Poincaré constraints on celestial amplitudes

The functional structure of celestial amplitudes as constrained by Poincare symmetry is investigated in 2, 3, and 4-point cases for massless external particles of various spin, as well as massive external scalars. Functional constraints and recurrence relations are found (akin to the findings in [24]) that must be obeyed by the respective permissible correlator structures and function coefficients. In specific three-point cases involving massive scalars the resulting recurrence relations can be solved, e.g., reproducing purely from symmetry a three-point function coefficient known in the literature. Additionally, as a byproduct of the analysis, the three-point function coefficient for gluons in Minkowski signature is obtained from an amplitude map to the celestial sphere.


Introduction
Recent interest in flat space holography has been enhanced by the concrete proposal by Pasterski, Shao and Strominger (PSS) in [1] to map flat space plane wave field solutions to conformal primary wave functions on the celestial sphere via an integral transform, allowing to express four-dimensional scattering amplitudes in terms of two-dimensional correlator like objects (celestial amplitudes). The singularity structure of higher-point celestial amplitudes was considered in [2]. The basis of conformal primary wave functions for massless particles of spins zero, one, and two was derived in [3], establishing the map to the celestial sphere in these cases, after gauge fixing, to be given by Mellin transform. Following the PSS prescription, explicit examples of amplitudes were mapped to the celestial sphere for scalar scattering [1,5,6,17], gluon scattering [4,8], and stringy/graviton scattering [9,19,20]. Modification of the PSS prescription, which makes the action of space-time translation simpler, has been proposed and investigated in [7,10,11]. Conformally soft behavior of operators on the celestial sphere was considered in [13][14][15][16][17][18][19][20][21]. Conformal partial wave decomposition of some four-point amplitudes on the celestial sphere was discussed in [5,17]. The representation of Poincaré symmetry generators for massless particles on the celestial sphere was presented in [12]. BMS symmetry in the language of flat space holography was recently considered in [22]. And the OPE of the energy-momentum tensor (see [23,24]) with gauge boson operators on the celestial sphere was obtained in [25], demonstrating that gauge boson operators in fact transform as Virasoro primaries.
Since the integral transform of amplitudes to the celestial sphere is in general hard to perform, only a limited number of examples has been calculated so far. In this work we take a parallel approach and instead consider constraints imposed on generic celestial amplitudes by the underlying Poincaré symmetry. Our main motivations are two-fold: • First, we are looking to constrain the general non-perturbative structure of celestial amplitudes in order to aid future calculations of amplitudes on the celestial sphere.
• Second, we are interested in determining which features of the currently known celestial amplitude examples stem from the particular types of particles and theories under consideration, and which features are more generally to be attributed to the overarching symmetry.
While in Minkowski space the manifestation of Poincaré symmetry in amplitudes may naively be summarized by the facts that all amplitudes must be proportional to momentum conservation delta functions and all Lorentz indices must be contracted, on the celestial sphere the symmetry is realized more non-trivially and imposes functional constraints. To constrain the amplitude structures, we employ the condition that celestial amplitudes must be annihilated by all Poincaré symmetry generators (Ward identities). Parametrization of Poincaré symmetry generators for massless particles is provided in [12], while we additionally derive momentum generator parametrization for massive scalars (3.2), which allows us to investigate amplitudes with massless external particles of various spin, as well as massive external scalars. The results we discover are as follows.
In case when all external particles are massless, two-, three-and four-point structures are required by Poincaré symmetry to be distribution-valued. In fact, the two-and threepoint structures are then constrained to vanish.
Nevertheless, in the three-point case (4.19) Poincaré symmetry implies that a nontrivial three-point function coefficient must exist, which must satisfy the recurrence relation (4.25) under shifts in conformal dimensions. For instance, the gluon three-point function coefficient was previously inferred from OPE considerations to involve the Euler beta function [25]. As a byproduct of our investigation we calculate the gluon three-point function coefficient directly from an amplitude map to the celestial sphere (A.9) and confirm that the beta function recurrence identity in this context is a consequence of the global translation invariance constraint (4.25), as mentioned in [26].
Perhaps the most interesting constraint we find, is the fact that a massless four-point amplitude on the celestial sphere can always be written in terms of a function of conformal cross ratio which may only depend on the overall sum of all conformal weights of the external particles, with appropriate fully determined pre-factors, and the conformal cross ratio constrained to be real (4.36).
In case when massive external scalars are present, the Lorentz subgroup constrains the two-, three-and four-point structures to have the same familiar form as usual CFT correla-tors. Momentum generators then impose further constraints on the coefficients depending on conformal weights, as well as on the function of cross-ratio in the four-point case.
The two-point structure of two massive scalars is only non-vanishing when both masses are equal, and the two-point function coefficient is uniquely constrained to be given by (5.7). The two-point structure of a massless particle with a massive scalar is ruled out.
The three-point structure of two massless particles with a massive scalar only exists if the two massless particles have the same spin. The recurrence relations imposed on the three-point function coefficient by Poincaré symmetry can then be solved under mild assumptions (5.14), which e.g. reproduces the three-point function coefficient of two massless scalars with a massive scalar [5] purely from symmetry considerations.
The three-point function coefficient for a massless scalar with two massive scalars of different mass must obey the recurrence relations (5.16); while the three-point structure of a massless spinning particle with two massive scalars only exists if the massive scalars have the same mass, and the symmetry imposed recurrence relations again can be solved generally (5.23).
The three-point function coefficient for three massive scalars of different mass must obey recurrence relations (5.24).
In case of four-point structures with at least one massive scalar, the function of crossratios must satisfy second order differential equations, additionally subject to recurrence shifts in the conformal weights of external particles.
This work is organized as follows. In section 2 we recall the formalism for mapping amplitudes of massless and massive particles to the celestial sphere. In section 3 we list Poincaré symmetry generators and algebra acting on the celestial sphere. Section 4 details the derivation of Poincaré symmetry constraints on two-, three-and four-point massless correlator structures, while section 5 repeats that exercise for correlator structures also involving massive scalars. We offer some discussion in section 6. Finally, appendix A describes the calculation of the gluon three-point celestial amplitude starting with Minkowski signature in the bulk.

