Missing Corner in the Sky: Massless Three-Point Celestial Amplitudes

We present the first computation of three-point celestial amplitudes in Minkowski space of massless scalars, photons, gluons, and gravitons. Such amplitudes were previously considered to be zero in the literature because the corresponding scattering amplitudes in the plane wave basis vanish for finite momenta due to momentum conservation. However, the delta function for the momentum conservation has support in the soft and colinear regions, and contributes to the Mellin and shadow integrals that give non-zero celestial amplitudes. We further show that when expanding in the (shadow) conformal basis for the incoming (outgoing) particle wave functions, the amplitudes take the standard form of correlators in two-dimensional conformal field theory. In particular, the three-point celestial gluon amplitudes take the form of a three-point function of a spin-one current with two spin-one primary operators, which strongly supports the relation between soft spinning particles and conserved currents. Moreover, the three-point celestial amplitudes of one graviton and two massless scalars take the form of a correlation function involving a primary operator of conformal weight one and spin two, whose level-one descendent is the supertranslation current.


Introduction
Scattering amplitudes presented in the conformal primary basis in four-dimensional Minkowski space are naturally viewed as correlators on the celestial two-sphere at null infinity, and referred to as the celestial amplitudes [1][2][3][4][5]. The connection between four and two dimensions provides a concrete dictionary for celestial holography, which aims to reformulate the flat space quantum gravity as a celestial conformal field theory (CCFT). The viewpoint from the celestial sphere has brought us a lot of new insights. For instance, under the celestial holography dictionary, massless spinning particles in the soft (or conformally soft) limit are dual to currents that generate infinite-dimensional symmetries in the CCFT, and the corresponding soft theorems are recast as Ward identities [6][7][8][9][10][11][12][13][14][15][16][17][18][19].
From the analogy with two-dimensional conformal field theory (CFT), one expects the structure constants and central charges of this infinite-dimensional algebra are contained in the three-point correlators of the currents. However, three-point scattering amplitudes of massless particles are commonly regarded to vanish, because in the plane wave basis the momentum conservation cannot be satisfied for generic momentum. This problem is usually avoided in the literature by going to the Klein space with the unphysical (2, 2) split signature [11,12,14,[20][21][22][23][24][25][26]. Even so, the three-point celestial gluon and graviton amplitudes still do not take the standard form of the current three-point functions in 2d CFT [14,20].
In this paper, we overcome this problem and present the first computation of threepoint scattering amplitudes of massless particles in the Lorentzian (− + ++) signature. The computation relies on an important observation that solutions to the total momentum conservation of three massless particles in Minkowski space do exist in two special regions, and give nontrivial contributions to the massless three-point celestial amplitudes. They are the soft region where one of the particles has zero energy, and the colinear region where the momenta of all the incoming and outgoing particles are colinear. The three-point celestial amplitudes with only massless scalars receive contributions from both limits regions (see Section 3). The three-point celestial amplitudes involving massless spinning particles vanish in the colinear region, due to the tensor structure in the cubic vertex. Non-zero amplitudes in the (conformally) soft region are extracted using the logarithmic conformal primary wave functions obtained in [9] (see Section 4 and 5).
The results are best presented when the incoming (outgoing) particles are in the (shadow) conformal primary basis. In this prescription, scattering amplitudes are given by the shadow products between the incoming and outgoing states, which correspond to the Belavin-Polyakov-Zamolodchikov (BPZ) inner products in the CCFT [27]. The celestial amplitudes in this prescription are summarized in Table 1. We can see that all of them take the standard form of the correlators of primary operators in 2d CFT. In particular, the soft photon and gluon correspond to the spin-1 conserved currents, and the soft graviton corresponds to a primary operator of dimension ( 3 2 , − 1 2 ), whose level-1 descendent is the supertranslation current [9]. particles celestial amplitudes operator dimensions scalar-photon-scalar scalar-graviton-scalar A 1 0 →2 −− soft 3 0 ∝z 12 z 2 13 z 12 z 2 23 1 three gluons Table 1: The celestial amplitudes of two massless scalars with a photon or a graviton, and the three-point celestial gluon amplitudes. The logarithmic conformal primary wave functions were used for the soft (second) particles. The conformal dimensions of the corresponding CCFT operators are given in the last column.
The remainder of this paper is organized as follows. Section 2 reviews the celestial amplitude and the conformal primary basis of low spin, gives a general formula for massless and massive spin-conformal primary basis, and discusses the massless limit of the massive conformal primary basis. Section 3 computes the three-point celestial amplitudes of massless scalars. Section 4 computes the celestial amplitudes of two massless scalars with a photon or a graviton. Section 5 computes the three-point celestial gluon amplitudes. Section 6 ends with a summary and future directions.

