# Celestial double copy from the worldsheet

Eduardo Casali  and Atul Sharma

###### Abstract

Using the ambitwistor string, we compute tree-level celestial amplitudes for
biadjoint scalars, Yang-Mills and gravity to all multiplicities. They are
presented in compact CHY-like formulas with operator-valued scattering
equations and numerators acting on a generalized hypergeometric function. With
these we extend the celestial double copy to tree-level amplitudes with
arbitrary number of external states. We also show how color-kinematics duality
is implemented in celestial amplitudes and its interpretation in terms of a
generalized twisted cohomology theory.

##  1 Introduction

The perturbative S-matrix of quantum field theories can have many interesting
properties and relations. If we take these relations to reflect some
fundamental property of the respective QFTs, we expect them to survive in some
form after a change of basis in the Hilbert space. The S-matrix is usually
computed in a basis of plane waves for external particles due to its
simplicity, but other bases might be more useful in answering certain
questions. In [1, 2, 3, 4] it was argued that a basis of external particles
transforming as conformal primaries on the sphere at the null boundary of
Minkowski space is the most appropriate one to study a potential holographic
CFT living on this sphere. Amplitudes with conformal wavefunctions as external
states were dubbed celestial amplitudes. These can be obtained from amplitudes
computed in the usual plane wave basis by a Mellin transform. This has the
effect of mixing the UV and the IR. Yet universal properties of amplitudes,
such as soft limits and IR factorization [1, 5, 6, 7, 8, 9, 10, 11, 12, 13,
14, 15, 16], survive this change of basis albeit in a different guise.
Recently, it was shown in [17] that relations between the S-matrices of Yang-
Mills and gravity known as the double copy also survives this change of basis,
at least up to four point amplitudes.

While an all-multiplicity proof could be found using the methods employed in
[17], which relied on position space Feynman rules, they soon become
cumbersome. From our experience with the tree-level amplitudes in the usual
plane wave basis we know that there should be more compact descriptions of the
S-matrix. For certain classes of celestial amplitudes in four dimensions all-
multiplicity formulas already exist in the literature: tree-level Yang-Mills
MHV and NMHV amplitudes were computed in [18]. These computations are done by
performing Mellin transforms on the energies of the external particles which,
while doable for some low point amplitudes including loop [19, 16] and string
[20] amplitudes, become highly involved for generic amplitudes. CFT inspired
methods have also been used to holographically constrain MHV amplitudes [21,
22] and it might be that further understanding of the celestial OPEs [23, 13,
24, 25] will lead to better methods.

Here we choose to use a true and tested method to generate compact expressions
for massless S-matrices, the ambitwistor string [26]. This is a worldsheet
theory from which the CHY formulas [27, 28] can be obtained as correlation
functions of vertex operators in a chiral CFT. These are a generalization of
the original Berkovits-Witten twistor string [29, 30]. Being string theories,
the sum over Feynman diagrams is replaced by an integral over the moduli space
of punctured spheres providing a very compact representation of the tree-level
S-matrix. A Mellin transform of the CHY formula was used in [8] to study
celestial soft theorems, but the Mellin integrations were not carried out
explicitly. Here we carry those out explicitly and check it against a first
principles derivation of it using the ambitwistor string with vertex operators
in the conformal basis. This gives a framework for celestial massless
S-matrices making use of the celestial scattering equations in analogy with
the usual CHY formula. The amplitudes are written as operators acting on the
Mellin transform of a scalar contact vertex which we carry out explicitly in
terms of generalized hypergeometric functions.

Another important feature of the CHY formulas and the ambitwistor string is
that the double copy relation between Yang-Mills and gravity amplitudes is
made manifest. The double copy is a procedure to obtain gravitational
amplitudes from a “square” of Yang-Mills amplitudes [31]. While in the
ambitwistor string the double copy amounts to a substitution rule, using
traditional methods it relies on kinematical numerators satisfying the so-
called color-kinematics duality [32], that is, they obey relations analogous
to Jacobi identities among color factors. Using the formulas computed using
the ambitwistor string for celestial amplitudes we show that tree-level
celestial Yang-Mills and gravity amplitudes are related by a double copy,
generalizing the procedure given for low points in [17]. Moreover, by
interpreting the numerators given by the ambitwistor string as cohomology
classes in a generalization of twisted cohomology, we can uplift the results
of [33] to the celestial case. In doing this we find a natural generalization
of color-kinematics duality to celestial amplitudes and show how to obtain
color-kinematical dual celestial numerators from the ambitwistor string.

An extra motivation to study celestial amplitudes is as a toy model for
understanding double copy and color-kinematics duality for amplitudes around
curved backgrounds. Low multiplicity amplitudes have been computed for plane
wave backgrounds in [34, 35, 36] and shown to have a double copy structure,
with some higher point results recently given in [37, 38, 39]. But
computations still relied heavily on the leftover momentum conservation and
special properties of plane wave backgrounds. Ambitwistor string computations
have also provided new formulas for amplitudes in AdS spacetimes [40, 41]
which have a structure very close to the one we find for celestial amplitudes.
We would like to find a notion of double copy that can be more easily
generalized to curved spaces and valid for a larger class of spacetimes
including asymptotically AdS spacetimes.

#### Summary of results:

  * •

We compute compact, $n$-point formulas for tree-level celestial amplitudes of
biadjoint scalars, gluons and gravitons. They are given in a universal form as
operator-valued numerators acting on the Mellin transform of the scalar
contact vertex. Analogous to the case with external plane waves we can
represent the amplitudes either as a sum over trivalent graphs (3.19), or as
integrals over the moduli space of $n$-punctured Riemann surfaces localized to
the operator-valued celestial scattering equations (3.10).

  * •

As a consequence of the computation given above, we prove that the celestial
double copy introduced in [17] is valid for amplitudes at all multiplicities.

  * •

We introduce a generalization of twisted cohomology to operator-valued twisted
forms which are the relevant objects in the CHY formulas for celestial
amplitudes. We use this, together with a straightforward generalization of the
results in [33], to define color-kinematics duality for celestial amplitudes
and show how to obtain color-kinematical dual numerators from the ambitwistor
string numerators.

This paper is organized as follows: we start in section 2 by reviewing some
facts about celestial amplitudes, CHY formulas, double copy and twisted
homology to make this paper more self-contained. The reader familiar with
these topics can safely skip these sections. Section 3 contains the formulas
for celestial amplitudes with explanations about its constituents. Next, in
section 4, we introduce the ambitwistor model and go through the calculation
used to obtain the formulas introduced previously. In section 5 we introduce a
generalization of the twisted cohomology to operator-valued forms, give a
definition of color-kinematics duality for celestial amplitudes and show how
numerators obtained from the ambitwistor string naturally obey this duality.
We finish with some discussion about this framework and possible further
generalizations in section 6.

##  2 Review

In order to make this paper self-contained we quickly review in this section
some background on recent technology used in the study of amplitudes. We keep
the reviews short and focused on what is needed in the rest of the paper. In
subsection 2.1 we review some facts about celestial amplitudes, in subsection
2.2 CHY formulas and the scattering equations are introduced. In subsection
2.3 we recall the basics of color-kinematics duality, and in subsection 2.4 we
review how the CHY formulas and double copy are to be interpreted in light of
twisted cohomology on the moduli space of punctured Riemann spheres.

###  2.1 Celestial amplitudes

Celestial amplitudes are obtained by Mellin transforming the usual momentum
space amplitudes to make manifest their transformations under the conformal
group of the celestial sphere at null infinity $\mathscr{I}^{+}$. This is
essentially a change of basis [3] on the Fock space from the plane wave basis
into a basis of conformal primary wavefunctions111There’s evidence that more
states than the ones obtained from the Mellin transform are necessary to
describe the quantum theory on $\mathscr{I}$, see [23].

Let $X$ be a point in $\mathbb{R}^{1,d+1}$, with $D:=d+2$, and denote the
massless scalar plane wave as $\mathrm{e}^{\mathrm{i}s\omega q\cdot X}$ where
$\omega$ denotes its energy, $q\in S^{d}$ is a direction on the celestial
sphere, and $s=\pm 1$ denotes if the particle is outgoing or incoming. The
scalar conformal primary wavefunction is obtained by Mellin transforming the
energy $\omega$,

$\phi_{\Delta}^{s}(X;q)=\mathcal{M}(\mathrm{e}^{\mathrm{i}s\omega q\cdot
X-\varepsilon\omega})=\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\Delta}\,\mathrm{e}^{\mathrm{i}s\omega
q\cdot
X-\varepsilon\omega}=\frac{(-\mathrm{i}\,s)^{\Delta}\,\Gamma(\Delta)}{(-q\cdot
X-\mathrm{i}\,s\,\varepsilon)^{\Delta}}\,,$ (2.1)

giving an external wavefunction that lives on the celestial sphere with a new
quantum number: the conformal dimension $\Delta$. Conformal wavefunctions for
external gluons and gravitons also exist but here we will work directly with
the Mellin transform of the plane waves,

$\begin{array}[]{cc}a^{\mathsf{a},s}_{\mu,\Delta}=\mathsf{T^{a}}\,\epsilon_{\mu}\,\phi_{\Delta}^{s}(X;q)&h^{s}_{\mu\nu,\Delta}=\epsilon_{\mu}\,\tilde{\epsilon}_{\nu}\,\phi_{\Delta}^{s}(X;q)\,,\end{array}$
(2.2)

which are given in terms of the scalar wavefunctions, the color generators
$\mathsf{T^{a}}$, and polarizations $\epsilon^{\mu},\tilde{\epsilon}^{\mu}$
that only depend on $q^{\mu}$ [4]. These wavefunctions transform as conformal
primaries up to gauge transformations [3]. Wavefunctions for massive particles
have also been worked out [42, 43] but we won’t have anything to say about
massive particles. Amplitudes are in principle computed using (2.1) and (2.2)
as external wavefunctions, but it is more convenient to simply Mellin
transform the usual amplitudes computed with plane waves. A celestial
amplitude is formally given by

$\mathcal{A}(\\{\Delta_{i},q_{i},s_{i}\\})=\int_{\mathbb{R}_{+}^{n}}\prod_{i=1}^{n}\frac{\mathrm{d}\omega_{i}}{\omega_{i}}\,\omega^{\Delta_{i}}\,A(\\{\omega_{i},q_{i},s_{i}\\})\,.$
(2.3)

Explicitly carrying out the Mellin transforms can quickly become unwieldy as
the number of external particles increases. In the following section we’ll
introduce the CHY representation of amplitudes which gives a very compact
formula for massless $n$-point amplitudes.

