Celestial two-point functions and rectified dictionary

A naive celestial dictionary causes massless two-point functions to take the delta-function forms in the celestial conformal field theory (CCFT). We rectify the dictionary, involving the shadow transformation so that the two-point functions follow the standard power-law. In this new definition, we can smoothly take the massless limit of the massive dictionary. We also compute a three-point function using the new dictionary and discuss the OPE in CCFT.


Introduction
The holographic principle [1,2] states the duality between a quantum gravity in some spacetime and a non-gravitational theory in a lower-dimensional spacetime.Representative examples are the AdS/CFT correspondence [3,4,5], and also the dS/CFT correspondence [6,7].Recently, many researchers have sought to establish a holographic correspondence in an asymptotic flat spacetime, celestial holography [8,9,10,11,12,13,14]. Celestial holography is a conjecture about a correspondence between a quantum gravity in a four-dimensional asymptotically flat spacetime and a conformal field theory on a two-dimensional sphere (celestial sphere CS 2 ).This holography relies on the fact that the symmetry of a gravitational theory on asymptotically flat spacetime seems to contain the Virasoro symmetry [15,16].
The central object of the celestial holography is the relation between scattering amplitudes of a quantum theory in a four-dimensional asymptotically flat spacetime and correlation functions of a conformal field theory on the celestial sphere (celestial conformal field theory; CCFT), which is schematically expressed as follows: ( These operators O in CCFT are constructed from field operators ϕ(X) in four-dimensional spacetime so that O transform appropriately under the two-dimensional conformal transformations.This construction has been achieved by extracting components for a specific basis called the conformal primary wave function Φ ∆ (X; z) (see e.g.[13,17,18]) as O ∆ (z) := (Φ ∆ (X; z) , ϕ (X)) KG . (1.2) However, the proposed dictionary in the literature leads to a peculiar form for the two-point functions for massless fields.Indeed, the two-point functions take the following delta-functional behavior if we use the proposed dictionary: On the other hand, three-point functions take the standard forms determined by the conformal transformation (e.g.[18,21] ).It looks hard to reconcile these behaviors with the operator product expansion (OPE).
In this paper, we rectify the dictionary by adopting alternative conformal wave functions for massless incoming particles.This rectification is equivalent to interpreting operators constructed from the conventional massless conformal wave functions as the conformal shadow operators for incoming particles.In our dictionary, the celestial two-point functions follow the standard power-law: (1.4) Furthermore, our rectification does not affect the form of three-point functions.
Here we note that a similar idea is proposed in [20].However, their approach is different from ours.In [20] the shadow transformation is taken for all the particles including outgoing particles, and then two-point functions still result in the delta functions.It is important to adopt the shadow transformation only for incoming particles to obtain the power-law for the two-point functions. 2ur rectification is supported by the fact that our altered choice of the conformal wave functions for massless fields is smoothly connected to the massive wave functions (used in the literature) in the massless limit.It is further supported by the crossing symmetry in the context of scattering amplitudes.
This paper is organized as follows.In Sec. 2, we propose a rectified celestial dictionary.By considering the two-point scattering amplitudes for one particle passing through, we redefine operators in CCFT to achieve a desirable form for the two-point functions.Specifically, we alter the treatment of incoming particles.We consider the massless limit and see the consistency of the dictionary for massless and massive fields.We further discuss this modification from the viewpoint of the crossing symmetry.In Sec. 3, we discuss the validity of our definitions using OPE.In addition, we reevaluate the celestial three-point correlator for one massive and two massless particles, which was computed previously, using our new definitions, confirming that the form of the three-point function is appropriate.Sec. 4 is devoted to a summary.
Notation Here we summarize the notation used in this paper.
Massless on-shell momenta p µ will be parameterized by ω (0 ≤ ω < ∞) and angle coordinates, which we will use complex coordinates z ∈ C, as, We define the integral measure on C as as in [23].For massive on-shell momenta p µ with mass m, we introduce pµ := m −1 p µ .pµ satisfies p2 = −1 and it represents the embedding coordinates of the three-dimensional unit hyperbolic space H 3 to R 1,3 .pµ can be parameterized by the Poincare coordinates (y, z) as We define the integral measure on H 3 as 2 Rectified holographic dictionary and two-point functions In the celestial holography, the following relation between scattering amplitudes of massless fields and correlation functions is proposed (e.g.[17]): It has a problem, especially in the two-point functions as follows.The massless two-point scattering amplitude is given by From it, we can calculate the celestial two-point correlators as Thus, the two-point functions for massless scalar fields show a delta-function behavior.The conformal symmetry may allow this delta-function behavior of the two-point functions.However, as we will discuss in Sec. 3, it is problematic for the property of the CCFT.Therefore, in this section, we rectify the dictionaries for celestial holography in order to make two-point functions an appropriate form.Our rectification is natural in terms of the massless limit of massive dictionaries and a crossing symmetry for scattering amplitudes.