Conformal primary basis
Celestial amplitudes are obtained by expanding the position space amplitudes with respect to the conformal primary wavefunctions [1,28] instead of the plane-waves, i.e., 1 where M(X j ) is the scattering amplitude in position space, and φ ± ∆,a (z; X), φ ± ∆,a (z; X) are the conformal primary and shadow conformal primary wave functions. ∆ and a are the conformal dimension and the spin of the wave functions, respectively. The coordinates (z,z) on the celestial sphere are related to the massless on-shell momenta q µ through q µ (ω, z) = ωq µ (z) = ω(1 + zz, z +z, −i(z −z), 1 − zz) (2.2) and to the massive on-shell momenta p µ through Here, we choose to expand the wave functions of the incoming particles in the conformal primary basis while the outgoing particles in the shadow conformal primary basis. This prescription is equivalent to expanding the wave functions of both incoming and outgoing particles in the same conformal primary basis but modifying the inner product in the definition of the S-matrix from the Klein-Gordon inner product to the shadow product [27]. Finally, note that while the helicity in four dimensions aligns with the spin on the celestial sphere for conformal primary basis φ ± ∆,a , the helicity and spin have opposite signs for the shadow conformal primary basis φ ± ∆,a due to the shadow transform.

Scalar
The conformal primary wave functions φ ± ∆ (z; X) for massless and massive scalars with mass m are given by and respectively. Here, G ∆ (z,z;p) is the bulk-to-boundary propagator which takes the form as in terms ofq andp in (2.2) and (2.3). 2 Starting with (2.4) and (2.5), one can obtain another set of conformal primary basis by performing the shadow transformations. As shown in [28], the shadow transformation of massive conformal primary basis (2.5) takes the same form as (2.5) up to a change of conformal dimension from ∆ to 2 − ∆. On the other hand, the shadow transformation of massless conformal primary basis (2.4) is The integral over d 2 z can be computed, leading to the following relation [28] φ ± ∆ (z; X) =

Massless spin-one
The conformal primary basis for massless spin-one particles is [9,28] A ∆,± µa (q; X) = (2.9) The corresponding shadow conformal primary wave function is related to A ∆,± µa by leading to From (2.9), it is easy to check that A ∆,± µa (q; X) is gauge equivalent to the Mellin transformation of the plane-wave basis up to a constant when ∆ = 1, i.e.
On the other hand, in the conformally soft region ∆ = 1, both A ∆,± µa and A ∆,± µa reduce to pure gauge and lead to vanishing celestial amplitude due to the gauge invariance. The conformal primary wave functions that are not pure gauge can be constructed from the combination of A ∆,± µa and A ∆,± µa : With the help of (2.10), the log mode A log,± µa can be further written as (2.14) In the later computations, we will also use the wave functions in momentum space. Since A ∆,± µa satisfies the massless Klein-Gordon equation A ∆,± µa = 0, it admits the Fourier expansion in the on-shell momentum space as where the integral is over all null momenta q µ . The Lorentz gauge condition ∂ µ A ∆,± µa (q; X) = 0 in the momentum space becomes the transverse condition q µ A ∆,± µa (q; q ) = 0. Similarly, one has the Fourier modes α ∆,± a (q; q ), V ∆,± µa (q; q ), A ∆,± µa (q; q ) and A log,± µa (q; q ), which are functions of the on-shell momentum q µ , and the latter three satisfy the transverse condition. However, V log,± µa and α log,± a do not admit an expansion in the on-shell momentum space like (2.15), because they do not satisfy the massless Klein-Gordon equation. After defining the Fourier modes, (2.9) can be translated into the momentum space, giving

