The light-ray OPE and conformal colliders

We derive a nonperturbative, convergent operator product expansion (OPE) for null-integrated operators on the same null plane in a CFT. The objects appearing in the expansion are light-ray operators with fixed spin and bounded transverse spin, whose matrix elements can be computed by the generalized Lorentzian inversion formula. For example, a product of average null energy (ANEC) operators has an expansion in spin-3 light-ray operators. An important application is to collider event shapes. The light-ray OPE gives a nonperturbative expansion for event shapes in special functions that we call celestial blocks. As an example, we apply the celestial block expansion to energy-energy correlators in N=4 Super Yang-Mills theory. Using known OPE data, we find perfect agreement with previous results both at weak and strong coupling, and make new predictions at weak coupling through 4 loops (NNNLO).


Introduction
In this work, we study a product of null-integrated operators on the same null plane in a conformal field theory (CFT) in d > 2 dimensions (figure 1): Here, we use lightcone coordinates (1. 2) The operators are located at different transverse positions y 1 , y 2 ∈ R d−2 , and their spin indices are aligned with the direction of integration (the v direction). As an example, when O 1 and O 2 are stress-tensors, (1.1) is a product of average null energy (ANEC) operators. In [1], we established sufficient conditions for the existence of the product (1.1). Such null-integrated operators arise naturally in "event shape" observables in collider physics [2][3][4][5][6]. They also appear in shape variations of information-theoretic quantities in quantum field theory [7][8][9], as generators of asymptotic symmetry groups [10], and in studies of positivity and causality [1,[11][12][13][14][15][16][17]. We review event shapes and null-integrated operators in section 2.
Each null-integrated operator is pointlike in the transverse plane R d−2 , so it is natural to ask whether there exists an operator product expansion (OPE) describing the behavior of the product (1.1) at small | y 12 |:  Here, the objects O i have dimensions δ i and the powers of | y 12 | are fixed by dimensional analysis.
The OPE for local operators is a powerful tool in CFT. It allows one to compute correlation functions and to formulate the bootstrap equations [18,19]. A similar OPE for null-integrated operators (1.1) could have myriad applications. Thus, we would like to ask whether (1.3) exists, whether it is convergent or asymptotic, and what the objects O i are.
Hofman and Maldacena analyzed this question in N = 4 SYM and found the leading terms in the small-| y 12 | expansion where O 1 , O 2 are stress tensors and currents [5]. At weakcoupling, the leading objects are certain integrated Wilson-line operators. At strong coupling, the leading objects can be described using string theory in AdS: they are certain stringy shockwave backgrounds. What is the analog of these results in a general nonperturbative CFT? Can we extend the leading terms to a systematic convergent expansion?
There is a simple and beautiful argument for the existence of an OPE of local operators in a nonperturbative CFT (see e.g. [20] On the left, we show a conformal frame where the null plane is u = 0, and the operators are at different transverse positions y 1 , y 2 ∈ R d−2 . On the right, we show a conformal frame where the null plane is future null infinity I + and the null-integrated operators are separated by an angle θ 12 on the celestial sphere. We give the relationship between θ 12 and y 12 in (1.10). Note that the entire circle at spatial infinity is really a single point i 0 . Thus, the operators become coincident at the beginnings and ends of their integration contours.
signature. We surround the operators with a sphere S d−1 (assuming all other operator insertions are outside the sphere) and perform the path integral inside the sphere. This produces a state |Ψ on the sphere. In a scale-invariant theory, |Ψ can be expanded in dilatation eigenstates (1.4) By the state-operator correspondence, these eigenstates are equivalent to insertions of local operators at the origin |O i = O i (0)|0 . Thus (1.4) is the desired OPE. Unfortunately, this argument does not work for the product (1.1). There is no obvious way to surround the null-integrated operators with an S d−1 such that other operators are outside the sphere. The structure of (1.3) suggests that perhaps we should surround the null-integrated operators with an S d−3 in the transverse space R d−2 . However there is no obvious Hilbert space of states associated with such an S d−3 . Nevertheless, using different technology, we will show that a convergent OPE (1.3) for null-integrated operators does exist in a general nonperturbative CFT. The objects appearing on the right-hand side are the light-ray operators O ± i,J defined in [22] with a particular spin i,J is obtained by smearing a pair of local operators in a special way in the neighborhood of a light-ray. We review this construction in section 3.2. The matrix elements of O ± i,J can be computed via a generalization of Caron-Huot's Lorentzian inversion formula [22,23]. The spectrum of operators O ± i,J is related to the spectrum of local operators by analytic continuation in spin J; i labels different Regge trajectories. For example, if O 1 = O 2 = T , then J = 3 and we obtain an OPE in terms of O + i,3 , see figure 2. Our strategy to establish the OPE (1.3) is as follows. First, in section 3.3 we decompose the left-hand side of (1.3) into conformal irreps by smearing the transverse coordinates y 1 , y 2 , using harmonic analysis for the transverse conformal group SO(d − 1, 1). In section 3.4, we focus on a single irrep and compute its matrix elements. Such matrix elements can be written in terms of an integral of a double commutator. After some manipulation, we express this integral as a linear combination of the generalized Lorentzian inversion formula of [22], i.e. as a sum of matrix elements of O ± i,J 's. Thus, the original product of operators is a sum of O ± i,J 's. The light-ray OPE (1.3) and the construction of light-ray operators in [22] give two different ways of creating light-ray operators, and it is not obvious a priori that they should be related. For example, the light-ray operators of [22] involve smearing a pair of local operators in a region off the null plane orthogonal to the light-ray. By contrast, in the light-ray OPE, we move operators in the y directions, keeping them on the null plane. Even though the smearing kernels are very different, the resulting operators turn out to be related, essentially that acting with (1.1) on the vacuum immediately gives zero (as we will review shortly). Instead, we would like to act on nontrivial states, and then the theorem of [21] does not apply. due to analyticity properties of conformal correlators. The fact that the light-ray operators of [22] can be obtained in two very different ways suggests that they may represent some kind of complete set of observables associated to a light ray, in the same sense that local operators represent a complete set of observables associated to a point.
As an example, consider the case where O 1 = φ 1 and O 2 = φ 2 are scalars, so that J 1 + J 2 − 1 = −1. 2 Following the procedure above, we find the OPE 3 (1.5) Here, C δ ( y, ∂ y ) is the same differential operator that appears in an OPE of local primary scalars in d − 2 dimensions. It has an expansion (1.7) Here, λ is an SO(d − 2) representation encoding spin in the transverse plane, s = ± is a signature, (a) labels conformally-invariant three-point structures, and D (a),s δ,λ is a differential operator that generalizes C δ .
In section 6 we find that the light-ray OPE also carries information about contact terms in the y 1 → y 2 limit. These contact terms are important in at least two aspects. First they are a part of the physical information present in event shape observables. Second, they arise in commutators of null-integrated operators [10], leading to an interesting algebra.
An interesting property of the light-ray OPE is that the transverse spins that appear are bounded. Specifically, the possible SO(d − 2) representations appearing in dv 1 O 1 dv 2 O 2 are given by listing all SO(d − 1, 1) representations in the local OPE O 1 × O 2 , and removing the first rows of their Young diagrams. (We give a simpler version of this rule in (3.99).) For example, in the OPE of null-integrated scalars (1.5), the maximal transverse spin is zero (since only traceless symmetric tensors appear in φ 1 × φ 2 ). In the OPE of ANEC operators (1.7), the possible transverse representations are •, , , , , , . This is very different from the naïve expectation that an OPE of point-like objects can contain objects with arbitrarily high spin. Ultimately, it is a consequence of the same analyticity properties that relate different smearings of local operators.

