra, amounts to introducing the minimal possiblenumber of states and, in principle, allows us to manifest a larger global symmetry in thenon-supersymmetric theory.If the desired supergravity theory is of the Maxwell-Einstein class, the non-supersymmetricgauge theory needs to be a YM-scalar theory with extra fermions, so that additional vectormultiplets are obtained as double copies involving one gauge-theory hypermultiplet and onefermion. Because of Working Rule 4, we will take the additional fermions to transform in81 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES D D D D fermions n F ( D, P, ˙ P ) conditions flavor group4 1 P R or W SU ( P )5 1 P R SO ( P )6 1 P + ˙ P RW SO ( P ) × SO ( ˙ P )7 2 2 P R SO ( P )8 4 4 P R or W U ( P )9 8 8 P PR U Sp (2 P )10 8 8 P +8 ˙ P PRW
U Sp (2 P ) × U Sp (2 ˙ P )11 16 16 P PR U Sp (2 P )12 16 16 P R or W U ( P ) k +8 16 D k r ( k, P, ˙ P ) as for k as for k Table 7: Parameters in the double-copy construction for homogeneous supergravities [121, 240]. n F ( D, P, ˙ P ) is the number of 4 D irreducible spinors in the non-supersymmetric gauge theory,which can obey a reality (R), pseudo-reality (PR) or Weyl (W) conditions. Note that the patternrepeats itself with periodicity 8. the same pseudo-real representation R used for the supersymmetric theory. The Lagrangianis then written as L = − F ˆ aµν F ˆ aµν + 12 ( D µ φ a ) ˆ a ( D µ φ a ) ˆ a + i λ α D µ γ µ λ α + g φ a ˆ a Γ a βα λ α γ t ˆ a R λ β − g f ˆ a ˆ b ˆ e f ˆ c ˆ d ˆ a φ a ˆ a φ b ˆ b φ a ˆ c φ b ˆ d , (5.41)where ˆ a, ˆ b are adjoint indices of the gauge group and α, β = 1 , . . . , n F and a, b = 1 , . . . , ( D − a βα in the global indices need to be constrained by imposingthe duality between color and kinematics at four points.Imposing the duality on amplitudes between two adjoint scalars and two matter fermionsgives the constraint [121] { Γ a , Γ b } = − δ ab , (5.42)that is, the matrices Γ a are gamma matrices which belong to a ( D − D dimensions. A secondparameter, P , will count the number of irreducible fermions in D dimensions.Exercise 5.6: Show that imposing CK duality on amplitudes the two-scalars two-fermionamplitudes given by the Lagrangian (5.41) yields the relation (5.42).An important difference with the standard treatment of D -dimensional spinors is thefact that fermions transform in pseudo-real representations of the gauge group. To obtainirreducible spinors, a case-by-case analysis is necessary. Depending on the value of the82 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES parameter D , one can impose reality (R) or pseudo-reality (PR) conditions [121] λ = λ t C CV ,
R : C = C D − , PR : C = C D − Ω , (5.43)where C D − and C are the internal and spacetime charge-conjugation matrices, respectively.They obey the relations C D − Γ a C − D − = − ζ (Γ a ) t , C γ µ C − = − ζ ( γ µ ) t , ζ = ± V is theunitary antisymmetric matrix entering the pseudo-reality condition for the gauge-group rep-resentation matrices, V t ˆ a R V † = − ( t ˆ a R ) ∗ . Ω is an antisymmetric real matrix acting on indiceswhich run over the number P of irreducible spinors. Alternatively, if D is even, one canimpose Weyl conditions. If we have more than one irreducible spinor, an extra flavor sym-metry is present (either U ( P ), SO ( P ) or U Sp ( P ), depending on whether Weyl, Reality orpseudo-Reality conditions were employed). A separate treatment is needed when D = 6 , P, ˙ P which count the number of each.Explicit computations reveal that soft-scalar limits vanish for amplitudes constructedby the double copy [121]. Hence, this generalized construction yields supergravities withhomogeneous scalar manifolds. The dimension-by-dimension analysis is given in Table 7.The number of vector multiplets in the four-dimensional supergravity is equal to ( D − n F ),where n F ( D, P, ˙ P ) is the number of 4 D fermions in the non-supersymmetric gauge theory;the supergravity bosonic states are obtained as double copies in the following way [121]: A − − = ¯ φ ⊗ A − , h − = A − ⊗ A − , A − = φ ⊗ A + , h + = A + ⊗ A + ,A − = φ ⊗ A − , i ¯ z = A + ⊗ A − , A = ¯ φ ⊗ A + , − iz = A − ⊗ A + ,A A − = A − ⊗ φ A , i ¯ z A = ¯ φ ⊗ φ A , A A + = A + ⊗ φ A , − iz A = φ ⊗ φ A ,A α − = χ − ⊗ λ α − , i ¯ z α = χ + ⊗ λ α − , A α + = χ + ⊗ λ α + , − iz α = χ + ⊗ λ α − . (5.44)Exercise 5.7: Show that the amplitude M tree5 (cid:16) z , ¯ z , z α , ¯ z β , z (cid:17) has vanishing single-soft lim-its for all external scalars.Exercise 5.8: Show that the amplitude M tree3 (cid:16) A a − , A α − , z β (cid:17) can be expressed as M tree3 (cid:16) A a − , A α − , z β (cid:17) = κ √ h i (cid:16) U t Γ a C − (cid:17) αβ . (5.45)A remarkable result is that the theories listed in Table 7 reproduce the complete classifica-tion of homogeneous supergravities by de Wit and van Proeyen [282]. Theories obtained withthis construction include some classic examples. Specifically, for P = 1 and D = 7 , , , M R D = Sp (6 , R ) U (3) , M C D = SU (3 , S ( U (3) × U (3)) , M H D = SO ∗ (12) U (6) , M O D = E − E × U (1) . (5.46)83 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
An important property is that Magical theories are unified, that is there exists a symmetrywith respect to which all vector fields transform in a single irreducible representation. Phys-ically, this implies that fields from different matter multiplets have the same properties. Incontrast, vectors in generic homogeneous theories typically have different interactions accord-ing to whether they are obtained as vector-scalar or as fermion-fermion from a double-copyperspective. The construction of these theories from a supergravity perspective relies ondegree-three Jordan algebras which have as elements 3 × R , C , H , O ). Reviewing the supergravity construction is beyond the scopeof this review; we refer the reader to Ref. [329] for details.Aside from the Magical Supergravities, there is another class of examples of unifiedtheory in four dimensions. They are obtained by choosing D = 4 and P is arbitrary. Itis not difficult to see that the construction exhibits a global U ( P ) flavor symmetry, as wellas that the resulting supergravity theory will have ( P + 1) complex scalars in its spectrum.Putting together this information results in the scalar manifold [121, 240] M D = U ( P + 1 , U ( P + 1) × U (1) , (5.47)which is the complex projective space CP P +1 . Theories in this family are also referredto as minimally-coupled or the Luciani model. The analysis can be repeated in dimensionsdifferent from four. In five and six dimensions we find exactly one infinite-dimensional familyof unified theories:5 D : Generic non-Jordan family M D = SO ( P + 1 , SO ( P + 1) , (5.48)6 D : Generic Jordan family M D = SO ( P + 1 , SO ( P + 1) . (5.49)In both cases, the double-copy construction is similar to the one in four dimensions: the non-supersymmetric gauge theory is a YM theory in the appropriate dimension with an arbitrarynumber of fermions and no additional scalars. While the parameter P is by construction non-negative, it should be noted that pure supergravities in dimensions 4 , , P = −
1. This observation will be consequential in formulatingdouble-copy constructions for pure supergravities with N = 2 in various dimensions. Theconstruction outlined in this section has been used to compute one-loop matter amplitudesin these theories and to analyze their UV properties at that order, see Ref. [294]. Pure supergravities with N = 1 , , N ≥ N <
4, however, the gravity multiplet does not contain any scalarfield so the complex scalar under consideration belongs to a matter multiplet. Hence, to84
A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES N tensoring vector states ghosts = matter ⊗ matter0 + 0 A µ ⊗ A ν = h µν ⊕ φ ⊕ a ( ψ + ⊗ ψ − ) ⊕ ( ψ − ⊗ ψ + ) = φ ⊕ a V N =1 ⊗ A µ = H N =1 ⊕ Φ N =2 (Φ N =1 ⊗ ψ − ) ⊕ ( ¯Φ N =1 ⊗ ψ + ) = Φ N =2 V N =2 ⊗ A µ = H N =2 ⊕ V N =2 (Φ N =2 ⊗ ψ − ) ⊕ ( ¯Φ N =1 ⊗ ψ + ) = V N =2 V N =1 ⊗ V N =1 = H N =2 ⊕ N =2 (Φ N =1 ⊗ ¯Φ N =1 ) ⊕ ( ¯Φ N =1 ⊗ Φ N =1 ) = 2Φ N =2 V N =2 ⊗ V N =1 = H N =3 ⊕ V N =4 (Φ N =2 ⊗ ¯Φ N =1 ) ⊕ ( ¯Φ N =2 ⊗ Φ N =1 ) = V N =4 V N =2 ⊗ V N =2 = H N =4 ⊕ V N =4 (Φ N =2 ⊗ ¯Φ N =2 ) ⊕ ( ¯Φ N =2 ⊗ Φ N =2 ) = 2 V N =4 Table 8: Pure gravities constructed as double copies [188]. The construction necessitates ghostsfrom matter-antimatter double copies. Barred multiplets transform in the anti-fundamental rep-resentation. For compactness, graviton and vector supermultiplets H , V N < include the CPT-conjugate states. Pairs of chiral/antichiral N = 1 supermultiplets are grouped in N = 2 hypermul-tiplets, denoted as Φ N =2 . obtain pure supergravities with N <
4, one needs to modify the construction and removethe contributions to amplitudes of the unwanted scalar. At tree level, one can always projectout the unwanted scalars from the amplitudes by judiciously choosing the asymptotic states.Special care is however necessary for loops.A solution to the problem was first outlined in Ref. [188]. The first step is to introducean additional matter representation (the fundamental representation, without any loss ofgenerality) in both gauge theories. The precise map depends on the desired amount of su-persymmetry and is listed in Table 8. Since the various graphs contributing to the amplitudecarry representation information associated to all internal and external lines, we can organizethe graphs with no external matter according to the number of matter loops. We can thentreat the additional matter as a ghost multiplet by associating an extra minus sign to eachmatter loop (as with Faddeev-Popov ghosts). More explicitly, loop-level pure-supergravityamplitudes are constructed using the prescription M ( L ) m = i L − (cid:18) κ (cid:19) m +2 L − X i ∈ cubic Z d LD ‘ (2 π ) LD ( − | i | S i n i ˜ n i D i , (5.50)where | i | denotes the number of matter loops in the i -th graph. It has been shown by explicitcalculation through two loops and argued to all loop orders that amplitudes obtained withthis prescription have the same unitarity cuts as the ones of the pure supergravities listedin Table 8. It is interesting to note that N = 2 ghost multiplets are constructed as doublecopies involving fermions in the non-supersymmetric theory. One can in principle consideran analogous construction involving scalars, but amplitudes constructed in this way wouldhave unitarity cuts which are different from those of pure supergravities. This observationprovides a clue on the meaning of the construction: formally, the prescription above isequivalent to considering one of the unified infinite families of supergravities described atthe end of the last subsection and setting P = − [341, The interpretation of the divergence is rather subtle because of its dependence on evanescent operators A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES N = 1 , , N < N = 2 supergravity. Ingeneral, supergravities with hypermultiplets are less constrained than theories with vectormultiplets. A subset of such theories, however, appears to be closely related to theories withvector multiplets through a procedure known as c -map [346].Starting from a Maxwell-Einstein theory in four dimensions, one first reduces the theoryto three dimensions. After dualization of the vector field, each supermultiplet in the three-dimensional theory contains four real scalars and four Majorana fermions, which is the fieldcontent corresponding to a hypermultiplet. Since the hypermultiplet action is the same inany dimension up to six, the three-dimensional theory obtained with this procedure can beuplifted to higher dimension, leading to the image of the original Maxwell-Einstein theoryunder the c -map.The basic double-copy construction for theories with hypermultiplets was mentioned inRef. [121] and further detailed in Ref. [240]. It relies on taking as one copy N = 2 SYMtheory with matter hypermultiplets and, as the second copy, a YM theory with extra matterscalars. The simplest realization of this construction involves a SYM theory with a singlehypermultiplet in a real representation and a YM theory with m real scalars. A Lagrangianfor the latter theory is: L = − F ˆ aµν F ˆ aµν + 12 D µ ϕ I D µ ϕ I + g (cid:16) ϕ [ I t ˆ a R ϕ J ] (cid:17)(cid:16) ϕ [ I t ˆ a R ϕ J ] (cid:17) . (5.51)Scalar fields ϕ I are labeled by flavor indices I, J = 1 , . . . , m , which refer to the global SO ( m )symmetry, and gauge-group representation indices for some real representation R , which wedo not display; t ˆ a R are the gauge-group generators in this representation and ˆ a, ˆ b are adjointindices. One can verify that this theory obeys CK duality at four points. The computationis identical to the one for a higher-dimensional YM theory reduced to four dimensions, withthe only difference being related to representation of the scalar fields.Based on the symmetry SO ( m ) × SO (4) which is manifest in the construction, the 4 m realhypermultiplet scalars in the theory together with the universal dilaton-axion parametrizethe scalar manifold M D = SU (1 , U (1) × SO ( m, SO ( m ) × SO (4) . (5.52)The second term in the product manifold is the special quaternionic-Kähler manifold whichis the image of the generic Jordan family scalar manifold under the c -map. From the point and choice of fields [339, 340]. In principle, it is possible to choose a different coefficient for the quartic scalar coupling while preservingCK duality. Indeed the scalar sector of this theory is the same as the theory discussed at the end of Sec. 5.2.The theory given here can also be constructed as a field-theory orbifold of an adjoint YM theory in higherdimension. See also Ref. [122] for a similar discussion. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES of view of scattering amplitudes, the relation between the two classes of theories is a conse-quence of the fact that the kinematic numerator factors from the non-supersymmetric gaugetheory are identical in the two constructions. The differences relate to the pairing betweenthe kinematic numerators of the two gauge theories, which is now different because of thedifferent gauge-group representations and color factors.Several additional constructions for ungauged supergravities with various matter contentsdeserve mention:• Various ( N = 1) × ( N = 1) double copies were studied in Refs. [239, 241, 295, 347].In this case, at least one hypermultiplet is present. Since N = 1 gauge theoriesdo not generically uplift to higher dimensions, the construction does not manifestlygive a supergravity which can be written in five dimensions and, hence, three-pointamplitudes cannot be used to specify the theory completely. Instead, the identificationrelies on symmetry consideration and on the possibility of embedding the theory intoa supergravity with extended supersymmetry.• Several examples of N = 1 supergravities constructed as double copies are known. Theknown examples can often be seen as truncations of theories with a larger number ofsupersymmetries [239, 241, 295, 347].• Various examples of non-supersymmetric gravities constructed as double copies areknown [1, 188]. Among these, the simplest example is Einstein gravity with a scalarand antisymmetric tensor, which we have already encountered in Sec. 2. This theory isconstructed as the square of YM theory. In four-dimensions, the antisymmetric tensoris dual to an axion. The scalar-sector Lagrangian is then identical to the one in (5.30).• An interesting version of the construction applies to the so-called twin supergravi-ties [348]. These are pairs of supergravities with different amounts of supersymmetrieswhich share the same bosonic Lagrangian but have different fermionic field contentand interactions.Exercise 5.9: Consider two supergravities constructed as field theory orbifolds of N = 8supergravity with the following generators:Theory 1 : R = diag (cid:16) , e πi , e πi , e πi , , , , (cid:17) , R = diag (cid:16) , , , , − , − , − , − (cid:17) , Theory 2 : R = diag (cid:16) , i, i, i, − , − , − , − i (cid:17) . Find the corresponding spectra and, using the manifest symmetries of the construction, finda candidate for the scalar manifolds.
YME theories are supergravities that involve nonabelian gauge interactions among (someof) the vector fields. Surprisingly, they admit a very simple double-copy realization, whichrelies on the following principle [120]: 87
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Gravity coupled to YM Gauge theory 1 Gauge theory 2 N = 4 YME supergravity N = 4 SYM YM + φ N = 2 YME supergravity (gen.Jordan) N = 2 SYM YM + φ N = 1 YME supergravity N = 1 SYM YM + φ N = 0 YME + dilaton + B µν YM YM + φ N = 0 YM DR -E + dilaton + B µν YM DR YM + φ Table 9: Amplitudes in YME gravity theories for different number of supersymmetries, corre-sponding to different choices for the left gauge-theory factor entering the double copy [120]. YM DR denotes the YM-scalar theory obtained from dimensional reduction. To introduce nonabelian gauge interactions in a gravitational theory from the dou-ble copy, it is sufficient to add trilinear couplings among adjoint scalar fields in oneof the gauge theories entering the construction.The relevant Lagrangian was introduced in (5.1). The effect of the trilinear coupling is tointroduce nonzero supergravity amplitudes of the form M (1 A A − , A B − , A C + ) = iA (1 A − , A − , A + ) A (1 φ A , φ B , φ C )= − κ √ λF ABC h i h ih i = i κ λ ˜ F ABC h i h ih i , (5.53)i.e. amplitudes between three spin-1 fields which are proportional to an antisymmetric tensorobeying Jacobi relations and have the same momentum dependence as the three-gluon am-plitudes from the supergravity Lagrangian. In particular, the supergravity gauge couplingconstant g s is related to the parameter λ in (5.1) as g s = (cid:18) κ (cid:19) λ , (5.54)where we have temporarily re-introduced κ . In this construction, the global-symmetry tensor F ABC , which obeys the Jacobi identity (5.4), is identified with the structure constants ofthe supergravity gauge group. Hence, a global symmetry in one of the two gauge theoriesbecomes a local symmetry in the resulting double-copy gravity theory.We note that this approach gives, by construction, gauge groups which are subgroups ofthe manifest isometry group of the corresponding Maxwell-Einstein theory. These groupsare necessarily compact. Gauging a subgroup of the R symmetry, a construction whichresults in the so-called gauged supergravities, requires a more involved procedure which willbe outlined in Sec. 5.3.8. The double-copy construction for YME theories can be adapted tosupergravities with various amounts of supersymmetry, which are listed in Table 9 [120].Aside from spelling out the construction at the level of the gauge-theory Lagrangian, itis interesting to consider the implications of the double-copy structure on YME amplitudes.We start from the double copy (cid:16) YM + φ (cid:17) = (cid:16) YM + φ (cid:17) ⊗ (cid:16) φ theory (cid:17) , (5.55)88 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES i.e. we note that the double copy between the YM+ φ theory and the bi-adjoint φ theorygives amplitudes from the YM+ φ theory itself. By choosing numerator factors correspond-ing to the DDM basis [180], we then write a color-ordered tree amplitude between k gluonsand m ≥ φ theory as follows A YM+ φ k,m (1 , . . . , k | k + 1 , . . . , k + m ) = − i X w ∈ σ ...k N k ( w ) A φ k + m ( w ) + Perm(1 , . . . , k ) . (5.56) A φ k + m ( w ) are amplitudes in bi-adjoint φ theory that are color-ordered only with respect toone of the two colors. In the above formula we are summing over all color orderings w thatbelong to the set σ ··· k ; which is explicitly constructed using a shuffle product (cid:1) , as σ ··· k = (cid:26) { k + 1 , γ, k + m } (cid:12)(cid:12)(cid:12)(cid:12) γ ∈ α (cid:1) β (cid:27) , where α = { , , , . . . , k } and β = { k + 2 , . . . , k + m − } . (5.57)The set σ ··· k contains all shuffles of the gluon ( α ) and scalar ( β ) sets that respect theordering within each set, with the additional constraint that the first and last scalars areheld fixed. We will refer to the elements of this set as “words” w . A remarkable observationis that gauge invariance is sufficient to fix the numerators N k ( w ) in the expression above.Color-ordered single-trace YME amplitudes are obtained by replacing the φ partial am-plitudes with partial amplitudes belonging to pure YM theory (or its supersymmetric rela-tives, depending on the target gravitational theory) [156], M YME(SG) k,m (1 , . . . , k | k + 1 , . . . , k + m ) = X w ∈ σ ...k N k ( w ) A (S)YM k + m ( w ) + Perm(1 , . . . , k ) . (5.58)Since the partial amplitudes A (S)YM k + m ( w ) obey the same relations as A φ k + m ( w ) (including inparticular the BCJ relations), the YME amplitudes given by this formula will be by con-struction gauge invariant. The numerators N k ( w ) are obtained by imposing gauge invarianceon (5.56), that is by imposing that the amplitude vanishes after the replacement ε i → p i . (5.59)Along the same lines of Sec. 3.2, we construct the numerators N k ( w ) from the followingindependent Lorentz invariants: n ( ε i · z i ) , ( p i · z i ) , ( ε i · ε j ) , ( ε i · p j ) , ( p i · p j ) o , ( i, j = 1 , . . . , k ) (5.60)where p i denote only the momenta of the gluons. The momenta of the scalars will onlyappear implicitly through the region momenta z i = z i ( w ) that we define as z i ( w ) = X ≤ j ≤ lw l = i p w j , (5.61)which give the sum of the momenta of all the particles to the left of the i -th gluon in themultiperipheral graph corresponding to the word w (including the gluon momentum p i , seeFig. 19). 89 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES · · · k + 2 · · · · · ·· · · k k + m − k + m z z z kk + 1 · · · Figure 19: Half-ladder graph for the YM+ φ theory. The gluons are denoted with 1 , , . . . , k andthe remaining particles are scalars. z i denote the momentum of the internal scalar to the right ofgluon i . We consider the case of one external gluon ( k = 1). By dimensional analysis, each termin N ( w ) will need to contain a single factor of momentum. σ is the set of external-legorderings in which the order of the scalars is preserved and the single external gluon isinserted in different positions (leaving a scalar as the first and last entry). A natural guessfor the numerator is given by N = 2( ε · z ) . (5.62)We can check gauge invariance with the replacement (5.59); the gauge variation of theamplitude becomes X w ∈ σ ( p · z ) A (S)YM m +1 ( w ) = 0 , (5.63)which is zero as a consequence of the BCJ relations. Indeed, (5.63) is precisely the funda-mental BCJ relation (2.26) [1, 113]. That this BCJ relation can be obtained from YMEamplitudes with a single graviton, using the numerator (5.62), was first shown in Ref. [133].The next-simplest example has two external gluons. Now σ will be the set of external-leg orderings in which the two gluons are inserted in different positions while leaving theorder of scalars and gluons unchanged (and keeping scalars as the first and last entries). Thelast term in (5.58) is obtained by exchanging the two external gluons. The numerators are N = 4( ε · z )( ε · z ) + 2( p · z )( ε · ε ) , (5.64)where the last (contact) term is fixed by imposing gauge invariance. Taking a gauge variation ε → p of the amplitude we obtain:4 (cid:18) X w ∈ σ + X w ∈ σ (cid:19) ( p · z )( ε · z ) A (S)YM m +2 ( w )+ 2( ε · p ) (cid:18) X w ∈ σ ( p · z ) A (S)YM m +2 ( w ) + X w ∈ σ ( p · z ) A (S)YM m +2 ( w ) (cid:19) . (5.65)Using the fundamental BCJ relation (2.26), the reader can verify that the above gaugevariation reduces to( ε · p ) X w ∈ σ (cid:16) ( p · z ) + ( p · z ) (cid:17) A (S)YM m +2 ( w ) = 0 . (5.66)90 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
This is equivalent to the sum of two BCJ relations and thus vanishes for all amplitudes thatsatisfy CK duality.Exercise 5.10: Verify (5.66) starting from (5.65) and using the fundamental BCJ relation.Semi-recursive expressions for YME amplitudes with up to five external gravitons weregiven in Ref. [156]. General expressions for any multiplicity based on BCFW recursion weregiven in Ref. [214] in the single-trace case and in Ref. [216] in the multi-trace case. Thereader may also consult [212, 289] for alternative expressions obtained through the CHYformalism and [307, 349] for loop-level amplitudes in YME theory.
