Cosmological Scattering Equations at Tree-level and One-loop

We recently proposed a formula for tree-level $n$-point correlators of massive $\phi^4$ theory in de Sitter momentum space which consists of an integral over $n$ punctures on the Riemann sphere and differential operators in the future boundary dubbed the cosmological scattering equations. This formula was explicitly checked up to six points via a map to Witten diagrams using the global residue theorem. In this work we provide further details of these calculations and present an alternative formulation based on a double cover of the Riemann sphere. This framework can be used to derive simple graphical rules for evaluating the integrals more efficiently. Using these rules, we check the validity of our formula up to eight points and sketch the derivation of $n$-point correlators. Finally, we propose a similar formula for 1-loop $n$-point correlators in terms of an integral over $(n+2)$ punctures on the Riemann sphere, which we verify at four points. The 1-loop formula holds for small masses in de Sitter space and arbitrary masses satisfying the Breitenlohner-Freedman bound after Wick-rotating to Anti-de Sitter space.


Introduction
The scattering equations of Cachazo, He, and Yuan (CHY) [1][2][3] revolutionized the study of scattering amplitudes, revealing new perturbative dualities [4][5][6], and new techniques for computing loop amplitudes [7][8][9][10][11] and soft limits [12][13][14][15][16]. These equations take a very compact and universal form, describing a broad range of quantum field theories in flat space. In a nutshell, this approach is used to map scattering amplitudes to an integration over punctures on a Riemann sphere. The role of the scattering equations is to constrain the locations of punctures corresponding to the external legs of the scattering amplitude. The specific details of the interactions are encoded by the integrand which can be constructed from a simple set of building blocks. It is fair to say that the CHY formulae were the culmination of decades of research on perturbative scattering amplitudes that led to a wealth of many other powerful techniques such as twistor string theory [17][18][19][20][21][22], recursion relations [23][24][25], unitarity methods [26,27], the double copy relating guage theory to gravity [28,29], and new geometric formulations [30][31][32].
When Fourier transformed to momentum space, cosmological correlators develop singularities as the energy (defined as the sum of the magnitudes of the boundary momenta) goes to zero. This is interpreted as a flat space limit, such that the coefficients of these singularities correspond to scattering amplitudes in one higher dimension [50]. The relation between scattering amplitudes and conformal correlators opens up the possibilty of importing amplitude methods to cosmology.
In a recent publication [76], we conjectured a worldsheet formula for tree-level cosmological correlators of massive φ 4 theory, which is a toy model for inflation [77]. This formula was inspired by previous proposals for massless bi-adjoint scalar theories in AdS embedding space [78,79].
One of the key insights of these papers was to promote the flat space scattering equations to differential operators built out of conformal generators acting on the boundary. In our work, the scattering equations are formulated in de Sitter momentum space, which is the natural language for cosmology and makes the flat space limit much more transparent, so they are referred to as the cosmological scattering equations (CSE). The CSE do not trivially follow from the scattering equations in AdS embedding space, and our construction introduces an operatorial Pfaffian which is the core ingredient for describing more general interactions in de Sitter space. Potential ambiguities that could arise in curved background were shown to be absent and we explicitly checked the formula up to six points by using the global residue theorem (GRT) [80] to map it to a sum of Witten diagrams.
In this paper we will provide detailed derivations of the above results and extend them in several directions. In particular, we will present an alternative worldsheet formula based on the double cover formulation developed in [81] for flat space. The basic idea is to consider the Riemann sphere as a quadratic curve embedded in a two-dimensional complex projective space, promoting the branch-cut parameter to a new variable. By integrating this variable, we can access different factorization channels. This formulation streamlines the derivation of simple graphical rules for a more efficient evaluation of the worldsheet integrals, which can also be applied to the single-cover approach. Using these rules, we then verify our proposal for φ 4 correlators up to eight-points, which include diagrams with non-ladder topology. While ladder diagrams can be computed recursively, non-ladder ones have a more complicated structure and we will present a systematic way to evaluate them. With the insights we gain at eight points, we then sketch the calculation of n-point correlators.
Finally, we also propose a formula for 1-loop correlators of massive φ 4 theory in de Sitter space.
The basic idea is to consider a tree-level correlator with two auxiliary punctures. They encode the loop momentum and have unfixed scaling dimensions, which must be integrated over. As we show explicitly at 4-points, the bulk-to-boundary propagators for the auxiliary punctures get glued together to form a bulk-to-bulk propagator after integrating over their scaling dimensions, giving rise to 1-loop Witten diagrams. This gluing relies on a split representation for the bulk-tobulk propagator first proposed in AdS d+1 [61,82], which we Wick-rotate to dS d+1 . The resulting dS propagator is only valid for masses 0 ≤ m ≤ d/2, or equivalently scaling dimensions in the complementary series. On the other hand, after Wick-rotating back to AdS, our 1-loop formula is valid for any mass satisfying the Breitenlohner-Freedman bound m 2 ≥ −d 2 /4 [83].
The structure of this paper is as follows. In section 2 we review some basic facts about the cosmological wavefunction and its perturbative calculation using Witten diagrams. We also review the worldsheet formula for tree-level cosmological correlators proposed in [76]. In section 3 we describe a new formula in terms of the double cover and use it to derive simple graphical ruels for evaluating worldsheet integrals. In section 4 we show that the formula reproduces the expected Witten diagram expansion at four and six points, and in section 5 we extend this to eight-points and comment on n points. In section 6 we propose a new worldsheet formula for 1loop correlators and verify it at 4-points. Finally in section 7 we present the concluding remarks.
We also have a number of Appendices giving more details about dS isometries in momentum space, SL(2, C) symmetry of the worldsheet formula, the double cover formalism, and 6-point correlators.
Note added: While completing this manucript, a proposal for constructing 1-loop Witten diagrams from tree-level ones in AdS embedding space appeared in [84] which has some overlap with the results in section 6 of this paper. Note that there are some important differences in the two approaches. For example, our 1-loop formula is for correlators in de Sitter momentum space and makes use of the CSE. Moreover, while [84] focuses on φ 3 theory, we focus on φ 4 theory, although we believe both approaches can be extended to more general theories in (A)dS.