Formalism and conventions
We recall the formalism of mapping a massive scalar, or massless scalar, gluon or graviton plane wave solution to the celestial sphere.
In the case of the massive scalar, the map involves an integral transform over a hyperbolic slice H 3 of Minkowski space corresponding to the constant mass squared p 2 j = −m 2 j hyper surface for external particle momenta [1]. An H 3 slice in Poincaré coordinates y, z,z has the metric where a, b, c, d ∈ C and ad − bc =ād −bc = 1. The transformation parameters a, b, c, d are the same that enter the corresponding SL(2, C) Möbius transformations acting on complex coordinates w i ,w i on the celestial sphere. We also recall the embedding map for the (unit) momentum of a particle living on the hyperboloidp µ ∶ H 3 → R 1,3 to bê In [1] the plane wave basis of a mass m scalar (outgoing or incoming ±) was mapped to the conformal primary wave function basis with conformal dimension ∆ on the celestial sphere (in the continuous series representation ∆ = 1 + iλ with λ ∈ R) via the integral transform where the terms to the power ∆ are the scalar bulk to boundary propagator on H 3 . The bulk to boundary propagator can be written as (−2q ⋅p) −∆ , while q µ j is a null direction pointing to the celestial sphere. The map of a Minkowski space amplitude of massive scalars A n to the celestial sphere is then given by (2.7) Similarly, in [3] making use of a specific gauge, it was shown that amplitudes A n of massless particles of spin zero, one, and two are mapped to the celestial sphere via the Mellin transform where incoming or outgoing massless particle momenta are p µ j = ±ω j q µ j . Prescriptions (2.7) and (2.8) are to be mixed appropriately, depending on the spin and mass of the respective external particles in amplitude A n .

