CPHT-RR049.072020

A Double Copy for Celestial Amplitudes

Eduardo Casali� and Andrea Puhm♦

� Center for Quantum Mathematics and Physics (QMAP)and Department of Physics, University of California,

Davis, CA 95616 USA

♦ CPHT, CNRS, Ecole polytechnique, IP Paris,F-91128 Palaiseau, France

Abstract

Celestial amplitudes which use conformal primary wavefunctions rather than plane waves

as external states offer a novel opportunity to study properties of amplitudes with manifest

conformal covariance and give insight into a potential holographic celestial CFT at the null

boundary of asymptotically flat space. Since translation invariance is obscured in the con-

formal basis, features of amplitudes that heavily rely on it appear to be lost. Among these

are the remarkable relations between gauge theory and gravity amplitudes known as the

double copy. Nevertheless, properties of amplitudes reflecting fundamental aspects of the

perturbative regime of quantum field theory are expected to survive a change of basis. Here

we show that there exists a well-defined procedure for a celestial double copy. This requires

a generalization of the usual squaring of numerators which entails first promoting them

to generalized differential operators acting on external wavefunctions, and then squaring

them. We demonstrate this procedure for three and four point celestial amplitudes, and

comment on the extension to all multiplicities.

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Contents

1 Introduction 1

2 Background 3

2.1 Momentum Basis vs Conformal Basis . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.2 Celestial Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.3 Double Copy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Double Copy Revisited 7

3.1 Three-Point Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Four-Point Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

4 Double Copy for Celestial Amplitudes 11

4.1 Three-Point Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4.2 Four-Point Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Discussion 16

1 Introduction

Scattering amplitudes are usually calculated using asymptotic states in a plane waves basis. In

the case of massless particles, these states are labeled by null momenta which, in four dimensions,

are described by three parameters, a point on a two-sphere and a scaling given by its energy. By

performing a Mellin transform on the energy we can interpret these parameters as coordinates

on the null boundary of Minkowski space. From the point of view of the amplitudes, applying a

Mellin transform to each external state amounts to a change in basis of the asymptotic states.

In this basis, introduced in [1], the four-dimensional Lorentz group SL(2,C) acts manifestly

as the group of conformal transformations of the two-sphere at null infinity, called the celestial

sphere. Amplitudes in which asymptotic states are in this conformal basis are then called celestial

amplitudes [2–5]. Because they make conformal covariance manifest, celestial amplitudes offer a

novel opportunity to study properties of amplitudes that may be obscured in a plane wave basis.

Moreover, it is hoped that studying celestial amplitudes will shed light on aspects of a possible

flat space holographic principle, where a CFT living on the celestial sphere would be dual to a

bulk theory living on asymptotically flat spacetime.

Recently, much work has been done in studying celestial amplitudes. In particular, the con-

nection between the Ward identities of asymptotic symmetries of asymptotically flat spacetime

and the well-known soft theorems of gauge and gravity amplitudes [6–8] has provoked a body

1

of work on the conformal analog of soft theorems [9–19]. This has revealed an interesting map

between the energetically soft expansion in the plane wave basis and the residues of poles of ce-

lestial amplitudes that arise from conformally soft limits of the scaling dimensions of the external

states under the SL(2,C) Lorentz group.

Due to their complicated analytic structure (even at tree-level) progress in understanding

detailed properties of celestial amplitudes has been relatively slow though compared to their

counterparts in the plane wave basis. Given that the Mellin transform to the conformal basis

mixes the UV and the IR, it is curious yet reassuring that universal aspects of amplitudes such

as soft factorization theorems survive in the conformal basis. In general, if we expect some

feature of amplitudes to reflect fundamental properties of the perturbative regime of quantum

field theory, then these features should survive a change of basis.

One such remarkable feature are the double copy relations [20], which states that gravitational

amplitudes can be obtained by a well-defined “squaring” of gauge theory amplitudes. These

relations are known to hold to all multiplicities at tree-level [21–24] and can be seen as implied

by string theory relations [21, 22, 25–34] in their low energy limit. They also hold at loop-level

in a plethora of cases and for many pairs of field theories, see [35] for a comprehensive review.

Remarkably, the double copy has been shown to hold even for amplitudes around some curved

backgrounds [36–41], and there are also versions of the double copy relating full non-linear

solutions of gauge theory and gravity [42–55]. Thus it is reasonable to expect the double copy

to also hold in some form after a change of basis for the asymptotic particles.

In this paper we show that this is indeed the case for celestial amplitudes by providing a

celestial double copy prescription. The main issue to be addressed is how to extricate the features

of the double copy that survive in a setting where momentum conservation is not manifest. This

is not dissimilar to the situation for amplitudes around a curved background, and as shown

in [36–40] there a notion of the double copy persists. Manifest momentum conservation is also

lost in the conformal basis, even though the background is still flat1. In this sense, celestial

amplitudes may be viewed as sitting between the usual amplitudes in a plane wave basis in

flat backgrounds and amplitudes around non-trivial backgrounds. This provides yet another

incentive to study the double copy for the celestial amplitudes, as it presents aspects of curved

space amplitudes without the extra complications of non-trivial backgrounds.

We will show below that there exists a well-defined procedure for a celestial double copy. This

requires a generalization of the usual squaring of numerators to first promoting these numerators

to differential operators acting on external scalar wavefunctions and then squaring them. We

show explicitly how this procedure works for three and four-point celestial amplitudes, and

comment on extensions to all multiplicities.