Massless spin-two
The conformal primary basis for massless spin-two particles has been studied in [9,28]. It takes the form as where the terms on the second line are total derivatives. The corresponding shadow conformal primary wave function is related to h ∆,± µνzz by We note that when ∆ = 0 and ∆ = 1, h ∆ µνzz (q, X) is proportional to the Mellin transformation of the plane-wave basis up to pure diffeomorphism: In the conformally soft limit ∆ → 1, the conformal primary wave function (2.18) and its shadow (2.19) coincide and both reduce to pure diffeomorphism. In this limit, a new conformal primary wave function h log,± µνzz , which is not pure diffeomorphism, can be constructed from Armed with (2.18) and (2.19), h log,± µνzz can be re-written as .
(2.23) and α log,± µzz is just some function ofq and X.

General spin
In this subsection, we give a general formula for the massless and massive conformal primary basis of arbitrary integer spin, which is derived from a covariant formalism introduced in Appendix A. Under specialization, one can get various conformal primary basis including those reviewed in the previous subsections, and the massive spin-conformal primary basis obtained [29].
The spin-massless conformal primary basis is 3 24) where N ∆, is a normalization constant 25) and P I is the spin-projection operator of SO(2) (2.26) In Appendix A.1, we show that the specialization of (2.26) to = 0 and 1 are proportional to the shadow conformal primary basis φ ± ∆ and A ∆,± µa as (2.27) The spin-massive conformal primary basis is (2.28) where J = − , − + 1, · · · , is the eigenvalue of the Cartan generator of the SO(3) little group and corresponds to the spin on the celestial sphere. The tensor structures P P and P Q are given by (2.26) with I replaced by the projectors (2.29) For the J = = 0 case, (2.28) coincides with (2.5) up to a constant factor (2.30)

Massless limit
One advantage of (2.28) is that it enjoys a nice behaviour under massless limit. Particularly, (2.28) with |J| = directly reduces to the corresponding massless spin-conformal primary basis in the massless limit m → 0. To see this, we note that setting J = in (2.28) leads to Using the transversality condition (A.24) and the fact that P Q is traceless with respect to the metric g, we can remove the projection operator P P and get (2.32) and where we used the fact that which produces the massless scalar conformal primary wave function (2.4). This formula follows directly from the limit which holds when Re(∆) ≥ 1 and can be proved as follows. We note that the massless limit m → 0 in (2.38) vanishes unless z = z . Thus the left-hand side of (2.38) must be proportional to δ (2) (z − z ). 4 The proportional constant can be fixed by noting that which proves (2.38).
The two massless limits for the scalar wave functions can be summarized as (2.41)

Three-point celestial amplitudes of massless scalars
In this section, we will compute the three-point celestial amplitudes involving three massless scalars. We will use the shadow basis (2.7) for outgoing particles and conformal primary basis (2.4) for incoming particles. The computations of the one-to-two and the two-toone amplitudes are in Section 3.1 and 3.2, respectively. For each amplitude, we give two computations that give the same results. The first computation is by directly performing the Mellin transform and shadow transform on the scattering amplitude in the plane wave basis, and the second computation is by taking the massless limit of the known celestial or shadow celestial amplitudes of one massive and two massless scalars.

