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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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