Four-point correlators of light-ray operators in CCFT

We compute the four-point correlator of two gluon light-ray operators and two gluon primaries from the four-gluon celestial amplitude in $(2,2)$ signature spacetime. The correlator is non-distributional and allows us to verify that light-ray operators appear in the OPE of two gluon primaries. We also carry out a conformal block decomposition of the terms involving the exchange of gluon operators.


Introduction
Correlation functions in conformal field theories contain a wealth of information. In generic CFTs, the OPE coefficients are related to three-point functions and four-point functions contain information about the spectrum of the theory, which can be deduced by means of the conformal block decomposition. Celestial conformal field theories (CCFTs), whose three-and (tree-level) four-point correlators are easily computed via Mellin 1 transforms of momentum space scattering amplitudes [1][2][3], make these relationships opaque due to the distributional nature of their correlators. Nevertheless, several attempts have been made to deduce the spectrum and the OPE coefficients. These include direct analysis of the distributional correlators using conventional CFT techniques [4][5][6][7], asymptotic symmetries [8][9][10], and representation theory [11][12][13][14]. The analysis has revealed that the CCFT spectrum typically contains light-ray operators and shadows in addition to the usual primaries.
Therefore the computation of correlators involving shadow or light-ray operators is essential to gain a better understanding of the role played by these operators in CCFT. At four points, the additional integrals inherent in the definitions of the shadow or light transforms render the correlators non-distributional, thereby making the analysis more straightforward (or, at least, more "traditional"). The shadow transform has been employed in [15,16] to obtain, amongst other things, a non-distributional four-point correlator and its conformal block decomposition. The light transform is most naturally defined in a Lorentzian CFT, 2 which requires an analytic continuation of the Euclidean CFT on the celestial sphere. It has been shown that such an analytically continued CFT lives on a Lorentzian torus [18] and the correlators have a natural interpretation as Mellin transforms of scattering amplitudes in (2, 2) signature bulk spacetime. These techniques have been employed in [19] to produce non-distributional three-point functions.
There are other motivations for studying light-ray operators in CFT beyond the desire to better understand their role in OPEs. It was noted in [19] that the light transform is an analogue of Witten's half-Fourier transform to twistor space [20], and the generalization where one transforms each operator on z i orz i depending on helicity is similarly analogous to the "link representation" of [21,22]; these developments have had enormous impact on the study of amplitudes. Moreover, in [23] Strominger recognized a universal symmetry algebra based on w 1+∞ in the gravitational S-matrix by carrying out an appropriate light transform on the result of [24].
The central result of our paper is the following formula for the tree-level correlator of two gluon light-ray operatorsL[O ∆,J ] and two gluon primaries O ∆,J : In the region z,z > 1, the function F is given by A complete description of F for all values of z,z can be found in Section 3. The paper is organized as follows. In Section 2 we discuss the tree-level four-gluon amplitude in (2, 2) signature spacetime and compute the corresponding celestial correlator. We highlight key differences compared to its (3, 1) analogue. In Section 3 we compute its double light transform and derive the result (1.1) for all values of z,z. We extract information about the OPE of gluon primaries from the correlator in Section 4, and we study its conformal block decomposition in Section 5.
. (2.6) Note that the terms in the product are required by conformal invariance, which does not fix the overall dependence on the cross-ratio z. We emphasize that the absolute values in (2.6) follow directly from our starting point (2.2) in (2, 2) signature; they are not imposed by hand. However, it is worth pointing out that the absolute values obscure all information about causality. Indeed the causal structure of correlation functions is encoded in branch cuts which arise as we cross the light-cone singularities at z ij = 0 orz ij = 0. We hope to analyze these issues in more detail in the future.

The light transform
The "anti-holomorphic" light transform of an operator O ∆,J with conformal weight ∆ and spin J is defined asL It is easy to check thatL[O ∆,J ](z,z) transforms as an operator with conformal weight 1 + J and spin ∆−1 6 . A similar definition exists for the "holomorphic" light transform with respect to z, which we will denote by L[O ∆,J ](z,z). For more details, we refer the reader to [27,28]. We now compute the light transforms of the correlator (2.6). The computation of the first light transform is straightforward due to the presence of the delta function, which we write as It is worthwhile to pause here to draw attention to the bulk point singularity located at z =z. While such singularities have been shown to absent in correlation functions of local operators in [29], their presence in CCFT has already been hinted at in [30]. We can proceed with the computation of the second light transform in a similar manner. We choose to light transform the remaining negative helicity gluon w.r.tz 2 . Making use of (3.3) we find where, with the help of the change of variable to t =z 12 z 34 /z 13z2 4 , we have which is an integral over four marked points (one of which is at infinity). We relegate the details of the evaluation of this integral to Appendix A. The result depends on the relative positions of z,z and 1. If they are on opposite sides of 1 (z < 1 <z orz < 1 < z) then the result can be written as where C is defined in (1.4). On the other hand, if they are on the same side of 1 (either z,z > 1 or z,z < 1) then the result takes the form shown in (1.3).