###  2.2 CHY formulas

The CHY formulas present the $D$-dimensional, tree-level massless S-matrix of
several quantum field theories [27, 28, 44] as an integral formula. A generic
CHY formula is written as

$A_{n}=\delta^{D}\left(\sum_{i}^{n}k_{i}\right)\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\mathcal{I}(\\{g,k,\epsilon,z\\})\;\tilde{\mathcal{I}}(\\{g,k,\tilde{\epsilon},z\\})\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(E_{i})\,.$
(2.4)

The integral is taken over the moduli space of $n$-punctured Riemann spheres
$\mathcal{M}_{0,n}$ which carries an action of the group
$\text{SL}(2,\mathbb{C})$ denoted by the factor of $1/\text{vol
SL}(2,\mathbb{C})$ in the measure. Concretely, this implies we can fix the
position of any 3 punctures, for example $\\{z_{1},z_{2},z_{3}\\}$, removing
the associated differentials $\mathrm{d}z_{i}$ and introducing the Jacobian
$(z_{1}-z_{2})(z_{2}-z_{3})(z_{3}-z_{1})$ in the integral. The final result is
invariant under the choice of which particles are fixed. The numerators
$\mathcal{I}$ $(\tilde{\mathcal{I}})$ are rational functions of the external
data $\\{k,\epsilon,\tilde{\epsilon}\\}$, couplings $g$ and coordinates
$z_{i}$ on $\mathcal{M}_{0,n}$. Different choices of numerators correspond to
different choices of theories. For example, taking

$\mathcal{I}=\tilde{\mathcal{I}}=\text{PT}_{n}=\frac{\text{Tr}(\mathsf{T}^{\mathsf{a}_{1}}\mathsf{T}^{\mathsf{a}_{2}}\cdots\mathsf{T}^{\mathsf{a}_{n}})}{(z_{1}-z_{2})(z_{2}-z_{3})\cdots(z_{n}-z_{1})}+\text{perm.}$
(2.5)

gives the S-matrix for a cubic biadjoint scalar. Another interesting numerator
is the reduced Pfaffian,

$\text{Pf}^{\prime}\Psi_{n}=2\frac{(-1)^{i+j}}{(z_{i}-z_{j})}\text{Pf
}(\Psi^{ij}_{ij})\,,$ (2.6)

with Pf the Pfaffian of the $2n\times 2n$ matrix,

$\Psi=\left(\begin{array}[]{cc}A&-C^{T}\\\ C&B\end{array}\right)\,,$ (2.7)

with two lines and two columns removed, denoted by $\Psi^{ij}_{ij}$. Its
components are given by the matrices

$\begin{array}[]{ccc}A_{ij}=\begin{cases}\dfrac{k_{i}\cdot
k_{j}}{z_{i}-z_{j}}&i\neq j\\\
0&i=j\end{cases}\,,&\,B_{ij}=\begin{cases}\dfrac{\epsilon_{i}\cdot\epsilon_{j}}{z_{i}-z_{j}}&i\neq
j\\\ 0&i=j\end{cases}\,,&\,C_{ij}=\begin{cases}\dfrac{\epsilon_{i}\cdot
k_{j}}{z_{i}-z_{j}}&i\neq j\\\ -\sum_{l\neq i}\dfrac{\epsilon_{i}\cdot
k_{l}}{z_{i}-z_{l}}&i=j\end{cases}\end{array}\,.$ (2.8)

Taking $\mathcal{I}=\text{Pf}^{\prime}\Psi_{n}$ and
$\tilde{\mathcal{I}}=\text{PT}_{n}$ as numerators the CHY formula gives the
S-matrix for external gluons in Yang-Mills. Using a Pfaffian for both,
$\mathcal{I}=\text{Pf}^{\prime}\Psi(k,\epsilon)$ and
$\tilde{\mathcal{I}}=\text{Pf}^{\prime}\Psi(k,\tilde{\epsilon})$, gives the
S-matrix for gravitational amplitudes in NS-NS gravity.

The universality of the CHY representation is due to the presence of the
scattering equations $E_{i}$ [45], which are the last ingredient of (2.4) to
be explained. They appear as the arguments of the delta functions in
$\prod^{\prime}_{i}\bar{\delta}(E_{i})$. These delta functions are taken as
holomorphic delta functions, that is
$\bar{\delta}(E_{i})=\bar{\partial}(z)^{E_{i}}$, effectively fixing the
contour of integration to the solutions of the scattering equations. The
symbol $\prod^{\prime}$ means that delta functions for three equations should
be omitted from the product and another Jacobian of the form
$(z_{i}-z_{j})(z_{j}-z_{k})(z_{k}-z_{i})$ should be added. The scattering
equations themselves are

$E_{i}=\sum_{j\neq i}\frac{k_{i}\cdot k_{j}}{z_{i}-z_{j}}\,.$ (2.9)

The delta functions impose them as constraints on the $z_{i}$’s, completely
localizing the integration over $\mathcal{M}_{0,n}$. There are generically
$(n-3)!$ points in $\mathcal{M}_{0,n}$ which solve the scattering equations.
Denoting the solutions of the scattering equations by $\sigma_{i}$, the
amplitudes above can be written as

$A_{n}=\delta^{D}\left(\sum_{i=1}^{n}k_{i}\right)\sum_{i=1}^{(n-3)!}\left.\frac{\mathcal{I}\;\tilde{\mathcal{I}}}{\Phi}\right|_{\sigma_{i}}$
(2.10)

with $\Phi=\text{det }\partial_{j}E_{i}$, the Jacobian coming from the delta
functions.

In the CHY formulas the sum over trivalent graphs is replaced by the fully
localized integral over $\mathcal{M}_{0,n}$, making manifest properties which
are hard to see from the Feynman diagrammatic expansion. CHY formulas also
exist for many theories beyond the ones reviewed above [44, 46]; many can be
seen as originating from an unconventional string theory, called the
ambitwistor string [26, 47]. The advantage of having a worldsheet theory like
the ambitwistor string is the ease of generalizations to other settings, e.g.
higher loops [48, 49, 50, 51, 52, 53] and curved backgrounds [54, 35, 55, 56,
57, 58], which in turns leads to new formulas for amplitudes in these
settings. In section 4 we’ll show how CHY-like formulas for celestial
amplitudes originate from the ambitwistor string.

###  2.3 Color-kinematics duality and double copy

The original double copy [31] is a prescription to obtain gravitational
amplitudes from suitable squares of Yang-Mills amplitudes. There are now many
pairs of theories known to have amplitudes related by a double copy
prescription, as well as several proposals for double copy of non-linear
solutions. See [59] for a recent review of the field. Here we’ll present the
aspects of the original double copy which we’ll need in the rest of the paper.
Given a tree-level Yang-Mills amplitude it can be represented as a sum over
trivalent graphs by opening up four point interactions in the Feynman
diagrams,

$A^{\text{YM}}_{n}=\delta^{D}\left(\sum_{i=1}^{n}k_{i}\right)\sum_{\gamma\in\Gamma}\frac{c_{\gamma}\,n_{\gamma}}{\prod_{e\in\gamma}P_{e}}\,.$
(2.11)

Here $\Gamma$ is the set of trivalent graphs, $c_{\gamma}$ are color
numerators carrying the gauge group information, $n_{\gamma}$ are kinematical
numerators which are polynomials in the external momenta and polarizations,
and $P_{e}$ are the propagators associated to each edge $e$ of the graph
$\gamma$. Color numerators corresponding to graphs which differ only by BCJ
moves on some subgraph, see figure 1, obey identities inherited from the usual
Jacobi identity,

$c_{\gamma_{s}}+c_{\gamma_{t}}+c_{\gamma_{u}}=0\,.$ (2.12)

Figure 1: Graphs related by BCJ moves

If the four point vertices are opened up appropriately then, at tree-level,
kinematical numerators can always be found such that they satisfy identities
analogous to the ones satisfied by the color numerators. That is they satisfy
the Jacobi-like relation,

$n_{\gamma_{s}}+n_{\gamma_{t}}+n_{\gamma_{u}}=0\,.$ (2.13)

These kind of numerators are called color-kinematical dual numerators [32].
Substituting in (2.11) the color numerators by a set of color-kinematical dual
numerators $\tilde{n}_{\gamma}$ (obtained from $n_{\gamma}$ by replacing
$\epsilon$ with $\tilde{\epsilon}$) yields another amplitude,

$A^{\text{G}}_{n}=\delta^{D}\left(\sum_{i=1}^{n}k_{i}\right)\sum_{\gamma\in\Gamma}\frac{n_{\gamma}\,\tilde{n}_{\gamma}}{\prod_{e\in\gamma}P_{e}}\,,$
(2.14)

which turns out to be a gravitational amplitude. The requirement that
numerators obey color-kinematics duality ensures that (2.14) is invariant
under linear diffeomorphisms. This is the double copy prescription.

The CHY formulas reviewed in the previous section also have a double copy
structure. There, the analogue of color numerators is the Parke-Taylor factor
PT and the analogue of kinematical numerators is the Pfaffian
$\text{Pf}^{\prime}\Psi$. The sum over trivalent graphs is replaced by an
integral over the moduli space and double copy becomes a simple substitution
rule. Color-kinematics seems to be disconnected from the double copy in this
case, but they turn out to be intimately related through twisted cohomology on
$\mathcal{M}_{0,n}$. We review this in the following subsection.

###  2.4 Twisted cohomology

The ingredients in the CHY formula have an interesting interpretation in terms
of a cohomological theory on the moduli space of punctured Riemann spheres
$\mathcal{M}_{0,n}$. Here we’ll quickly go over the relevant details of the
constructions in [60, 33] that we’ll generalize in section 5. Proofs of the
statements presented below, as well as details of the computations in the
context of amplitudes can be found in [60, 33, 61]. Other mathematical details
can be found in the original mathematical literature [62, 63, 64, 65].

We start by defining a meromorphic one-form in $\mathcal{M}_{0,n}$ using the
scattering equations:

$\omega=\sideset{}{{}^{\prime}}{\sum}_{i}E_{i}\,\mathrm{d}z_{i}\,,\qquad
E_{i}=\sum_{j\neq i}\frac{k_{i}\cdot k_{j}}{z_{ij}}\,,$ (2.15)

where the prime in $\sum^{\prime}$ denotes that three points have been fixed
to account for the $\text{SL}(2,\mathbb{C})$ invariance. Using momentum
conservation one can show that the form (2.15) doesn’t depend on which three
points were fixed and that it only has simple poles along the boundaries of
$\mathcal{M}_{0,n}$. With this in hand we define the twisted de Rham operators

$\nabla_{\pm}=\mathrm{d}\pm\omega\,.$ (2.16)

Both are flat connections $[\nabla_{\pm},\nabla_{\pm}]=0$ on certain line
bundles $\mathcal{L}$ and its dual $\mathcal{L}^{\vee}$ over
$\mathcal{M}_{0,n}$ called local systems. Sections of $\mathcal{L}$
$(\mathcal{L}^{\vee})$ are given by functions on $\mathcal{M}_{0,n}$ that are
locally covariantly constant in $\nabla_{+}$ ($\nabla_{-}$). The operator
$\nabla_{+}$ acts naturally on
$\Omega^{\bullet}(\mathcal{M}_{0,n},\mathcal{L})$, the complex of differential
forms on $\mathcal{M}_{0,n}$ with coefficients in $\mathcal{L}$, and squares
to zero $(\nabla_{+})^{2}=0$. We use it to define twisted cohomology groups,

$H^{\bullet}(\mathcal{M}_{0,n},\mathcal{L})=\frac{\text{Im
}\nabla_{+}}{\text{Ker }\nabla_{+}}\,.$ (2.17)

Generically, i.e. for generic momenta, the only non-vanishing group is the
middle-dimensional one $H^{n-3}(\mathcal{M}_{0,n},\mathcal{L})$ with $(n-3)!$
independent generators. There is an analogous construction for cohomology
groups with values on $\mathcal{L}^{\vee}$ which we omit.

Taking $\mathrm{d}\mu$ a top holomorphic form on $\mathcal{M}_{0,n}$, the
numerators $\varphi_{+}=\mathcal{I}\,\mathrm{d}\mu$ and
$\varphi_{-}=\tilde{\mathcal{I}}\,\mathrm{d}\mu$ can be interpreted as
elements of the twisted cohomology groups $\varphi_{+}\in
H^{n-3}(\mathcal{M}_{0,n},\mathcal{L})$ and $\varphi_{-}\in
H^{n-3}(\mathcal{M}_{0,n},\mathcal{L}^{\vee})$. Moreover, amplitudes in the
CHY formalism can be interpreted as a bilinear pairing between these
cohomology groups:

$A_{n}=\langle\varphi_{+}|\varphi_{-}\rangle=\delta^{D}\left(\sum_{i=1}^{n}k_{i}\right)\sum_{i=1}^{(n-3)!}\left.\frac{\mathcal{I}\;\tilde{\mathcal{I}}}{\Phi}\right|_{\sigma_{i}}\,,$
(2.18)

giving the intersection number of $\varphi_{+}$ and $\varphi_{-}$.

An alternative evaluation of this pairing can be given where the contours
around the scattering equations are deformed, picking up contributions only
from the boundaries of $\mathcal{M}_{0,n}$. A codimension 1 boundary is
reached when the $n$-punctured sphere degenerates into a nodal surface given
by two punctured spheres connected by a node. Higher codimension boundaries
are reached with further dengenerations of these spheres into surfaces with
more nodes. The deepest boundaries arise when all that is left is a nodal
surface composed of spheres with three marked points. These spheres are
connected through these marked points in such a way that no closed cycle can
be drawn that passes through the nodal points. That is, the nodal surface
resembles a tree-graph with only trivalent vertices. In fact, one can label
these deepest boundaries by trivalent trees whose punctures are distributed
along the external edges. Different boundaries correspond to different
assignments of labels for the external punctures.

