# Effective Field Theories and Cosmological Scattering Equations

C. Armstrong<EMAIL_ADDRESS>Department of Mathematical
Sciences, Durham University,
Stockton Road, DH1 3LE, Durham, United Kingdom  H. Gomez
<EMAIL_ADDRESS>Department of Mathematical Sciences, Durham
University,
Stockton Road, DH1 3LE, Durham, United Kingdom  Facultad de Ciencias Basicas,
Universidad Santiago de Cali,
Calle 5 $N^{\circ}$ 62-00 Barrio Pampalinda, Cali, Valle, Colombia  R.
Lipinski Jusinskas<EMAIL_ADDRESS>Institute of Physics of the Czech Academy
of Sciences & CEICO,
Na Slovance 2, 182 21, Prague, Czech Republic  A. Lipstein
<EMAIL_ADDRESS>Department of Mathematical Sciences, Durham
University,
Stockton Road, DH1 3LE, Durham, United Kingdom  J. Mei<EMAIL_ADDRESS>Department of Mathematical Sciences, Durham University,
Stockton Road, DH1 3LE, Durham, United Kingdom

###### Abstract

We propose worldsheet formulae for correlators of the massive non-linear sigma
model (NLSM), scalar Dirac-Born-Infeld (DBI), and special Galileon (sGal)
theories in de Sitter momentum space in terms of the recently proposed
cosmological scattering equations constructed from conformal generators in the
future boundary. The four-point integrands are constructed from simple
building blocks and we propose a generalised double copy which maps the NLSM
correlator to the DBI and sGal correlators, including mass deformations and
curvature corrections. Finally, we compute the soft limits of these
correlators and find that they can also be written in terms of boundary
conformal generators acting on contact diagrams.

###### Contents

  1. 1 Introduction
  2. 2 Effective Field Theory Amplitudes
    1. 2.1 Lagrangians
    2. 2.2 Worldsheet formulae
  3. 3 de Sitter Correlators from Effective Actions
    1. 3.1 Witten Diagrams
    2. 3.2 Four-point correlators
  4. 4 de Sitter Correlators from the Worldsheet
    1. 4.1 Cosmological Scattering Equations
    2. 4.2 Building Blocks
    3. 4.3 Generalised Double Copy
  5. 5 Soft Limits
    1. 5.1 Contact diagram
    2. 5.2 Two Derivatives
    3. 5.3 Four Derivatives
    4. 5.4 Six Derivatives
  6. 6 Conclusion
  7. A Six-point NLSM correlators from Witten diagrams
  8. B Six-point NLSM from the worldsheet
  9. C Four-point diagram with double pole
  10. D Six-derivative results

##  1 Introduction

In recent years, there has been exciting progress in adapting methods used to
compute scattering amplitudes, which probe physics at its shortest distances,
to cosmological observables, which probe physics at its largest distances. In
the latter case, we have in mind boundary correlators in four dimensional de
Sitter space (dS4), which provides an approximate description of the early
Universe according to the inflationary paradigm [1, 2, 3, 4]. The cosmological
observables we will focus on are known as coefficients of the wavefunction of
the universe [5], which can be computed via a Wick rotation of Witten diagrams
in Anti de Sitter (AdS) and can be treated like conformal field theory (CFT)
correlators in the future boundary of dS [6, 7, 8, 9, 10, 11, 12]. In this
paper we will interchangeably refer to these observables as wavefunction
coefficients and correlators. In-in correlators [13] can then be computed by
squaring of the wavefunction and computing expectation values [7].

Recent developments include geometric approaches [14, 15], methods based on
factorisation [16, 17, 18, 19, 20], Mellin-Barnes representations [21, 22],
unitarity methods [23, 24, 25, 26, 27, 28, 29, 30], color/kinematic duality
[31, 32, 33, 34, 35, 36, 37], the double copy [38, 39, 40], and the
cosmological scattering equations (CSE) [41, 42]. See also [43, 44] for other
recent work adapting amplitude ideas to (A)dS correlators. This paper will
focus mostly on the CSE and double copy. In flat space, the double copy
relates graviton amplitudes to the square of gluon amplitudes [45, 46].
Moreover, the scattering equations of Cachazo, He, and Yuan (CHY) provide a
universal framework for describing scattering amplitudes in terms of
worldsheet integrals [47, 48, 49] and reveal a vast web of relations among
quantum field theories such as the non-linear sigma model (NLSM), Dirac-Born-
Infeld (BDI), and special Galileon (sGal) theories, extending the scope of the
double copy [50, 51, 52]. The CSE represent an extension of the flat space
scattering equations to dS momentum space and were initially proposed to
describe massive $\phi^{4}$ theory as a toy model of inflation, building on
earlier work on bi-adjoint scalar scalar theory in AdS position space [53,
54]. One of the key properties of this proposal is an operatorial integrand
constructed from conformal generators in the future boundary of dS which acts
on a contact diagram. An important open question is how to extend this
structure to more general scalar theories, and in this paper we will take the
first steps to address it.

Towards this goal, we will extend the CSE framework to the NLSM, scalar DBI,
and sGal theories in dS space. When lifting derivative interactions to curved
background, new subtleties arise. First of all, the spacetime Lagrangians
receive curvature corrections whose coefficients cannot be fixed by the flat
space limit. At four points, we find that such corrections can be described
using simple building blocks at the level of the worldsheet integrand. We can
also deform the integrand to describe bulk scalar fields with arbitrary mass.
The other class of subtleties is related to more technical aspects of the
problem. Becasue of derivative interactions in the curved background, the
worldsheet formulae can have ordering ambiguities involving differential
operators with nontrivial commutators. Such ambiguities can arise in the NLSM
at six points and the DBI and sGal theories at four points, although we find a
simple prescription for avoiding them at four points.

Our worldsheet formulae give rise to four-point correlators in terms of
boundary conformal generators acting on contact diagrams. We also identify a
simple prescription for mapping the NLSM correlator into the DBI and sGal
correlators at the level of the worldsheet integrand, which we refer to as a
generalised double copy. After specifying a prescription for evaluating the
worldsheet integrals to avoid ordering ambiguities, this provides a systematic
way to implement the double copy of four-point correlators in these theories.
Finally, we study the soft limits of the four-point correlators and find that
they can once again be written in terms of boundary conformal generators
acting on certain three-point contact diagrams. In the flat space limit, the
scattering amplitudes of these theories are well-known to exhibit vanishing
soft limits [55, 56] as a result of underlying hidden symmetries [57, 58, 59].
In our treatment, the coefficients of curvature corrections are left unfixed
and the soft limits we derive are completely general. Lagrangians for the DBI
and sGal theories with certain hidden symmetries in dS were recently proposed
in [60]. It would be interesting to compute their four-point correlators in
terms of the building blocks we derive in this paper and check if their soft
limits exhibit any additional simplicity. Soft limits of flat space
correlators in these theories were also recently analysed in [61].

This paper is organised as follows. In section 2 we review the NLSM, DBI, and
sGal theories in flat space and the CHY representation for their amplitudes.
In section 3, we then discuss how to lift these theories to dS space up to
4-point interactions, and compute their four-point wavefunction coefficients
in terms of Witten diagrams. In section 4, we then propose worldsheet formulae
for these correlators in terms of CSE, and a generalised double copy which
maps the NLSM correlator into the DBI and sGal correlators, including
curvature corrections and mass deformations. In section 5, we analyse the soft
limits of these four-point correlators and show that they can be written in
terms of boundary conformal generators acting on certain three-point contact
diagrams. We present our conclusions in section 6. We also have a number of
Appendices in which we compute wavefunction coefficients for the six-point
NLSM in dS using Witten diagrams and worldsheet methods, and provide more
details on 4-point correlators.

##  2 Effective Field Theory Amplitudes

In this work we focus on certain scalar effective field theories (EFT), namely
the non-linear sigma model (NLSM), scalar Dirac-Born-Infeld theory (DBI), and
special Galileon theory (sGal). Their scattering amplitudes take a
particularly simple form in the CHY approach [50].

###  2.1 Lagrangians

We start with a quick review of the scalar EFT Lagrangians.

The NLSM Lagrangian is given by

$\begin{split}\mathcal{L}_{\mathrm{NLSM}}&=\frac{1}{8\lambda^{2}}\mathrm{Tr}(\partial_{\mu}U^{\dagger}\partial^{\mu}U),\\\
&=-\mathrm{Tr}\left[\tfrac{1}{2}\partial_{\mu}\Phi\partial^{\mu}\Phi+\lambda^{2}\Phi^{2}\partial_{\mu}\Phi\partial^{\mu}\Phi+\lambda^{4}\left(\Phi^{4}\partial_{\mu}\Phi\partial^{\mu}\Phi+\tfrac{1}{2}\Phi^{2}\partial_{\mu}\Phi\Phi^{2}\partial^{\mu}\Phi\right)+\ldots\right]\end{split}$
(2.1)

where $U=(\mathbb{I}+\lambda\Phi)(\mathbb{I}-\lambda\Phi)^{-1}$, $\Phi$ is in
the adjoint representation of SU$(N)$, and the ellipsis denotes higher-point
interactions. For the DBI theory we have

$\begin{split}\mathcal{L}_{\mathrm{DBI}}&=\frac{1}{\lambda}\left(\sqrt{1-\lambda\left(\partial\phi\right)^{2}}-1\right),\\\
&=-\frac{1}{2}\partial\phi\cdot\partial\phi-\frac{\lambda}{8}\left(\partial\phi\cdot\partial\phi\right)^{2}+\ldots.\end{split}$
(2.2)

Finally, in $d=4$ the special Galileon theory is given by [58]

$\mathcal{L}_{\mathrm{sGal}}=-\frac{1}{2}\left(\partial\phi\right)^{2}-\frac{\lambda}{8}\left(\partial_{\mu}\partial_{\nu}\phi\right)^{2}\left(\partial\phi\right)^{2}.$
(2.3)

Note that the four-point interactions in the DBI and sGal theories contain
four derivative and six-derivatives, respectively. They are unique up to
integration by parts and equations of motion. Their lift to curved
backgrounds, however, is not unique because covariant derivatives no longer
commute and there are curvature corrections, as we describe later on.

###  2.2 Worldsheet formulae

The CHY formulae express tree-level scattering amplitudes as integrals over
the Riemann sphere, which localise onto solutions of the scattering equations
(SE):

$S_{a}=\sum_{a\neq b}\frac{2\,k_{a}\cdot
k_{b}}{\sigma_{ab}}=0,\qquad\sigma_{ab}\equiv\sigma_{a}-\sigma_{b},$ (2.4)

where $\sigma_{a}$ is the holomorphic coordinate of the $a$-th puncture, and
the punctures are in one-to-one correspondence with the external legs of the
amplitude. The details of the theory appear only in the worldsheet integrand,
and a generic $n$-point tree level amplitude takes the form

$\mathcal{A}_{n}=\int_{\gamma}\prod_{a=1\atop a\neq
b,c,d}^{n}\text{d}\sigma_{a}\,(S_{a})^{-1}\,(\sigma_{bc}\sigma_{cd}\sigma_{db})^{2}\,\mathcal{I}_{n},$
(2.5)

where $\left\\{b,c,d\right\\}$ are fixed punctures. The integration contour is
defined by the intersection $\gamma=\bigcap_{a\neq b,c,d}\gamma_{S_{a}}$,
where $\gamma_{S_{a}}$ encircles the poles of $S_{a}$.

The CHY integrands for the scalar EFTs of interest are defined using simple
building blocks, with various operations to transform between different
theories [50]. They are summarised in Table 1 and we will briefly review their
construction.

Theory | Integrand
---|---
NLSM | $\mathrm{PT}(\mathrm{Pf}^{\prime}A)^{2}$
DBI | $\mathrm{Pf}X(\mathrm{Pf}^{\prime}A)^{3}$
sGal | $(\mathrm{Pf}^{\prime}A)^{4}$
Table 1: A summary of CHY integrands for a selection of scalar EFTs.

The integrand for color-ordered amplitudes of the NLSM is

$\mathcal{I}^{\mathrm{NLSM}}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})=\rm{PT}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})(\mathrm{Pf}^{\prime}A)^{2},$
(2.6)

where $\alpha=(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})$ denotes a specific
ordering, and $\rm{PT}$ is the Parke-Taylor factor

${\rm
PT}(\alpha_{1},\alpha_{2},\ldots,\alpha_{n})=\left(\sigma_{\alpha_{1}\alpha_{2}}\sigma_{\alpha_{2}\alpha_{3}}\cdots\sigma_{\alpha_{n}\alpha_{1}}\right)^{-1}.$
(2.7)

In practice we will use the canonical ordering, i.e. ${\rm PT}(1,2,\ldots,n)$,
so for simplicity we introduce the notation

${\rm PT}\equiv{\rm PT}(1,2,\ldots,n).$ (2.8)

The reduced Pfaffian ${\rm Pf}^{\prime}A$ is given by

$\displaystyle{\rm Pf}^{\prime}A$ $\displaystyle=$
$\displaystyle\frac{(-1)^{c+d}}{\sigma_{cd}}{\rm Pf}A^{cd}_{cd},$ (2.9)
$\displaystyle{\rm Pf}A^{cd}_{cd}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{r_{1}s_{1}\ldots
r_{p-1}s_{p-1}}(A^{cd}_{cd})_{r_{1}s_{1}}\cdots(A^{cd}_{cd})_{r_{p-1}s_{p-1}}}{2^{p-1}(p-1)!},$
(2.10)

where the matrix $A^{cd}_{cd}$ is obtained from the $n\times n$ matrix

$\displaystyle A_{rs}$
$\displaystyle=\begin{cases}\displaystyle\frac{2k_{r}\cdot
k_{s}}{\sigma_{rs}},&r\neq s,\\\ \displaystyle\leavevmode\nobreak\
\leavevmode\nobreak\ 0,&r=s,\end{cases}$ (2.11)

by removing any pair of rows and columns $\\{c,d\\}$, with $n=2p$.
Alternatively the Pfaffian can be computed from the the square root of the
determinant.