1Nevertheless, the action of the Poincare group still gives non-trivial constraints for celestial amplitudes [16,56].

2

The paper is organized as follows. In section 2 we introduce celestial amplitudes and the

conformal basis of wavefunctions obtained from a Mellin transform of the plane wave basis. We

review the basic aspects of the double copy and how it is intimately tied to the plane wave basis

before generalizing in section 3 the squaring procedure to more general backgrounds. In section 4

we discuss our proposal for the celestial double copy which requires promoting particle momenta

to differential operators acting on external wavefunctions. We apply this procedure to three-

point amplitudes in section 4.1 and to four-point amplitudes in section 4.2. Finally we comment

about its generalization to higher points in section 5 and end with a set of open questions.

2 Background

This section provides a brief review of some of the salient features of the double copy and of

celestial amplitudes. In section 2.1 we discuss the map between the plane wave basis usually

employed in studying scattering amplitudes to the basis of so called conformal primary wave-

functions used for celestial amplitudes which we introduce in section 2.2. This offers a novel

opportunity to study properties of amplitudes for which conformal covariance is manifest but

translation symmetry is obscured. The double copy relation between gauge theory and gravity is

typically studied in the plane wave basis which we briefly review in section 2.3 before discussing

in section 3 its generalization to settings where momentum conservation is obscured or lost. We

work in four-dimensional Minkowski space R1,3 with spacetime coordinates Xµ with µ = 0, 1, 2, 3

and denote by z, z a point on S2. We consider massless scattering processes where states are

labeled by null momentum four-vectors kµ, such that k2 = 0.

2.1 Momentum Basis vs Conformal Basis

Scattering problems of gauge bosons are usually studied in the plane wave basis which consists

of spin-one wavefunctions

εµ;`(k)e±ik·X , ` = ±1 (2.1)

in Lorenz gauge, and spin-two wavefunctions

εµν;`(k)e±ik·X , ` = ±2 (2.2)

in harmonic (de Donder) gauge. Here εµ;`(k) are the polarization vectors for helicity ` one-particle

states which satisfy ε`(k) ·k = 0, ε±(k)∗ = ε∓(k), and ε`(k) ·ε`′(k)∗ = δ``′ . The polarization tensor

is taken to be εµν;`(k) = εµ;`(k)εν;`(k). Plane wave solutions are thus labeled by the spatial

momentum ~k, the four-dimensional helicity ` and a sign distinguishing incoming from outgoing

states. The plane wave basis makes translation symmetry and momentum conservation manifest

3

but obscures other features. This includes transformation properties of amplitudes under the two-

dimensional global conformal group which arises as the action of the four-dimensional Lorentz

group SL(2,C) on the celestial sphere at null infinity.

To make conformal symmetry manifest in scattering amplitudes, an alternative basis of mass-

less wavefunctions was constructed in [1], called conformal primary wavefunctions. These are

labeled by a point (z, z) on the celestial sphere, and the quantum numbers under the conformal

group, namely the conformal dimension ∆ and the two-dimensional spin J , as well as a sign

distinguishing incoming from outgoing states. Writing the momentum four-vector as

kµ = ωqµ(z, z) , (2.3)

we define the null vector2

qµ(z, z) = (1 + zz, z + z,−i(z − z), 1− zz) , (2.4)

which under an SL(2,C) transformation z → z′ = (az + b)/(cz + d), z → z′ = (az + b)/(cz + d)

transforms as a vector up to a conformal weight, qµ → qµ′ = |cz + d|−2Λµνqν . Here a, b, c, d ∈ C

with ad − bc = 1 and Λµν is the associated SL(2,C) group element in the four-dimensional

representation. The derivatives with respect to z and z correspond to the polarization vectors

of, respectively, helicity +1 and −1 one-particle states propagating in the qµ direction, namely

∂zqµ =√

2εµ+(q) and ∂zqµ =√

2εµ−(q). The polarization tensors of helicity +2 and −2 states are

obtained as above by the product of same helicity polarization vectors.

The defining property of massless conformal primary wavefunctions is that they transform as

two-dimensional conformal primaries with spin J and conformal dimension ∆ under the SL(2,C)

Lorentz group and satisfy of course the bulk equations of motion. Massless spin-zero conformal

primaries satisfying the scalar wave equation ∂µ∂µφ∆,± = 0 are related to the plane wavefunctions

by a Mellin transform3

φ∆,±(X; z, z) =

∫ ∞0

dωω∆−1e±iωq·X−εq0ω =

(∓i)∆Γ(∆)

(−q ·X ∓ iεq0)∆, (2.5)

where the ± label denotes outgoing (+) and incoming (−) wavefunctions. Under SL(2,C)

transformations (2.5) behaves as a two-dimensional scalar conformal primary with dimension ∆

φ∆,±(

ΛµνX

ν ;az + b

cz + d,az + b

cz + d

)= |cz + d|2∆φ∆,±(Xµ; z, z) . (2.6)

2We use −+ ++ signature here while we switch to −+−+ signature for three-point amplitudes in which the

null vector is given by qµ(z, z) = (1 + zz, z + z, z − z, 1 − zz). We will not need the explicit expressions for qµ

though, only the properties of the polarization vectors stated above.3The iε prescription is introduced to circumvent the singularity at the light sheet where q ·X = 0.