Commutators and superconvergence
Our analysis does not assume or require that null-integrated operators commute. Indeed, we can write an expression for a commutator of null-integrated operators using the OPE. For example, the commutator of ANEC operators is given by the odd-signature terms in (1.7), In [1], we showed that a commutator of ANEC operators vanishes if J 0 < 3, where J 0 is the Regge intercept of the theory, and furthermore J 0 ≤ 1 in unitary CFTs. It is interesting to understand how vanishing occurs on the right-hand side of (1.8). Note that the operators on the right-hand side are light-ray operators with spin 3 and odd signature. We show in section 4.1 that if J 0 < 3, such operators must be null integrals of local spin-3 operators. 5 However, local spin-3 operators are not allowed in the T × T OPE by conservation conditions and Ward identities [24]. Thus, the commutator vanishes. As we explain in section 4.1, this argument generalizes to establish vanishing of a commutator of null-integrated operators whenever J 1 + J 2 > J 0 + 1. It turns out that even if local operators with signature (−1) J 1 +J 2 −1 and spin J 1 +J 2 −1 do appear in the local O 1 ×O 2 OPE, they do not survive in the light-ray OPE. This provides another (somewhat circuitous) way to derive the commutativity conditions of [1]. An exception can occur at vanishing transverse separation y 12 = 0. In that case, the commutator may contain contact terms, which can be computed by our light-ray OPE formula. As an example, in section 6.1, we describe how to compute contact terms in a commutator of null-integrated nonabelian currents (assuming J 0 < 1), reproducing results of [10].
Vanishing of the commutator of ANEC operators means that the odd-signature terms in (1.7) disappear, and the OPE of ANEC operators can be simplified to a sum of even-signature light-ray operators with spin 3. This generalizes the results of [5].
Despite the fact that local spin-3 operators are not allowed in the T × T OPE, we can try to compute their OPE data with the Lorentzian inversion formula. This is equivalent to evaluating matrix elements of the right-hand side of (1.8). The result must be zero. However, if we plug the OPE in a different channel (the "t-channel") into the inversion formula, we obtain sums that are not zero term by term. The conditions that these sums vanish are precisely the "superconvergence" sum rules of [1]. As we explain in section 4.3, in this language it is simple to argue that (suitably-smeared) superconvergence sum rules have a convergent expansion in t-channel blocks.

Celestial blocks and event shapes
An important application of the light-ray OPE is to event shapes [2][3][4][5][6]. For example, to compute a two-point event shape, we place a pair of null-integrated operators ("detectors") along future null infinity (right half of figure 1) and evaluate a matrix element in a momentum eigenstate |O(p) . By applying the OPE (1.5), we obtain a sum of matrix elements of individual light-ray operators O ± i,J in momentum eigenstates |O(p) , (1.9) The quantity (1.9) is fixed by conformal symmetry up to a constant. It plays a role for event shapes analogous to the role that conformal blocks play in the usual OPE expansion of local 4-point functions. It is proportional to a function of a single cross-ratio ζ = 1 − cos θ 12 2 = y 2

12
(1 + y 2 1 )(1 + y 2 2 ) ∈ [0, 1], (1.10) where θ 12 is the angle between detectors on the celestial sphere. We have also written ζ in terms of the transverse positions y 1 , y 2 in the conventions of [5]. In an event shape, ζ → 0 is the collinear limit, while ζ → 1 corresponds to back-to-back detectors. We call (1.9) a "celestial block." In section 5, we compute celestial blocks by solving an appropriate conformal Casimir equation. For example, when O is a scalar, the result is 6 Note that f ∆ 1 ,∆ 2 ∆ becomes a pure power ζ ∆−∆ 1 −∆ 2 +1 2 in the collinear limit ζ → 0. The light-ray OPE thus yields an expansion for two-point event shapes in celestial blocks. For example, using (1.5) and superconformal symmetry [27,28], an energy-energy correlator (EEC) in N = 4 SYM can be written as (1.12) where ∆ i runs over dimensions of Regge trajectories at spin J = −1, and p ∆ i are squared OPE coefficients of operators in the 105 representation of SO (6) in the O 20 × O 20 OPE, analytically continued to spin J = −1. The state ψ(p) carries momentum p = (p 0 , 0, 0, 0) and is created by acting with an O 20 operator on the vacuum. The angle between energy detectors is cos θ = n 1 · n 2 , and ζ is defined by (1.10). The coupling-independent contact terms 1 4 (2δ(ζ) − δ (ζ)) are related to the contribution of protected operators to the EEC. Thus, (1.12) expresses the EEC in N = 4 SYM in terms of OPE data. This formula holds nonperturbatively in both the size of the gauge group N c and the 't Hooft coupling λ. In section 7, we check it against previous results at weak and strong coupling and find perfect agreement. Using known results for leading-twist OPE data in N = 4 SYM, we use (1.12) to make new predictions for the small-angle limit of N = 4 energy-energy correlators through 4 loops (NNNLO).
We conclude in section 8 with discussion and future directions. In appendix A we summarize our notation, in appendix B we review general representations of orthogonal groups, and in appendix C we clarify some points about analytic continuation in spin. Appendices D, E and F contain details of the calculations described in the main text.
Note added: During the last stages of this work we learned about [29] and [30] which have some overlap with our analysis. Let us briefly describe the results of [29] and [30] in relation to our work.
In [29] the EEC in N = 4 SYM was analyzed using the Mellin space approach of [6]. We analyze N = 4 SYM in section 7. It was shown in [29] how the back-to-back ζ → 1 limit of the EEC is captured by the double light-cone limit of the correlation function studied in [31]. It led to the derivation of (F.1) and identification of the coefficient function H(a) with a certain spin-independent part of the three-point functions of large spin twist-2 operators. We do not analyze ζ → 1 limit of the EEC in a generic CFT in the present paper. Similarly, a leading small angle asymptotic of the EEC in N = 4 SYM, the small ζ limit of (7.97), was rederived in [29]. 7 Based on (7.97), the four-loop small angle asymptotic was worked out in [29], we do it in section 7.7. This represents the leading small-angle asymptotic of our complete, non-perturbative OPE formula (1.12).
In [30] a factorization formula describing the small ζ → 0 limit for the EEC was derived in a generic massless QFT, conformal or asymptotically free, in terms of the time-like data of the theory. The authors [30] applied their results to QCD, N = 1, and N = 4 SYM, in particular they analyzed the effects of a non-zero β-function which goes beyond our considerations in the present paper. In the conformal case of N = 4 SYM which is relevant to our analysis, the leading small-angle asymptotic was derived in [30] through three loops.
In addition, both [29,30] emphasized the importance of contact terms in the EEC (we compute these using the OPE in section 6.2), the way to compute them from the small angle and back-to-back limits, see appendix F, and their importance to the Ward identities (7.17,7.18). In particular, [29,30] checked that the N = 4 SYM NLO result [32] satisfies Ward identities, we do this in section 7.5.4. In [29] it was also checked that the NNLO result [33] satisfies Ward identities, which we do in section 7.6.