Two-to-one amplitude
In this subsection, we consider the 1 0 2 0 → 3 0 scattering. Here We use the superscript 0 to indicate that the corresponding particle is a scalar.
Direct computation in momentum space The corresponding celestial amplitude A ∆ i 1 0 2 0 →3 0 is given by where we have used (2.4), (2.24), and (2.27) in the second equality. By noting the identity and using the delta-function δ (4) (q 1 + q 2 − q 3 ), we can compute the integral over d 4 q 3 leads to The delta-function δ(−2ω 1 ω 2q1 ·q 2 ) can be rewritten as We note that the first two terms in (3.4) correspond to the soft region and the last term corresponds to the colinear region. Using (3.4) and evaluating the integral over ω 1 and ω 2 , we get and We mention here that the soft part A soft,∆ i 1 0 2 0 →3 0 (and the celestial amplitude A ∆ i 1 0 2 0 →3 0 ) converge only when ∆ 1 ≥ 1 and ∆ 2 ≥ 1.
Computation using massless limit (3.5), (3.6) and (3.7) can also be derived by taking the massless limit of the celestial amplitudes A ∆ i ,m 1 0 2 0 →3 0 which involve two incoming massless scalars and one outgoing massive scalar with mass m. Specifically, according to (2.27), (2.30) and (2.35), we have We stress here that to get finite must hold. 5 Furthermore, to have well-defined massless limit, we need to assume Re(a) ≥ 0. Then taking the massless limit forces −iν ≡ a = 0 with ν ∈ R.
For generic ∆ i s which satisfy (3.11), we can use 5 These two conditions have to be satisfied such that the integral over p 3 is finite in the computation of where on the first equality we expanded (|z 12 (3.14) Substituting (3.12) into (3.9) then produces the colinear part A colinear,∆ i On the other hand, for ∆ 1 = 1 or ∆ 2 = 1, we use the following formula which agrees with the soft part A soft,∆ i 1 0 2 0 →3 0 (3.6). The massless limit of A ∆ i ,m 1 0 2 0 →3 0 then can be obtained by adding the contribution from generic ∆ i s and the contribution from ∆ 1 = 1 and

One-to-two amplitude
In this subsection, we consider the 3 0 → 1 0 2 0 scattering with three massless scalars.
Direct computaion in the momentum space The corresponding celestial amplitude A ∆ i 3 0 →1 0 2 0 is given by where we have used (2.4), (2.24), and (2.27) in the second equality. Defining p ≡ q 1 +q 2 ≡ Mp with M ≥ 0 andp 2 = −1 and changing integral variables then lead to [31] This follows from the distributional formula which is understood as follows. The LHS, when integrated against a test function, has poles on the complex a-plane at a = −n for n ∈ Z ≥0 with residues given by the residue of the RHS integrated against the same test function. More detailed discussions can be found in Appendix B of [30] and references within.
where we defined Y µ ≡ 2(−q 1 ·p)p µ −q µ 1 . Using the following expansion (3.20) Defining λ, y, z, andz through leading to Rewriting the delta-function as and evaluating the remaining integrals leads to Computation using massless limit (3.24) can also be derived by taking the massless limit of the celestial amplitudes A ∆ i ,m 1→2 which involve two outgoing massless particles and one incoming massive particle with mass m. Specifically, according to (2.37), we have