Poincaré generators and algebra
Translation generators (momenta) for massless particles act on massless states on the celestial sphere as follows [12]: where ∂ x = ∂ ∂x , and h j = are the holomorphic and anti-holomorphic conformal weights of the respective particle j on the celestial sphere (with spin J j ).
Analogously to the massless case (3.1), we provide a representation of massive momentum generators dependent only on celestial sphere coordinates w j ,w j , and conformal dimensions. As can be directly verified, the following operator representation indeed has the correct momentum component eigenvalues m jp µ j when acting on the massive scalar conformal primary wave function (2.5): with m j being the mass of the j-th particle. Lorentz generators M µν j = −M νµ j acting on particle j are given by (again consistent with [12]) where ∂ j = ∂ ∂w j and∂ j = ∂ ∂w j . It is straightforward to verify explicitly that the M µν j satisfy the Lorentz algebra where η µν = diag(−1, 1, 1, 1), and adding P µ j the Poincaré algebra is completed by It is also straightforward to check explicitly that the operators (3.1) or (3.2) correctly close the Poincaré algebra (3.8), (3.9). However, in the massive case (3.2) this is only true if the conformal weights have no spin h j =h j = ∆ j 2 . This demonstrates that the massive momentum generator representation on the celestial sphere is spin dependent.

Poincaré constraints on massless 2,3 and 4-point structures
Poincaré symmetry implies that all algebra generators have to annihilate the n-point amplitude structures on the celestial sphere by the Ward identity constraints: where j = ±1 depending on whether particle j is outgoing or incoming.
In the following we consider the cases n = 2, 3, 4, for which the symmetry produces special constraints.
It turns out that the action of the massless momentum generators forces the n = 2, 3, 4 celestial amplitudes to be distribution-valued. To see that, note that the equation ∑ j j P µ j A = 0 can be understood as a linear set of equations for n unknowns X j ≡ j exp( 1 2 ∂ h j + 1 2 ∂h j )A, which in matrix notation can be written as As is well known, such a linear set of equations can have non-trivial solutions for n ≤ 4 only if the 4 × n matrix Q has appropriately reduced rank. As a consequence, determinants of all maximal minors of matrix Q must vanish. Such constraints reduce the regions of values the coordinates w j ,w j can take on, which will be parametrized by Dirac delta distributions.

Two-point structure
Minor determinant constraints It is easy to check that the determinants of all six 2 × 2 minors of matrix Q = (q µ 1 q µ 2 ) vanish only if the two points on the celestial sphere coincide This is in line with momentum conservation expectation in Minkowski space, stating that ingoing and outgoing states in a two point process should be collinear. Therefore, any valid two-point structure A 2 on the celestial sphere must be proportional to a product of delta functions imposing the above constraints: 1 where g is so far a generic function of its arguments. 1 Since the delta function arguments are complex conjugates of each other, they are to be understood as setting the real and the imaginary part to zero, which may be more transparent in the alternative linear combination of arguments:

Lorentz invariance constraints
Next we impose the Lorentz symmetry constraints. As one part of Lorentz invariance, the function g should be invariant under global 2D translations on the celestial sphere. Therefore, we conclude that g actually depends on differences w 1 − w 2 andw 1 −w 2 only, so that we take the general ansatz is a two-point function coefficient. On the celestial sphere, the annihilation of a correlation function of primary fields by symmetry generators in the context of Ward identities is equivalent to the condition that the correlator must have the correct transformation weight under Möbius transformation (2.3). Under this transformation the two-point structure A 2 is expected to transform as It is straightforward to see that this transformation weight can be realized by our ansatz A 2 only if we demand so that analogously to the usual CFT case [28] the w i ,w i dependence is completely fixed by symmetry, and A 2 becomes where we define a finite valued delta function to ensure the conformal weight equality h 1 = h 2 ,h 1 =h 2 . As a cross-check we verify that Lorentz generators annihilate the two-point structure as well: ∑ i M µν i A 2 = 0. Since we have established that A 2 is a distribution, it is clear that derivatives hitting the delta functions will have to be evaluated in a distributional sense . To arrive at a correct result after such partial integration, it is important to choose a proper normalization for the operator ∑ i M µν i . To that end, we define the action of the operator on A 2 to be where we divide out the continuous dependence on w 1 , w 2 that appears in A 2 , so that the equation isolates only the eigenvalue of ∑ i M µν i next to the delta functions. Resolving delta function derivatives in a distributional sense as mentioned above: and similarly for δ ′ (w 1 −w 2 ), we confirm that the two-point structure is properly annihilated.