Null integrals and symmetries
Let us begin by examining the symmetries of a product of light-ray operators (1.1). This analysis will already give a hint why the objects O ± i,J appear in the OPE. Firstly, consider a boost (u, v, y) → (λ −1 u, λv, y), λ ∈ R + . (2.1) Each null-integrated operator dv i O i;v···v has boost eigenvalue 1 − J i , where 1 comes from the measure dv i and −J i comes from the lowered v-indices. Thus, the product (1.1) has boost . In other words, it transforms like the null integral of an operator with spin J 1 + J 2 − 1 [5]. Another important symmetry is CRT, which is an anti-unitary symmetry taking Combining CRT with Hermitian conjugation, we obtain a linear map on the space of operators. It is easy to check that We call the eigenvalue with respect to the combination of CRT and Hermitian conjugation the "signature" of the operator. Applying CRT and Hermitian conjugation to (1.1), we find where [ , ] and { , } denote a commutator and anticommutator, respectively. The extra minus sign in the commutator appears because Hermitian conjugation reverses operator ordering. It often happens (under circumstances described in [1] and discussed in section 4.1) that the commutator (2.4) vanishes. For example, a commutator of ANEC operators on the same null plane vanishes. For simplicity, suppose that the commutator vanishes. In this case, the product (1.1) is the same as the anticommutator (2.5). Thus, (1.1) transforms like the nullintegral of an operator with spin J 1 + J 2 − 1 and signature (−1) J 1 +J 2 . An integrated local operator can never have these quantum numbers. This shows that the OPE (1.3) cannot be computed by simply performing the usual OPE between O 1 and O 2 inside the integral.
In the embedding formalism, Minkowski space is realized as a subset of the projective null cone in R d,2 . Let us choose coordinates X = (X + , X − , X µ ) = (X + , X − , X 0 , · · · , X d−1 ) on R d,2 , with metric (2.6) The projective null cone is the locus X · X = 0, modulo positive rescalings X ∼ λX (λ ∈ R + ). This space is topologically S 1 × S d−1 . Lorentzian CFTs live on the universal cover of the projective null cone M d , which is topologically R × S d−1 -sometimes called the Lorentzian cylinder. The conformal group SO(d, 2) is the universal cover of SO(d, 2). Minkowski space corresponds to the locus X = (X + , X − , X µ ) = (1, 2 , where x ∈ R d−1,1 . Spatial infinity i 0 is obtained by taking x → ∞ in a spacelike direction and rescaling X so it remains finite, yielding X i 0 = (0, 1, 0). Timelike infinity i ± corresponds to X i ± = (0, −1, 0). (Note that future and past infinity i ± correspond to the same embeddingspace vector, but they are distinguished on the universal cover of the projective null cone.) Finally, null infinity corresponds to the points X I ± (σ, z) = (0, −2σ, z), z = (±1, n), where n ∈ S d−2 is a point on the celestial sphere and σ is retarded time.
The Lorentzian cylinder M d is tiled by Minkowski "patches" (figure 3). To every point p ∈ M d , there is an associated point T p obtained by shooting light rays in all future directions from p and finding the point where they converge in the next patch. In embedding coordinates, T takes X → −X. For example, T takes spatial infinity i 0 to future infinity i + . We sometimes write p + ≡ T p and p − ≡ T −1 p.
To describe operators with spin, it is helpful to introduce index-free notation. Given a traceless symmetric tensor O µ 1 ···µ J (x), we can contract its indices with a future-pointing null polarization vector z µ to form (2.7) When O µ 1 ···µ J (x) is an integer-spin local operator, O(x, z) is a homogeneous polynomial of degree J.
In the embedding formalism, the operator O(x, z) gets lifted to a homogeneous function O(X, Z) of coordinates X, Z ∈ R d,2 , subject to the relations X 2 = X · Z = Z 2 = 0 [40]. It is defined by (2.11) Index-free notation and the procedure of lifting operators to the embedding space can be generalized to arbitrary representations of the Lorentz group. We describe this construction in appendix B.

Review: the light transform
Null-integrated operators like those in (1.1) can be understood in terms of a conformallyinvariant integral transform called the "light-transform" [22]. In embedding-space language, the light-transform is defined by This transform is invariant under SO(d, 2) because (2.12) only depends on the embeddingspace vectors X, Z. It respects the gauge redundancy (2.9) because a shift Z → Z + βX can be compensated by shifting α → α + β in the integral. The initial point of the integration contour in (2.12) is X, since Z − (−∞)X is projectively equivalent to X. Furthermore, if O(X, Z) has homogeneity (2.10), then its light-transform has homogeneity transforms like a primary at X with dimension 1 − J and spin 1 − ∆: (2.14) Note that the light-transform naturally gives rise to operators with non-integer spin.
In Minkowski coordinates, L becomes In the second line above, we used gauge invariance (2.9) to shift −X → −X − (Z − αX)/α = −Z/α and then homogeneity (2.10) to pull out factors of (−α). In the third line, we used (2.11 ∈ Z ≥0 , which is not allowed in a unitary CFT [41]. One can also verify that L[O]|Ω = 0 by deforming the α contour in the complex plane inside a Wightman correlation function [22].
Let us now return to the boost selection rule J = J 1 + J 2 − 1 from section 2.1. To recover the setup of that section, we can set where 0, y ∈ R d−2 . Note that these satisfy the conditions X 2 0 = X 0 · Z 0 = Z 2 0 = 0. The light-transform becomes Thus, we should think of dv O v···v as a primary operator associated to the point X 0 at past null infinity. Consider now a product of null-integrated operators This is a product of primaries associated to the same point X 0 at past null infinity (with different polarization vectors Z 0 , Z 0 ). Thus, the dimension of the product (assuming it is well-defined) is the sum of the dimensions: . 9 This is the same as the dimension of the light-transform of an operator with spin J 1 + J 2 − 1. Hence, we have recovered the selection rule from section 2.1.
The relationship between this argument and the one in section 2.1 is that the dilatation generator that measures dimension around the point X 0 is the same as the boost generator discussed in section 2.1.
An important observation is that the product (2.18) transforms like a primary operator at the point X 0 . This is because both factors L[O 1 ](X 0 , Z 0 ) and L[O 2 ](X 0 , Z 0 ) are killed by the special conformal generators that fix X 0 . (Alternatively, we can simply observe that (2.18) is a homogeneous function of X 0 on the null cone in the embedding space, which again implies that it transforms like a primary.) Thus, when we consider the OPE expansion of (2.18) in the limit Z 0 → Z 0 , the only terms appearing will be other primary operators at the point X 0 .