Massless scalar
The celestial holographic dictionary gives us a translation between scattering amplitudes in the Minkowski space and the correlation functions on the CCFT.In this subsection, we consider massless scalar fields.For the scattering amplitude A({p out i }, {p in j }), we propose the following dictionary: (2.4) Here conformal dimension ∆ i takes principal series It is essentially the Mellin transformation of the scattering amplitudes through the parameterization (1.5) as proposed in the literature.The difference from the literature is that we interpret the operators corresponding to the incoming particles as the shadow-transformed operators O ∆ defined by (for details of shadow operators, see, e.g., [24]) (2.5) The reason why we take this rectification of the dictionary is that the two-point functions of (non-shadow) operators take the conventional form in CFT as we see below.
Let us consider the two-point functions obtained from the above dictionary.The scattering amplitude is trivial as Using the parameterization (1.5), the delta function can be rewritten as: Thus, we have Using the dictionary (2.4), the obtained two-point function is (2.9) Performing the inverse shadow transformation on O + (z 2 ), we can obtain the two-point function of non-shadow operators as 4 (2.12) Therefore, the above dictionary leads to the standard power-law form of the conformal two-point functions.If we do not interpret the operators corresponding to the incoming particles as the shadow operators, the two-point functions become the delta function δ (2) (z 1 − z 2 ) as (2.9).The dictionary can be interpreted as the relation between the creation-annihilation operators in the Minkowski space and the operators in the correlation function as follows: Here, the bulk field ϕ(x) is related to the asymptotic fields ϕ in/out in the Heisenberg picture: where they have the mode expansion For free fields, a in = a out , and we have (2.17) Thus, the Hermitian conjugation with respect to the Klein-Gordon inner product in the Minkowski space corresponds to the shadow transformation on the celestial sphere.This is different from [19] where the Hermitian conjugation with respect to "the shadow product" (which is different from the usual Klein-Gordon product) corresponds to the two-dimensional shadow transformation.Accordingly, in [19], it is argued that O is the conjugation of O with respect to the BPZ inner product because ⟨O O⟩ follows the power-law form in their formulation.In our formulation, 4 We use the formula and then we have O is not the conjugation of O w.r.t. the BPZ inner product because ⟨O O⟩ is a delta function.
The relation (2.13) can be rewritten as where the symbol (, ) KG denotes the Klein-Gordon inner product defined by φ − 1+iλ (x, z) and φ + 1+iλ (x, z) are massless conformal primary wave functions defined by Computing the integrals, we obtain Here, we note that this φ + is defined so that it gives (2.18), and is proportional to the standard conformal wave functions (non-shadow one) in, e.g., [17,18].From (2.18), we also obtain the formula directly computing O + 1+iλ (non-shadow one) as where φ + 1+iλ is an inverse shadow transformed function from φ + 1+iλ defined by

.24)
This φ + 1+iλ is a so-called shadow conformal wave function, up to a factor, in the literature (see [11]).We have to use it for incoming particles to obtain the power-law two-point functions (2.12).

Massive scalar
Next, we consider the massive scalars.Our dictionary for the massive scalars is the same as that in literature, up to multiplicative factors.However, we will see that the massless limit of the conformal wave functions in this conventional dictionary reproduces our massless conformal wave functions for the rectified dictionary given in the previous subsection 2.1.This fact supports our rectified dictionary.
The holographic dictionary for massive scalars is as follows: where c(m, λ) 2 which is chosen so that the two-point functions are normalized as we will see below.Here, G ∆ (p; z) is a bulk-boundary propagator used in the Euclidean AdS 3 /CFT 2 correspondence: (2.26) We should note that the principal series of massive scalar fields is ∆ ∈ 1 + iλ where λ > 0. Up to the multiplicative factors, this dictionary is the same as that in literature (see, e.g., [11]).
To compare the massive dictionary with the massless one, we also write down the dictionary involving the shadow operator.The above dictionary (2.25) leads to (2.28) Let us confirm that this definition leads to the standard form of conformal two-point func- (2.27) tions.For the Minkowski two-point amplitude the above dictionary leads to (2.30) (2.31) Thus, for massive scalars, the dictionary proposed in the literature leads to the standard form of the two-point functions.
As in the massless fields, we can relate the CCFT operator to the creation/annihilation operators as follows: a out (mp), ( a † in (mp). (2.33) For free fields, a in = a out , we have

.34)
This relation is the same as the massless case.