For the scalar DBI theory, the worldsheet integrand is given by

$\mathcal{I}^{\mathrm{DBI}}=\mathrm{Pf}X(\mathrm{Pf}^{\prime}A)^{3},$ (2.12)

where

$\displaystyle X_{rs}$
$\displaystyle=\begin{cases}\displaystyle\frac{1}{\sigma_{rs}},&r\neq s,\\\
\displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\ 0,&r=s.\end{cases}$
(2.13)

The $X$-matrix arises from dimensional reduction of the CHY formula for Yang-
Mills amplitudes [50]. These reason for this will be further explained below.
Finally, the integrand for the sGal theory is given by

$\mathcal{I}^{\mathrm{sGal}}=(\mathrm{Pf}^{\prime}A)^{4}.$ (2.14)

Observe that the integrands of both DBI and sGal can be obtained from NLSM via
the following substitutions, respectively:

$\displaystyle{\rm PT}$ $\displaystyle\rightarrow$
$\displaystyle\mathrm{Pf}X(\mathrm{Pf}^{\prime}A),$ (2.15) $\displaystyle{\rm
PT}$ $\displaystyle\rightarrow$ $\displaystyle\left({\rm
Pf^{\prime}}A\right)^{2},$ (2.16)

which encode the double copy structure of these theories. Roughly speaking,
$\rm{sGal}=\rm{NLSM}^{2}$ and $\rm{DBI}=\rm{NLSM}\times\rm{YM}$, where
$\rm{YM}$ corresponds to the dimensional reduction of Yang-Mills theory. More
precisely, $\mathrm{Pf}X(\mathrm{Pf}^{\prime}A)$ can be written as a linear
combination of $(n-2)!$ Parke-Taylor factors, where the coefficients are the
Yang-Mills BCJ master numerators after the identification
$\epsilon_{i}\cdot\epsilon_{j}=1$ and $\epsilon_{i}\cdot k_{j}=0$. [48, 50,
62, 63, 64, 65, 52].

##  3 de Sitter Correlators from Effective Actions

In this section we review the computation of field theory observables in de
Sitter. For convenience, we use the Poincaré patch with radius set to one:

$ds^{2}=\frac{1}{\eta^{2}}(d\vec{x}^{2}-d\eta^{2}),$ (3.1)

where $-\infty<\eta<0$ is the conformal time, and $\vec{x}$ denotes the
boundary coordinate, with individual components $x^{i}$, $i=1,..,d$. In
practice, we set the dimension of the boundary $d=3$.

In-in correlators [13] can be computed from a cosmological wavefunction as
follows [7]:

$\langle\phi(\vec{k}_{1})\ldots\phi(\vec{k}_{n})\rangle=\frac{\int\mathcal{D}\phi\,\phi(\vec{k}_{1})...\phi(\vec{k}_{n})\left|\Psi[\phi]\right|^{2}}{\int\mathcal{D}\phi\left|\Psi[\phi]\right|^{2}}.$
(3.2)

The scalars $\phi$ are taken to be in the future boundary, Fourier transformed
to momentum space. The functional $\Psi[\phi]$ is the cosmological
wavefunction, which can be perturbatively expanded as

$\ln\Psi[\phi]=-\sum_{n=2}^{\infty}\frac{1}{n!}\int\prod_{i=1}^{n}\frac{{\rm
d}^{d}k_{i}}{(2\pi)^{d}}\Psi_{n}(\vec{k}_{1},\ldots,\vec{k}_{n})\phi(\vec{k}_{1})\ldots\phi(\vec{k}_{n}).$
(3.3)

The wavefunction coefficients $\Psi_{n}$ can be treated as $n$-point CFT
correlators in the future boundary, so we will interchangeably refer to them
as correlators. In momentum space, they can be expressed as

$\Psi_{n}=\delta^{d}(\vec{k}_{T})\langle\langle\mathcal{O}(\vec{k}_{1})...\mathcal{O}(\vec{k}_{n})\rangle\rangle,$
(3.4)

where $\vec{k}_{T}=\vec{k}_{1}+\ldots+\vec{k}_{n}$, and the double brackets
denote a CFT correlator on the boundary. The scalar operators $\mathcal{O}$
have scaling dimension $\Delta$, and are dual to scalar fields $\phi$ in the
bulk with mass

$m^{2}=\Delta(d-\Delta).$ (3.5)

Note that $\Delta=d$ describes minimally coupled scalars while
$\Delta=(d+1)/2$ describes conformally coupled scalars.

The correlators $\Psi_{n}$ satisfy conformal Ward identities (CWIs), which are
a consequence of the de Sitter isometries. The conformal generators are $D$
(dilatation), $P_{i}$ (translations), $K_{i}$ (special conformal
transformations), and $M_{ij}$ (rotations). The CWIs can be cast as

$\sum_{a=1}^{n}P_{a}^{i}\Psi_{n}=\sum_{a=1}^{n}D_{a}\Psi_{n}=\sum_{a=1}^{n}K_{a}^{i}\Psi_{n}=\sum_{a=1}^{n}M_{a}^{ij}\Psi_{n}=0,$
(3.6)

where $a,b,...$ are particle labels and

$\begin{split}P^{i}&=k^{i},\\\ D&=k^{i}\partial_{i}+(d-\Delta),\\\
K_{i}&=k_{i}\partial^{j}\partial_{j}-2k^{j}\partial_{j}\partial_{i}-2(d-\Delta)\partial_{i},\\\
M_{ij}&=\left(k_{i}\partial_{j}-k_{j}\partial_{i}\right),\end{split}$ (3.7)

with $\partial_{i}=\tfrac{\partial}{\partial k^{i}}$. Observe here that the
conformal dimension of the scalars appear explicitly in generators. Boundary
vector indices will be freely raised and lowered here using a flat metric.

###  3.1 Witten Diagrams

From a more traditional field theory perspective, the cosmological correlators
can be computed through Witten diagrams [7, 11, 10]

The scalar equation of motion is given by

$[\eta^{2}\partial_{\eta}^{2}+(1-d)\eta\partial_{\eta}-\eta^{2}\partial^{i}\partial_{i}+m^{2}]\phi=0.$
(3.8)

We will consider momentum eigenstates, such that
$\phi=\mathcal{K}_{\nu}(k,\eta)e^{i\vec{k}\cdot\vec{x}}$, with
$\mathcal{K}_{\nu}(k,\eta)$ denoting the bulk-to-boundary propagator:

$\mathcal{K}_{\nu}(k,\eta)=\mathcal{N}k^{\nu}\eta^{d/2}H_{\nu}(-k\eta).$ (3.9)

Here, $\nu=\Delta-d/2$, $k=|\vec{k}|$, $H_{\nu}$ is a Hankel function of the
second kind, and we will leave the normalisation $\mathcal{N}$ unspecified for
now. $\mathcal{K}_{\nu}(k,\eta)$ satisfies
$(\mathcal{D}_{k}^{2}+m^{2})\mathcal{K}_{\nu}=0$, with

$\mathcal{D}_{k}^{2}\equiv\eta^{2}\partial_{\eta}^{2}+(1-d)\eta\partial_{\eta}+\eta^{2}k^{2}.$
(3.10)

Contact diagrams are given by a product of bulk-to-boundary propagators
integrated over the bulk, expressed as

$\displaystyle\mathcal{C}^{\Delta}_{n}$ $\displaystyle\equiv$
$\displaystyle\int\frac{d\eta}{\eta^{d+1}}U_{1,n}(\eta),$ (3.11)
$\displaystyle U_{m,n}(\eta)$ $\displaystyle=$
$\displaystyle\prod_{a=m}^{n}\mathcal{K}_{\nu}(k_{a},\eta).$ (3.12)

More general Witten diagrams involve also bulk-to-bulk propagators,
$G_{\nu}(k,\eta,\tilde{\eta})$, satisfying

$(\mathcal{D}_{k}^{2}+m^{2})G_{\nu}=\eta^{d+1}\delta(\eta-\tilde{\eta}).$
(3.13)

As it turns out, however, any tree level diagram can be obtained from contact
diagrams through certain differential operations.

In order to see this, let us first consider the action of the boundary
generators in (3.7) on bulk-to-boundary propagators. It can be expressed in
terms of derivatives with respect to conformal time

$\begin{array}[]{ccc}D\mathcal{K}_{\nu}=\eta\tfrac{\partial}{\partial\eta}\mathcal{K}_{\nu},&&P^{i}\mathcal{K}_{\nu}=k^{i}\mathcal{K}_{\nu},\\\
K_{i}\mathcal{K}_{\nu}=\eta^{2}k_{i}\mathcal{K}_{\nu},&&M_{ij}\mathcal{K}_{\nu}=0.\end{array}$
(3.14)

Now consider the following operator

$\mathcal{D}_{a}\cdot\mathcal{D}_{b}=\frac{1}{2}(P_{a}^{i}K_{bi}+K_{ai}P_{b}^{i}-M_{a,ij}M_{b}^{ij})+D_{a}D_{b}.$
(3.15)

Using (3.14) one finds that

$(\mathcal{D}_{a}\cdot\mathcal{D}_{b})\mathcal{K}_{\nu}^{a}\mathcal{K}_{\nu}^{b}=\eta^{2}[\partial_{\eta}\mathcal{K}_{\nu}^{a}\partial_{\eta}\mathcal{K}_{\nu}^{b}+(\vec{k}_{a}\cdot\vec{k}_{b})\mathcal{K}_{\nu}^{a}\mathcal{K}_{\nu}^{b}],$
(3.16)

with shorthand notation $\mathcal{K}_{\nu}^{a}=\mathcal{K}_{\nu}(k_{a},\eta)$.
This observation can then be used to show that

$(\mathcal{D}_{1\ldots
p}^{2}U_{1,p})U_{p{+}1,n}=(\mathcal{D}_{1}+\ldots+\mathcal{D}_{p})^{2}U_{1,n}.$
(3.17)

where in the left hand side $\mathcal{D}_{1\ldots p}^{2}$ is defined in (3.10)
with $k=|\vec{k}_{1}+\ldots+\vec{k}_{p}|\equiv k_{1\ldots p}$ and $p<n$, and
the right hand side is built using the boundary conformal generators in
momentum space (3.7), satisfying $\mathcal{D}_{a}\cdot\mathcal{D}_{a}=-m^{2}$.
In particular, we can derive the following identity,

$[(\mathcal{D}_{1}+\ldots+\mathcal{D}_{p})^{2}+m^{2}]^{-1}\mathcal{C}^{\Delta}_{n}=\int\frac{d\eta}{\eta^{d+1}}\frac{d\tilde{\eta}}{\tilde{\eta}^{d+1}}U_{p+1,n}(\eta)G_{\nu}(k_{1...p},\eta,\tilde{\eta})U_{1,p}(\tilde{\eta}),$
(3.18)

which can be easily demonstrated using (3.17) and the equation of motion of
the bulk-to-bulk propagator (3.13). Therefore, any tree level exchange diagram
can be recast as a differential operator given in terms of the boundary
momenta acting on a contact diagram.

Finally, observe that acting with $\mathcal{D}_{a}\cdot\mathcal{D}_{b}$ on a
pair of bulk-to-boundary propagators is equivalent to acting with
$\nabla_{a}\cdot\nabla_{b}$, where $\nabla_{a}$ is a bulk covariant derivative
acting on leg $a$. This identity will be very useful for computing Witten
diagrams for derivative interactions containing terms of the form
$\nabla\phi\cdot\nabla\phi$. Another useful identity is

$\displaystyle[(\mathcal{D}_{a}\cdot\mathcal{D}_{b}),(\mathcal{D}_{b}\cdot\mathcal{D}_{c})]\mathcal{C}_{n}^{\Delta}$
$\displaystyle=2(K_{a}\cdot P_{c}-P_{a}\cdot
K_{c})D_{b}\mathcal{C}_{n}+\mathrm{cyc}(abc),$ $\displaystyle=0.$ (3.19)

The commutator is not zero but vanishes when acting on a contact diagram. This
was first derived in the embedding space formalism [34], and its
generalization to momentum space is straightforward. Using (3.14) we can
easily show that the right hand side vanishes.

###  3.2 Four-point correlators

Our strategy in this section will be to lift the effective actions in section
2.1 to de Sitter space up to four point interactions, use them to compute the
four-point boundary correlators using Witten diagrams, and express the result
in terms of boundary conformal generators acting on a contact term. It will be
convenient to define the following operators:

$\begin{array}[]{ccccc}\hat{s}=\mathcal{D}_{1}\cdot\mathcal{D}_{2},&&\hat{t}=\mathcal{D}_{1}\cdot\mathcal{D}_{4},&&\hat{u}=\mathcal{D}_{1}\cdot\mathcal{D}_{3},\end{array}$
(3.20)

which satisfy

$\hat{s}+\hat{t}+\hat{u}=m^{2},$ (3.21)

when acting on contact diagrams. This can be seen using the CWIs in (3.6).

In de Sitter space, the expansion of the NLSM action up to four points can be
cast as

$S_{4}^{NLSM}=-\int
d^{4}x\sqrt{-g}\mathrm{Tr}\\{\tfrac{1}{2}\partial\Phi\cdot\partial\Phi+\tfrac{1}{2}m^{2}\Phi^{2}+\lambda^{2}\Phi^{2}\partial_{\mu}\Phi\partial^{\mu}\Phi+\tfrac{1}{4}C\Phi^{4}\\},$
(3.22)

where we have included a mass term and a possible quartic interaction coming
from a curvature correction (recall we have set the dS radius to one). The
mass is also proportional to the curvature and vanishes in the flat space
limit. Using the identity (3.16), it is straightforward to show that the four-
point correlator obtained from Witten diagrams is given by

$\Psi_{4}^{NLSM}=\delta^{3}\left(\vec{k}_{T}\right)\left[2\lambda^{2}\left(\hat{s}+\hat{t}\right)-C\right]\mathcal{C}_{4}^{\Delta}=-\delta^{3}\left(\vec{k}_{T}\right)\left(2\lambda^{2}\hat{u}+C\right)\mathcal{C}_{4}^{\Delta}.$
(3.23)

Up to quartic vertices and six-derivative interactions, the most general
effective action for a scalar field in (A)dS is given by [66]

$S^{(6)}_{4}=-\int
d^{4}x\sqrt{-g}\\{\tfrac{1}{2}\nabla\phi\cdot\nabla\phi+\tfrac{1}{2}m^{2}\phi^{2}+\tfrac{1}{8}A(\nabla_{\mu}\nabla_{\nu}\phi)^{2}\nabla\phi\cdot\nabla\phi+\tfrac{1}{8}B(\nabla\phi\cdot\nabla\phi)^{2}+\tfrac{1}{4!}C\phi^{4}\\},$
(3.24)

where $A$, $B$, and $C$ are undetermined numerical coefficients. Other
possible interactions are related by integration by parts or the free equation
of motion $\nabla^{2}\phi=-m^{2}\phi.$ For the sGal theory, the 6-derivative
interaction is the naive uplift of the one in (2.3) while the lower-derivative
interactions correspond to curvature corrections and a mass term. In the flat
space limit, these are subleading and the action reduces to (2.3) for
$A=\lambda$. Hence, (3.24) represents the uplift of the special Galileon
theory to a curved background, with unfixed coefficients corresponding to
curvature corrections. Additional data must be specified in order to fix them,
such as soft limits, and we will explore this section 5. For the DBI theory,
we set $A=0,B=\lambda$, and there is a single curvature correction with
unfixed coefficient $C$. In the flat space limit, the action reduces to (2.2)
up to quartic interactions.