Solving momentum annihilation equations
The equations ∑ j j P µ j A 2 = 0 (with 1 = − 2 ) create two independent constraints for A 2 in (4.8), since each operator e (∂ h j +∂h j ) 2 shifts the arguments of the modified delta functions such that terms proportional toδ(h 12 ± 1 2 ) appear, which cannot be canceled against each other. The powers in w 12 andw 12 do not develop a positive real part overall, so that δ(w 12 )δ(w 12 ) does not reduce the result to zero, thus the only way to satisfy the annihilation constraint is to demand When a shift of the argument of a function makes it vanish and the function domain is unbounded, it means that the function itself is zero. Therefore, we conclude that in general such that a massless two-point structure is ruled out by Poincaré symmetry.

Three-point structure
Demanding that determinants of all four 3 × 3 minors of matrix Q = (q µ 1 q µ 2 q µ 3 ) vanish, and fully reducing these constraints leads to four branches of solutions. The first branch demands the coincidence of all three complex points, e.g. (4.14) while three more branches demand the coincidence of only pairs of points From momentum conservation we have the intuition that once two out of three momenta become collinear, the third must become collinear as well; so that we pick the first branch to proceed. Therefore, we conclude that a generic massless three-point structure A 3 on the celestial sphere must be proportional to a product of delta functions imposing the first branch constraints, e.g. 2 where g for now is an arbitrary function of its arguments.
2 Once again, to make sense of the complex valued arguments of the delta functions, we may alternatively consider the real valued linear combinations of arguments

Lorentz invariance constraints
We proceed to impose the Lorentz symmetry constraints. As before, due to 2D translation invariance on the celestial sphere, the three-point structure may depend only on differences of coordinates w i − w j andw i −w j . This leads to the general ansatz with a three-point function coefficient C. Demanding the three-point structure to have the correct primary-field transformation weight under Möbius transformation fixes the powers p i ,p i uniquely analogously to the usual CFT case [28], so that we obtain the final result As a cross-check we verify that Lorentz generators annihilate the three-point structure: Since A 3 is a distribution as in the previous subsection, delta function derivatives will have to be evaluated in a distributional sense. To facilitate that, as previously we define the action of the operator ∑ i M µν i on A 3 to have the normalization which isolates the eigenvalue of ∑ i M µν i next to the delta functions. Pairs of delta functions have overlapping dependence on w 1 andw 1 in this case, 3 so that delta function derivatives are resolved in a distributional sense as follows and similarly for δ ′ (w 1 −w 2 )δ(w 1 −w 3 ) and δ(w 1 −w 2 )δ ′ (w 1 −w 3 ). Here the extracted differential operators are fixed such that each derivative has equal weight up to an overall sign. With this, Lorentz generators readily annihilate the three-point structure.