The end result is that the contour given by the scattering equations can be
deformed to pick contributions from the deepest boundaries of the moduli
space. This presents the amplitude as a sum over the boundaries of
$\mathcal{M}_{0,n}$ labelled by trivalent graphs,

$A_{n}=\delta^{D}\left(\sum_{i=1}^{n}k_{i}\right)\sum_{\gamma\in\Gamma}\frac{(\varphi_{+})_{\gamma}\,(\varphi_{-})_{\gamma}}{P_{\gamma}}\,.$
(2.19)

In this expression,

$(\varphi_{\pm})_{\gamma}=\underset{v_{\gamma}}{\text{Res}}\,\varphi_{\pm}$
(2.20)

are residues of the top meromorphic forms $\varphi_{\pm}$ along the boundary
divisor $v_{\gamma}$ labelled by the trivalent graph $\gamma$. This
representation is similar to the field theory one (2.11) and (2.14) but here
the numerators are guaranteed to obey color-kinematics. This follows from
global residue theorems on $\mathcal{M}_{0,n}$ as explained in [33].

The general argument can be summarized as follows: take a triple of trivalent
graphs $\gamma_{s},\gamma_{t},\gamma_{u}$ differing only on a subgraph
connecting four edges as shown in figure 1. These three divisors can be seen
as arising from the same corner of $\mathcal{M}_{0,n}$ where a four-punctured
sphere degenerates as one of its punctures, $z$, approaches one of the other
three punctures, $z_{s},z_{t},z_{u}$, generating the three boundary divisors
associated to the trivalent graphs $\gamma_{s},\gamma_{t},\gamma_{u}$, see
figure 2.

Figure 2: Neighborhood of the three degenerations related by BCJ moves.

We model the neighborhood of these degenerations by a thrice-punctured sphere
$\Sigma_{3}$ with coordinate $z$ and the three marked points
$z_{s},z_{t},z_{u}$ fixed to some values using the $\text{SL}(2,\mathbb{C})$
symmetry. Near these degenerations the differential form $\varphi_{+}$
restricts to a 1-form on $\Sigma_{3}$ with poles along the marked points.
Kinematical numerators are given by residues of this form

$n_{\gamma_{a}}=\underset{z=z_{a}}{\text{Res}}\,\varphi_{+}\,,\qquad
a\in\\{s,t,u\\}\,.$ (2.21)

and are related by a linear identity due to the global residue theorem on
$\Sigma_{3}$,

$n_{\gamma_{s}}+n_{\gamma_{t}}+n_{\gamma_{u}}=0\,.$ (2.22)

While this argument would work for any regular differential forms on
$\mathcal{M}_{0,n}$, only forms which are elements of the twisted cohomology
groups actually generate field theory amplitudes. Moreover, many
simplifications occur by choosing good representatives for these cohomologies.

##  3 Celestial scattering equations

The CHY formulas reviewed in the previous section have a natural celestial
analogue. These are most straightforwardly – if somewhat formally – derived by
directly Mellin transforming the momentum space expressions. We start this
section by performing these Mellin transforms and writing down explicit
formulas for celestial amplitudes. These take the form of Gelfand
A-hypergeometric functions in all dimensions, now governed by the _celestial
scattering equations_. Some example computations are also provided.

###  3.1 Amplitude formulas

The $n$-point celestial amplitude can be written down as a Mellin transform of
the momentum space formula (2.4) [8],

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\int_{\mathbb{R}^{n}_{+}}\prod_{j=1}^{n}\frac{\mathrm{d}\omega_{j}}{\omega_{j}}\,\omega_{j}^{\Delta_{j}}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}\biggl{(}\sum_{j\neq
i}\frac{s_{i}\,s_{j}\,\omega_{i}\,\omega_{j}\,q_{i}\cdot
q_{j}}{z_{i}-z_{j}}\biggr{)}\\\
\times\mathcal{I}(\\{g,s\,\omega\,q,\epsilon,z\\})\;\tilde{\mathcal{I}}(\\{g,s\,\omega\,q,\tilde{\epsilon},z\\})\,\delta^{D}\left(\sum_{i=1}^{n}s_{i}\,\omega_{i}\,q_{i}\right).$
(3.1)

The primary trick to simplify this is to perform as many
$\omega_{j}$-integrals as possible against the momentum conserving delta
function. We can already do this at the level of (3.1), but we will find it
much more illuminating to first manipulate it a bit.

For this we make use of the momentum generators in Mellin variables. The
action of the momentum operator $P_{\alpha\dot{\alpha}}$ on a function $A(k)$
of null momentum $k=s\,\omega\,q$ is given by a trivial multiplication

$P_{\mu}\cdot A(k)=k_{\mu}\,A(k)\,.$ (3.2)

Its action on the Mellin transform of $A(k)$ is then easily expressed as

$P_{\mu}\cdot\int_{\mathbb{R}_{+}}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\Delta}\,A(k)=s\,q_{\mu}\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\Delta+1}\,A(k)=s\,q_{\mu}\,\mathrm{e}^{\partial_{\Delta}}\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\Delta}\,A(k)\,.$
(3.3)

This dictates the definition of the celestial translation symmetry generators
[66],

$\mathcal{K}_{\mu}:=s\,q_{\mu}\,\mathrm{e}^{\partial_{\Delta}}\,,$ (3.4)

which are operator-valued null vectors, $\mathcal{K}_{i}^{2}=0$. Momentum
conservation is then equivalent to invariance under the diagonal translation
generator, that is, $n$-point celestial amplitudes should be annihilated by

$\sum_{i=1}^{n}\mathcal{K}_{i}^{\mu}=\sum_{i=1}^{n}s_{i}\,q_{i}^{\mu}\,\mathrm{e}^{\partial_{\Delta_{i}}}\,.$
(3.5)

Indeed, acting with this on (3.1) produces a factor of
$\sum_{i}s_{i}\,\omega_{i}\,q_{i}^{\alpha\dot{\alpha}}$ inside the Mellin
transforms which vanishes by momentum conservation. So, at least whenever the
Mellin transforms converge (or are understood distributionally), celestial
amplitudes are invariant under diagonal translations.

Hence we can make the formal replacements,

$\omega_{i}\mapsto\mathrm{e}^{\partial_{\Delta_{i}}}\,,\qquad
k_{i}\mapsto\mathcal{K}_{i}\,,$ (3.6)

inside the scattering equations and the CHY integrands. By converting them
into operators we can take these objects outside the Mellin integrals.
Moreover, the various $\mathcal{K}_{i}$’s clearly commute with each other and
there is no operator ordering ambiguity. We thus find the following
expression,

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\;\mathcal{I}(\\{g,\mathcal{K},\epsilon,z\\})\;\tilde{\mathcal{I}}(\\{g,\mathcal{K},\tilde{\epsilon},z\\})\;\mathcal{S}_{n}(\\{\Delta,q,s\\})\,,$
(3.7)

having defined the _celestial scattering equations_ ,

$\mathcal{E}_{i}:=\sum_{j\neq
i}\frac{\mathcal{K}_{i}\cdot\mathcal{K}_{j}}{z_{i}-z_{j}}\,,$ (3.8)

and the Mellin transformed $n$-point contact diagram,

$\mathcal{S}_{n}:=\int_{\mathbb{R}^{n}_{+}}\prod_{j=1}^{n}\frac{\mathrm{d}\omega_{j}}{\omega_{j}}\,\omega_{j}^{\Delta_{j}}\,\delta^{D}\left(\sum_{i=1}^{n}s_{i}\,\omega_{i}\,q_{i}\right)\,.$
(3.9)

Rigorously speaking, replacements like (3.6) are supposed to be made inside
analytic functions. In section 4, we will justify performing these
replacements inside the delta functions $\bar{\delta}(E_{i})$ by deriving it
from the ambitwistor worldsheet CFT.

$\mathcal{S}_{n}$ is the contribution of a $\phi^{n}$ contact term to the
$n$-point celestial amplitude of a theory of massless scalars $\phi$. In
appendix A we explicitly evaluate these contact diagrams for arbitrary
multiplicity and dimension, citing here only the final expression for the
celestial CHY formulas,

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\;\mathcal{I}(\\{g,\mathcal{K},\epsilon,z\\})\;\tilde{\mathcal{I}}(\\{g,\mathcal{K},\tilde{\epsilon},z\\})\\\
\times
F_{n}(\\{\Delta,q,s\\})\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\sum_{i}\Delta_{i}-D}\,.$
(3.10)

In particular, the replacement (3.6) applied to the Parke-Taylor factor (2.5)
and the Pfaffian (2.6) provides the celestial equivalent of kinematical
numerators in (3.7) for biadjoint scalar, Yang-Mills and gravity celestial
amplitudes. Using the celestial scattering equations, these take the form of
operator-valued numerators acting on the same universal function

$F_{n}(\\{\Delta,q,s\\})\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\sum_{i}\Delta_{i}-D}\,.$
(3.11)

The function $F_{n}(\\{\Delta,q,s\\})$ takes different forms depending on
whether $n>D$ or $n\leq D$.

#### $\bm{n>D:}$

In this case, there are more $\omega_{j}$ integrals than momentum conserving
delta functions. This allows us to perform $D$ of the Mellin integrals – say
those over $\omega_{1},\dots,\omega_{D}$ – and the rest simplify to

$F_{n}(\\{\Delta,q,s\\})=\frac{1}{u}\prod_{l,r}\Theta(u_{lr})\\\
\times\int_{[0,1]^{n-D}}\prod_{r}\frac{\mathrm{d}\xi_{r}}{\xi_{r}}\,\xi_{r}^{\Delta_{r}}\,\prod_{l}\biggl{(}\sum_{r^{\prime}}u_{lr^{\prime}}\,\xi_{r^{\prime}}\biggr{)}^{\Delta_{l}-1}\delta\biggl{(}1-\sum_{r^{\prime\prime}}\xi_{r^{\prime\prime}}\biggr{)}\,.$
(3.12)

Here, the various indices run over $l=1,2,\dots,D$ while
$r,r^{\prime},r^{\prime\prime}=D+1,\dots,n$. The coefficients $u$ and $u_{lr}$
in the integrand can be expressed in terms of $D\times D$ minors of the
$D\times n$ matrix $(q_{1}^{\mu},q_{2}^{\mu},\dots,q_{n}^{\mu})$ of celestial
positions. Define the determinants,

$(i_{1}\,i_{2}\dots\,i_{D}):=\varepsilon_{\mu_{1}\mu_{2}\dots\mu_{D}}\,q_{i_{1}}^{\mu_{1}}\,q_{i_{2}}^{\mu_{2}}\cdots
q_{i_{D}}^{\mu_{D}}\,,$ (3.13)

where $\varepsilon_{\mu_{1}\mu_{2}\dots\mu_{D}}$ is the $D$-dimensional Levi
Civita symbol. Then we can express them quite compactly as

$u=|(1\,2\dots D)|\,,\qquad u_{lr}=-s_{l}\,s_{r}\,\frac{(1\,2\dots
l-1\;r\;l+1\dots D)}{(1\,2\dots D)}\,.$ (3.14)

The Heaviside step functions constrain the $u_{lr}$ to be positive. The
integrals over the $\xi_{r}$’s produce Aomoto-Gelfand hypergeometric functions
over the Grassmannian Gr$(n-D,n)$ (or equivalently on Gr$(D,n)$) [67, 68].
Such integrals were first identified in the context of celestial amplitudes in
[18].

#### $\bm{n\leq D:}$

In this case, the result is distributional with $D-n+1$ leftover delta
functions,

$F_{n}(\\{\Delta,q,s\\})=\delta^{D-n+1}\biggl{(}\sum_{l}s_{l}\,u_{ln}\,q_{l}+s_{n}\,q_{n}\biggr{)}\;\frac{1}{u}\prod_{l}\Theta(u_{ln})\,u_{ln}^{\Delta_{l}-1}\,.$
(3.15)

Here, $l=1,2,\dots,n-1$. The spacetime index $\mu$ has also been partitioned
into two sets: $\mu=(r,a)$, where $r=0,1,\dots,D-n$ while $a=D-n+1,\dots,D-1$.
Then we have localized the Mellin integrals on the delta functions imposing
$\sum_{i}s_{i}\,\omega_{i}\,q_{i}^{a}=0$ by solving for the $\omega_{l}$ in
terms of $\omega_{n}$. The various coefficients in the above expression are

$u=|(1\,2\dots n-1)|\,,\qquad u_{ln}=-s_{l}\,s_{n}\,\frac{(1\,2\dots
l-1\;n\;l+1\dots n-1)}{(1\,2\dots n-1)}\,,$ (3.16)

where the determinants $(i_{1}\,i_{2}\dots i_{n-1})$ are now
$(n-1)\times(n-1)$ minors of the $(n-1)\times n$ matrix
$(q_{1}^{a},q_{2}^{a},\dots,q_{n}^{a})$. Again, these coefficients satisfy
positivity constraints $u_{ln}>0$ for all $l$. The remaining delta functions
impose momentum conservation in the “transverse” $D-n+1$ dimensions. The
result is still Lorentz invariant, the Mellin transforms themselves preserve
the symmetry but in performing them explicitly, non-Lorentz covariant choices
had to be made.