The four-point correlator obtained from (3.24) is

$\Psi^{(6)}_{4}=\delta^{3}\left(\vec{k}_{T}\right)[A(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})+(dA-B)(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})-C]\mathcal{C}_{4}^{\Delta}.$
(3.25)

We can illustrate this derivation using Witten diagrams. For example, let’s
consider the the six-derivative interaction term

$\displaystyle\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{2}\nabla\phi\cdot\nabla\phi$
$\displaystyle=$
$\displaystyle\eta^{6}\eta^{\mu\nu}\eta^{\rho\sigma}\eta^{\kappa\lambda}(\nabla_{\mu}\phi)(\nabla_{\nu}\phi)(\nabla_{\rho}\nabla_{\kappa}\phi)(\nabla_{\sigma}\nabla_{\lambda}\phi),$
(3.26) $\displaystyle=$
$\displaystyle\eta^{6}(\partial_{\mu}\phi\partial^{\mu}\phi)\eta^{\rho\sigma}\eta^{\kappa\lambda}(\partial_{\rho}\partial_{\kappa}-\Gamma^{\alpha}_{\rho\kappa}\partial_{\alpha}\phi)(\partial_{\sigma}\partial_{\lambda}\phi-\Gamma^{\beta}_{\sigma\lambda}\partial_{\beta}\phi).$

The corresponding tree-level Witten diagrams are given by

$\begin{split}&\int_{-\infty}^{0}\frac{d\eta}{\eta^{4}}\eta^{6}\bigg{[}(\vec{k}_{1}\cdot\vec{k}_{2})^{2}\mathcal{K}_{1}\mathcal{K}_{2}+2\vec{k}_{1}\cdot\vec{k}_{2}\dot{\mathcal{K}}_{1}\dot{\mathcal{K}}_{2}+\ddot{\mathcal{K}}_{1}\ddot{\mathcal{K}}_{2}\\\
&\qquad+\frac{1}{\eta}\left(2\vec{k}_{1}\cdot\vec{k}_{2}(\mathcal{K}_{1}\dot{\mathcal{K}}_{2}+\dot{\mathcal{K}}_{1}\mathcal{K}_{2})-k_{1}^{2}\mathcal{K}_{1}\dot{\mathcal{K}}_{2}-k_{2}^{2}\dot{\mathcal{K}}_{1}\mathcal{K}_{2}+\dot{\mathcal{K}}_{1}\ddot{\mathcal{K}}_{2}+\ddot{\mathcal{K}}_{1}\dot{\mathcal{K}}_{2}\right)\\\
&\qquad+\frac{2}{\eta^{2}}\left(\vec{k}_{1}\cdot\vec{k}_{2}\mathcal{K}_{1}\mathcal{K}_{2}+2\dot{\mathcal{K}}_{1}\dot{\mathcal{K}}_{2}\right)\bigg{]}\left[\vec{k}_{3}\cdot\vec{k}_{4}\mathcal{K}_{3}\mathcal{K}_{4}+\dot{\mathcal{K}}_{3}\dot{\mathcal{K}}_{4}\right]+...,\end{split}$
(3.27)

where $\mathcal{K}_{a}$ are bulk-to-boundary propagators,
$\dot{\mathcal{K}}_{a}$ denotes a derivative with respect to conformal time,
and the ellipsis denotes the $t$ and $u$ channels. If we recast this result in
terms of differential operators acting on a contact diagram we see that

$-\frac{1}{8}\left(\nabla_{\mu}\nabla_{\nu}\phi\right)^{2}\nabla\phi\cdot\nabla\phi\rightarrow\left[\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3}+d\left(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2}\right)\right]\mathcal{C}_{4}^{\Delta}.$
(3.28)

The other terms in (3.25) can be directly derived from (3.24) using (3.16). In
Appendix A we evaluate a 6-point NLSM correlator using Witten diagrams.

##  4 de Sitter Correlators from the Worldsheet

In this section we will use the cosmological scattering equations introduced
in [41] to compute the four-point correlators obtained from Witten diagrams in
the previous section.

###  4.1 Cosmological Scattering Equations

In order to lift the scattering equations to de Sitter space, we replace
$k_{a}\cdot k_{b}$ in (2.4) with differential operators acting in the future
boundary, given by (3.16), and introduce a mass deformation. The scattering
equations then have an operatorial character,

$\displaystyle S_{a}=\sum_{b=1\atop b\neq a}^{n}\frac{2\,({\cal
D}_{a}\cdot{\cal D}_{b})+\mu_{ab}}{\sigma_{ab}}\equiv\sum_{b=1\atop b\neq
a}^{n}\frac{\alpha_{ab}}{\sigma_{ab}}\,\,,$ (4.1)

where $\mu_{a\,a\pm 1}=-m^{2}$ modulo $n$ and zero otherwise. This mass
deformation is analogous to the flat space one in [67] and assumes canonical
ordering of the external legs $\mathbb{I}_{n}=(1,2,\ldots,n)$. These are
referred to as cosmological scattering equations (CSE). The CWIs in (3.6) can
be recast as

$\sum_{a\neq b}\alpha_{ab}\Psi_{n}=0,$ (4.2)

which imply an underlying ${\rm SL}(2,\mathbb{C})$ symmetry in the scattering
equations, just like in flat space. This symmetry can be used to fix the
location of three punctures. For more details, see [42].

The worldsheet formula in (2.5) can then be lifted to de Sitter space as
follows:

$\Psi_{n}=\delta^{d}(\vec{k}_{T})\int_{\gamma}\prod_{a\neq
b,c,d}^{n}\text{d}\sigma_{a}\,S_{a}^{-1}(\sigma_{bc}\sigma_{cd}\sigma_{db})^{2}\,\,{\cal
I}_{n}{\cal C}_{n}^{\Delta},$ (4.3)

where the integrand may also contain boundary conformal generators. For
theories with $\phi^{n}$ interactions, we are free to shuffle the CSE with
other terms in the integrand $\mathcal{I}_{n}$ [42]. On the other hand, this
may not be the case for derivative interactions, and we believe this issue
calls for a more systematic investigation in the future. In this work we fix
the ordering with CSE appearing to the left of the integrand.

The flat space integrands were constructed from Pfaffians defined in (2.9).
Their obvious uplift to de Sitter is through the matrix

$\displaystyle A_{rs}$
$\displaystyle=\begin{cases}\displaystyle\frac{\alpha_{rs}}{\sigma_{rs}},&r\neq
s,\\\ \displaystyle\leavevmode\nobreak\ \leavevmode\nobreak\
0,&r=s.\end{cases}$ (4.4)

As discussed previously, theories with higher-derivative interactions can have
curvature corrections that are absent in flat space. Therefore, we cannot
simply lift the flat space integrands in section 2.2 to dS by replacing
kinematic invariants with differential operators. This procedure has to be
supplemented by curvature corrections and mass deformations, which we will
describe in the following subsections.

###  4.2 Building Blocks

There are four basic building blocks for the four-point integrands. Here we
will describe each of them, explaining how to evaluate the corresponding
worldsheet integrals.

The simplest building block is

$\mathcal{I}^{\phi^{4}}={\rm PT}\left.{\rm Pf}X\right|_{{\rm{conn}}}{\rm
Pf^{\prime}}A,$ (4.5)

where the matrix $X$ is defined in (2.13) and

$\displaystyle{\rm Pf}\,X$ $\displaystyle=$
$\displaystyle\frac{1}{\sigma_{12}\sigma_{34}}-\frac{1}{\sigma_{13}\sigma_{24}}+\frac{1}{\sigma_{23}\sigma_{14}},$
(4.6) $\displaystyle\left.{\rm Pf}\,X\right|_{\rm{conn}}$ $\displaystyle=$
$\displaystyle-\frac{1}{\sigma_{13}\sigma_{24}}.$ (4.7)

The integrand (4.5) describes a contact diagram for a $\phi^{4}$ interaction
[50, 41]. Note that at four-points, $\rm{Pf}\,X$ can be written as a sum of
three terms which correspond to perfect matchings and $\left.{\rm
Pf}X\right|_{\rm{conn}}$ refers to the connected perfect matching with respect
to the ordering of the Parke-Taylor factor (see Figure 1).

Figure 1: Graphic representation of ${\rm PT}\,{\rm Pf}X|_{\rm conn}$. The
circle refers to the ${\rm PT}$ factor while the intersecting lines correspond
to ${\rm Pf}X|_{\rm conn}$.

Fixing legs $\left\\{1,2,4\right\\}$ and deleting legs $\left\\{2,4\right\\}$
from the $A$-matrix in the Pfaffian leads to the correlator

$\Psi_{4}^{\phi^{4}}=-\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\left(\sigma_{41}\sigma_{12}\sigma_{24}\right)^{2}\left[\frac{1}{\sigma_{12}\sigma_{23}\sigma_{34}\sigma_{41}}\frac{\sigma_{13}\sigma_{23}\sigma_{43}}{\hat{S}_{3}}\frac{\alpha_{13}}{\left(\sigma_{13}\sigma_{24}\right)^{2}}\right]\mathcal{C}_{4}^{\Delta},$
(4.8)

where the contour $\gamma_{3}$ encircles the pole arising from $S_{3}$. The
first term in the integrand is the standard Jacobian associated with the ${\rm
SL}(2,\mathbb{C})$ fixing, and we have rescaled the third scattering equation
to

$\hat{S}_{3}=\alpha_{13}\sigma_{23}\sigma_{43}+\alpha_{23}\sigma_{13}\sigma_{43}+\alpha_{43}\sigma_{13}\sigma_{23}.$
(4.9)

After some cancellations we see there is only a simple pole at $\sigma_{13}=0$
so we wrap the contour around this pole to obtain

$\Psi_{4}^{\phi^{4}}=\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\left[\frac{\sigma_{41}\sigma_{12}}{\sigma_{13}}\frac{1}{\hat{S}_{3}}\alpha_{13}\right]\mathcal{C}_{4}=-\delta^{3}(\vec{k}_{T})\int_{\sigma_{13}}d\sigma_{3}\left[\frac{\sigma_{41}\sigma_{12}}{\sigma_{13}}\frac{1}{\hat{S}_{3}}\alpha_{13}\right]\mathcal{C}_{4}^{\Delta}.$
(4.10)

Computing the residue then gives

$\Psi_{4}^{\phi^{4}}=-\delta^{3}(\vec{k}_{T})\left[\left.\frac{\sigma_{12}\sigma_{41}}{\hat{S}_{3}}\right|_{\sigma_{13}=0}\alpha_{13}\right]\mathcal{C}_{4}^{\Delta}=\delta^{3}(\vec{k}_{T})\,\mathcal{C}_{4}^{\Delta}.$
(4.11)

Next we consider the naive uplifts of the flat space integrands of section
2.2. Note that these uplifts will not describe the full 4-point correlators in
dS because they will be missing mass deformations and curvature corrections.
We will explain how to encode these additional terms in the worldsheet
integrands in the next subsection. The naive uplift of the NLSM integrand in
(2.6) is given by

$\mathcal{I}^{NLSM}={\rm PT}\left({\rm Pf}^{\prime}A\right)^{2}.$ (4.12)

As before, we will fix legs $\left\\{1,2,4\right\\}$ and delete legs
$\left\\{2,4\right\\}$ from the $A$-matrix to obtain

$\Psi_{4}^{NLSM}=\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\left(\sigma_{41}\sigma_{12}\sigma_{24}\right)^{2}\left[\frac{1}{\sigma_{12}\sigma_{23}\sigma_{34}\sigma_{41}}\frac{\sigma_{13}\sigma_{23}\sigma_{43}}{\hat{S}_{3}}\left(\frac{\alpha_{13}}{\sigma_{13}\sigma_{24}}\right)^{2}\right]\mathcal{C}_{4}^{\Delta}.$
(4.13)

After simplifying the integrand there is once again a simple pole at
$\sigma_{13}=0$, so we wrap the contour around this pole:

$\Psi_{4}^{NLSM}=-\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\left[\frac{\sigma_{12}\sigma_{41}}{\sigma_{13}}\frac{1}{\hat{S}_{3}}\alpha_{13}^{2}\right]\mathcal{C}_{4}^{\Delta}=\delta^{3}(\vec{k}_{T})\int_{\sigma_{13}=0}d\sigma_{3}\left[\frac{\sigma_{12}\sigma_{41}}{\sigma_{13}}\frac{1}{\hat{S}_{3}}\alpha_{13}^{2}\right]\mathcal{C}_{4}^{\Delta}.$
(4.14)

Evaluating the residue of this pole finally gives

$\Psi_{4}^{NLSM}=\delta^{3}(\vec{k}_{T})\left[\left.\frac{\sigma_{12}\sigma_{41}}{\hat{S}_{3}}\right|_{\sigma_{13}=0}\alpha_{13}^{2}\right]\mathcal{C}_{4}^{\Delta}=-\delta^{3}(\vec{k}_{T})\alpha_{13}\,\mathcal{C}_{4}^{\Delta}.$
(4.15)

In Appendix B, we generlise this computation to 6-points. This will illustrate
a number subtleties that can arise for worldsheet descriptions of theories
with derivative interactions, such as the presence of higher-order poles and
potential ordering ambiguities.