Scalar-photon-scalar amplitude
Let us first compute the celestial amplitude A ∆ i 1 0 →2 − 3 0 of one incoming massless scalar, one outgoing photon, and one outgoing massless scalar. 7 Using (2.4), (2.12), (2.7) and (4.1), we find that the celestial amplitude where we used the momentum conservation to rewrite q µ 1 + q µ 3 as 2q µ 1 − q µ 2 and dropped q µ 2 since ∂ z 2q 2 ·q 2 = 0. As we have seen in (3.4), the supports of momentum conservation contain the soft regions and the colinear region: However, since the colinear region demandsq 1 =q 2 =q 3 and the conformal primary basis is transverse with respect to the momentum, the colinear region has no contribution to the celestial amplitude. Moreover, we note that the soft region ω 3 = 0 implies q µ 1 = q µ 2 , which leads to vanishing celestial amplitude. Thus, only the soft region ω 2 = 0 has non-vanishing contribution to the celestial amplitude A ∆ i 1 0 →2 + 3 0 and we can write the delta-function as This leads to (4.6) To have convergent ω 2 -integral, we have to assume Re(∆ 2 ) ≥ 1. Then the ω 2 -integral forces ∆ 2 = 1 which vanishes due to the prefactor ∆ 2 −1. Thus we conclude that A ∆ i 1 0 →2 − 3 0 vanishes. The vanishing of A ∆ i 1 0 →2 − 3 0 is due to the soft mode of photon with ∆ 2 = 1. As we discussed in Section 2, instead of using (2.9), we should use (2.14) as the spinone conformal primary basis for the soft mode with ∆ 2 = 1. In other words, the celestial Minkowski space. On the other hand, for outgoing particles, the helicity in two-dimensional CCFT get flipped comparing with the helicity in four-dimensional Minkowski space. This is because that the shadow transformation flips the helicity. 8 As we will see later, only the soft mode with ∆ 2 = 1 has contribution. Since A ∆2,± µz = A ∆2,± µz when ∆ 2 = 1, we can use the conformal primary basis A ∆2,± µz although photon is outgoing. (4.7) Here A log,− µz (q 2 ; q 2 ) is A log,− µz (q 2 ; X) (2.14) in momentum space, defined similar to (2.15). Rewriting the delta-function δ (4) (q 1 − q 2 − q 3 ) as Computing the integral over dω 1 , d 3 q 2 and d 3 q 3 , we get the integral representation of where we used (2.14), and use the abbreviations ∂ µ ≡ ∂ ∂X µ , A log,− 2,µz ≡ A log,− µz (q 2 ; X) and similarly for other fields. Using the fact that ∂ 2 φ = ∂ 2 φ = 0, we have This leads to (4.12) 9 In the rest of paper, we do not write (q; X) explicitly for conformal primary basis in position space. For example, φ ± ∆ is used to denote φ ± ∆ (q; X).
Using (2.4), (2.8) and (2.14), A ∆ i 1→2,z can be written as (4.13) We recognize that the X-integral is just the celestial amplitude A ∆ 3 ∆ 2 ∆ 1 1 0 →2 0 3 0 which can be obtained from (3.24) by switching 1 and 3, i.e., (4.14) Now we reach a subtlety. Substituting the expression (3.24) into the above equality, we will get something that does not transform correctly under the conformal symmetry. This is because (3.24) contains the delta-function δ(∆ 1 + ∆ 2 − ∆ 3 ) that is singular and needs to be regularized while preserving the conformal symmetry. One resolution is to turn on a small mass m for the incoming particle, which smooths out the delta-function while keeping the conformal symmetry manifestly. We would take the massless limit m → 0 at the end of the computation. In other words, we have where we used (3.25) and A is obtained from (3.26) by switching 1 and 3. Substituting (3.26) and (3.27) into the above equality and using (3.15)

leads to
which has the standard form of scalar-current-scalar three-point functions in 2d CFT.
We stress here that to get (4.16), we introduced a mass regulator. In Appendix B, we give another way to compute (4.16) by performing the shadow transformation on the shadow celestial amplitude A ∆ i

Scalar-graviton-scalar amplitude
Let us compute the celestial amplitude A ∆ i 1 0 →2 −− 3 0 of one incoming massless scalar, one outgoing graviton, and one outgoing massless scalar. Like the case of scalar-photon-scalar, only the soft region ω 2 = 0 contributes to the celestial amplitude A ∆ i 1 0 →2 −− 3 0 . Again the deltafunction δ(ω 2 ) forces ∆ 2 = 1 which implies that we should use h log,− 2,µνzz in (2.21) to compute Following the computation in Section 4.1, we get Since φ + 1,∆ 1 and φ − 3,∆ 3 satisfy the massless Klein-Gordon equation, the term including ∂ (µ α log,− 2,ν)zz vnishes after integrating by parts, leading to Following the computations in Section 4.1 with the mass regulator, we finally get that which is a three-point function of primary operators with conformal weights ( . Following [9], we can further define an operator P z with conformal weight ( 3 2 , 1 2 ), which was referred to as the supertranslation current [32,33]. By acting ∂z 2 on (4.20), we obtain the three-point function (4.21) Thus the OPE between P z and O + ∆ 1 (z 1 ) takes the form as where C ∆ 1 +1,∆ 3 is the coefficient of the two-point celestial amplitude of massless scalars, which contains a delta function δ(∆ 3 − ∆ 1 − 1) that cancels the delta function in the numerator. (4.22) agrees with the result in [9].
Finally, by a computation similar to the one in Section 3, one may alternatively obtain the scalar-photon-scalar and the scalar-graviton-scalar celestial amplitudes by taking the massless limit of the massive spinning particle amplitudes in [34]. We leave this to future work.