Review
We will work in the Poincaré patch of (d + 1)-dimensional de Sitter with radius R and metric , and x denotes the Euclidean boundary directions, with individual components x i . For simplicity, we will set R = 1. In this section we will define the correlators we wish to compute, and describe in detail how to do it perturbatively using Witten diagrams.

Cosmological Correlators
In-in correlators in de Sitter can be computed from a cosmological wavefunction as follows: where φ are values of the fields in the future boundary Fourier transformed to momentum space, and Ψ [φ] is the cosmological wavefuntion, which is a functional of φ. In principle, we should integrate over the boundary values of all the fields, including the metric, but perturatively we can restrict to matter fields and for simplicity we will only consider scalar fields. In more detail, the wavefunction can be perturbatively expanded as follows: where the wavefunction coefficients Ψ n can be treated as an n-point CFT correlators in the future boundary, our main focus here. We will refer to them as cosmological correlators.
The n-point cosmological correlator can be expressed in momentum space as where k T = k 1 + ... + k n and the object in double brackets can be treated as a CFT correlator in the future boundary. We will work with scalar operators O of scaling dimension ∆, dual to massive scalar fields φ in the bulk satisfying The mass is related to the scaling dimension by For a given m 2 , the two linearly independent solutions of (2.4) have conformal dimensions which are related by a shadow symmetry ∆ ± = (d − ∆ ∓ ). In practice we choose the scaling dimension with relative plus sign. The light solutions (m 2 ≤ d 2 /4) are parametrized by ν ∈ R, while for heavy solutions (m 2 > d 2 /4) ν is imaginary. The former is known as the complementary series, while the latter is the principal series. Note that ∆ = d and ∆ = (d + 1)/2 correspond to bulk scalars which are minimally or conformally coupled, respectively.
The conformal Ward identities (CWI) for the cosmological correlator can be expressed as where a, b, ... are particle labels and the conformal generators in momentum space are with ∂ i = ∂ ∂k i . A derivation of these generators can be found in Appendices A and B. We will not need to consider rotation generators L ij since we focus on correlators of scalar operators.

de Sitter propagators in momentum space
It is convenient to consider Fourier modes of the scalar field φ along the boundary, where k = | k|, ν = ∆ − d/2. Plugging this into (2.4) then implies The solutions K ν (k, η) are related to the Hankel functions of order ν. Here we are going to work with the bulk-to-boundary propagators in the following form: where K ν is the modified Bessel function of the second kind and H (2) ν denotes the Hankel function of the second kind. Note that the first line was obtained by Wick rotating the standard bulk-toboundary propagator in AdS [61] according to z → −iη, and the second line was obtained using the identity for x ∈ R − . The asymptotic behaviour of K ν when η → −∞ has a positive-frequency Minkowski parameter λ. In momentum space, this is translated to (η, k i ) → (λη, k i /λ), and K ν has scaling dimension d − ∆: The other linearly independent solution of (2.10) is denoted by P ν (k, η), and will be specified below.
1 More details on the boundary conditions involved can be found in e.g. [65].
The bulk-to-bulk propagators G ν (η,η; k) satisfy where we take the limit η 0 → 0 as explained in the beginning of the section. The solution is given by [85] where θ(η −η) is the Heaviside step function. In order to show that this expression satisfies (2.16), recall that the Wrosnkian of the two linearly independent solutions of (2.10) satisfies for a convenient normalization of P ν (k, η). More explicitly, the time-ordered bulk-to-bulk propagator in dS can be cast as for η >η. If η <η, we simply exchange η andη. Here, H ν is the Hankel function of the first kind, and the limit η 0 → 0 is implicit.
Let us briefly comment on the relation to the bulk-to-boundary propagator in AdS, which has the following split representation [61,82]:

Witten diagrams
Cosmological correlators admit a perturbative expansion in terms of bulk Witten diagrams ending on the future boundary. Here we take the bulk theory to be a scalar with mass m and quartic self-interaction.
The bulk-to-boundary propagators are the building blocks of the contact diagrams C ∆ n : where momentum conservation in the boundary directions and the integration over η ∈ {−∞, 0} are implicit. As we will see, all tree-level Witten diagrams can be obtained from contact diagrams by acting with certain differential operators.
A central object in our analysis is the action of the operator on the product K ν (k a , η)K ν (k b , η) ≡ K a ν K b ν . When acting on K ν , the boundary generators in (2.8) can be written in terms of derivatives with respect to conformal time This can be demonstrated by choosing a convenient integral representation for K ν , e.g.
which comes from a Fourier transformation of the bulk-to-boundary propagator in position space.
Therefore, we obtain Now let us consider the action of the operator D 2 ab ≡ D 2 k , with k = | k a + k b |, on the product K a ν K b ν : The generalization to n-particles is straightforward. Let us define where k 1...n = | k 1 + . . . + k n |. Then, using the results in (2.24) and (2.26), we obtain with p < n. The left-hand-side is the bulk Casimir in (2.29) and the right-hand-side is written in terms of the boundary conformal generators in momentum space (2.8).
In practice we will encounter the inverse of boundary differential operators constructed from those in (2.23) acting on the product of bulk-to-boundary propagators. Using (2.31) we can then replace them with the inverse of the bulk differential operator in (2.11) leading to bulk-to-bulk propagator insertions: Note that we are not explicitly inverting the differential operator on the left-hand side, a much more involved problem that is not being addressed here. Instead, the equation above is a formal equivalence that can be immediately verified by acting on both sides with (D 2 1...p + m 2 ). On the left-hand side we use the first line of (2.31). As for the right-hand side we simply recall the equation of motion (2.16). While the inverse operator is naively blind to boundary conditions, we promptly observe that the right-hand side of (2.32) carries this information through the bulkto-bulk propagator. This construction leads to a fundamental result expressing exchange Witten diagrams in terms of differential operators acting on contact diagrams: which will be very useful later on. Explicit formulae for the bulk-to-bulk propagator are given in the previous subsection.

Worldsheet Formula
In [76] we proposed a new formula for computing tree-level correlators of massive φ 4 theory in de Sitter momentum space based on a curved space analogue of the scattering equations. In more detail, the correlators are expressed as integrals over the Riemann sphere, mapping each external leg to a puncture, and the contour of integration is defined to encircle the points where the following differential operators vanish when acting on the rest of the integrand: where σ a is the holomorphic coordinate of the a'th puncture and µ ab is a mass deformation equal to −m 2 when a and b are adjacent and zero otherwise. This mass deformation assumes canonical ordering of the external legs, i.e. (1, 2, . . . , n) ≡ I n [89]. Different orderings are obtained by permutations.
We refer to the equations which define the contour of integration as the cosmological scattering equations. In flat space, the scattering equations can be explicilty solved and the worldsheet integral can be evaluated by summing over solutions to the scattering equuations. Due to the operatorial nature of (2.34), it is not yet known how to do this in de Sitter space, so we instead deform the contour of integration in order to convert the worldsheet integral to a sum of Witten diagrams. This approach was inspired by the ambistwistor string formulae in AdS position space first proposed in [78,79].
In Appendix B we show that the CSE enjoy an SL(2, C) symmetry: where {σ b , σ c , σ d } denote the fixed punctures. The integration contour is defined by the intersection γ = a =b,c,d γ Sa , where γ Sa encircles the pole where S a vanishes when acting on the theory-dependent integrand I n . Following similar steps to [78,79], it is simple to check that the differential operators in (2.34) commute, so the measure, is well-defined and SL(2, C) invariant.
Using the CSE defined above, we can now define a worldsheet formula for n-point correlators of massive φ 4 theory in dS momentum space [76]: where n = 2p ∈ even, S n−1 is the permutation group and Here PT(I n ) = (σ 12 σ 23 ...σ n1 ) −1 is the Parke-Taylor factor, cp(I n ) denotes all connected perfect matchings related to the ordering (1, 2, ..., n) [4], and the reduced Pfaffian Pf A is given by where The matrix A cd cd is obtained from the n × n matrix by removing any pair of rows and columns {c, d}. From the commutation relations one can see that the reduced Pfaffian Pf A is well defined. In addition, it is straightforward to which imply that Ψ n satisfies the CWI. Note that A(I n ) is independent of the choice of rows and columns one removes from the A-matrix. In Appendix D we explicitly show this with a six-point example.
Finally, notice that each connected perfect matching term in A(I n ) has a natural graph representation. For example, the graph correponding to (σ 1,p−2 σ 2,p+2 σ 3,n−1 σ 4,n−2 · · · σ n,p ) −1 is given by Fig. 1. The external circle represents the Parke-Taylor factor, PT(I n ), the black lines depict the perfect matching, and the red line indicates the rows and columns removed from the A-matrix. The underlined labels {n−1, n, 1} are the coordinates fixed by the SL(2, C) symmetry, i.e. the punctures {σ n−1 , σ n , σ 1 } are not integrated. The notation we are going to use for this type of graph is A(I n : 1 p − 2, 2 p + 2, 3 n − 1, . . . , n p), where the argument following the double dots denotes the connected perfect matching.
The flat space limit of (2.39) can accessed by taking η → −∞ in the conformal time integrals.
Using the asymptotic form of the bulk-to-boundary propagators and (2.26), it is not difficult to show that our proposal has the expected flat space limit [76].