A key feature of the double-copy construction for YME theories is that it can be generalizedto cases in which the nonabelian gauge supersymmetry of the supergravity theory is spon-taneously broken. Since YME theories possess a moduli space in which the unbroken-gaugephase is given by a single isolated point, the fact that the double-copy construction admitsan extension of this sort gives a strong hint of its applicability for generic gravity theories.At the same time, the construction we review here is one of the simplest examples in whichsome of the fields are massive. The double-copy construction for a Higgsed supergravity hasthe schematic form (cid:18)
Higgsed YME SG (cid:19) = (cid:18) Coulomb-branch SYM theory (cid:19) ⊗ (cid:18) YM + massive scalars (cid:19) . (5.67)Schematically, amplitudes for the first gauge-theory factor can be obtained with a two-stepprocess: (1) break the gauge-group down to a subgroup (see Sec. 5.2.1); (2) assign masses(seen as compact momenta) to fields transforming in matter representations of the unbrokensubgroup (see Sec. 5.2.3). We have seen in Sec. 5.3.6 that, with the appropriate choicesof gauge theories, the global symmetry in one of the gauge-theory factors becomes a non-abelian gauge symmetry in (super)gravity. The scenario discussed here extends this propertyby showing that an explicitly-broken symmetry in one of the two gauge theories becomes,through the double copy, a spontaneously-broken gauge symmetry in (super)gravity.To avoid a notationally-heavy discussion, we will review here the simplest example of theHiggsed double-copy construction. We start from a N = 2 SYM theory with SU ( N + M )gauge group and decompose it with respect to the SU ( M ) × SU ( N ) × U (1) subgroup. Thedirect sum of the adjoint representations of the unbroken gauge-group factors is denoted as G ; the corresponding fields are left massless. In addition, there will be two vector multipletstransforming in the bifundamental R = ( M , N ) and anti-bifundamental R = ( M , N )representations. These fields are made massive by a suitable assignment of momenta alongone single compact dimension. This relies, implicitly, on the fact that the theory can beuplifted to higher dimension. The resulting bosonic states are given in Table 10 (whilefermionic states are not displayed). As discussed in Sec. 5.2.3, this is equivalent to giving ascalar VEV h φ i = V t ˆ0 , (5.68)91 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Fields Representation Mass( A µ , φ, ¯ φ ) G W µ , ϕ ) R m ( W µ , ϕ ) R − m Fields Representation Mass( A µ , φ, ¯ φ ) G ϕ R mϕ R − m Table 10: Bosonic fields in gauge-theory factors for the example of double-copy construction forHiggsed supergravities. where t ˆ0 =diag (cid:16) M I M , − N I N (cid:17) and V is a real parameter (since our assignment of compactmomenta only involves a single compact dimension).At this stage, we need to examine the constraints coming from the duality between colorand kinematics at four points. CK duality for amplitudes between two adjoint and twomatter fields is automatically satisfied. This is a consequence of the fact that the theorycan be obtained by assigning compact momenta to a higher-dimensional massless theory,as explained in Sec. 5.2.3. Alternatively, one could adopt a bottom-up approach and startfrom a Lagrangian involving massive vectors and scalars and leave free parameters in theinteraction terms. Imposing CK duality would fix the interaction terms to be the ones ofthe Coulomb-branch theory.Amplitudes involving four matter fields require a more detailed analysis. In particular,scattering amplitude of four massive scalars can be cast in the form A (cid:16) ϕ ˆ α , ϕ ˆ β , ϕ ˆ γ , ϕ ˆ δ (cid:17) = − ig (cid:18) n t c t D t + n u c u D u + n s c s D s (cid:19) , (5.69)where the color factors are c t = ˜ f ˆ a ˆ δ ˆ α ˜ f ˆ a ˆ γ ˆ β , c u = ˜ f ˆ a ˆ γ ˆ α ˜ f ˆ a ˆ δ ˆ β , c s = ˜ f ˆ γ ˆ δ ˆ (cid:15) ˜ f ˆ (cid:15) ˆ α ˆ β , (5.70)while the inverse propagators are D t = ( p + p ) , D u = ( p + p ) , D s = ( p + p ) − (2 m ) . (5.71)To understand the mass (2 m ) in the massive channel it is useful to recall that masses havebeen assigned as momenta in some additional dimensions. Because of this, masses areconserved at each vertex. Since the color factor c s contains two fields of the same complexrepresentation of masses m meeting at a vertex, the third field must necessarily have mass2 m . The kinematic numerators are given by: n t = − p · p + p · p + 2 m , n u = − p · p + m − p · p ,n s = p · p + 2 p · p + m . (5.72)These numerators can be obtained from (1.12) by assigning momenta along a single compactdimension or, alternatively, from the YM-scalar Lagrangian (5.5) with a = 0.Exercise 5.11: Modify the example discussed above by introducing a VEV that correspondsto compact momenta along two compact dimensions. Write the numerators for four-scalaramplitudes. As discussed in Sec. 2, it is convenient to write the color factors in terms of ˜ f ABC = √ if ABC . A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
We note that the s -channel color factor is zero because there does not exist an invariantgauge-group object with two bifundamental and one anti-bifundamental indices. How-ever, the corresponding numerator factor is nonzero. Alternatively stated, the color factorsobey two-term algebraic relations, while numerator factors obey three-term relations. Thisobservation affects the choice of the second gauge-theory factor entering the double-copyconstruction, which must have an identically-vanishing s -channel numerator.We choose a non-supersymmetric theory with one complex massive scalar transformingin the representation conjugate to the one of the Coulomb-branch theory (see Table 10). ItsLagrangian is L = −
14 ˜ F ˆ aµν ˜ F ˆ aµν + 12 D µ φ a ˆ a D µ φ a ˆ a + D µ ϕD µ ϕ − m ϕϕ − g f ˆ a ˆ b ˆ e f ˆ c ˆ d ˆ e φ a ˆ a φ b ˆ b φ a ˆ c φ b ˆ d − g ϕt ˆ a R ϕ )( ϕt ˆ a R ϕ ) + g φ a ˆ a φ a ˆ b ϕt ˆ a R t ˆ b R ϕ + gλφ a ϕt ˆ a R ϕ , (5.73)where a, b = 1 , t R are gauge-group representation matrices for the massive scalars, andonly φ a enters the trilinear scalar couplings. The reason for this latter choice is that we wanta construction which manifestly uplifts to five dimensions. Without any loss of generalitywe can rotate the other scalars which appear in the trilinear couplings into φ a . One cancheck that numerators of this theory obey a two-term relation.Putting all together, the spectrum of the resulting supergravity theory is given by [122]: A − − = ¯ φ ⊗ A − , h − = A − ⊗ A − , A − = φ ⊗ A + , h + = A + ⊗ A + ,A − = φ ⊗ A − , i ¯ z = A + ⊗ A − , A = ¯ φ ⊗ A + , − iz = A − ⊗ A + ,A A − = A − ⊗ φ A , i ¯ z A = ¯ φ ⊗ φ A , A A + = A + ⊗ φ A , − iz A = φ ⊗ φ A ,W i = W i ⊗ ϕ , ϕ = ϕ ⊗ ϕ . (5.74)with massive fields given by R ⊗ R bilinears (the index i runs over the massive-vector threephysical polarizations). Note that this construction has two free parameters: the mass m and the constant λ in the trilinear scalar couplings. Comparison with amplitudes fromthe Higgsed supergravity Lagrangian leads to the identification (5.54). The masses ofsupergravity fields are the same as those of the gauge-theory fields from which they areconstructed. In turn, this determines the choice of scalar base-point for the supergravityperturbative expansion that matches the result of the double copy. Given the presenceof two massive W bosons in the supergravity spectrum, the supergravity gauge-symmetrybreaking is SU (2) → U (1).This is arguably the most straightforward example of Higgsed supergravity constructedas double copy. In the general case, we need to consider a generic breaking of the Coulomb-branch theory. The structure constants, generators and Clebsch-Gordan coefficients obeyrelations inherited from the Jacobi relations of the original gauge group. A first set of In a standard formulation of the Higgs mechanism, this channel does not appear in the amplitude becausethe necessary vertices are absent from the Lagrangian. To obtain a Higgsed supergravity, we take λ > A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES relations is f ˆ d ˆ a ˆ c f ˆ c ˆ b ˆ e − f ˆ d ˆ b ˆ c f ˆ c ˆ a ˆ e = f ˆ a ˆ b ˆ c f ˆ d ˆ c ˆ e ,f ˆ a ˆ β ˆ γ f ˆ b ˆ γ ˆ α − f ˆ b ˆ β ˆ γ f ˆ a ˆ γ ˆ α = f ˆ a ˆ b ˆ c f ˆ c ˆ β ˆ α ,f ˆ a ˆ γ ˆ (cid:15) f ˆ (cid:15) ˆ β ˆ δ − f ˆ a ˆ β ˆ (cid:15) f ˆ (cid:15) ˆ γ ˆ δ = f ˆ a ˆ (cid:15) ˆ δ f ˆ γ ˆ β ˆ (cid:15) . (5.75)These relations are necessary to ensure gauge invariance. The Clebsch-Gordan coefficients f ˆ γ ˆ β ˆ (cid:15) need to obey additional identities: f ˆ α ˆ γ ˆ (cid:15) f ˆ (cid:15) ˆ β ˆ δ − f ˆ α ˆ β ˆ (cid:15) f ˆ (cid:15) ˆ γ ˆ δ = f ˆ α ˆ (cid:15) ˆ δ f ˆ γ ˆ β ˆ (cid:15) , (cid:18) f ˆ β ˆ (cid:15) ˆ γ f ˆ α ˆ (cid:15) ˆ δ + f ˆ α ˆ (cid:15) ˆ δ f ˆ β ˆ (cid:15) ˆ γ + f ˆ a ˆ β ˆ γ f ˆ a ˆ α ˆ δ (cid:19) − ( ˆ α ↔ ˆ β ) = f ˆ α ˆ β ˆ (cid:15) f ˆ (cid:15) ˆ δ ˆ γ . (5.76)The seven-term identity is to be thought of as a compact way of writing a set of three-and two-term identities. The general construction for Higgsed supergravities proceeds asfollows [122]:• One introduces a non-supersymmetric gauge theory with massive scalars and imposesthe identities (5.75) and (5.76) on its numerator factors. Note that the numerators ofthe Coulomb-branch theory need not obey the same identities.• Masses need to be matched on both gauge-theory factors. For gaugings that upliftto five dimensions, the Higgs mechanism requires that the Coulomb-branch theorymasses be proportional to a preferred U (1) gauge generator (given by the direction ofthe VEV). Imposing CK duality results in demanding than the masses in the explicitly-broken massive-scalar theory also be proportional to a preferred U (1) global generator(in our example, the U (1) acting as a phase rotation on the complex scalars).• In general, the symmetry-breaking pattern (number of factors in the gauge group, num-ber of matter representation, existence of Clebsch-Gordan coefficients correspondingto a given triplet of representations) from the Coulomb-branch gauge theory matchesboth that of the explicitly-broken theory and that of the supergravity theory.• Identification of the supergravity relies on the unbroken limit (setting all masses tozero), as well as on the symmetry breaking information encoded in the trilinear scalarcouplings.A list of constructions for Higgsed supergravities with various amounts of supersymmetrycan be found in Table 11.Exercise 5.12: What would happen if we attempted to double copy two Coulomb-branchtheories realized both in terms of compact momenta? Find out as much information aspossible on the resulting gravity theory. 94 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Gravity coupled to (cid:8)(cid:8)(cid:8)
YM Left gauge theory Right gauge theory N = 4 (cid:8)(cid:8)(cid:8) YME supergravity N = 4 S (cid:8)(cid:8)(cid:8) YM YM + (cid:0)(cid:0) φ N = 2 (cid:8)(cid:8)(cid:8) YME supergravity (gen.Jordan) N = 2 S (cid:8)(cid:8)(cid:8) YM YM + (cid:0)(cid:0) φ N = 0 (cid:8)(cid:8)(cid:8) YM DR -E + dilaton + B µν (cid:8)(cid:8)(cid:8) YM DR YM + (cid:0)(cid:0) φ Table 11: New double-copy constructions corresponding to spontaneously-broken YME gravitytheories for different amounts of supersymmetry [122]. The dimensionally-reduced YM DR theorymust have at least one scalar to provide the VEV responsible for spontaneous symmetry breaking. Fields Representation Mass( A µ , ¯ φ a ) G W µ , ϕ s ) R m ( W µ , ϕ s ) R − m Fields Representation Mass( A µ , ϕ α ) G χ R mχ R − m Table 12: Fields in gauge-theory factors for a simple example of a double-copy construction for N = 2 gauged supergravities. In this subsection, we consider an important variant of the construction for Higgsed super-gravities. In a sense, the construction outlined in the previous subsection can be regarded asthe simplest double-copy prescription which produces a gravity with massive vector fields.Various details of the construction can then be traced back to the requirement that suchmassive vectors obey the relevant Ward identities corresponding to spontaneous symmetrybreaking.Along similar lines, we might want to consider double copies leading to massive spin-3 / R symmetry is promoted to a gauge symmetry under whichgravitini are charged. In a gauged supergravity with a Minkowski vacuum, minimal couplingbetween gravitini and gauge vector produces a nonzero amplitude of the form M (cid:16) ψ i , ψ j , A a (cid:17) = ig R t aij ¯ v µ (cid:1) ε v µ + O (cid:16) ( g R ) (cid:17) . (5.77) g R is the coupling constant and v lµ ( l = 1 ,
2) are the gravitini’s polarization spinor-vectors.The matrices t aij generate the gauged R -symmetry subgroup acting nontrivially on the grav-itini. The above amplitude does not vanish with the replacement v lµ → v lµ + k lµ (cid:15) , (cid:19)(cid:19) k l (cid:15) = 0 . (5.78)Since this replacement correspond to an linearized supersymmetry transformation, the pres-ence of a nonzero amplitude of the form (5.77) signifies that supersymmetry is spontaneouslybroken. Indeed, R -symmetry gauging and spontaneous supersymmetry breaking go hand inhand for supergravities which admit Minkowski vacua. In turn, the fact that supersymme-try is spontaneously broken results in (some) massive gravitini. This can be understood by95 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES comparing the number of physical polarizations of our gravitini; because some of the super-symmetry generators are broken, they cannot be used to eliminate components of gravitini.Some of the gravitini will have four physical polarizations and must therefore become mas-sive.This observation provides a hint on how to find a double-copy construction for amplitudesof gauged supergravities with Minkowski vacua. In analogy with the previous subsection,we start by seeking a construction that has the following two properties:1. contains massive spin-3 / W bosonsin one gauge theory with massive fermions in the other;2. reduces to the construction of the corresponding ungauged supergravity in the masslesslimit.The simplest realization with these properties has the schematic form (cid:18) Gauged Supergravity (cid:19) = (cid:18) Coulomb-branch YM (cid:19) ⊗ (cid:18) s (cid:8)(cid:8)(cid:8) uper YM (cid:19) , (5.79)where the second factor stands for a theory with explicit supersymmetry breaking and mas-sive fermions.Next, we discuss the two gauge theories separately, focusing on the particular case of N = 2 supersymmetry and using the toolbox introduced in Secs. 5.2.2 and 5.2.3. Unlikethe case of Higgsed YME theories, the Coulomb-branch theory is non-supersymmetric; wewill take it to be a pure YM theory coupled with n scalars, obtained from dimensionalreduction from D = ( n + 4) dimensions. The corresponding VEV will be parameterized by a n -dimensional vector which we will denote as V a . The theory with supersymmetry explicitlybroken by fermion masses is obtained by starting with four-dimensional SU ( N + M ) N = 2SYM theory and spontaneously breaking the gauge group to G = SU ( N ) × SU ( M ) × U (1)by introducing a VEV h ϕ α i = e V α × Diag (cid:18) N I N , − M I M (cid:19) . (5.80)We then orbifold the theory by a Z generated by γ = diag( I N , − I M ): A µ γA µ γ − , χ
7→ − γχγ − , ϕ γϕγ − . (5.81)Note that, as explained in Sec. 5.2.2, this operation preserves CK duality. The VEVs in boththeories are chosen to have the same magnitude ( V a ) = ( e V α ) , so that the two theories havecommon mass spectra. The explicitly-broken theory has Lagrangian L N (cid:26)(cid:26) = 2 = 1 g Tr (cid:20) − F µν F µν − D µ ϕ α D µ ϕ α + 14 [ ϕ α , ϕ β ] + i χ Γ µ D µ χ + 12 χ Γ α [ ϕ α + h ϕ α i , χ ] (cid:21) , (5.82)where χ is a six-dimensional Weyl fermion and α, β = 5 ,
6. The fields in the gauge-theoryfactors are listed in Table 12. Denoting with ξ µ the massive gravitino field on the supergravityside, the fermionic states have the following double-copy origin: ξ µ = W µ ⊗ χ − W ν ⊗ (cid:18) γ µ − ip µ m (cid:19) γ ν χ ,ξ = W ν ⊗ γ ν χ , ( U λ ) s = ϕ s ⊗ χ . (5.83)96 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
The combination on the first line is manifestly transverse and γ -traceless. U is a unitarymatrix diagonalizing the spin-1 / U (1) R gauge vector is: A U (1) R + = − A + ⊗ ϕ ± φ ⊗ A + . (5.84)We note that the massless limit leads an ungauged theory belonging to the Generic Jordanfamily discussed in Sec. 5.3.2. The freedom of choosing the U (1) R gauge group correspondsto the choice of VEVs in the two gauge theories entering the construction. As for Higgsedsupergravities, this is the simplest example of the double-copy construction. However, it isimmediate to generalize the construction reviewed here to U (1) R gaugings of N = 4 , , p i = p ⊥ i − m p i · q q . Here q is a reference momentum and p ⊥ i , q are both massless. Write massivespinor polarizations as v t + = (cid:16) | i ⊥ ] , m | q i / h i ⊥ q i (cid:17) and v t − = (cid:16) m | q ] / [ i ⊥ q ] , | i ⊥ i (cid:17) . Show that anamplitudes involving massive gravitini with ± polarizations and the A − vector field can bewritten as M tree3 (cid:16) ξ + , ξ − , A − (cid:17) = −√ im Ω h ⊥ q ih ⊥ q i , Ω = [3 ⊥ ⊥ ] [1 ⊥ ⊥ ][2 ⊥ ⊥ ] . (5.85)The construction outlined above can be generalized to allow gauging of nonabelian sub-groups of the R symmetry. To do so, we need to consider double copies that [284]:1. contain massive spin-3 / N = 8 supergravity and start bywriting both copies of N = 4 SYM as the dimensional reduction of SYM theories in tendimensions. For the left gauge-theory factor, we choose undeformed N = 4 SYM theory onthe Coulomb branch. In the right gauge-theory factor, we introduce a massive deformationwhich involves trilinear scalar couplings, L = −
14 ( F ˆ aµν ) + 12 ( D µ φ ˆ aI ) − m IJ φ ˆ aI φ ˆ aJ − g f ˆ a ˆ b ˆ e f ˆ c ˆ d ˆ e φ ˆ aI φ ˆ bJ φ ˆ cI φ ˆ dJ − gλ f ˆ a ˆ b ˆ c F IJK φ ˆ aI φ ˆ bJ φ ˆ cK + i ψ (cid:0)(cid:0) Dψ −
12 ¯ ψM ψ + g φ ˆ aI ¯ ψ Γ I t ˆ a R ψ . (5.86)97 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
CK duality of the two-scalar-two-fermion four-point amplitude demands that the fermionicmass matrix M obey the relation h Γ I , n Γ J , M oi + iλF IJK Γ K = 0 , (5.87)where Γ I are the Dirac matrices in higher dimensions and F IJK are related to the struc-ture constants of the supergravity gauge group. The right gauge theory is then Higgsed andorbifolded, following the same strategy outlined in the N = 2 example. Double copies involv-ing theories obtained with this prescription need however to satisfy additional consistencyrequirements.Referring to the literature for the general construction [284], we consider the action (5.86)with an SU (3 N ) gauge group and the deformation M = i g , λF = g . (5.88)This deformation breaks ten-dimensional Lorentz invariance to SO (3) × SO (6 ,
1) and canbe uplifted to seven dimensions. Starting from D = 7, we take a a Z orbifold which acts as ψ → e π Γ g † ψg , φ I → R IJ (cid:16) π (cid:17) g † φ J g, g = diag (cid:16) I N , e i π I N , e i π I N (cid:17) (5.89)where I, J = 5 , R generates a rotation in the 5-6 plane. We also take the scalarmass-matrix to be m = m = m , m IJ = 0 otherwise . (5.90)After the projection, the fields of the theory are organized schematically as: A µ , φ i ψ r φ + ˜ ψ r A µ , φ i ψ r φ − ˜ ψ r A µ , φ i , (5.91)where i = 4 , , , r = 1 , r = 3 ,
4, and φ ± = φ ± iφ . In the above equation, we representthe fields surviving the projection as entries the in 3 N × N matrices originating from theparent theory; each entry is an N × N block. To obtain a number of states that reproducesthe spectrum of N = 8 supergravity, we need to combine the representations ( N , ¯ N , ) with( , N , ¯ N ) and the representation ( ¯ N , N , ) with ( , ¯ N , N ) into a (reducible) representationwhich is denoted as R . This can be realized by rewriting the Lagrangian in a way that onlyrepresentation matrices for R appear explicitly.In the left theory, we take a a VEV of the form h φ i = diag (cid:16) u I N , u I N , u I N (cid:17) , u + u + u = 0 . (5.92)Since the two irreducible representations that have been combined into R need to have thesame mass, we get a condition involving the VEV parameters, u − u = u − u → u = u + u . (5.93)98 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
Rep. R L Supergravity fields mass G V N =4 A µ ⊕ φ i H N =4 ⊕ V N =4 R V m N =4 ψ r m N =4 u ¯ R V m N =4 ˜ ψ r m N =4 u R V m N =4 φ + V m N =4 u ¯ R V m N =4 φ − V m N =4 u Table 13: Fields and mass spectra for gauging of N = 8 supergravity with N = 4 residual super-symmetry [284]. In addition, we get the following conditions by matching the mass spectra of the two theories: M = − u m = 4 u . (5.94)We list the fields from the double copy with their respective mass spectra in Table 13.The vacuum of this theory has an unbroken SU (2) × U (1) gauge group which is reflectedby the F IJK tensors in (5.88). N = 4 unbroken supersymmetry is inherited from theCoulomb-branch gauge-theory factor. Many additional examples can be worked out alongsimilar lines. A complete classification of double-copy-constructible gaugings is currently anopen problem. A double-copy construction for conformal gravity was set forth in Ref. [152] and furtherinvestigated in Ref. [153]. Before we get into the details of that construction, let us reviewsome general properties of conformal gravity. The simplest model is that of Weyl gravity,which has the four-derivative action S = − κ Z d x √− g ( W µνρσ ) , (5.95)where W µνρσ is the Weyl curvature tensor, and κ is a dimensionless coupling. The actionis invariant under local rescaling of the metric, g µν → Ω( x ) g µν ; more generally the theorypossesses local conformal symmetry at the classical level. The symmetry can be extended tolocal superconformal symmetry by considering supergravity formulations of the Weyl theory.It is believed that N = 4 is the maximum allowed supersymmetry. In contrast to expec-tations from SYM and ordinary two-derivative supergravity, the maximally supersymmetrictheory is not unique, in fact it has an infinite number of free parameters [350]. The free pa-rameters are encoded in a free holomorphic function that multiplies the square of the Weyltensor, − κ √− g − L N =4 = f ( τ )( W + µνρσ ) + f ( τ )( W − µνρσ ) + . . . , (5.96)where W ± µνρσ = W µνρσ / ± ( i √− g/ W λκµν (cid:15) λκρσ is the (anti-)selfdual Weyl tensor and thecomplex scalar τ = ie − φ + χ is the dilaton-axion field. The ellipsis denotes additional terms Note that compared to Ref. [152] we are using a convention where we have swapped i ¯ τ with − iτ . Thischanges the sign of the axion field, which is physically unobservable. A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES that are fully constrained by the superconformal symmetry. The choice f ( τ ) = 1 correspondsto the supersymmetrization of the Weyl theory, and it is usually called minimal conformalsupergravity. When f ( τ ) is not constant, the theory corresponds to non-minimal conformalsupergravity. The double-copy constructions that we will consider corresponds to the twocases [153]: f ( τ ) = − iτ ( N = 4 Berkovits-Witten theory) ,f ( τ ) = 1 ( N = 4 minimal conformal supergravity) . (5.97)These two cases are special. The Berkovits-Witten theory [351] corresponds to the uniqueconformal supergravity theory that has an uplift to 10 dimensions [145, 152, 153, 352]. Attree level, the minimal theory has the same SU (1 ,
1) electromagnetic duality symmetry as N = 4 supergravity, and certain all-multiplicity tree-level amplitudes are the same as inthat theory. All N = 4 conformal supergravities are expected to be anomalous at loop levelunless they are coupled to four vector multiplets [353, 354].For reasons of conciseness, we will restrict the discussion in this section to scatteringamplitudes where the external states are plane waves. As is well known, the four-derivativeaction of conformal gravity also permits other types of asymptotic states, see e.g. Refs. [153,355] for further details. The double copy that gives amplitudes in the Berkovits-Wittenconformal supergravity theory has the schematic form (cid:16) Berkovits-Witten CSG (cid:17) = (cid:16) SYM (cid:17) ⊗ (cid:16) ( DF ) -theory (cid:17) , (5.98)where SYM is the maximally supersymmetric Yang-Mills theory, and the ( DF ) theory is abosonic gauge theory with dimension-six operators which has the following Lagrangian [152]: L ( DF ) = 12 ( D µ F a µν ) − g F + 12 ( D µ ϕ α ) + g C αab ϕ α F aµν F b µν + g d αβγ ϕ α ϕ β ϕ γ . The vector A aµ transforms in the adjoint representation of a gauge group G with indices a, b, c . ϕ α are additional scalars transforming in a real representation for which C αab and d αβγ are invariant tensors. We have used the short-hand notation F = f abc F aνµ F bλν F cµλ . Itshould be noted that C αab , T a R and d αβγ are implicitly defined through the two relations: C αab C αcd = f ace f edb + f ade f ecb , (5.99) C αab d αβγ = ( T a R ) βα ( T b R ) αγ + C βac C γcb + ( a ↔ b ) , which are sufficient relations for expressing tree-level gluon amplitudes in terms only f abc tensors.A massive deformation of this theory was also introduced in Ref. [152] and is defined bythe Lagrangian: L ( DF ) +YM = 12 ( D µ F a µν ) − g F + 12 ( D µ ϕ α ) + g C αab ϕ α F aµν F b µν + g d αβγ ϕ α ϕ β ϕ γ − m ( ϕ α ) − m ( F aµν ) . (5.100)100 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
This theory interpolates between the ( DF ) theory and a pure YM theory and has the massas a free parameter. Along similar lines, the theory (5.100) can be further augmented byintroducing adjoint scalars φ aA which are also charged under a global group and appearin trilinear couplings which are analogous to the ones introduced for YME theories andnonabelian gauged supergravities: L ( DF ) +YM+ φ = 12 ( D µ F a µν ) − g F + 12 ( D µ ϕ α ) + g C αab ϕ α F aµν F b µν + g d αβγ ϕ α ϕ β ϕ γ − m ( ϕ α ) − m ( F aµν ) + 12 ( D µ φ aA ) + g C αab ϕ α φ aA φ bA (5.101)+ gλ f abc F ABC φ aA φ bB φ cC . These deformations of the ( DF ) theory will also play an important role for double-copyconstructions involving various string theories, which are reviewed in the next subsections.We also note that the ( DF ) theory is just a representative of a large class of gauge theorieswith higher-dimension operators. An investigation of their amplitudes in the general caseis an open problem; we refer the reader to [296] for a similar construction of supergravitieswith higher-dimension operators and to [145] for a study of the ( DF ) theory from the pointof view of ambitwistor strings.Exercise 5.14: Show that three- and four-gluon color-ordered amplitudes in the ( DF ) the-ories have the expressions A ( DF ) (1 , ,
3) = − ε · p )( ε · p )( ε · p ) , (5.102) A ( DF ) (1 , , ,
4) = 4 s s s (cid:18) p · ε s − p · ε s (cid:19)(cid:18) p · ε s − p · ε s (cid:19)(cid:18) p · ε s − p · ε s (cid:19)(cid:18) p · ε s − p · ε s (cid:19) , and that they obey color-kinematics duality. Note that the products between polarizationvectors, ε i · ε j , always cancel out (this is a special property of the ( DF ) theory).Finally, we will consider amplitudes in the minimal N = 4 conformal supergravity theory.For external plane waves at tree level, the relation is (cid:16) minimal CSG (cid:17) = (cid:16) SYM (cid:17) ⊗ (cid:16) minimal ( DF ) -theory (cid:17) , (5.103)where we have truncated the bosonic gauge theory to a “minimal” version, L min . ( DF ) = 12 ( D µ F a µν ) . (5.104)However, as the reader may confirm, the all tree-level plane-wave amplitudes in this theoryvanish—a property that is also true of minimal conformal supergravity. In order to havesomething nonvanishing to compare with, we must deform the two theories by a mass term, L min . ( DF ) +YM = 12 ( D µ F a µν ) − m ( F aµν ) . (5.105)101 A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES
The resulting double copy (cid:16) mass-deformed minimal CSG (cid:17) = (cid:16) SYM (cid:17) ⊗ (cid:16) minimal ( DF ) + YM (cid:17) , (5.106)gives amplitudes in a mass-deformed minimal N = 4 theory that interpolates between(Weyl) and a Ricci scalar term − κ √− g − L N =4 = ( W µνρσ ) − m R + . . . (5.107)where the ellipsis are additional terms fixed by supersymmetry. The tree amplitudes, forexternal plane waves, in the mass-deformed theories, are proportional to the correspondingamplitudes in ordinary YM and supergravity [153], A min . ( DF ) +YM = m A YM ,M mass-def. min. CSG = m M SG . (5.108)In addition to considering N = 4 conformal supergravity, the corresponding theories withreduced supersymmetry N = 0 , , N = 4 SYM factor inthe double copies (5.98), (5.103) and (5.106) by N = 0 , , N = 0 , , N = 4 theory, in close analogy to the case of ordinary two-derivative supergravitytheories. In refs. [109, 356], disk integrals that appear in open-string amplitudes were organized interms of building blocks Z σ ( ρ (1 , , . . . , n )) = (2 α ) n − Z σ {−∞≤ z ≤ z ≤ ... ≤ z n ≤∞} dz dz . . . dz n vol(SL(2 , R )) Q ni With these building blocks, the open-superstring amplitudes with massless external statescolor-ordered, with respect to the Chan-Paton factors, can be expressed directly in terms ofYang-Mills scattering amplitudes [109], and written in terms of a field theoretic double-copyfactorization in [356], A treeOS ( σ (1 , , , . . . , n )) = X τ,ρ ∈ S n − (2 ,...,n − Z σ (1 , τ, n, n − S [ τ | ρ ] A SYM (1 , ρ, n − , n ) , (5.112)where the ( n − × ( n − S [ τ | ρ ] = S [ τ (2 , . . . , n − | ρ (2 , . . . , n − introduced in Sec. 2.3.1. It is fascinating to note that a suggestive hint of thistype of field-theoretic double-copy factorization was identified in Ref. [358].Rather than focusing on the partially-ordered open string amplitudes (5.112), considerinstead the content of the full Chan-Paton-dressed open supersymmetric string amplitude.Dressing Z σ with all relevant ( n − Z tree (1 , ..., n ) ≡ X σ ∈ S n − (2 ,...,n ) Tr [ T a T a σ (2) · · · T a σ ( n − T a σ ( n ) ] X ρ ∈ S n − Z σ ( 1 , ρ, n − , n ) , (5.113)which obeys only the field-theory amplitude relations (i.e. Eq. (5.110) with the replacement Z σ → Z tree ). The full Chan-Paton-dressed open superstring amplitude, A OS = X σ ∈ S n − Tr [ T a T a σ (2) · · · T a σ ( n − T a σ ( n ) ] A treeOS (1 , σ ) , (5.114)can also be written entirely as a field-theory double copy A OS = X τ,ρ ∈ S n − Z tree (1 , τ, n, n − S [ τ | ρ ] A SYM (1 , ρ, n − , n ) . (5.115)An interesting open problem is the physical interpretation of the above building blocks.Given the adjoint field-theory relations obeyed by the ordered Z ( ρ ), it is natural to con-sider the orderless-functions resulting by dressing the ρ ordering with adjoint f abc structureconstants as per a DDM basis. This yields a fully dressed function that can be expressedin terms of cubic graphs dressed with two factors that both satisfy Jacobi identities andantisymmetry: Z = X i z i c i D i = X ρ ∈ S n − c | ρ | n Z tree (1 , ρ, , (5.116)such that: Z tree (1 , ρ (2) , . . . ρ ( m − , m ) = − i X i ∈ planar b i ρ z i D i , (5.117)where z i are Jacobi satisfying functions of both higher-derivative scalar kinematics andstring Chan-Paton factors, D i are the propagators of the graph, and b i ρ ∈ { , ± } areinteger coefficients that depend on the ordering ρ . Both Z and Z can be be derived as thetree-level amplitudes, color-dressed and ordered respectively, of an effective field theory of Note that the α -dependent KLT kernel, given in Ref. [24] (and its inverse in Ref. [357]), needs notfeature in the factorization of tree-level string amplitudes, cf. Eq. (5.121). A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES double-colored scalar fields in which the scalars obey an equation of motion of the schematicform [171] (cid:3) Φ = Φ + α ζ (cid:16) ∂ Φ + Φ (cid:17) + α ζ (cid:16) ∂ Φ + ∂ Φ + Φ (cid:17) + O ( α ) . (5.118)This theory was named Z-theory in refs. [169, 171, 310]. It is worth noting that the colorstructure of the leading term in the equation of motion is the same as the bi-adjoint φ theory.The entire tower of higher derivative operators relevant to the open-string are encoded inthis effective scalar theory, whose double copy with the supersymmetric gauge theory yieldsthe supersymmetric open string. Schematically, the formula (5.115) can be rewritten withthe short-hand notation(massless open superstring) = (Z-theory) ⊗ (SYM) . (5.119)The simplest set of Z-theory amplitudes arise when one trivializes the string Chan-Patonfactors, taking all the generators to be the identity, corresponding to a U (1) group. Thisoperation on the Chan-Paton dressed open string results in a symmetrization over all ordersreferred to as the abelian or photonic open-string whose low-energy limit yields amplitudesin maximally supersymmetric DBI theory, where the fermionic sector is of Volkov-Akulovtype [302–305, 359–363]. Abelian Z amplitudes yield in the low-energy limit NLSM ampli-tudes in the α → N = 4 SYM in four dimensions to generate DBI-VA amplitudes [125, 307].A closed-string version of the Z-theory amplitudes involves integrals over the Modulispace of punctured Riemann spheres [155, 364–366]sv Z ( τ | σ ) = α π ! n − Z d z d z . . . d z n vol(SL(2 , C )) Q ni A WEB OF DOUBLE-COPY-CONSTRUCTIBLE THEORIES string ⊗ QFT SYM ( DF ) + YM ( DF ) + YM + φ Z-theory open superstring open bosonic string compactified openbosonic stringsv(open superstring) closed superstring heterotic (gravity) heterotic (gauge/gravity)sv(open bosonic string) heterotic (gravity) closed bosonic string compactified closedbosonic stringTable 14: Various known double-copy constructions of string amplitudes [371]. The single-valuedprojection sv( • ) converts disk to sphere integrals. We should note that these constructions are of the generic form (5.112), i.e. they involvethe field-theory KLT kernel, and apply at tree level and with massless external states. Re-markably, the free mass parameter in the ( DF ) +YM theory is related to the inverse stringtension α as m = − α . (5.125)Various relations between Z-theory and string amplitudes are summarized in Table 14. Someextensions to loop level can be found in refs. [372–378]. Additionally, a double-copy con-struction for string amplitude in terms of field-theory amplitudes in the CHY formalism wasobtained in Refs. [379, 380]. We conclude the section by listing further examples of double-copy constructions.