Now we analyse the naive uplift of the DBI integrand,

$\mathcal{I}^{DBI}={\rm Pf}X\left({\rm Pf}^{\prime}A\right)^{3}.$ (4.16)

Fixing legs $\left\\{1,2,4\right\\}$ as above, we find

$\Psi_{4}^{DBI}=\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}(\sigma_{14}\sigma_{12}\sigma_{24})^{2}\left[{\rm
Pf}X\frac{1}{S_{3}}\left({\rm
Pf}^{\prime}A\right)^{3}\right]\mathcal{C}_{4}^{\Delta}.$ (4.17)

Recall from (4.6) that $\rm{Pf}X$ has three terms. To evaluate the worldsheet
integral containing the first term, it is convenient to choose $({\rm
Pf}^{\prime}A)^{3}=({\rm Pf}A_{23}^{23})^{2}({\rm Pf}A_{24}^{24})$ such that

$\Psi_{4}^{12,34}=\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\frac{\left(\sigma_{14}\sigma_{12}\sigma_{24}\right)^{2}}{\sigma_{12}\sigma_{34}}\left[\frac{\sigma_{13}\sigma_{23}\sigma_{43}}{\hat{S}_{3}}\left(\frac{\alpha_{14}}{\sigma_{14}\sigma_{23}}\right)^{2}\frac{\alpha_{13}}{\sigma_{13}\sigma_{24}}\right]\,\mathcal{C}_{4}^{\Delta},$
(4.18)

where the superscript on $\Psi_{4}$ denotes the contribution from the first
term in (4.6). The contour integral can be evaluated as above to obtain

$\Psi_{4}^{12,34}=\delta^{3}(\vec{k}_{T})\left[\left.\frac{\sigma_{12}\sigma_{24}}{\hat{S}_{3}}\right|_{\sigma_{23}=0}\alpha_{14}^{2}\alpha_{13}\right]\mathcal{C}_{4}^{\Delta}=-\delta^{3}(\vec{k}_{T})\alpha_{14}\alpha_{13}\mathcal{C}_{4}^{\Delta},$
(4.19)

where we wrapped the contour around the pole $\sigma_{23}=0$, evaluated the
residue, and used the CWI to cancel $\alpha_{23}$ in the denominator with
$\alpha_{14}$ in the numerator. Note that the ordering of $\alpha_{14}$ and
$\alpha_{13}$ in the final result is not important due to (3.1). For the
remaining two terms in (4.17) we choose $({\rm Pf}^{\prime}A)^{3}$ to be
$({\rm Pf}A_{14}^{14})^{2}({\rm Pf}A_{12}^{12})$ and $({\rm
Pf}A_{13}^{13})^{2}({\rm Pf}A_{12}^{12})$, respectively. Using similar
manipulations, we finally obtain

$\Psi_{4}^{DBI}=-\delta^{3}(\vec{k}_{T})\left(\alpha_{12}\alpha_{14}+\alpha_{14}\alpha_{13}+\alpha_{13}\alpha_{12}\right)\mathcal{C}_{4}^{\Delta}.$
(4.20)

This choice of Pfaffians avoids higher-order poles in the worldsheet
coordinates which are more subtle to evaluate. We describe such an example in
Appendix C. It will also be useful to consider the following integrand, whose
corresponding correlator follows trivially from the above calculation:

$({\rm Pf}^{\prime}A)^{3}\left.{\rm Pf}X\right|_{{\rm
conn}}\rightarrow-\delta^{3}(\vec{k}_{T})\alpha_{12}\alpha_{14}\mathcal{C}_{4}^{\Delta}.$
(4.21)

Finally, let us consider the naive uplift of the special Galileon integrand

$\mathcal{I}^{sGal}=\left({\rm Pf}^{\prime}A\right)^{4}.$ (4.22)

In this case, the four-point correlator is given by

$\Psi_{4}^{sGal}=\delta^{3}(\vec{k}_{T})\int_{\gamma_{3}}d\sigma_{3}\left(\sigma_{14}\sigma_{12}\sigma_{24}\right)^{2}\left[\frac{1}{S_{3}}\left({\rm
Pf}^{\prime}A\right)^{4}\right]\mathcal{C}_{4}^{\Delta}.$ (4.23)

As in the DBI case, there are Pfaffian choices exclusively leading to simple
poles. The following choice leads to a permutation invariant result:

$({\rm Pf}^{\prime}A)^{4}=\frac{1}{3}\left\\{\frac{1}{\sigma^{2}_{34}}({\rm
Pf}A_{34}^{34})^{2}\,\frac{(-1)}{\sigma_{23}}({\rm
Pf}A_{23}^{23})\frac{1}{\sigma_{24}}\,({\rm Pf}A_{24}^{24})+{\rm
cyclic}(2,3,4)\right\\}.$ (4.24)

Note that other choices can give different results due to non-trivial
commutators which only vanish in the flat space limit. Hence we must specify a
choice of Pfaffian. Following closely the computations above, we obtain

$\Psi_{4}^{sGal}=\frac{1}{3}\delta^{3}(\vec{k}_{T})\left(\alpha_{12}\alpha_{14}\alpha_{13}+\alpha_{14}\alpha_{13}\alpha_{12}+\alpha_{13}\alpha_{14}\alpha_{12}\right)\mathcal{C}_{4}^{\Delta}.$
(4.25)

Using (3.1), it is not difficult to see that the above expression is
permutation invariant.

###  4.3 Generalised Double Copy

Using the building blocks in the previous subsection (in particular (4.11) and
(4.15)), we see that the four-point NLSM correlator (including a curvature
correction) in (3.23) follows from the following integrand:

$\mathcal{I}_{4}^{NLSM}=\lambda^{2}{\rm PT}\left({\rm
Pf}^{\prime}A\right)^{2}+c{\rm PT}\left.{\rm Pf}X\right|_{\rm{conn}}{\rm
Pf^{\prime}}A,$ (4.26)

if we identify the unfixed parameter $c$ with $-C$. Now consider the following
shift of the first term in (4.26):

$\lambda^{2}{\rm PT}\rightarrow a{\rm Pf^{\prime}}A\left({\rm
Pf^{\prime}}A+m^{2}\left.{\rm Pf}X\right|_{{\rm conn}}\right)+b\left({\rm
Pf^{\prime}}A{\rm Pf}X+m^{2}{\rm PT}\right),$ (4.27)

where $a$ and $b$ are also free coefficients. We then obtain the following
integrand:

$\mathcal{I}_{4}^{(6)}=a({\rm Pf}^{\prime}A)^{3}({\rm
Pf^{\prime}}A+m^{2}\left.{\rm Pf}X\right|_{{\rm conn}})+b({\rm
Pf}^{\prime}A)^{2}({\rm Pf^{\prime}}A{\rm Pf}X+m^{2}{\rm PT})+c{\rm
PT}\left.{\rm Pf}X\right|_{\rm{conn}}{\rm Pf^{\prime}}A.$ (4.28)

Using the prescription for evaluating worldsheet integrals described the
previous subsection (in particular (4.11), (4.15), (4.20), (4.21), and
(4.25)), we see that the corresponding correlator is

$\Psi_{4}^{(6)}=\delta^{3}(\vec{k}_{T})\left[\tfrac{8}{3}a(\tilde{s}\tilde{t}\tilde{u}+\tilde{t}\tilde{u}\tilde{s}+\tilde{u}\tilde{s}\tilde{t})-4b(\tilde{s}\tilde{t}+\tilde{t}\tilde{u}+\tilde{u}\tilde{s})+c\right]\mathcal{C}_{4}^{\Delta},$
(4.29)

where $\tilde{s}$, $\tilde{t}$, and $\tilde{u}$ are related to their hatted
counterparts (3.20) as

$\begin{array}[]{ccccc}\tilde{s}=\hat{s}-\tfrac{1}{2}m^{2},&&\tilde{t}=\hat{t}-\tfrac{1}{2}m^{2},&&\tilde{u}=\hat{u}-\tfrac{1}{2}m^{2}.\end{array}$
(4.30)

It is then straightforward to match (4.29) with (3.25) via the following
identification of unfixed coefficients:

$\begin{array}[]{ccccc}A=\tfrac{8}{3}a,\,\,\,B=2a\left(m^{2}+\frac{4}{3}d\right)-2b,\,\,\,C=-\tfrac{1}{3}am^{6}+bm^{4}-c.\end{array}$
(4.31)

Hence, the worldsheet integrand (4.28) encodes the effective action in (3.24).
Moreover, (4.27) can be thought of as a double copy procedure encoding mass
deformations and curvature corrections. In particular, it reduces to (2.16)
and (2.15) in the flat space limit for $a\neq 0$ and $a=0$, respectively
(recalling that the mass is measured in units of the inverse dS radius). We
therefore refer to this as a generalised double copy. Note that this proposal
leaves the curvature corrections and mass deformations unfixed. To fix these
coefficients, we must specify additional data beyond the flat space limit. We
will suggest a strategy for doing this in the next section.

##  5 Soft Limits

In the previous section, we found the building blocks of four-point
correlators of scalar EFTs in dS to be:

$\mathcal{C}_{4}^{\Delta},\,\,\,\hat{u}\mathcal{C}_{4}^{\Delta},\,\,\,\left(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2}\right)\mathcal{C}_{4}^{\Delta},\,\,\,\left(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3}\right)\mathcal{C}_{4}^{\Delta}.$
(5.1)

For notational simplicity, we will leave out the delta function imposing
momentum conservation along the boundary. The first one is a $\phi^{4}$
contact diagram, while the remaining building blocks arise from the action on
this contact diagram with the differential operators defined in (3.20). The
first two terms contribute when lifting NLSM to dS, while the last two arise
when lifting the DBI and sGal theories. In this section, we will compute these
objects explicitly in the cases $\Delta=2,3$ and analyse their soft limits. In
practice, all the integrals we encounter will be of the form
$\int^{0}_{-\infty}\eta^{\alpha}e^{-iE\eta}d\eta$. To evaluate them, we rotate
the contour clockwise onto the complex plane so that the integrand is
exponentially damped as $\eta\to-\infty$ [10]. To simplify our expressions we
will set the normalisation of the bulk-to-boundary propagators in (3.9) to be
$\mathcal{N}=\left(-1\right)^{\nu-\frac{1}{2}}\sqrt{\frac{\pi}{2}}$.

We will show that the soft limits of the building blocks in (5.1) are given in
terms of boundary conformal generators acting on certain three-point contact
diagrams, extending previous results in the context of cosmology [7, 68, 69]
to more general effective field theories. Soft limits of DBI and sGal
correlators in flat space were recently analysed in [61]. Note that the soft
limit of the four-point correlators in (3.23) and (3.25) can be obtained from
linear combinations of soft limits we derive in this section. By appropriately
choosing the coefficients of curvature corrections in the corresponding EFTs
in (3.22) and (3.24) we can set many of these terms to zero, which may signal
the existence of hidden symmetries. It would then be interesting compare the
resulting EFTs to those recently constructed in [60].

###  5.1 Contact diagram

Let us first consider the four-point contact diagram. For conformally coupled
scalars, we find

$\mathcal{C}_{4}^{\Delta=2}=\int\frac{d\eta}{\eta^{4}}\left(\mathop{\prod}\limits_{i=1}^{4}\mathcal{K}_{1/2}^{i}\right)=\frac{1}{E}.$
(5.2)

It is trivial to see that the soft limit of this correlators corresponds to a
three-point contact diagram for conformally coupled scalars with a time-
dependent interaction:

$\lim_{\vec{k}_{1}\rightarrow
0}C_{4}^{\Delta=2}=\mathcal{C}_{3,\eta}^{\Delta=2},$ (5.3)

where

$\mathcal{C}_{3,\eta}^{\Delta=2}=\int\frac{d\eta}{\eta^{4}}\left(\eta\,\mathop{\prod}\limits_{i=2}^{4}\mathcal{K}_{1/2}^{i}\right)=\frac{1}{E}.$
(5.4)

This three-point correlator will also arise in the soft limit of more
complicated four-point correlators of conformally coupled scalars. In
particular, we will obtain conformal generators acting on (5.4).

For minimally coupled scalars, the integrals need to be regulated. We will use
the prescription $d\rightarrow d+2\epsilon$,
$\Delta\rightarrow\Delta+\epsilon$, which leaves the spectral parameter
$\nu=\Delta-d/2$ unchanged [70, 71, 72, 73], such that

$\displaystyle\mathcal{C}_{4}^{\Delta=3+\epsilon}$ $\displaystyle=$
$\displaystyle\int\frac{d\eta}{\eta^{4+2\epsilon}}\mathop{\prod}\limits_{i=1}^{4}\mathcal{K}_{3/2}^{i},$
$\displaystyle=$
$\displaystyle\frac{1}{E^{2\epsilon}}\big{\\{}E^{3}[\Gamma(-2+2\epsilon)+\Gamma(-3+2\epsilon)]+\Gamma(2\epsilon)(k_{1}k_{2}k_{3}+\ldots)$
$\displaystyle+\Gamma(-1+2\epsilon)E\left(k_{1}k_{2}+\ldots\right)+\frac{k_{1}k_{2}k_{3}k_{4}}{E}\Gamma(1+2\epsilon)\big{\\}},$

where the ellipsis inside each parenthesis denotes all permutations involving
the four legs. We then find the following soft limit:

$\displaystyle\lim_{\vec{k}_{1}\rightarrow
0}\mathcal{C}_{4}^{\Delta=3+\epsilon}=$
$\displaystyle\frac{1}{E^{2\epsilon}}\\{E^{3}[\Gamma(-2+2\epsilon)+\Gamma(-3+2\epsilon)]+\Gamma(2\epsilon)k_{2}k_{3}k_{4}$
$\displaystyle+\Gamma(-1+2\epsilon)E(k_{2}k_{3}+k_{2}k_{4}+k_{3}k_{4})\\}.$
(5.6)

We can now compare this with the three-point contact diagram of minimally
coupled scalars:

$\begin{split}\mathcal{C}_{3}^{\Delta=3+\epsilon}=\int\frac{d\eta}{\eta^{4+2\epsilon}}\mathop{\prod}\limits_{i=2}^{4}\mathcal{K}_{3/2}^{i}.\end{split}$
(5.7)

Unlike the contact diagram in (5.4), this contact diagram does not contain a
time-dependent interaction. After performing the integral in (5.7) and
changing $\epsilon\rightarrow 2\epsilon$, we finally obtain (5.6):

$\lim_{\vec{k}_{1}\rightarrow
0}C_{4}^{\Delta=3+\epsilon}=\mathcal{C}_{3}^{\Delta=3+2\epsilon},$ (5.8)

which, for clarity, can be easily expanded in $\epsilon$:

$\mathcal{C}_{3}^{\Delta=3+\epsilon}=\frac{k_{2}^{3}+k_{3}^{3}+k_{4}^{3}}{3}\left(\frac{1}{\epsilon}-\gamma_{E}-\ln
E+1\right)+\frac{1}{9}E^{3}-k_{2}k_{3}k_{4}+\mathcal{O}(\epsilon),$ (5.9)

where $\gamma_{E}$ is the Euler-Mascheroni constant. As we will see, (5.7)
will continue to play a role in the soft limit of more complicated four-point
correlators of minimally coupled scalars.