We also remark that the leftover Mellin integral over $\omega$ in (3.10) is
generically divergent. An interpretation of such integrals was given in [69]
in terms of “generalized delta functions”, allowing one to declare

$\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\sum_{i}\Delta_{i}-D}\equiv
2\pi\,\delta\\!\left(\mathrm{i}\bigl{(}\textstyle\sum_{i}\Delta_{i}-D\bigr{)}\right)\,.$
(3.17)

Formally, we see for instance from (3.15) that we indeed need the condition
$\sum_{i}\Delta_{i}=D$ to hold so that it has conformal weight $\Delta_{n}$ in
$q_{n}^{\mu}$. If the final amplitude is finite or marginally convergent, this
awkwardness gets resolved by the application of the $\mathcal{K}_{i}$’s on
$\mathcal{S}_{n}$ as this shifts the conformal weights appropriately. We
discuss such shifts below.

Let us now study some salient features of our formulas. First note that the
various scattering equations $\mathcal{E}_{i}$ trivially commute since the
$\mathcal{K}_{i}$ commute. Moreover, even though momentum conservation is
absent, they are still $\text{SL}(2,\mathbb{C})$ invariant when acting on
translation invariant objects. This is a consequence of the fact that the
residue of any $\mathcal{E}_{i}$ at $z_{i}=\infty$ is given by

$\underset{z_{i}=\infty}{\text{Res}}\,\mathcal{E}_{i}\,\mathrm{d}z_{i}=\mathcal{K}_{i}\cdot\sum_{j=1}^{n}\mathcal{K}_{j}\,,$
(3.18)

were we used $\mathcal{K}_{i}^{2}=0$. This residue is proportional to the
diagonal translation generator which annihilates the contact diagram.

Having evaluated the Mellin transform of $\mathcal{S}_{n}$, we can use (3.10)
to solve the scattering equations and obtain the celestial S-matrix of a
variety of theories. Since the operators $\mathcal{K}_{i}$ formally behave as
commuting numbers, the scattering equations are solved by the same algebraic
expressions coming from the $(n-3)!$ solutions in momentum space. The
worldsheet path integral implements a sum over all of these solutions and
generates trivalent graphs $\gamma$ contributing to the $n$-point amplitude.
The resulting celestial amplitude has the following general structure,

$\mathcal{A}_{n}=\sum_{\gamma\in\Gamma}\frac{\mathcal{N}_{\gamma}\,\tilde{\mathcal{N}}_{\gamma}}{\prod_{e\in\gamma}P_{e}}\cdot\mathcal{S}_{n}\,.$
(3.19)

Each internal edge $e$ of a graph $\gamma$ now comes with an operator-valued
propagator denominator,

$P_{e}=\left(\sum_{i\in e}\mathcal{K}_{i}\right)^{2}=2\sum_{i,j\in
e}s_{i}\,s_{j}\,q_{i}\cdot
q_{j}\,\mathrm{e}^{\partial_{\Delta_{i}}+\,\partial_{\Delta_{j}}}\,.$ (3.20)

The numerators $\mathcal{N}_{\gamma}$ and $\tilde{\mathcal{N}}_{\gamma}$ are
also theory dependent operators given by residues of the CHY numerators
$\mathcal{I}$ and $\tilde{\mathcal{I}}$ analogous to eq. (2.20), see also
section 5, which act on the celestial contact amplitude.

For $n>D$, one can take these operators inside the Mellin integrals in
$\mathcal{S}_{n}$. For $i=r$, one straightforwardly replaces
$\mathrm{e}^{\partial_{\Delta_{r}}}\mapsto\xi_{r}$, while for $i=l$ one
replaces $\mathrm{e}^{\partial_{\Delta_{l}}}\mapsto\sum_{r}u_{lr}\,\xi_{r}$ as
expected. Simultaneously, one also needs to appropriately shift
$\sum_{i}\Delta_{i}$ occurring in the distributional factor (3.17), and this
is most easily seen by working through the examples given in the next
subsection. The resulting integrals yield Gelfand A-hypergeometric functions
[70, 71]. Similarly, for $n\leq D$, one can replace factors
$\mathrm{e}^{\partial_{\Delta_{l}}}\mapsto u_{ln}$ and
$\mathrm{e}^{\partial_{\Delta_{n}}}\mapsto 1$, while again shifting
$\sum_{i}\Delta_{i}$ appropriately.

###  3.2 Examples

To illustrate our methods, we work out some examples of biadjoint scalar
amplitudes in different dimensions. Generalizing to gluons and gravitons is
computationally tedious but straightforward.

####  $\bm{4}$ points.

In this case, we only have one scattering equation. We choose to fix the three
points $z_{1},z_{2},z_{3}$ so that the remaining scattering equation
corresponds to $z_{4}$:

$\mathcal{E}_{4}\equiv\frac{\mathcal{K}_{4}\cdot\mathcal{K}_{1}}{z_{41}}+\frac{\mathcal{K}_{4}\cdot\mathcal{K}_{2}}{z_{42}}+\frac{\mathcal{K}_{4}\cdot\mathcal{K}_{3}}{z_{43}}=0\,.$
(3.21)

Fixing the global conformal symmetry also introduces a Faddeev-Popov
determinant $(z_{12}z_{23}z_{31})^{2}$ into the worldsheet integrals, where
$z_{ij}:=z_{i}-z_{j}$. Using
$2\pi\mathrm{i}\,\bar{\delta}(\mathcal{E}_{i})=\bar{\partial}_{i}(1/\mathcal{E}_{i})$,
we then need to evaluate

$\mathcal{A}_{4}=\oint\displaylimits_{\mathcal{E}_{4}=0}\frac{\mathrm{d}z_{4}}{2\pi\mathrm{i}\,\mathcal{E}_{4}}\;z_{12}^{2}\,z_{23}^{2}\,z_{31}^{2}\,\mathcal{I}\,\tilde{\mathcal{I}}\cdot\mathcal{S}_{4}\,.$
(3.22)

For convenience, we take the fixed points to be
$z_{1}=0,z_{2}=1,z_{3}\to\infty$. As explained before, since the
$\mathcal{K}_{i}$’s commute, we can formally solve the scattering equations as
in momentum space. We find

$\frac{\mathcal{K}_{4}\cdot\mathcal{K}_{1}}{z_{4}}+\frac{\mathcal{K}_{4}\cdot\mathcal{K}_{2}}{z_{4}-1}=0\implies
z_{4}=z^{*}_{4}=\frac{\mathcal{K}_{1}\cdot\mathcal{K}_{4}}{\mathcal{K}_{4}\cdot(\mathcal{K}_{1}+\mathcal{K}_{2})}\,.$
(3.23)

More precisely, this makes sense via its action
$z^{*}_{4}\cdot\mathcal{S}_{4}$ on the contact diagram which produces an
ordinary number. To obtain particular amplitudes we can insert various
different numerators. For example, we can compute the biadjoint scalar
amplitude in the color ordering (1234|1234). This is done by substituting the
(color-stripped) Parke-Taylor factors,

$\frac{1}{z_{12}\,z_{23}\,z_{34}\,z_{41}}\cdot\frac{1}{z_{12}\,z_{23}\,z_{34}\,z_{41}}\,,$
(3.24)

for $\mathcal{I}\,\tilde{\mathcal{I}}$. Using
$\sum_{i=1}^{4}\mathcal{K}_{i}\cdot\mathcal{S}_{4}=0$, one easily finds

$\begin{split}\mathcal{A}_{4}(1234|1234)&=-\left(\frac{1}{\mathcal{K}_{1}\cdot\mathcal{K}_{2}}+\frac{1}{\mathcal{K}_{1}\cdot\mathcal{K}_{4}}\right)\mathcal{S}_{4}\\\
&=-\left(s_{1}\,s_{2}\,\frac{\mathrm{e}^{-(\partial_{\Delta_{1}}+\,\partial_{\Delta_{2}})}}{q_{1}\cdot
q_{2}}+s_{1}\,s_{4}\,\frac{\mathrm{e}^{-(\partial_{\Delta_{1}}+\,\partial_{\Delta_{4}})}}{q_{1}\cdot
q_{4}}\right)\mathcal{S}_{4}\,.\end{split}$ (3.25)

Clearly, all that the derivatives do is shift some of the weights by $-1$.
This phenomenon is also clear from the perspective of the plane wave basis.
There the propagator denominators contain factors of $\omega_{i}$’s which
induce precisely these shifts.

Eq. (3.25) can be further simplified on each channel. When $D<4$, using (3.12)
and (3.17), the $s$-channel contribution becomes

$s_{1}\,s_{2}\,\frac{\mathrm{e}^{-(\partial_{\Delta_{1}}+\,\partial_{\Delta_{2}})}}{q_{1}\cdot
q_{2}}\,\mathcal{S}_{4}=2\pi\,\delta\\!\left(\mathrm{i}\bigl{(}\textstyle\sum_{i=1}^{4}\Delta_{i}-D-2\bigr{)}\right)\frac{s_{1}\,s_{2}}{q_{1}\cdot
q_{2}}\\\ \times
F_{4}(\Delta_{1}-1,q_{1},s_{1};\Delta_{2}-1,q_{2},s_{2};\Delta_{3},q_{3},s_{3};\Delta_{4},q_{4},s_{4})\,,$
(3.26)

with $F_{4}$ containing a Gr$(4-D,D)$ Aomoto-Gelfand hypergeometric integral.
More interesting is the case $D\geq 4$. Here, one finds

$s_{1}\,s_{2}\,\frac{\mathrm{e}^{-(\partial_{\Delta_{1}}+\,\partial_{\Delta_{2}})}}{q_{1}\cdot
q_{2}}\,\mathcal{S}_{4}=2\pi\,\delta\\!\left(\mathrm{i}\bigl{(}\textstyle\sum_{i=1}^{4}\Delta_{i}-D-2\bigr{)}\right)\frac{s_{1}\,s_{2}}{q_{1}\cdot
q_{2}}\\\
\times\delta^{D-3}\biggl{(}\sum_{l=1}^{3}s_{l}\,u_{l4}\,q_{l}+s_{4}\,q_{4}\biggr{)}\;\frac{u_{34}}{u}\prod_{l=1}^{3}\Theta(u_{l4})\,u_{l4}^{\Delta_{l}-2}\,.$
(3.27)

Other channels contribute similar terms.

#### Propagators at higher points.

At $n$ points, each Feynman diagram of the cubic biadjoint scalar theory comes
with $n-3$ propagators and trivial numerators. Clearly, each propagator
denominator induces a shift $\sum_{i}\Delta_{i}\mapsto\sum_{i}\Delta_{i}-2$ in
the Mellin integral (3.17). Altogether, we find a factor,

$\delta\\!\left(\mathrm{i}\bigl{(}\textstyle\sum_{i=1}^{n}\Delta_{i}-D-2\,(n-3)\bigr{)}\right)\,,$
(3.28)

in the final amplitude. With $D=d+2$, the on-shell phase space of massless
particles in $\mathbb{R}^{1,D-1}$ is spanned by conformal basis states on the
principal continuous series $\Delta_{i}\in\frac{d}{2}+\mathrm{i}\,\mathbb{R}$
[3]. So the distribution (3.28) is a true delta function precisely for $d=4$,
i.e., in six dimensions. It is curious to note that $D=6$ is precisely the
dimension in which the biadjoint scalar theory is a classical
CFT.222Similarly, it is known from [4] that Mellin transforms of gluon
amplitudes are marginally convergent precisely for $D=4$: the dimension in
which Yang-Mills is classically a CFT.