• The non-gravitational (supersymmetric) DBI theory was constructed in Ref. [125] usingthe scattering equation formalism (see also [285, 301]). It can be regarded as thedouble copy of (S)YM theory and the NLSM. It should be noted that the NLSM canbe obtained in the α → φ theory [306]. The latter gauge-theory factor can be obtained from the α → φ theory and the NLSM [156].• Volkov-Akulov theory has tree-level amplitudes that can be obtained from supersym-metric DBI by restricting the external states to be fermions. Since DBI only hasnonvanishing even-point amplitudes, and internal bosons would require tree-level fac-torization with an odd number of particles (2 × fermions + 1 boson), this restrictiongives a consistent truncation of the theory. The double copy for Volkov-Akulov theory105 BCJ DUALITY AT LOOP LEVEL can thus be inferred to be a product between NLSM and SYM with only externalfermions.• Two copies of the NLSM give the so-called special-Galileon theory [125, 301].• In three dimensions, two copies of the BLG theory [245, 246] yield an alternativeconstruction for maximal three-dimensional supergravity [119, 243, 244, 297, 381]. Thethree-dimensional version of CK duality relevant to this construction is based on a so-called three-algebra. The three-algebra for BLG theory is introduced formally using atotally antisymmetric triple product [ X, Y, Z ]. Using a basis of generators the tripleproduct can be expressed using rank-four structure constants, h T a , T b , T c i = f abcd T d . (5.126)Consistency of the algebra requires that the structure constants satisfy the four-termidentity 0 = f abcl f dleg + f bael f dlcg + f ceb l f dalg + f eca l f dblg , (5.127)which plays the same role as the standard Jacobi identity for a Lie two-algebra. Itturns out that the only nontrivial compact three-algebra is SO(4) [382], where f abcd = (cid:15) abcd . However, for color-kinematics duality to work, it is sufficient to impose the four-term identity, whereas identities specific to SO(4) should be ignored. Finally, we maynote that the closely-related ABJM theory [383] appears to not have similarly niceproperties under color-kinematics duality. While the tree-level ABJM amplitudes upto six points obey the duality and their double copy gives three-dimensional maximalsupergravity, at eight points the double copy does not reproduce the correspondingamplitude in maximal supergravity [243, 244]. Since there is no dynamical gravitonin three dimension, this mismatch is not forbidden by the diffeomorphism symmetryargument in Sec. 2.5.Additional theories for which a double-copy construction has been proposed involve massivehigher-spin N = 7 W-supergravity theories [384, 385]; this amount of supersymmetry hasnot been accessible through different constructions. Chiral higher-spin theories have beenshown to obey generalized BCJ relations in Ref. [386]. Theories with gravitationally-coupledfermions have been discussed in Ref. [387]. A construction of the free spectrum of D = 3supergravities in terms of SYM theories with fields valued in the four division algebraswas given in [388]. Further examples of constructions in higher dimensions include half-maximal supergravity in six dimensions [389] and the so-called (4 , 0) theory in six dimensions[268, 390], which can be seen as the double copy of two (2 , 0) theories, at least at the levelof the free spectrum [391, 392]. In this section, we describe loop-level examples of BCJ duality and the associated double-copyconstruction. Whenever a gauge-theory integrand can be found in a form that manifests theduality between color and kinematics, corresponding gravity integrands can be immediately106 BCJ DUALITY AT LOOP LEVEL n 12 34 ℓ = n 12 34 ℓ − n ℓ Figure 20: A BCJ kinematic numerator relation at one loop. When the external particles are gluonsthis holds just as well for adjoint or fundamental representation particles circulating in the loop.The shaded (red) line differs between the diagrams, but the others are identical. written down via the double-copy procedure. This procedure enormously simplifies theconstruction of gravity loop integrands and has been successful for carrying out a variety ofloop-level studies in perturbative quantum gravity theories (see e.g. Refs. [15, 17, 18, 23, 31–33, 36, 292, 293]). As explained in Sec. 5, the precise gravity theory to which the integrandsbelong depends on the choice of input gauge theories. We start by briefly recalling thedefinition and the main points of the duality and of the double-copy construction, discussedat length in Sec. 2. With the appropriate separation of diagrams’ symmetry factors andjudicious choice of loop momenta, they are essentially the same as at tree level.Similarly to tree-level amplitudes, loop-level amplitudes in a gauge theory coupled tomatter fields can be organized as a sum over diagrams with only cubic (trivalent) verticesby multiplying and dividing by appropriate propagators to absorb contact diagrams intodiagrams with only cubic vertices. If all fields are in the adjoint representation of the gaugegroup, this rearrangement puts the amplitude in a form equivalent to Eq. (2.1), A L - loop m = i L − g m − L X S m X j Z L Y l =1 d D ‘ l (2 π ) D S j c j n j ( ‘ ) D j , (6.1)where the c i are color factors obtained by assigning structure constant factors ˜ f abc = i √ f abc to each cubic vertex. The first sum runs over the set S m of m ! permutations of the exter-nal legs. The second sum runs over the distinct L -loop m -point diagrams with only cubicvertices. As at tree level, by multiplying and dividing by propagators, it is trivial to ab-sorb contribution from higher-than-three-point vertices into numerators of diagrams withonly cubic vertices. The symmetry factor S j counts the number of automorphisms of thelabeled diagram j from both the permutation sum and from any internal automorphismsymmetries. This symmetry factor should not be included in the kinematic numerator.The nontrivial conjecture is that, as at tree level, for every loop-level color Jacobi identitythere is a matching kinematic numerator identity (2.7). c i − c j = c k ⇔ n i ( ‘ ) − n j ( ‘ ) = n k ( ‘ ) . (6.2)However, unlike at tree level, one has to be cautious with the treatment of degrees of free-dom that are not fixed by the external states. This includes proper accounting of the loop Note that this symmetry factor is different from the symmetry factor in Eq. (2.1), where S j counts theautomorphisms of graphs with fixed external legs. BCJ DUALITY AT LOOP LEVEL 12 34 Figure 21: A one-loop box integral, I ( s, t ), appearing in the one-loop four-point N = 4 SYM and N = 8 supergravity amplitudes. The three independent relabelings of external legs appear in theamplitudes. momenta of the numerators, generically called ‘ , as well as being careful to not set to zerocolor factors that vanish when summing over internal indices.We can change the signs of the color factors using the antisymmetry of the f abc s, butany relative signs between color factors in the Jacobi relation are then inherited by thecorresponding relation between the kinematic numerator factors. A simple example of suchloop-level relations is illustrated in Fig. 20 for the case of a one-loop amplitude. At loop-level,the duality between color and kinematics (2.7) remains a conjecture [2], although evidencein its favor continues to accumulate [4–6, 9–23, 156].Just as for tree-level numerators, once gauge-theory numerator factors which satisfy theduality are available, replacing the color factors by the corresponding numerator factors, c i → n i yields the double-copy form of gravity loop integrands (2.11), M L - loop m = i L − (cid:18) κ (cid:19) m − L X S m X j Z L Y l =1 d D ‘ l (2 π ) D S j ˜ n j ( ‘ ) n j ( ‘ ) D j , (6.3)where ˜ n j and n j are gauge-theory numerator factors, which can come from distinct gaugetheories and κ is the gravitational coupling defined below Eq. (1.5). The duality needs tobe manifest in only one of the two gauge-theory amplitudes for the double-copy formula tohold. N = 4 SYM theory The simplest example that illustrates CK duality at loop level is the one-loop four-pointsuperamplitude of N = 4 SYM theory. These amplitudes are remarkably simple, makingthem very useful for this purpose.The Jacobi identity obeyed by the structure constants of any Lie algebra guaranteesthat, in any gauge theory with all fields in the adjoint representation of the gauge group,any one-loop four-point amplitude can be organized as A - loop4 (1 , , , 4) = g (cid:18) c A - loop4 (1 , , , 4) + c A - loop4 (1 , , , 3) + c A - loop4 (1 , , , (cid:19) , (6.4)where the color factor c in Eq. (6.4) corresponds to the one of the box diagram in Fig. 21and is given by dressing each three-point vertex with an f abc structure constant, and summingover all repeated indices, c = 4 f ba c f ca d f da e f ea b . (6.5)108 BCJ DUALITY AT LOOP LEVEL The other two color factors are obtained by relabeling and we normalized c following stan-dard conventions [88]. Passing to a trace basis for the color factors identifies A - loop (1 , , , N = 4 SYM theory. Each color-ordered superamplitude in Eq. (6.4) is especially simple and given by [393] A - loop N =4 (1 , , , 4) = istA tree N =4 (1 , , , I ( s, t ) , (6.6)where I ( s, t ) is the box integral illustrated in Fig. 21, s = ( p + p ) and t = ( p + p ) are the usual Mandelstam invariants and A tree N =4 (1 , , , n = 4 case ofEq. (B.17), is the color-ordered four-point tree superamplitude.Since the diagram structure of the kinematic propagators in the three color-ordered am-plitudes entering Eq. (6.4) matches that of their color factors, the kinematic numeratorsof the representation (6.1) of the one-loop amplitude can be straightforwardly identified.The combination stA tree N =4 (1 , , , 4) is fully crossing-symmetric, as a consequence of the BCJfour-point tree-level amplitude relations (1.28), so all three numerators are the same, n = n = n = istA tree N =4 (1 , , , 4) = [12][34] h ih i δ (8) ( X i =1 λ i η Ii ) , (6.7)where we have specialized to four-dimensional external kinematics in the last equality.Because triangle and bubble diagrams do not appear in the N = 4 SYM amplitude (6.6)(or, alternatively, they enter with vanishing numerators), it is straightforward to check, usingEq. (6.7), that the BCJ duality relation illustrated in Fig. 20 holds. The remaining kinematicJacobi relations are also satisfied for similar reasons.The corresponding N = 8 supergravity amplitude follows immediately from the basicdouble-copy substitution (2.10), replacing color factors by numerators and compensating forthe change in coupling. This gives M - loop N =8 (1 , , , 4) = − istu M tree N =8 (1 , , , (cid:18) I ( s, t ) + I ( s, u ) + I ( t, u ) (cid:19) , (6.8)where we used (1.4), (cid:18) κ (cid:19) (cid:16) stA tree N =4 SYM (1 , , , (cid:17) = stu M tree N =8 (1 , , , , (6.9)109 BCJ DUALITY AT LOOP LEVEL to replace the square of the N = 4 SYM four-point tree-level amplitude with the N = 8supergravity four-point tree-level amplitude. This is a consequence of the KLT relations(1.31) and the BCJ amplitude relation (1.28). The amplitude in Eq. (6.8) reproduces theknown N = 8 supergravity four-point tree-level amplitude [194, 393].The explicit value of the massless scalar box integral I ( s, t ) appearing in both the N = 4SYM and N = 8 supergravity one-loop four-point amplitudes is I ( s, t ) = Z d D ‘ (2 π ) D ‘ ( ‘ − p ) ( ‘ − p − p ) ( ‘ + p ) , (6.10)where the p i ’s are the external momenta and the Feynman iε prescription, not includedexplicitly, is used to define the propagators. In dimensional regularization, we take D = 4 − (cid:15) with (cid:15) small. The explicit functional form of I ( s, t ) is (see e.g. Refs. [394, 395]) I ( s, t ) = i c Γ st (cid:20) (cid:15) (cid:18) ( − s ) − (cid:15) + ( − t ) − (cid:15) (cid:19) − ln (cid:18) − s − t (cid:19) − π (cid:21) + O ( (cid:15) ) , (6.11)with c Γ = (4 π ) (cid:15) π Γ(1 + (cid:15) )Γ(1 − (cid:15) ) Γ(1 − (cid:15) ) . (6.12)The other box integrals can be obtained from this one by relabeling. Using these explicitexpressions one can verify general properties of (super)gravity amplitudes, such as existenceof only soft infrared (IR) divergences.We can use Eq. (6.4), together with the N = 4 SYM numerators (6.7), to immediatelyobtain the four-point amplitudes of any 4 ≤ N ≤ N = 4 four-point SYM kinematic numerators (6.7) are independent ofthe loop momentum, they come out of the integral as in Eq. (6.8) and behave essentiallythe same way as color factors. Thus, to obtain results for N ≥ N ≤ N = 4 SYM numerators in Eq. (6.7). This gives us a general representation of the four-pointamplitudes of all N ≥ M - loop N +4 susy (1 , , , 4) = (cid:18) κ (cid:19) istA tree4 (1 , , , (cid:18) A - loop N susy (1 , , , 4) + A - loop N susy (1 , , , A - loop N susy (1 , , , (cid:19) . (6.13)As explained above, A N susy are one-loop color-ordered gauge-theory amplitudes after loopintegration for a theory with N (including zero) supersymmetries (c.f. Eq. (6.4)). Thisexpression applies just as well for external matter multiplets in N = 4 supergravity. Theneeded integrated gauge-theory amplitudes may be found in Refs. [270, 394].The double copy of amplitudes of gauge theories with N < BCJ DUALITY AT LOOP LEVEL To illustrate CK duality and the double-copy construction at one loop, we considerthe one-loop identical-helicity four-gluon amplitude in QCD with N f quark flavors in thefundamental representation, originally constructed in Ref. [270]. It is A - loopQCD (1 + , + , + , + ) = 2 g [1 2] [3 4] h i h i (cid:18) ( c − N f c f ) I ( s, t )[ µ ] (6.14)+ ( c − N f c f ) I ( s, u )[ µ ] + ( c − N f c f ) I ( t, u )[ µ ] (cid:19) , where the color factor associated with the quark loop is c f = Tr[ T a T a T a T a ] + Tr[ T a T a T a T a ] . (6.15)For simplicity, we have assumed that the quarks are massless. Here µ is the ( − (cid:15) )-dimensionalcomponent of loop momentum, so ‘ = ‘ (4) + µ , ‘ = ( ‘ (4) ) − µ , (6.16)and I ( s, t )[ µ ] is the integral corresponding to the diagram in Fig. 21 with a µ numeratorfactor. As required by Bose symmetry, the prefactor is fully cross symmetric, i.e.[1 2] [3 4] h i h i = [2 3] [4 1] h i h i = [1 3] [2 4] h i h i , (6.17)and, up to the supermomentum conservation delta function, it is the same as in Eq. (6.7).Exercise 6.2: Show the prefactor in Eq. (6.17) is crossing symmetric. Spinor properties maybe found in Appendix B and in various reviews [88, 89, 91].It is not difficult to check that the amplitude in Eq. (6.14) obeys CK duality. Considerthe duality relation in Fig. 20: because the triangle diagrams have vanishing numerators inEq. (6.14), the duality requires the different box integrals to have an identical numerator,which follows from Eq. (6.17) and the integrals’ numerators being crossing symmetric.Exercise 6.3: Make the quarks massive. For the identical helicity case, the integral numeratoris obtained with the replacement µ → ( µ + m q ) [270] while the loop propagators becomemassive with mass m q . Do the BCJ relations hold? What does the double-copy theorycorrespond to?Consider now the double-copy construction with one of the two amplitude factors beingEq. (6.14) with N f = 0. Taking the second amplitude to be the four-gluon superamplitudeof N = 4 SYM theory given in Eqs. (6.4),(6.6) leads to an anomalous superamplitude in N = 4 supergravity [260]: M - loop (1 , , , N =4 = 2 (cid:18) κ (cid:19) [1 2] [3 4] h i h i ! δ (8) ( X i =1 λ i η Ii ) × (cid:18) I ( s, t )[ µ ] + I ( s, u )[ µ ] + I ( t, u )[ µ ] (cid:19) . (6.18) It may also be obtained via the dimension-shifting relation [396] from the four-gluon superamplitude in N = 4 SYM theory (6.4),(6.6). BCJ DUALITY AT LOOP LEVEL As outlined in Sec. 4, this amplitude breaks the U (1) duality symmetry of this theory [260]and is the amplitude-level manifestation of the duality anomaly identified in [263] from aLagrangian perspective.Another example is the double copy in which both amplitudes are given by Eq. (6.14).In D dimensions, the double copy of a gluon has a total for ( D − states, corresponding toa graviton ( D ( D − / D − D − / N f = 0leads to the four-graviton amplitude in a theory with a dilaton and antisymmetric tensor, M - loop (1 + , + , + , + ) = 4 (cid:18) κ (cid:19) [1 2] [3 4] h i h i ! (cid:18) I ( s, t )[ µ ] + I ( s, u )[ µ ] + I ( t, u )[ µ ] (cid:19) . (6.19)The polarization vectors in the spinor-helicity basis used in Eq. (6.14) project out the dilatonand antisymmetric tensor asymptotic states from this amplitude. BCJ duality and thedouble copy for general helicity have been described in Refs. [12, 397]. For a theory withonly gravitons and no anti-symmetric tensor or dilaton, the result for the identical helicityfour-graviton amplitude is the same as in Eq. (6.19), except that the overall factor of 4becomes a factor of 2. This can be proven by inserting graviton physical-state projectorsinto the unitarity cuts, as described in Appendix C.The integrals in the gauge-theory and gravity amplitudes in Eqs. (6.14), (6.18) and (6.19), I ( s, t )[ µ k ] = Z d D ‘ (2 π ) D µ k ‘ ( ‘ − p ) ( ‘ − p − p ) ( ‘ + p ) , (6.20)evaluate to I ( s, t )[ µ ] = − i (4 π ) 16 + O ( (cid:15) ) , I ( s, t )[ µ ] = − i (4 π ) s + 2 t + st ) . (6.21)Their finite values arise due to a cancellation of the O ( (cid:15) ) numerator factors and O ( (cid:15) − ) IRdivergences. From this perspective, the nonvanishing amplitude (6.14) may be interpretedas a self-duality anomaly [264]. The integrals (6.20) may also be interpreted in terms ofhigher-dimensional integrals [396].Exercise 6.4: Consider the double copy of amplitudes in QCD with N f > N f ? Answer the same questions if the N f > N = 4 SYM theory, but for now the discussion is quite general. Weonly need to discuss the maximally-helicity-violating (MHV) amplitude, as the only othernonvanishing one, the MHV amplitude, can be obtained by hermitian conjugation. Five-point amplitudes with other external states can be obtained through a suitable sequence ofsupersymmetry transformations. This amplitude was constructed in Refs. [163, 398] in a112 BCJ DUALITY AT LOOP LEVEL Figure 22: Pentagon and box integrals appearing in the N = 4 SYM five-point one-loop amplitudes.The complete set of such integrals is generated by permuting external legs and removing overcounts. color-trace basis. Here we rearrange it slightly and write it in the structure-constant basis, A - loop5 (1 , , , , 5) = g X S / ( Z × Z ) c A - loop5 (1 , , , , , (6.22)where A - loop5 on the right-hand side are the five-point color-ordered partial amplitudes.The sum runs over the distinct permutations of the external legs: this is the set of all 5!permutations, S , but with cyclic, Z , and reflection symmetries, Z , removed, leaving 12distinct permutations. The color factor c is the one obtained from the pentagon diagramshown in Fig. 22, with legs following the cyclic ordering, by dressing each vertex with an˜ f abc . This color decomposition holds for any gauge-theory amplitude with only adjoint-representation particles and can be reached by starting from a generic color decompositionin terms of products of structure constants and repeatedly using the Jacobi identity to favorstructure constants with a single external color index.Exercise 6.5: By starting from Feynman diagrams, apply color Jacobi identities to expressall color factors in terms of those of pentagon diagrams. What is the generalization for an ar-bitrary number of external legs? (Feynman diagrams can helpful proving various properties,even if not useful for high-multiplicity explicit calculations.)Exercise 6.6: Generalize Eq. (6.22) to include quarks in the fundamental representation inthe loop. (See Eq. (6.14) at four points).For N = 4 SYM theory, the color-ordered one-loop five-point amplitudes in Eq. (6.22)are [163, 398], A - loop N =4 (1 , , , , 5) = 12 A tree5 (1 , , , , (cid:18) s s I (12)3454 + s s I + s s I + s s I + s s I (cid:19) + O ( (cid:15) ) , (6.23)where A tree5 (1 , , , , 5) is the color-ordered MHV tree-level amplitude. We may obtain theentire one-loop five-point MHV superamplitude by replacing A tree5 (1 , , , , 5) with the five-point tree-level MHV superamplitude in Eq. (B.17). The external kinematic invariants are113 BCJ DUALITY AT LOOP LEVEL s ij = ( p i + p j ) . The I abc ( de )4 are scalar box integrals where the legs in parenthesis connect tothe same vertex, e.g. I (12)3454 is the box diagram in Fig. 22. This representation (6.23) ofthe amplitude does not manifestly satisfy the duality. An alternative representation of theMHV superamplitude, which manifests the duality between color and kinematics, is [4]: A - loop N =4 (1 , , , , 5) = g (cid:18) X S / ( Z × Z ) c n I + X S / Z c [12]345 n [12]345 s I (12)3454 (cid:19) . (6.24)Each of the two sums runs over the distinct permutations of the external legs of the integrals.For I , the set S / ( Z × Z ) denotes all permutations but with cyclic and reflectionsymmetries removed, leaving 12 distinct permutations. For I (12)3454 the set S / Z denotes allpermutations but with the two symmetries of the one-mass box removed, leaving 30 distinctpermutations. The pentagon numerator for this representation of the superamplitude is n = − δ (8) ( Q ) [1 2] [2 3] [3 4] [4 5] [5 1]4 i(cid:15) (1 , , , , (6.25)where 4 i(cid:15) (1 , , , 4) = 4 i(cid:15) µνρσ k µ k ν k ρ k σ = [1 2] h i [3 4] h i − h i [2 3] h i [4 1]. With thispentagon numerator, the box numerators that manifest the kinematic Jacobi relations illus-trated in Fig. 23 are n [12]345 = n − n . (6.26)Other box numerators are obtained by relabeling. It is not difficult to see that the diagramswith triangle or bubble integrals have vanishing numerators. For example, the numerator ofthe triangle diagram with momenta p + p at one vertex and p + p at another is n [12]345 − n [12]354 = n − n − n + n (6.27)= − δ (8) ( Q )4 i(cid:15) (1 , , , (cid:26) [1 2] [2 3] [3 4] [4 5] [5 1] + [2 1] [1 3] [3 4] [4 5] [5 2]+ [1 2] [2 3] [3 5] [5 4] [4 1] + [2 1] [1 3] [3 5] [5 4] [4 2] (cid:27) , where we used momentum conservation to relate all Levi-Civita symbols contracted withfour external momenta. Upon use of the Schouten identities,[5 1] [2 3] − [5 2] [1 3] = [1 2] [3 5] , [2 3] [4 1] − [1 2] [3 4] = [1 3] [4 2] , (6.28)the term in brackets in Eq. (6.27) vanishes, so n [12]345 − n [12]354 = n − n − n + n = 0 . (6.29)The first of the identities (6.28) is used to combine the first two terms in Eq. (6.27) and thesecond identity shows that the remaining term cancel.Exercise 6.7: Show that all kinematic numerator relations hold for the amplitude given inEq. (6.24). 114 BCJ DUALITY AT LOOP LEVEL n 12 3 45 = n − n Figure 23: A BCJ kinematic numerator relation between a diagram containing a box integral andtwo pentagon diagrams. The shaded (red) line differs between the diagrams, but the others areidentical. A nice feature of this representation is that the numerator factors of both the pentagonand box integrals do not depend on loop momentum. This greatly simplifies the constructionof the corresponding supergravity amplitudes.Given that the duality holds for the representation (6.24) of the five-point one-loop MHV N = 4 SYM superamplitude, we can immediately obtain the corresponding N = 8 ampli-tude. We replace the color factors with a numerator factor (2.10), c → n , c [12]345 → n [12]345 , (6.30)as well as the gauge coupling with the gravitational one. The resulting five-graviton one-loopMHV superamplitude in N = 8 supergravity reads (2.11) M N =8 (1 , , , , 5) = (cid:18) κ (cid:19) (cid:18) X S / ( Z × Z ) ( n ) I + X S / Z ( n [12]345 ) s I (12)3454 (cid:19) , (6.31)where the sums run over the same permutations as in Eq. (6.24) and, as discussed in Sec. 4,the δ (16) ( Q ) should be understood as containing eight different η parameters for each externalparticle.The scalar pentagon integral and the one external-mass box integral have been computedin Ref. [395]. We include them here for convenience: I (12)3454 = − ic Γ s s − (cid:15) (cid:20) ( − s ) − (cid:15) + ( − s ) − (cid:15) − ( − s ) − (cid:15) (cid:21) + Li (cid:18) − s s (cid:19) + Li (cid:18) − s s (cid:19) + 12 ln (cid:18) s s (cid:19) + π + O ( (cid:15) ) , (6.32) I = X Z − ic Γ ( − s ) (cid:15) ( − s ) (cid:15) ( − s ) (cid:15) ( − s ) (cid:15) ( − s ) (cid:15) " (cid:15) + 2 Li (cid:18) − s s (cid:19) + 2 Li (cid:18) − s s (cid:19) − π + O ( (cid:15) ) , (6.33)where Li ( x ) is the dilogarithm function and c Γ is defined in Eq. (6.12).Exercise 6.8: Show that the double copy of the one-loop five-point amplitude where onecopy is of an MHV amplitude and the second an MHV amplitude vanishes. The MHV115 BCJ DUALITY AT LOOP LEVEL amplitude is obtained from the MHV one by parity which amounts to replacing h a b i ↔ [ a b ]and flipping the overall sign of the amplitude. Do you expect a similar property to hold at n points or at higher loops?As discussed above, the double-copy construction works even if the duality is manifest inonly one gauge-theory factor. Starting with the color decomposition in Eq. (6.22) and usingthe fact that for the one-loop five-point N = 4 SYM amplitude the pentagon numeratorsare independent of loop momentum, we immediately obtain five-point superamplitudes for( N + 4)-extended supergravities. By taking the second copy to be any pure SYM theory,with color-ordered one-loop five-point amplitudes A - loop N (1 , , , , M - loop N (1 , , , , 5) = (cid:18) κ (cid:19) X S / ( Z × Z ) n A - loop N (1 , , , , . (6.34)Here n is given in Eq. (6.25) and the sums run, as in the case of the N = 4 amplitude,over all the permutations which are not related to each other by cyclic permutations orreflections. Gauge theories with reduced supersymmetry provide an opportunity to discuss the con-struction of duality-satisfying (loop-level) scattering amplitudes with fields in representa-tions other than the adjoint. A simple example, which we will review here in some detail,is the one-loop four-matter-field superamplitude in N = 2 SYM theory with a single hy-permultiplet in a complex representation R [30, 294]. The color factors c j in Eq. (6.1) arenow constructed by dressing every vertex of every diagram with a gauge-group generator inthe appropriate representation. This more complicated color structure is a consequence ofreduced supersymmetry, which allows for matter fields in non-adjoint representations. Tokeep supersymmetry manifest, we organize the hypermultiplet asymptotic states as on-shellsuperfields and their CPT-conjugates, which are treated as distinct:Φ N =2 ˆ α = χ +ˆ α + η i ϕ i ˆ α + η η ˜ χ − ˆ α Φ N =2 ˆ α = ˜ χ ˆ α + + η i ϕ ˆ αi + η η χ ˆ α − . (6.35)The lower and upper ˆ α is the R and ¯ R representation indices, respectively. As outlined inSec. 5.2.2, such superfields with reduced supersymmetry can in principle be obtained fromthe ones of N = 4 SYM theory by an orbifold truncation.At one loop, a duality-satisfying representation of the four-hypermultiplet superamplitudecan be constructed in terms of two master numerators, which can be chosen to belong totwo box diagrams. Adopting the standard notation for theories with matter (super)fields,we denote the adjoint vector multiplet with a curly line and the complex-representationhypermultiplet with a solid line with an arrow. The master numerator factors and the116 BCJ DUALITY AT LOOP LEVEL corresponding diagrams are [30, 294]: n 12 34 = s h ih i δ (4) (cid:16) Q (cid:17) n 14 23 = − st h ih i δ (4) (cid:16) Q (cid:17) , (6.36)where δ (4) (cid:16) Q (cid:17) = δ (4) (cid:16) P n η in λ n (cid:17) . The other box integrals can then be obtained by permuta-tion, keeping in mind that the overall superamplitude possesses a Z × Z Fermi symmetryunder the exchange of hypermultiplet superfields. For example, a third box-integral numer-ator is n 13 24 = − su h ih i δ (4) (cid:16) Q (cid:17) . (6.37)Numerators for the triangle and bubble diagrams can be obtained via the kinematicnumerator relations. They can be organized in two distinct sets: (1) those that mirrorrelations between color factors which are a consequence of the defining commutation relationsof the color Lie algebra and (2) those that corresponding to color relations that hold only forcertain groups and representations but are nonetheless required for the consistency of thedouble copy of a hypermultiplet with a vector multiplet. An example of numerator relationsfrom the first group is n 12 34 − n 13 24 = n 12 34 . (6.38)From the double-copy perspective, following the argument presented in Sec. 2.5, these re-lations are required for obtaining gravity amplitudes invariant under linearized diffeomor-phisms. An example of color relations that hold only for certain groups and representationsis T ˆ a ˆ γ ˆ α T ˆ a ˆ δ ˆ β = T ˆ a ˆ δ ˆ α T ˆ a ˆ γ ˆ β . (6.39)The corresponding numerator relations include, for example, n 14 23 = n . (6.40)While these color relations are satisfied only for certain choices of gauge group and rep-resentations, the fact that the form (6.1) is independent of such choices suggests that one117 BCJ DUALITY AT LOOP LEVEL may choose, as we do here, to always impose the corresponding numerator relations. Onemay easily convince oneself that these numerator relations are required by consistency of thedouble-copy construction in case the hypermultiplet fields are combined with spin one fieldsin the conjugate matter representation. In other cases they may be regarded as “bonus”relations; it is not clear a priori that there exist solutions to the numerator relations in thesecond group even when solutions to the numerator relations in the first group do.In Sec. 5.3.3, we have reviewed the double-copy construction for homogeneous supergrav-ities, and showed that it reproduces the existing classification of such theories. An importantingredient of the construction are matter fields in pseudo-real representations. It is thereforeinstructive to see how our one-loop numerators described above are modified in this case.To enforce the pseudo-reality of the gauge group representation (i.e. the equivalence of theupper and lower ˆ α indices in Eq. (6.35)), the on-shell superfields Φ N =2 and Φ N =2 are iden-tified. Consequently, the superamplitude needs to acquire a complete Fermi symmetry forall of its external legs. Drawing from this observation, the numerators for a theory withpseudo-real half-hypermultiplets can be obtained as the unique set of numerators which areboth invariant under the permutation of all external legs and reduce to the numerators forthe complex case whenever the corresponding color factors are nonzero. More concretely, inthe pseudo-real case we have only one master numerator: n 12 34 = s h ih i δ (4) (cid:16) Q (cid:17) , (6.41)and all the other numerators are obtained either from permutation symmetry or from the nu-merator relations (6.38) and (6.40). For half-hypermultiplets in pseudo-real representations,solid lines no longer carry an arrow since the matter half-hypermultiplets are CPT-self-conjugate.Exercise 6.9: Given the master numerator (6.41), use numerator relations to generate allnonzero numerators (up to permutation symmetry).We emphasize that, as in the case of the one-loop four- and five-point superamplitudesof N = 4 SYM theory, the duality-satisfying kinematic numerators of the superamplitudereviewed here are independent of the loop momentum. Consequently, the physical proper-ties of the double-copy supergravity theory can be directly related to properties of the othergauge theory entering the construction. Consider, for example, the construction for homo-geneous Maxwell-Einstein supergravities explained in Sec. 5.3.3. We can relate the one-loopdivergences of supergravity amplitudes with four vector superfields constructed as hyper-multiplet × fermion to a linear combination of various parts of the one-loop beta functionof the non-supersymmetric gauge theory, M (cid:12)(cid:12)(cid:12)(cid:12) div = − i (4 π ) s δ (4) ( Q ) h ih i (cid:18) κ (cid:19) ( sA tree s,φ (cid:18) β φ (cid:12)(cid:12)(cid:12) T ( G ) − β φ (cid:12)(cid:12)(cid:12)(cid:12) T ( R ) (cid:19) + sA tree s,A (cid:18) β A (cid:12)(cid:12)(cid:12) T ( G ) − β A (cid:12)(cid:12)(cid:12)(cid:12) T ( R ) (cid:19)) (cid:15) +Perms . (6.42)118 BCJ DUALITY AT LOOP LEVEL 12 3412 34 Figure 24: Diagrams for the two-loop integrals appearing in the two-loop four-point N = 4 and N = 8 supergravity amplitudes. Here β φ , β A are the beta-functions for the gauge coupling and the Yukawa interactions.We use the notation β (cid:12)(cid:12)(cid:12) T ( G ) ,T ( R ) to label the parts of the relevant beta functions that areproportional to the index of the adjoint, T ( G ),and pseudo-real, T ( R ), matter representations. A tree s,A and A tree s,φ are, respectively, the s -channel gluon and scalar exchange parts of the gaugetheory tree-level amplitudes. Finally, it should be noted that compact expressions for two-loop amplitudes for N = 2 gauge theories with matter can be found in Ref. [399].Exercise 6.10: Use the result of Exercise 7.9 to verify equation (6.42). If the duality between color and kinematics holds at tree-level in D dimensions, then it alsoholds on all D -dimensional generalized cuts that decompose a loop amplitude into a sum ofproducts of tree amplitudes. Thus, barring anomalies, it is expected to hold beyond one-looplevel. As an illustrative example, consider the two-loop four-point amplitude of N = 4 SYMtheory. This amplitude, originally constructed in Refs. [194, 400], is A - loop4 (1 , , , 4) = − g st A tree4 (1 , , , (cid:18) c P1234 s I - loop , P4 ( s, t ) + c P3421 s I - loop , P4 ( s, u )+ c NP1234 s I - loop , NP4 ( s, t ) + c NP3421 s I - loop , NP4 ( s, u ) + cyclic (cid:19) , (6.43)where “+ cyclic” indicates that one should add the two cyclic permutations of (2 , , f abc factor.As the diagrams appearing in the amplitude are already cubic, we can read off thekinematic numerators for each diagram. They are: n P1234 = n NP1234 = is tA tree4 (1 , , , , n P3412 = n NP3412 = is tA tree4 (1 , , , ,n P1342 = n NP1342 = iustA tree4 (1 , , , , n P4213 = n NP4213 = iustA tree4 (1 , , , ,n P1423 = n NP1423 = ist A tree4 (1 , , , , n P2314 = n NP2314 = ist A tree4 (1 , , , . (6.44)The factor stA tree4 (1 , , , , , 4) are added.119 BCJ DUALITY AT LOOP LEVEL n p q = n 12 34 p q − n 12 34 p q n 12 34 p q = n 12 34 p q − n 12 34 qp n 12 34 qp = n 13 24 qp − n 23 14 pq Figure 25: Examples of BCJ kinematic numerator relations at two loops. Because of the limited set of nonvanishing diagrams, it is straightforward to check thatthis amplitude satisfies all duality relations. Three of them are shown in Fig. 25. Thecomplete set may be obtained by starting with the diagrams in Fig. 24 and systematicallygenerating the duality relations.Following the double-copy prescription (2.10), we obtain the corresponding N = 8 super-gravity amplitude by replacing the color factor with a numerator factor, c P1234 → is tA tree (1 , , , , c NP1234 → is tA tree (1 , , , , (6.45)including relabelings and then swapping the gauge coupling for the gravitational one. Indeed,this gives the correct N = 8 supergravity amplitude, as first noted in Ref. [194] which alsoverified it against the direct construction from unitarity cuts.As mentioned in Sec. 2, generalized gauge invariance implies that only one of the twocopies must be in a form manifestly satisfying the duality (2.7); for the second copy, such aform should exist but its use is not required. The color Jacobi identity allows us to express anyfour-point color factor of an adjoint representation in terms of the ones in Fig. 24 [180]. If theduality and double-copy properties hold and because of the independence of the momentumof the N = 4 SYM numerator factors (6.44), it is possible to obtain integrated N ≥ N ≤ N = 4 SYM [4].Due to the high degree of supersymmetry, one can express the amplitude in terms of onlysix contributing diagrams shown in Fig. 26. To be concise, we will only quote the duality-satisfying numerator of diagram (a); it is n (a)12345 ( p, q ) = 14 (cid:18) γ (2 s − s + τ p − τ p ) + γ ( s + 2 s − τ p + τ p )+ 2 γ ( τ p − τ p ) + γ ( s + s − τ p + τ p ) (cid:19) , (6.46)120 BCJ DUALITY AT LOOP LEVEL p q p q p q (c) 4 p q p q (b)3 4512 p q Figure 26: The six nonzero diagrams that contribute to two-loop five-point amplitude in N = 4SYM and N = 8 supergravity. where the two independent loop momenta are called p and q . The Lorentz invariants are τ ip = 2 p i · p , τ iq = 2 p i · q and s ij = ( p i + p j ) . The external state dependence for the MHVamplitude is captured by the γ ij , where γ ≡ n [12]345 = δ (8) ( Q ) [1 2] [3 4] [4 5] [5 3]4 i(cid:15) (1 , , , 4) (6.47)is the one-loop box numerator given in Eq. (6.26), and the other γ ij are given by S permu-tations of this expression. Note that the γ ij satisfy the relations γ ij = − γ ji , X i =1 γ ij = 0 , (6.48)from which it follows that there are only six independent variables of this type. The diagramnumerators of the MHV amplitude are obtained by replacing γ ij by their CPT conjugates.Exercise 6.11: Show that the kinematic numerators corresponding to diagrams (b)–(f) inFig. 