Double Cover
In order to further test our construction, we will show that it reproduces the correct Witten diagram expansion for tree-level correlators up to eight points, with a sketch of the proof for n points. In practice, we will evaluate the worldsheet integrals in (2.40) using certain graphical rules which make calculations much more efficient. These rules can be conveniently derived by reformulating the worldsheet formula using a double cover formalism in which the worldsheet is represented by a two-sheeted Riemann surface connected by a cut which naturally encodes factorization [81]. A detailed review of this formalism can be found in Appendix C, but we will state some basic definitions below.
The idea of the double cover is to consider the Riemann sphere embedded in CP 2 by the quadratic curve y 2 = z 2 − Λ 2 , where Λ = 0 is a constant and (z, y) ∈ C 2 . The points (Λ, 0) and (−Λ, 0) are branch points. Since the curve is quadratic, there are only two-sheets. Thus, without loss of generality, we can say that "y" tells us on which of the two sheets the point is and "z" gives us the position on the sheet. For example, we state the puncture (z a , y a ) = (z a , z 2 on the lower one. There are 2n − 3 parameters to be integrated in this double cover approach. This in contrast to the single-cover approach described in section 2.4, which involves the integration over n − 3 complex parameters. In particular, one starts with 2n + 1 integration variables, namely (z 1 , . . . , z n , y 1 , . . . , y n ) and Λ.
The SL(2, C) symmetry and the additional scaling symmetry of the CP 2 space can then be used to gauge-fix four of them. In practice, we will fix four of the z punctures.
Let us now sketch how the building blocks in the single cover approach can be adapted to the double cover language. The Parke-Taylor factor and the CSE become Noting that T ab is antisymmetric, the A-matrix and the reduced Pfaffian are then given by With these definitions, we can now explain how to adapt the worldsheet formulae in (2.39) to the double cover approach and use it to derive some useful integration rules. More details are presented in Appendix C.

Alternative Worldsheet Formula
Using the ingredients above, the corresponding version of the tree-level proposal in (2.40) is given by where 4) and the measure dµ Λ n has the form In the double cover approach Λ is treated like an integration variable and we are able to gauge away four punctures. The Faddeev-Popov determinants for this gauge fixing are ∆ (pqr) = (τ p:q τ q:r τ r:p ) −1 , where {z p , z q , z r , z m } are the gauge fixed punctures. The integration contour γ is determined by the (n − 3) scattering equations (S Λ a ) −1 and the hypersurfaces defined by the n quadratic curves, Notice that the first part of the measure, a ya dya Ca , sums over all possible ways to place the punctures (z a , y a ) on the different sheets (y a is thought of as an independent variable), and dΛ Λ is the scale invariant measure over Λ.
Using the global residue theorem (GRT) [80], one of the scattering equations can be swapped by the contour |Λ| = ε so, A(I n ) can be written as where, without loss of generality, Γ is defined by the (n − 4) scattering equations, (S Λ a ) −1 , a = p, q, r, m, and the solution of the (n + 1) equations As in the single cover approach, each connected perfect matching in A(I n ) has a natural graph representation. The only difference with Fig. 1 is that there are four fixed punctures, i.e.
four underlined labels. For example, if we consider the same perfect matching term as in Fig.   1, T 1,p−2 T 2,p+2 T 3,n−1 T 4,n−2 · · · T n,p , then the graph related to this term is given by the Fig. 2, where we have chosen {z n−2 , z n−1 , z n , z 1 } as the four fixed punctures.