###  5.2 Two Derivatives

Next, we will analyse the soft limit of $\hat{u}\mathcal{C}_{4}^{\Delta}$. In
the conformally coupled case, it can be cast as

$\hat{u}\mathcal{C}_{4}^{\Delta=2}=-\frac{2}{E^{3}}[\vec{k}_{1}\cdot\vec{k}_{3}-k_{1}k_{3}-\tfrac{E}{2}(k_{2}+k_{4})],$
(5.10)

with soft limit

$\lim_{\vec{k}_{1}\to
0}\hat{u}\mathcal{C}_{4}^{\Delta=2}=\frac{1}{E}-\frac{k_{3}}{E^{2}}.$ (5.11)

We recognise this expression as the dilatation operation:

$D_{3}\left(\frac{1}{E}\right)=\frac{1}{E}-\frac{k_{3}}{E^{2}}.$ (5.12)

Hence, we find that the soft limit of (5.10) can be obtained by acting with a
dilatation on the three-point contact diagram of (5.4):

$\lim_{\vec{k}_{1}\to
0}\hat{u}\mathcal{C}_{4}^{\Delta=2}=D_{3}\,\mathcal{C}_{3,\eta}^{\Delta=2}.$
(5.13)

In the minimally coupled case, we find

$\begin{split}\hat{u}\mathcal{C}_{4}^{\Delta=3}&=\frac{\vec{k}_{1}\cdot\vec{k}_{3}}{E}\left[-E^{2}+\sum_{i<j}k_{i}k_{j}+\frac{k_{1}k_{2}k_{3}k_{4}}{E}\left(\frac{1}{k_{1}}+\frac{1}{k_{2}}+\frac{1}{k_{3}}+\frac{1}{k_{4}}+\frac{2}{E}\right)\right]\\\
&\qquad-2\frac{k_{1}^{2}k_{3}^{2}}{E}\left(1+\frac{k_{2}+k_{4}}{E}+\frac{2k_{2}k_{4}}{E^{2}}\right),\end{split}$
(5.14)

where requires dimensional regularisation in intermediate steps but has a
finite output. Taking the soft limit then gives

$\lim_{\vec{k}_{1}\to
0}\hat{u}\mathcal{C}_{4}^{\Delta=3}=\frac{\vec{k}_{1}\cdot\vec{k}_{3}}{E}\left(k_{2}^{2}+k_{3}^{2}+k_{4}^{2}+k_{2}k_{3}+k_{3}k_{4}+k_{4}k_{2}-\frac{k_{2}k_{3}k_{4}}{E}\right),$
(5.15)

which can be obtained by acting with a conformal boost on the three-point
contact diagram in (5.7):

$\lim_{\vec{k}_{1}\to 0}\hat{u}\mathcal{C}_{4}^{\Delta=3}=(\vec{k}_{1}\cdot
K_{3})\,\mathcal{C}_{3}^{\Delta=3}.$ (5.16)

Note that the divergences in (5.9) are removed by the action of conformal
generators so we set $\epsilon=0$. This will continue to hold for higher-
derivative correlators so we will set $\epsilon=0$ in those cases as well.

###  5.3 Four Derivatives

We now consider the term
$(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}_{4}^{\Delta}$. For
conformally coupled scalars, we obtain

$(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}^{\Delta=2}_{4}=\frac{24}{E^{5}}[(\vec{k}_{1}\cdot\vec{k}_{2}-k_{1}k_{2})(\vec{k}_{3}\cdot\vec{k}_{4}-k_{3}k_{4})+\mathrm{cyc}(234)]-\frac{8}{E^{3}}\sum_{i<j}k_{i}k_{j}+\frac{4}{E}.$
(5.17)

In this case the soft limit is simply

$\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}^{\Delta=2}_{4}=\frac{4}{E^{3}}(k_{2}^{2}+k_{3}^{2}+k_{4}^{2}),$
(5.18)

which can be recast as a second order combination of dilatation operators
acting on the three-point contact diagram of (5.4):

$\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}_{4}^{\Delta=2}=2(D_{2}^{2}+D_{3}^{2}+D_{4}^{2})\mathcal{C}_{3,\eta}^{\Delta=2}.$
(5.19)

In the minimally coupled case, we obtain

$(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}^{\Delta=3}_{4}=\frac{24k_{1}k_{2}k_{3}k_{4}}{E^{5}}(\vec{k}_{1}\cdot\vec{k}_{2}-k_{1}k_{2})(\vec{k}_{3}\cdot\vec{k}_{4}-k_{3}k_{4})\\\
+\frac{2}{E^{3}}(\vec{k}_{1}\cdot\vec{k}_{2})(\vec{k}_{3}\cdot\vec{k}_{4})\big{(}E^{2}+E\sum_{i<j}k_{i}k_{j}+\sum_{i<j<l}k_{i}k_{j}k_{l}\big{)}\\\
-\frac{2}{E^{4}}[(\vec{k}_{1}\cdot\vec{k}_{2})k_{3}^{2}k_{4}^{2}(E+2(k_{1}+k_{2})+(\vec{k}_{3}\cdot\vec{k}_{4})k_{1}^{2}k_{2}^{2}(E+2(k_{3}+k_{4}))+\mathrm{cyc}(234)].$
(5.20)

We then find that the soft limit is given by

$\begin{split}\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}_{4}^{\Delta=3}&=\frac{\vec{k}_{1}\cdot\vec{k}_{2}}{E^{3}}\big{[}4k_{2}^{2}k_{3}k_{4}+k_{2}E(k_{3}k_{4}+2k_{2}(k_{3}+k_{4}))\\\
&\qquad\qquad+E^{2}(2k_{2}^{2}+k_{2}k_{3}+k_{3}k_{4}+k_{4}k_{2})-E^{4})\big{]}+\mathrm{cyc}(234),\end{split}$
(5.21)

which corresponds to applying the following quadratic combination of conformal
generators to a three-point contact diagram:

$\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}_{4}^{\Delta=3}=-2\big{[}D_{2}(\vec{k}_{1}\cdot
K_{2})+\mathrm{cyc}(234)\big{]}\mathcal{C}_{3}^{\Delta=3}.$ (5.22)

In practice, the soft limits in (5.19) and (5.22) are most easily derived at
the level of the integrand. In particular, this requires taking the soft limit
of bulk-to-boundary propagators and their derivatives, and then using
equations of motion to remove derivatives acting on the bulk-to-boundary
propagator for the soft leg as well as factors of $k_{a}^{2}$. For example, in
the conformally coupled case we have

$\begin{split}\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2})\mathcal{C}_{4}^{\Delta=2}&=\int
d\eta\left(\frac{1}{\eta}(-k_{2}^{2}\dot{\mathcal{K}}_{1}\mathcal{K}_{2}+\dot{\mathcal{K}}_{1}\ddot{\mathcal{K}}_{2})-\frac{1}{\eta^{2}}\dot{\mathcal{K}}_{1}\dot{\mathcal{K}}_{2}\right)\mathcal{K}_{3}\mathcal{K}_{4}+\mathrm{cyc}(234),\\\
&=\int\frac{d\eta}{\eta^{3}}[2\eta^{2}\ddot{\mathcal{K}}_{2}-\eta\dot{\mathcal{K}}_{2}-2\mathcal{K}_{2}]\mathcal{K}_{3}\mathcal{K}_{4}+\mathrm{cyc}(234),\\\
&=2\left(D_{2}^{2}+D_{3}^{2}+D_{4}^{2}\right)\int\frac{d\eta}{\eta^{3}}\mathcal{K}_{2}\mathcal{K}_{3}\mathcal{K}_{4}.\end{split}$
(5.23)

The analogous construction in the minimally coupled case is given by

$\begin{split}\lim_{{\vec{k}_{1}}\to
0}\left(\hat{s}^{2}+\hat{t}^{2}+\hat{u}^{2}\right)\mathcal{C}_{4}^{\Delta=3}&=(\vec{k}_{1}\cdot\vec{k}_{2})\int
d\eta\left(\frac{2}{\eta}\mathcal{K}_{1}\dot{\mathcal{K}}_{2}+\frac{1}{\eta^{2}}\mathcal{K}_{1}\mathcal{K}_{2}\right)\mathcal{K}_{3}\mathcal{K}_{4}+\mathrm{cyc}(234),\\\
&=-2[D_{2}(\vec{k}_{1}\cdot
K_{2})+\mathrm{cyc}(234)]\int\frac{d\eta}{\eta^{4}}\mathcal{K}_{2}\mathcal{K}_{3}\mathcal{K}_{4}.\end{split}$
(5.24)

Using this method, one can also derive the soft limits in (5.13) and (5.16).

###  5.4 Six Derivatives

Finally we consider the six derivative interaction
$(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta}$. In the
conformally coupled case we obtain

$(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=2}=\Big{\\{}\frac{90}{E^{7}}[(\vec{k}_{1}\cdot\vec{k}_{2})+(\vec{k}_{3}\cdot\vec{k}_{4})-k_{1}k_{2}-k_{3}k_{4}]^{3}+\frac{156}{E^{5}}[(\vec{k}_{1}\cdot\vec{k}_{2})-k_{1}k_{2}][(\vec{k}_{3}\cdot\vec{k}_{4})-k_{3}k_{4}]\\\
\qquad\qquad-\frac{54}{E^{5}}(k_{1}k_{2}+k_{3}k_{4})[(\vec{k}_{1}\cdot\vec{k}_{2})+(\vec{k}_{3}\cdot\vec{k}_{4})-k_{1}k_{2}-k_{3}k_{4}]-\frac{79}{E^{3}}(k_{1}k_{2}+k_{3}k_{4})+\mathrm{cyc}(234)\Big{\\}}+\frac{93}{4E},$
(5.25)

with soft limit

$\lim_{{\vec{k}_{1}}\to
0}(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=2}=-\frac{108}{E^{4}}k_{2}k_{3}k_{4}+\frac{52}{E^{3}}(k_{2}k_{3}+k_{3}k_{4}+k_{4}k_{2})-\frac{12}{E}.$
(5.26)

Using the methods described above, we can obtain this by acting with the
following combination of conformal generators on a three-point contact
diagram:

$\lim_{\vec{k}_{1}\to
0}(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=2}=\left(6(D_{2}^{3}+D_{3}^{3}+D_{4}^{3})-22(D_{2}^{2}+D_{3}^{2}+D_{4}^{2})+20\right)\mathcal{C}_{3,\eta}^{\Delta=2}.$
(5.27)

In the minimally coupled case, the expression for the integrated correlator
can be found in appendix D. The soft limit is given by

$\lim_{\vec{k}_{1}\to
0}(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=3}=\frac{(\vec{k}_{1}\cdot\vec{k}_{2})}{E^{4}}\big{[}3k_{2}^{5}+12k_{2}^{4}(k_{3}+k_{4})+k_{2}^{3}(k_{3}^{2}+36k_{3}k_{4}+k_{4}^{2})\\\
-k_{2}^{2}(k_{3}+k_{4})(11k_{3}^{2}+21k_{3}k_{4}+11k_{4}^{2})-4k_{2}(k_{3}+k_{4})^{2}(k_{3}^{2}+k_{3}k_{4}+k_{4}^{2})\\\
-(k_{3}+k_{4})^{3}(k_{3}^{2}+k_{3}k_{4}+k_{4}^{2})\big{]}+\mathrm{cyc}(234).$
(5.28)

This expression can also be recast in terms of conformal generators acting on
a three-point contact diagram. For example, we can write

$\begin{split}\lim_{\vec{k}_{1}\to
0}(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=3}=[2(3D_{2}^{2}-11D_{2})(\vec{k}_{1}\cdot
K_{2})+\mathrm{cyc}(234)]\mathcal{C}_{3}^{\Delta=3}.\end{split}$ (5.29)

We note that these expressions are not unique. For example, the operators
$K_{2}$ and $D_{2}$ do not commute and so we could choose to order them
differently and pick up different coefficients.

Let us close this section with some general comments. In general, we find that
soft limits require both special conformal and dilatation operators acting on
a contact diagram. In the conformally coupled case, it is always possible to
express them in terms of only dilatation operators but at the level of the
integrand this is not the whole story; when considering how the integrand
behaves in the soft limit we find that both types of conformal generators
appear. Another interesting point is that we can obtain particularly simple
soft limits by tuning coefficients in the effective action in (3.24). For
example, if we choose the mass to be that of a conformally coupled scalar and
set the coefficients
$\left\\{A,B,C\right\\}=\lambda\left\\{1,-8,-20\right\\}$, then the soft limit
of the corresponding 4-point correlator in (3.25) is simply given by

$\lim_{\vec{k}_{1}\to
0}\Psi_{4}^{(6)}=6(D_{2}^{3}+D_{3}^{3}+D_{4}^{3})\mathcal{C}_{3,\eta}^{\Delta=2}.$
(5.30)

It would be interesting to see how the corresponding EFT is related to the
ones constructed in [60] or if it has any hidden symmetries.

##  6 Conclusion

The study of correlation functions in (Anti) de Sitter space using curved-
space analogues of the scattering equations is still an ongoing endeavour.
Prior to this paper, they have only been formulated for scalar theories with
polynomial interactions, notably $\phi^{3}$ [53, 54] and $\phi^{4}$ [41, 42].
We have now initiated the study of scalar theories with derivative
interactions using this framework. In particular, we study the wave function
coefficients of the NLSM, DBI, and sGal theories in dS using the cosmological
scattering equations. These effective scalar theories are of particular
interest in flat space since their scattering amplitudes have a very elegant
description and are related to each other in a simple way in terms of CHY
formulae [50].

In this paper we have proposed new formulae for four-point wavefunction
coefficients of different scalar EFTs in the form of worldsheet integrals
which encode both curvature corrections and mass deformations. We showed that
the DBI and sGal integrands can be obtained from the NLSM integrand by
replacing a Parke-Taylor factor with a linear combination of simple building
blocks involving Pfaffians of certain operatorial matrices. Because the
integrands are constructed from differential operators which do not generally
commute, this leads to potential ordering ambiguities that are absent in
theories with polynomial interactions. Such ambiguities can occur in the DBI
and sGal theories at four points and the NLSM at six points. At four points we
introduced a simple prescription for defining the worldsheet integrals such
that the final results are permutation-invariant free of ordering ambiguities.
We have also studied the soft limits of the resulting four-point correlators
and derived formulae in the form of differential operators acting on three-
point contact diagrams. For conformally coupled scalars, the three-point
contact diagrams involve a time-dependent interaction, while for minimally
coupled scalars the contact diagram is divergent and needs to be regulated.
However, all divergences cancel out after acting with the appropriate
combinations of boundary conformal generators.

There are a number of directions for future investigation. For example, it
would be of interest to generalise our formulae to any number of points. In
order to do so, there are several technical questions that need to be
addressed. First of all, we need to develop a systematic classification of
higher-derivative corrections to scalar EFTs analogous to the one in [66]
beyond four-points, which to our knowledge has not been carried out yet. It
would then be interesting to see if there is a simple generalisation of the
double copy prescription in (4.27) which encodes such corrections at higher
points. Secondly, we need a systematic understanding of ordering ambiguities
that could arise in the corresponding worldsheet integrals and a prescription
for removing them. A natural starting point along these lines would be to
analyse potential ordering ambiguities in the six-point NLSM calculation in
Appendix B.