26 can be obtained from n (a)12345 ( p, q ) using kinematic Jacobi relations. Which numeratorshappens to be independent of loop momenta? Which numerators are identical to each other(due to the Jacobi relation collapsing to a two-term identity)?The two-loop N = 4 SYM amplitude is given by the sum over the six diagrams (a)–(f)in Fig. 26, together with the sum over the S permutations over the external legs, A - loop5 = ig X S X j ∈{ a ,... f } Z d D p d D q (2 π ) D S j c ( j )12345 n ( j )12345 ( p, q ) Q α j p α j . (6.49)121 BCJ DUALITY AT LOOP LEVEL The corresponding N = 8 supergravity amplitude is obtained by the double-copy replace-ments c ( j )12345 → n ( j )12345 ( p, q ) and g → κ/ S j thatappear in Eq. (6.49).The two-loop five-point amplitudes of both N = 4 SYM and N = 8 supergravity, aspresented above, were integrated in D = 4 − (cid:15) dimensions in refs. [327, 401–403]. So far, we illustrated various one- and two-loop amplitudes that manifest the color-kinematicsduality. To be more concrete, in this subsection we go through in detail how to constructduality-satisfying amplitudes when the system of numerators is quite large. As a sophisti-cated example—though still quite manageable—consider the three-loop four-point amplitudeof the N = 4 SYM and N = 8 supergravity theories [2].Apart from the duality and unitarity constraints, it is beneficial to systematically imposevarious other constraints which become more important as the complexity of the problemincreases. Although not required, such auxiliary constraints, when appropriately chosen, cangreatly facilitate the construction. If a constraint is too strong and leads to an inconsistencywith unitarity, then one may relax or modify it as needed. This strategy is especially effectivefor theories with high degrees of supersymmetry, because of their restricted power counting.For the three-loop four-point N = 4 SYM amplitudes, a natural set of constraints is asfollows.1. One-loop tadpole, bubble and triangle subdiagrams do not appear in any diagram [222,404, 405].2. A one-loop n -gon subdiagram carries no more than n − stA tree4 , the numerators are polynomials in D -dimensional Lorentz scalar products of the independent loop and external momenta.4. Numerators carry the same relabeling symmetries as the diagrams (cf. discussion inSec. 3).In general, the choice of auxiliary constraints depends on the problem at hand. For exam-ple, the third constraint above is specific to the four-point amplitude, and should be modifiedfor higher-point amplitudes because of their more complicated external-state structure. Asdescribed in the previous subsection, a relatively simple generalization has been found forthe five-point (super)amplitude [4], involving prefactors that are proportional [161, 406, 407]to linear combinations of five-point color-ordered tree-amplitudes. For amplitudes in lesssupersymmetric theories, all but the fourth condition must also be relaxed, because theirpower counting is such that one-loop triangle and bubble subdiagrams do appear; this isrelated to e.g. the running of their couplings. The above constraints also work well for the122 BCJ DUALITY AT LOOP LEVEL (c)5 6 7 J b J a (a) 321 4 7 J i J h (h) 41 357 667 J g J l (l)12 34675 J k (k)21 345 76 J j (j)12 4 (e) 412 35 67 7 6 J f (f)12 3455 6 7 J d J c Figure 27: The diagrams for constructing the N = 4 SYM and N = 8 supergravity three-loopfour-point amplitudes. The shaded (red) lines indicate the application of the duality relation. Theexternal momenta are outgoing and the arrows indicate the directions of the labeled loop momenta.Diagram (e) is the master diagram. four-loop four-point amplitudes of N = 4 SYM [6], but fail at five loops. A procedure whichworks for this case is described in Sec. 8.Because the duality imposes stringent relations between diagrams’ numerators, a remark-ably small subset of generalized unitarity cuts is then sufficient to completely determine theintegrand. Of course, to confirm that it is correct, it is necessary to verify that it reproducescorrectly a spanning set of unitarity cuts that fully determine the amplitude. Quite gener-ally, one expects that a problem with a generalized cut can be addressed by relaxing someof the auxiliary constraints.Let us return now to the three-loop four-point amplitudes of N = 4 SYM theory andillustrate these ideas. A straightforward enumeration shows that there are 17 distinct cubicdiagrams with three loops and four external legs, which do not have one-loop triangle, bubbleor tadpole subdiagrams. It turns out that the twelve diagrams shown in Fig. 27 are sufficientfor finding a solution to the duality and unitarity cut constraints, as shown in Ref. [2]. Hadwe kept all 17 diagrams, the construction would be slightly more involved, with the result123 BCJ DUALITY AT LOOP LEVEL n = n 412 35 67 − n 421 35 67 n = n 412 35 67 Figure 28: Examples of a BCJ kinematic numerator relation at three loops for N = 4 SYM theory.In the two term relations one of the three numerators in a Jacobi triplet of diagrams vanishes. that the numerators of the additional diagrams vanish identically.The four-point amplitudes of N = 4 SYM theory are special. Applying the third condi-tion above we write the numerator as n ( x ) = − istA tree4 (1 , , , N ( x ) , (6.50)where ( x ) refers to the label for each diagram in Fig. 27 and N ( x ) are scalar functions whichdepend on three independent external momenta, labeled by p , p , p , and on (at most) threeindependent loop momenta, labeled by ‘ , ‘ , ‘ , N ( x ) ≡ N ( x ) ( p , p , p , ‘ , ‘ , ‘ ) . (6.51)The coefficient stA tree4 (1 , , , 4) is fully crossing symmetric, as noted in Eq. (6.7).Next, consider the duality relations. We need to discuss first those that allow us to expressthe complete set of numerators N ( x ) in terms of a small subset—the master numerators.Some of them are shown in Fig. 28. The remaining relations are subsequently verified oncethe former are solved together with the constraints imposed by the unitarity cuts. For thethree-loop four-point N = 4 SYM amplitude, a simple restricted set of duality relations124 BCJ DUALITY AT LOOP LEVEL is [6, 196]: N (a) = N (b) ( p , p , p , ‘ , ‘ , ‘ ) ,N (b) = N (d) ( p , p , p , ‘ , ‘ , ‘ ) ,N (c) = N (a) ( p , p , p , ‘ , ‘ , ‘ ) ,N (d) = N (h) ( p , p , p , ‘ , ‘ , p , − ‘ + ‘ − ‘ ) + N (h) ( p , p , p , ‘ , ‘ , p , + ‘ − ‘ ) ,N (f) = N (e) ( p , p , p , ‘ , ‘ , ‘ ) ,N (g) = N (e) ( p , p , p , ‘ , ‘ , ‘ ) ,N (h) = − N (g) ( p , p , p , ‘ , ‘ , p , − ‘ − ‘ ) − N (i) ( p , p , p , ‘ − ‘ , ‘ − ‘ + ‘ − p , , ‘ ) ,N (i) = N (e) ( p , p , p , ‘ , ‘ , ‘ ) − N (e) ( p , p , p , − p − ‘ − ‘ , − ‘ − ‘ , ‘ ) ,N (j) = N (e) ( p , p , p , ‘ , ‘ , ‘ ) − N (e) ( p , p , p , ‘ , ‘ , ‘ ) ,N (k) = N (f) ( p , p , p , ‘ , ‘ , ‘ ) − N (f) ( p , p , p , ‘ , ‘ , ‘ ) ,N (l) = N (g) ( p , p , p , ‘ , ‘ , ‘ ) − N (g) ( p , p , p , ‘ , ‘ , ‘ ) , (6.52)where p i,j ≡ p i + p j . To simplify the notation, we have suppressed the canonical arguments( p , p , p , ‘ , ‘ , ‘ ) of the numerators on the left-hand side of the equations (6.52). Eachrelation specifying an N ( x ) is generated by considering the kinematic Jacobi relations dualto the color Jacobi relations corresponding to the shaded (red) line and labeled J x in Fig. 27.In general, duality relations relate triplets of numerators; if however one of the diagrams isnot present in Fig. 27, e.g. because it has a one-loop triangle subdiagram, then we obtain atwo-term relation. Five of the equations above are of this type and they result in pairs ofnumerators being equal.The system (6.52) can be used to express any kinematic numerator factor as a combi-nation of the numerator N (e) with various different arguments. Thus, diagram (e) can betaken as the sole master diagram. This is a convenient choice, but not the only possible one;for example, either diagram (f) or (g) can also be used as a single master diagram. None ofthe remaining nine diagrams, however, can act alone as a master diagram.The numerator factor of diagram (e) is constructed such that the unitarity cuts are sat-isfied simultaneously with the duality constraints. An expression that satisfies the maximalcuts is given by the so-called “rung-rule” numerator [400], N (e)rr = s ( ‘ + p ) , (6.53)which follows from the general features of iterated two-particle cuts.We wish to find a modification N (e)rr → N (e) such that all the other numerators determinedfrom it via Eq. (6.52) are consistent with the unitarity cuts. We start by requiring that themaximal-cut of diagram (e) is correct (see Appendix C.3 for a description of the maximalcuts), and that the auxiliary constraints above are satisfied. That is, the departure from N (e)rr vanishes on the maximal cut, the numerator N (e) has mass dimension four and possessesthe symmetry of the diagram; no loop momentum for any box subdiagram in (e) appears init (ruling out ‘ and ‘ ), and N (e) is at most quadratic in the pentagon loop momenta ‘ .The last condition is a little weaker than the second auxiliary condition listed earlier, whichdemands linearity in ‘ ; we relax it slightly to make it easier to find deformations that vanish125 BCJ DUALITY AT LOOP LEVEL on maximal cuts, and impose later that the ‘ terms cancel out. The symmetry conditionimplies that N (e) is invariant under { p ↔ p , p ↔ p , ‘ → p + p − ‘ } . (6.54)The most general polynomial consistent with these constraints is N (e) = s ( ‘ + p ) + ( αs + βt ) ‘ + ( γs + δt )( ‘ − p ) + ( αs + βt )( ‘ − p − p ) , (6.55)where the four parameters α, β, γ, δ are to be determined by further constraints. All addedterms are proportional to inverse propagators and therefore vanish on the maximal cut.Thus, given that Eq. (6.53) is consistent with the maximal cuts, so is Eq. (6.55).The second auxiliary constraint above demands that the numerator of a pentagon subdi-agram be at most linear in the corresponding loop momentum, ‘ , not quadratic as assumedabove. Therefore we impose that the coefficient of ‘ in Eq. (6.55) vanishes. This yields therelation γ = − − α and δ = − β , which simplifies Eq. (6.55) to N (e) = s ( τ + τ ) + ( αs + βt )( s + τ − τ ) , (6.56)where we use the notation, τ ij ≡ p i · ‘ j , ( i ≤ , j ≥ . (6.57)We are therefore left with two undetermined parameters, α and β .We determine the remaining parameters by imposing that the numerators of other di-agrams determined through Eq. (6.52) are consistent with the auxiliary constraints andunitarity cuts. A convenient starting point is the numerator of diagram (j), N (j) , which isdetermined in terms of N (e) by the 9th duality constraint in Eq. (6.52). Inserting Eq. (6.56)into this relation leads to N (j) = s (1 + 2 α − β )( τ − τ ) + βs ( t − u ) . (6.58)Because the smallest loop in diagram (j) carrying ‘ is a box subdiagram, our auxiliaryconstraints require that this momentum be absent from N (e) . Setting the first term inEq. (6.58) to zero implies that β = 1 + 2 α , which in turn leads to N (e) = s ( τ + τ ) + ( α ( t − u ) + t )( s + τ − τ ) , (6.59) N (j) = (1 + 2 α )( t − u ) s , (6.60)leaving undetermined a single parameter α .To obtain the value of the final parameter we use the numerator of diagram (a) expressedin terms of N (e) by Eq. (6.52). Because every loop in diagram (a) is part of a box, the auxiliaryconstraint that a one-loop box subdiagram cannot carry loop momentum then implies that N (a) cannot contain loop momentum. By solving the duality relations (6.52), the numerator N (a) is given by N (a) = N (e) ( p , p , p , − p + ‘ − ‘ + ‘ , ‘ − ‘ , − ‘ )+ N (e) ( p , p , p , − p − ‘ + ‘ , − ‘ , ‘ − ‘ ) − N (e) ( p , p , p , ‘ − ‘ , ‘ , ‘ − ‘ ) − N (e) ( p , p , p , ‘ − ‘ , ‘ , − ‘ ) − N (e) ( p , p , p , ‘ , ‘ , ‘ − ‘ ) − N (e) ( p , p , p , ‘ , ‘ , − ‘ ) . (6.61)126 BCJ DUALITY AT LOOP LEVEL Plugging in the value of the numerator factor N (e) in Eq. (6.59), and simplifying we obtain N (a) = s + (1 + 3 α ) (cid:18) ( τ − τ ) s − τ + τ ) s + ( τ − τ − τ + 2 τ ) t + 4 ut (cid:19) . (6.62)Demanding that this expression is independent of loop momenta, fixes the final parameterto be α = − / N (e) = s ( τ + τ ) + 13 ( t − s )( s + τ − τ ) . (6.63)With a proposed expression for N (e) in hand, Eq. (6.52) then determines all other numer-ators and thus the complete amplitude. The resulting numerators are collected in Tab. 15.To confirm that this is indeed the correct amplitude, it is necessary to verify a complete setof unitarity cuts. The three-loop four-gluon amplitude in N = 4 SYM theory is determinedonly by its maximal and next-to-maximal cuts, so it is relatively straightforward to checkthem all. As a highly-nontrivial test, one can also check the next-to-next-to-maximal cuts.The resulting cuts match those of previous expressions of the amplitude [408, 409] on all D -dimensional unitarity cuts. Thus, the amplitude is complete. We stress again that it ishighly-nontrivial that there exists a solution to all duality relations which is consistent withall unitarity cuts and exhibits all the diagram symmetries.Squaring the numerators n ( x ) = stA tree4 (1 , , , n ( x ) , using Eq. (6.3), yields the numera-tors for the three-loop four-point N = 8 supergravity superamplitude. This form has beenconfirmed against previous expressions [408, 409] on a spanning set of D -dimensional unitar-ity cuts [2]. Using as the second copy the three-loop four-point numerator factors of N < N )-extended supergrav-ity theories. The case N = 0 was discussed at length in Refs. [33, 36, 37], where it was usedto explore the UV properties of half-maximal supergravities and demonstrate the absence ofUV divergences at this loop order in four dimensions.Exercise 6.13: Work through the entries in Tab. 15 to explicitly confirm that they do indeedsatisfy BCJ duality.The strategy followed above generalizes straightforwardly to the four-loop four-point [6]and two-loop five-point amplitudes of N = 4 SYM and supergravity. It has also been testedin a variety of other cases, including the one- and two-loop amplitudes in various theories withfewer supersymmetries [241], and nonsupersymmetric gauge and gravity theories [12, 397]as well as to the construction of form factors in N = 4 SYM through five loops [9, 17].While the application of the double-copy construction to gauge-theory form factors yieldsquantities consistent with the linearized diffeomorphism invariance of a gravity theory, theirprecise physical interpretation is currently an open question. The examples described above are but a sample of the many loop-level amplitudes that haverepresentations that manifest the duality between color and kinematics. Among them are127 BCJ DUALITY AT LOOP LEVEL diagram N = 4 SYM ( r N = 8 supergravity) numerator(a)–(d) s (e)–(g) (cid:16) s ( − τ + τ + t ) − t ( τ + τ ) + u ( τ + τ ) − s (cid:17) / (cid:16) s (2 τ − τ + 2 τ − τ + 2 τ + τ + τ − u )+ t ( τ + τ − τ + 2 τ − τ − τ − τ − τ ) + s (cid:17) / (cid:16) s ( − τ − τ − τ + τ + τ + 2 t )+ t ( τ + τ + 2 τ + 2 τ + 3 τ ) + u τ + s (cid:17) / s ( t − u ) / Table 15: The numerator factors for diagrams in Fig. 27 [2]. The first column labels the diagram,the second column the relative numerator factor for N = 4 SYM theory. The square of this is therelative numerator factor for N = 8 supergravity. The momenta are labeled as in Fig. 27 and the τ ij are defined Eq. (6.57). various examples of supersymmetric [4, 6–8, 11, 14, 16, 19, 20, 399] and nonsupersymmet-ric [10, 12, 15, 156, 349] gauge-theory amplitudes, form factors [9, 17, 18, 22, 23], stringtheory amplitudes, and their field theory limits [13, 372, 410–412]. Additionally, a system-atic method to determine BCJ numerators for one-loop amplitudes which makes use of theglobal constraints on the loop-momentum dependence of the numerators imposed by thekinematic Jacobi identities was introduced in Ref. [11].It has moreover been shown that the leading and subleading [5, 413] factorization theo-rems of gauge and gravity theories are consistent with the double-copy procedure to all ordersin perturbation theory, thus providing some all-loop-levels evidence for this conjecture. Inanother interesting example, the duality has been applied to QCD scattering amplitudes, inorder to find hidden relations between coefficients of loop integrals [414, 415].The multitude of nontrivial examples suggests that the duality between color and kine-matics does extend to loop amplitudes, even though no proof exists as yet. Finding a proofwould likely provide a guide towards more systematic constructions of representations ofamplitudes that manifest the duality.Even when they are expected to exist, the construction of duality-satisfying amplitudesrepresentations is not always straightforward. An alternative, discussed and illustrated onthe two-loop four-point all-plus pure-YM amplitude in Ref. [416], is to relax the demand thatthe duality be manifest off shell and impose instead that it be manifest only on a spanningset of generalized unitarity cut. The double-copy construction then yields an expressionwhich coincides with the corresponding supergravity amplitude on a spanning set of cuts;the two must therefore be the same.It has proven difficult to find representations of the five-loop four-point N = 4 SYM am-128 GENERALIZED DOUBLE COPY plitude for which the duality between color and kinematics is manifest. In particular, theexpected power-counting constraints suggested by supersymmetry appear to not be com-patible with duality and D -dimensional unitarity cuts. To find the corresponding N = 8supergravity amplitude the generalized double-copy construction provides an efficient ap-proach, as it uses gauge-theory amplitudes’ representations that should exhibit the dualitybut do not manifest it; we review it in the next section. The success of the generalized dou-ble copy, using the five-loop amplitudes’ representations constructed in Refs. [38, 218, 417],strongly suggests that it should be possible to manifest the duality for the five-loop four-point N = 4 SYM amplitude. Presumably, this will require integrands that relax some thesimplifying assumptions, such as locality or manifest relabeling symmetry of the diagrams.Using string theory to define global diagram labels in the field-theory limit, there has beensome very interesting progress on finding integrands that manifest CK duality [418]. Whenever gauge-theory amplitudes are available in a form that manifests the duality betweencolor and kinematics, the BCJ double-copy construction provides the most efficient meansfor obtaining the corresponding gravity integrands. However, in some cases, such as thefive-loop four-point amplitude of N = 8 supergravity, it has proven difficult to find suchrepresentations. In other cases, such as the all-plus two-loop five-gluon amplitude in pure-YM theory, the BCJ form of the amplitude has a superficial power-count much worse thanthat of Feynman diagrams [15] and thus an analysis of UV properties of its double copyis cumbersome at best. It can therefore be advantageous to have a double-copy methodfor converting generic representations of gauge-theory amplitude to gravity ones, withoutfirst constructing BCJ representations for them. Such a procedure has been developed inRef. [417] and applied in Refs. [38, 218] to construct the five-loop four-point integrand of N = 8 supergravity and to extract its UV properties after integration. If we start with a generic representation of a gauge-theory amplitude where BCJ duality isnot manifest and apply the double-copy substitution rule (2.10), in general, we do not obtaina correct gravity amplitude. Nevertheless, this “naive double copy” can be systematicallycorrected to give the desired amplitude. As we summarize below, the correction terms havea regular pattern reminiscent of the KLT tree-level amplitudes relations [86], allowing us toobtain the most complicated corrections directly from gauge theory. To start the generalized double-copy construction we first need to reorganize slightly the two(possibly distinct) gauge-theory amplitudes that comprise the two sides of the double copy.Starting with any local representations of the amplitudes, which may include four- or higher-point contact terms, we reorganize them into a format that has only three-point vertices and Another possible method, proposed in Ref. [416] and illustrated on the two-loop four-point pure-YMamplitude in D dimensions, is to demand that the duality between color and kinematics holds only onunitarity cuts. This provides a straightforward construction of the generalized unitarity cuts of the double-copy theory, which need to be subsequently assembled into the complete gravity amplitude. GENERALIZED DOUBLE COPY 412 3 412 3412 3412 3 412 3 412 3412 3412 3 ⇒ 412 3 n , n , n , n , n , n , n , n , n , Figure 29: An example illustrating the notation in Eq. (7.1). Expanding each of the two four-pointblob gives a total of nine diagrams. The n i,j correspond to labels used in the generalized unitaritycut. The shaded thick (blue and red) lines are the propagators around which BCJ discrepancyfunctions are defined. the maximum number of propagators. If a given term has fewer propagators we multiply anddivide by the propagators needed to form diagrams with only cubic vertices that correspondto the color factor of the given term. Once the gauge-theory amplitudes are written in thisformat the next step is to apply the double-copy substitution (2.10) to these amplitudes,despite neither gauge theory manifesting the BCJ duality between color and kinematics. Asalready mentioned, this so-constructed naive double-copy expression is, in general, not acorrect (super)gravity amplitude. Nonetheless, it is a good starting point for obtaining thefull gravity amplitude as, by construction, it reproduces the maximal and next-to-maximalcuts of the desired (super)gravity amplitude. (See Appendix C for a description of themethod of maximal cuts.) In maximal cuts, where all propagators are cut, the amplitudeis reduced to a sum of products of gauge-theory three-point tree amplitudes. Because on-shell gravity three-vertices are products of gauge-theory ones, maximal cuts trivially satisfythe double-copy property for any representation of the single copy amplitudes. The nextto maximal cuts, where one of the propagators are left uncut, also automatically give thecorrect gravity expressions because, if present, the duality between color and kinematics isautomatic for on-shell four-point tree amplitudes [1].Beyond the next-to-maximal cuts, the naive double copy will generally not give correctunitarity cuts, and nontrivial corrections are necessary. These required corrections can beorganized into contact terms via the method of maximal cuts described in Appendix C.However, for complicated problems, such as N = 8 supergravity [218] at five-loops it becomescumbersome to use the method of maximal cuts to obtain the missing terms.130 GENERALIZED DOUBLE COPY Instead, it turns out that it is possible to construct general formulae that relate thenecessary cut-correction terms to the violations of the kinematic Jacobi relations (2.7) inthe gauge-theory amplitudes. The derivation of such formulae relies only on the existence ofduality-satisfying representations for all tree-level amplitudes.Indeed, the existence of BCJ representations at tree level implies that such representationsshould also exist for all cuts of gauge-theory amplitudes that decompose the loop integrandinto products of tree amplitudes. This further implies that the corresponding generalizedunitarity cuts of the gravity amplitude can be expressed in double-copy form, C GR = X i ,...,i q n BCJ i ,i ,...i q ˜ n BCJ i ,i ,...i q D (1) i . . . D ( q ) i q , (7.1)where the n BCJ and ˜ n BCJ are the BCJ numerators associated with each of the two single-copy parent theories. In this expression the cut conditions are understood as being imposedon the numerators. Each sum runs over the diagrams of each tree amplitude composingthe generalized cut and D ( m ) i m are the products of the uncut propagators associated to eachdiagram of m -th tree amplitude. This notation is illustrated in Fig. 29 for an N MC atthree loops. In this figure, each of the two four-point blobs is expanded into three diagrams,giving a total of nine diagrams. For example, the combination of indices i = 1 and i = 1refers to the three-loop diagram obtained by taking the first diagram from each blob andconnecting it to the three-point vertices; the result, in the ordering of diagrams chosen foreach of the two four-point amplitude, is the first cubic diagram on the first line of Fig. 29.The denominators in Eq. (7.1) correspond to the thick (colored) lines in the diagrams.The BCJ numerators in Eq. (7.1) are related [2, 41] to those of an arbitrary representationby a generalized gauge transformation which shifts the numerators subject to the constraintthat the amplitude is unchanged; the shift parameters follow the same labeling scheme asthe numerators themselves, n i ,i ,...i q = n BCJ i ,i ,...i q + ∆ i ,i ,...i q . (7.2)The shifts ∆ i ,i ,...i q are constrained to leave the corresponding cuts of the gauge-theoryamplitude unchanged. Using such transformations we can reorganize a gravity cut in termsof cuts of a naive double copy and an additional contribution, C GR = X i ,...,i q n i ,i ,...i q ˜ n i ,i ,...i q D (1) i . . . D ( q ) i q + E GR (∆) , (7.3)where the cut conditions are imposed on the numerators. Rather than expressing the cor-rection E GR in terms of the generalized-gauge-shift parameters, it is useful to re-expressthe correction terms as bilinears in the violations of the kinematic Jacobi relations (2.7) bythe generic gauge-theory amplitude numerators. These violations are referred to as BCJdiscrepancy functions.As an example, the generalized unitarity cut in Fig. 29 is composed of two four-pointtree amplitudes and the rest are three-point amplitudes. For any cut of this structure, twofour-point trees connected to any number of three-point trees, the correction has a simpleexpression, E × = − d (1 , d (2 , (cid:18) J • , ˜ J , • + J , • ˜ J • , (cid:19) , (7.4)131 GENERALIZED DOUBLE COPY where d ( b,p ) i is the p -th propagator of the i -th diagram inside the b -th amplitude factor and J • ,i ≡ X i =1 n i i , J i , • ≡ X i =1 n i i , ˜ J • ,i ≡ X i =1 ˜ n i i , ˜ J i , • ≡ X i =1 ˜ n i i , (7.5)are BCJ discrepancy functions . Our notation is to label the type of cut by m × m ×· · · m k where each m i specifies the number of legs on each tree amplitude with m i ≥ E × = − X i ,i =1 d (1 , i d (2 , i (cid:18) J • ,i ˜ J i , • + J i , • ˜ J • ,i (cid:19) . (7.6)Similarly, a cut with a single five-point tree amplitude and the rest three-point treeamplitudes is given by C = X i =1 n i ˜ n i d (1 , i d (1 , i + E with E = − X i =1 J { i, } ˜ J { i, } + J { i, } ˜ J { i, } d (1 , i d (1 , i , (7.7)where J { i, } and J { i, } are BCJ discrepancy functions associated with the first and secondpropagator of the i -th diagram. (See Ref. [218] for further details.)As the cut level k increases, the formulae relating the amplitudes’ cuts with the cuts ofthe naive double copy become more intricate, but the basic building blocks remain the BCJdiscrepancy functions. Formulas like (7.6), (7.7) and their generalizations can enormouslystreamline the computation of the contact term corrections and are especially helpful at fiveloops at the N MC and N MC level, where calculating the contact terms via the maximal-cutmethod can be rather involved. Beyond this level, the contact terms become much simplerdue to a restricted dependence on loop momenta and are better dealt with using the methodof maximal cuts and KLT relations [86], as described in Ref. [218]. To illustrate the discussion above, we now present a relatively simple though nontrivial con-struction of the three-loop four-point amplitude of N = 8 supergravity, which was studiedin several other different approaches [2, 6, 408, 409]. As described in Sec. 6 the most efficientway to construct it is to first obtain a BCJ representation of corresponding N = 4 SYM am-plitude and then apply the double-copy construction. Instead, we construct it here throughthe generalized double copy, from a non-BCJ form of the N = 4 SYM amplitude of Ref. [409]whose numerators are included in Tab. 16 with the momentum labeling in Fig. 27(a)-(i), cor-responding to the one of Ref. [2]. An overall factor of stA tree4 is not included in Tab. 16. We will sometimes omit the second argument, p , when an amplitude factor has a single propagator. We will sometimes denote the BCJ discrepancy function with either • in the position i or by { i, } whenthe i -th amplitude factor has a single propagator (i.e. it is a four-point amplitude). GENERALIZED DOUBLE COPY n , n , 12 36 1 3 n , 32 46 Figure 30: The three diagrams whose kinematic numerators contribute to J { , } , . The thick shaded(red) line marks the off-shell legs participating in the dual Jacobi relation. The shaded (red) dotindicates the off-shell leg of the second amplitude factor. Diagram N = 4 SYM numerators.