Review: event shapes and the celestial sphere
The symmetries of light-ray operators on a null plane are easiest to understand when we take the null plane to be I + . This corresponds to choosing the embedding-space coordinates where z ∈ R d−1,1 is a future-pointing null vector. The integration contour for the lighttransform now lies inside I + , running from i 0 to i + along the z direction (figure 4). 9 Ordinarily in CFT, we do not consider a product of operators at coincident points. Instead, we place them at separated points and study the singularity as they approach each other, for example The dimensionful factor x ∆ k −∆ 1 −∆ 2 allows the scaling dimension ∆ k to be different from ∆1 + ∆2. However, if the coincident limit x → 0 is nonsingular, the only operators that survive the limit must obey the selection rule ∆ k = ∆1 + ∆2. Figure 4: A one-point event shape [42]. The detector O = O EHT is integrated along a null line (blue) along future null infinity, starting at spatial infinity i 0 and ending at future timelike infinity i + . (Note that the circle at spatial infinity is really a single point.) The red wavy lines indicate energy propagating from the interior of Minkowski space out to null infinity, created by the insertion of the source φ 1 (p).
transforms like a primary inserted at spatial infinity, which means it is killed by momentum generators Consequently, its matrix elements with other operators are translationally-invariant, for example (Throughout this work, we use φ to denote scalar operators and O to denote operators in general Lorentz representations.) Thus, it is natural to go to momentum space, where Note that |φ i (p) vanishes unless p is inside the forward lightcone p > 0, by positivity of energy. 10 We often abuse notation by writing where it is understood that we have stripped off the momentum-conserving factor (2π) d δ (d) (p+ q). More generally, we can consider a product of light-transforms along I + , inserted between momentum eigenstates (2.26) Following [43], we call such matrix elements "event shapes." This terminology comes from interpreting (2.26) as the expectation value of a product of "detectors" O 1 , · · · , O n in a "source" state |φ 1 (p) and "sink" state φ 2 (p)|. The detectors sit at points on the celestial sphere and are integrated over retarded time to capture signals that propagate to null infinity. In addition to being translationally-invariant, L[O](∞, z) transforms in a simple way under d-dimensional Lorentz transformations SO(d − 1, 1): they act linearly on the polarization vector z. The Lorentz group in d-dimensions is the same as the Euclidean conformal group on the (d−2)-dimensional celestial sphere. Indeed, we can think of z ∈ R d−1,1 as an embeddingspace coordinate for the celestial sphere S d−2 . Furthermore, L[O](∞, z) is homogeneous of degree 1 − ∆ in z, due to (2.13). Thus, L[O](∞, z) transforms like a primary operator on the celestial sphere with dimension δ = ∆ − 1.
In the previous coordinates (2.16), the group SO(d − 1, 1) acted by conformal transformations on the transverse direction y. The coordinates y are stereographic coordinates on S d−2 . Thus, we have proven the claim from section 2.1 that dv O v···v transforms as a primary in the transverse space.
The event shape (2.26) is similar to a correlator of operators with dimensions δ i = ∆ i − 1 in a Euclidean (d−2)-dimensional CFT. However, the presence of a timelike momentum p breaks the Lorentz group further to SO(d − 1). In the language of (d−2)-dimensional CFT, this is similar to the symmetry-breaking pattern that occurs in the presence of a spherical codimension-1 boundary or defect [25,26]. This fact will play an important role in section 5. We can choose a center-of-mass frame p = (p 0 , 0, . . . , 0) and write z i = (1, n i ) with n i ∈ S d−2 . The dependence on p 0 is fixed by dimensional analysis, so we can additionally set p 0 = 1. The event shape then becomes a nontrivial function of dot-products n i · n j .
In addition to respecting symmetries, event shapes are useful for studying positivity conditions. For example, consider the average null energy operator E = 2L[T ], where T µν is the stress tensor. E is positive-semidefinite [5,7,12]. To test this, we could compute the expectation value of E in several different states (primaries and descendents at different points, etc.) and then aggregate the resulting positivity conditions. However, it is sufficient to study event shapes O i (p)|E|O j (p) for the following reason. The Hilbert space of a CFT position-dependence for simplicity, we have φ(t) = e iHt φ(0)e −iHt . The minus sign is in the right-hand exponential e −iHt because that operator generates Schrodinger time-evolution. Acting on the vacuum, we obtain e iHt φ(0)|Ω , which is a sum of positive-imaginary exponentials e iEt . To get a nonzero result under the Fourier transform, we must multiply by e −iEt , which is contained in the factor e ip·x . is densely spanned by states of the form (2.27) where O i are primary operators and f i (x) are test functions. Positivity of E is thus equivalent to the statement that for any collection of test functions f i (x), This is the same as saying that K ij (x 1 − x 2 ) is a positive-semidefinite integral kernel. To determine whether a kernel is positive-semidefinite, we should compute its eigenvalues and check that they are positive. Because K ij (x 1 − x 2 ) is translation-invariant, it can be partially diagonalized by going to Fourier space. Thus, E is positive-semidefinite if and only if its one-point event shapes are positive-semidefinite.

1-point event shapes
As an example, let us compute a one-point event shape φ 2 |L[O]|φ 1 , where O has dimension ∆ and spin J, and φ 1 , φ 2 are scalars. We start from the Wightman function 11 (2.31) In (2.30), we have written the i prescription appropriate for the given operator ordering. This is obtained by introducing small imaginary parts x 0 i → x 0 i − i i with 2 > 3 > 1 in the same order as the operators appearing in the Wightman function. We often omit explicit i 's, restoring them only when necessary during a computation. In these cases, the i prescription should be inferred from the ordering of the operators in the correlator. The light-transform of (2.30) is [22] .
This indeed has the form of a conformally-invariant three-point function with an operator with dimension 1 − J and spin 1 − ∆. The notation i > j means "x i is inside the future lightcone of x j ." Below, we will also use the notation i ≈ j to indicate that x i is spacelike from x j . We have written (2.32) in the kinematics 2 > 3 > 1 − (figure 5), where all the quantities in parentheses are positive. This time, we have left the i prescription implicit.
We should now take x 3 to spatial infinity. Keeping track of i prescriptions, we find This is indeed translation-invariant. It is straightforward to compute the Fourier transform where . (2.36) The theta function θ(p) ≡ θ(−p 2 )θ(p 0 ) restricts p to lie in the forward lightcone. Overall, the one-point event shape is given by . (2.37) Note that this is consistent with dimensional analysis in p, homogeneity in z, and Lorentz invariance. In [1] we describe an algorithm for computing more general one-point event shapes.