Three-point celestial gluon amplitudes
In this section, we first compute the celestial amplitude A ∆ i 1 a →2 b 3 c of three gluons with helicities (a, b, c) in Minkowski space. Here, we omit the color indices. After that, we compare our results with the existing three-gluon celestial amplitude in the Klein space [20].

Computation of
We start with the corresponding scattering amplitude M µνρ which takes the form as 10 Since z i andz i are not independent in Minkowski space, the colinear region dose not contribute to the celestial amplitude A ∆ i 1 a →2 b 3 c . However, we still have contribution from the soft region. Indeed, we can write the delta-function for momentum conservation as Following the steps in Section 4.1, it is easy to check that the delta-function δ(ω i ) forces ∆ i = 1 with i = 2, 3. In the rest of this section, we will focus on the celestial amplitude In other words, we have where A ∆,± µa (q; q ), A ∆,± µa (q; q ), and A log,± µa (q; q ) are the Fourier transforms of (2.9), (2.11), and (2.14). Equipped with the momentum conservation and the fact that A ∆ i ,+ µz (q i ; q i ) and Following the steps in Section 4.1, soft 3 c can be recast into the form as Integrating by parts and using the fact that ∂ ρ A ∆ 3 ,− 3,ρc = ∂ µ A ∆ 1 ,+ 1,µa = 0, one can show that the pure gauge in A log,− 2,νb does not contribute to A ∆ 2 =1 1 a →2 b 3 c . This leads to Since V log,− 2,νb ∝ X ν and X µ A ∆ 3 ,− 3,µc = X µ A ∆ 1 ,+ 1,µa = 0, we find that leading to Using (2.9), (2.11) and (2.14) we get (5.14) The support of delta-function δ (4) (q 1 − q 2 − q 3 ) contains soft region and colinear region in both Minkowski space and Klein space. The colinear region demands that In Minkowski space, (5.15) implies z 1 = z 2 = z 3 andz 1 =z 2 =z 3 , leading to vanishing M −−+ . However, since z i andz i are independent real variables in Klein space, (5.15) implies z 1 = z 2 = z 3 orz 1 =z 2 =z 3 . Indeed, the momentum conservation in Klein space has support at 11z