(a)-(d) s (e)-(g) s ( p + τ )(h) s ( τ + τ ) − t ( τ + τ ) + st (i) s ( p + τ ) − t ( p + τ + p ) − ( s − t ) p / Table 16: A non-BCJ form of the three-loop four-point N = 4 SYM diagram numerators fromRef. [409]. We define τ ij = 2 p i · p j , s = ( p + p ) , t = ( p + p ) and u = ( p + p ) . Following the generalized double-copy construction, the N = 8 supergravity numeratorsof diagrams (a)–(i) are squares of the corresponding N = 4 SYM ones: N N =8( x ) = n x ) , (7.8)where x ∈ { a , . . . , i } . This defines the naive double copy. This is not the complete supergrav-ity amplitude given that the gauge-theory numerators do not satisfy the BCJ relations (2.7),as can be confirmed by checking its generalized unitarity cuts. To complete the supergravityamplitude we need to find the missing contact terms.Given that the N MC-level contact terms are automatically accounted for in the naivedouble copy, contact terms first appear at the N MC level. There are a total of 62 possibleindependent such contact diagram, corresponding to diagrams obtained by starting from thefirst nine diagrams in Fig. 27 and collapsing all pairs of propagators. Of these, all but thefour diagrams (j)-(m) in Fig. 31 vanish.As an example, consider the contact diagram in Fig. 31(l), composed of two four-pointvertices. We obtain it from Eq. (7.4). First, we identify the nine cubic diagrams thatcontribute to it (some are vanishing) and pick one whose numerator we label as n , ; we choosediagram (c) in Fig. 27. The two J -functions are calculated by relabeling the appropriatenumerators to the labels of Fig. 30. For example, J { u , } , is obtained from the N = 4 SYM133 GENERALIZED DOUBLE COPY (j) (k) (l) (m)31 2 4 112 34 2 34 12 34 Figure 31: Nonvanishing contact terms appearing in the generalized double copy construction ofthe three-loop four-point amplitude of N = 8 supergravity. numerators of the three diagrams shown in Fig. 30, n , = s , n , = s ( t + τ + τ ) , n , = s ( u − τ ) , (7.9)corresponding to relabeling of diagrams (c) and (g) in Fig. 27. Summing and applyingmomentum conservation gives J { , } , = sτ . Similarly, J , { , } = sτ . With these labels,the two off-shell inverse propagators are τ and τ , so that from Eq. (7.4) the N = 8supergravity contact term numerator for diagram (l) is N N =8(l) = − J { , } , J , { , } τ τ = − s . (7.10)The other three independent contact terms corresponding to diagrams (j), (k) and (m), cansimilarly be obtained from Eq. (7.7), with the result N N =8(j) = − ( s − t ) , N N =8(k) = N N =8(m) = − s . (7.11)All other nonvanishing contact terms are relabelings of these. This generalized double-copy procedure has been systematically used to obtain the five-loopfour-point integrand of N = 8 supergravity [218], which was then used to analyze the UVproperties of this theory at five loops [38]. In this case, it was sufficient to work out formulaefor the extra corrections up to the N MCs, because beyond this the missing contact termsare simple enough to straightforwardly obtain by numerical analysis.As discussed before, Eqs. (7.4) and (7.7) can be used for all N MCs in any double-copytheory. These are sufficient to determine the three-loop four-point amplitude in N = 8supergravity, because of its low power count. Beyond this order the corresponding formulaefor E depend on the detailed labeling of the corresponding cut. We include here E × × and E × and comment on E given as an ancillary file in Ref. [218].The additional terms that promote a cut composed of three four-point amplitude factorsof the naive double copy to the cut of the corresponding double-copy theory [218, 417] areobtained by following the steps detailed in Sec. 7.1. It is convenient to organize then intothe contribution of single- and double-discrepancy functions: E × × = T + T . (7.12)134 GENERALIZED DOUBLE COPY They are T = − X i =1 J • , ,i ˜ J , • ,i d (1)1 d (2)1 d (3) i − X i =1 J • ,i , ˜ J ,i , • d (1)1 d (2) i d (3)1 − X i =1 J i , • , ˜ J i , , • d (1) i d (2)1 d (3)1 + { J ↔ ˜ J } ,T = J • , , ˜ J , • , • d (1)1 d (2)1 d (3)1 + J , • , ˜ J • , , • d (1)1 d (2)1 d (3)1 + J , , • ˜ J • , • , d (1)1 d (2)1 d (3)1 + { J ↔ ˜ J } , (7.13)where e.g. ˜ J , • , • is defined as ˜ J i , • , • = X i =1 3 X i =1 ˜ n i ,i ,i , (7.14)with n i ,i ,i being the numerators of the cut of the naive double copy. As mentioned ear-lier, we dropped the second upper label in d ( b,p ) i defined below Eq. (7.4) because four-pointdiagrams have only a single propagator, so b = 1 for all terms in Eq. (7.13).To simplify T we used the relations J , • , • d (1)1 = J , • , • d (1)2 = J , • , • d (1)3 , J • , , • d (2)1 = J • , , • d (2)2 = J • , , • d (2)3 , J • , • , d (3)1 = J • , • , d (3)2 = J • , • , d (3)3 , (7.15)which identify various double-discrepancy functions.The additional terms that promote a cut composed of one five-point and one four-pointamplitude factors of the naive double copy to the cut of the corresponding double-copytheory can be organized as E × = X i =1 3 X j =1 d (1 , i d (1 , i d (2 , j " − J { i, } ,j ˜ J { i, } ,j − (cid:18) − (cid:19) × (cid:18) J { i, } ,j ˜ J { i, } , • + J { i, } ,j ˜ J { i, } , • (cid:19) − J { i, } ,j ˜ J i, • − J { i, } ,j ˜ J i, • + 130 X k ∈J i σ k,i J { k, } ,j ˜ J i, • + 130 X k ∈J i σ k,i J { k, } ,j ˜ J i, • + { J ↔ ˜ J } , (7.16)where J i is the set of five diagrams connected to diagram i through Jacobi relations on the twopropagators, including diagram i which appears once, and σ k,i are the signs with which theircolor factors enter in the color Jacobi relations, with the normalization that σ i,i = 1. WhileEq. (7.16) is quite different from the corresponding E × in Ref. [218], the two expressionsare in fact equivalent, as can be shown by reducing to a basis of BCJ discrepancy functions,or by directly evaluating the additional terms for a choice of representation of the five-pointamplitude. The essential advantage of Eq. (7.16) is that it does not make reference either toa specific ordering of the diagrams of the five-point amplitude or to a specific choice of orderof propagators for each diagram. These features may be the key to extending Eq. (7.16) tocuts with higher-point tree-level amplitude factors. That is, the color factors of the corresponding diagrams obey the relation c i + X k ∈J i σ k,i c k = 0 , which is just the sum of the two Jacobi relations on the two propagators of diagram i . GENERALIZED DOUBLE COPY Similarly to E × and E , both E × × and E × are not local. To extract their corre-sponding contact terms it is necessary to subtract the contribution of the 4 × 4- and 5-contactterms which contribute to the 4 × × × ExtraJ_6pt.m of Ref. [218]. Unlike E × , E , E × × and E × above however, E is presented in terms of a basis of independentdiscrepancy functions, obtained by solving the constraints they obey due to their definitionin terms of the cut kinematic numerators of the single-copy parent theories. The expression isalso not manifestly organized in terms of the kinematic denominators of the 105 diagrams ofthe six-point tree-level diagram. While, for these reasons, the available E is not manifestlycrossing symmetric, it is sufficient for greatly simplifying the analytic structure of N MC witha single six-point tree amplitude, compared to the direct construction of such cuts via e.g.the KLT relations.A feature of the nonsymmetric correction terms E GR expressed in terms of the some basisof BCJ discrepancy functions is that, when evaluated on a cut, they may lead to termsthat behave as 0 / 0. These are harmless when the 0 in the numerator is manifest, sinceit corresponds to an absent diagram. Sometimes, however, the 0 in the numerator is notmanifest and arises due to a cancellation between distinct terms, that can leave behind anontrivial finite piece. When this occurs, the simplest strategy is to take advantage of theasymmetry in the formula, to relabel it to avoid such problematic cases.Generalized double-copy formulae such as those reviewed here, give the cuts of anydouble-copy theory in terms of generic representations of the amplitudes of the single-copyparent theories. It is therefore an interesting problem to find similar general formulae formore complicated—perhaps all—cuts at any loop order. We can argue based on the gaugeinvariance of the single-copy theories that the correction terms must be linear in the BCJdiscrepancy functions of each of the single-copy theories [218]. That is, E = X i,j M ij J i ˜ J j , (7.17)for some appropriate matrix M ij whose entries are rational functions of the kinematic in-variants of the cut. This structure is compatible with the fact that the corrections shouldall vanish if the duality between color and kinematics were manifest in either one of thetwo single copies [41]. A further heuristic argument for the general form (7.17) of the cor-rection terms E relies on an understanding of the structure of the terms that need to beadded to cuts of the naive double copy in order to restore the linearized diffeomorphisminvariance expected of the cuts of amplitudes of a gravitational theory. As we saw in Secs. 1and 2, a gauge transformation of tree-level amplitudes—and thus also of the cuts of a loopamplitude—is given by a sum of terms each of which is proportional to some linear combi-nation of color Jacobi relations. Consequently, a linearized diffeomorphism transformationof the naive double copy yields a sum of terms each containing a BCJ discrepancy functionfrom either one of the two single copies. To restore diffeomorphism invariance these termsmust be cancelled by the transformation of further terms that are added to the cuts of the136 CLASSICAL DOUBLE COPY naive double copy. Assuming that the structure of these terms is the same for all doublecopy theories, they must be of the form (7.17). See Ref. [218] for more details.While the generalized double-copy method has already been successful for the highlynontrivial case of N = 8 supergravity at five loops [218, 417], its development is only at thebeginning. Having a general tool for converting gauge-theory amplitudes in any represen-tation to gravity ones is clearly useful and important. A good starting point would be toderive general formulae for tree-level amplitudes [218, 419–421] in terms of a naive doublecopy, plus corrections in terms of the BCJ discrepancy functions. At present such formulaeare known only through six points. If an elegant solution to the tree-level problem can befound, it should be immediately applicable to finding a general solution to the loop-levelone. One obvious application would be towards a definitive resolution of the UV behaviorof extended supergravity theories. This would require calculations beyond those that havealready been carried out (see e.g. Refs. [38, 292]), and would likely need a version of thegeneralized double copy to be practical. N = 5 supergravity at five loops is an especiallyinteresting case for future study, given that at four loops it exhibits an enhanced cancellationof UV divergences [292]. It is important to know whether this continues at higher loops. As we have seen at length, the duality between color and kinematics and the double-copyconstruction are essential tools in the construction of gauge and (super)gravity scattering am-plitudes at higher-loop orders and/or at higher multiplicity. In close analogy with tree-levelscattering amplitudes, the perturbative construction of solutions of the classical equationsof motion of a field theory (perhaps in the presence of sources) also exhibits an expansionin tree-level diagrams. One may consequently expect that, with an appropriate definition,some version of double-copy construction may lead to a construction of solutions of Ein-stein’s equations (perhaps also in the presence of other fields) in terms of solutions of YMequations of motion (perhaps also in the presence of other fields). If one could turn thedouble copy into a systematic tool for analyzing classical solutions one could hope for newadvances analogous to the ones that have occurred for scattering amplitudes.As we shall discuss below, such a relation between classical solutions is not withoutsubtleties and comes with quantifiable differences from the case of flat-space scattering am-plitudes. Flat-space scattering amplitudes carry an inherent simplicity in that they arecompletely independent of gauge and field variable choices. However, in contrast to scatter-ing amplitudes, generic classical solutions change nontrivially under gauge transformationsand, moreover, they are sensitive to the nonlinear terms in the gauge transformations. Thus,to relate gauge and gravity solutions it is necessary to make correlated gauge choices in thetwo theories; the principles for making such choices are unclear. Related to this, the form ofthe equations of motion depends strongly on the choice of field variables. Thus, any naiveextension of the scattering-amplitudes’ double copy of fields can be completely obscuredby nonlinear coupling-dependent terms that depend on some a priori chosen form of theequations of motion.As yet, no coherent set of rules for the construction of double copies for generic classicalsolutions in gravity theories has been formulated, though a variety of nontrivial tantalizing137 CLASSICAL DOUBLE COPY examples have been found. (See e.g. Refs. [50–77].) Ideally, any such rules should smoothlygeneralize those of scattering amplitudes and reduce to them in the appropriate limits. Theclasses of examples that have been constructed and analyzed emphasize both the similar-ities and the differences between classical solutions and scattering amplitudes, and exposethe subtleties that need to be addressed in order to formulate a general framework. Theirexistence, however, suggests that it may be possible to find generic solutions of a gravitytheory in terms of solutions of the two gauge theories that give its scattering amplitudes.The most obvious application of these ideas are towards improving calculations of as wellas calculations in post-Newtonian expansion of gravitational interaction potentials as wellas calculations potentially relevant to gravitational-wave detection. These type of calcula-tions can be phrased in terms of scattering amplitudes [78, 79, 422–427] and therefore arelikely to lead to useful new results, such as the computation of the third post-Minkowskiancontribution to the conservative two-body potential [80, 82].In this section we describe the known constructions of gravity classical solutions in termsof gauge-theory solutions, commonly referred to as “classical double copies”. We outlinetheir relation and similarities with the double copy of scattering amplitudes and summarizethe examples that have been discussed in this framework. We start with a description ofperturbative solutions in gravity before turning to complete double copies. There is a close relation between solutions of classical equations of motion of some fieldtheory and the Green’s functions of that theory. The classical field generated by an arbitrarysource is the generating functional for the tree-level connected Green’s functions. Given afield theory of some field φ with Lagrangian L , a solution of the equation of motion withgeneral sources, δ L δφ = ζ , (8.1)is given in terms of the generating functional of connected tree-level Green’s functions by[428] φ [ x, ζ ] = δW [ ζ ] tree δζ , (8.2)and moreover W [ ζ ] = Z d D x (cid:16) L [ φ [ x, ζ ]] − ζφ [ x, ζ ] (cid:17) . (8.3)The relation between Green’s functions and scattering amplitudes given by the LSZ reduc-tion implies in turn that, by amputating the sources, W becomes the generating functional oftree-level S-matrix elements. This may be realized by taking the source to be the quadraticoperator acting on an on-shell wave solution of the free equation of motion. The solution(8.2) with such sources is the generating function of Berends-Giele currents—i.e. Green’sfunctions of fundamental fields with exactly one leg off shell ; it therefore may also be inter-preted as a solution of the Berends-Giele off-shell recursion relation [431]. This idea was used Green’s functions with two legs off shell have been constructed in gauge theories coupled to fundamentalmatter in [429, 430]. CLASSICAL DOUBLE COPY φ [ x, ζ ] = + + + + . . . Figure 32: The first few terms in the expansion of a classical solution in terms of sources. Eachheavy dot represents a source ζ . The free end is at position x . The weight of each vertex is notspecified and may contain derivatives acting on the propagators connecting it to other vertices,sources or the point x . in Refs. [432, 433] to construct an implicit representation (referred to as the “perturbiner”)of gluon scattering amplitudes in four-dimensional YM theory and the gravitational dress-ing of certain classes of such amplitudes. Tree-level amplitudes of higher-dimensional andsupersymmetric YM theories have been constructed using this method in Refs. [114, 434]and in certain effective field theories and deformations of YM theories in Refs. [172, 435]. Itwas also was used in Ref. [43] to construct the kinematic algebra dual to the color algebrain self-dual YM theory. Solutions for the supersymmetric versions of Berends-Giele currentthat manifest CK duality were given in Ref. [436].Thus, Eqs. (8.1) and (8.2) allow us to construct perturbative approximations of solutionswith the appropriate source in terms of the scattering amplitudes of the theory. More-over, should it be possible to resum the scattering amplitudes into a generating functional,Eq. (8.2) provides an exact solution of the equation of motion with the appropriate sources.Depending on the chosen sources, the construction can be carried out either in momentumspace (if the sources are momentum eigenstates) or in position space.Introducing sources in gauge and gravity theories can be confusing for at least two reasons.First, fixed sources coupling to vector fields or with the graviton may break gauge invariance.A resolution of this would-be problem is the gauge-fixing that is necessary for any (tree-level)computation, which already breaks gauge invariance. One then adds sources in the gauge-fixed theory, in which the question of gauge invariance should not arise. Second, related,nonabelian vector fields and gravitons self-interact and consequently they can self-source.Examples are all solutions of vacuum Einstein’s equations as well as solutions of classical YMequations such as the instanton. For a stable configuration the matter stress tensor should becovariantly constant with respect to the metric that it sources; thus, it has some knowledgeof the solution. This implies that the perturbative construction of such solutions requires ajudicious choice of source which may itself receive corrections order by order in perturbationtheory. Examples were discussed in e.g. Refs. [437] and [438] for the Schwarzschild andReissner-Nordström black holes, respectively.Unlike scattering amplitudes, solutions of the classical field equations can be changedby (1) field redefinitions (2) coordinate changes and (3) gauge transformations (if gauge139 CLASSICAL DOUBLE COPY symmetries are present). 46 47 As yet, Lagrangians that manifest CK duality are known toonly a few perturbative orders [41, 42, 150, 151]. It is natural to expect that, if one hadsuch a complete Lagrangian, classical solutions constructed through a classical double copywould solve its equations of motion. It is natural to expect that nontrivial field redefinitionsand coordinate transformations are necessary to map such a solution to the field variables ofa more standard Lagrangian. In fact, the perturbative Lagrangians manifesting the double-copy properties of gravity require this as well as elimination of auxiliary fields.To illustrate the perturbative construction of solutions of supergravity equations ofmotion we outline here the derivation of the first terms [438] of the Reissner-Nordströmsolution—a charged black hole of (super)gravity coupled with a vector field A µ of fieldstrength F µν . The vanishing-charge limit leads to the corresponding (first) term(s) in theSchwarzschild solution, discussed in Ref. [437]. The relevant action is S = S G + S EM + S gauge fixing + S ζ , L G = 1 κ √− gg µν R µν , L EM = 116 π √− gg µρ g νσ F µν F ρσ , L ζ = 12 g µν ( ζ Mµν + ζ EMµν ) + A µ ζ µ ≡ √− gg µν ( T Mµν + T EMµν ) + √− gA µ j µ , L gauge fixing = − π ( ∂ µ A µ ) + 12 ( ∂ µ ( √− gg µν )) . (8.4)To construct a perturbative solution around Minkowski space the metric is assumed of theform g µν = η µν + κh µν . (8.5)There are several sources that lead to the desired solution. One may choose, for example,the stress tensor of a charged point particle. Alternatively, one may choose an extendedsource—a sphere of radius (cid:15) of uniform mass density ρ and uniform charge density σ . Thegeneral form of the stress tensor is T ν µ = ( ρ + p ) u ν u µ + pδ ν µ , g µν u µ u ν = 1 , (8.6)where ρ is the mass density function and p the (potentially phenomenological) pressure. Inthe case of a “ball of dust” with uniform mass and charge densities, the components of thesource turn out to be (after choosing u = (1 , , , 0) and imposing covariant constancy of thestress tensor) [438] ζ M = ρθ ( (cid:15) − r ) = 3 m π(cid:15) θ ( (cid:15) − r ) , ζ Mij = p (0) η ij = 3 Q π(cid:15) ( r − (cid:15) ) δ ij θ ( (cid:15) − r ) ,ζ µ = σδ µ θ ( (cid:15) − r ) = 3 Q π(cid:15) θ ( (cid:15) − r ) δ µ , (8.7) Symmetries of the equations of motion which are not symmetries of the action, such as parts of the U-duality symmetry of four-dimensional supergravity theories, may be used to generate inequivalent solutionsfrom known ones. See e.g. Ref. [439] for a review. The same choices also affect Feynman rules; however, when Feynman rules are combined into a scatteringamplitude there is no dependence upon these choices, although solutions of the classical equations of motion(and also Green’s functions) depend on them. Note that this choice is different form the one typically used for perturbative S-matrix calculations andin later subsections, but it is useful here as it avoids nonlinear terms involving the metric fluctuation andthe sources. CLASSICAL DOUBLE COPY where m and Q are the total mass and charge, respectively. The pressure p (0) is chosen suchthat this configuration of mass and charge densities is static under Newtonian gravitationalattraction and Coulomb repulsion. As the metric receives κ n corrections, so will the pressurefunction (hence the upper label “(0)” in ζ Mij above).The first correction to the flat space metric and the electromagnetic field due to thesources (8.7), given by the first two diagrams in Fig. 32, is h A µ ( x ) i ζ = Z d d y ∆ µν ( x − y ) ζ ν ( y ) + . . . ,κ h h µν ( x ) i ζ = κ Z d d y ∆ µν,ρσ ( x − y ) ζ Mρσ ( y )+ κ Z d d yd d x d d x ∆ µν,ρσ ( x − y ) γ ρσ,ητ ( ∂ y ) h A η ( y ) ih A τ ( y ) i . (8.8)Here ∆ µν,ρσ is the graviton propagator in the chosen de-Donder gauge (cf. L gauge fixing ), ∆ µν isthe photon propagator in Lorentz gauge and γ ρσ,ητ ( ∂ y ) describes the graviton-photon three-point interaction. We note that, due to the κ dependence in expansion of the metric (8.5),the trilinear graviton-photon vertex contributes before the three-graviton vertex.The extended nature of the source implies that the vector potential is different for r < (cid:15) and r > (cid:15) . Denoting by tilde the Fourier-transform of the source, h A µ ( y ) i = δ µ Z d p e ip · x − p ˜ ζ ( p ) = δ µ Qr θ ( r − (cid:15) ) + Q (cid:15) − Qr (cid:15) ! θ ( (cid:15) − r ) ! ≡ δ µ U , (8.9)which is just the Coulomb potential of the assumed charge distribution. Defining similarlythe Newtonian potential of the given mass distribution, W ≡ Z d p e ip · x − p ˜ ζ M ( p ) = ρ πr θ ( r − (cid:15) ) + ρ π(cid:15) − ρr π(cid:15) ! θ ( (cid:15) − r ) , (8.10)and the action of the inverse Laplace operator on a time-independent function F ( x ) as1 ∇ F ( x ) ≡ π Z d y F ( y ) | x − y | , (8.11)the components of the metric fluctuations around flat Minkowski space are κ h h i ζ = 8 πG (cid:18) W + 3 1 ∇ p (0) − η kl π ∇ ∂ k U ∂ l U (cid:19) ,κ h h ij i ζ = 8 πG (cid:18) W − ∇ p (0) − η kl π ∇ ∂ k U ∂ l U (cid:19) δ ij − G ∇ ∂ i U ∂ j U ,κ h h i i ζ = 0 . (8.12)Evaluating the integrals and defining the physical mass M = m + 35 Q (cid:15) , (8.13)141 CLASSICAL DOUBLE COPY it follows [438] that for r > (cid:15) the metric components are g = 1 + 2 M Gr − Q Gr + O ( G ) ,g ij = − (cid:18) − M Gr (cid:19) δ ij + Q Gr x i x j + O ( G ) ,g i = 0 . (8.14)This matches the Reissner-Nordström solution in Cartesian coordinates and de Dondergauge [438] : ds = r + Q G − M G ( r + M G ) dt − (cid:18) M Gr (cid:19) ( dx i ) + ( Q G − M G )( r + M G ) r ( r + Q G − M G ) ( x i dx i ) . (8.15)We note that the Q → M for nonvanishing electric charge. We also notethat Eq. (8.13) is a reflection of the field backreaction on sources. In fact, the redefinition(8.13) is necessary for the solution to have a smooth limit to a point source. The relationbetween the physical mass M and the “free mass” m receives further corrections as higherorders are included. The double-copy formulation of classical gravity calculations has the potential to streamlinecalculations such as those outlined in the previous subsection by exploiting the close relationbetween the tree expansion in Fig. 32 and that of tree-level S-matrix elements. Given thatwe do not as yet have a general framework for applying the double copy to perturbativesolutions, detailed analyses of specific examples, as we do below, help identify the correctphysical extension of the amplitudes double-copy rules to this setting. Before we proceed tosummarize the various options and illustrate their application to this problem, we begin withseveral comments which connect it to some of the calculations above and alert the reader topoints that will arise.As noted in Sec. 2, the double-copy spectrum naturally contains a dilaton and a two-index antisymmetric tensor (or equivalently a pseudo-scalar in four dimensions). As fortree-level scattering amplitudes where these unwanted states can be projected out at treelevel by a suitable choice of asymptotic states, solutions of Einstein’s equation may be foundby choosing gauge-theory sources such that their double copy does not source the dilatonand/or the anti-symmetric tensor [57, 58]. Choosing gauge-theory sources that are then usedin the double copy appears to bypass the need for a judicious choice a matter stress tensor assource for the gravity solution; however, prescribed properties of supergravity solutions and While this is different from the standard form of the Reissner-Nordström solution, it can be mapped toit by a coordinate transformation and field redefinition. CLASSICAL DOUBLE COPY their corresponding sources undoubtedly translate into properties of gauge-theory sources.At the time of this writing, a complete dictionary has not yet been formulated.CK duality as defined for scattering amplitudes in Sec. 2, requires that external lines areon the free mass shell. Thus, in the tree expansion in Fig. 32 the duality can be expectedto hold only up to terms that vanish if the sources obeyed free-field equations of motion.The discussion in Sec. 2 then implies that such a feature leads to breaking of linearizedgauge (diffeomorphism) invariance in the double-copy theory due to the presence of sources.This may be interpreted as the double-copy realization of the fact that gravity sources breakdiffeomorphism invariance. For the same reason, gravity field equations can be satisfied by adouble-copy field configuration only up to terms proportional to the free equations of motionof the sources. Thus, for a comparison with a direct solution of supergravity equationsof motion, such terms must be eliminated by field, coordinate and source redefinitions.This mirrors the backreaction of gravitational field on its source, illustrated in the previoussubsection. It is not a priori obvious that gauge-theory classical solutions which differ bygauge transformations lead through the double copy to gravity solutions that differ by fieldredefinitions and coordinate transformations.Perturbative spacetimes and their relation to perturbative solutions of the YM equationsof motion were discussed in Ref. [58]. Below we outline their construction. As in thecalculation of scattering amplitudes, we begin with the YM action (see Eq. (1.5)), whoseequations of motion in the presence of sources are ∂ µ F aµν + gf abc A bµ F cµν = ζ aµ , (8.16)where g is the coupling constant and the field-strength tensor F aµν is F aµν = ∂ µ A aν − ∂ ν A aµ + gf abc A bµ A cν . (8.17)It is straightforward to include matter fields. Apart from their equations of motion, inclusionof the matter fields also gives specific expressions for the sources ζ . We will not discuss thispossibility any further, choosing ζ to be non-dynamical and focusing on the gauge sector.The goal, following Ref. [58], is to solve perturbatively Eq. (8.16), A aµ = A (0) aµ + gA (1) aµ + g A (2) aµ + · · · , (8.18)and construct from it a solution of the double-copy theory.The action for Einstein gravity coupled with a dilaton and an antisymmetric tensor,which is the double copy of two pure D -dimensional gauge theories, is given in Eq. (2.47).For the construction of a perturbative solution of its equations of motion, the fields areexpanded as h µν = h (0) µν + κ h (1) µν + (cid:18) κ (cid:19) h (2) µν + · · · ,B µν = B (0) µν + κ B (1) µν + (cid:18) κ (cid:19) B (2) µν + · · · ,φ µν = φ (0) µν + κ φ (1) µν + (cid:18) κ (cid:19) φ (2) µν + · · · . (8.19)143 CLASSICAL DOUBLE COPY We can combine these different fields into a single field H . Since the asymptotic values ofthese fields are all obtained by projection from the tensor product of the two asymptoticgauge fields, it is convenient to have the field H which has this property at every order in κ .That is, in its expansion in κ , H µν = H (0) µν + κ H (1) µν + (cid:18) κ (cid:19) H (2) µν + · · · , (8.20) H ( n ) is the double copy of the n -th order term in the expansion of the gauge-theory field.There are no cross terms between different orders in the vector field expansion (8.18). Thisis a consequence of the fact that different orders are given by different tree configurations inFig. 