It would also be of great interest to apply our approach for computing
correlators to the EFTs recently constructed in de Sitter space based on
hidden symmetries [60]. Note that the effective actions we consider in this
paper have general massess and curvature corrections up to 4-points, so the
actions derived in [60] should correspond to a particular choice of these
parameters. It would then be interesting to compute the corresponding four-
point wavefunction coefficients and their soft limits in terms of the building
blocks derived in this paper and investigate the interplay of hidden
symmetries with soft limits, generalising the analysis of [57, 58, 59] to dS.
Note that our double copy proposal does not fix massess or coefficients of
curvature corrections, but these can be constrained by imposing simplicity of
the soft limits, as we illustrate in the end of section 5. It would also be
interesting to relate our results to those recently obtained in flat space
[61] using the methods recently developed in [24].

Finally, it is important to extend the techniques of this paper to more
realistic models by considering different worldsheet integrands encoding more
general interactions and the breaking of conformal symmetry by inflation [74,
75, 76]. Moreover inflationary three-point correlators can be obtained by
giving a small mass to one of the legs of a four-point function in de Sitter
(proportional to the slow-roll parameter) and then taking a soft limit [77,
16, 78]. Our results in section 5 should be useful in this regard.

Acknowledgements

We thank Paul McFadden, Allic Sivaramakrishnan, and Fei Teng for useful
discussions. CA, HG, and AL are supported by the Royal Society via a PhD
studentship, PDRA grant, and a University Research Fellowship, respectively.
JM is supported by a Durham-CSC Scholarship. RLJ acknowledges the Czech
Science Foundation - GAČR for financial support under the grant 19-06342Y.

##  Appendix A Six-point NLSM correlators from Witten diagrams

In this appendix we extend the calculations in section 3.2 to six-points for
the NLSM. At this level, we have the following Lagrangian,

$\begin{split}\mathcal{L}_{\mathrm{NLSM}}^{\textrm{6-pt}}={\rm
Tr}\left[-\frac{1}{2}\partial\Phi\cdot\partial\Phi-\frac{1}{2}m^{2}\Phi^{2}-\lambda^{2}\Phi^{2}\partial_{\mu}\Phi\partial^{\mu}\Phi-\frac{1}{4}C\Phi^{4}\right.\\\
\quad\left.-\lambda^{4}\left(\Phi^{4}\partial_{\mu}\Phi\partial^{\mu}\Phi+\frac{1}{2}\Phi^{2}\partial_{\mu}\Phi\Phi^{2}\partial^{\mu}\Phi\right)-\frac{1}{6}F\Phi^{6}\right],\end{split}$
(A.1)

where $m$ is the mass, and $C$ and $F$ are free coefficients coming from
possible curvature corrections. For simplicity, we will set $m=C=F=0$. We will
write the result in terms of the boundary differential operators in (3.16)
acting on a contact diagram, which we will compare against the results of a
worldsheet calculation in Appendix B. The final result will be free of
ambiguities since all the differential operators which appear commute. A
similar formula was previously derived in [34] using AdS embedding
coordinates.

The correlator is given by a sum of exchange and contact diagrams shown in
Figure 2.

Figure 2: Witten diagrams for the six-point NLSM amplitude. The exchange
diagram is summed over the three inequivalent cyclic permutations.

To obtain the desired form, we must write the four-point vertices in the
exchange diagrams in such a way that derivatives only act on bulk-to-boundary
propagators.

Figure 3: Four-point vertex appearing on the left-hand-side of a 6-point NLSM
exchange diagram.

Let us consider the four-point vertex appearing on the left-hand-side of an
exchange diagram, illustrated in Figure 3. Using the Feynman rules derived
from (A.1) and the identity in (3.16) we find

$\begin{split}\mathcal{A}_{L}&=\int\frac{d\eta}{\eta^{d+1}}\eta^{2}\big{\\{}\mathcal{K}_{1}\mathcal{K}_{2}\mathcal{K}_{3}\mathcal{K}_{L}[(\vec{k}_{1}\cdot\vec{k}_{2})+(\vec{k}_{2}\cdot\vec{k}_{3})+(\vec{k}_{3}\cdot\vec{k}_{L})+(\vec{k}_{L}\cdot\vec{k}_{1})]\\\
&\qquad\qquad+\dot{\mathcal{K}}_{1}\dot{\mathcal{K}}_{2}\mathcal{K}_{3}\mathcal{K}_{L}+\mathcal{K}_{1}\dot{\mathcal{K}}_{2}\dot{\mathcal{K}}_{3}\mathcal{K}_{L}+\mathcal{K}_{1}\mathcal{K}_{2}\dot{\mathcal{K}}_{3}\dot{\mathcal{K}}_{L}+\dot{\mathcal{K}}_{1}\mathcal{K}_{2}\mathcal{K}_{3}\dot{\mathcal{K}}_{L}\big{\\}},\end{split}$
(A.2)

where $\mathcal{K}_{L}$ is associated to the bulk-to-bulk propagator and
$\dot{\mathcal{K}}=\partial_{\eta}\mathcal{K}$. Using integration by parts to
remove the derivatives acting on $\mathcal{K}_{L}$ gives

$\begin{split}\mathcal{A}_{L}&=\int\frac{d\eta}{\eta^{d-1}}\mathcal{K}_{L}\Big{[}(-2\vec{k}_{1}\cdot\vec{k}_{3}\mathcal{K}_{1}\mathcal{K}_{2}\mathcal{K}_{3}-2\dot{\mathcal{K}}_{1}\mathcal{K}_{2}\dot{\mathcal{K}}_{3}-k_{1}^{2}\mathcal{K}_{1}\mathcal{K}_{2}\mathcal{K}_{3}-k_{3}^{2}\mathcal{K}_{1}\mathcal{K}_{2}\mathcal{K}_{3}\\\
&\qquad\qquad-\mathcal{K}_{1}\mathcal{K}_{2}\ddot{\mathcal{K}}_{3}-\ddot{\mathcal{K}}_{1}\mathcal{K}_{2}\mathcal{K}_{3}+\frac{1-d}{\eta}\left(\dot{\mathcal{K}}_{1}\mathcal{K}_{2}\mathcal{K}_{3}+\mathcal{K}_{1}\mathcal{K}_{2}\dot{\mathcal{K}}_{3}\right)\Big{]},\end{split}$
(A.3)

where we have also used momentum conservation at the vertex giving
$\vec{k}_{L}=-(\vec{k}_{1}+\vec{k}_{2}+\vec{k}_{3})$. Using the equations of
motion for $\mathcal{K}_{1}$ and $\mathcal{K}_{3}$, we are left with

$\mathcal{A}_{L}=-2\int\frac{dz}{z^{d+1}}\left[\mathcal{K}_{2}\mathcal{K}_{L}\eta^{2}\left(\vec{k}_{1}\cdot\vec{k}_{3}\mathcal{K}_{1}\mathcal{K}_{2}\mathcal{K}_{3}+\dot{\mathcal{K}}_{1}\mathcal{K}_{2}\dot{\mathcal{K}}_{3}\right)\right].$
(A.4)

Performing similar manipulations on the other four-point vertices, we finally
obtain

$\Psi_{6}^{NLSM}=\lambda^{4}\delta^{3}\left(\vec{k}_{T}\right)\bigg{[}\left(\alpha_{13}\alpha_{123}^{-1}\alpha_{46}-\alpha_{24}\right)+\mathrm{Cyc}[i\to
i{+}2]\bigg{]}\mathcal{C}_{6}^{\Delta}$ (A.5)

where
$\alpha_{ijk}=\left(\mathcal{D}_{i}+\mathcal{D}_{j}+\mathcal{D}_{k}\right)^{2}$.
Note that the first term comes from the exchange diagrams and the second term
is from the contact diagram. These can be seen from the Witten diagrams in
Figure 2. This is manifestly free of any ordering ambiguities since the
products of operators that appear can be written in any order. Note that this
result can be obtained from flat space Feynman diagrams simply by replacing
$2k_{i}\cdot k_{j}$ with $\alpha_{ij}$. Similarly, the flat space limit of
(A.5) is equivalent to the result given in [79].

##  Appendix B Six-point NLSM from the worldsheet

In this Appendix we will compute the six-point correlator of the NLSM in dS
using the CSE. For simplicity, we will set the mass and curvature corrections
to zero as we did in the previous Appendix. The worldsheet calculation turns
out to be very intricate and we obtain an expression involving boundary
differential operators which don’t commute, in contrast to the result of the
previous Appendix. If we ignore commutators it is easy to match the two
expressions. We therefore expect that the additional terms coming from taking
commutators should cancel out although we have not verified this due to the
complexity of the expression. We leave a systematic study of such commutators
and potential ordering ambiguities to future work.

Lifting the flat space formula in (2.6) to dS as explained in section 4, the
six-point NLSM correlator in the CSE framework is written as
$\Psi_{6}^{NLSM}=\delta^{3}\left(\vec{k}_{T}\right)A_{{\rm NLSM}}{\cal
C}_{6}^{\Delta}$ where

$\displaystyle A_{{\rm NLSM}}=\int_{\gamma}\,\prod_{i=2}^{4}{\rm
d}\sigma_{i}\,(S_{i})^{-1}\left(\sigma_{15}\sigma_{56}\sigma_{61}\right)^{2}{\rm
PT}\,\frac{1}{\sigma_{15}^{2}}\,({\rm Pf}A_{15}^{15})^{2}$ (B.1)

where we have fixed legs $1,5,6$ and removed the rows and columns $\\{1,5\\}$
from the $A$-matrix, and the contour encircles the poles corresponding to the
CSE for legs $2,3,4$ i.e.
$\gamma\equiv\gamma_{S_{2}}\cap\gamma_{S_{3}}\cap\gamma_{S_{4}}$.

In the following, we will focus our attention on the object $A_{{\rm NLSM}}$,
which is understood to act on the contact diagram, and impose momentum
conservation along the boundary. In order to evaluate the worldsheet integral,
we expand $({\rm Pf}A^{15}_{15})^{2}$

$\displaystyle({\rm
Pf}A^{15}_{15})^{2}=\frac{\alpha_{a}}{\sigma_{23}^{2}\sigma_{46}^{2}}+\frac{\alpha_{b}}{\sigma_{24}^{2}\sigma_{36}^{2}}+\frac{\alpha_{c}}{\sigma_{26}^{2}\sigma_{34}^{2}}-{\rm
PT}(2,3,6,4)\,\alpha_{d}-{\rm PT}(2,3,4,6)\,\alpha_{e}-{\rm
PT}(2,4,3,6)\,\alpha_{f},$ (B.2)

and evaluate the integral for each term separately, where the “$\alpha_{i}$”
operators are given by the expressions

$\displaystyle\alpha_{a}\equiv\alpha_{23}^{2}\,\alpha_{46}^{2}\,,\quad\alpha_{b}\equiv\alpha_{24}^{2}\,\alpha_{36}^{2}\,,\quad\alpha_{c}\equiv\alpha_{26}^{2}\,\alpha_{34}^{2}\,,\quad\alpha_{d}\equiv\alpha_{23}\,\alpha_{46}\,\alpha_{24}\,\alpha_{36}+\alpha_{24}\,\alpha_{36}\,\alpha_{23}\,\alpha_{46}\,,$
$\displaystyle\alpha_{e}\equiv\alpha_{23}\,\alpha_{46}\,\alpha_{26}\,\alpha_{34}+\alpha_{26}\,\alpha_{34}\,\alpha_{23}\,\alpha_{46}\,,\quad\alpha_{f}\equiv\alpha_{24}\,\alpha_{36}\,\alpha_{26}\,\alpha_{34}+\alpha_{26}\,\alpha_{34}\,\alpha_{24}\,\alpha_{36}\,.$
(B.3)

Therefore, $A_{\rm NLSM}$ turns into

$\displaystyle A_{\rm NLSM}={\cal A}^{(a)}\alpha_{a}+{\cal
A}^{(b)}\alpha_{b}+{\cal A}^{(c)}\alpha_{c}-{\cal A}^{(d)}\alpha_{d}-{\cal
A}^{(e)}\alpha_{e}-{\cal A}^{(f)}\alpha_{f},$ (B.4)

where we have denoted

${\cal A}^{(i)}=\int_{\gamma}\,\prod_{i=2}^{4}{\rm
d}\sigma_{i}\,(S_{i})^{-1}\left(\sigma_{15}\sigma_{56}\sigma_{61}\right)^{2}{\cal
I}^{(i)},\qquad i\in\\{a,b,c,d,e,f\\},$ (B.5)

and defined the integrands

$\displaystyle{\cal I}^{(a)}={\rm
PT}\,\frac{1}{\sigma_{15}^{2}\,\sigma_{23}^{2}\,\sigma_{46}^{2}},\quad{\cal
I}^{(b)}={\rm
PT}\,\frac{1}{\sigma_{15}^{2}\,\sigma_{24}^{2}\,\sigma_{36}^{2}},\quad{\cal
I}^{(c)}={\rm
PT}\,\frac{1}{\sigma_{15}^{2}\,\sigma_{26}^{2}\,\sigma_{34}^{2}},$
$\displaystyle{\cal I}^{(d)}={\rm PT}\,\frac{{\rm
PT}(2,3,6,4)}{\sigma_{15}^{2}},\quad{\cal I}^{(e)}={\rm PT}\,\frac{{\rm
PT}(2,3,4,6)}{\sigma_{15}^{2}},\quad{\cal I}^{(f)}={\rm PT}\,\frac{{\rm
PT}(2,4,3,6)}{\sigma_{15}^{2}}.$ (B.6)

We will evaluate the worldsheet integrals using the graph representation and
integration rules introduced in [41, 42]. The graphs corresponding to each
term in $A_{\rm NLSM}$ are shown in Fig. 4. The chords in each circle encode
the poles in worldsheet coordinates not contained in the Parke-Taylor factor.
For example, ${\cal A}^{(a)}$ contains double poles $\sigma_{15}$,
$\sigma_{23}$, and $\sigma_{46}$ so we draw double lines between these points
on the circumference. When evaluating contour integrals, the worldsheet will
factorise when a subset of punctures collide. We then draw closed loops around
the punctures that collided and refer to them as factorisation cuts. There is
a simple set of rules for determining which factorisation cuts contribute:

  * •

Rule I. All factorization cuts with fewer than two fixed marked-points (labels
with underlines) vanish.

  * •

Rule II. If the factorization cuts more than four lines in the corresponding
graph then, this contribution vanishes.

In practice, these rules make calculations much more efficient.

$\,\alpha_{a}\,+\\!\\!\\!\\!\\!\\!\\!$ $\,\alpha_{b}\,+\\!\\!\\!\\!\\!\\!\\!$
$\,\alpha_{c}\,-\\!\\!\\!\\!\\!\\!\\!$ $\,\alpha_{d}\,-\\!\\!\\!\\!\\!\\!\\!$
$\,\alpha_{e}\,-\\!\\!\\!\\!\\!\\!\\!$ $\,\alpha_{f}$

Figure 4: All diagrams for $A_{\rm NLSM}$.