Massless limit
Let us consider the massless limit of the massive conformal wave functions.Near the massless limit, the Bessel function is expanded as We thus have, near m = 0,

Comments on crossing symmetry
Crossing symmetry relates amplitudes involving an incoming positive energy particle to ones involving an outgoing negative energy particle.However, we have not explicitly established a dictionary for negative energy particles.It would be natural to treat all outgoing particles on an equal footing regardless of whether they have positive or negative energy, as below: Here, p 2 is a future-directed vector, and thus −p 2 is a past-directed one representing negative energy.We note that the superscript "−" on operators is used here to denote outgoing, irrespective of whether it carries positive or negative energy, and "m" represents that the operators correspond to massive particles in the Minkowski spacetime.O − is interpreted as operators for particles with outgoing positive energy, while O ′− is interpreted as those for particles with outgoing negative energy.
For comparison, we can write the amplitude again from the viewpoint of interpretation which includes one incoming positive energy particle and one outgoing positive energy particle, O + is interpreted as an operator for particles with incoming positive energy.From crossing symmetry, the following equation holds: From this relation, comparing (2.47) with (2.48), we can establish the relation between O ′− and O + as follows (see Fig. 1): (2.50) The above discussion holds even when considering O − 1 as more general operators in CCFT.In other words, regarding O 2 , the above relation holds at the operator level.
In our rectified dictionary, we determine that (2.47) takes the form of the standard two-point function in CFT in both massive and massless cases.In the massive case, (2.47) and (2.48) take an ordinary form of two-point functions in two-dimensional CFT.This is the reason why we do not have to modify the dictionary for massive fields up to normalization factors.Since we Figure 1: The relationship within the operator set in CCFT.Crossing symmetry establishes the equivalence between outgoing positive energy operators and incoming negative energy operators.can smoothly take the massless limit as we have seen, a similar discussion should hold even for massless fields. 9

Operator product expansion
The previous proposed celestial dictionary leads to a peculiar property in that the two-point functions take the delta function form while higher-point functions follow the power-law.In this section, we focus on OPE, making ensure the consistency of our definition in the dictionary.Three-point functions in two-dimensional CFT take the form where f 123 is a structure constant.In two-dimensional CFT, the OPE is schematically expressed as (see e.g.[25]): 9 A situation is more subtle for massless scalars.A naive extension of the massless dictionary to the negative energy is (2.51)However, this naive extension does not work well, and we need a more careful extension so that it is consistent with the massless limit.
where k runs over the primary operators in CCFT. 10 Utilizing this equation, the three-point function can be reformulated as follows: In the case ⟨O k (z 1 )O 3 (z 3 )⟩ ∝ 1 |z 13 | 2∆ 3 , we can write three-point function as follows: On the other hand, in the case ⟨O k (z 1 )O 3 (z 3 )⟩ ∝ δ (2) (z 1 − z 3 ), the three-point function becomes In (3.1), there is no distributional behavior.Thus we should have (3.4) rather than (3.5).In the next section, we compute a concrete example for three-point functions.

Two-massless and one-massive three-point function
Here, we consider a tree level scattering amplitude for two incoming massless and one outgoing massive scalar fields with the coupling constant g (considered in [26,18]), and reevaluate it by our dictionary.The scattering amplitude is given by A(1, 2, 3) = −i(2π) 4 gδ (4) (ω 1 q1 + ω 2 q2 − mp 3 ).(3.6)In our dictionary, the corresponding celestial three-point function becomes

.7)
10 More precisely, the sum over k should be an integral because the CCFT has a continuous spectrum.
Taking the inverse shadow transformation for the incoming particles, 11 we obtain where (3.12) On the other hand, if we take OPE for O + ∆ 2 O −,m ∆ 3 , we have We should note that the sum over k runs over all operators for both massless and massive particles.However, from the discussion on the two-point functions, only massless operators contribute to this expansion because O + ∆ 1 is the operator for the massless field.If the twopoint function takes the delta function form, the three-point function has to be proportional to δ (2) (z 1 − z 2 ) in the OPE limit |z 23 | → 0 which is inconsistent with (3.12).Thus, the previous celestial dictionary in the literature may have inconsistency.In our rectified dictionary, the two-point function follows the standard power-law and it is consistent with the behavior of the 11 We use the following formula [24]: where z ij ≡ z i − z j .It leads to (3.9) three-point function (3.12) as in the usual CFTs.

Four-point function
As stated by [27,20], the four-point celestial correlators for massless particles result in delta functions if we use the previous dictionary.It is caused by the delta function form of the two-point functions from the following OPE:

Summary
In this paper, we have rectified the holographic dictionary in the celestial holography so that the celestial two-point functions follow the standard power-law instead of the delta function.
The dictionary for massive scalars is smoothly connected to the one for massless scalars in the massless limit.We have also discussed that the dictionary is natural, as it is suggested by massless limit and crossing symmetry.In our definition, the hermitian conjugate with respect to the four-dimensional Klein-Gordon inner product is the same as the shadow transformation in CCFT.Furthermore, the behavior of the OPE according to this definition is consistent with the results of the two-point function.Our redefinition has led to a substantial reevaluation of properties in CCFT, specifically in OPEs and conformal block expansion [26,28,29].