Collinear limits and the OPE
Four-point correlators in any CFT contain information about the OPE of the operators they involve. A direct computation of the OPE from the four-gluon correlator in CCFT is usually hindered by the fact that the correlator is distributional, proportional to δ(z −z) 7 . In this section, we exploit the non-distributional nature of the correlator involving two light-ray operators (3.4) to obtain information about the OPE between gluon primaries. To that end we consider the collinear limit as both z 34 andz 34 approach zero. In this limit the cross-ratio z approaches zero, and from the appropriate expression for F(z,z) given in (1.3) we begin by reading off the leading term as z 34 → 0: where we have retained only the leading O(1/z 34 ) singular terms. It is now straightforward to take the limitz 34 → 0. The cross-ratioz becomes 0 in this limit, so the hypergeometric functions approach 1. In terms of three-point functions with two or three light-ray operators, computed in (B.2) and (B.3) of Appendix B, the leading terms can be written as By reinstating color indices and structure constants f abc of the gauge group in the obvious way, we infer from this collinear limit the OPE 3) The first term involves a gluon primary of weight ∆ i + ∆ j − 1 and has been computed from various methods [8,9,[33][34][35][36], while the second term involves a light-ray operator and was conjectured in Section 5 of [6]. The appearance of the second term is also consistent with the fact that the conformal block decomposition of four-point correlators involves the exchange of light-ray operators [5].
We pause here to point out that the OPE coefficient involving one "incoming" and one "outgoing" gluon computed in [8] by Mellin transforming the splitting function is proportional The relative minus sign between the two beta functions is apparently at odds with our result. However, the splitting function obtained from the (2, 2) signature amplitude (2.6) involves an absolute value and its Mellin transform is in agreement with (4.3).
The four-point correlator can also be used to compute the OPE between two light-ray operators and the OPE between one light-ray operator and a primary; we comment on these in Appendix C.

Conformal block decomposition
In the previous section we showed directly from the four-point correlator (3.4) that the OPE of two primaries involves a linear combination of a primary and a light-ray operator. In this section we perform a conformal block decomposition of the term in (3.4) corresponding to the exchange of gluon operators (meaning gluon primaries and their descendants, as opposed to light-ray operators and their descendants).
The SL(2, R) × SL(2, R) conformal symmetry allows us to set z 1 = ∞, z 2 = 1, z 4 = 0 (and similarly forz i ). Then z = z 3 andz =z 3 , and we can extract where (we assume that 0 < z,z < 1) We focus on the first term, which corresponds to the exchange of gluon operators, and leave the decomposition of the second term, which corresponds to the exchange of light-ray operators, to future work. We proceed along the lines of [15] in order to massage I 1 (z,z) into a form from which the conformal block decomposition can be read off. First we use the identity and then the Burchnall-Chaundy expansion [37,38] of the Appell function x n y n 2 F 1 (a + n, b 1 + n, c + 2n, x) 2 F 1 (a + n, b 2 + n, c + 2n, y) , along with the Gauss recursion relations for 2 F 1 , to express Here the coefficients are In the present application, operators 3 and 4 are gluon primaries with J = +1 and for these we should take . On the other hand, for the light-transformed operators 1 and 2 we need to take h 12 = ∆ 1 −∆ 2 2 = −h 12 in terms of the original weights ∆ 1 , ∆ 2 (the latter are the ones that appear in (5.6)).
We can read off the spectrum of exchanged states in (5.5) to be which interestingly indicates that only positive helicity operators are exchanged.
Besides the singular point at |t| = ∞, the integrand exhibits three singular points at t = 1, z,z. Thus there are seemingly six different configurations which need to be analyzed: I :z < 1 < z, II : z < 1 <z III : 1 < z <z, IV : 1 <z < z V :z < z < 1, VI : z <z < 1 We will demonstrate that the result of the integral (A.1) can be brought to a form where there are only two distinct configurations. To see this, let us first evaluate F in configuration I :z < 1 < z. The integral then breaks up into four regions All of these four integrals are Gauss hypergeometric functions; explicitly where we have maintained the ordering between the four regions in (A.2) and the four terms in (A.3) to indicate to the reader which term arises from which region. Using standard identities and Euler transformations we can write the result more compactly as (3.6). An explicit computation of the integral in configuration II and subsequent use of the abovementioned identities reveals the identical result. Similar techniques can be applied to show that in regions III − VI the integral takes the form (1.3).

B Three-point functions and their light transforms
In this Appendix we present expressions for three-point functions involving two and three light-ray operators. These expressions serve as a point of comparison for the OPE relations (4.3).
We begin by adapting the three gluon amplitude in (2, 2) signature spacetime presented in (3.5) of [3] to the context of our paper by performing a sum over the i . This results in A direct application of the definition of the light transform in (3.1) yields where C is defined in (1.4).

C Further OPE limits of the four-point amplitude
In this Appendix we examine the z 12 → 0 and z 13 → 0 collinear limits of the correlator (3.4). First consider the limit z 12 → 0. In this regime, the correlator becomes This can be expressed in terms of generic three-point correlators by noting that the z ij and z ij dependence of each term reveals which types of operators must appear. Schematically, we must have where ρ 1 and ρ 2 are functions independent of z,z, and S[O m ∆ 1 +∆ 2 −1,− ] and L[O m ∆ 1 +∆ 2 −1,− ] are the shadow and light transforms of some massive operator. Since three-point correlators of massless operators are always distributional, by "massive operator" we simply mean one whose three-point functions are non-distributional. The appearance of such operators is necessary to account for the structures in (C.1). It would be interesting to understand the physical content of these terms and make connections to the existing literature on light-ray OPEs (see for example [40]). Finally, we consider the limit z 23 → 0, which corresponds to z → 1. First it is helpful to use hypergeometric identities to rewrite the function F (z,z) as which simplifies the limit z → 1, giving with two constants ρ 1 and ρ 2 . The introduction of the massive light-ray operator is again necessary to produce a three-point correlator with the correct structure.