Let us illustrate in detail how to evaluate the first diagram in Figure 4.
From the integration rules, it is simple to see that there are two
factorisation contributions for this diagram, which have been illustrated in
Fig. 5.

$(I)$  $(II)$

Figure 5: Factorization contributions for ${\cal A}^{(a)}$.

The contribution $({I})$, is given when
$\sigma_{2}\rightarrow\sigma_{3}\rightarrow\sigma_{4}\rightarrow\sigma_{6}$.
To perform this computation, we use the parametrisation, $\sigma_{a}=\epsilon
x_{a}+\sigma_{6}$, with $a=2,3,4,6$, $x_{4}=\text{constant}$, $x_{6}=0$,
$\sigma_{6}=\sigma_{L}$, and expand around $\epsilon=0$,

$\displaystyle\text{d}\sigma_{2}\wedge\text{d}\sigma_{3}\wedge\text{d}\sigma_{4}=\epsilon^{2}\,x_{46}\,\text{d}x_{2}\wedge\text{d}x_{3}\wedge\text{d}\epsilon$
$\displaystyle(\sigma_{15}\sigma_{56}\sigma_{61})^{2}\,{\rm
PT}\,\frac{1}{\sigma_{15}^{2}\,\sigma_{23}^{2}\,\sigma_{46}^{2}}=\frac{1}{\epsilon^{6}}\,\frac{(\sigma_{15}\sigma_{5L}\sigma_{L1})^{2}}{(\sigma_{L1}^{2}\,\sigma_{5L}^{2}\,\sigma_{15}^{2})}\,\frac{1}{(x_{23}^{2}\,x_{34}\,x_{46}^{2})}+{\cal
O}(\epsilon^{-5}),$ (B.7)

and

$\displaystyle S_{2}=\frac{1}{\epsilon}\left[\hat{S}_{2}+{\cal
O}(\epsilon)\right],\qquad S_{3}=\frac{1}{\epsilon}\left[\hat{S}_{3}+{\cal
O}(\epsilon)\right],\qquad S_{4}=\frac{1}{\epsilon}\left[\hat{S}_{4}+{\cal
O}(\epsilon)\right],$
$\displaystyle\hat{S}_{2}=\frac{\alpha_{23}}{x_{23}}+\frac{\alpha_{24}}{x_{24}}+\frac{\alpha_{26}}{x_{26}}+\frac{\alpha_{2R}}{x_{2R}},\,\,\hat{S}_{3}=\frac{\alpha_{32}}{x_{32}}+\frac{\alpha_{34}}{x_{34}}+\frac{\alpha_{36}}{x_{36}}+\frac{\alpha_{3R}}{x_{3R}},\,\,\hat{S}_{4}=\frac{\alpha_{42}}{x_{42}}+\frac{\alpha_{43}}{x_{43}}+\frac{\alpha_{46}}{x_{46}}+\frac{\alpha_{4R}}{x_{4R}},$
(B.8)

where $x_{R}=\infty$, and $\alpha_{aR}=\alpha_{a1}+\alpha_{a5}$, $a=2,3,4$.
Now we can deform the contour $\gamma=\gamma_{\hat{S}_{2}+{\cal
O}(\epsilon)}\cap\gamma_{\hat{S}_{3}+{\cal
O}(\epsilon)}\cap\gamma_{\hat{S}_{4}+{\cal O}(\epsilon)}$ into
$\tilde{\gamma}=\gamma_{\epsilon}\cap\gamma_{\hat{S}_{2}+{\cal
O}(\epsilon)}\cap\gamma_{\hat{S}_{3}+{\cal O}(\epsilon)}$, where
$\gamma_{\epsilon}=\\{|\epsilon|=\delta\\}$.

After integrating over $\gamma_{\epsilon}$, the full integral is split into
two parts: one with $\\{\sigma_{1},\sigma_{5},\sigma_{L}\\}$ and the other
with $\\{x_{2},x_{3},x_{4},x_{6},x_{R}\\}$. Moreover, using the identity

${\rm PT}(4,6,R)\,\hat{S}_{4}+{\rm PT}(3,6,R)\,\hat{S}_{3}+{\rm
PT}(2,6,R)\,\hat{S}_{2}={\rm
PT}(6,R)\,(\alpha_{23}+\alpha_{24}+\alpha_{26}+\alpha_{34}+\alpha_{36}+\alpha_{46}),$
(B.9)

we find that on the support of $\gamma_{\hat{S}_{2}}\cap\gamma_{\hat{S}_{3}}$,
$\hat{S}_{4}$ reduces to

$\hat{S}_{4}\Big{|}_{\gamma_{\hat{S}_{2}}\cap\gamma_{\hat{S}_{3}}}=\frac{{\rm
PT}(6,R)}{{\rm PT}(4,6,R)}\,(\alpha_{15}),$ (B.10)

where we have used the CWI. Putting everything together, the contribution
$(I)$ for ${\cal A}^{(a)}$ is given by

$\displaystyle{\cal A}^{(a)}\Big{|}_{(I)}$
$\displaystyle=\frac{(\sigma_{15}\sigma_{5L}\sigma_{L1})^{2}}{(\sigma_{L1}^{2}\sigma_{5L}^{2}\sigma_{15}^{2})}\,(\alpha_{15})^{-1}\int_{\hat{\gamma}}\prod_{i=2}^{3}\text{d}x_{i}(\hat{S}_{i})^{-1}(x_{46}x_{6R}x_{R4})^{2}\frac{{\rm
PT}(2,3,4,6,R)}{x^{2}_{23}\,x_{46}\,x_{6R}\,x_{R4}}$
$\displaystyle=(\alpha_{15})^{-1}\int_{\hat{\gamma}}\prod_{i=2}^{3}\text{d}x_{i}(\hat{S}_{i})^{-1}(x_{46}x_{6R}x_{R4})^{2}\,\frac{{\rm
PT}(2,3,4,6,R)}{x^{2}_{23}\,x_{46}\,x_{6R}\,x_{R4}},$ (B.11)

with $\hat{\gamma}=\gamma_{\hat{S}_{2}}\cap\gamma_{\hat{S}_{3}}$.

= $(\alpha_{15})^{-1}$

Figure 6: Diagrammatic representation of ${\cal A}^{(a)}\Big{|}_{(I)}$.

The expression in (B) is depicted in Fig. 6. This diagram has two
factorisation contributions which are illustrated in Fig. 7.

$(I\,A)$  $(I\,B)$

Figure 7: Factorization contributions for the five-point diagram obtained in
${\cal A}^{(a)}\Big{|}_{(I)}$.

The contribution $({I\,A})$ arises when $x_{2}\rightarrow x_{3}\rightarrow
x_{4}$. In this case we use the parametrisation $x_{a}=\epsilon\,y_{a}+x_{4}$,
$a=2,3,4$, $y_{2}=$constant, $y_{4}=0$ and $x_{4}=x_{L}$. We then have

$\displaystyle\text{d}x_{2}\wedge\text{d}x_{3}=\epsilon\,y_{24}\,\text{d}\epsilon\wedge\text{d}y_{3}$
$\displaystyle(x_{46}x_{6R}x_{R4})^{2}\,\frac{{\rm
PT}(2,3,4,6,R)}{x^{2}_{23}\,x_{46}\,x_{6R}}=\frac{1}{\epsilon^{4}}\,\frac{(x_{L6}x_{6R}x_{RL})^{2}}{(x_{L6}x_{6R}x_{RL})^{2}}\,\frac{1}{(y_{23}^{3}\,y_{34})}+{\cal
O}(\epsilon^{-3}),$ (B.12)

and

$\displaystyle\hat{S}_{2}=\frac{1}{\epsilon}\left[\tilde{S}_{2}+{\cal
O}(\epsilon)\right],\qquad\hat{S}_{3}=\frac{1}{\epsilon}\left[\tilde{S}_{3}+{\cal
O}(\epsilon)\right],$
$\displaystyle\tilde{S}_{2}=\frac{\alpha_{23}}{y_{23}}+\frac{\alpha_{24}}{y_{24}}+\frac{\alpha_{2\hat{R}}}{y_{2\hat{R}}},\qquad\tilde{S}_{3}=\frac{\alpha_{32}}{y_{32}}+\frac{\alpha_{34}}{y_{34}}+\frac{\alpha_{3\hat{R}}}{y_{3\hat{R}}},$
(B.13)

where $y_{\hat{R}}=\infty$, and
$\alpha_{a\hat{R}}=\alpha_{a1}+\alpha_{a5}+\alpha_{a6}$, $a=2,3$. By the GRT,
we deform the contour $\hat{\gamma}=\gamma_{\tilde{S}_{2}+{\cal
O}(\epsilon)}\cap\gamma_{\tilde{S}_{3}+{\cal O}(\epsilon)}$ into
$\gamma^{\prime}=-\gamma_{\epsilon}\cap\gamma_{\tilde{S}_{3}+{\cal
O}(\epsilon)}$. Thus, integrating around $\epsilon=0$, the five-point integral
breaks into two parts: one with $\\{x_{6},x_{R},x_{L}\\}$ and the other with
$\\{y_{2},y_{3},y_{4},y_{\hat{R}}\\}$. Using the identity

${\rm PT}(2,4,\hat{R})\,\tilde{S}_{2}+{\rm
PT}(3,4,\hat{R})\,\tilde{S}_{3}={\rm PT}(4,\hat{R})\,\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal D}_{4})^{2}+m^{2}\right],$ (B.14)

we find that on the support of $\gamma_{\tilde{S}_{3}}$, $\tilde{S}_{2}$
reduces to

$\tilde{S}_{2}\Big{|}_{\gamma_{\tilde{S}_{3}}}=\frac{{\rm PT}(4,\hat{R})}{{\rm
PT}(2,4,\hat{R})}\,\left[({\cal D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right].$ (B.15)

The contribution $({I\,A})$ is therefore given by

$\displaystyle{(I\,A)}$
$\displaystyle=\frac{(x_{L6}x_{6R}x_{RL})^{2}}{(x_{L6}x_{6R}x_{RL})^{2}}\,\left[({\cal
D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}\\!\\!\int_{\gamma_{\tilde{S}_{3}}}\\!\\!\\!\text{d}y_{3}(\tilde{S}_{3})^{-1}(y_{24}y_{4\hat{R}}y_{\hat{R}2})^{2}\,\frac{{\rm
PT}(2,3,4,\hat{R})}{y_{23}^{2}\,y_{4\hat{R}}^{2}}$ $\displaystyle=\left[({\cal
D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}\\!\\!\int_{\gamma_{\tilde{S}_{3}}}\\!\\!\\!\text{d}y_{3}(\tilde{S}_{3})^{-1}(y_{24}y_{4\hat{R}}y_{\hat{R}2})^{2}\,\frac{{\rm
PT}(2,3,4,\hat{R})}{y_{23}^{2}\,y_{4\hat{R}}^{2}}\,.$ (B.16)

The contribution in (B.16) is represented in Fig. 8.

$=\left[({\cal D}_{2}+{\cal D}_{3}+{\cal D}_{4})^{2}+m^{2}\right]^{-1}$

Figure 8: Diagramatic representationn for the contribution $({I\,A})$.

Since the four-point diagram in Fig. 8 has a double pole, it can potentially
give rise to ambiguities associated with the ordering of non-commuting
differential operators. In appendix C, we compute this contribution in detail
and show that it is well defined. In particular, we find that

$\displaystyle\int_{\gamma_{\tilde{S}_{3}}}\\!\\!\\!\text{d}y_{3}(\tilde{S}_{3})^{-1}(y_{24}y_{4\hat{R}}y_{\hat{R}2})^{2}\,\frac{{\rm
PT}(2,3,4,\hat{R})}{y_{23}^{2}\,y_{4\hat{R}}^{2}}=\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}\right)\left(\alpha_{23}\right)^{-1}.$
(B.17)

Plugging this into (B.16) then gives

$\displaystyle{(I\,A)}=\left[({\cal D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}\right)\left(\alpha_{23}\right)^{-1}.$
(B.18)

Similarly, we find the $({\rm I\,B})$ contribution in Figure 7 to be

$\displaystyle{(I\,B)}=(\alpha_{1235}-\alpha_{15})^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}+\alpha_{36}\right)\left(\alpha_{23}\right)^{-1}$
(B.19)

where
$\alpha_{1235}\equiv\alpha_{12}+\alpha_{13}+\alpha_{15}+\alpha_{23}+\alpha_{25}+\alpha_{35}$.
Hence, the factorization contribution $({\rm I})$ in Fig. 5 is given by

$\displaystyle{\cal A}^{(a)}\Big{|}_{({I})}$
$\displaystyle=(\alpha_{15})^{-1}\left\\{\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}\right)\left(\alpha_{23}\right)^{-1}\right.$
$\displaystyle+\left.(\alpha_{46}-\alpha_{15})^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}+\alpha_{36}\right)\left(\alpha_{23}\right)^{-1}\right\\}.$
(B.20)

The computation for the contribution $({\rm II})$ in Fig. 5 is completely
analogous and we obtain

$\displaystyle{\cal A}^{(a)}\Big{|}_{({II})}\\!$
$\displaystyle=\\!(\alpha_{46})^{-1}\\!\left\\{\left[({\cal D}_{1}+{\cal
D}_{2}+{\cal
D}_{3})^{3}+m^{2}\right]^{-1}\\!\left(\alpha_{23}\right)^{-1}\\!\left(-\alpha_{13}\right)\left(\alpha_{23}\right)^{-1}\right.$
$\displaystyle+\left.(\alpha_{15}-\alpha_{46})^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}+\alpha_{36}\right)\left(\alpha_{23}\right)^{-1}\right\\}.$
(B.21)

Putting it all together, we finally obtain the following formula for the first
diagram in Figure 4:

$\displaystyle{\cal A}^{(a)}$ $\displaystyle={\cal
A}^{(a)}\Big{|}_{({I})}+{\cal A}^{(a)}\Big{|}_{({II})}$
$\displaystyle=\left[({\cal D}_{1}+{\cal D}_{2}+{\cal
D}_{3})^{2}+m^{2}\right]^{-1}(\alpha_{46})^{-1}\left(\alpha_{23}\right)^{-1}\left(-\alpha_{13}\right)\left(\alpha_{23}\right)^{-1}$
$\displaystyle+\left[({\cal D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}\right)\left(\alpha_{23}\right)^{-1}$
$\displaystyle+(\alpha_{15})^{-1}(\alpha_{46})^{-1}\left(\alpha_{23}\right)^{-1}\left(\alpha_{23}+\alpha_{34}+\alpha_{36}\right)\left(\alpha_{23}\right)^{-1}.$
(B.22)