32 and thus do not mix in the double copy.On shell, at the linearized level and in the appropriate gauges it is possible [58] toformulate the equations of motion in terms of a linear combination of the three fields: H (0) µν = h (0) µν + B (0) µν + P qµν φ (0) . (8.21)In the absence of sources they are ∂ ρ ∂ ρ H (0) µν = 0 . (8.22)A source modifies the right-hand side appropriately and must have the transversality andtrace properties of H (0) µν . The field (8.20) has been referred to in Ref. [58] as the “fat graviton”,in contrast with the “skinny graviton”, h µν . In Eq. (8.21) P qµν is a projector, which dependson a fixed null vector q , defining the physical dilaton. In position space it is P qµν = 1 D − η µν − q µ ∂ ν − q ν ∂ µ q · ∂ ! . (8.23)Conversely, the three physical fields can be extracted from H (0) by projection: φ (0) = η µν H (0) µν , B (0) µν = 12 ( H (0) µν − H (0) νµ ) , h (0) µν = 12 ( H (0) µν + H (0) νµ ) − P qµν H (0) ρρ . (8.24) Following Ref. [58], to solve the YM equations (8.16) we choose the Lorenz gauge, ∂ µ A aµ = 0,and to leading order in the coupling the equation becomes ∂ A (0) aµ = ζ aµ . (8.25)Consistency with the gauge condition requires that ζ be transverse. To start instead witha scattering state it suffices to replace ζ aµ → ε µ c a ∂ µ ∂ µ exp( ip · x ) where c a is some colorwave function and take the limit p → A (0) aµ while not using itsspecific form, though in specific examples it may be necessary to be more specific.Wave solutions of the YM free-field equations, A (0) aµ = X j c aj ε jµ ( p ) e ip · x , (8.26) The de Donder gauge for the graviton and the Lorentz gauge for the tensor field. CLASSICAL DOUBLE COPY with little-group indices j , p · ε j = 0 = q · ε j and c aj color wave functions, can be straightfor-wardly double-copied to wave solutions of the free field equation of the action (2.47). In theabsence of sources this is just a reorganization of the usual double copy of scattering states.The gravity solution, H (0) µν ( x ) = h ij ε iµ ( p ) ε jν ( p ) e ip · x , (8.27)can be decomposed into the graviton, B -field and dilaton using (8.24). The constant factor h ij is arbitrary and can be chosen such that the gravity and gauge-theory asymptotic waveshave the same normalization. It can also be used to project out the B -field and dilaton andobtain a linearized solution of Einstein’s equations, ıe.g. c aj = c a a j , h ij = a i a j , a · a = 0 , (8.28)where a i are “kinematic gauge-theory wave functions”.In general, whether or not the B -field and dilaton can be turned off depends on thegauge-theory sources and on their relation to gravity sources. The relevant solutions of the(8.25), in position and momentum space, for an arbitrary source is A (0) aµ ( x ) ∝ Z d D y ζ aµ ( y ) | x − y | D − , F [ A (0) aµ ]( p ) = F [ ζ aµ ]( p ) p , (8.29)where F is the Fourier-transform operator. The rules for constructing the correspondinglinearized gravity solution and sources are yet to be completely clarified. Here we attemptto formalize several possibilities, while leaving others for future development.In identifying suitable relations between gauge and gravity sources it is important thatthe result can be interpreted as the linearized stress tensor of some field theory and thus thatit conforms with energy conditions expected of such a stress tensor [53]. Not every possibleconstruction has this property; indeed, it was shown in Ref. [53] that, while the source forKerr-Schild solutions (whose linearized approximation is exact and will be discussed in somedetail in Sect. 8.3) can be obtained by specifying the charge distribution sourcing the corre-sponding gauge-theory solutions and imposing ∇ µ T µν = 0, they do not obey simultaneouslythe weak and strong energy conditions. While discussions of energy conditions have appearedin the literature (see below for references), a thorough analysis is currently absent and wewill refrain from attempting one here. We emphasize that the classical double copy is best defined so that it yields a solution of (8.22); moreover, its sources should be constructed outof the gauge-theory sources such that they do not have any unphysical features. In general, itis necessary to verify whether the resulting source obey reasonable energy conditions beforeattempting to promote it from a non-dynamical source to a dynamical one, realized in termsof the fundamental fields of a quantum theory. These requirements may be used to identifysome of the rules of the construction.We begin with a source of the type ζ aµ = c a ζ µ , (8.30)with constant color factor c a and transverse ζ µ . Even though c a need not have any particularalgebraic properties, it is natural to take at face value the fact that the solution (8.29)is given by the first Feynman diagram in Fig. 32 and apply the usual double-copy rules:145 CLASSICAL DOUBLE COPY c a → ˜ ζ µ . Even though nonlinear corrections to a YM solution with this source vanishbecause f abc c b c c = 0, nonlinear corrections to its gravity counterpart may be present; weshall see this explicitly in Sec. 8.2.2. Similarly to scattering-amplitudes double copy, it isvery important to not discard color factors that vanish due to the summation over the colorindices. (Examples where this is crucial are found in Refs. [6, 69].)Gauge-theory sources may exhibit a less transparent separation of color and kinemat-ics, e.g. ζ ( x ) aµ = X i c ai ζ ( x ) iµ , (8.31)with several distinct independent color factors c ai and transverse (position-dependent) ζ iµ .Similarly to the case of asymptotic scattering states, we may still apply the (color factor) → (kinematic factor) replacement in momentum space with the same twist as in that case(and in the case of a wave solution) of allowing for a constant relative rotation of sources.Formally X i c ai F [ ζ iµ ]( p ) −→ X i,j h ij F [ ζ iµ ]( p ) F [ ˜ ζ iµ ]( p ) , (8.32)where F is the Fourier-transform operator. In general it may be possible to allow h ij to bea function of momentum; Lorentz invariance demands that it should be a function of p andthus it can only lead to shifts of H (0) µν by local functions. From this perspective, h ij → h ( p ) ij should be equivalent to field and/or coordinate redefinition in the gravity theory.In both this case and in the simpler previous case (which may be obtained by taking theindices i and j to take a single value), the resulting space linearized solution is H (0) µν ( p ) = P ij h ( p ) ij F [ ζ iµ ]( p ) F [ ˜ ζ jν ]( p ) p . (8.33)Comparing this the general solution of Eq. (8.22) with a source, we identify the numeratoras the Fourier-transform of that source. Transforming back to position space, it follows thatthe gravity source is given by the convolution of the two YM sources with a kernel definedby the matrix h ij : ζ µν ( x ) = Z d D yd D z ˜ h ( | x − y − z | ) ij ζ iµ ( y ) ˜ ζ jµ ( z ) . (8.34)Such a relation between gauge and gravity sources was discussed in Ref. [54] and is reminis-cent of the off-shell definition of the linearized fat graviton in Eq. (4.6).Gauge transformations, whose linearized form is A aµ → A aµ + ∂ µ χ a , can map a solutionsuch as (8.29) into one that has less straightforward identification of a momentum space“kinematic numerator”. To explore this possibility let us assume that A aµ ( x ) has the generalform A (0) aµ ( x ) = 1 x X i c ai n iµ ( x ) , (8.35) Time-independent vector potentials, of the form A aµ ( ~x ) = P i c ai µ i ( ~x ) / | ~x | , can be treated similarly. Theapparent difference in the engineering dimension between the expression of A aµ ( ~x ) here and that in Eq. (8.35)stems from the difference in the dimension of the measure of the three-dimensional and four-dimensional(inverse) Fourier-transform operator. CLASSICAL DOUBLE COPY where n ( x ) may contain terms that either eliminate the overall factor or introduce strongersingularities. The Fourier transform of this vector potential can be defined formally as F [ A (0) aµ ]( p ) = X i c ai ˆ n iµ ( i∂/∂p ) F (cid:20) x (cid:21) ( p ) , (8.36)where the operators ˆ n iµ ( i∂/∂p ) are obtained from n iµ ( x ) by the formal replacement x µ i∂/∂p µ . This operation is to be understood in the sense of distributions, i.e. the Fouriertransform is taken in the presence of a test function that falls off sufficiently fast so integrationby parts does not yield any boundary terms. Interpreting the operators n iµ ( i∂/∂p ) as thekinematic numerators, the linearized double copy may be defined as F [ H (0) µν ]( p ) = X ij h ij ˆ n iµ ( i∂/∂p ) ˆ˜ n jν ( i∂/∂p ) F (cid:20) x (cid:21) ( p ) . (8.37)The commutation properties of the operators ˆ n and ˆ˜ n together with the properties of h ij determine whether or not this double copy yields a purely gravitational solution or thesolution also contains nontrivial dilaton and/or anti-symmetric tensor. Fourier-transformingback to position space for a constant matrix h ij suggests a (linearized) gravitational source(in de Donder gauge) ζ µν ( x ) = X ij h ij n i ( x )˜ n j ( x ) . (8.38)See Ref. [54] for a further discussion on the relation of gauge and gravity sources in atime-dependent setting and Ref. [63] for examples where symmetries help identify the ap-propriate sources. The above construction is related to the position-space replacement rulesof Refs. [51, 63].A non-dynamical source can also be interpreted in the spirit of a (spontaneous) break-ing of the gauge group and thus apply the corresponding double-copy rules discussed inSec. 5 together with the fact that the linearized solution (8.29) is given by the first Feyn-man diagram in Fig. 32. That is, the source is decomposed in irreducible representations ofthe unbroken (global part of the) gauge group and the double copy amounts to construct-ing gauge-invariant bilinears. This interpretation should also be subject to the consistencyconditions discussed in Sec. 5 regarding the spectrum of the double-copy theory.Ultimately gravitational sources should be dynamical (we shall review this in Sec. 8.4); asa step in this direction while eschewing the full dynamics of matter fields one may demand,as was done in Ref. [57], that the gauge-theory source obeys covariant current conservation, D µ ζ aµ = 0 . (8.39)Imposing it anticipates that ζ aµ can be realized in terms of some other fields, in a gaugeinvariant Lagrangian without settling on a specific realization.The previous discussion and examples above refer to cases in which the sources of atleast one of the two gauge theories are smooth functions, perhaps with compact support. Ifboth momentum-space sources contain singular distributions their product requires a careful This construction may in principle be generalized to ˆ n = ˆ n ( p, i∂/∂p ). We leave this to the readers whoread this footnote. CLASSICAL DOUBLE COPY definition, especially if their product is ill-defined, such as a product of Dirac δ -functions.A physical perspective together with the expectation that there exists a Lagrangian thatmanifests the double-copy properties of Eq. (2.47) suggests a natural prescription. Becausethe momenta of the two gauge theories are identified through the double copy, it is naturalthat constraints on it be imposed only once. Thus, if overlapping constraints are imposed bythe gauge-theory sources, they should be included only once in the double copy of the source.It is perhaps interesting that this prescription yields identical (linearized) gravity solutionsfrom distinct (linearized) gauge-theory solutions—e.g. two point-like sources present for alltimes vs. one point-like source present for all time and one instantaneous source.To illustrate this, let us consider the field of a static point-like charge. The four-currentis proportional to u = (1 , , , 0) and the vector potential is [58] A (0) aµ ( x ) = gc a u µ πr , A (0) aµ ( p ) = gc a u µ δ (1) ( p ) p . (8.40)Consequently, H (0) µν is H (0) µν ( p ) = κ M u µ u ν δ (1) ( p ) p , (8.41)which can be easily Fourier-transformed to position space. In writing this expression wemade certain identifications between the gauge coupling and constants in the gravity theory.Since H (0) µν is symmetric, b µν = 0; it is not traceless, so there is a nontrivial dilaton φ = H (0) µµ = + κ M πr . (8.42)Using Eq. (8.24) and the projector (8.23), the correction to the metric is h µν = κ M πr (cid:18) u µ u ν + 12 ( η µν − q µ l ν − q ν l µ ) (cid:19) , with l = 1 r + z (0 , x, y, r + z ) . (8.43)Running a similar construction in the opposite direction, shock-wave solutions of Ein-stein’s equations which are also solutions of linearized Einstein’s equations were shown inRef. [50] to be related, through a double-copy procedure, to certain wave solutions of YMtheory. The relevant gravitational source ζ µν is identified such that the scattering of someparticle off a high-energy graviton is equivalent to all orders in perturbation theory to thescattering off ζ µν ; the gravitational shock wave, given by the Aichelberg-Sexl [440], is thesolution of (linearized) Einstein’s equation with this source. The corresponding gauge-theorysource ζ aµ was similarly constructed, i.e. such that the scattering of some particle off a highenergy gluon is equivalent to all orders in perturbation theory to the scattering off ζ aµ . Thesource turned out to be of the type (8.30) and the gauge-theory shock wave is the solu-tion of (linearized) YM equations with this source. The two waves are related by the usualcolor → kinematics replacement. By construction, scattering off the gravitational wave canalso be obtained though this replacement from scattering off the gauge-theory wave, to allorders in perturbation theory. 148 CLASSICAL DOUBLE COPY With a linearized solution in hand, nonlinear corrections can be computed directly, byevaluating increasingly higher orders in the tree expansion in Fig. 32. The goal however itto explore the realization of nonlinear corrections to the gravity solutions as a double copyof the nonlinear corrections to the YM solutions. We will review this here, loosely followingRef. [58]. As we shall see, this comparison will emphasize the importance of the choice offields, a feature that will be further discussed for complete solutions.Nonlinear corrections to a linearized solution are expressed, through the tree expansionin Fig. 32, as convolutions of the linearized solution with kernels given by Feynman vertices.Since however, the gravity source depends on the metric it sources, one may either includeexplicitly such modifications (as O ( κ n ≥ ) corrections to the source) or ignore them andobtain a solution for a choice of fields such that source changes are absent. These twoperspectives have an analog in the two YM theories, where sources may either be correctedorder by order in perturbation around the trivial solution such that they are e.g. covariantlyconstant, D · ζ = 0, or they are fixed, respectively.The first nonlinear correction to some solution A (0) cµ follows easily in terms of the standardthree-point vertex. To utilize the same rules as for amplitudes double copy, it is convenientto present it in momentum space: A (1) aµ ( − p ) = i p f abc Z d D p (2 π ) D d D p (2 π ) D (2 π ) D δ ( p + p + p ) × h ( p − p ) γ η µβ + ( p − p ) µ η βγ + ( p − p ) β η γµ i A (0) bβ ( p ) A (0) cγ ( p ) . (8.44)The factor in the square parenthesis is the usual kinematic part of the off-shell three-gluonvertex and has the same antisymmetry properties as the color factor. While this expressionmay be simplified somewhat by making use of the transversality of A (0) aµ , we will choose notto do so.Taking two configurations like (8.44) and replacing the color factors of one with thekinematics of the other while leaving the propagators untouched (which, apart form usingthe same double-copy rules for amplitudes also includes the application of the results of theprevious subsection A (0) aµ ( p ) ˜ A (0) bν ( p ) → H (0) µν ( p )) leads to H (1) µµ ( − p ) = 14 p Z d D p (2 π ) D d D p (2 π ) D (2 π ) D δ ( p + p + p ) × h ( p − p ) γ η µβ + ( p − p ) µ η βγ + ( p − p ) β η γµ i (8.45) × h ( p − p ) γ η µ β + ( p − p ) µ η β γ + ( p − p ) β η γ µ i H (0) ββ ( p ) H (0) γγ ( p ) . This expression has the same structure as the first nonlinear correction to the solutions of theequations of motion of the action (2.47) except that the trilinear interaction of gravitons, B fields and dilatons was replaced by the factorized integrand kernel above. This factorizationis the same as that of the three-point amplitudes from Eq. (2.47). It can be seen explicitlyby starting from the complete three-point vertices and using transversality and the on-shellcondition for the external states. While H (0) γγ is transverse by construction, it obeys, in149 CLASSICAL DOUBLE COPY general, a free-field equation with a source. Thus, H (1) µµ ( − p ) given above represents thefirst correction to a gravity solution for the choice of a fluctuations such that the trilinearvertex is free of terms that vanish on the free mass shell. This vertex is related to the onefollowing from the expansion of the Lagrangian by a field redefinition.There exists further freedom in the relation between H (1) µµ ( − p ) and the fluctuations ofthe metric, B field and dilaton. At the linearized level the later are given by the decompo-sition (8.21). For higher-order corrections however this decomposition may be modified. Asdiscussed in the beginning of this section, the amplitudes double copy guarantees only thatthe asymptotic states—or linearized solutions—double copy. At higher orders in κ theremay exists further terms in the relation between gauge-theory and gravity fields which areprojected out when the LSZ reduction is applied to a Green’s function. At the first nonlinearorder this is H (1) µν = h (1) µν + B (1) µν + P qµν φ (1) + T (1) µν ( h (0) , b (0) , φ (0) ) , (8.46)and at arbitrary order H ( n ) µν = h ( n ) µν + B ( n ) µν + P qµν φ ( n ) + T ( n ) µν ( h ( m ) , b ( m ) , φ ( m ) , m < n ) . (8.47)Such terms may be interpreted as field redefinitions connecting the initial choice of gravityfields (8.19) to the ones “chosen” by the double copy. They also capture various choicesthat can be made during the calculation, such as gauge choices and—highlighted by theirappearance in the first nonlinear correction—use of the free/lower order equations of motionin the definition of vertices. Terms of this type may be eliminated by nontrivial choicesof the kernel in Eq. (8.33). These “transformation functions” [58] may be determined bycomparing the perturbative solution of the equations of motion of the action (2.47) with theresult of the double copy. The main physical information they contain is that they providethe connection between the fields natural from a double-copy perspective and the naturalfluctuations in the gravity Lagrangian. In the special case of the self-dual theory, it is knownhow to choose a parametrization of the metric perturbation such that the double copy ismanifest [43]. For these field variables T µν = 0 to all orders in the tree diagram expansionof self-dual spacetimes.An example illustrating this discussion and dramatically emphasizing the relevance ofthe choice of field variables was given in Ref. [58] using the linearized gravity solution inEq. (8.41) and its gauge-theory counterpart in Eq. (8.40). This example also emphasizesthe importance of not dropping terms whose color factors vanish after summation over colorindices. The first nonlinear correction H (1) to Eq. (8.41) was obtained in Ref. [58]; it is H (1) µν ( x ) = − (cid:18) κ (cid:19) M πr ) ˆ r µ ˆ r ν , (8.48)where ˆ r µ = (0 , x /r ). It turns out that a nontrivial transformation function is necessary toturn H = η + κH (0) + κ H (1) into a solution of the equations of motion to O ( κ ) in the150 CLASSICAL DOUBLE COPY variables (8.19). It is given by [58] T (1) µν ( − p ) = Z d D p (2 π ) D d D p (2 π ) D (2 π ) D δ ( p + p + p ) 14 p ( H (0)2 αβ H (0) αβ p µ p ν + 8 p α H (0)3 αβ H (0) β ( µ p ν )1 + 8 p · p H (0) µα H (0) ν α − η µν p · p H (0)2 αβ H (0) αβ + 4 η µν p α H (0)3 αβ H (0) βγ p γ + P µνq h D − p · p H (0)2 αβ H (0) αβ − D − p α H (0)3 αβ H (0) βγ p γ i) , (8.49)where we used the shorthand notation H (0) i µν ≡ H (0) µν ( p i ) , and p ( µ q ν ) ≡ 12 ( p µ q ν + p ν q µ ) . (8.50)This first transformation function T (1) µν holds for all cases that have symmetric and trans-verse H (0) µν and h (0) µν .Since Eq. (8.40) is an exact solution of the YM equations of motion, one may wonderwhether it is possible that it has some other, physically equivalent form which can be double-copied to an exact solution of dilaton-axion-gravity in some field variables. To this end, itis necessary that the first correction to this equivalent form of Eq. (8.40) vanishes beforesummation over color indices. We shall see in Sec. 8.3 that this is indeed possible.Proceeding to higher orders is in principle straightforward, but quite tedious in practice.The new features compared to the discussion above relates to the need of a representationof the corrections to the YM equations which manifest CK duality up to terms that areprojected out by the LSZ reduction. Since the only difference between the asymptotic statesof scattering amplitudes and A a (0) µ is that the latter obey an on-shell condition with sources,CK duality can be satisfied only up to such terms. Similarly to scattering amplitudes,a generic perturbative classical solution is related to one that exhibits the duality (in thisrestricted sense) by generalized gauge transformations. As in that case, such transformationsare not always easy to find. As in that case, a Lagrangian whose Feynman rules lead tomanifestly CK-dual representation or the use of the generalized double-copy constructioncan alleviate this issue.To quintic order in fields, the Lagrangian in Ref. [41] provides the requisite Feynmanrules to obtain the gauge-theory perturbative classical solution in a form that can be doublecopied directly. This was exploited in Ref. [58], where the second nonlinear correction wasdiscussed. As explained there, the quartic YM vertex does not contribute to a symmetricdouble copy and the second term in the perturbative solution of YM equations is givenentirely in terms of the three-point vertex: A (2) aµ ( − p ) = ip f abc Z d D p (2 π ) D d D p (2 π ) D (2 π ) D δ ( p + p + p ) (8.51) × h ( p − p ) γ η µβ + ( p − p ) µ η βγ + ( p − p ) β η γµ i A (0) bβ ( p ) A (1) cγ ( p ) . CLASSICAL DOUBLE COPY It leads to the second correction H (2) in the gravitational solution H (2) µµ ( − p ) = 12 p Z d D p (2 π ) D d D p (2 π ) D (2 π ) D δ ( p + p + p ) × h ( p − p ) γ η µβ + ( p − p ) µ η βγ + ( p − p ) β η γµ i (8.52) × h ( p − p ) γ η µ β + ( p − p ) µ η β γ + ( p − p ) β η γ µ i H (0) ββ ( p ) H (1) γγ ( p ) . The graviton, antisymmetric tensor and dilaton components can be easily extracted usingthe projectors; to connect this general expression to a solution with specific sources in specificcoordinates T (2) must be computed as well. We refer to Ref. [58] for details.Exercise 8.1: Explore the possibility of using a quasi-classical solution obtained by foldingscattering amplitudes in BCJ representation against external sources to construct solutionsfor the gravity field equations. This is equivalent to removing terms proportional to thefree-field equations from Green’s functions and using the result to construct an ansatz for aclassical solution. The resulting double-copy field configuration should be correct—for somechoice of field variables—up to terms that are proportional to the free field equations, i.e.up to field redefinitions.Steps towards the double copy of nonlinear classical solutions beyond second order weretaken in Refs. [172, 435] for the special case of perturbiners or Berends-Giele currents. Start-ing from the perturbiners of certain effective field theories [435] and F and F -deformedYM theory, perturbiners of the corresponding gravity theories were constructed using theKLT relations. Because only one leg of the Berends-Giele current is off shell, the relationbetween the objects thus constructed and the “true” gravitational perturbiner is simplerthan in the most general case: it consists only of a gauge transformation and involves nofield redefinition.The need for a Lagrangian yielding CK-satisfying Feynman rules or, more generally, ofGreen’s functions manifesting CK duality on all of their internal lines may be circumventedthrough the generalized double-copy construction discussed in Sec. 7. Generalizing slightlyto Green’s functions, the starting point is any general perturbative expressions for the gauge-theory solutions expressed in terms of cubic diagrams; quartic vertices, if present, are resolvedin the usual way. Because of lack of manifest CK duality, their double copy does not yieldsolutions of the equations of (2.47) up to field redefinitions. The formulae discussed inSec. 7 provide the correction terms. As in the examples discussed earlier in this section,transformation functions are probably necessary to relate the result of the generalized doublecopy to a solution in some chosen coordinates. It remains an open problem to have an a prioriunderstanding of the choice of fields in the gravitational theory that set all transformationfunctions to zero. In the discussion of perturbative construction of gravity solutions in Sec. 8.2 we encountered,following Ref. [58], linearized solutions which are exact solutions of YM equations—such as152 CLASSICAL DOUBLE COPY that in Eq. (8.40)—which double copy to linearized solutions of gravity which receive higher-order corrections. While, as emphasized there, this can be understood as a consequence ofthe special properties of the color factors of the YM solution, it is important to understandwhether there exists a choice of field variables for which these contributions to not arise atall and consequently the transformation functions vanish identically to all orders in classicalperturbation theory. The general expectation is that if a gauge-theory solution does notreceive corrections beyond n -th order in perturbation theory, then its corresponding gravitysolution will also be exact beyond that order.As pointed out in Ref. [51], following Ref. [441], a particular ansatz for the metric lin-earizes the source-free Einstein’s equations and thus can potentially give these metrics asdouble copies of solutions of YM equations which do not receive nonlinear corrections. Theyare know as Kerr-Schild metrics; the ansatz is given in terms of a scalar function φ (whichis not the dilaton) and a vector k which is null and geodesic with respect to the backgroundmetric ¯ g µν : g µν = ¯ g µν + κh µν ≡ ¯ g µν + κ φ k µ k ν , ¯ g µν k µ k ν = 0 , ( k · ¯ ∇ ) k µ = 0 . (8.53)The background (or fiducial) metric ¯ g µν is also used to raise and lower indices on the metricfluctuation h and ¯ ∇ µ is the corresponding background-covariant derivative. One componentof k can be set to unity, thus absorbing its dynamics in φ . The Kerr-Schild form is special inthat the metric perturbation—or the graviton—explicitly decomposes into a direct productof the vector k µ with itself. The remarkable property of this ansatz is that it linearizesthe Ricci tensor and reduces Einstein’s equations to a single nontrivial relation between thefunction φ and the source. The components of the Ricci tensor are R µν = ¯ R µν + κ (cid:20) − h µρ ¯ R ρν + 12 ¯ ∇ ρ (cid:16) ¯ ∇ ν h µρ + ¯ ∇ µ h ρν − ¯ ∇ ρ h µν (cid:17)(cid:21) , (8.54)where ¯ R µν is the Ricci tensor associated with the background metric ¯ g µν . We emphasizethat the linear dependence on the metric fluctuation h in Eq. (8.53) holds only for the indexpositions in Eq. (8.54).A simple choice of background metric is ¯ g µν = η µν (with a mostly-minus signature), usedat length in this context in Ref. [51]. For this choice the background-covariant derivativesbecome regular derivatives. Further choosing k = 1, the components of the Ricci tensor are R = 12 ∂ i ∂ i φ ,R i = − ∂ j h ∂ i (cid:16) φk j (cid:17) − ∂ j (cid:16) φk i (cid:17)i ,R ij = 12 ∂ l h ∂ i (cid:16) φk l k j (cid:17) + ∂ j (cid:16) φk l k i (cid:17) − ∂ l (cid:16) φk i k j (cid:17)i ,R = ∂ i ∂ j (cid:16) φk i k j (cid:17) . (8.55)All Latin indices run over the space-like directions. Thus, the scalar function φ is determinedby a Poisson-type equation.A generalization of the Kerr-Schild ansatz in Eq. (8.53) is the double-Kerr-Schild ansatz [442],which is given in terms of two scalar functions and two null, geodesic and mutually orthogonal153 CLASSICAL DOUBLE COPY vectors: g µν = ¯ g µν + κh µν = ¯ g µν + κ ( φ k µ k ν + ψ l µ l ν ) , (8.56) k = l = k · l = 0 , ( k · ¯ ∇ ) k µ = 0 , ( l · ¯ ∇ ) l µ = 0 , where as before ¯ g µν is a background metric and is used in all index contractions andbackground-covariant derivatives ¯ ∇ . For these field variables the Ricci tensor is R µν = ¯ R µν + κ (cid:20) − h µρ ¯ R ρν + 12 ¯ ∇ ρ (cid:16) ¯ ∇ ν h µρ + ¯ ∇ µ h ρν − ¯ ∇ ρ h µν (cid:17)(cid:21) + R µν, non − lin . , (8.57) R µν, non − lin . = − κ (cid:20) 12 ¯ ∇ µ h ( k ) ρδ ¯ ∇ ν h ( l ) δρ + h ( l ) µδ ¯ ∇ ρ ¯ ∇ ν h ( k ) ρδ + ¯ ∇ ρ (cid:16) h ( l ) ρδ ¯ ∇ δ h ( k ) µν + 2 h ( l ) ρδ ¯ ∇ ( ν h ( k ) µ ) δ − h ( l ) µδ ¯ ∇ [ ρ h ( k ) δ ] ν (cid:17)(cid:21) + ( k ↔ l ) , (8.58)where h ( k ) µν = φk µ k ν , h ( l ) µν = ψl µ l ν . (8.59)The linearity of Einstein’s equations in Kerr-Schild variables implies that any single Kerr-Schild metric can also be thought of as a double Kerr-Schild metric.In higher dimensions further generalizations are possible, involving up to D − k and l . Additionally,it was argued in Ref. [442] that in the so-called Plebansky coordinates, the nonlinear partof the Ricci tensor, R µν, non − lin . , vanishes identically and solutions of the linearized Einstein’sequations are also exact solutions. In this section we shall review the double-copy interpretation of the Schwarzschild solution,emphasizing its realization vis-à-vis the discussion in the previous section. We will thensummarize and comment on generalizations of this approach to other spacetimes.The Kerr-Schild form of the Schwarzschild solution is (see e.g. Ref. [443]) g µν = η µν + κ π Mr k µ k ν , (8.60)where M is positive and the null four-vector k is chosen such that the line element is rota-tionally invariant: k µ = (1 , x i /r ) , r = X i =1 x i x i . (8.61)The double-copy form of this solution was discussed at length in Ref. [51]. With the def-inition of the linearized double copy for singular sources discussed in Sec. 8.2.1 and up toidentification of parameters, it can be seen that the departure of the metric (8.60) fromMinkowski space is given by the double copy of A aµ = gc a k µ πr . (8.62)154 CLASSICAL DOUBLE COPY It can be straightforwardly verified that A aµ satisfies Maxwell’s equations with a point-likesource at the origin. Eq. (8.62), however, is not the standard potential of such a source.Rather, as shown in Ref. [51], it is related to the standard potential of a point-like charge(8.62) by a gauge transformation with parameter Λ a : A aµ = A aµ + ∂ µ Λ a ,A aµ = gc a u µ πr , u µ = (1 , , , , Λ a = gc a π log r , (8.63) j aµ = − gc a u µ δ (3) ( x ) . It is interesting to contrast the two gauge-equivalent vector potentials A aµ and A aµ . Bothare proportional to a single color vector c a and because of this they both are formally exactsolutions of the nonlinear YM equations. For the latter one, A , the vanishing color factorsof the corrections are multiplied by nontrivial kinematic dependence and thus, as discussedin Sec. 8.2.2, there are nonlinear corrections that should be included which are crucial fortransforming its double copy into a solution of the full Einstein’s equations. For the former, A , one can check that the vanishing color factors come together with vanishing kinematicdependence. Therefore, the corrections to the double copy of two A vectors also vanish andthus the linearized double copy does not receive nonlinear corrections. This underscores theimportance of the gauge choice for the gauge-theory solutions that participate in the classicaldouble copy.To recover the Schwarzschild solution the parameters of the two theories are replaced as κ ↔ g , M ↔ | c | . (8.64)We note that the norm of the color vector c a , which may be identified as the charge of thesource under the sole Cartan generator of the gauge group that is nontrivial, correspondsto the mass of the Schwarzschild black hole. This seems to suggest a relation between theuniqueness of the Coulomb-like solution and Birkhoff’s theorem.It is interesting and important to note that, despite the gauge-theory solutions beingsourced by the same charge distributions and being gauge-equivalent, their classical doublecopies as defined here are inequivalent. Indeed, while the solution constructed in this sectionhas only a nontrivial metric, the one constructed perturbatively in Sec. 8.2.2 stating fromEq. (8.40) also has a nontrivial dilaton which cannot be removed while preserving a nontrivialmetric. With the current understanding of the classical double copy, the fact that gauge-equivalent gauge-field configurations lead to inequivalent gravitational field configurationsappears to be an unavoidable feature. At this juncture, it seems best to start with validgravitational solutions and work backwards to gauge theory.Considerations similar to the ones outlined above have been used to give a double-copyinterpretation to the Kerr back hole, black brane solutions, shock-wave and plane-wave solu-tions [51] and to the (anti) de Sitter spaces in Ref. [52]. In the latter cases the cosmologicalconstant is related to the charge density of a uniform charge distribution. Gravity solu-tions with additional matter fields turned on, such as the Taub-NUT space, have a doubleKerr-Schild form and, as argued in Ref. [52], have a double-copy interpretation (in the samesense as discussed above) in terms of a dyon solution whose electric and magnetic chargesare related to the mass and the NUT charge.155 CLASSICAL DOUBLE COPY Kerr-Schild solutions with time-dependent sources, describing accelerating black holes,have been discussed in Ref. [54] where a relation was constructed between the electromagneticradiation of an accelerating charge and the gravitational radiation of an accelerating pointmass and thus represents an effective description of the complete vacuum solution. Thecontraction of the corresponding sources with gluon and graviton polarization vector/tensorgives the amplitude for the Bremsstrahlung process. Other gravitational wave solutions,including vacuum solutions, which are of the double-Kerr-Schild type, were discussed inRef. [74].All YM solutions that appeared in the constructions reviewed here are also solutions ofMaxwell’s equations. Using the fact, discussed in Sec. 5, that YM theory can be interpretedas a double copy of itself with a theory of a bi-adjoint scalar field, more complicated solutionscan be constructed by taking the double copy of e.g. a Maxwell solution with a solution ofthe bi-adjoint scalar theory. This observation was explored in Refs. [52, 62], while solutionsof bi-adjoint scalar theory were constructed in Refs. [55, 73] and [61]. The details pertainingto the relation between the sources of various solutions remain to be fully worked out. Thisperspective also makes contact with the off-shell Lagrangian double copy of Refs. [56, 255].Following Ref. [53], the gravitational stress tensor of certain double-copy Kerr-Schild so-lutions was expressed linearly in terms of the current sourcing the gauge-theory solution.With this relation, in most cases they are not stress-energy tensor of a perfect fluid andcontains shear stresses and, moreover, they do not obey the weak-energy condition. It ispossible that other choices of coordinates and field variables display double-copy behaviorthat simultaneously map YM solutions to gravitational ones and satisfy the energy condi-tions.Further generalizations, involving a nontrivial fiducial metric ¯ g in the Kerr-Schild ansatz,were discussed in Refs. [52, 62, 63]. As discussed in Refs. [52, 62], if the fiducial metric isof Kerr-Schild type, then every such solution can also be interpreted as a (multiple) Kerr-Schild metric with Minkowski space as fiducial metric (referred to as Type A constructions inRef. [62]). Among the examples discussed are the de Sitter and anti de Sitter generalizationsof the Schwarzschild black hole.Solutions with a non-Kerr-Schild background metric have been discussed in Ref. [62](referred to there as Type B constructions) and in Ref. [63]. They are realized in terms ofsolutions of gauge theory on a space with the fiducial metric. The classical scale invariance ofYM theories implies that, for a fiducial metric is conformally Minkowski, the gauge theoryis effectively in flat space (up to a curvature-dependent scalar mass term). Examples ofthis type were discussed in Ref. [62]. Apart from black holes in asymptotically maximally-symmetric spaces which are also treated in this framework, Ref. [63] also gives double-copyinterpretations to black strings, black branes, and various types of gravitational waves. Thecorresponding localized sources for the YM and scalar theories, for both stationary andtime-dependent examples, are also identified and examples are given in terms of Kerr-Schildvectors k .While a coherent picture for the classical double copy of exact gauge-theory solutionsto exact (matter-coupled) gravity solutions is still to be formulated, the examples discussedin the literature and summarized here give hope that such a relation may be genericallypossible. 