Solving momentum annihilation equations
On the support of the delta functions in A 3 the equations ∑ j j P µ j A 3 = 0 (where i = − j = − k = ±1 with i, j, k ∈ {1, 2, 3} all different) can be simplified to where we use the symbol ε = w ij =w ij = 0 to count formal powers of zero on the support of the delta functions. The action of the total momentum squared (∑ j j P j ) 2 A 3 = 0 leads to For all other structures the total momentum squared operator annihilation constraint is redundant, in this case however it is satisfied subtly differently. Note that for any values of spins of the three particles we have A 3 ∼ ε 1−i ∑ j λ j , so that (4.24) is automatically satisfied by ε = 0. On the other hand, in (4.23) the overall power of ε does not possess a positive real part, so that (despite the fact that A 3 itself vanishes) it leads to a non-trivial constraint on the three-point function coefficient: We will see that this constraint is properly satisfied e.g. by the gluon three-point function coefficient derived in appendix A.

Four-point structure
Maximal minor determinant constraints For n = 4 the matrix Q = (q µ 1 q µ 2 q µ 3 q µ 4 ) is square, so that its only maximal minor is Q itself. Demanding the vanishing of the determinant of Q and fully reducing this constraint leads to several branches of solutions such that different points become degenerate (w i = w j ). However, these collinear configurations describe special kinematics, while for generic nondegenerate kinematics there exists only exactly one solution that can be summarized by the constraint z =z, (4.26) where we have defined the cross-ratios Intuitively, this can be understood as the momentum conservation constraint forcing a fourth momentum to point to the same celestial circle the other three momenta happen to point to. 4 Therefore, we conclude that a generic massless four-point structure on the celestial sphere must be proportional to a delta function imposing the above constraint: where g at this point is an arbitrary function of its arguments.

Lorentz invariance constraints
Since the delta function only depends on conformal cross-ratios z,z, in this case all delta function derivatives cancel out of the action of Lorentz generators ∑ i M µν i . Thus the delta function becomes a spectator and the four-point structure can essentially be treated as a function instead of a distribution.
Considering that the Lorentz part of Poincaré invariance implies the same constraints as the usual conformal covariance of the four-point structure on the celestial sphere, a convenient way to write a generic conformally covariant four-point structure of primary fields is by using the conventional pre-factor (see e.g. [27]) so that , is a function of the cross ratios z,z. It is then straightforward to verify that the above structure indeed is properly annihilated by all Lorentz generators ∑ i M µν i (in this case the normalization is irrelevant). Since we are making use of the factor F n=4 in our parametrization, Möbius transformations lead to correct primary field transformation weights per definition.

Solving momentum annihilation equations
As a next step, we proceed to solve the equations ∑ j j P µ j A 4 = 0 to find further constraints on f h 1 ,h 2 ,h 3 ,h 4 h 1 ,h 2 ,h 3 ,h 4 (z,z). We recall the following cross ratio regions for different incoming and outgoing particle configurations, which always arise in 4-pt celestial amplitudes due to the presence of total momentum conservation delta functions in Minkowski space amplitudes regardless of any other amplitude features (see e.g. example in [4]): Additionally making use of the cross ratio relation z =z, the equations are solved by The only way to reconcile these three equations is to conclude 5 where f h 1 +h 2 +h 3 +h 4 h 1 +h 2 +h 3 +h 4 (z,z) is a new generic function that may now depend only on the sums of all conformal weights collectively. Note that despite the z =z constraint, we maintain a complex notation to emphasize the single-valuedness of the result.
Collecting everything together, Poincaré invariance ensures that the celestial four point structure for massless particles can always be written as where especially the momentum generator constraints have led to special features that are not familiar from usual CFT. All celestial amplitudes with four massless external particles must obey this pattern.

Poincaré constraints on 2,3 and 4-point structures with massive scalars
Since the operators (3.2) involve different types of derivatives and shifts, the action of the total momentum generator cannot be thought of as an overdetermined linear set of equations. Therefore, in the massive case there are no minor determinant constraints and all the n-point structures are regular functions instead of distributions. With this, the Lorentz symmetry imposes familiar conformal symmetry constraints on the celestial sphere, such that conformally covariant two-, three-and four-point structures can be written as [27,28] with the finite-valued delta functionδ(x) as defined in (4.9), and conformal cross ratios z,z defined in (4.27). We expect to find further constraints on coefficients C and the function of cross ratio f from the total momentum generator annihilation condition.