Discussions
In this paper, we gave the first computation of the celestial amplitudes of three massless particles in Minkowski space. We showed that due to the existence of soft and colinear regions in the support of the delta-function for momentum conservation, the celestial amplitudes of three massless particles in Minkowski space do not vanish and take the standard form of correlation functions in CFT. Focusing on the specific amplitudes of scalar-scalar-scalar, scalar-photon-scalar, scalar-graviton-scalar, and gluon-gluon-gluon, we found that the massless three-point celestial amplitudes of scalars receive contributions from both the soft and colinear regions, while the massless three-point celestial amplitudes of gluon and graviton receive contribution only from the soft region. Moreover, by looking at the celestial amplitudes of scalar-photon-scalar and gluon-gluon-gluon, we found that the scattering amplitudes involving a soft spin-one particle are mapped to be conformal correlators involving a spin-one current on the celestial sphere. These two examples directly confirmed the relation between soft spin-one particles in Minkowski space and conserved currents on the celestial sphere at the level of three-point amplitudes. In addition, we also found that the soft graviton with positive helicity in the scalar-graviton-scalar scattering is mapped to be a primary operator with conformal weight ( 3 2 , − 1 2 ). By further taking az-derivative on this primary operator, we obtained the supertranslation current. We derived the OPE between the supertranslation current and scalar primaries and found the OPE matches with the one found in [9].
Several interesting open questions ensue from our work. First, it would be interesting to extend our analysis to higher-point celestial amplitudes to further examine the relation between soft spinning particles and conserved currents. Similar to the three-point case, we expect our prescription (2.1) for the celestial amplitudes and the logarithmic conformal primary wave functions (2.13) and (2.21) should play an important role in the higher-point case. While only the soft region has contributed in the three-point celestial amplitudes with spinning particles, the higher-point celestial amplitudes receive contributions from other solutions to the momentum conservation. Thus, to study the conserved currents in higherpoint celestial amplitudes, one must extract their contributions from taking the (conformally) soft limit or projecting the wave function onto the logarithmic conformal primary wave functions.
Another avenue would be to explore how the stress tensor emerges from the soft region at the level of three-point celestial amplitudes. The celestial stress tensor is usually constructed as a shadow of the subleading conformally soft graviton [8,35,36]. The obstruction of this way in the three-point celestial amplitude is that the energy ω is strictly equal to zero. Then it is unclear how to obtain the subleading conformally soft graviton.
Finally, it would be of great interest to study the celestial amplitudes of three conserved spin-one currents as well as of three stress tensors. These two correlators are important as they encode the information of the level and the central charge. To get these two correlators, one must take double or triple (conformally) soft limits of celestial amplitudes. The double soft limit which is necessary to consider T T OPE was studied in [37]. However, there is an obstruction in [37] to reproducing the T T OPE by taking the double soft limit. 12 A possible resolution to this obstruction was proposed very recently in [39] for amplitudes in the Klein space, which used the ambidextrous basis [26,40]. Their basis involves the light transformation, and hence cannot be defined in Minkowski space, which has an Euclidean celestial sphere. It would be interesting to develop techniques in Minkowski space that allow us to take the double soft limit. Perhaps, one can naively use the shadow transformation to replace the light transformation. 13 where denotes the spin and the normalization factor N ∆, is defined in (2.25). Explanations are made here. µ, ν, . . . and a, b, . . . were used to denote indices in R 1,3 and R 2 , respectively. P was introduced to projects the embedding space operator O N ∆{µ} (q) into the position space conformal primary operator and Π(X, q ) {µ} {ν} was introduced to make the conformal primary basis satisfy the equation of motion. We also note that the projector Π(X, q) Next, we prove that (A.1) is conformally covariant. The conformal covariance of (A.1) is ensured by the insertion of the conformal two-point function O N ∆ {ν} (q )(PO N ) ∆{a} (q) . Indeed, under the Lorentz transformation X → ΛX andq →q , we have 15 Changing q to Λq and using the fact that (Λq ) · (ΛX) = q · X as well asq = Λq leads to we then get Since (PO ∆ )(q) projects O ∆ (q) into a conformal primary operator in two-dimensional space, the following identity must hold 14 The embedding space formalism used in this paper can be found in [42][43][44] and the references therein. There is a different CFT embedding space formalism which was studied in [45,46]. 15 To simplify the notation, we omit the labeling N in the rest of this paper.
where D(Λ) {a} {b} is the matrix element implementing the action of Lorentz group SO (2).
This leads to which verify the transformation property.
Let us focus on massless spin-particles. The conformal primary wave function (A.1) should satisfy the Fronsdal equation, After imposing the Lorentz gauge ∂ ν Φ ∆,±

Relation to shadow conformal primary basis
In this appendix, we will specialize the formula (2.24) to the = 0 and 1, and show that they are proportional to the shadow conformal primary basis φ ± ∆ and A ∆,± µa , respectively. First, for = 0, it is straightforward to see that where φ ± ∆ (z; X) is the shadow conformal primary wave function (2.8). Next, for = 1, we have where we defined D 2q ≡ d 2 z . The term containing η µν is proportional to the scalar shadow conformal primary basis (A.14) To compute the remaining part, we note that the Lorentz invariance and scaling behaviour fix that To determine the proportional constant N , we contract both sides with −q ν , giving