2-point event shapes
A two-point event shape is constrained by dimensional analysis, homogeneity, and Lorentz invariance to take the form ζ takes values between 0 and 1. In the last step of (2.39) we evaluated ζ in a center-of-mass frame where p = (p 0 , 0) and z i = (1, n i ). The limit ζ → 0 corresponds to the detector directions z 1 and z 2 becoming parallel, which is described by the light-ray-light-ray OPE discussed in section 3. The limit ζ → 1 corresponds to the detectors becoming back-to-back in the frame of p.
3 The light-ray-light-ray OPE

Summary of computation
In this section, we compute an expansion for Here, we summarize the key steps of the computation. Our summary will be schematic. We omit details and illustrate calculations using diagrams (which do not capture some subtleties).
The first step is to decompose (3.1) into irreducible representations of the conformal group. As discussed in section 2.3, (3.1) transforms like a primary at the point x with scaling dimension (1 − J 1 ) + (1 − J 2 ). However, it does not transform irreducibly under the Lorentz group SO(d−1, 1) that fixes x. The appropriate set of irreducible representations are principal series representations labeled by δ ∈ d−2 2 + iR. To obtain such a representation, we smear the polarizations z 1 , z 2 against a kernel t δ where Dz is a measure on the projective null cone defined in (3.36). We write t δ explicitly in (3.57).
On the second line of (3.2), we implicitly defined a kernel L δ that combines the light transforms with smearing in z 1 , z 2 . We can represent L δ pictorially by The reason for this split is to accommodate for the next two operations, which act only on either Minkowski or celestial coordinates. The blue triangle represents making the points x i coincident. The red three-point kernel represents smearing polarization vectors with t δ . The next step is to compute matrix elements of W δ . Because a light-transformed operator kills the vacuum, we have The appearance of the double commutator suggests that we could relate the matrix elements of W δ to the Lorentzian inversion formula. To see this relation, first note that by conformal invariance we have where 0|O 4 OO 3 |0 (b) are conformally-invariant three-point structures for the given representations, and in (3.5) we have their light-transforms. The different structures have a label b, and summation over b is implicit. Diagrammatically, we can express (3.4) and (3.5) as where "dDisc" indicates the double-commutator. The function A b (δ) contains the matrix elements we are interested in. To extract it, we pair with a dual structure (the pairing will be defined in (3.44)) (3.7) The dual structure ( We denote the operation of inverting a structure by an enclosing green circle, −1 , suggestively labeled by a green inverse ( −1 ). In pictorial language, (3.7) is This is a four-point pairing between the double-commutator and a particular conformal block, as can be seen by cutting along the lines of the operators 1, 2, 3, and 4: The generalized Lorentzian inversion formula [22] also has this form, Therefore, we can relate A b (δ) to C ± ab (δ + 1, J 1 + J 2 − 1) by relating the two conformal blocks in (3.10) and (3.11), (3.12) Both conformal blocks are obtained by gluing three-point structures. The structure appearing on the right is the same for both blocks, so we only need to relate the structures on the left, The inverse of the cross on the right-hand side of (3.12) is integration against a two-point structure. 13 Here, the two-point structure is indicated by a dot on the left-hand side of (3.13). The operation of integrating against a two-point structure is a Lorentzian shadow transform, which changes the quantum numbers from (1 − J, 1 − ∆) (labeled as O L with an outgoing arrow) to (J + d − 1, ∆ − d + 1) (labeled as O L with an ingoing arrow). Thus, we can compute γ a by pairing both sides of (3.13) with the structure (3.14) Here, we rearranged our diagram into a pairing of two-point structures. Finally, we must compute the bubble diagram on the right-hand side. After substituting the definition of L δ (3.3), we obtain an expression involving a triple light transform of the three-point structure a, (3.15) The superscripts L ± are related to a subtlety not captured in the diagrams. The doublediscontinuity produces additional θ-functions in the expression for the block on the righthand side of (3.10). On the left-hand side of (3.13), these theta functions modify the kernel L δ so that the light-transforms become "half light-transforms" L ± , i.e. null integrals over semi-infinite lines. These are what appear in (3.15). 14 It turns out that the result of (3.15), and therefore also γ a , is remarkably simple. In section 3.4.4, we conjecture a formula for it in the case of an arbitrary three-point structure 0|O 1 OO 2 |0 (a) of operators in arbitrary representations. Putting everything together, we obtain which can be written This expresses matrix elements of the smeared product W δ in terms of matrix elements of light-ray operators. The smearing can be undone by suitably integrating over δ, where C δ is a differential operator. Lifting this to an operator equation, we have Finally, the δ-contour can be closed to the right, picking up a sum over light-ray operators, as discussed in section 5.2.

Review: light-ray operators and the Lorentzian inversion formula
Let us now proceed with the detailed computation. The objects that will ultimately appear in the OPE expansion of L[O 1 ](x, z 1 )L[O 2 ](x, z 2 ) are light-ray operators [22]. In this section, we collect some facts about these operators that will be needed below. For simplicity, consider first the case where O 1 = φ 1 and O 2 = φ 2 are scalars. Light-ray operators are defined by starting with a bi-local object that transforms as a primary under the conformal group SO(d, 2), The object O ± ∆,J has dimension 1 − J and spin 1 − ∆, which are the quantum numbers of the light-transform of an operator with dimension ∆ and spin J. The ± sign is the signature, which is the eigenvalue under a combination of CRT and Hermitian conjugation, as discussed in section 2.1.
The object O ± ∆,J is meromorphic in ∆ and J and has poles of the form Its residues O ± i,J are light-ray operators. Light-ray operators are analytic continuations in spin of light-transforms of local operators. When J is a nonnegative integer, we have Here, O i,J is a spin-J operator appearing in the φ 1 × φ 2 OPE with coefficient f 12O i,J , and i labels different Regge trajectories. Note that the even-signature light-ray operators O + i,J are analytic continuations in J of light-transformed even-spin operators, while O − i,J are analytic continuations in J of light-transformed odd-spin operators.
Matrix elements of light-ray operators can be computed via a Lorentzian inversion formula. Let φ 3 , φ 4 be primary scalars for simplicity. A time-ordered correlator involving the object O ± ∆,J is given by We use the shorthand notation that φ i is at position x i unless otherwise specified. We also use the notation from [22] where correlators in the state |Ω are physical, while correlators in the state |0 are conformally-invariant structures for the given representations. The structure on the right-hand side of (3.23) is the light-transform of the standard three-point structure for two scalars and a spin-J operator, analytically continued in J, (3.24) ) is given in (2.33).
In (3.23), the time-ordering acts on φ 1 , φ 2 inside O ± ∆,J . Thus the object O ± ∆,J is not really an operator. However, its singularities as a function of ∆ come only from the region where φ 4 acts on the future vacuum and φ 3 acts on the past vacuum, so upon taking residues, we obtain a genuine operator The coefficient function C ± (∆, J) is given by Caron-Huot's formula [23] C ± (∆, J) = κ ∆+J 4 .
Here, we have defined a stripped four-point function g(z, z), which is a function of conformal cross-ratios 15 (3.28) The t-channel double-discontinuity dDisc t is defined by where g or g indicates we should take z around 1 in the direction shown, leaving z held fixed. Similarly, (3.30) where now g or g indicates we should take z around −∞ in the direction shown, leaving z held fixed. Finally, G ∆ i ∆,J (z, z) denotes a conformal block for external scalars with dimensions ∆ i ≡ d − ∆ i , exchanging an operator with dimension ∆ and spin J. In our conventions, it behaves as z for negative cross-ratios satisfying |z| |z| 1. In Caron-Huot's formula (3.26), G and G appear with dimension and spin swapped according to (∆, J) → (J + d − 1, ∆ − d + 1).