In general, we will have propagator denominators acting on the celestial
contact diagram. As a simple example, consider the action of a single
propagator acting on $\mathcal{S}_{n}$ for $n>D$,

$\frac{1}{(\sum_{i\in e}\mathcal{K}_{i})^{2}}\cdot\mathcal{S}_{n}\,,$ (3.29)

for an internal edge $e$ of a Feynman graph. When $n>D$, $F_{n}$ from (3.12)
changes to

$\frac{1}{u}\prod_{l,r}\Theta(u_{lr})\int_{[0,1]^{n-D}}\prod_{r}\frac{\mathrm{d}\xi_{r}}{\xi_{r}}\,\xi_{r}^{\Delta_{r}}\,\prod_{l}\biggl{(}\sum_{r^{\prime}}u_{lr^{\prime}}\,\xi_{r^{\prime}}\biggr{)}^{\Delta_{l}-1}\delta\biggl{(}1-\sum_{r^{\prime\prime}}\xi_{r^{\prime\prime}}\biggr{)}\\\
\times\left[\sum_{l,l^{\prime}\in e}s_{l}\,s_{l^{\prime}}\,q_{l}\cdot
q_{l^{\prime}}\sum_{r,r^{\prime}}u_{lr}\,u_{l^{\prime}r^{\prime}}\,\xi_{r}\,\xi_{r^{\prime}}+2\sum_{l,r\in
e}s_{l}\,s_{r}\,q_{l}\cdot
q_{r}\sum_{r^{\prime}}u_{lr^{\prime}}\,\xi_{r}\,\xi_{r^{\prime}}+\sum_{r,r^{\prime}\in
e}s_{r}\,s_{r^{\prime}}\,q_{r}\cdot
q_{r^{\prime}}\,\xi_{r}\,\xi_{r^{\prime}}\right]^{-1}\,.$ (3.30)

Due to the new quadratic polynomial in the integrand, the result now produces
a Gelfand A-hypergeometric function [70, 71].

Every propagator is taken care of by insertion of such quadratic polynomials.
We may also find the insertion of higher degree polynomials through the
numerators in gluon and graviton amplitudes, which are also handled by the
same theory of hypergeometric integrals. Similar integrals have also recently
occurred in the physics of loop level Feynman integrals [72] as well as
stringy canonical forms [73].

##  4 Ambitwistor strings

###  4.1 Models

In this section, we derive our worldsheet formulas for celestial amplitudes
using ambitwistor string theories [26]. These are chiral string theories with
target the space of null geodesics in Minkowski space. The worldsheet action
of a large class of such models takes the general form,

$S=\frac{1}{2\pi}\int_{\Sigma}P_{\mu}\bar{\partial}X^{\mu}-\mu\,T-e\,H+S_{\text{matter}}\,.$
(4.1)

Here, $X^{\mu}$ and $P_{\mu}$ are fields of conformal weight $(0,0)$ and
$(1,0)$ respectively on the string worldsheet $\Sigma$, and
$S_{\text{matter}}$ is the action of a pair of auxiliary worldsheet CFTs. $T$
denotes the weight $(2,0)$ stress tensor, while $\mu$ and $e$ are weight
$(-1,1)$ Beltrami differentials gauging the constraints $T=0$ and
$H:=P^{2}/2=0$ respectively. The latter generates the gauge symmetry

$\delta X^{\mu}=\alpha P^{\mu}\,,\qquad\delta P_{\mu}=0\,,\qquad\delta
e=\bar{\partial}\alpha\,,$ (4.2)

where $\alpha$ is a weight $(1,0)$ vector field on $\Sigma$. Thus, the
constraint $P^{2}=0$ and associated gauge redundancy $X\sim X+\alpha P$ reduce
our target space to the space of null geodesics: ambitwistor space.

The choice of matter action $S_{\text{matter}}$ gives rise to amplitudes of a
variety of theories:

Biadjoint scalar $\displaystyle:\quad
S_{\text{matter}}=S_{\mathfrak{g}}+S_{\tilde{\mathfrak{g}}}\,,$ (4.3) Yang-
Mills $\displaystyle:\quad
S_{\text{matter}}=S_{\mathfrak{g}}+\frac{1}{2\pi}\int_{\Sigma}\frac{1}{2}\,\psi_{\mu}\bar{\partial}\psi^{\mu}-\chi\,\psi_{\mu}P^{\mu}\,,$
(4.4) Gravity $\displaystyle:\quad
S_{\text{matter}}=\frac{1}{2\pi}\int_{\Sigma}\frac{1}{2}\,\psi_{\mu}\bar{\partial}\psi^{\mu}+\frac{1}{2}\,\tilde{\psi}_{\mu}\bar{\partial}\tilde{\psi}^{\mu}-\chi\,\psi_{\mu}P^{\mu}-\tilde{\chi}\,\tilde{\psi}_{\mu}P^{\mu}\,.$
(4.5)

$S_{\mathfrak{g}}$ and $S_{\tilde{\mathfrak{g}}}$ denote actions of current
algebra CFTs corresponding to Lie algebras $\mathfrak{g}$ and
$\tilde{\mathfrak{g}}$. We will denote their weight $(1,0)$ currents as
$j^{\mathsf{a}}$ and $\tilde{\jmath}^{\tilde{\mathsf{a}}}$ respectively. In
the Yang-Mills and gravitational cases, the fields $\psi^{\mu}$,
$\tilde{\psi}^{\mu}$ both denote weight $(\frac{1}{2},0)$ worldsheet fermions,
while $\chi$ and $\tilde{\chi}$ are weight $(-\frac{1}{2},1)$ fermionic
Lagrange multipliers gauging fermionic worldsheet symmetries akin to
supersymmetry. Worldsheet correlators of the current algebra systems give rise
to the Parke-Taylor type numerators (2.5), while those of the fermions produce
the Pfaffian type numerators (2.6).

Since both the $XP$ system and matter actions are free CFTs, it is easy to
find their fundamental OPEs. In the former case, one finds

$X^{\mu}(z)\,P_{\nu}(w)\sim\frac{\delta_{\nu}^{\mu}}{z-w}\,.$ (4.6)

Similarly, the fermionic OPEs read,

$\psi^{\mu}(z)\,\psi^{\nu}(w)\sim\frac{\eta^{\mu\nu}}{z-w}\,,\qquad\tilde{\psi}^{\mu}(z)\,\tilde{\psi}^{\nu}(w)\sim\frac{\eta^{\mu\nu}}{z-w}\,.$
(4.7)

Lastly, the current algebra OPEs are the standard ones,

$j^{\mathsf{a}}(z)\,j^{\mathsf{b}}(w)\sim\frac{f^{\mathsf{abc}}}{z-w}\,,\qquad\tilde{\jmath}^{\tilde{\mathsf{a}}}(z)\,\tilde{\jmath}^{\tilde{\mathsf{b}}}(w)\sim\frac{f^{\tilde{\mathsf{a}}\tilde{\mathsf{b}}\tilde{\mathsf{c}}}}{z-w}\,,$
(4.8)

where $f^{\mathsf{abc}}$ and
$f^{\tilde{\mathsf{a}}\tilde{\mathsf{b}}\tilde{\mathsf{c}}}$ are the structure
constants of $\mathfrak{g}$ and $\tilde{\mathfrak{g}}$ respectively. The
levels of the current algebras can be non-zero, producing multi-trace
amplitudes, but since we’re only going to be interested in the single trace
contributions we omit terms proportional to the level.

As in standard string theory, on gauge fixing we will also add ghost fields
for each of the gauge symmetries. For the bosonic symmetries generated by $T$
and $H$, one adds two $bc$ ghost systems consisting of fermionic ghosts
$b,\tilde{b}$ with weights $(2,0)$ and $c,\tilde{c}$ with weights $(-1,0)$.
For the supersymmetries generated by $\psi\cdot P$ and $\tilde{\psi}\cdot P$,
one adds $\beta\gamma$ systems with bosonic ghosts $\beta,\tilde{\beta}$ of
weights $(\frac{3}{2},0)$ and $\gamma,\tilde{\gamma}$ of weights
$(-\frac{1}{2},0)$. These fields also have the well-known OPEs,

$b(z)\,c(w)\sim\frac{1}{z-w}\,,\qquad\tilde{b}(z)\,\tilde{c}(w)\sim\frac{1}{z-w}\,,$
(4.9)

and

$\beta(z)\,\gamma(w)\sim\frac{1}{z-w}\,,\qquad\tilde{\beta}(z)\,\tilde{\gamma}(w)\sim\frac{1}{z-w}\,.$
(4.10)

The expression for the BRST operator will vary depending on the matter content
of the ambitwistor string. For the tree models we discuss, biadjoint scalar,
Yang-Mills and gravity, their BRST operators square to zero with appropriate
choices of dimensions and current algebra level. When setting up the model we
implicitly assume to be working with choices that render the BRST charge
nilpotent, but our final expressions after all the worldsheet calculations
have been performed are actually valid in any spacetime dimension. They are
after all, tree-level amplitudes of field theories, which are the same
independent of the spacetime dimension. More details on the BRST charges can
be found in [26].

Next, we construct vertex operators for these theories in the conformal
primary basis of states. It is easiest to start with the biadjoint scalar
states. Their fixed vertex operators are given by

$V_{i}=c\,\tilde{c}\,j\cdot\mathsf{T}\,\tilde{\jmath}\cdot\tilde{\mathsf{T}}\,\phi_{i}(X)\,,$
(4.11)

where $i$ is a particle label and
$\phi_{i}(X)\equiv\phi_{\Delta_{i}}^{s_{i}}(X;q_{i})$ denotes the scalar
conformal primary wavefunctions of (2.1). Generators of the Lie algebras
$\mathfrak{g}$ and $\tilde{\mathfrak{g}}$ are denoted by $\mathsf{T^{a}}$ and
$\tilde{\mathsf{T}}^{\tilde{\mathsf{a}}}$ respectively. Integrated vertex
operators are obtained form the usual descent procedure.

Similarly, the fixed vertex operator of a conformal primary gluon external
state in Yang-Mills is

$V_{i}^{-1}=c\,\tilde{c}\,\delta(\gamma)\,j\cdot\mathsf{T}\,\epsilon_{i}\cdot\psi\,\phi_{i}(X)\,,$
(4.12)

where the superscript $-1$ stands for picture number. To construct the
corresponding picture number 0 vertex operator, we descend by computing the
OPE of $V_{i}^{-1}$ with the picture changing operator
$\Upsilon=\delta(\beta)\,\psi\cdot P$. Using (4.6) and the scalar wavefunction
(2.1), it is easily seen that

$P_{\mu}(z)\,\phi_{i}(X(w))\sim\frac{\mathcal{K}_{i\,\mu}\,\phi_{i}(X(w))}{z-w}\,.$
(4.13)

Such OPEs are the means by which the celestial translation generators
$\mathcal{K}_{i\,\mu}=s_{i}\,q_{i\,\mu}\,\mathrm{e}^{\partial_{\Delta_{i}}}$
of (3.4) will enter our analysis. They act on the wavefunctions to their
right. This produces the picture number 0 vertex operators,

$V_{i}^{0}=c\,\tilde{c}\,j\cdot\mathsf{T}\left(\epsilon_{i}\cdot
P+\epsilon_{i}\cdot\psi\,\mathcal{K}_{i}\cdot\psi\right)\phi_{i}(X)\,.$ (4.14)

A similar analysis holds for gravitons, yielding the conformal basis vertex
operators,

$V_{i}^{-1,-1}=c\,\tilde{c}\,\delta(\gamma)\,\delta(\tilde{\gamma})\,\epsilon_{i}\cdot\psi\,\epsilon_{i}\cdot\tilde{\psi}\,\phi_{i}(X)\,,$
(4.15)

at picture number $-1$, and

$V_{i}^{0,0}=c\,\tilde{c}\left(\epsilon_{i}\cdot
P+\epsilon_{i}\cdot\psi\,\mathcal{K}_{i}\cdot\psi\right)\bigl{(}\tilde{\epsilon}_{i}\cdot
P+\tilde{\epsilon}_{i}\cdot\tilde{\psi}\,\mathcal{K}_{i}\cdot\tilde{\psi}\bigr{)}\,\phi_{i}(X)\,,$
(4.16)

at picture number 0. We reiterate that, just like ordinary momenta, the
$\mathcal{K}_{i}$’s commute with each other and there is no ordering ambiguity
here. BRST closure of these operators follows immediately from the conformal
primary representatives being on-shell.