4

We can then construct massless spin-one wavefunctions as

V ∆,±µ;J (Xµ; z, z) = εµ;J(z, z)φ∆,±(X; z, z) , J = ±1 , (2.7)

and spin-two wavefunctions as

V ∆,±µν;J (Xµ; z, z) = εµν;J(z, z)φ∆,±(X; z, z) , J = ±2 , (2.8)

where we identified the two-dimensional spin J with the four-dimensional helicity `. These sat-

isfy, respectively, the Maxwell equations and the linearized Einstein equations. Up to pure gauge

terms4, the wavefunctions (2.7)-(2.8) transform covariantly under SL(2, C) Lorenz transforma-

tions and as two-dimensional conformal primaries with, respectively, spin J = ±1 and J = ±2

and conformal dimension ∆ [1,9]. In [1] they were shown to form a complete δ-function normal-

izable basis for ∆ ∈ 1 + iR on the principal continuous series of the SL(2,C) Lorentz group5.

Under the Mellin transform, the basis of finite energy ω > 0 plane wavefunctions is thus mapped

to a basis of conformal primary wavefunctions with continuous complex conformal dimension

∆ ∈ 1 + iR. The two bases are summarized in Table 1.

Bases Plane Waves Conformal Primary Wavefunctions

Notation εµ;`(k)e±ik·X , εµν;`(k)e±ik·X V ∆,±µ;J (Xµ; z, z), V ∆,±

µν;J (Xµ; z, z)

Continuous Labels ~k ∆, z, z

Discrete Labels 4d Helicity ` 2d Spin J

incoming vs outgoing incoming vs outgoing

Table 1: Notation and labels for the bases of plane waves and conformal primary wavefunctions.

2.2 Celestial Amplitudes

Conformal properties of massless scattering amplitudes are made manifest in celestial amplitudes

which are obtained with external states in a conformal basis rather than a plane wave basis. We

consider scattering processes of massless particles with momenta6

kµj = sjωjqµj (zj, zj) , (2.9)

4For certain values of ∆ these correspond to large gauge transformations of asymptotically flat spacetime.5A subtlety arises for zero-modes with ∆ = 1 which was addressed in [10].6Here and in the following we absorb the sign distinguishing incoming from outgoing states into the definition

of the four-momentum.

5

with ωj > 0 and null vectors qµj directed at points (zj, zj) on the celestial sphere at null infinity

where the particles cross, with sj = ± labeling outgoing/incoming states. Given an amplitude

in the plane wave basis

An({ωj, `j; zj, zj}) = An({ωj, `j; zj, zj}) δ(4)(∑n

j=1 sjωjqj

), (2.10)

the celestial amplitude is obtained by a Mellin transform on each of the external particles as

An({∆j, Jj; zj, zj}) =n∏j=1

(∫ ∞0

dωjω∆j−1j

)An({ωj, `j; zj, zj}) . (2.11)

Under SL(2,C) Lorentz transformations celestial amplitudes transform as [1, 3, 5]

An({

∆j, Jj;azj + b

czj + d,azj + b

czj + d

})=

n∏j=1

((czj +d)∆j+Jj(czj + d)∆j−Jj

)An({∆j, Jj; zj, zj}) , (2.12)

where ∆j are the conformal dimensions and Jj ≡ `j the spins of operators inserted at points

(zj, zj) ∈ S2. Celestial amplitudes thus share conformal properties with correlation functions on

the celestial sphere. This offers a novel opportunity to study scattering amplitudes in a basis that

makes conformal symmetry manifest. At the same time translation invariance becomes obscured

in the conformal basis. Hence, properties of amplitudes that make heavy use of momentum

conservation appear to be lost in celestial amplitudes. An example of this is the perturbative

double copy relating Einstein gravity amplitudes to the “square” of Yang-Mills amplitudes [35].

2.3 Double Copy

Let Γn be the set of trivalent graphs with n external legs and no internal closed loops, corre-

sponding to tree-level amplitudes7. Yang-Mills amplitudes can be presented as a sum over these

trivalent graphs as

AYMn = δ(4)( n∑j=1

kµj

) ∑γ∈Γn

cγnγΠγ

, (2.13)

where, for a particular trivalent graph γ, the numerators cγ are its color factors, that is, traces

of the Lie algebra matrices of the external particles. The numerators nγ are the kinematical

factors given by polynomials in the momenta kj and polarizations εj of the external particles.

The denominator Πγ is a product of the scalar propagators associated with the graph γ. The

color factors cγ are not all independent due to the Lie algebra Jacobi identity. For a triple of

graphs related by a BCJ move [57], see figure 1, the color factors obey

cs − ct + cu = 0 . (2.14)

6

− +

cs ct cu

= 0

1

2 3

4 1

32

41

2 3

4

Figure 1: Triple of graphs related by a BCJ move.