More general representations
Before generalizing to non-scalar O 1 , O 2 , we must establish some notation for conformal representations. A primary operator O is labeled by a dimension ∆ and a representation ρ of SO(d − 1, 1), which we can think of as a list of weights under the Cartan subalgebra of SO(d − 1, 1).
When O is local, ρ is finite-dimensional. In this case, we define shadow and Hermitian conjugate representations to have weights where ρ R denotes the reflection of ρ and (ρ R ) * is the dual of ρ R . The conjugate shadow representation O † has weights and thus admits a conformally-invariant pairing with O: For continuous-spin operators, ρ is no longer finite-dimensional. It has weights ρ = (J, λ), where J ∈ C is spin and λ is a finite-dimensional representation of SO(d − 2). We can think of J as the length of the first row of the Young diagram of ρ, while λ encodes the remaining rows. Altogether, we specify the multiplet of O by a triplet (∆, J, λ).
Operators with non-integer J admit a different kind of conformally-invariant pairing Here, O S † has weights In (3.34), we implicitly contract the SO(d − 2) indices in the representations λ and λ * . The measure D d−2 z is defined by is homogeneous of degree 0 in z, so that the integral is well-defined. Using the pairings (3.33) for integer-spin operators and (3.34) for continuous-spin operators, we can construct conformally-invariant pairings between two-and three-point structures, as we will see below.
In the diagrams in section 3.1 and below, we use an outgoing arrow labeled O to denote a representation O, and an ingoing arrow labeled O to denote the dual representation, either O † or O S † as appropriate to O. Joining lines represents the conformally-invariant pairing appropriate for the representations. When (3.37) (Sums over a, b are implicit.) Following the notation of [22] (see also appendix A), we use the subscript Ω to distinguish physical correlators from conformally-invariant structures. Thus, when O 1 , O 2 are not scalars, O ± ∆,J gets generalized to have an additional SO(d − 2) representation label λ and structure label a: O ± ∆,J,λ(a) . It has residues which for integer J and signature ± = (−1) J become light-transforms of local operators: Let O 3 , O 4 be primary operators (not necessarily scalars). Three-point functions containing O ± i,J,λ(a) are given by (3.41) A cartoon diagram for the first integral on the right hand side is given in (3.11). Let us describe the ingredients in (3.41) in detail. Again, we use the shorthand notation that O i is at position x i . The integral is over a Lorentzian configuration where 4 > 1, 2 > 3, and all other pairs of points are spacelike separated. In terms of cross-ratios, this is the same as the integration region 0 < z, z < 1 in (3.26).
The object in the second line of (3.41) is schematic notation for a conformal block obtained by merging the two three-point structures (It is not simply a ratio of three-point and twopoint structures.) The precise merging procedure is described in [22] -it is essentially the usual procedure of summing over products of descendent three-point functions to obtain a conformal block, generalized to continuous spin. We will see some examples below. Pictorially, the block is The three-point structures making up the conformal block are defined by where T i is translation to the next Minkowski patch discussed in section 2.2. Here, (·, ·) L is a conformally-invariant pairing defined by using (3.44) The notation 1/ vol( SO(d, 2)), means that the integral should be gauge-fixed using the Fadeev-Popov procedure. To obtain the last line, we used SO(d, 2) transformations to gauge-fix where e 0 is a unit-vector in the time direction. Finite-dimensional Lorentz indices are implicitly contracted between the two three-point structures.
The two-point structure in the denominator of (3.41) is defined by is the double light-transform of a time-ordered two-point structure. Even though the light-transform of an operator annihilates the vacuum, the light-transform of a time-ordered structure is delta-function supported. After light-transforming again, we obtain a two-point structure that is nonzero at separated points. These details are explained in [22]. The Lorentzian two-point pairing is given by In the last line, we gauge-fixed The last line of (3.41) includes a three-point structure that has been acted on by a combination of CRT and Hermitian conjugation, The role of this term is to ensure that O ± has the correct signature ±1. We give more details on this term in appendix C.

Harmonic analysis on the celestial sphere
Consider a product of light-transforms of local operators, placed at the same spacetime point For simplicity, we take O 1 , O 2 to be traceless symmetric tensors. Each light-transformed operator has dimension 1 − J i , and thus the product (3.48) transforms like an operator with We would like to additionally decompose (3.48) into irreducible representations of the Lorentz group that fixes x. To do so, we can use harmonic analysis [44] (or "conglomeration" [45]) for SO(d−1, 1), treating it as a Euclidean conformal group in d−2 dimensions. Harmonic analysis for SO(d + 1, 1) was reviewed in [46]. In this section, we collect some of the needed ingredients from [46], replacing d → d − 2.
The SO(d − 1, 1) representations that will appear are d−2-dimensional operator representations P δ,λ with scaling dimension δ and finite-dimensional SO(d − 2)-representation λ. We write P δ when λ is trivial. We can think of the null vectors z i ∈ R d−1,1 as embedding-space coordinates for the celestial sphere S d−2 . In this language, for example, we have a celestial three-point structure Here, P δ are not physical operators -they label representations of SO(d − 1, 1), and (3.49) denotes the unique three-point structure (up to normalization) for the given representations. We will also use the notation [22] where λ R is the reflected representation and λ * is the dual representation to λ. 16 We will be particularly interested in principal series representations of SO(d−1, 1), which have δ ∈ d−2 2 + iR. Their significance is that they furnish a complete set of irreducible representations for decomposing objects that transform under SO(d−1, 1). 17 For example, consider a function f (z 1 , z 2 ) that transforms like a product of scalar operators with dimensions δ 1 , δ 2 on S d−2 . It can be decomposed into traceless-symmetric-tensor principal series representations, i.e. representations where λ is the spin-j traceless symmetric tensor representation of SO(d − 2). We denote these by P δ,j .
Let us define the "partial wave" The integration measure in (3.51) is given by (3.36). The quantities in (3.53) are the Plancherel measure µ (d−2) (δ, j) for SO(d−1, 1), a shadow transform factor S , and a three-point pairing ( . Explicit definitions and formulas for all of these quantities are available in [46]. We will not need them here, since these factors will ultimately cancel. The only formula we will need is the "bubble" integral [46] t δ,j Here P δ,j (z)P † δ,j (z ) is a two-point structure on the celestial sphere. 18 The infinite factor vol SO(1, 1) will cancel in all calculations below. In the notation of section 3.1, we have (3.57) The function f (z 1 , z 2 ) can be expanded in partial waves [44,46] The differential operator C δ,j (z 1 , z 2 , ∂ z ) is defined by This is simply the d−2-dimensional version of the usual differential operator appearing in an OPE of conformal primaries. Thus, (3.58) takes the form of an OPE in d − 2 dimensions, where we have a contour integral over the principal series δ ∈ d−2 2 + iR instead of a sum over δ. The contour can sometimes be deformed to give a sum, as we will see below.
Several objects above carry indices, and we are leaving the contraction of indices between dual objects implicit. For example, P δ,j (z) carries j traceless-symmetric indices for the tangent bundle of S d−2 , and consequently W δ,j (z) does too. The differential operator C δ,j also carries these indices, and they are contracted in (3.58).
When f (z 1 , z 2 ) transforms like a product of more general operators in representations P δ 1 ,λ 1 and P δ 2 ,λ 2 , then there can be multiple celestial three-point structures labeled by an index α. Consequently, the partial wave W (α) δ,λ (z) and differential operator C δ,λ,α carry additional structure labels, and we have a more general expansion