###  4.2 Worldsheet correlators

We start this section by computing the correlators of the biadjoint scalar
vertex operators (4.11) in detail and deriving the celestial scattering
equations. After this we give the generalization to Yang-Mills and gravity.

To compute $n$-particle amplitudes we quantize these models on a $n$-punctured
Riemann sphere with punctures located at $z_{1},\dots,z_{n}$. We work in
conformal gauge $\mu=0$ and also gauge fix $e$ to be an element of the
$(n-3)$-dimensional Dolbeault cohomology group
$H^{0,1}(\Sigma,T_{\Sigma}(z_{1}+\cdots+z_{n}))$. The gauge freedom (4.2) is
precisely the freedom in choosing a representative of its cohomology class.
Explicitly, we pick the gauge fixing condition,

$e=\sum_{i=4}^{n}r_{i}\,e_{i}\,,$ (4.17)

having chosen the punctures $z_{4},\dots,z_{n}$ as our moduli without loss of
generality. The $\\{e_{i}\\}$ denote a standard basis of this cohomology
group. They act against quadratic differentials like $H$ by picking
$2\pi\mathrm{i}$ times their residues at the $z_{i}$. So for instance we will
find an insertion of

$\exp\left(-\frac{1}{2\pi}\int_{\Sigma}e\,H\right)=\exp\left(-\mathrm{i}\sum_{i=4}^{n}r_{i}\,\underset{z=z_{i}}{\text{Res}}\,H(z)\right)$
(4.18)

inside the path integral for any correlator, coming from the action (4.1). The
$r_{i}$ provide $n-3$ moduli that are left to be integrated over after the
gauge fixing.

The gauge fixed action of the biadjoint scalar ambitwistor string is given by

$S=\frac{1}{2\pi}\int_{\Sigma}P\cdot\bar{\partial}X+b\,\bar{\partial}c+\tilde{b}\,\bar{\partial}\tilde{c}+S_{\mathfrak{g}}+S_{\tilde{\mathfrak{g}}}\,.$
(4.19)

Using this action, the $n$-particle celestial amplitude is computed by the
correlator,

$\mathcal{A}_{n}=\int\displaylimits_{\Gamma\subset
T^{*}\mathcal{M}_{0,n}}\mathrm{d}^{n-3}z\;\mathrm{d}^{n-3}r\;\left\langle\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}\;\prod_{i=4}^{n}B_{i}\,\tilde{B}_{i}\;\prod_{j=1}^{n}V_{j}\right\rangle\,,$
(4.20)

where the integral is performed over an appropriate middle-dimensional contour
$\Gamma$ in $T^{*}\mathcal{M}_{0,n}$ [74].333The exact definition of this
contour will not be needed in what follows. In this expression, we have
inserted a product of $n-3$ picture changing operators,

$B_{i}=\frac{1}{2\pi}\int_{\Sigma}e_{i}\,b\,,\qquad\tilde{B}_{i}=\frac{1}{2\pi}\int_{\Sigma}e_{i}\,\tilde{b}\,,$
(4.21)

needed to soak up fermionic zero modes and produce the measure on
$\mathcal{M}_{0.n}$. Their OPEs with fixed vertex operators produce integrated
vertex operators corresponding to a choice of $n-3$ $z_{i}$’s. The other three
puncture locations $z_{1},z_{2},z_{3}$ have been fixed using the residual
$\text{SL}(2,\mathbb{C})$ symmetry.

One computes the products of $B_{i}$ and $\tilde{B}_{i}$ with $V_{i}$ using
the OPEs (4.9) to find

$B_{i}\,\tilde{B}_{i}\cdot
V_{i}=j\cdot\mathsf{T}\,\tilde{\jmath}\cdot\tilde{\mathsf{T}}\,\phi_{i}(X(z_{i}))\,.$
(4.22)

This strips off the $c$ and $\tilde{c}$ ghosts from $n-3$ of the vertex
operators, and correlators of the remaining ghost zero modes produce a factor
of $(z_{12}z_{23}z_{31})^{2}$ that can be accommodated by inserting a factor
of $1/\mathrm{vol}\,\mathrm{SL}(2,\mathbb{C})^{2}$ in the sense of Faddeev-
Popov. Next, the current algebra correlators generate a pair of Parke-Taylor
numerators PTn and $\widetilde{\mathrm{PT}}_{n}$ of the form (2.5)
($\widetilde{\mathrm{PT}}_{n}$ contains traces over products of
$\tilde{\mathsf{T}}^{\tilde{\mathsf{a}}_{i}}$). We throw out the multitrace
terms since we’re interested only in amplitudes where Yang-Mills states are
exchanged. All these simplifications leave us with a correlator of the $XP$
CFT,

$\mathcal{A}_{n}=\int_{\Gamma}\frac{\mathrm{d}^{n}z\;\mathrm{d}^{n}r}{\mathrm{vol}\,\mathrm{SL}(2,\mathbb{C})^{2}}\,\mathrm{PT}_{n}\;\widetilde{\mathrm{PT}}_{n}\;\left\langle\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}\;\prod_{j=1}^{n}\phi_{i}(X(z_{i}))\right\rangle_{XP}\,.$
(4.23)

We evaluate the last correlator by utilizing the OPE (4.13) computed before.

It is easily shown that $H(z)=\frac{1}{2}P^{2}(z)$ acts on a product of scalar
wavefunctions $\phi_{i}(X)$ by the OPE,

$H(z)\,\prod_{i=1}^{n}\phi_{i}(X(z_{i}))\sim\sum_{j}\sum_{k\neq
j}\frac{\mathcal{K}_{j}\cdot\mathcal{K}_{k}}{(z-z_{j})(z-z_{k})}\;\prod_{i=1}^{n}\phi_{i}(X(z_{i}))\,.$
(4.24)

As a result, on performing the $XP$ path integral, $H(z)$ is frozen to its
“classical” value,

$H(z)=\sum_{i}\sum_{j\neq
i}\frac{\mathcal{K}_{i}\cdot\mathcal{K}_{j}}{(z-z_{i})(z-z_{j})}\,.$ (4.25)

Note that the double poles dropped out in this computation due to
$\mathcal{K}_{i}^{2}=0$. Finally, the exponential
$\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}$ can be brought outside the
correlator. Using (4.18), the $r$-integrals subsequently give rise to the
$n-3$ scattering equations,

$\int\frac{\mathrm{d}^{n}r}{\mathrm{vol}\,\mathrm{SL}(2,\mathbb{C})}\;\exp\left(-\mathrm{i}\sum_{i=4}^{n}r_{i}\,\underset{z=z_{i}}{\text{Res}}\,H(z)\right)=\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\,,$
(4.26)

where

$\underset{z=z_{i}}{\text{Res}}\,\sum_{i}\sum_{j\neq
i}\frac{\mathcal{K}_{i}\cdot\mathcal{K}_{j}}{(z-z_{i})(z-z_{j})}=\sum_{j\neq
i}\frac{\mathcal{K}_{i}\cdot\mathcal{K}_{j}}{z_{i}-z_{j}}\equiv\mathcal{E}_{i}\,.$
(4.27)

This justifies the replacements
$\omega_{i}\mapsto\mathrm{e}^{\partial_{\Delta_{i}}}$ done within the
scattering equations in section 3. The result is a top-form integrated over
$\mathcal{M}_{0,n}$.

The remaining correlator over the $XP$ system contains only the insertion
$\prod_{i}\phi_{i}(X(z_{i}))$. Performing the path integral over $P_{\mu}$
imposes its equation of motion following from the gauge fixed action (4.19),
$\bar{\partial}X^{\mu}=0$. On $\Sigma=\mathbb{CP}^{1}$, this reduces the path
integral over $X^{\mu}$ to an integral over its constant zero mode. Denoting
this zero mode again by $X^{\mu}$, we find

$\left\langle\prod_{i=1}^{n}\phi_{i}(X(z_{i}))\right\rangle_{XP}=\int_{\mathbb{R}^{1,D-1}}\mathrm{d}^{D}X\;\prod_{i=1}^{n}\phi_{i}(X)\equiv\mathcal{S}_{n}\,.$
(4.28)

This is precisely the scalar contact celestial amplitude of (3.9), as can be
seen by expressing the scalar conformal primary wavefunctions in terms of
momentum eigenstates via (2.1). We thus arrive at the CHY formula for
celestial amplitudes of the biadjoint scalar theory,

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\;\mathrm{PT}_{n}\;\widetilde{\mathrm{PT}}_{n}\;\mathcal{S}_{n}(\\{\Delta,q,s\\})\,.$
(4.29)

Translation invariance of $\mathcal{S}_{n}$ and commutativity of the
$\mathcal{K}_{i}$’s guarantees permutation invariance of the scattering
equations, so all choices of $n-3$ scattering equations are equivalent.

An analogous story holds for Yang-Mills and gravitational amplitudes. For
instance, in the former case we need to compute a correlator of the form,

$\mathcal{A}_{n}=\int\displaylimits_{\Gamma\subset
T^{*}\mathcal{M}_{0,n}}\mathrm{d}^{n-3}z\;\mathrm{d}^{n-3}r\;\left\langle\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}\;\prod_{i=4}^{n}B_{i}\,\tilde{B}_{i}\;V_{1}^{-1}\,V_{2}^{-1}\,\prod_{j=3}^{n}V_{j}^{0}\right\rangle\,,$
(4.30)

having chosen to insert picture number $-1$ vertex operators for particles 1
and 2 without loss of generality. The path integrals over the fermionic ghosts
is done as usual and we pull out the insertion of
$\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}$ from the correlator as before
leading to the celestial scattering equations. The path integral over the
currents $j^{\mathsf{a}}$ generates a single Parke-Taylor factor plus
multitrace terms we once again ignore. This leaves the correlators of the $XP$
and $\psi$ systems along with the $\beta\gamma$ ghosts to be performed,

$\langle\delta(\gamma(z_{1}))\,\delta(\gamma(z_{2}))\rangle_{\beta\gamma}\left\langle\prod_{i=3}^{n}\left(\epsilon_{i}\cdot
P+\epsilon_{i}\cdot\psi\,\mathcal{K}_{i}\cdot\psi\right)\\!(z_{i})\;\prod_{j=1}^{n}\phi_{i}(X(z_{j}))\right\rangle_{XP\psi}\,.$
(4.31)

Following [26], these yield a Pfaffian-type numerator (2.6), now containing
the operators $\mathcal{K}_{i}$ in place of ordinary momenta $k_{i}$. The
final result is the formula,

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\;\mathrm{PT}_{n}\;\mathrm{Pf}^{\prime}\Psi_{n}(\\{\epsilon,\mathcal{K}\\})\;\mathcal{S}_{n}(\\{\Delta,q,s\\})\,,$
(4.32)

encoding gluon celestial amplitudes in arbitrary dimensions.

For gravity, the only new ingredient is the replacement of the current algebra
system with a second fermionic system. The correlator of interest is given by

$\mathcal{A}_{n}=\int\displaylimits_{\Gamma\subset
T^{*}\mathcal{M}_{0,n}}\mathrm{d}^{n-3}z\;\mathrm{d}^{n-3}r\;\left\langle\mathrm{e}^{-\frac{1}{2\pi}\int_{\Sigma}e\,H}\;\prod_{i=4}^{n}B_{i}\,\tilde{B}_{i}\;V_{1}^{-1,-1}\,V_{2}^{-1,-1}\,\prod_{j=3}^{n}V_{j}^{0,0}\right\rangle\,.$
(4.33)

Using the same calculations as above it reduces to the graviton celestial
amplitude formula

$\mathcal{A}_{n}=\int_{\mathcal{M}_{0,n}}\frac{\mathrm{d}^{n}z}{\text{vol
SL}(2,\mathbb{C})}\;\sideset{}{{}^{\prime}}{\prod}_{i}\bar{\delta}(\mathcal{E}_{i})\;\mathrm{Pf}^{\prime}\Psi_{n}(\\{\epsilon,\mathcal{K}\\})\;\mathrm{Pf}^{\prime}\Psi_{n}(\\{\tilde{\epsilon},\mathcal{K}\\})\;\mathcal{S}_{n}(\\{\Delta,q,s\\})\,.$
(4.34)

There are two unifying features of all these formulas. The first is the
presence of the same contact amplitude $\mathcal{S}_{n}$ and celestial
scattering equations governing the three expressions. The second is the
manifest double copy structure at the level of the CHY integrands given by a
simple replacement rule. In the next section, we return to this point with the
machinery of twisted cohomology as applied to these operator-valued
integrands.