Mellin basis and light ray basis
We note that besides the standard power-law form, the scalar two-point conformal correlations can also be a delta-function when ∆ + ∆ = 2, i.e., which is exactly the massless scalar conformal primary wave function given in (2.4).
Before going to spinning conformal primary basis, we mention that the formula (A.1) can also be applied to Klein space. In Klein space, since z andz become two independent real variables, the correlation functions in two-dimensional Lorentzian CFT can be written as a product of correlation functions in two one-dimensional CFTs. In this case, we can choose one of the two one-dimensional correlation functions to be one-dimensional delta function and another to be the power-law two-point function: ( Here we embed holomorphic coordinates z and anti-holomorphic coordinatesz into the onshell momenta q µ through q µ = ωq µ = ω(1 + zz, z +z, z −z, 1 − zz) .

A.2 Massive conformal primary basis
With the help of spinning AdS bulk-to-boundary propogator, an index-free expression for massive spin-conformal primary basis has been obtained in [29]. In this subsection, we will using a different to construct the massive spin-conformal primary basis which satisfies the massive Klein-Gordon equation and transforms covariantly under SL(2, C). Our conformal primary basis reproduces the results in [29].
To construct the massive spin-conformal primary basis, we first note that massive spinbosons are in the spin-representation of the little group SO(3). This representaion can be labeled by two parameters and J with J = − , − + 1, · · · , − 1, which are eigenvalues of the spin operatorsŜ 2 andŜ z . On the other hand, the spin-operator of SO(2) only has two components. To match the degree of freedoms, a massive spin-operator in Minkowski space should be mapped into a set of spin-s operator O s on the celestial sphere with |s| = |J|.
To implement this map, we introduce the following two metrics built from the on-shell momenta p and celestial coordinatesq: (A.23) By construction, P µν and Q µν are symmetric with respect to µ and ν and satisfy the following transversality conditionp µ P µν =p µ Q µν =q µ Q µν = 0 . (A.24) Using the transversality condition, it is easy to check that both P µν and Q µν are idempotent, i.e., P · P = P , Q · Q = Q. (A. 25) Combining the idempotent with the fact that Tr(P ) = 3 and Tr(Q) = 2, we conclude that P µν is a projector projecting a four-dimensional Lorentzian space into a three-dimensional subspace that is orthogonal top µ , while Q µν is a projector projecting a four-dimensional Lorentzian space into a two-dimensional subspace that is orthogonal to bothp µ andq µ . Since P and Q are metrics in three-and two-dimensional subspaces inside four-dimensional Lorentzian space, we can construct two spin-projection operator P P and P Q of subgroups SO(3) and SO(2) inside SO(1, 3) through simply replacing the metric I in P I by P and Q, respectively.
Armed with P P and P Q , the massive spin-conformal primary basis Φ ∆, ,J,± {µ }{a | J|};m (q; X) takes the form as (2.28) in Section 2.4. The explanation is made here. First, we insert the projection operator P P to get spin-representation of the little group SO (3). After that, we contract with − |J|qs to get the representation with |Ŝ z | = |J|. Finally we contract with P |J| Q · (∂ zq ) |J| to get the spin-|J| representation of SO(2) on the celestial sphere. Since every step is conformally covariant, the conformal covariance of (2.28) is manifested.

B Shadow celestial amplitude
In this appendix, we compute the shadow celestial amplitudes A ∆ i 3 0 →1 0 2 0 of three massless scalars. After that, We perform the shadow transformations on A ∆ i 3 0 →1 0 2 0 to reproduce (3.24). Moreover, armed with A ∆ i 3 0 →1 0 2 0 , we also compute the shadow celestial amplitude A ∆ i z 1−ā 3 We can obtain A ∆ i Substituting (B.10) into the above equality and restricting the prefactors to the support at ∆ 3 = ∆ 1 then reproduces (4.16).