Light-ray OPE from the Lorentzian inversion formula
Applying (3.51) and (3.58) to the product (3.48), we have where the partial waves are given by (3.63) and the kernel L δ,j is given by (3.64) Here, we have defined δ i = ∆ i − 1. We are taking O 1 , O 2 to be traceless symmetric tensors for simplicity, so that the partial wave expansion (3.62) only includes traceless symmetric tensors of spin-j on the celestial sphere. We remove this restriction in section 3.4.4. We can represent the kernel L δ,j pictorially as: The blue solid and red dashed lines represent the Minkowski and celestial coordinates, respectively. We will not need to plug in the definition of L δ,j until the very end of our computation, accordingly we will omit the blue and red lines until necessary. The object W δ,j (x, z) is a bilocal integral that transforms like a primary of SO(d, 2) with is another bilocal integral that transforms in the same way. However, the integration kernels that define W δ,j and O ± ∆,J,j(a) are very different, so it is not immediately clear what the relationship is between them. For example, the kernel L δ,j has δ-function support when x 1 , x 2 lie on the future null cone of x. By contrast, the kernel used to define O ± ∆,J,j(a) has support off the null cone of x.
Another puzzle is that L δ,j is nonvanishing for all j ∈ Z ≥0 . By contrast, the object Despite these puzzles, W δ,j will actually be a linear combination of O ± ∆,J,j(a) 's. A necessary condition for this to be true is that exotic values of j (i.e. values that aren't allowed in the O 1 × O 2 OPE) lead to vanishing W δ,j , even though L δ,j is nonzero. We will see that this is indeed true.

Matrix elements of W δ,j
To determine W δ,j (x, z), it suffices to study its matrix elements in states created by local primary operators O 3 and O 4 : In the last line, we introduced θ-functions θ(4 > 1)θ(2 > 3) that remove the regions where x 1 is spacelike from x 4 and x 2 is spacelike from x 3 . They are redundant because commutators vanish at spacelike separation. However, they will play an important role later, so we include them. Pictorially, we have . (3.68) To avoid visual clutter, we will omit theta functions in our diagrams. The fact that (3.67) is the integral of a double commutator suggests that we should relate it to the Lorentzian inversion formula. In fact, the proof of the generalized Lorentzian inversion formula in [22] proceeds from an expression similar to (3.67). We now follow the steps of that derivation.
First note that conformal invariance implies is a basis of structures for the given representations, and A b (δ, j) are coefficients we would like to determine. A sum over b is implicit. In terms of diagrams, that is Following [22], it is useful to act on both sides with T 4 (equivalently relabel x 4 → T 4 x 4 = x + 4 ), giving Note that T 4 simply acts on three-point structures by multiplication by a phase. Nevertheless, it is useful to keep the abstract notation in (3.71). This relabeling turns the causal relationship 4 > x > 3 − into 3 > 4 and 3 ≈ x and 4 ≈ x (see figure 7). Here i ≈ j means x i is spacelike from x j , see appendix A. We write these relationships compactly as (3 > 4) ≈ x. Our Lorentzian pairing (3.44) is defined for this type of causal relationship. Thus, to isolate A b (δ, j), we can take the Lorentzian pairing of both sides with a dual structure This gives outside the brackets, the configuration of points inside the brackets (figure 7) is obtained from figure 6 by relabeling 2 → 2 + and 4 → 4 + . Note that the bracketed quantity is a conformally-invariant function of x 1 , x 2 , x 3 , x 4 that is an eigenfunction of the conformal Casimirs acting simultaneously on points 1, 2 (or equivalently 3, 4). Hence it is a linear combination of conformal blocks. To compute it, we can follow the computation in appendix H of [22]. The kernel T 2 L δ,j forces x to lie on the past lightcone of x 1 and the future lightcone of x 2 (see figure 7). Thus, as x 1 → x 2 (equivalently x 3 → x 4 ) the integration point x is forced to stay away from x 3 , x 4 . This means we can compute the integral by taking an OPE-type limit x 3 , x 4 → x inside the integrand (figure 8).
j" means i is on the future null cone of j. Let us imagine that 1, 2, 3, 4 are fixed and ask where x can be. We see that x is spacelike from 3, 4 and 3 > 4, equivalently Furthermore, x is constrained to lie on the S d−2 given by the intersection of the past lightcone of 1 and future lightcone of 2. We show lightlike segments between x and 1 (solid blue) and between 2 + and x + (solid purple), which are subsets of the light-transform contours from figure 6. The image of the solid purple segment under T −1 is shown in dotted purple.
Here, O F † has the weights of something that can be paired with We must also replace The two-point function in (3.74) is then integrated against the 12 three-point structure, giving a Lorentzian shadow transform The result is the conformal block In the second line, we use the notation for a conformal block where the three-point structures in the numerator should be merged using the two-point function in the denominator. The precise meaning of this notation is the first line of (3.77). Pictorially, the block can be represented as At this point, we can understand why A b (δ, j) vanishes for exotic j not allowed in the O 1 × O 2 OPE. Recall that L δ,j does not vanish for exotic j. This is possible essentially because L δ,j involves δ-functions, and the presence of these δ-functions changes the space of conformally-invariant three-point structures. However, the shadow-transformed structure S[T 2 L δ,j ] does not contain any δ-functions because they are integrated over in (3.76). Thus, it is subject to the usual classification of conformally-invariant three-point structures. It transforms like a three-point function