##  5 Celestial color-kinematics duality

We have seen in the previous section how the celestial versions of the CHY
formulas still manifest a double copy structure for the operator-valued
numerators. This can either be stated as a substitution rule for ambitwistor
integrands or in terms of their residues and trivalent graphs. In the latter
method, gauge invariance of the double copied amplitude is not manifest even
though we know it is gauge invariant by construction. As reviewed in section 2
gauge invariance of the double copied amplitude is guaranteed if the
kinematical numerators obey color-kinematics duality. We show below that an
analogous requirement holds for the operator-valued numerators of celestial
amplitudes obtained from the ambitwistor string. The proof will closely follow
[33] by recasting the operator-valued numerators as elements of a generalized
twisted cohomology.

Let $\mathcal{C}_{n}$ denote the configuration space for the celestial data
$\\{\Delta_{i},q_{i},s_{i}\\}$ of $n$-particle celestial amplitudes. The
operators $\mathcal{K}_{i}$ defined in (3.4) map smooth functions on
$\mathcal{C}_{n}$ to themselves by shifting their conformal dimensions. These
operators are not derivations as they don’t satisfy Leibniz rule, but they are
linear and commute with each other. Denote the
$C^{\infty}(\mathcal{C}_{n})$-algebra generated by the $\mathcal{K}_{i}$’s by
$\mathfrak{C}$. It forms a subalgebra of
$\mathrm{Hom}(C^{\infty}(\mathcal{C}_{n}),C^{\infty}(\mathcal{C}_{n}))$. Even
though translation invariance is not manifest it is still a symmetry of
celestial amplitudes, so our operators $\mathcal{K}_{i}$ will only ever act on
functions $\mathcal{S}$ satisfying

$\sum_{i=1}^{n}\mathcal{K}_{i}\mathcal{S}=0\,.$ (5.1)

Without loss of generality, we quotient $\mathfrak{C}$ by the ideal generated
by $\sum_{i}\mathcal{K}_{i}$ to define

$\mathfrak{K}:=\mathfrak{C}/\langle\textstyle{\sum}_{i}\mathcal{K}_{i}\rangle\,$
(5.2)

as the reduced kinematical space where celestial amplitudes live. In what
follows, we will treat the operator-algebras $\mathfrak{C}$ and $\mathfrak{K}$
as infinite-dimensional vector spaces spanned by monomials in
$\mathcal{K}_{i}$’s.

With some abuse of notation, we also denote by $\mathfrak{K}$ the infinite-
dimensional trivial vector bundle $\mathcal{M}_{0,n}\times\mathfrak{K}$. In
analogy with the usual twisted cohomology, we define the twisted de Rham
operators

$\nabla_{\pm}=\mathrm{d}\pm\omega\,,$ (5.3)

with $\mathrm{d}$ the exterior derivative on $\mathcal{M}_{0,n}$ and the
connection one-form built out of the celestial scattering equations,

$\omega=\sideset{}{{}^{\prime}}{\sum}_{i}\mathcal{E}_{i}\,\mathrm{d}z_{i}\,,\qquad\mathcal{E}_{i}=\sum_{j\neq
i}\frac{\mathcal{K}_{i}\cdot\mathcal{K}_{j}}{z_{ij}}\,.$ (5.4)

The operator-valued connections $\nabla_{+}$ and $\nabla_{-}$ act via
straightforward multiplication on forms valued in $\mathfrak{K}$, denoted by
$\Omega^{\bullet}(\mathcal{M}_{0,n},\mathfrak{K})$.444The images of
$\sum_{i}\mathcal{K}_{i}$ under $\nabla_{\pm}$ are again in
$\langle\sum_{i}\mathcal{K}_{i}\rangle$ since the $\mathcal{K}_{i}$ operators
multiply as usual, commute with each other and with $\mathrm{d}$. We also
formally define the line bundles $\mathfrak{L}$ and $\mathfrak{L}^{\vee}$
whose sections are $\mathfrak{K}$-valued functions on
$\mathcal{M}_{0,n}$.555More precisely, $\mathfrak{L}$ and
$\mathfrak{L}^{\vee}$ are local systems given by an abelian operator-valued
representation of $\pi(\mathcal{M}_{0,n})$. We leave the study of analytic
aspects of these definitions to future work. The connections $\nabla_{\pm}$
square to zero so we use them to define twisted de Rham cohomologies with
coefficients in $\mathfrak{L}$ and $\mathfrak{L}^{\vee}$, denoted by
$H^{\bullet}(\mathcal{M}_{0,n},\mathfrak{L})$ and
$H^{\bullet}(\mathcal{M}_{0,n},\mathfrak{L}^{\vee})$.

The operators $\mathcal{K}_{i}$, viewed as elements of $\mathfrak{K}$, obey
the same algebraic identities as the usual momenta for plane waves $k_{i}$.
That is, they are null, commute with each other and obey a “momentum
conservation” identity. Due to this, the analysis in [60, 33] carries over to
the celestial case with the appropriate substitutions. Let $\varphi_{\pm}$ be
two $\mathfrak{K}$-valued meromorphic top forms on $\mathcal{M}_{0,n}$ with
singularities only along its boundary divisor. Take these to also obey
$\nabla_{\pm}\varphi_{\pm}=0$ so that they are representatives of cohomology
classes in $H^{n-3}(\mathcal{M}_{0,n},\mathfrak{L})$ and
$H^{n-3}(\mathcal{M}_{0,n},\mathfrak{L}^{\vee})$. Explicit examples are the
ambitwistor numerators (2.5) and (2.6) with the replacement
$k_{i}\mapsto\mathcal{K}_{i}$, multiplied by the top holomorphic form
$\mathrm{d}\mu$ on $\mathcal{M}_{0,n}$.

We define intersection numbers $\langle\varphi_{+}|\varphi_{-}\rangle$ in
analogy with [60] but in our case this pairing is not a number but an
operator. There are two common ways to compute these intersection numbers: one
is given by the CHY formulas, as a moduli space integral localized to the
solutions of the scattering equations; the other by evaluating the paired
forms $\varphi_{\pm}$ near the highest codimension boundaries of
$\mathcal{M}_{0,n}$666Explicitly, this requires a finding a compactly
supported cohomological representative of one of the forms $\varphi_{\pm}$.
This coincides with the formula obtained by deforming the contours given by
the scattering equations. labelled by trivalent graphs. The first method
recovers the formula (3.10) for numerators
$\varphi_{+}=\mathcal{I}\,\mathrm{d}\mu$ and
$\varphi_{-}=\tilde{\mathcal{I}}\,\mathrm{d}\mu$. The latter gives the
amplitude as a sum over trivalent graphs $\gamma$,

$\langle\varphi_{+}|\varphi_{-}\rangle=\sum_{\gamma\in\Gamma}\frac{\mathrm{Res}_{v_{\gamma}}(\varphi_{+})\,\mathrm{Res}_{v_{\gamma}}(\varphi_{-})}{\prod_{e\in\gamma}P_{e}}\,,$
(5.5)

with edges denoted by $e$. Residues are taken along the boundary divisor
$v_{\gamma}$ associated to the trivalent graph $\gamma$. Each edge has an
associated denominator $P_{e}=(\sum_{i\in e}\mathcal{K}_{i})^{2}$ analogous to
propagators acting as inverse operators. This is nothing other than eq. (3.19)
which was obtained by deforming the contours prescribed by the scattering
equations.

With the framework introduced above it is straightforward to adapt the
arguments of [33] to show how color-kinematics is implemented for celestial
amplitudes. The set up is analogous to section 2.4. Take a triple of trivalent
graphs $\gamma_{s},\gamma_{t},\gamma_{u}$ differing only on a subgraph
connecting four edges as shown in figure 1. The boundary divisors for these
graphs arise from a four-punctured sphere degenerating as one of its punctures
$z$ approaches one of the other punctures $z_{s},z_{t},z_{u}$. The
neighborhood of this degeneration is modelled by a sphere with three
punctures, $\Sigma_{3}$, with coordinate $z$ and fixed marked points
$z_{s},z_{t},z_{u}$, see figure 2. Take $\varphi_{M}$ as a
$\mathfrak{K}$-valued 1-form on $\Sigma_{3}$ with poles along the marked
points. The operator-valued numerators,

$\mathcal{N}_{\gamma_{a}}=\underset{z=z_{a}}{\text{Res}}\,\varphi_{M}\,,\qquad
a\in\\{s,t,u\\}\,,$ (5.6)

are then related by a linear identity due to the global residue theorem on
$\Sigma_{3}$,

$\mathcal{N}_{\gamma_{s}}+\mathcal{N}_{\gamma_{t}}+\mathcal{N}_{\gamma_{u}}=0\,,$
(5.7)

valid in $\mathfrak{K}$, i.e. the sum of operators on the left vanishes
precisely when it acts on translationally invariant functions. In
$\mathfrak{C}$, that is, before taking the quotient, the statement of color-
kinematics duality is

$\mathcal{N}_{\gamma_{s}}+\mathcal{N}_{\gamma_{t}}+\mathcal{N}_{\gamma_{u}}=\mathcal{O}\cdot\sum_{i=1}^{n}\mathcal{K}_{i},$
(5.8)

for some operator $\mathcal{O}^{\mu}\in\mathfrak{C}$.

As an example, we illustrate (5.8) for the four-point Yang-Mills amplitude.
The operator-valued numerators are easily read off from their momentum space
counterparts. We have

$\mathcal{N}_{s}=\left[\epsilon_{1}\cdot\epsilon_{2}\,(\mathcal{K}_{1}-\mathcal{K}_{2})^{\mu}+2\,\epsilon_{1}\cdot\mathcal{K}_{2}\,\epsilon_{2}^{\mu}-2\,\epsilon_{2}\cdot\mathcal{K}_{1}\,\epsilon_{1}^{\mu}\right]\eta_{\mu\nu}\\\
\times\left[\epsilon_{3}\cdot\epsilon_{4}\,(\mathcal{K}_{4}-\mathcal{K}_{3})^{\nu}-2\,\epsilon_{3}\cdot\mathcal{K}_{4}\,\epsilon_{4}^{\nu}+2\,\epsilon_{4}\cdot\mathcal{K}_{3}\,\epsilon_{3}^{\nu}\right]\\\
-\left(\epsilon_{1}\cdot\epsilon_{3}\,\epsilon_{2}\cdot\epsilon_{4}-\epsilon_{1}\cdot\epsilon_{4}\,\epsilon_{2}\cdot\epsilon_{3}\right)\left(\mathcal{K}_{1}\cdot\mathcal{K}_{2}+\mathcal{K}_{3}\cdot\mathcal{K}_{4}\right)\,$
(5.9)

for the $s$-channel. Other channels are obtained from permutations of this:
$\mathcal{N}_{t}$ by exchanging $2\leftrightarrow 3$ with an overall minus
sign, and $\mathcal{N}_{u}$ by the permutation $(2,3,4)\rightarrow(4,2,3)$.
Making use of $q_{i}^{2}=\epsilon_{i}\cdot q_{i}=0$, these three numerators
are easily shown to obey

$\mathcal{N}_{s}+\mathcal{N}_{t}+\mathcal{N}_{u}=\left(\mathcal{R}_{1234}^{\mu}+\mathcal{R}_{1342}^{\mu}+\mathcal{R}_{1423}^{\mu}\right)\sum_{i=1}^{4}\mathcal{K}_{i\,\mu}\in\langle\textstyle\sum_{i}\mathcal{K}_{i}\rangle\,,$
(5.10)

where

$\mathcal{R}_{ijkl}^{\mu}=2\,\bigl{(}\epsilon_{i}^{\nu}\,\epsilon_{j}^{\mu}-\epsilon_{j}^{\nu}\,\epsilon_{i}^{\mu}\bigr{)}\,(\mathcal{K}_{k}-\mathcal{K}_{l})_{\nu}+2\left(\epsilon_{k}^{\nu}\,\epsilon_{l}^{\mu}-\epsilon_{l}^{\nu}\,\epsilon_{k}^{\mu}\right)(\mathcal{K}_{i}-\mathcal{K}_{j})_{\nu}\in\mathfrak{C}\,.$
(5.11)

As claimed, the kinematic Jacobi identity is violated only up to something
proportional to the diagonal translation generators.