156 CLASSICAL DOUBLE COPY To further illustrate the importance of the choice of field variables for the interpretation ofthe result of classical double-copy constructions as exact solutions of Einstein’s equations(perhaps coupled to additional matter) let us briefly discuss the charged black hole solutionin the presence of an additional scalar field. (See also Ref. [62] for a discussion of chargedblack holes). The equations of motion are standard R µν − g µν R = 12 φ ( g αβ F µα F νβ − g µν F · F ) + 1 φ ( ∇ µ ∇ ν φ − g µν ∇ · ∇ φ ) , ∇ · ∇ φ = 14 φ F · F , ∇ α F αµ = − ∇ α φφ F αµ , (8.65)and can be obtained by Kaluza-Klein reduction from Einstein’s equations in five dimensionsthrough the usual ansatz g = g + φ A ⊗ A φ Aφ A φ ! . (8.66)As we shall see, in these field variables the charged black hole solution does not a clear classi-cal double-copy interpretation; we will identify the field variables in which it does, parallelingthe smooth relation of double copy between theories related by dimensional reduction.The four-dimensional charged black hole can be obtained via Kaluza-Klein reductionfrom a five-dimensional black string. In Kerr-Schild form, it is g = η + ϕ ˆ k ⊗ ˆ k , η µν ˆ k µ ˆ k ν = 0 , ϕ = Mr , (8.67)where r is the radial coordinate in the three coordinates transverse to the string. A suitablesolution of the constraints constants defining ˆ k is that it is a boost of the vector (1 , ˆ r , r is the unit vector in three dimensions orthogonal to the string:ˆ k = ( γ, ˆ r , βγ ) ≡ ( k, βγ ) , γ = 11 − β . (8.68)Using the reduction ansatz in Eq. (8.66), the four-dimensional fields are: g = η + ϕ β γ ϕ k ⊗ k , k = ( γ, ˆ r ) , k = 1 − γ = − β − β ,φ = q β γ ϕ , A = βγ ϕ β γ ϕ k . (8.69)It is not difficult to check that this field configuration is a solution of Eqs. (8.65).It is also not difficult to see that this field configuration departs from the Kerr-Schildansatz in that the vector k defining the departure of the metric from Minkowski space is The corresponding action is in the string frame, and may be mapped to the Einstein frame by a rescalingof the five-dimensional metric. CLASSICAL DOUBLE COPY time-like rather than null. Moreover, the dependence on ϕ suggests that all fields are givenby a nontrivial resummation of tree diagrams.Another choice of field variables, g = ˜ g ˜ A ˜ A φ ! , (8.70)which is closely related to the dimensional reduction of asymptotic states of scatteringamplitudes, is more suitable for a classical double-copy interpretation. Indeed, the four-dimensional fields are˜ g = η + ϕk ⊗ k , k = ( γ, ˆ r ) , k = 1 − γ = − β − β , ˜ φ = β γ ϕ , ˜ A = ϕk , (8.71)which are related in the sense described in Sec. 8.2.1 to the following solution of gauge theorycoupled to a scalar field: A a YM = ϕk c a , φ a YM = N ϕ c a , (8.72)where c a is some color vector.We note that the field configuration (8.71) is not a solution of Eq. (8.65), but it is asolutions of the equations obtained from them through the field redefinition mapping thefields in Eq. (8.66) to those in Eq. (8.70). Moreover, while structurally similar to the Kerr-Schild ansatz, it is not of the same type because the vector k is time-like. The violationof the null condition is compensated by the contribution of the vector and scalar fields.While the relation between (8.72) and (8.71) is linear, the fact that k is not null allowsin principle for a nonvanishing kinematic part in the nonlinear corrections to (8.72) andthus to potential nonlinear corrections to their double copy, cf. Sec. 8.2.2. The fact that(8.71) is an exact solution suggests absence of the nonlinear corrections to the scalar andvector fields in Eq. (8.72). These features may allow further generalization of the classicaldouble-copy interpretation of solutions of Kerr-Schild type. An alternative constructionof the charged dilatonic black hole solution discussed here, which uses the standard four-dimensional equations of motion (8.65) and a generalization of the double-Kerr-Schild ansatz(8.56) which also includes certain internal dimensions, was discussed in [444].Exercise 8.2: Show that the first nonlinear correction to (8.71) vanishes by evaluating it interms of the kinematic factors of the corrections to the gauge-theory solution (8.72). Earlier in this section we have reviewed and illustrated various possible definitions of thedouble copy of classical solutions gauge theories to solutions of Einstein’s equations coupledperhaps with additional matter and summarized the existing results. One of the fundamentalresults of general relativity (and, in fact, of any gravity theory) is the emission of gravitationalwaves—classical gravitational radiation emitted in processes involving massive astrophysical158 CLASSICAL DOUBLE COPY bodies such as neutron stars or black holes, perhaps with macroscopic intrinsic angularmomentum.From the perspective of general relativity such calculations have a long history, withnumerical and perturbative results in various approximation schemes, which we will notreview here; see Refs. [129, 130, 445] for reviews. They have been stunningly confirmedthrough the direct experimental detection of gravitational waves by the LIGO and Virgocollaborations [446]. We expect that the double-copy approach to such calculations will leadto important technical simplifications and bring new insight into these problems.The first nontrivial contribution to the radiation process involves five particles: the twoincoming and outgoing massive bodies and the outgoing graviton. To evaluate this, it isnecessary to fix a model for the massive bodies that can be included in the double-copyconstruction. In Ref. [57] they were represented in terms of gauge fields, effectively as thelinearized solutions in Eq. (8.40). The classical double copy then (effectively) yields a gravitysolution whose linearized form is (8.41) and thus the massive objects being scattered sourceboth gravitons and dilatons. It moreover appears that the double-copy rules used in Ref. [57]assume that a Lagrangian that manifests CK duality is available. Indeed, the gauge-groupgenerators are replaced with the kinematic dependence of the off-shell three-point vertex,thus assuming that the latter have the same algebraic properties as the former. While fora Lagrangian that manifests CK duality this replacement is, of course, equivalent to theusual rules in Sec. 2, for a general Lagrangian, however, further terms may be necessary.Nevertheless, the result reproduces direct calculations [57, 59] in dilaton-coupled gravity.Similar techniques have been used to obtain the corresponding results in Einstein-Yang-Mills theory [447].In a different approach, suggested in Ref. [425] based on earlier ideas of Refs. [448, 449],incoming and outgoing spinless massive bodies are represented as double copies of minimally-coupled massive scalar Φ. The Lagrangian is L = − 12 Tr F µν F µν + X i h ( D µ Φ i ) † ( D µ Φ i ) − m i | Φ | i , (8.73)with the scalar field in some (complex) representation of the gauge group and D µ the cor-responding covariant derivative. Then, a certain classical limit is taken to ensure that, asfor classical particles, masses are parametrically larger than their spatial momenta. In thisapproach one can choose the couplings of these particles such that, on the one hand, CKduality is present and on the other their double copy does not yield a dilaton source. Dueto Birkhoff’s theorem, this model is sufficient describe the gravitational wave emission farfrom the horizon of black holes, where the large (classical) masses ensures that the linearizedemission is captured accurately. We outline the relevant calculation, following Ref. [425].The five diagrams contributing to the scattering process Φ i Φ j → Φ i Φ j h µν are shown inFig. 33 and the corresponding amplitude is A = − i (cid:18) n a c a D a + n b c b D b + n c c c D c + n d c d D d + n e c e D e (cid:19) . (8.74)The denominators D a , . . . , D e and the color factors c a , . . . , c e are easily read from the dia-grams in Fig. 33, taking into account that the scalar Φ i is in some complex representation159 CLASSICAL DOUBLE COPY p p kp − q p − q (a) p p kp − q p − q (b) p p kp − q p − q (c) p p kp − q p − q (d) p p kp − q p − q (e)Figure 33: The five cubic diagrams for inelastic scalar scattering with gluon production in gaugetheory. The legs carrying momenta p and p are incoming and the remaining ones are outgoing. R i with generators T aR i . The kinematic numerators follow from the Feynman rules of theLagrangian (8.73). They are: n a = (2 p + q ) · (2 p − q ) ε · (2 p + 2 q ) − (2 p · q + q ) ε · (2 p − q ) ,n b = (2 p − k − q ) · (2 p − q ) 2 ε · p + 2 p · k ε · (2 p − q ) ,n c = (2 p − q ) µ (2 p − q ) ρ [( k + q ) µ η νρ + ( q − q ) ν η ρµ − ( k + q ) ρ η µν ] ε ν , (8.75) n d = (2 p − q ) · (2 p + q ) ε · (2 p + 2 q ) − (2 p · q + q ) ε · (2 p − q ) ,n e = (2 p − q ) · (2 p − k − q ) 2 ε · p + 2 p · k ε · (2 p − q ) , where ε is the gluon polarization vector. The color identities that are important for thegauge invariance of A in Eq. (8.74) are c a − c b = c c c d − c e = c c . (8.76)It can be easily checked that the numerators (8.4) obey the corresponding kinematic relations.The double-copy amplitude follows from the usual rules, see Sec. 2: M = − i (cid:18) n a n a D a + n b n b D b + n c n c D c + n d n d D d + n e n e D e (cid:19) . (8.77)The tensor product of the two outgoing gluon polarization vectors can be projected onto agraviton state. For internal lines a more involved projection is necessary [188]. We shallreturn to it shortly.To relate the amplitude just constructed to the classical scattering of massive bodies itis necessary to focus on the classical kinematic regime. There exists many “classical limits” We note that, as discussed in previous sections, a quartic scalar term is not necessary for CK dualitybecause the scalar fields are taken in a complex representation of the gauge group. CLASSICAL DOUBLE COPY of a field theory and all of them involve the limit of vanishing Planck’s constant, which musttherefore be restored (on dimensional grounds) in the field theory expressions. The limitwe are interested in is also the one in which masses and other quantum numbers, such asexternal momenta and charges, are parametrically large compared to the momenta exchangedbetween particles. Thus, the classical limit is equivalent with a large mass expansion [422]: m i → m i (cid:126) , g → g (cid:126) , (cid:126) → , p µi → m i v µi , v i = 1 . (8.78)Because the coupling (charges) and masses are scaled simultaneously, this limit makes partsof tree-level and loop-level diagrams of the same order and consequently all such contributionsenter nontrivially in this classical limit [422].The limit (8.78) must be taken while enforcing the exact on-shell condition for all externalparticles. In particular( p i − q i ) = m i − m i v i · q i + q i = m i ⇒ m i v i · q i = q i . (8.79)Thus, if external momenta are parametrically larger than the exchanged ones, this equationcan be satisfied only if v i · q i ∼ O ( m − i ) . (8.80)This condition must be enforced when taking the classical limit of Eq. (8.77). Defining thevariables P µ ≡ k · v v µ − k · v v µ ,Q µ ≡ ( q − q ) µ − q k · v v µ + q k · v v µ , (8.81)this classical limit of the amplitude (8.77) is M cl = − im m ε µν " P µ P ν q q + 2 v · v q q ( Q µ P ν + Q ν P µ )+( v · v ) Q µ Q ν q q − P µ P ν ( k · v ) ( k · v ) ! , (8.82)where ε µν ≡ ε µ ε ν . As pointed out in Ref. [425], there is a close relation between thisamplitude and the metric perturbation (i.e. radiation field) constructed in Ref. [57]. Themetric perturbation is given by the Fourier transform to position space of M cl with respect tothe incoming scalar momenta, subject to the constraints imposed by the on-shell conditionsfor all external momenta. We note that the mass of the particles enters only as an overallfactor in the amplitude (8.82) and, consequently, in the associated metric perturbation. Thisproperty, implying that the features of the metric are essentially independent of a (spinless)source, may be interpreted physically as a reflection of Birkhoff’s theorem.To obtain the analogous results in Einstein’s gravity theory it is necessary to projectout the dilaton and antisymmetric tensor field from all diagrams. As reviewed in Sec. 5 Other formulations of the classical limit, leading to the same result but with a different physical reasoning,were discussed in [78, 80, 82]. CLASSICAL DOUBLE COPY following Ref. [188], this can be done by introducing further “ghost” fields whose couplingsare adjusted such that they remove the (un)desired degrees of freedom. For the case athand the relevant Lagrangian is [425] L = − 12 Tr F µν F µν + Tr D µ χD µ χ + X i h ( D µ Φ i ) † D µ Φ i − m i Φ † i Φ i − Xm i Φ † i χ Φ i i , (8.83)where D µ is the gauge-covariant derivative, χ is the adjoint ghost, X is its coupling to bedetermined. The mass factors are included such that the ghost field has canonical dimensionand X is dimensionless. While in the adjoint representation, the ghost field is allowed todouble copy only with itself.The unknown coupling X can be determined by comparing the 2 → X = 1 D − , (8.84)where D is the spacetime dimension. The Lagrangian (8.83) can then be used to evaluate theadditional Feynman diagrams that remove the dilaton and axion contribution to Eq. (8.82).The complete amplitude in scalar-coupled pure gravity is M GR = − im m ε µν " P µ P ν q q + 2 v · v q q ( Q µ P ν + Q ν P µ )+ (cid:18) ( v · v ) − D − (cid:19) Q µ Q ν q q − P µ P ν ( k · v ) ( k · v ) ! , (8.85)where the polarization tensor ε µν now includes only graviton degrees of freedom. As shownin Ref. [425], this reproduces the result of the (far more complicated) direct computation ingeneral relativity coupled to point particles.It is interesting to note that, repeating the calculation in Refs. [57, 59] such that thesources correspond to massive spinless bodies and removing the dilaton and axion contri-bution through a procedure similar to the one described above appears to lead [425] to adifferent result than (8.85). While the origin of the difference is not clear, it is possible thatthey are due to the unusual double-copy rules employed in Refs. [57, 59].Exercise 8.3: Evaluate the amplitude for the graviton production in the scattering of massivecharged spinless bodies YME theory and compare the result with that of Ref. [447]. The close relation between Green’s functions and scattering amplitudes of quantum field the-ories suggests that relations between scattering amplitudes of different theories may translate, An alternative possibility is to carry out the double copy while keeping track of the helicity of internalfields and making sure that only graviton modes appear on all internal lines of diagrams [80, 82]. CLASSICAL DOUBLE COPY in particular gauges and for special choices of field variables, into relations between classicalsolutions of the corresponding equations of motion. In this section we reviewed at lengthexamples in which this expectation is realized and certain solutions of gauge theories can beused to construct, through a classical double copy, certain solutions of gravity theories. Im-portant points—such as the relation between the gauge choice for the gauge-theory solutionand the properties of the corresponding gravity solution, or the identification of the bestchoice of gravity field variables such that no transformation functions are present—remainto be fully understood and the complete rules of the classical double-copy construction tobe spelled out. The examples we discussed, as well as the additional ones that may befound in the literature, show that such an approach can have useful applications to currentproblems in gravitational physics. Chief among them is precision predictions of gravitationalwaves; as we saw in Sec. 8.4, the (classical) double-copy construction may help streamlinethe evaluation of the expected signal from the relevant astronomical events.Further applications of the double copy to gravitational wave physics, which we didnot discuss in detail, relate to the calculation of gravitational interaction potential in thepost-Newtonian expansion. While standard methods, using the gravitation Lagrangian, arewell-developed and results through fourth post-Newtonian order are available [450–453],double-copy calculations such as in Refs. [454–456] may bring a novel perspective to thisproblem. Indeed, advances based on the double copy and new developments [78] in theeffective field approach [423, 457] resulted in a new state of the art result at the third orderin Newton’s constant [80].A common feature of the classical solutions constructed to date through such methodsis that, in the appropriate field variables, Einstein’s equations become linear. This includesthe case of the Kerr black hole which was shown in [458] to be related to a certain complexdeformation of the Coulomb potential. It is also shown that the change in momentum ina scattering event, known as “the impulse”, can be described via a double copy of a pointcharge. Progress towards further understanding some of the rules of the classical doublecopy may follow from finding examples where nonlinear contributions are nonvanishing.Perhaps the easiest approach to exploring such cases is the analysis of a Kerr-Schild solutionfor another choice of field variables; an example would be to repeat the calculations inRefs. [437, 438] using modern approaches.The construction of a gravitational Lagrangian whose fields are explicitly constructed interms of those of the two single-copy gauge theories may also lead to new ways of relatingthe solutions of the two theories. The linearized Lagrangians of certain N = 2 supergravitieshave been organized in this fashion in Refs. [56, 255].Another (notoriously difficult) problem which may benefit from the existence of a sys-tematic classical double copy is gravitational perturbation theory around a curved groundstate such as the (anti) de Sitter space, or the Schwarzschild black hole. The double copywill likely relate it to gauge-theory perturbation theory around a nontrivial classical solutionof YM equations of motion. Attempts in this direction have been discussed in Ref. [60] and[72] where, respectively, the three- and four-point amplitudes of gravitons around a gravi-tational plane wave were expressed in terms of three- and four-gluon amplitudes around aparticular gauge-theory plane wave. While developing general methods for such calculationsis an interesting problem in its own right, it is likely that their main applications will be togauge/string duality. 163 CONCLUSIONS The study of the classical double copy is in its infancy and many avenues remain tobe explored; we expect that the resulting methods will yield important new progress ingravitational physics, especially on the problem of gravitational radiation. The duality between color and kinematics and the double-copy construction offer a radically-different perspective on gravity theories compared to traditional Lagrangian or Hamiltonianapproaches. For the well-studied case of scattering amplitudes, the duality provides powerfulmeans for converting results in gauge theory to those of gravity. This has led to progress instudying the behavior of various gravitational theories at high perturbative orders, such asthe UV behavior of extended supergravity at four and five loops [38, 291–293] and the thirdpost-Minkowskian corrections to the classical Hamiltonian for compact binaries [80, 82]. Atpresent there are no other means to evaluate such high orders.Remarkably, the idea of CK duality and of the double-copy structure extends to theorieswith no obvious connection to gauge or gravity theories, as reviewed in Sec. 5. The fact thatthe scattering amplitudes of theories whose Lagrangians seem to have little to do with eachother contain the same kinematical objects is rather striking and points to new nontrivialconstraints shared by consistent theories. Additionally, by now the duality and double copyhave been established for a large number of examples of classical solutions [50–77].There are several areas where further progress would be welcomed. For example, itis not at the moment clear how far the notion of CK duality and the double copy canbe carried beyond scattering amplitudes. Many of the examples of classical solutions thatdisplay the duality make use of special properties, such as the existence of Kerr-Schildforms of the metric. It would be very important to find more general examples. Classicalsolutions are inherently more difficult to study because they depend on coordinate andgauge choices and, without the appropriate choices, the double-copy structure is obscured.This may be contrasted with scattering amplitudes, which are independent of the choice ofgauge and, to a large extent, field variables, making it much easier to formulate double-copyrelations. To avoid carrying out complicated case-by-case analyses, a key step is to findunderlying principles for choosing gauges and field variables in both single- and double-copytheories that make it more straightforward to identify relations between off-shell quantities.It would also be interesting to see if the more invariant color-trace-based formulation ofthe duality [48, 157–161] might shed light on extensions of CK duality beyond scatteringamplitudes.A possible path to unraveling the principles for choosing gauges and field variables maybe the study of correlation functions. The computation of correlation functions of gauge-invariant operators in gauge theories may be approached through generalized unitarity [459],which relates it to the construction of tree-level scattering amplitudes and form factors ofthe same gauge-invariant operators. In this respect, CK duality has been formulated andused for the form factors of certain operators in four- and five-loop calculations in N = 4SYM theory [9, 17, 18, 22]. Therefore, it seems plausible to extend CK duality to the cor-relation functions of these operators. However, a puzzle arises if one considers the naturalstep of constructing the double copy of such correlation functions. Since correlation func-164 CONCLUSIONS tions of gauge-invariant operators in gauge theories are gauge invariant, one may concludefollowing the discussion in Sec. 2 and Sec. 4 that the corresponding gravitational correlationfunctions are automatically diffeomorphism-invariant. It is well-known however that localdiffeomorphism-invariant operators do not exist in gravitational theories. Since correlatorsof gauge-dependent operators depend on choices of field variables, it appears that the grav-itational correlation functions obtained though the double copy should be understood asbeing given for a particular choice of field variables and perhaps also for particular choiceof gauge. A further puzzle originates from contrasting the results of the double copy forconformal gauge theories that admit a string-theory dual to the results of the correspondingstring theory in anti-de-Sitter space. On the one hand, gauge-theory correlation functionsare given by string-theory correlators with prescribed boundary conditions in anti-de-Sitterspace; on the other, the gauge-theory correlators can be used to construct correlators in agravitational theory ( not a string theory) in a Minkowski vacuum and with the same amountof supersymmetry as the AdS one. While technically difficult, it would clearly be interestingto understand the implications of such a relation.Apart from formal developments such as the ones described above, perturbative calcu-lations in curved spacetime are playing an increasingly-important role in the current de-velopment of our understanding of the universe. Initial attempts to use the double-copyconstruction in this context, involving calculations in certain plane wave spacetimes, havebeen discussed in [60, 62, 72, 460, 461]. As in flat space, the double-copy construction mayhelp by relating such calculations with simpler ones in gauge theory, especially for space-times which are themselves classical double copies. It is obviously nontrivial to extend theinsights of flat-spacetime scattering to the many conceptual and technical challenges posedby cosmological correlators in de Sitter, yet there is already a developing program [462–467] leveraging the identification of S-matrix elements emerging as residues of well-definedsingularities of such quantities.While CK duality and the associated double copy have been crucial for uncovering theUV properties of various supergravities [33, 34, 36, 38, 218, 293] and for identifying a newset of nontrivial enhanced UV cancellations [292], to move forward it is essential to gaina thorough grasp on the structures or symmetries that are responsible for the appearanceof the latter. Progress in this direction has been reported in [34, 40, 468], but much moreremains to be done to have a satisfactory understanding. Presumably, the duality and doublecopy play a key role in these cancellations.Another important topic is to expand the web of theories related by the duality anddouble-copy construction. As illustrated in Fig. 17 of Sec. 5, theories that may appear to beunrelated are bound together by double-copy relations. In many of these cases, the connec-tion is rather obscure from a Lagrangian perspective, e.g. that Dirac-Born-Infeld theory hasa relation to the special Galileon theory, by sharing the NLSM as a composite theory via thedouble copy. A crucial open question is whether it is possible to get a complete classificationof all double-copy-constructible theories. An equally important question is whether all su-pergravity theories can be expressed in a double-copy format [121, 240]. In this review, wediscuss a large number of examples, which are collected in Tables 4 and 5. It is a surprisingfact that the only known unitary UV completions of gravity and higher-dimensional YM, theclosed and open superstring, require their constituent effective field theories to be compatiblewith the field-theory adjoint double-copy to all orders of α at tree-level [169, 171, 310, 356].165 CONCLUSIONS Another basic research direction is to find the underlying algebra behind the dualitybetween color and kinematics. A natural expectation is that the kinematic Jacobi identitiesare due to an infinite-dimensional Lie algebra [43, 45, 47, 49]. Indeed, for the case of self-dualfield configurations, corresponding to amplitudes with identical helicity, the algebra has beenidentified as that of the area-preserving diffeomorphisms in one lightcone and one transversedirection [43]. However, extending this observation to general helicity or field configurationshas proven to be challenging.Constructing a Lagrangian that automatically generates Feynman rules that manifestthe duality would greatly help with finding double-copy relations between classical solutions.However, at present, only perturbative order-by-order constructions of such Lagrangians areknown [41, 42, 150, 151]. From the perspective of gravity theories, Lagrangians that displaythe required factorization of Lorentz indices [94] have been obtained to all orders [95, 469].However, as yet, it is unclear which all-orders gauge-theory Lagrangians can reproduce themthough double copy. One difficulty is that such Lagrangians would likely contain an infinitenumber of auxiliary fields to make them local.To further streamline higher-loop computations would be particularly desirable. Whilethere has been enormous progress in carrying out such computations to relatively high orders(see e.g., Refs. [6, 38, 293, 470] for four and five-loop calculations) we should always striveto go further. At high orders, it can be nontrivial to find representations of loop integrandsthat manifest CK duality [15, 416]. As described in Sec. 7, these difficulties can be bypassedvia a generalized double copy [218, 417] that can be used to convert any representation ofgauge-theory amplitudes to corresponding gravity ones, relying only on the proven existenceof the duality at tree level. Finding generalizations of this procedure for any number of loopsor legs would be important.Strengthening connections between the double copy and other advances in scatteringamplitudes would also be advantageous. In particular, the amplituhedron [471] gives novelgeometric descriptions of amplitudes. A detailed formulation has been given for the planarsector of N = 4 SYM theory. Making contact with CK duality requires extending theseresults to the nonplanar sector. Evidence suggests that this may be possible [195, 472].The double copy seems to hint at some kind of interpretation of gravitons as composedof spin-1 particles. Of course, these cannot be any kind of naive bound states, which areforbidden by the Witten-Weinberg theorem [473]. Still, the double copy strongly suggeststhat gravitons and gluons ultimately belong together, presumably along the lines realizedby string theory. (See Refs. [474–476] for steps in this direction.) Understanding any fun-damental physical implications of the way gauge and gravity theories are intertwined by thedouble copy is a key problem that deserves further attention.The application of the double copy to gravitational-wave physics [446] is currently thesubject of intense investigation, specifically regarding the post-Newtonian [477] and post-Minkowskian [478, 479] approaches to the inspiral phase of binary mergers (see the follow-ing reviews for details and references [129–132]). A nontrivial application of CK dualityto the study of gravitational radiation has been discussed with a worldline formulation inRef. [57], where the duality has been established through next-to-leading order [69]. Re-lated progress was also reported in Refs. [59, 64, 65, 447]. Other investigations relatedto gravitation-wave physics that directly draw from scattering-amplitudes methods can befound in Refs. [54, 79, 423–427, 480–483]. A systematic and scalable approach for obtaining166 CONCLUSIONS high-order corrections to conservative two-body potentials in the post-Minkowskian frame-work was presented in Ref. [78]. This has been successfully used to find the third post-Minkowskian corrections [80, 82], starting from two-loop amplitudes obtained via the doublecopy. It is noteworthy that this is one order beyond previous calculations [424, 484, 485].While their impact on improving templates for LIGO/Virgo is currently under study [81],these results should also offer new insights into the general structure of high-order two-bodyHamiltonians. Acknowledgments We thank T. Adamo, L. Borsten, E. Bjerrum-Bohr, J. Bourjaily, L. Dixon, M. Duff, A. Edi-son, S. Ferrara, M. Günaydin, S. He, E. Herrmann, Y.-t. Huang, G. Kälin, R. Kallosh,D. Kosower, A. Luna, D. Lüst, C. Mafra, G. Mogull, R. Monteiro, S. Nagy, H. Nico-lai, A. Ochirov, D. O’Connell, J. Parra-Martinez, L. Rodina, O. Schlotterer, C.-H. Shen,S. Stieberger, F. Teng, J. Trnka, A. Tseytlin, P. Vanhove, I. Vazquez-Holm, C. White andS. Zekioglu for helpful discussions, collaboration and comments during the course of writingthis review. Z.B. is supported by the U.S. Department of Energy (DOE) under grant no. DE-SC0009937. JJMC is grateful for the support of Northwestern University, CEA/CNRS-Saclay, and the European Research Council under ERC-STG-639729, Strategic Predictionsfor Quantum Field Theories . R.R. is supported by the U.S. Department of Energy (DOE)under grant no. DE-SC0013699. The research of M.C. and H.J. is supported by the Knutand Alice Wallenberg Foundation under grants KAW 2013.0235 and KAW 2018.0116 - FromScattering Amplitudes to Gravitational Waves , the Ragnar Söderberg Foundation (SwedishFoundations’ Starting Grant), and the Swedish Research Council under grant 621-2014-5722.This work was performed in part at the Munich Institute for Astro- and Particle Physics(MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe” and inpart at the Aspen Center for Physics, which is supported by National Science Foundationgrant PHY-1607611. In addition, Z.B. thanks Mani L. Bhaumik for generous support overthe years. 