Integration rules
In our previous work [76], we proposed a couple of rules for evaluating worldsheet integrals for the φ 4 model in a simple and straightforward way. They share a strong resemblance to the Yang-Mills and NLSM cases investigated in [86,87]. In this section we will prove these rules from the double cover point of view, which can then be applied to the single cover approach. They can be conveniently visualized using the graphs introduced in section 2.4 (see Figures 1 and 2 in the single and double cover formalisms, respectively). More specifically, the proposed rules are used to identify the vanishing graphs through factorization cuts, guiding a convenient choice of gauge (fixed punctures) and reduced Pfaffian.
Towards the first rule, we observe that the integration over the previously introduced y a variables localizes the integrand on the curves C a = 0, with the solutions y a = ± z 2 a − Λ 2 . The punctures are distributed among the two sheets in all 2 n possible combinations. Due to the Z 2 symmetry between the upper and lower sheets, only 2 n−1 of them are inequivalent. After computing the integration over |Λ| = ε, the two sheets factorize into two single-covers connected by a propagator corresponding to a differential operator such as in (2.33). On each of the two single-cover sheets, three punctures must be fixed due to the SL(2, C) redundancy. When Λ = 0, the branch-cut closes at a point giving two new punctures at the origin of each sheet. Each sheet must then have two more fixed punctures, which come from the fixed punctures in the original double cover prescription, i.e. the points (z p , z q , z r , z m ). If there are not exactly two of these marked-points on each of the single covers, the configuration vanishes trivially. We summarize this in the rule: • Rule I. All configurations (or factorization cuts) with fewer than two fixed marked-points vanish.
In Let us now prove another useful rule. Without loss of generality, we will choose (z p , z q , z r , z m ) = (z 1 , z 2 , z 3 , z 4 ). Following rule I, we will only be concerned with configurations where only two of these punctures are on the same sheet. We would like to determine the behaviour of the different terms of the integrand around Λ = 0. In order to evaluate the integration measure, the Faddeev-Popov determinant, and the term (S Λ 4 ) −1 , we consider a configuration where the punctures {z p+1 , . . . , z n , z 1 , z 2 } are located on the upper sheet, i.e. y a = + z 2 a − Λ 2 , and the punctures {z 3 , z 4 , . . . , z p } are located on the lower one, i.e. y a = − z 2 a − Λ 2 . By expanding around Λ = 0, we obtain where z L = z R = 0, α aL = α a3 + α a4 + · · · + α ap (a = p + 1, . . . , n) and α bR = α b p+1 + · · · + α bn + α b1 + α b2 (b = 5, 6, . . . , p). Next, we need to determine the leading order expansion of the Parke-Taylor factor, the reduced Pfaffian, and the connected perfect matching terms when Λ → 0. To do that, we are going to use the graph representation explained in Fig. 2. For a given configuration, with punctures distributed between the upper and the lower sheet, the lines that connect the vertices cross the banch-cut. After expanding the terms in the integrand around Λ = 0, it is straightforward to note that the leading order contribution is related to the number of lines cut by the branch-cut.
In Table 1 we have classified the Λ-behaviour of these integrands.

Factor
Lines cut by the branch-cut Table 1: Λ-dependence when terms in the integrand terms are expanded around Λ = 0. Empty entries mean that such terms do not occur. Now, combining the expansions (3.10), (3.11) and the Table 1, we state the following rule: • Rule II. If the branch-cut (factorization) cuts more than four lines for a given configuration in the corresponding graph then, this contribution vanishes.
In Fig. 4, this rule is illustrated through a simple example where the dashed black line (factorization cut) is cutting more than four lines. Now that the integration rules have been properly established, we are ready to demonstrate their use through a couple of examples and present some general results. This is our focus in the next two sections.

Four and Six Points
In this section we will evaluate the worldsheet integral in (2.39) at four and six points, showing that it generates the expected sum over Witten diagrams, and that there are no ambiguities in the integrand. This is a nontrivial feature given that we work with operatorial building blocks.
The four and six point examples are simpler because they only involve ladder diagrams. More general diagrams contribute above six-points, as we discuss in the next section. To make the calculations more efficient, we use the integration rules discussed above. They can be extended to the single-cover approach, where the role of the branch-cut variable Λ is given by the infinitesimal parameter that controls the rate at which punctures approach each other in a factorization cut.

Four points
Let us first consider the ordered correlator The corresponding graph of A(I 4 ) is drawn in Fig. 5(a). We see there is only one factorization contribution, σ 3 → σ 2 , which is given by the dashed black line. To perform the computation we choose the following parametrization: σ a = x a + σ 2 , a = 2, 3, with x 2 = 0, x 3 = constant and σ 2 ≡ σ L . The measure and integrand of (4.1) can be expanded as follows By the GRT, the contour γŜ 3 +O( ) can be deformed into γ = {| | = δ}. One then finds that and which is the four-point contact diagram illustrated in Figure 6.

Six points
From equation (2.40), we see that A(I 6 ) is encoded in the four diagrams given in Fig. 7.
As explained above, the integration over γ 26 vanishes so the only contribution comes from the second contourγ 2 .

Eight Points and Beyond
In this section we will use the proposed formula (2.39) to compute the tree-level eight-point correlator. This correlator involves perfect matchings with non-ladder topologies and therefore exhibits more generic structure than the lower-point correlators considered in the previous section. We then sketch how to extend our eight-point calculations to n-points.
From the integration rules of subsection 3.2, we find there are twelve non-zero contributions at eight-points, given in Fig. 10. Observe that there are only two different topologies: ladder and non-ladder diagrams, respectively first and second rows in Fig. 10. Furthermore, several graphs are simply related by relabelings, so it is enough to sample one in each row.