At tree-level we can always find kinematical numerators nγ that obey the same identities as

the color factors

ns − nt + nu = 0 . (2.15)

These are called color-kinematics dual numerators, or BCJ satisfying numerators [57]. Substi-

tuting the color factors in (2.13) by these numerators we obtain the expression

AGn = δ(4)( n∑j=1

kµj

) ∑γ∈Γn

n2γ

Πγ

, (2.16)

which, surprisingly, is a representation for an amplitude involving external gravitons - this is the

double copy procedure [24]. Since the numerators nγ originally came from Yang-Mills theory we

say that gravity is, in this sense, the square of gauge theory. Note that this presentation of the

double copy relies heavily on the fact that external particles are in the plane wave basis, and on

momentum conservation which appears as an overall factor in both amplitudes. Both of these

will require a generalization in order to arrive at a double copy for celestial amplitudes which we

discuss in section 4.

3 Double Copy Revisited

In this section we rewrite the Yang-Mills amplitude (2.13) and the gravitational amplitude (2.16)

in a form more suitable for the generalization to celestial amplitudes. This procedure, introduced

in [36] in the context of scattering on plane wave backgrounds, is the most natural generalization

when computing amplitudes using the usual Feynman rules in a setting that lacks momentum

conservation.

We consider a representation of the amplitudes with leftover spacetime integrals, that is, the

representation obtained from position space Feynman diagrams, and make use of the integral

7This representation also holds for loop integrands, but we restrict ourselves here to tree-level amplitudes.

7

representation of the δ-function

δ(4)( n∑j=1

kµj

)=

∫d4X

(2π)4e∑n

j=1 ikj ·X . (3.1)

Since we will be working with integrands and our manipulations are essentially algebraic, we

allow ourselves to use complexified momenta for the external particles being agnostic about

the signature of spacetime. We also note that, although we use four-dimensional notation, the

manipulations in this section are valid in arbitrary spacetime dimension.

3.1 Three-Point Amplitudes

To illustrate the procedure, consider first the three-point Yang-Mills amplitude. From the Yang-

Mills Lagrangian we extract the three-point vertex that contributes to this amplitude, represent-

ing it as

AYM3 =1

2Tr

∫d4X

(2π)4(aµ1a

ν2∂νa3µ − aµ2aν1∂νa3µ + cyclic) , (3.2)

with

aµ = T aεµeik·X (3.3)

the wavefunction for an external gauge theory state in the plane wave basis. It is then trivial to

write the three-point amplitude as

AYM3 = f a1a2a3i

2

∫d4X

(2π)4(ε1 · (k2 − k3) ε2 · ε3 + cyclic)

3∏j=1

eikj ·X . (3.4)

At this point not much seems to be gained since we could simply perform the integral over

spacetime to obtain the momentum conserving δ-function, but this is a luxury we lack in more

general settings. The double copy of (3.4) is immediate

− 1

4

∫d4X

(2π)4(ε1 · (k2 − k3) ε2 · ε3 + cyclic)2

3∏j=1

eikj ·X . (3.5)

For the gravitational amplitude we use

AG3 =1

2

∫d4X

(2π)4(hµν1 ∂µh2ρσ∂νh

ρσ3 − 2hρν1 ∂µh2ρσ∂νh

µσ3 + permutations) (3.6)

as three-point interaction. This is most easily obtained from the action introduced in [58, 59]

which was checked in [36] to match the three-point interaction from of the Einstein-Hilbert

Lagrangian. Using this interaction the amplitude is

AG3 = −1

2

∫d4X

(2π)4

((ε2 · ε3)2 ε1 · k2 ε1 · k3 − 2 ε1 · ε2 ε2 · ε3 ε3 · k2 ε1 · k3 + perm.

) 3∏j=1

eikj ·X .

(3.7)

8

The expressions (3.5) and (3.7) can be shown to be equal, up to a minus sign, by momentum

conservation, or equivalently, integration by parts.

From this simple example we see that the equivalent of extracting the overall momentum

conserving δ-function in the double copy is to consider the integrand of the spacetime integral.

Moreover, the full integrand is not doubled copied, we must first extract the scalar wavefunctions

eikj ·X as an overall factor. This uses the fact that an external gravitational state in the plane

wave basis is represented as

hµν = εµενeik·X , (3.8)

such that the scalar wavefunction is a common factor between the gravitational and Yang-Mills

states. This remains true for states in the conformal basis, and we will make use of it in our

celestial double copy prescription. Note that (3.5) is not manifestly permutation symmetric. In

order to see it a series of integration by parts have to be used. These integration by parts are

equivalent to using momentum conservation, and so are trivial to use in this example since there

is only one integral. The situation becomes more involved at higher points.

3.2 Four-Point Amplitudes

The representation for a generic amplitude can be exemplified by the four-point amplitude,

which we now turn to. The four-point amplitude in Yang-Mills theory has three contributions

from two three-point vertices tied together by a propagator (the three exchange channels) and a

contributions from the four-point contact interaction. We represent this decomposition by

AYM4 = As4 +At4 +Au4 +Acontact4 . (3.9)

The contribution from the s-channel is

As4 = −f a1a2bf a3a4b

∫d4X

(2π)4

d4Y

(2π)4[ε1 · ε2 (k1 − k2)µ + 2 ε1 · k2 ε

µ2 − 2 ε2 · k1 ε

µ1 ] Gµν(X, Y )

× [ε3 · ε4 (k4 − k3)ν − 2 ε3 · k4 εν4 + 2 ε4 · k3 ε

ν3] ei(k1+k2)·Xei(k3+k4)·Y , (3.10)

and other exchange channels are simply given by substitutions on the expression for the s-channel:

2↔ 3 for the t-channel, and (2, 3, 4)→ (4, 2, 3) for the u-channel. Here

Gµν(X, Y ) = ηµν G(X, Y ) = ηµν

∫d4k

eik·(X−Y )

k2(3.11)

9

is the usual propagator. The contact term can be read off directly from the four-point vertex in

the Lagrangian

Acontact4 =

(f a1a2bf a3a4b(ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

+ f a1a3bf a2a4b(ε1 · ε2 ε3 · ε4 − ε1 · ε4 ε2 · ε3)

+ f a1a4bf a2a3b(ε1 · ε2 ε3 · ε4 − ε1 · ε3 ε2 · ε4))∫ d4X

(2π)4

4∏j=1

eikj ·X .