With this we have shown that numerators given by
$\mathrm{Res}_{v_{\gamma}}(\varphi_{\pm})$ satisfy the celestial color-
kinematics duality and (5.5) makes manifest the the celestial double copy
structure for any number of external particles extending the results of [17].

##  6 Discussion

Ambitwistor strings are important tools for the study of flat space holography
as they provide a framework to study celestial amplitudes to any multiplicity
and in several dimensions. This is exemplified by our all-multiplicity
expressions (3.10), (3.12) and (3.15) which allowed us to generalize the
computations of [18] to general dimensions and polarizations. In fact, we
showed that for multiplicities with $n>D$, amplitudes in $D$ dimensions can be
uniformly expressed in terms of Gelfand A-hypergeometric functions. For $n\leq
D$, we found distributional expressions with $(D-n+1)$-dimensional delta
functions coming from residual momentum conservation. We also observed that
the Mellin transforms of biadjoint scalar amplitudes are marginally convergent
in $D=6$, just as the Yang-Mills ones in $D=4$ [4]. These happen to be the
dimensions in which the classical theories are conformal.

There are also worldsheet models specialized to $D=4$ with target space
$\mathscr{I}$ that have been shown to compute Yang-Mills and gravity
amplitudes in the plane wave basis [75, 76]. These models are adapted to the
spinor-helicity formalism and might be better adapted to the study of a
conjectural $2$d CFT on $\mathscr{I}$. It would be interesting to understand
how the computational methods we used above can be adapted to this case, and
to compute the celestial amplitudes in these $D=4$ models.

Beyond explicit expressions for the $n$-point celestial amplitudes, our
worldsheet formulas have provided a new outlook on their double copy.
Celestial color-kinematics duality and double copy depend on two properties of
the amplitudes: kinematical numerators can be represented as operators acting
on external kinematics, that is, acting only on quantities at the boundary of
spacetime; and that total derivatives can also be characterized as operators
acting only on the external kinematics. The latter is simply the statement
that amplitudes are translation invariant in the celestial case even if not
manifestly so.

These properties might be expected to hold only in the simpler case of flat
spacetime. But we know of at least one case where they also hold for
amplitudes in a curved background, namely, AdS. The works [41, 40] used the
ambitwistor string to write CHY-like formulas for amplitudes in AdS
spacetimes. These have similar structure to the celestial ones, with
kinematical numerators and the scattering equations acting as operators on the
scalar contact vertex. These amplitudes are not translation invariant.
Instead, the decoupling of total derivatives on the scalar contact diagram can
be identified with a quotient by the ideal of diagonal conformal
transformations in analogy with the celestial case. We then expect that a
similar notion of color-kinematics duality holds for AdS with kinematical
numerators obeying a relation like (5.8) up to some symmetry of the space of
external data.

Double copy in AdS can also be expected to hold in a similar fashion to the
celestial double copy. To show this we must first find the appropriate CHY
numerators taking into account that some terms which naively look like non-
zero contributions might decouple on top of the scattering equations. To
characterize such terms a generalization of twisted cohomology analogous to
the one we defined for celestial amplitudes would be very useful. We expect
that several acceptable numerators could be found using insights from the
ambitwistor string together with an interpretation of these numerators as a
generalized twisted cohomology.

An interesting question is whether the framework introduced above holds for
loop amplitudes. There are a couple of ambitwistor formulas for loop
amplitudes [48, 50], the most successful being the ones based on nodal
surfaces [49, 51, 52, 53]. The latter can be generalized to the celestial case
making use of our replacement rule (3.6) to extract numerators in front of the
Mellin transform of a scalar quantity. The outstanding issue is that the loop-
level scattering equations and the numerators can depend on the loop momentum
which we’d rather not have in a purely celestial description. If one goes back
to position space Feynman diagrams this problem is absent since we can leave
all the internal propagators in their position space representation.
Numerators that don’t depend on loop momentum can still be pulled out in front
as operators acting on a scalar loop diagram, but now there is a proliferation
of different topologies coming from the loop diagrams. It would be interesting
to see if there is a way to encode the effect of loop momentum in the
numerators in terms of operators acting on the external variables. Perhaps,
some insight could be gained from explicit Mellin transforms giving loop-level
celestial amplitudes [77, 19, 16].

## Acknowledgements

We thank Tim Adamo and Andrea Puhm for comments on the draft. EC’s research is
supported in part by U.S. Department of Energy grant DE-SC0009999 and by funds
provided by the University of California. AS is supported by a Mathematical
Institute Studentship, Oxford.

##  Appendix A Mellin transform of the contact diagram

In this appendix we evaluate the celestial $n$-point contact diagram

$\mathcal{S}_{n}=\int_{\mathbb{R}^{n}_{+}}\prod_{i=1}^{n}\frac{\mathrm{d}\omega_{i}}{\omega_{i}}\,\omega_{i}^{\Delta_{i}}\;\delta^{D}\biggl{(}\sum_{j=1}^{n}s_{j}\,\omega_{j}\,q_{j}\biggr{)}\,.$
(A.1)

There are two cases to consider: $n>D$ and $n\leq D$. In the former case, we
will find the structure of Aomoto-Gelfand hypergeometric integrals which also
appear in four dimensions [18]. In the latter case, we will be able to perform
all the Mellin integrals and the result will be distributional due to leftover
delta functions.

#### a) $\bm{n>D:}$

We begin by solving the delta functions for $\omega_{l}$, $l=1,2,\dots,D$, in
terms of $\omega_{r}$, $r=D+1,\dots,n$, and the other variables. To do this,
we use Cramer’s rule. Define the determinants

$(i_{1}\,i_{2}\dots
i_{D}):=\varepsilon_{\mu_{1}\mu_{2}\dots\mu_{D}}\,q_{i_{1}}^{\mu_{1}}\,q_{i_{2}}^{\mu_{2}}\cdots
q_{i_{D}}^{\mu_{D}}\,,$ (A.2)

where $\varepsilon_{\mu_{1}\mu_{2}\dots\mu_{D}}$ stands for the
$D$-dimensional Levi Civita symbol. Then first write

$\sum_{l}s_{l}\,\omega_{l}\,q_{l}^{\mu}=-\sum_{r}s_{r}\,\omega_{r}\,q_{r}^{\mu}\,.$
(A.3)

Contracting both sides with
$\varepsilon_{\mu\mu_{1}\mu_{2}\dots\hat{\mu}_{l}\dots\mu_{D}}\,q_{1}^{\mu_{1}}\,q_{2}^{\mu_{2}}\cdots\widehat{q_{l}^{\mu_{l}}}\cdots
q_{D}^{\mu_{D}}$ (where a hat denotes omission), we can solve for $\omega_{l}$
to find

$\omega_{l}=\sum_{r}u_{lr}\,\omega_{r}\,,\qquad
u_{lr}=-s_{l}\,s_{r}\,\frac{(1\,2\dots l-1\;r\;l+1\dots D)}{(1\,2\dots D)}\,.$
(A.4)

Note that demanding $\omega_{l}>0$ for all $\omega_{r}>0$ restricts all the
coefficients $u_{lr}$ to be positive. Accounting for this using heaviside step
functions, along with a Jacobian $u=|(1\,2\dots D)|$ coming from solving the
delta functions, our integral turns into

$\begin{split}\mathcal{S}_{n}&=\int_{\mathbb{R}^{n}_{+}}\prod_{i=1}^{n}\frac{\mathrm{d}\omega_{i}}{\omega_{i}}\,\omega_{i}^{\Delta_{i}}\;\frac{1}{u}\prod_{l}\left\\{\delta\biggl{(}\omega_{l}-\sum_{r^{\prime}}u_{lr^{\prime}}\,\omega_{r^{\prime}}\biggr{)}\prod_{r}\Theta(u_{lr})\right\\}\\\
&=\frac{1}{u}\prod_{l,r}\Theta(u_{lr})\int_{\mathbb{R}^{n-D}_{+}}\prod_{r}\frac{\mathrm{d}\omega_{r}}{\omega_{r}}\,\omega_{r}^{\Delta_{r}}\,\prod_{l}\left(\sum_{r^{\prime}}u_{lr^{\prime}}\,\omega_{r^{\prime}}\right)^{\Delta_{l}-1}\,.\end{split}$
(A.5)

The leftover integral has the standard form of an Euler-type integral.

To perform the $\omega_{r}$ integrals, define new integration variables
$\omega:=\sum_{r}\omega_{r}$ along with $\xi_{r}:=\omega_{r}/\omega$. Doing
this, we find

$\mathcal{S}_{n}=\frac{1}{u}\prod_{l,r}\Theta(u_{lr})\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\sum_{i}\Delta_{i}-D}\\\
\times\int_{[0,1]^{n-D}}\prod_{r}\frac{\mathrm{d}\xi_{r}}{\xi_{r}}\,\xi_{r}^{\Delta_{r}}\,\prod_{l}\left(\sum_{r^{\prime}}u_{lr^{\prime}}\,\xi_{r^{\prime}}\right)^{\Delta_{l}-1}\delta\biggl{(}1-\sum_{r^{\prime\prime}}\xi_{r^{\prime\prime}}\biggr{)}\,.$
(A.6)

The integral over $\omega$ is divergent but is usually interpreted as a
distribution [69]. Note that the degree of divergence will change on
application of the $\mathcal{K}_{i}$ operators in the scattering equations and
kinematical numerators. The integral over the $\xi_{r}$’s has the structure of
an Aomoto-Gelfand hypergeometric function.

#### b) $\bm{n\leq D:}$

In this case, the delta functions can be used to perform all the Mellin
transforms. To do this, we solve for $\omega_{l}$, $l=1,\dots,n-1$, in terms
of $\omega_{n}$. Divide the spacetime index $\mu$ into two sets: $\mu=(r,a)$,
where $r=0,1,\dots,D-n$ while $a=D-n+1,\dots,D-1$. We solve the last $n-1$
momentum conservation constraints,

$\sum_{l}s_{l}\,\omega_{l}\,q_{l}^{a}=-s_{n}\,\omega_{n}\,q_{n}^{a}\,,$ (A.7)

for the $\omega_{l}$. This yields

$\omega_{l}=u_{ln}\,\omega_{n}\,,\qquad u_{ln}=-s_{l}\,s_{n}\,\frac{(1\,2\dots
l-1\;n\;l+1\dots n-1)}{(1\,2\dots n-1)}\,,$ (A.8)

where the determinants $(i_{1}\,i_{2}\dots i_{n-1})$ are now defined as

$(i_{1}\,i_{2}\dots i_{n-1}):=\varepsilon_{a_{1}a_{2}\dots
a_{n-1}}\,q_{i_{1}}^{a_{1}}\,q_{i_{2}}^{a_{2}}\cdots q_{i_{n-1}}^{a_{n-1}}\,,$
(A.9)

with $\varepsilon_{a_{1}a_{2}\dots a_{n-1}}$ the $(n-1)$-dimensional Levi
Civita symbol. Again, these solutions are supplemented by the constraints
$u_{ln}>0$ for all $l$.

Renaming $\omega_{n}\equiv\omega$, we find the expression

$\mathcal{S}_{n}=\delta^{D-n+1}\biggl{(}\sum_{l}s_{l}\,u_{ln}\,q_{l}+s_{n}\,q_{n}\biggr{)}\;\frac{1}{u}\prod_{l}\Theta(u_{ln})\,u_{ln}^{\Delta_{l}-1}\int_{0}^{\infty}\frac{\mathrm{d}\omega}{\omega}\,\omega^{\sum_{i}\Delta_{i}-D}\,,$
(A.10)

where the Jacobian is now $u=|(1\,2\dots n-1)|$. The remaining delta functions
constrain the $q_{i}^{r}$ for $r=0,1,\dots,D-n$.

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