Non-Ladder Contributions
Now we are going to compute the contributions of the non-ladder diagrams given by the graphs on the bottom row of Fig. 10  , (5.14) with γ = 6 a=2 γ Sa . Again, we identify two factorization contributions via the integration rules, given by σ 2 → σ 8 and σ 3 → σ 4 → σ 5 → σ 6 → σ 7 . Using the contour γ S 6 for the GRT, the first factorization cut vanishes trivially. For the second contribution we use the parametrization given in (5.2), with σ a = x a + σ L , a = 3, 4, 5, 6, 7, x 6 = constant, x 7 = 0, and σ 7 = σ L . Next, we obtain a similar expansion to (5. where PfA 1L 1L and PfA R7 R7 are given in equation (  In order to compute A(R, 3, 4, 5, 6, 7 : R6, 35, 47) in (5.17) we will use the SL(2, C) symmetry to choose a more convenient puncture fixing. First we rewrite the integrand in way that makes this symmetry manifest. This can be accomplished using the following deformation of the CSE: , a = 4, 5, with ∆ ab = ∆ ba and Since α ab = α ba , then (5.18) is the only possible deformation.
It is simple to show that these scattering equations are SL(2, C) covariant. Note that the ∆ ab parameters were inspired by ones in [90][91][92].
The deformations in (5.18) correspond to the following deformed A-matrix of A(R, 3, 4, 5, 6, 7 : R6, 35, 47): On the support of the deformed CSE we then have the identity with Now that we have figured out how to rewrite the CHY integrand of A(R, 3, 4, 5, 6, 7 : R6, 35, 47) in a manifestly SL(2, C) invariant way, we can choose the previously used gauge Here we have Since the four point subdiagrams given in (5.16) which corresponds to the Witten diagram in Figure 15. The other non-ladder diagrams in Fig.   10 are obtained from A(I 8 : 16, 28, 35, 47) by relabelling. Given that all terms in (5.15) and (5.24) commute, we conclude that the integrand of the 8-point correlator is free of ambiguities.
We do not have yet a recursion for non-ladder diagrams, though the method we developed in section 5.2 can be systematically applied. Recall that in section 5.2, we decomposed the eightpoint non-ladder digram in Fig. 13 into a product of two ladder diagrams, one with four external legs and the other with six legs. We then deformed the CSE and Pfaffian of the 6-point diagram to make its SL(2, C) symmetry manifest and chose a different gauge fixing to obtain a graph (integrand) with the structure Fig. 16. The generalization of this procedure to higher-point non-ladder diagrams is straightforward. Consider the non-ladder diagram in Fig. 18. After applying the integration rules and GRT, we obtain just one factorization contribution, where with the CSE and elements of the Pfaffian, they do not introduce any ambiguity. This implies that after factorizing the non-ladder diagram all terms commute, providing an inductive proof that the worldsheet integrand is free of ambiguities.

One-loop Scattering Equations
In this section, we will generalize the tree-level construction in section 2.4 to 1-loop. Our proposal for the 1-loop n-point correlator is to take a tree-level (n+2)-point correlator, deform it, and paste together two of the legs. This is analogous to the Feynman tree theorem for scattering amplitudes [93,94] and similar formulae have appeared in the context of ambitwistor strings [8,11,95] and loop-level recursion [9,25]. More precisely, the 1-loop n-point cosmological correlator for φ 4 theory in dS is given by whereΨ is a deformed tree-level correlator which we will describe below, is the loop momentum projected onto the boundary, ν ± = ±iω, and ν 1 = .... = ν n = ν = ∆ − d/2. Legs 1 to n are external with fixed mass m 2 = ∆ (d − ∆), while legs ± are internal and their mass is integrated over: 3) The mass of the internal legs is given by As we will explicitly see in the next subsection, the integral over ω essentially connects the bulkto-boundary propagators of the ± legs into a bulk-to-bulk propagator. Note that the tree-level correlator encodes a sum over Witten diagrams with the internal legs appearing in different places, so after pasting them together this gives a sum over 1-loop Witten diagrams. As we will explain in the next subsection, we expect our formula to work for scalar fields with mass 0 ≤ m ≤ d/2 in dS, but after Wick rotation it should work for any m 2 ≥ −d 2 /4 in AdS.
In more detail, the deformed tree-level correlator in (6.1) is essentially the same as the one in (2.39) except for a few modifications: where ρ labels permutations of legs {1, ..., n, −} and p = n 2 + 1. The worldsheet integralÃ is similar to (2.40). However, we deform the differential operators and contact diagram as described below and discard perfect matchings in which the ± legs are attached to the same vertex. This is indicated by the subscript 1PI. The motivation for removing such contributions will become clear soon. Moreover, we fix the punctures {σ ± , σ 1 }. This can be motivated by 1-loop formulae arising from ambitwistor string theory, where one starts with a genus-one worldsheet with a single fixed puncture (σ 1 ) which degenerates to a spherical worldsheet with two more fixed punctures (σ ± ) [7,8]. Finally, note that (6.4) has an extra factor of 1 2 compared to (2.39), which is simply a symmetry factor of the 1-loop Witten diagrams that will arise after pasting the ± legs together.
Let us describe the integrand in more detail. The deformed contact diagram is a product of bulk-to-boundary propagators where two legs have a different scaling dimension than the others: (6.5) The differential operators in the integrand and CSE are deformed as follows: This was inspired by the deformed 1-loop SE proposed in [11]. Prior to the deformation, the CSE for particles {1, n, +, −} are given by It is not difficult to check that they are SL(2, C) invariant, and the deformation in (6.6) preserves this symmetry. Since we are fixing legs {1, +, −}, the deformed CSE for these legs will not be needed in practice.
In summary, a 1-loop correlator can essentially be obtained from a tree-level correlator with two additional legs. One important difference compared to the tree-level formula discussed earlier is that the spectral parameters of two bulk-to-boundary propagators in the contact term are different than the rest. Nevertheless, many previous formulas can be generalized. In particular, using (2.24) we find where the spectral parameters for the two propagators can be different. Using this identity, we also find where D 2 1...p is defined in (2.11) with k = k 1...p and the contact term consists of bulk-to-boundary propagators with different spectral parameters: K νa (k a , η). (6.10) Note that the differential operator on the left-hand-side of (6.9) does not depend on the spectral Figure 19: Evaluating the worldsheet formula for the perfect matching on the left gives the treelevel 6-point Witten diagram in the middle. Pasting together the ± legs using the procedure described in the text then gives the 1-loop 4-point Witten diagram on the right.
parameters of the bulk-to-boundary propagators on which it acts.
In the next subsection, we will verify that (6.1) gives the correct 1-loop 4-point correlator. It is also not difficult to see that it solves the CWI for any number of legs. In the limit k ± → ± , the momentum delta function becomes that of an n-point correlator. Since the conformal generators for each external leg can be commuted past the Pfaffian and CSE to act on the bulk-to-boundary propagator for that leg in the contact diagram, the CWI are equivalent to those of a tree-level n-point correlator.