(3.12)

Following [38] we split the contact term into three contributions according to the corresponding

color factor. For example, the term in (3.12) proportional to the structure constants cs =

f a1a2bf a3a4b corresponds to the s-channel, since it has the same color structure. This term can

be rewritten as

(ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

∫d4X

(2π)4

d4Y

(2π)4�XG(X, Y )ei(k1+k2)·Xei(k3+k4)·Y

= −(ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )ei(k1+k2)·Xei(k3+k4)·Y

× (k1 · k2 + k3 · k4) ,

(3.13)

where we used the fact that �XG(X, Y ) = (2π)4δ(4)(X − Y ) = �YG(X, Y ). Performing similar

manipulations in the other terms we can write the four-point amplitude (3.9) as

AYM4 =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y ) (csnsΦs + ctntΦt + cunuΦu) , (3.14)

with

cs = f a1a2bf a3a4b , ct = f a1a3bf a2a4b , cu = f a1a4bf a2a3b ,

Φs = ei(k1+k2)·Xei(k3+k4)·Y , Φt = ei(k1+k3)·Xei(k2+k4)·Y , Φu = ei(k1+k4)·Xei(k2+k3)·Y ,(3.15)

and

ns = −[ε1 · ε2 (k1 − k2)µ + 2 ε1 · k2 εµ2 − 2 ε2 · k1 ε

µ1 ] ηµν

× [ε3 · ε4 (k4 − k3)ν − 2 ε3 · k4 εν4 + 2ε4 · k3 ε

ν3]

− (ε1 · ε3 ε2 · ε4 + ε1 · ε4 ε2 · ε3)(k1 · k2 + k3 · k4) . (3.16)

Numerators for other channels are given by simple permutations. One can check that these

numerators obey color-kinematics duality using momentum conservation. To avoid using mo-

mentum conservation explicitly requires a series of integration by parts and the fact that

∂XG(X, Y ) = −∂YG(X, Y ). The double copy for the four-point amplitude is given by substitut-

ing the color factors by kinematical numerators c→ n, yielding the gravitational amplitude

AG4 =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )

((ns)

2Φs + (nt)2Φt + (nu)

2Φu

). (3.17)

10

This expression reduces to a sum over trivalent graphs given in (2.16) after integrating over the

spacetime points. It might seem that not much has been gained by writing the amplitudes in this

cumbersome form. But as pointed out in [38] this is essential to generalize the double copy to a

setting were momentum conservation is lost, and will be critical for the celestial double copy.

By similar arguments an n-point amplitude can be written as an integral over n−2 spacetime

coordinates, mimicking the decomposition into trivalent graphs. This is, of course, nothing

more than making use of the position space Feynman rules and decomposing contact terms into

trivalent vertices. In a plane wave background it was seen that the kinematical numerators

n depend on spacetime points, making identities obtained from integration by parts quite non-

trivial. This fact also complicates color-kinematics duality, as it is now not immediately clear how

to compare numerators at different points. This feature will also appear for celestial amplitudes,

and we will comment more about it in the next section.

4 Double Copy for Celestial Amplitudes

Using the framework reviewed in the previous section we consider a representation of the celestial

amplitudes (2.11) with leftover spacetime integrals. The celestial double copy for an n-point

amplitude will then be manifest at the level of integrands for n− 2 spacetime integrals.

4.1 Three-Point Amplitudes

To compute celestial gluon and graviton amplitudes we use as the external states the conformal

wavefunctions (2.7)-(2.8). For simplicity we omit the ± labels on the wavefunctions, as well as the

helicity index J since the manipulations in this section do not depend on these, and stop writing

explicitly the dependence on the conformal dimension ∆. Thus, we write the wavefunctions as

Vµ(X) = T aεµφ(X) , Vµν(X) = εµενφ(X) , (4.1)

where we added a color index for the spin-one wavefunction and emphasized the spacetime

dependence. In analogy to how spacetime derivatives act on wavefunctions in a momentum basis

we define the generalized celestial momentum Kµ as

∂µφ(X) =∆qµ

(−q ·X)φ(X) ≡ Kµ(X)φ(X) . (4.2)

The explicit dependence of Kµ on spacetime is to be contrasted with the spacetime independent

kµ of the plane wave basis and is a precursor for the failure of the naive squaring procedure

for celestial amplitudes. Note that this dependence on spacetime points is also present in the

generalized momenta around plane wave backgrounds [36].