167 NOTATION AND LIST OF ACRONYMS A Notation and list of acronyms In this appendix, we summarize our notation and conventions for the reader’s convenience.Throughout the review, we denote with A n (1 , · · · , n ) gauge-theory color-dressed ampli-tudes, while A n (1 , · · · , n ) is used to indicate color-stripped partial amplitudes. In somesections, where theories containing different fields are discussed, it is convenient to adoptthe notation A n (cid:16) , . . . , n Φ n (cid:17) , (A.1)which makes explicit which field is associated to each external leg of the amplitude. Super-amplitudes are denoted with the same symbol as the corresponding amplitudes, i.e. it shouldbe clear from the context whether A n (1 , · · · , n ) and A n (1 , · · · , n ) refer to an amplitude orto the corresponding superamplitude. Writing the S -matrix as S = 1 + iT , our amplitudescorrespond to the iT term, i.e. they give the output of the Feynman-diagram calculation.Amplitude in a gravitational theory are denoted as M n (1 , · · · , n ). For notational simplicity,we set κ = 2 in most formulas, where κ is the gravitational coupling constant.Our conventions for the phase in the BCJ representation of gauge-theory and gravityamplitudes follows the one in Ref. [156] and departs from the original BCJ papers [1, 2].With this choice, YM numerators are real when written in terms of polarization vectors.Calculations presented in this review involve a spacetime metric of mostly-minus signa-ture. Our spinor-helicity conventions are obtained from the ones of Ref. [91] by the minimalreplacement ( η µν ) E&H → − ( η µν ) our . (A.2)In particular, angle and square brackets are the same as in Ref. [91].Gauge-group fundamental fields are represented with high indices and anti-fundamentalfields are represented with low indices. For example, the generator for the fundamentalrepresentation is written as ( t a ) ji ≡ t ai ¯ . (A.3)Generators are normalized as Tr( t a t b ) = δ ab , (A.4)and obey commutation relations of the form[ t a , t b ] = if abc t c . (A.5)In amplitude calculations it is convenient to rescale the group-theory generators and structureconstants as T a ≡ √ t a , ˜ f abc ≡ i √ f abc , (A.6)so that we have the identity Tr( T a T b ) = δ ab . (A.7)In particular, color factors entering the formula (2.1) are written in terms of T a s and ˜ f abc s, i.e.are written in terms of hermitian objects carrying a factor of √ R , we denote the corresponding generators as t a R . In some sections of this review, for168 SPINOR HELICITY AND ON-SHELL SUPERSPACES example in Sec. 5, we frequently use hatted indices for gauge-group indices of the gaugetheories entering the double-copy construction to differentiate them from global indices.Finally, we conclude this appendix with a list of acronyms commonly used throughoutthe review:UV UltravioletIR InfraredKLT Kawai-Lewellen-Tye (formula/relations)CK duality Color/kinematics dualityBCJ Bern-Carrasco-Johansson (duality/relations)YM Yang-MillsSYM Super-Yang-MillsNLSM Nonlinear Sigma ModelDDM Del Duca-Dixon-Maltoni (amplitude representation)POP Partially-ordered permutationsBCFW Britto-Cachazo-Feng-Witten (relations/recursion in field theory)KK Kleiss-Kuijf (amplitude relations)QCD Quantum ChromodynamicsCPT Charge-Parity-Time reversal (transformations)MHV Maximally-helicity-violating (amplitudes)LSZ Lehmann-Symanzik-Zimmermann (reduction)1PI One particle irreducible (effective action)BMS Bondi-Metzner-Sachs (transformations)SUSY Supersymmetry (tables and figures only)CSG Conformal supergravityBLG Bagger-Lambert-Gustavsson (theory)ABJM Aharony-Bergman-Jafferis-Maldacena (theory)VEV Vacuum expectation valueYM DR Yang-Mills-scalar theory from dimensional reductionYME Yang-Mills-Einstein (theory)DBI Dirac-Born-Infeld (theories)CHY Cachazo-He-Yuan (formalism, also known as scattering equations)SG Supergravity (tables and figures only)MZVs Multiple zeta valuesQFT Quantum field theoryLIGO Laser Interferometer Gravitational-Wave Observatory B Spinor helicity and on-shell superspaces In explicit expressions for amplitudes, such as those in Sec. 6 or Appendix C, it is veryconvenient to adopt a helicity (circular polarization) basis for the asymptotic states of gluonsor gravitons. In this appendix, we summarize the spinor-helicity formalism [88, 89, 486–490], which offers a convenient Lorentz covariant formalism for describing helicity, leading to169 SPINOR HELICITY AND ON-SHELL SUPERSPACES remarkably compact expressions for scattering amplitudes. The resulting states fit naturallyinto on-shell supermultiplets [491]. B.1 Basics of spinor helicity The spinor-helicity formalism expresses the positive- and negative-helicity polarizations ofgluons (vectors) in terms of massless Weyl spinors ε + µ ( k ; q ) = h q | σ µ | k ] √ h q k i , ε − µ ( k ; q ) = [ q | σ µ | k i√ k q ] , (B.1)where q is an arbitrary null ‘reference’ momentum which can be chosen independently foreach external state of amplitudes and, because of gauge invariance , drops out of the finalexpressions. We use the compact notation h ij i ≡ 12 ¯ u ( k i )(1 + γ ) u ( k j ) , [ ij ] ≡ 12 ¯ u ( k i )(1 − γ ) u ( k j ) , h q | σ µ | k ] ≡ 12 ¯ u ( q ) γ µ (1 − γ ) u ( k ) , [ q | σ µ | k i ≡ 12 ¯ u ( q ) γ µ (1 + γ ) u ( k ) , (B.2)with u ( k ) following the standard textbook notation for solutions of the Dirac equation [100].The spinors | i i and | i ] transform in the ( , ) and ( , ) representations of the four-dimensionalLorentz group, respectively. The spinor products (B.2) are antisymmetric in their arguments.An important identity is the Schouten identity: h i j i h k l i = h i l i h k j i + h i k i h j l i , (B.3)which is a consequence of the vanishing of all three-index antisymmetric tensor with eachindex taking two values. The spinor products (B.2) are related to the usual scalar productsby h i j i [ j i ] = 2 k i · k j = s ij , (B.4)where the k i are null four momenta. The Fierz identity is in our conventions is h i | σ µ | j ] h k | σ µ | l ] = 2 h i k i [ l j ] . (B.5)Helicity amplitudes (that is, amplitudes with polarization vectors or tensors in helicitynotation) can be given a manifestly crossing symmetric representation. To this end it is nec-essary to assign all momenta to have the same orientation, either all outgoing or all incoming.When switching between the two different orientations the helicity label is reversed. This is,of course, natural: since the helicity measures the projection of the spin on the momentum,changing the orientation of the momentum reverses the helicity. Using spinor helicity we canobtain exceptionally compact expressions for gauge-theory scattering amplitudes [88].The physical graviton polarization tensors in helicity notation are direct products of thegluon ones, ε + µν ( k ; q ) = ε + ν ( k ; q ) ε + ν ( k ; q ) , ε + µν ( k ; q ) = ε + ν ( k ; q ) ε + ν ( k ; q ) . (B.6) Linearized gauge transformations, ε µ ( p ) → ε µ ( p ) + f ( p ) p µ , is realized as shifts of the spinors associatedto the reference vector, | q i → | q i + f ( p ) | p i , etc . SPINOR HELICITY AND ON-SHELL SUPERSPACES Their tracelessness , ε + µµ = 0, follows from the Fierz identity (B.5) with the appropriatechoice of spinors: h q | σ µ | k ] = 0 , [ q | σ µ | k i = 0 . (B.7)The relation (B.6) between graviton and gluon polarizations is the simplest manifestation ofthe double copy. Of course, the double copy holds for the full nonlinear theory, not just forpolarization tensors.Loop calculations require regularization; we will not discuss details of this issue hereexcept to note that to maximize the benefits of the spinor-helicity formalism, which is in-trinsically four-dimensional, it is necessary to choose a compatible version of dimensionalregularization [492, 493].Exercise B.1: Starting with Eqs. (1.10)-(1.13) apply Eq. (B.1) to obtain four-gluon ampli-tudes for the various helicity configurations. How clean can you make these expressions?See, for example, Ref. [88]. B.1.1 Massive spinor helicity As some of the theories described in this review involve massive fields, we will briefly outlinehow to adapt the spinor-helicity formalism to this case. A first possibility is to write amassive momentum k in terms of two massless momenta [494]: k = k ⊥ + m k · q q , (B.8)where k ⊥ is massless and we have also introduced a massless reference momentum q . Polar-izations for massive vectors are then written as ε µ + ( k ; q ) = h q | σ µ | k ⊥ ] √ h q k ⊥ i , ε µ − ( k ; q ) = [ q | σ µ | k ⊥ i√ k ⊥ q ] , ε µ ( k ; q ) = 1 m (cid:18) k µ ⊥ − q µ k · q (cid:19) , (B.9)where the first two physical polarizations reproduce (B.1) in the massless limit and ε µ ( k ; q )gives the longitudinal polarization. While this formalism allows us to find relatively compactexpressions for amplitudes with massive fields, the reference momentum q does not drop outfrom the final expressions. This is to be expected: for a massive particle helicity is not aLorentz-invariant quantity, so the decomposition (B.9) depends on the frame.A more elegant approach involves a doublet of spinors λ aα , ˜ λ a ˙ β which transform covariantlyunder the SO (3) ∼ = SU (2) little group appropriate for describing massive particles in fourdimensions. Massive momenta are then written as [495]: k α ˙ β = k µ σ µα ˙ β = (cid:15) ab | k a i α [ k b | ˙ β = (cid:15) ab λ aα ˜ λ b ˙ β , (B.10)where a, b are little group SU (2) indices and α, ˙ β are four-dimensional Weyl spinor indices.Massive vector polarizations are written n terms of the spinors λ aα , ˜ λ a ˙ β as ε abµ ( k ) = h k ( a | σ µ | k b ) ] √ m , (B.11)where the little group indices are symmetrized. This formalism can be straightforwardlyextended to construct polarization tensors for higher-spin massive fields [495] and presentsclose analogies with massless spinor-helicity in six dimensions[496].171 SPINOR HELICITY AND ON-SHELL SUPERSPACES B.2 On-shell superamplitudes For supersymmetric amplitudes, on-shell superspace provides a convenient organization ofamplitudes according to their physical helicity states which also tracks the relationshipsbetween the different component amplitudes. The power of such an on-shell superspacefollows from the fact that, for generic momentum configurations, scattering amplitudes areinsensitive to the nonlinear parts of (super)symmetry transformations (see Sec. 4 for moredetails.). This greatly simplifies the evaluation of state sums in both the on-shell recur-sion [162] and generalized unitarity [163, 164, 166] by allowing that all physical states betreated simultaneously.To illustrate the ideas we use N = 4 SYM theory [491] as an example. Similar con-structions exist in cases with less than maximal supersymmetry [497] as well as supergravitytheories [498]. These superspaces are obtained by extending the usual momentum space(parametrized in terms of spinor variables) with unconstrained Grassmann variables, η I with I = 1 , . . . , N , which transform in the fundamental representation of the R -symmetry groupand carry unit little group weight. The bosonic spinor variables carry kinematic information,while the Grassmann variables carry information on the helicity and R -symmetry represen-tation of the external states. On-shell superfields—i.e. fields defined on these superspaces—have a finite expansion in the fermionic variables, with each coefficient being a componentfields of definite helicity and R -symmetry representation. N = 4 SYM has a simple structurebecause all states can be assembled into a single CPT-self-conjugate on-shell superfield:Φ( η ) = g + + η I f + I + 12 η I η J φ IJ + 13! η I η J η K f − IJK + 14! η I η J η K η L g − IJKL , (B.12)where g + is the positive helicity gluon, f + I four positive helicity Majorana fermions, φ IJ six real scalars, f I − ≡ (cid:15) IJKL f − JKL four negative helicity Majorana fermions and g − is thenegative helicity gluons, for a total of 8 + 8 physical states (not including color degrees offreedom). The case of N = 8 supergravity is similar, with a four-dimensional CPT-self-conjugate on-shell superfield containing fields up to helicity ± η ) = h + + η I ψ + I + 12 η I η J A IJ + 13! η I η J η K χ + IJK + 14! η I η J η K η L φ IJKL + 15! η I η J η K η L η M χ − IJKLM + 16! η I η J η K η L η M η N A − IJKLMN (B.13)+ 17! η I η J η K η L η M η N η O ψ − IJKLMNO + 18! η I η J η K η L η M η N η O η P h − IJKLMNOP . Each supergravity state is a double copy of the gauge-theory states. The 256 physicalstates of N = 8 supergravity correspond to the 16 × 16 direct product of states of two N = 4SYM theories, as shown in Table 17.Supersymmetry transformations act on on-shell superfields (i.e. single-particle supersym-metry transformations) as Q ˙ αI = ˜ λ ˙ α ∂∂η I , Q I ˙ α = η I ∂∂ ˜ λ ˙ α , Q αI = λ α η I , Q αI = ∂ ∂λ α ∂η I ; (B.14) As we discussed in Sec. 5, supergravity states can more generally be understood as being in one-to-onecorrespondence with gauge-invariant bilinears of gauge-theory states. SPINOR HELICITY AND ON-SHELL SUPERSPACES g R + f R +˜ I φ R ˜ I ˜ J f R − ˜ I ˜ J ˜ K g R − ˜ I ˜ J ˜ K ˜ L g L + h + ψ +˜ I A +˜ I ˜ J χ +˜ I ˜ J ˜ K φ ˜ I ˜ J ˜ K ˜ L f L + I ψ + I A + I ˜ I χ + I ˜ I ˜ J φ I ˜ I ˜ J ˜ K χ − I ˜ I ˜ J ˜ K ˜ L φ LIJ A + IJ χ + IJ ˜ I φ IJ ˜ I ˜ J χ − IJ ˜ I ˜ J ˜ K A − IJ ˜ I ˜ J ˜ K ˜ L f L − IJKL χ + IJKL φ IJKL ˜ I χ − IJKL ˜ I ˜ J A − IJKL ˜ I ˜ J ˜ K ψ − IJKL ˜ I ˜ J ˜ K ˜ L g L − IJKL φ IJKL χ − IJKL ˜ I A − IJKL ˜ I ˜ J ψ − IJKL ˜ I ˜ J ˜ K h − IJKL ˜ I ˜ J ˜ K ˜ L Table 17: The states of N = 8 supergravity organized via the double copy. The SU (8) represen-tations are decomposed in representations of the SU (4) × SU (4) subgroup which is manifest inthe construction. For gauge theory ( g + , λ + , φ, λ − , g − ) carry helicity (+1 , , , − , − ± decorating theentries represent the sign of the helicity of the corresponding state. The double-copy states can bereorganized into the standard N = 8 multiplet containing 256 physical states, cf. eq. (B.13). the (linearized) supersymmetry transformations of component fields are extracted by actingon superfields with these generators and reading off the coefficient of the desired combina-tion of Grassmann variables η . Multi-particle supersymmetry generators are obtained bysumming the single-particle ones over all the particles; for example, Q ˙ αI acts on a product of n distinct fields as Q ˙ αI = n X i =1 Q i ˙ αI = n X i =1 ˜ λ ˙ αi ∂∂η Ii . (B.15)For supersymmetry algebras with less-than-maximal supersymmetry not all multiplets areCPT-self-conjugate; in such cases the fields of the theory form (perhaps several) CPT-conjugate pairs.Scattering amplitudes in supersymmetric field theories can be combined into superam-plitudes, defined as polynomials in Grassmann variables such that the coefficient of eachmonomial is a component amplitude whose helicity configuration is dictated by the η factorsthat multiply it and the structure of the superfields of the theory. The details depend onthe amount of supersymmetry; as above, we illustrate these ideas for N = 4 SYM theory.See Refs. [91, 497] for less supersymmetric cases. On-shell supersymmetry Ward identities,relating component amplitudes with different external field configurations, can be derivedby demanding that superamplitudes are annihilated by the multi-particle supersymmetrygenerators. A detailed descriptions may be found in Refs. [91, 498–500]. The unconstrainednature of the Grassmann variables makes it straightforward to translate summations of on-shell states needed in unitarity cuts or on-shell recursion into Grassmann integrations, whichtake care of the state bookkeeping. See Refs. [501, 502] for details. This procedure ensuresthat all generalized cuts are manifestly supersymmetric.The minimum number of Grassmann variables in superamplitudes enforces the conser-vation of the polynomial supercharge Q αI in Eq. (B.14), sometimes referred to as the “su-173 SPINOR HELICITY AND ON-SHELL SUPERSPACES permomentum”: δ (8) ( Q ) ≡ δ (8) n X j =1 λ αj η Ij = Y I =1 n X i GENERALIZED UNITARITY ℓ ℓ (b) 412 3 ℓ ℓ (a) 412 3 Figure 34: The (a) s and (b) t channel two-particle cuts of a one-loop four-point amplitude. Theexposed lines are all on-shell and the blobs represent tree amplitudes. construct the linearized supersymmetry transformations of the component fields of N = 4SYM theory. Derive the corresponding relations between the component MHV amplitudes. C Generalized unitarity In this appendix we give a brief summary of the modern generalized unitary method [163–166, 194, 217] used in multiloop calculations, focusing on their applications in double-copyconstructions. This provides some of the necessary background for our review of the gener-alized double-copy construction in Sec. 7. We will present several examples illustrating thebasic ideas and refer the reader to other reviews for further details [90, 196, 510, 511].The generalized-unitary method systematically builds complete loop-level integrands us-ing as input only on-shell tree-level amplitudes. A central feature is that simplifications andfeatures of the latter are directly imported into the former. In particular, with this methodwe can use tree-level double-copy relations to construct gravity loop integrands. We alsobriefly review a variant of generalized unitarity, known as the maximal-cut method [217],which meshes well with the generalized double copy [417] discussed in Sec. 7. A reorganiza-tion of the generalized-unitarity method that has various advantages is found in Ref. [512].Traditionally, unitarity of the scattering matrix is implemented at the integrated levelvia dispersion relations [513]. For our purposes, however, it is much more useful to use it atthe integrand level. We introduce the concept of a generalized cut that reduces an integrandto a sum of products of tree amplitudes A tree( j ) , X states A tree(1) A tree(2) A tree(3) · · · A tree( m ) . (C.1)Each cut propagator is replaced with a delta function enforcing on-shell constraint for thecorresponding momentum. The sum runs over all intermediate physical states that cancontribute given the external states of the amplitude being constructed. Some generalizedcuts of the one-loop four-point amplitude are shown in Fig. 34 and of the three-loop four-point amplitude in Fig. 35. In these figures the exposed lines are all on-shell delta functionsand the blobs represent on-shell tree amplitudes.Loop integrands are determined by spanning set of generalized cuts, i.e. a set of cuts whichreceive contributions from all the terms that could possibly be generated by the Feynmangraphs of the theory. Loop integrands are constructed by finding a single function whose176 GENERALIZED UNITARITY 412 3412 3412 3412 3 Figure 35: Examples of generalized cuts for a three-loop four-point amplitude. The exposed linesare all on-shell and the blobs represent tree amplitudes. cuts match all the products of tree amplitudes, summed over states corresponding to such aspanning set. Regardless of which set of cuts one uses to construct an integrand, one mustalways verify it on a minimal spanning set (i.e. a spanning set that contains the minimalnumber of cuts). To illustrate these ideas in practice we turn to a few simple unitarity cuts. C.1 One-loop example of unitarity cuts To illustrate the generalized unitarity method and how it meshes with double-copy ideasconsider the two-particle cuts of a one-loop four-point color-ordered gauge-theory amplitude.In these amplitudes the color factors are stripped away and the external legs follow a cyclicordering [88, 89]. The two-particle cuts of a one-loop four-point amplitude are obtained byputting two intermediate lines on shell, as illustrated in Fig. 34. For example, the s -channelcut in Fig. 34(a) is given by C gauge s = X states A tree4 ( − ‘ , , , ‘ ) A tree4 ( − ‘ , , , ‘ ) , (C.2)where the sum runs over all physical states in the theory. The cuts are evaluated usingmomenta that place all cut-line momenta on shell, i.e. ‘ i = 0 if the theory is massless.A particularly simple example is the color-ordered one-loop four-point amplitude in N = 4 SYM theory, which is useful for illustrating how gauge-theory unitarity cuts canbe converted to gravity ones. For this theory, after summing over all physical states thatcross the two-particle cut, the result takes a remarkably compact form [400], C gauge s == − istA tree4 (1 , , , 4) 1( ‘ − p ) ‘ − p ) . (C.3)All momenta are on shell. This expression is valid for any external states of the theory; thecut depends on them only through the overall factor A tree4 (1 , , , t -channel cut inFig. 34(b) is obtained by relabeling Eq. (C.3).The most straightforward way to verify these equations is by using four-dimensionalhelicity states and on-shell superspace, but they hold in D ≤ 10 dimensions as well (wheremaximal supersymmetric Yang-Mills theory is defined). Details may be found in Ref. [194].Putting back the cut propagators and loop integration we find a function with the correct s -channel cut, ist A tree4 (1 , , , I ( s, t ) (cid:12)(cid:12)(cid:12)(cid:12) s - cut , (C.4)177 GENERALIZED UNITARITY 12 34 ℓ ℓ ℓ ℓ Figure 36: The one-loop box integral and loop momentum labels used in the cut construction. where I ( s, t ) is the scalar box integral shown in Fig. 36 and given in Eqs. (6.10) and (6.11).By the cut operation in the s channel in Eq. (C.4) we mean to remove the integrationand to replace the two propagators 1 /‘ and 1 / ( ‘ − p − p ) with on-shell conditions,recovering Eq. (C.3) after identifying ‘ = ‘ and applying momentum conservation. Similarconsiderations show that the t channel cut can be written as ist A tree4 (1 , , , I ( s, t ) (cid:12)(cid:12)(cid:12)(cid:12) t - cut . (C.5)Once unitarity cuts are written in this way, as the cuts of a single function, it is easy tosee that the one-loop four-point amplitude, with no cut conditions, is obtained simply byremoving the cut conditions, A - loop N =4 (1 − , − , + , + ) = i st A tree4 (1 , , , I ( s, t ) , (C.6)matching the result in Eq. (6.6).These basic ideas generalize to any massless gauge theory and underpin many theoreticalstudies, including those for collider physics (see e.g. Refs. [165, 514, 515]). C.2 Converting gauge-theory unitarity cuts to gravity ones As discussed in Sec. 2, the BCJ forms of loop-level gauge-theory integrands can be directlyconverted to gravity ones. Because the unitarity-based construction uses tree amplitudesas input, we can also straightforwardly apply the KLT relations (2.45) to convert gravityunitarity cuts to sums of products of gauge-theory cuts. The BCJ form of the gauge-theorytree amplitudes (2.16) may also be used to apply the double copy to convert gauge-theorycuts to the corresponding gravity ones. The KLT form is especially useful when workingwith compact helicity amplitudes, while the BCJ form is helpful in D dimensions (i.e. whenusing dimensional regularization) with formal polarization vectors.Consider first the two-particle cut of a one-loop four-point amplitude show in Fig. 34(a)in a gravity theory. Using the KLT form of the double copy, it is given by C GR = X gravitystates M tree4 ( − ‘ , , , ‘ ) × M tree4 ( − ‘ , , , ‘ )= − s X gaugestates A tree4 ( − ‘ , , , ‘ ) × A tree4 ( − ‘ , , , ‘ ) ! × X gaugestates A tree4 ( ‘ , , , − ‘ ) × A tree4 ( ‘ , , , − ‘ ) ! , (C.7)178 GENERALIZED UNITARITY where we applied the KLT relation (1.31) to rewrite each gravity tree amplitude in terms of aproduct of two gauge-theory amplitudes. In this expression we have assumed that the gravitytheory of interest arises as a simple double copy, as it does for N = 8 supergravity. Thisallows us to decompose each state in the gravity theory into a “left” and a “right” gauge-theory state. For the case of N = 8 supergravity, summing over the states in the N = 8multiplet is equivalent to summing independently over the left and right N = 4 SYM gauge-theory multiplets. For theories which are not simple double copies, such as pure gravity, onemust remove unwanted states (i.e. dilaton and antisymmetric tensor) by inserting explicitphysical-state projectors into the cuts. These projectors have been used effectively at two-loops to study ultraviolet properties of various theories, including pure gravity [339] aswell as for computing the third post-Minkowskian correction to the conservative two-bodyHamiltonian [80, 82]. In some cases, it is sufficient to evaluate the generalized unitarity cutsin four dimensions, where we can simplify the input gauge-theory amplitudes enormouslyby using helicity states. In this case, a simple way to control which particles circulate inthe loops, is by correlating the state sum of the two gauge theories, For example, if wewant only gravitons to cross the cuts, then for each term we should have identical helicityfor the corresponding gluons in the state sum [80, 82]. A similar procedure works well forsupergravity theories which are obtained as orbifolds of e.g. N = 8 supergravity where theorbifold action cannot be decomposed into independent actions on the left and right N = 4SYM gauge theories [30].The one-loop four-point amplitude in N = 8 supergravity is an instructive illustration ofhow we can recycle gauge-theory unitarity cuts into gravity ones. For this case, Eq. (C.7)immediately collapses because, up to relabeling, each gauge-theory state sum is the N = 4SYM state sum in Eq. (C.3). Inserting the simplified N = 4 SYM cut (C.3) into (C.7) resultsin an equivalent relation for N = 8 supergravity, C GR = s ( st ) h A tree4 (1 , , , i ‘ − p ) ( ‘ − p ) ( ‘ + p ) ( ‘ + p ) = s [ st A tree4 (1 , , , ‘ − p ) ( ‘ − p ) ( ‘ − p ) ( ‘ − p ) = i stuM tree4 (1 , , , " ‘ − p ) + 1( ‘ − p ) ‘ − p ) + 1( ‘ − p ) . (C.8)To obtain this we partial fractioned the product of propagators and used the KLT relations(1.30) and the BCJ amplitude relations (1.28). As for gauge-theory cuts, the ‘ and ‘ are on shell. The t - and u -channel formulae are obtained by relabeling the external legs inEq. (C.8).Following the same strategy as for the reconstruction of the one-loop four-point N = 4SYM amplitude, it is then straightforward to obtain the N = 8 one-loop four-point ampli-tude, M - loop4 (1 , , , 4) = − i (cid:18) κ (cid:19) stuM tree4 (1 , , , (cid:18) I ( s, t ) + I ( s, u ) + I ( t, u ) (cid:19) . (C.9)Here I ( s, t ) is the box integrals defined in Eq. (6.10), while I ( s, u ) and I ( t, u ) are obtainedby appropriate relabeling of external legs. This agrees with the form obtained using the BCJ179 GENERALIZED UNITARITY double copy in Sec. 6 and agrees with the result first obtained by Brink, Green, Schwarz [393]in the field-theory limit of superstring theory.One can also use the BCJ double copy (2.11) for the component tree amplitudes of thegravity unitarity cuts. This is especially efficient when working in D dimensions (e.g. whenusing dimensional regularization), because compact helicity-based expressions for tree-levelamplitudes are no longer available and, consequently, the result of the KLT relations willbe cumbersome to use. In D dimensions, the BCJ form is a more natural form because itpreserves the diagram structure when converting from gauge theory to gravity. For example,the two-particle cut (a) in Fig. 34, can be evaluated using the form of the double copy inEq. (1.18), C (a)GR = X pols . M tree4 ( − ‘ , , , ‘ ) × M tree4 ( − ‘ , , , ‘ ) , (C.10)where the graviton tree amplitudes in the cut are obtained from the double copy formin Eq. (1.18) by simple relabelings. The sum over polarizations gives the physical-stateprojector. For gravitons in D dimensions the projector is P µνρσ ( p, q ) = X pols . ε µν ( − p ) ε ρσ ( p ) = 12 (cid:16) P µρ P νσ + P µσ P νρ (cid:17) − D s − P µν P ρσ , (C.11)where P µν is the gluon physical-state projector P µν ( p, q ) = X pols . ε µ ( − p ) ε ν ( p ) = η µν − q µ p ν + p µ q ν q · p , (C.12)with momentum p and a null reference momentum q . In some cases, terms that vanish on-shell can be added to tree amplitudes so that the dependence on the reference momentumdisappears because of the on-shell Ward identity for the gauge symmetry [80, 82, 516]. C.3 Method of Maximal Cuts A refinement of the unitarity method [163, 164], which is especially helpful at higher looporders, is the method of maximal cuts [217]. This method is not only a basic tool for checkingand building double-copy gravity integrands, but it also plays a central role in the generalized-double-copy construction described in Sec. 7. This construction was central to a computationdetermining the ultraviolet properties of N = 8 supergravity at five loops [38, 218, 417].In the method of maximal cuts, the unitarity cuts are clustered in levels according to thenumber k of internal propagators allowed to remain off shell, C N k MC = X states A tree m (1) · · · A tree m ( p ) , (C.13)where A tree m ( i ) are tree-level m ( i )-multiplicity amplitudes corresponding to the blobs, illustratedfor various cuts of a three-loop four-point amplitude illustrated in Fig. 35. The level k isrelated to the multiplicity of the various factors by k = p X i =1 ( m ( i ) − . (C.14)180 GENERALIZED UNITARITY 412 3 ⇒ ⇒ 41 341 23412 3 2 Figure 37: New contribution found via the method of maximal cuts can be assigned directly tocontact terms. In these diagrams all the exposed lines are on-shell, and propagators within a blobremain off shell. The cuts (C.13) can be applied to either gauge or gravity amplitudes. As illustrated in thefirst diagram in Fig. 35, at the maximal cut (MC) level the maximum number of propagatorsare replaced by on-shell conditions and all tree amplitudes appearing in Eq. (C.13) are three-point amplitudes. At the next-to-maximal-cut (NMC) level, illustrated in the second cut ofFig. 35, a single propagator is placed off shell and so forth.With this organization of generalized cuts, the integrands for L -loop amplitudes are ob-tained by first establishing an integrand whose maximal cuts are correct, then adding to itterms so that NMCs are all correct and systematically proceeding through the next k maxi-mal cuts (N k MCs), until no further contributions are found. Where this process completes isdictated by the power counting of the theory and by choices made at each level. For exam-ple, if minimal power counting is assigned to each contribution, for N = 4 SYM four-pointamplitudes, cuts through NMCs, N MCs and N MCs are sufficient at three [2], four [6] andfive loops [517], respectively.Most calculations (see e.g. Refs. [6, 9, 15, 17, 292, 293, 408, 409]) find it convenient toorganize the integrands in terms of diagrams with purely cubic vertices, such as the three-loop ones illustrated in Fig. 27. Representations with only cubic diagrams have certainadvantages: they are useful for establishing minimal power counting in each diagram, andthe number of diagrams used to describe the result proliferate minimally with the loop orderand multiplicity. A disadvantage is that ansätze are required for imposing various propertieson each diagram, including the desired power counting, symmetry, and the multiple unitaritycuts to which a given diagram contributes. As the loop order increases, it becomes cumber-some to solve the requisite system of equations that imposes these constraints. We can avoidthis in the generalized double-copy construction if we instead assign any new informationobtained in a N k MC to a contact diagram, as discussed in Sec. 7 and illustrated in Fig. 37.This is necessarily local because the nonlocal contributions are accounted for at previouslevels. C.3.1 Sewing superamplitudes We now briefly comment on the use of the on-shell superspace described in Appendix B forthe evaluation of the sums over the states crossing a unitarity cut. It turns out [501, 502, 518]that it can be conveniently expressed as an integration over the Grassmann parameters of181 GENERALIZED UNITARITY the cut legs. The generalized supercut in N -extended SYM theory is then given by C = Z (cid:20) k Y i =1 d N η i (cid:21) A tree(1) A tree(2) A tree(3) · · · A tree( m ) , (C.15)where A tree( j ) are the tree-level superamplitudes connected by k on-shell cut legs. For each cutleg, the integral over η selects all possible states on that leg and sums up their contribution tothe cut. These supercuts are functions on the on-shell superspace. For four and higher pointsthe tree amplitudes A tree( j ) are always proportional to a supermomentum delta function. Usingthe simple identity δ ( A ) δ ( B ) = δ ( A + B ) δ ( B ), this implies that all such cuts are proportionalto an overall supermomentum δ -function [501]. It turns out that such supercuts are sufficientfor determining massless superamplitudes. This then implies that the four-dimensional cutsof any loop amplitude with four or more external legs must be proportional to an overallsupermomentum conservation δ -function. Barring supersymmetry anomalies, this will alsobe the case for the corresponding superamplitudes.Fermionic integration provides one of the several different methods for the evaluationof supersums in unitarity cuts [222, 501, 502, 518]. There are two main approaches fororganizing the integration over the η parameters. In the first way, the fermionic δ -functionscan be used to localize the integration, so that the evaluation of the supersum amountsto solving a system of linear equations [501, 502]. In a second complementary approach,“index diagrams” are used to track the various contributions to the sum over states [502].This approach leads to a simple algorithm for reading off the contribution of the entiresupermultiplet from the purely gluonic ones and for reducing the number of supersymmetries.It was used in the construction of the complete four-loop four-point amplitude of N = 4 SYMtheory [3].The overall supermomentum-conserving δ -function has consequences on the ultravioletproperties of the theory akin to those of off-shell superspaces. In particular, in a theory with N -extended supersymmetry, it implies that at least N powers of momenta in the numeratorsof each diagram are external momenta. In turn, this implies that the superficial degree ofdivergence of each diagram is improved by N compared to that of the non-supersymmetrictheory. 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