4-point Example
Let us illustrate how this works at four points. First we consider the 6-point graph on the lefthand-side of Figure 19, which corredponds to a term in the deformed tree-level correlator in (6.4), where we fix the positions {1, +, −} and remove the rows and columns associated with legs {1, −} from the Pfaffian. After performing the deformation in (6.6) and summing over permutations, the deformed tree-level correlator is given bỹ where we sum over cyclic permuations of legs (2,3,4), which will ultimately correspond to the sum over the s, t, u channels of the 1-loop Witten diagrams. Note that the first term in (6.11) is just the deformed version of (4.9) with the relabeling {5, 6} → {−, +}. In particular, the deformed contact diagram is defined in (6.5) and the deformed Pfaffian is given by Following the same steps of the evaluation of the undeformed tree-level 6-point correlator in section 4.2, we findΨ Note that the 6-point contact diagram has been split in such a way that the ± legs are now on different vertices connected by a bulk-to-bulk propagator with boundary momentum k +12 , as illustrated in Figure 19. After taking the limit k ± → ± and performing the integral over ω in (6.1), the bulk-to-boundary propagators corresponding to legs ± will then be pasted together to form a bulk-to-bulk propagator, giving rise to the 1-loop Witten diagram illustrated on the right-hand-side of Figure 19. To see this, first note that the only ω dependence in (6.13) comes from the bulk-to-boundary propagators. The operator D 2 +12 does not depend on the spectral parameters of legs {+, 1, 2}, only on their momenta. As a result, the ω integral will only involve the bulk-to-boundary propagators for the ± legs and we are left with the following integral: Putting everything together, we finally obtain the 1-loop Witten diagram illustrated in Figure   20 summed over permutations: K ν ( k j , η )+perms. (6.15) This is just the expected expansion of the 1-loop 4-point cosmological correlator in terms of Witten diagrams. Finally, note that our definition of the deformed (n + 2)-point tree-level correlator in (6.4) discarded contributions that would give rise to 1-loop corrections to the bulk-to-boundary propagators illustrated in Figure 21. The motivation for discarding such contributions is that in the flat space limit one will get scattering amplitudes obtained by amputating the external legs and putting them on-shell. Alternatively, we can keep such contributions by removing the restriction on perfect matchings in (6.4) and including a factor of 2 for those where the ± legs are attached to the same vertex. + perms.