Two-point structure
The equations ∑ j j P µ j A m 1 ,m 2 2 = 0 for two massive scalar two-point structure, with momentum operators (3.2) and A 2 in (5.1) such that h j =h j = ∆ j 2 , lead to two independent 5 Up to redefinitions in terms of factors depending on ∑ i hi and ∑ ih i.
) respectively. On the support ofδ( ∆ 1 −∆ 2 −1 2 ), the vanishing condition reduces to where we abbreviate ∆ 2 = ∆, ∆ 1 = 1 + ∆. Similarly, on the support ofδ( ∆ 1 −∆ 2 +1 2 ), the vanishing condition reduces to where now we abbreviate ∆ 1 = ∆, ∆ 2 = 1 + ∆. It is clear that both resulting vanishing conditions can be reconciled only if we demand in which case the constraint is solved by with some ∆-independent constant c. As in the completely massless case, a two-point structure e.g., for m 1 = 0, m 2 = m > 0 does not exist since the two-point function coefficient is then constrained to vanish.

Three-point structure
In the three-point case we can consider configurations involving one, two, or three massive scalars. We study them in increasing order.
Two massless, one massive: To satisfy the equations ∑ j j P µ j A 0,0,m 3 = 0, we calculate the operator action, divide out the common dependence on w j ,w j in each term and demand that the coefficient of each resulting monomial in w j ,w j vanishes separately. With the third particle being a scalar of mass m, the vanishing of all coefficients can only be non-trivially satisfied when the two massless particles have the same spin and lead to the following conditions on the three-point function coefficient C: These constraints are properly satisfied e.g. by the three-point function coefficient for the case of two massless scalars and one massive scalar obtained in [5]. In fact, (5.9) is in general solved by (making use of Keeping in mind that kinematically the massive leg must always be outgoing while the two massless legs are incoming or vice versa 1 = 2 = − 3 , (5.10) and (5.9) then imply that the constants c 1 , c 2 must depend on sums of conformal weights Additionally, assuming that c 1 , c 2 depend on sums and differences of holomorphic and antiholomorphic conformal weights as an ansatz, (5.10) and (5.9) can be solved as well, with the full result being where we made use of 1 = 2 = − 3 to simplify, and c now are constants dependent on the spin J of particle 1 and 2. The three-point function coefficient for the case of two massless scalars and one massive scalar [5] is thus precisely found purely from symmetry in the case c 1 ≠ 0, c 2 = 0, J = 0, 1 = 2 = − 3 .
One massless, two massive: The equations ∑ j j P µ j A m 1 ,m 2 ,0 There are terms with shifts in different parameters in each of the three equations, which makes finding a general solution highly non-trivial. It is likely that possible solutions may involve hypergeometric functions.
On the other hand, in case when the massless particle has non-zero spin the overall constraints are given by (5.21) The first equation describes shifts in ∆ 1 only, so that a general solution can easily be found. Using this in the second equation specializes the general solution to account for ∆ 2 shifts. The third equation is then only satisfied if we demand 6 Finally, as before, we take an ansatz for the remaining unknowns to depend on combinations h 3 +h 3 , h 3 −h 3 , and obtain the overall general solution are arbitrary constants dependent on the spin of the third particle.
Zero massless, three massive: The equations ∑ j j P µ j A m 1 ,m 2 ,m 3 3 = 0, with all three particles being massive scalars of mass m 1 , m 2 , m 3 respectively, reduce to the following three constraints on the three-point function coefficient one equation for each triplet of indices (i, j, k) ∈ {(1, 2, 3), (2, 1, 3), (3, 1, 2)}. As in the case of one massless and two massive scalars, the above constraints are too complicated to be solved in general. However, they provide a set of conditions that necessarily have to be satisfied as a non-trivial cross-check, should the result be obtained by other means in the future.