11

Inserting the external states (4.1) into the Mellin transformed three-point vertices for Yang-

Mills (3.2) and gravity (3.6) we obtain the celestial three-gluon amplitude

AYM3 = f a1a2a3

1

2

∫d4X

(2π)4((ε1 · (K2 −K3) ε2 · ε3 + cyclic)

3∏j=1

φj

≡ f a1a2a3

∫d4X

(2π)4nYM

3∏j=1

φj ,

(4.3)

and the celestial three-graviton amplitude

AG3 =1

2

∫d4X

(2π)4

((ε2 · ε3)2 ε1 ·K2 ε1 ·K3 − 2 ε1 · ε2 ε2 · ε3 ε3 ·K2 ε1 ·K3 + perm.

) 3∏j=1

φj . (4.4)

In the plane wave basis we would proceed by squaring the Yang-Mills integrand and use momen-

tum conservation∑

j kµj = 0, as well as εj ·kj = 0 to show equality with the gravity integrand. In

the celestial case, the conformal wavefunctions still obey εj ·Kj = 0 due to our choice of gauge,

but because the generalized momenta are spacetime dependent momentum conservation is no

longer manifest in (4.3)-(4.4):8

3∑j=1

Kµj (X) =

3∑j=1

∆jqµj

(−qj ·X)6= 0 . (4.5)

A naive double copy of the celestial Yang-Mills numerator

(nYM)2 =1

4(ε1 · (K2 −K3) ε2 · ε3 + ε2 · (K3 −K1) ε3 · ε1 + ε3 · (K1 −K2) ε1 · ε2)2 , (4.6)

gives terms proportional to (εi · εj)2, such as

((ε1 · (K2 −K3) ε2 · ε3)2 = (ε2 · ε3)2((ε1 ·K2)2 − 2 ε1 ·K2 ε1 ·K3 + (ε1 ·K3)2) , (4.7)

and cross-terms like

2 (ε3 · (K1 −K2) ε1 · ε2)(ε2 · ε3 ε1 · (K2 −K3)) . (4.8)

One might hope to recover the corresponding terms in the gravity integrand (4.4) via integration

by parts as in the plane wave basis. However, it is easy to see that due to the spacetime

dependence of the celestial momenta, performing integration by parts on terms quadratic in the

same momenta, such as (ε1 ·K3)2 or (ε1 ·K2)(ε3 ·K2), produces extra terms which are not present

in the gravity amplitude. This results in an obstruction to the naive double copy even in the

simplest example of three-point amplitudes. It is clear that we need a generalization of the naive

squaring procedure to have any chance of double copying celestial amplitudes.

8While in (4.3) we still have∑3j=1K

µj (X) ' 0 via integration by parts, this is no longer true after squaring.

12

Celestial double copy: Our proposal for a celestial double copy is to introduce formal differ-

ential operators Ki that replace the generalized celestial momenta Ki in the numerators. These

operators are defined to act on the conformal wavefunctions φi as

Kµi φj(X) = Kµi (X)φi(X)δij . (4.9)

Notice that these differential operators keep track of particle labels and thus are not equivalent

to the usual partial derivatives. Instead they admit a representation as

Kµi = ∆iqµi e

∂∆i , (4.10)

and are thus related, but not equal, to the action of the translation generator of the Poincare

algebra acting on the coordiantes of the i-th particle [56]9. Promoting the celestial momenta Ki

in (4.3) to the differential operators Ki and squaring gives

(NYM)2 =1

4(ε1 · (K2 −K3)(ε2 · ε3) + cyclic)2

=1

4

(ε1 · (K2 −K3) ε1 · (K2 −K3) (ε2 · ε3)2

+2 ε1 · ε2 ε2 · ε3 ε3 · (K1 −K2) ε1 · (K2 −K3) + cyclic) ,

(4.11)

which acts on the external wavefunctions in the three-point amplitude. Evaluating this action

explicitly, the term proportional to 14

(ε2 · ε3)2 is(1 +

1

∆2

)(ε1 ·K2)2 − 2 ε1 ·K2 ε1 ·K3 +

(1 +

1

∆3

)(ε1 ·K3)2 , (4.12)

while the term proportional to 12ε1 · ε2 ε2 · ε3 is

ε3 ·K2 ε1 ·K3 − ε3 ·K1 ε1 ·K3 + ε3 ·K1 ε1 ·K2 −(

1 +1

∆2

)ε3 ·K2 ε1 ·K2 , (4.13)

and similar expressions for their permutations. This differs from the corresponding terms (4.7)

and (4.8) of the naive double copy by factors of 1 + 1∆i

in terms involving the same celestial

momenta. These arise from

Kµi Kνi φi(X) =

(1 +

1

∆i

)Kµi (X)Kν

i (X)φi(X) , (4.14)

and are precisely what is needed to cancel the additional terms that get generated when integrat-

ing by parts to match (4.11) to the gravity integrand. To see this explicitly we integrate by parts

9We thank Atul Sharma for pointing out this relation.

13

all the terms in the (4.12) and (4.13) that are not already present in the gravity integrand (4.4).

This yields the following relations, which hold inside the spacetime integral:10

ε1 ·K2 ε1 ·K3 ' −(

1 +1

∆2

)(ε1 ·K2)2 ' −

(1 +

1

∆3

)(ε1 ·K3)2 , (4.15)

and

ε3 ·K2 ε1 ·K3 ' −ε3 ·K1 ε1 ·K3 ' ε3 ·K1 ε1 ·K2 ' −(

1 +1

∆2

)ε1 ·K2 ε3 ·K2 , (4.16)

along with similar expressions for their permutations. This yields the final result

(NYM)2

3∏i=1

φi = −(ε2 · ε3)2 ε1 ·K2 ε1 ·K3 + 2 ε1 · ε2 ε2 · ε3 ε3 ·K2 ε1 ·K3 + cyclic)3∏i=1

φi , (4.17)

which is, up to a sign, exactly the gravitational numerator in (4.4). Hence, the celestial three-

gluon amplitude double copies into the celestial three-graviton amplitude.