The other diagrams in Figure 4 can be computed using similar methods, so we
just state the final results:

$\displaystyle{\cal A}^{(b)}$ $\displaystyle=-\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}(\alpha_{24})^{-1}-(\alpha_{15})^{-1}(\alpha_{24})^{-1}(\alpha_{36})^{-1},$
$\displaystyle{\cal A}^{(c)}$ $\displaystyle=\left[({\cal D}_{3}+{\cal
D}_{4}+{\cal
D}_{5})^{2}+m^{2}\right]^{-1}(\alpha_{26})^{-1}\left(\alpha_{34}\right)^{-1}\left(-\alpha_{15}\right)\left(\alpha_{34}\right)^{-1}$
$\displaystyle+\left[({\cal D}_{2}+{\cal D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}\left(\alpha_{34}\right)^{-1}\left(\alpha_{23}+\alpha_{34}\right)\left(\alpha_{34}\right)^{-1}$
$\displaystyle+(\alpha_{15})^{-1}(\alpha_{26})^{-1}\left(\alpha_{34}\right)^{-1}\left(\alpha_{23}+\alpha_{34}+\alpha_{36}\right)\left(\alpha_{34}\right)^{-1},$
$\displaystyle{\cal A}^{(d)}$ $\displaystyle=\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}\left(\alpha_{23}\right)^{-1},$
$\displaystyle{\cal A}^{(e)}$ $\displaystyle=-\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}\left\\{\left(\alpha_{23}\right)^{-1}+\left(\alpha_{34}\right)^{-1}\right\\},$
$\displaystyle{\cal A}^{(f)}$ $\displaystyle=\left[({\cal D}_{2}+{\cal
D}_{3}+{\cal
D}_{4})^{2}+m^{2}\right]^{-1}(\alpha_{15})^{-1}\left(\alpha_{34}\right)^{-1}\,.$
(B.23)

The factorisation cuts from which these were derived are illustrated in Figure
9. The complete result is then obtained by combining (B) with (B). Neglecting
commutators, it is not difficult to see that this agrees with (A.5).

(a)  $\rightarrow$   = $(\alpha_{15})^{-1}$
(b)  $\rightarrow$   = $(\alpha_{15})^{-1}$
(c)
$\rightarrow$      =$(\alpha_{15})^{-1}$
$\rightarrow$    = $(\alpha_{15})^{-1}$
$\rightarrow$    = $(\alpha_{15})^{-1}$

Figure 9: (a) Computing the ${\cal A}^{(b)}$ diagram. (b) Computing the ${\cal
A}^{(c)}$ diagram. (c) Computing the ${\cal A}^{(d)}$, ${\cal A}^{(e)}$, and
${\cal A}^{(f)}$ diagrams.

##  Appendix C Four-point diagram with double pole

In this Appendix, we will illustrate how to evaluate a worldsheet integral
with a double pole. This integral arises in the 6-point NLSM computation as
explained in the previous Appendix. Non-simple poles can also appear in four-
point worldsheet formulae of higher-derivative theories like DBI and sGal, as
explained in section 4, although they can be avoided using an appropriate
choice of Pfaffians. Let us consider the integral

${\cal
A}_{4}=\int_{\gamma_{S_{d}}}\text{d}\sigma_{d}\,(S_{d})^{-1}(\sigma_{ab}\sigma_{bc}\sigma_{ca})^{2}\,{\rm
PT}(a,b,c,d)\,\frac{1}{\sigma_{ab}^{2}\sigma_{cd}^{2}},$ (C.1)

where

$S_{d}=\frac{\alpha_{da}}{\sigma_{da}}+\frac{\alpha_{db}}{\sigma_{db}}+\frac{\alpha_{dc}}{\sigma_{dc}},$
(C.2)

and $\alpha_{da}+\alpha_{db}+\alpha_{dc}=0$ when acting on a 4-point contact
diagram with ${\cal D}_{a}^{2}\neq{\cal D}_{b}^{2}\neq{\cal
D}_{c}^{2}\neq-m^{2}$ (see section 6.2 of [42] for a discussion of how CWI and
${\rm SL}(2,\mathbb{C})$ symmetry are realised in this context).

In the following, it will be understood that $\mathcal{A}_{4}$ acts on a
contact diagram and that momentum in conserved along the boundary. In Fig. 10,
we give the graph representation for ${\cal A}_{4}$ and its factorisation
contribution after applying the integration rules reviewed in the previous
Appendix.

$\rightarrow$

Figure 10: Four-point diagrams with double pole and its factorisation
contribution.

This factorisation contribution is given when
$\sigma_{d}\rightarrow\sigma_{c}$. To carry out this computation we use the
parametrisation $\sigma_{i}=\epsilon x_{i}+\sigma_{c}$, with $i=c,d$,
$x_{d}=\text{constant}$, $x_{c}=0$, $\sigma_{c}=\sigma_{L}$, and expand around
$\epsilon=0$:

$\displaystyle\text{d}\sigma_{d}=x_{dc}\,\text{d}\epsilon$
$\displaystyle(\sigma_{ab}\sigma_{bc}\sigma_{ca})^{2}\,{\rm
PT}(a,b,c,d)\,\frac{1}{\sigma_{ab}^{2}\,\sigma_{cd}^{2}}=\frac{1}{\epsilon^{3}}\,\frac{(\sigma_{ab}\sigma_{bL}\sigma_{La})^{2}}{(\sigma_{ab}^{3}\,\sigma_{bL}\,\sigma_{La}\,x^{3}_{cd})}\,\left(1+\epsilon\frac{x_{cd}}{\sigma_{La}}+{\cal
O}(\epsilon^{2})\right),$ (C.3)

and

$\displaystyle S_{d}=\frac{1}{\epsilon}\left[\hat{S}_{d}(\epsilon)+{\cal
O}(\epsilon^{2})\right],\qquad\hat{S}_{d}(\epsilon)\equiv\frac{\alpha_{dc}}{x_{dc}}+\epsilon\left(\frac{\alpha_{da}}{\sigma_{La}}+\frac{\alpha_{db}}{\sigma_{Lb}}\right).$
(C.4)

Using the global residue theorem, the integral becomes an integration around
$\epsilon=0$, i.e.

${\cal
A}_{4}=\int_{\gamma_{\epsilon}}\frac{\text{d}\epsilon}{\epsilon}\,\left[\hat{S}_{d}(\epsilon)+{\cal
O}(\epsilon^{2})\right]^{-1}\frac{\sigma_{bL}}{(\sigma_{ab}\,x_{cd})}+\int_{\gamma_{\epsilon}}\frac{\text{d}\epsilon}{\epsilon^{2}}\,\left[\hat{S}_{d}(\epsilon)+{\cal
O}(\epsilon^{2})\right]^{-1}\frac{(\sigma_{bL}\sigma_{La})}{(\sigma_{ab}\
\,x^{2}_{cd})}.$ (C.5)

The first term has a simple pole and is simple to evaluate

$\int_{\gamma_{\epsilon}}\frac{\text{d}\epsilon}{\epsilon}\,\left[\hat{S}_{d}(\epsilon)+{\cal
O}(\epsilon^{2})\right]^{-1}\frac{\sigma_{bL}}{(\sigma_{ab}\,x_{cd})}=-\frac{\sigma_{bL}}{\sigma_{ab}}\,(\alpha_{dc})^{-1}.$
(C.6)

Since the second term in (C.5) has a double pole, we must compute the
derivative $\frac{\text{d}[(\hat{S}_{d}(\epsilon))^{-1}]}{\text{d}\epsilon}$,
which must be done with care since $\hat{S}_{d}$ is a differential operator.
Using the identity

$\hat{S}_{d}(\epsilon)\,[\hat{S}_{d}(\epsilon)]^{-1}=\mathbb{I},$ (C.7)

it is straightforward to arrive at the following result:

$\frac{\text{d}[(\hat{S}_{d}(\epsilon))^{-1}]}{\text{d}\epsilon}\Big{|}_{\epsilon=0}=-[\hat{S}_{d}(0)]^{-1}\hat{S}^{\prime}_{d}(0)\,[\hat{S}_{d}(0)]^{-1}=-x_{dc}^{2}\,(\alpha_{dc})^{-1}\left(\frac{\alpha_{da}}{\sigma_{La}}+\frac{\alpha_{db}}{\sigma_{Lb}}\right)(\alpha_{dc})^{-1}.$
(C.8)

Therefore, the second integral in (C.5) is given by

$\int_{\gamma_{\epsilon}}\frac{\text{d}\epsilon}{\epsilon^{2}}\,\left[\hat{S}_{d}(\epsilon)+{\cal
O}(\epsilon^{2})\right]^{-1}\frac{(\sigma_{bL}\sigma_{La})}{(\sigma_{ab}\
\,x^{2}_{cd})}=\frac{1}{\sigma_{ab}}(\alpha_{dc})^{-1}\left(\sigma_{Lb}\,\alpha_{da}+\sigma_{La}\,\alpha_{db}\right)(\alpha_{dc})^{-1}.$
(C.9)

Finally, we obtain

$\displaystyle{\cal A}_{4}$
$\displaystyle=\frac{1}{\sigma_{ab}}(\alpha_{dc})^{-1}\left(\sigma_{Lb}\,\alpha_{dc}+\sigma_{Lb}\,\alpha_{da}+\sigma_{La}\,\alpha_{db}\right)(\alpha_{dc})^{-1}$
$\displaystyle=-(\alpha_{cd})^{-1}(\alpha_{db})(\alpha_{cd})^{-1}$
$\displaystyle=(\alpha_{cd})^{-1}(\alpha_{ad}+\alpha_{cd})(\alpha_{cd})^{-1},$
(C.10)

where we have used the CWI in (C.2) in the last two lines.

##  Appendix D Six-derivative results

For minimally coupled scalars we find that

$\begin{split}(\hat{s}^{3}+\hat{t}^{3}+\hat{u}^{3})\mathcal{C}_{4}^{\Delta=3}&=\frac{360}{E^{7}}k_{1}k_{2}k_{3}k_{4}k_{1}^{\mu}k_{2,\mu}k_{3}^{\nu}k_{4,\nu}\left(k_{1}^{\sigma}k_{2,\sigma}+k_{3}^{\sigma}k_{4,\sigma}\right)\left(1+\frac{E}{6}\sum_{i=1}^{4}\frac{1}{k_{i}}+\frac{E^{2}}{30k_{1}k_{2}k_{3}k_{4}}\sum_{i<j}k_{i}k_{j}\right)\\\
&\qquad+\frac{60}{E^{6}}k_{1}k_{2}k_{3}k_{4}\left((k_{3}+k_{4})(k_{1}^{\mu}k_{2,\mu})^{2}+(k_{1}+k_{2})(k_{3}^{\mu}k_{4,\mu})^{2}\right)\\\
&\qquad+\frac{24}{E^{5}}\bigg{(}10k_{1}k_{2}k_{3}k_{4}k_{1}^{\mu}k_{2,\mu}k_{3}^{\nu}k_{4,\nu}\\\
&\qquad\qquad+\frac{1}{2}\sum_{i<j}k_{i}k_{j}\Big{(}2k_{1}k_{2}k_{3}k_{4}\left(k_{1}^{\mu}k_{2,\mu}+k_{3}^{\mu}k_{4,\mu}-k_{1}k_{2}-k_{3}k_{4}\right)-2k_{3}^{2}k_{4}^{2}k_{1}^{\mu}k_{2,\mu}\\\
&\qquad\qquad\qquad-2k_{1}^{2}k_{2}^{2}k_{3}^{\mu}k_{4,\mu}+2(k_{1}k_{2}+k_{3}k_{4})k_{1}^{\mu}k_{2,\mu}k_{3}^{\nu}k_{4,\nu}+k_{3}k_{4}(k_{1}^{\mu}k_{2,\mu})^{2}\\\
&\qquad\qquad\qquad+k_{1}k_{2}(k_{3}^{\mu}k_{4,\mu})^{2}\Big{)}+\left(1+\sum_{i\neq
j}\frac{k_{i}}{k_{j}}\right)k_{1}k_{2}k_{3}k_{4}\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}\\\
&\qquad\qquad-(k_{1}^{2}k_{2}^{2}+k_{3}^{2}k_{4}^{2})\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}+k_{3}^{2}k_{4}^{2}\vec{k}_{1}\cdot\vec{k}_{2}\left(-k_{1}^{2}-k_{2}^{2}+2(k_{1}k_{2}+k_{3}k_{4})\right)\\\
&\qquad\qquad+k_{1}^{2}k_{2}^{2}\vec{k}_{3}\cdot\vec{k}_{4}\left(-k_{3}^{2}-k_{4}^{2}+2(k_{1}k_{2}+k_{3}k_{4})\right)\\\
&\qquad\qquad-\frac{1}{2}\left(4k_{1}^{2}k_{2}^{2}k_{3}^{2}k_{4}^{2}+k_{1}^{2}k_{2}^{2}+(\vec{k}_{3}\cdot\vec{k}_{4})^{2}+(\vec{k}_{1}\cdot\vec{k}_{2})^{2}\right)\bigg{)}\\\
&\qquad+\frac{6}{E^{4}}\Bigg{(}\left(k_{1}k_{2}k_{3}k_{4}\sum_{i=1}^{4}\frac{1}{k_{i}}+\sum_{i\neq
j}k_{i}k_{j}^{2}\right)\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}\\\
&\qquad\qquad-2k_{3}^{2}k_{4}^{2}(k_{3}+k_{4})\vec{k}_{1}\cdot\vec{k}_{2}-2k_{1}^{2}k_{2}^{2}(k_{1}+k_{2})\vec{k}_{3}\cdot\vec{k}_{4}\Bigg{)}\\\
&\qquad+\frac{2}{E^{3}}\Big{(}2\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}(\vec{k}_{1}\cdot\vec{k}_{2}+\vec{k}_{3}\cdot\vec{k}_{4})-\sum_{i<j}k_{i}k_{j}\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}\\\
&\qquad\qquad+2k_{3}^{2}k_{4}^{2}\vec{k}_{1}\cdot\vec{k}_{2}+2k_{1}^{2}k_{2}^{2}\vec{k}_{3}\cdot\vec{k}_{4}\Big{)}\\\
&\qquad+\frac{4}{E}\vec{k}_{1}\cdot\vec{k}_{2}\vec{k}_{3}\cdot\vec{k}_{4}+\mathrm{Cyc.}[234].\end{split}$
(D.1)

This expression mixes different types of dot products in order to obtain a
more compact expression.

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