Four-point structure
In the four-point case we can consider configurations involving one, two, three, or four massive scalars. We take a closer look at the first case and summarize the features of the remaining cases.
Three massless, one massive: The equations ∑ j j P µ j A 0,0,0,m 4 = 0, with the fourth particle being a scalar of mass m, lead to four different consistency conditions. One of the conditions involves only shifts in operator dimensions 7 while the remaining three conditions involve shifts in operator dimensions as well as derivatives with respect to the holomorphic and anti-holomorphic cross ratios (5.28) where for brevity we suppressed the cross ratio dependence on function f = f (z,z). Having a mix of recurrence and differential equations in two dimensions makes these constraints hard to solve. However, they may be a valuable aid to cross-check results when performing the map to the celestial sphere on explicit examples. Since only one massive particle is participating in the scattering process, we can arrange momentum conservation to create an additional constraint This constraint could be particularly useful, since it only involves shifts in conformal weights of the massless particles and no z,z derivatives.
Two massless, two massive and beyond: In four-point cases when two or more of the scattering particles are massive scalars, the equations ∑ j j P µ j A 4 = 0 always lead to four constraints involving shifts in conformal weights as well as derivatives with respect to z andz; one constraint for each component µ. This is conceptually different from the two-and three-point cases in which all coefficients of resulting monomials in w j ,w j had to vanish separately, and it was not clear apriori how many constraints will arise. Since the four constraints can be generated straightforwardly from the explicit action of momentum generators ∑ j j P µ j A 4 = 0 and are rather unwieldy, we omit them here.

Discussion
Considering that irreducible representations of the conformal group are labeled by conformal dimension ∆ and spin J, different values of ∆ and J represent different operators in a CFT. On the other hand, irreducible representations of the Poincaré group are labeled by mass m and spin J, so that e.g. the label ∆ in this case can be varied for the same particle and therefore should be understood as a coordinate, or an energy reading, instead of a quantum number. This suggests that a Poincaré invariant theory on the celestial sphere should be noticeably different from a CFT. At the level of correlator structures we can observe the biggest differences for massless two-, three-and four-point structures where the structures become distribution-valued, while massive structures more readily resemble CFT correlators with a few extra conditions on the constants and parameters involved.
Additionally, we note that shifts in conformal dimensions ∆, such as induced by the Poincaré momentum operator, take the dimension value off the complex line of continuous series representation ∆ = 1 + iλ with λ ∈ R. This breaks some of the CFT specific formalism that is employed in the literature. For instance, the formalism of conformal partial wave decomposition for four-point correlators [5,17] relies on the convergence and orthogonality of the inner product employed, as well as the hermiticity of conformal Casimirs with respect to that inner product -moving the conformal dimensions off the continuous series line generically breaks these properties. In future work it may be interesting to instead derive a relativistic partial wave decomposition on the celestial sphere.
With these remarks we aim to emphasize that the applicability of CFT specific techniques in a Poincaré invariant theory has limitations.
where the delta functions are now in line with (4.19).
The ω 3 integration sets ω 3 = ω 1 + ω 2 , while ω 1 , ω 2 integrations proceed as where B(x, y) is the Euler beta function. Overall, the celestial amplitude result amounts to Replacing each w i − w j on the support of the delta functions by a parameter ε to count the formal powers of zero, we observe a match between the current result and (4.19) if we take δ(∑ j λ j ) into account .
(A. 8) This implies that the three-point gluon celestial amplitude vanishes for ε = 0, as expected. However, with this we are still able to identify the corresponding three-point function coefficient for gluons Poincaré invariance demands that the three-point function coefficient satisfies the constraint (4.25), where here we have 1 = 2 = − 3 . In this case, the constraint boils down to the relation B(x, y) = B(x + 1, y) + B(x, y + 1), which indeed is an identity exactly satisfied by the Euler beta function. The calculation for the helicity flipped case A ++−