4.2 Four-Point Amplitudes

We now discuss the generalization of the celestial double copy to four-point tree-level amplitudes.

The new ingredients appearing in the four-point amplitude are enough to motivate a conjecture

a higher-point extension, which we come back to at the end. These new ingredients are the

presence of the four-point contact term and the exchange diagrams.

As in the plane wave basis, we split the Yang-Mills amplitude into separate contributions

AYM4 = As4 + At4 + Au4 + Acontact4 , (4.18)

withAcontact4 =

(f a1a2bf a3a4b(ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

+ f a1a3bf a2a4b(ε1 · ε2 ε3 · ε4 − ε1 · ε4 ε2 · ε3)

+ f a1a4bf a2a3b(ε1 · ε2 ε3 · ε4 − ε1 · ε3 ε2 · ε4))∫ d4X

(2π)4

4∏j=1

φj(X)

(4.19)

the contribution from the contact term, and the s-channel contribution is

As4 = f a1a2bf a3a4b

∫d4X

(2π)4

d4Y

(2π)4[ε1 · ε2 (K1 −K2)µ

+ 2 ε1 ·K2 εµ2 − 2 ε2 ·K1 ε

µ1 ](X) Gµν(X, Y )

× [ε3 · ε4 (K4 −K3)ν − 2 ε3 ·K4 εν4 + 2 ε4 ·K3 ε

ν3](Y ) φ1(X)φ2(X)φ3(Y )φ4(Y ) . (4.20)

10E.g.∫

d4X(2π)4 φ1φ2φ3 ε1 ·K2 ε1 ·K3 = −

∫d4X(2π)4 ε1 · ∂(ε1 ·K2 φ1φ2)φ3 = −

(1 + 1

∆2

) ∫d4X(2π)4 φ1φ2φ3 (ε1 ·K2)2.

14

The difference to the discussion in section 3.2 is that the plane wavefunctions in (3.10) and (3.12)

are replaced by the scalar conformal primary wavefunctions φj in the celestial case via the Mellin

transform, and the momenta kµj are replaced by the celestial momenta Kµj . For the purpose of

the argument we only need two properties of the propagator Gµν(X, Y ), namely

Gµν(X, Y ) = ηµν G(X, Y ) , (4.21)

and

2XG(X, Y ) = (2π)4δ(4)(X − Y ) = 2YG(X, Y ) . (4.22)

Now, due to the spacetime-dependence of the celestial momenta, we cannot easily perform

the spacetime integrals, hence the necessity of introducing this spacetime representation for the

double copy. We again split the contact term into three contributions according to the color

factors. The term proportional to f a1a2bf a3a4b in (4.19) can be written as

(ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

∫d4X

(2π)4

d4Y

(2π)4�XG(X, Y )φ1(X)φ2(X)φ3(Y )φ4(Y )

= (ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )φ1(X)φ2(X)φ3(Y )φ4(Y )

× (K1 ·K2(X) +K3 ·K4(Y )) , (4.23)

where we used (4.22) and the fact that the celestial momenta are null, K2i (X) = 0. Performing

similar manipulations in the other terms we write the four-point amplitude (4.18) as

AYM4 =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )

(csnsΦs + ctntΦt + cunuΦu

), (4.24)

with

cs = f a1a2bf a3a4b , ct = f a1a3bf a2a4b , cu = f a1a4bf a2a3b ,

Φs = φ1φ2(X)φ3φ4(Y ) , Φt = φ1φ3(X)φ2φ4(Y ) , Φu = φ1φ4(X)φ2φ3(Y ) ,(4.25)

mimicking our earlier expressions in the plane wave basis. The numerators are given by

ns = [ε1 · ε2 (K1 −K2)µ + 2 ε1 ·K2 εµ2 − 2 ε2 ·K1 ε

µ1 ](X) ηµν

× [ε3 · ε4 (K4 −K3)ν − 2 ε3 ·K4 εν4 + 2 ε4 ·K3 ε

ν3](Y )

+ (ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)(K1 ·K2(X) +K3 ·K4(Y )) , (4.26)

and appropriate permutations of it.