Conclusion
We have provided further details on the recently proposed worldsheet formula for tree-level correlators of massive φ 4 theory in de Sitter momentum space [76], which is a toy model for inflationary cosmology. This formula is based on cosmological scattering equations defined in terms of conformal generators acting in the future boundary. Another key ingredient of the formula is a Pfaffian of the conformal generators which appears in the integrand. In principle there could be ambiguities arising from the operatorial nature of the building blocks involved, but they are nontrivially absent in all the examples we have considered. In [76] we verified the proposal up to six points and showed that in the flat space limit it reduces to the CHY formula for φ 4 amplitudes for any number of legs [4]. In this paper, we derived all of these results in detail and extended them up to eight points, using simple graphical rules for evaluating the worldsheet integrals which we derived from a double cover formulation. We also proposed a  [46,58,88].
While we have focused on one of the simplest models of inflationary cosmology, several steps can be taken to make it more realistic. For example, one could consider more general mass deformations which allow particles with different masses to propagate [90]. Another interesting direction would be to break some symmetries of the cosmological scattering equations, since cosmological surveys measure correlators of curvature perturbations which become nontrivial when de Sitter boosts are broken [96][97][98]. Moreover, it would be of great interest to explore more complicated Pfaffian structures corresponding to more general effective scalar theories, such as the non-linear sigma model, scalar DBI, and special Galileon theories, whose flat space amplitudes have a very elegant description in terms of CHY formulae [4]. When lifting to curved background, new subtleties can arise such as curvature corrections to the effective actions and ordering ambiguities in the worldsheet formulae associated with the operatorial nature of the integrands. It would also be interesting to generalize the construction to spinning correlators in order to make the double copy in (A)dS more systematic [66][67][68][69][70][71][72][73], which would in turn streamline the calculation of gravitional correlators in this background. Finally, it would be very desirable to evaluate the worldsheet formula by directly solving the cosmological scattering equations rather than mapping the formula to a sum of Witten diagrams using the global residue theorem. It may be possible to do this analytically using techniques from integrability [78,79], but a numerical approach may ultimately be needed. Indeed, even the scattering equations in flat space require numerical solution above five points for generic kinematics [99][100][101]. In summary, the study of the cosmological scattering equations is still in its infancy, and there are many exciting directions to be explored.

Acknowledgements
We thank Charlotte Sleight

A dS isometries in momentum space
The operator D 2 k in equation (2.11) is the Casimir of the isometry group of de Sitter space, defined as where the bulk generators for dS d+1 can be realized in momentum space as where ∂ η = ∂/∂η and ∂ i = ∂/∂k i . Plugging the bulk generators into (A.1) then leads to Moreover they satisfy the standard algebra of the Euclidean conformal group SO(1, d + 1): When acting on boundary correlators, we can work with the boundary counterparts of the above generators. The only condition is that the eigenvalue equation (2.10) holds at the future boundary as well, i.e. with η → 0 − . Noting that the asymptotic form of the eigenfunctions is η ∆ , the generators P i and L ij are unchanged while D and K i become The boundary conformal generators are also derived in the next Appendix by Fourier transforming the standard expressions in position space.

B Proof of SL(2, C)
In this Appendix, we will derive the conformal generators in momentum space given by (2.8) and prove the SL(2, C) symmetry of the CSE.
Let us begin by Fourier transforming a scalar correlator: Using translational invariance, we can write .., n. Changing integration variables then gives Now consider dilatations: Fourier transforming according to (B.1) then gives Moreovoer if we plug in the decomposition in (B.4) and integrate deriviatives of the delta function against a test function, we find Let us derive a slightly different form of the dilation Ward identity. First note that translational invariance implies that Replacing Next let's look at conformal boosts: After Fourier transforming, we find that the boost acts trivially on the delta function (after integrating against a test function and using rotational invariance), so the Ward identity reduces We can obtain another useful form of this Ward identity by substituting ∂ prove the SL(2, C) symmetry of the CSE. In particular we must show that the following sum vanishes:

C Review of Double Cover Formalism
In this Appendix, we will review of the double cover (DC) formalism introduced in the beginning of section 3. Recall that in this approach, the one represents the worldsheet by the quadratic curve y 2 a = z 2 a − Λ 2 , where Λ = 0 is a constant and (z a , y a ) ∈ C 2 . One then promotes Λ, which encodes factorization of the worldsheet, to an integration variable along with the 2n variables (z 1 , . . . , z n , y 1 , . . . , y n ). Below, we will describe some useful building blocks for constructing the integrand in the double cover formalism and show that it enjoys an SL(2, C) along with a scaling symmetry which can be used to fix four of the integration variables.
Thus, in order to define the A-matrix and the reduced Pfaffian we rewrite τ a:b in the following way: where T ab is clearly antisymmetric, T ab = −T ba . In terms of T ab , the Parke-Taylor factor becomes, PT Λ (I n ) = τ 1:2 τ 2:3 · · · τ n:1 = n a=1 (yz) a y a T 12 T 23 · · · T n1 . (C.7) Now, we are able to define the A-matrix and the reduced Pfaffian, where the prefactor a [(yz) a /y a ] in Pf A must be introduced (as in (C.7)) to have SL(2, C) invariance.

D More on 6 points
In this Appendix, we will carry out the 6-point calculation in section 4. By the integration rules, it is simple to see there are three factorization contributions given in Fig. 22. We will expand the reduced Pfaffian and compute each term separately. factorization contributions (notice that they are consistent with Fig. 22).