We now follow the same procedure as in the three-point case and promote all celestial mo-

menta Ki to differential operators acting on the scalar wavefunctions

Ns = [ε1 · ε2 (K1 −K2)µ + 2ε1 · K2 εµ2 − 2 ε2 · K1 ε

µ1 ] ηµν

× [ε3 · ε4 (K4 −K3)ν − 2 ε3 · K4 εν4 + 2 ε4 · K3 ε

ν3]

+ (ε1 · ε3 ε2 · ε4 − ε1 · ε4 ε2 · ε3)(K1 · K2 +K3 · K4) . (4.27)

15

The gravitational amplitude is obtained by substituting each color factor by its corresponding

kinematical operator

AG4 =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )

((Ns)2Φs + (Nt)2Φt + (Nu)2Φu

). (4.28)

To check it, we rely on the fact that any set of numerators of a trivalent graph decomposition

of a four-point Yang-Mills amplitude in the plane wave basis obeys color-kinematics. Hence, the

Yang-Mills numerators square to a gravitational amplitude which we repeat here for convenience

AG4 =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )

((ns)

2Φs + (nt)2Φt + (nu)

2Φu

), (4.29)

with the numerators given by (3.16) as well as its permutations, and

Φs = ei(k1+k2)·Xei(k3+k4)·Y Φt = ei(k1+k3)·Xei(k2+k4)·Y Φu = ei(k1+k4)·Xei(k2+k3)·Y . (4.30)

To show that (4.28) is the Mellin transform of (4.29) we make use of a few facts about the Mellin

transform. Note that (4.29) depends on the energy of each particle k0i in a polynomial fashion.

In fact, this dependence is no higher than quadratic on each energy, as can be seen from the

square of (3.16). When performing the Mellin transform there are only two cases to consider:

the numerator is linear in the energy∫dωiω

∆i−1i (ikµi )eiki·X = Kµ

i φi(X) , (4.31)

and the numerator is quadratic in the same energy∫dωiω

∆i−1i (ikµi )(ikνi )eiki·X =

(1 +

1

∆i

)Kµi K

νi φi(X) . (4.32)

Since this mirrors the action of the Ki operators on the conformal wavefunctions, Mellin trans-

forming (4.29) gives precisely (4.28) after the action of the Ki operators. Hence the celestial

four-gluon amplitude double copies into the celestial four-graviton amplitude.

5 Discussion

The construction of the celestial double copy demonstrated above for three and four particles

generalizes to higher point amplitudes. It entails first rewriting the amplitude in terms of n− 2

spacetime integrals by opening up four-point contact terms, and then promoting the celestial

momenta to differential operators. In this representation, the celestial double copy is given in

terms of squares of differential operators acting on the conformal wavefunctions.

16

In fact, this gives an interesting presentation for the amplitudes where, using (4.10), we

can express both gauge and gravity amplitudes as operators acting on integrals of the scalar

wavefunctions. For example, the three-point gauge and gravity amplitudes are given by operators

acting on the scalar three-point amplitude:

AYM3 = f a1a2a3NYM

∫d4X

(2π)4

3∏j=1

φ∆j

j (X) , AG3 = (NYM)2

∫d4X

(2π)4

3∏j=1

φ∆j

j (X). (5.1)

At higher points we dress each analog of the scalar trivalent graphs with operators. For example,

we write the four-gluon amplitude as

AYM4 = cs Ns s + ct Nt t + cu Nu u , (5.2)

and four-graviton amplitude as

AG4 = (Ns)2 s + (Nt)2 t + (Nu)2 u , (5.3)

where we write the s-channel contribution to the cubic four-point scalar amplitude as

s(∆1,∆2,∆3,∆4) =

∫d4X

(2π)4

d4Y

(2π)4G(X, Y )φ∆1

1 (X)φ∆22 (X)φ∆3

3 (Y )φ∆44 (Y ) , (5.4)

with the other channels, t and u, obtained in an alogous way.

In the expressions above we reinstated the dependence of the scalar wavefunctions φi on the

conformal dimensions ∆i to stress that those are the variables on which the N operators act.

This way of presenting the amplitude, using differential operators on the external wavefunctions,

is very reminiscent of recent results on Ambitwistor strings in AdS spacetime [60, 61], where

numerators are given by operators which seem to obey an operator version of the double copy.

The difference here is that we had to construct the propagators by hand using the cubic scalar

amplitudes, while in the Ambitwistor string one can use the scattering equations to do so.

Ambitwistor string theories also give formulas for amplitudes which manifest a double copy

structure [62]. These have been used to study some properties of celestial amplitudes in [13,63,64].

Amplitudes for an arbitrary number of external states are also given by these theories in very

compact forms and, perhaps, the best way to present the celestial double copy for arbitrary

multiplicities would be in the Ambitwistor setting.

Related to this is the question of how our proposal for the celestial double copy of the

four-dimensional spacetime integrands translates into a statement about the two-dimensional

correlators directly in the putative celestial CFT. In the plane wave basis, the correct way to

open up the four-point contact terms is to enforce that the numerators obey color-kinematics.

Translating these BCJ satisfying numerators into differential operators in the conformal basis

17

then guarantees that the square of these operators will correspond to celestial gravitational

amplitudes. Our celestial double copy construction therefore still makes use of constructions in

the plane wave basis which we would rather move away from. This requires finding structures

that do not heavily rely on the use of momentum conservation. We anticipate a generalization

of the color-kinematics duality for the plane wave basis numerators to constrain which sets of

differential operators are allowed in celestial amplitudes. This could shed light on aspects of the

kinematical algebra [65] in the celestial setting, and is a first step in moving away from the plane

wave basis. We leave this very interesting problem to future work.

Acknowledgements

We would like to thank Tim Adamo, Agnese Bissi, Cindy Keeler, Sebastian Mizera, Sabrina

Pasterski and Piotr Tourkine for discussions, and the Centro de Ciencias de Benasque Pedro

Pascual where this work was initiated for its hospitality. This research is supported in part by

U.S. Department of Energy grant DE-SC0009999 and by funds provided by the University of

California.

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