Relating to the inversion formula
After writing the quantity in brackets in (3.73) as a conformal block, our formula for A b (δ, j) looks extremely similar to the Lorentzian inversion formula (3.41). There are two main differences. Firstly, our formula for We need to express the former as a linear combination of the latter, and this is achieved by pairing with The second difference is that (3.73) involves an integral only over the double-commutator |Ω , corresponding to the "t-channel" term in (3.41). It does not include a contribution from the "u-channel" term. This is accounted for by averaging over even and odd spins.
In summary, comparing (3.77) and (3.41), we find Note that in this formula, the ratio of two-point structures is simply a number -it does not refer to the formation of a conformal block. The three-point pairing can be simplified further by rewriting it as a two-point pairing: In full detail, we have (3.81) In the last equality, we made the change of variables x 2 → T −1 2 x 2 = x − 2 and recognized the integrals over x, z, x , z as a Lorentzian two-point pairing (3.46). The infinite factors vol SO(1, 1) 2 will cancel shortly. Plugging in the definition of L δ,j (3.64), we have δ,j . With hindsight, we have included a factor vol SO(1, 1) −1 on the left-hand side so that q (a) δ,j is finite. Applying the bubble formula (3.56), we find Meanwhile, Lorentz invariance and homogeneity imply Finally, combining (3.62), (3.69), (3.40), and writing δ = ∆ − 1, we have (3.88) The differential operator C δ,j is defined by (3.59). Here, j max is the maximum "non-exotic" value of j -specifically, the maximum length of the second row of the SO(d − 1, 1) Young diagrams associated to operators appearing in the As an example, consider the case where O 1 = φ 1 and O 2 = φ 2 are scalars. 20 We have Let us compute q δ,0 and r δ,0 . The unique Wightman structure for two scalars and a spin J = −1 operator is The light-transform of O is given by (2.32) with the relabeling (1, 2, 3) → (2, 1, 0), and J = −1.
In embedding-space language, we find We should now specialize X 0 = (1, 0, 0) and compute the remaining light transforms and integrate To get the third line, we integrated over α 1 . The infinite factor vol SO(1, 1) cancels against the unbounded integral over α 2 . Alternatively, we could have used SO(1, 1)gauge invariance to fix α 2 = 1. Thus, we find Meanwhile, the quantity r ∆−1,0 was computed in [22] to be The ratio q ∆−1,0 /r ∆−1,0 is simply 1! We find (3.96)

Generalization and map to celestial structures
Let us summarize our result so far in slightly different language. In addition, we will generalize to the case where O 1 , O 2 are not necessarily traceless symmetric tensors.
transform as tensors in the representation λ i on the celestial sphere. To describe them, we can use the notation of appendix B. We write L[O i ](∞, z, w), where w = w 1 , . . . , w n−1 ∈ C d is a collection of null polarization vectors orthogonal to z, encoding rows in the Young diagram of λ i . The light-ray operators appearing in the OPE may also have nontrivial λ. In what follows, O stands for the representation with weights (∆, J, λ) = (δ + 1, In the notation of section 3.4.2, when λ is the spin-j representation of δ,j /r δ,j )C δ,j . The derivation of section 3.4.2 generalizes straightforwardly to give 21 where Res G H denotes restriction of a representation of group G to its subgroup H. One can derive this rule by considering the three-point structure O 1 (x 1 )O(x 0 , z)O 2 (x 2 ) as a function of x 1 , x 2 , x 0 , and z. It furthermore carries indices for ρ 1 , ρ 2 and λ which we have suppressed. Using conformal invariance, we can fix x 1 , x 0 , x 2 to lie on a line in the time direction and z to be (1, 1, 0, . . . ). The stabilizer group of this configuration is SO(d − 2), and the correlator must be invariant under this stabilizer group. This leads to (3.99). The result is equivalent to the rule stated in the introduction, which implies that the λ that appear are exactly those for which a Lorentzian inversion formula exists. Equation (3.97) essentially defines a map from a three-point structure scalars, this map is surprisingly simple: it takes the standard Wightman structure (3.89) to the standard differential operator C δ,0 . In fact, this map turns out to be simple in general. We claim that D (3.100) 21 As we discuss in section 4.1, only the term with signature (−1) J 1 +J 2 contributes at z1 ∝ z2. Here, we use embedding-space language, as explained in appendix B. The objects V i,jk and H ij are defined in appendix D, see also [40]. The two-point and three-point structures above are each specialized to the "celestial" locus . (3.101) This corresponds to placing all three operators on the celestial sphere given by the intersection of the future lightcone of the origin and the future lightcone of spatial infinity (figure 10). It is easy to check that the three-point function on the right-hand side of (3.100), after restricting to the celestial locus, has homogenity −δ which easily gives D δ,0 = C δ,0 . We have checked that (3.100) is equivalent to (3.97) for arbitrary traceless symmetric tensor representations by explicit calculation, see appendix D. It can also be justified by examining the limit as z 1 → z 2 in (3.97). It would be nice to prove (3.100) more directly.
One important caveat to this discussion is that it only applies for separated points, i.e. when z 1 is not proportional to z 2 . As we will see in section 6, this map has to be modified in some special cases if one wishes to study z 1 ∝ z 2 contact terms. It is instructive to see how commutativity appears from the point of view of the light-ray OPE. This will lead to nontrivial consistency conditions on the space of light-ray operators. In the remainder of this section, we assume the light-ray operators L[O 1 ] and L[O 2 ] are not coincident z 1 ∝ z 2 . We discuss how our arguments should be modified for coincident lightrays in section 6.
We derived an expression for Taking the difference with (3.98), we get the commutator In the last line, we have assumed that the behavior of the integrand at large ∆ is such that we can deform the ∆-contour to pick up poles on the positive real axis, obtaining a sum over Regge trajectories i. For more detail on deforming the ∆ contour, see section 5.2. The operators on the right-hand side of (4.2) have spin J = J 1 + J 2 − 1 and signature (−1) J . For example, when J 1 ≡ J 2 mod 2, the commutator is given by a sum of light-ray operators with odd J and odd signature. This is easy to understand from symmetries: the light-transforms L[O i ] have signature (−1) J i , and the commutator introduces an additional −1, since Hermitian conjugation reverses operator ordering.
These quantum numbers are exactly the ones needed for O (−1) J 1 +J 2 −1 i,J 1 +J 2 −1,λ i to be the lighttransform of a local operator. Let us assume this is the case (we return to this assumption in section 4.2). Using (3.39), we have where each O i has quantum numbers (∆, J, λ) = (δ i + 1, There are now two slightly different cases. In the first case, the local operators that would appear in the right hand side of (4.3) are not allowed to appear in the Euclidean OPE. 22 In other words, f 12O † i (a) are zero by selection rules. In this case we immediately find that the commutator is identically zero.
The second case is when f 12O † i (a) are not forbidden by Euclidean selection rules. To see that the commutator vanishes in this case, recall that the differential operator D (a) δ,λ is nonzero only if the three-point structure · · · (a) survives the map to celestial structures (3.100). However, the structure · · · (a) cannot survive this map if it also appears in a three-point function of local operators, modulo a small subtlety to be discussed below. The reason is that V 0,21 | celestial = 0, so the right-hand side of (3.100) vanishes unless · · · (a) contains a pole V −1 0,21 that can cancel this zero. Such poles are not allowed in three-point functions of local operators (which must be polynomial in polarization vectors z i ). It follows that There is a small subtlety in the above argument, which is due to the fact the statements about the map to celestial structures are correct for separated points only. As we will show in section 6, it does sometimes happen that tensor structures appearing in three-point functions of local operators map to contact terms on the celestial sphere.
The above argument was somewhat abstract, so let us give a concrete example. Consider the case O 1 = O 2 = T , where T is the stress tensor in a 3d CFT. The commutator [L[T ], L[T ]] is a sum of spin-3 light-ray operators on odd-signature Regge trajectories. By our assumption above, such operators are light-transforms of local spin-3 operators. However, the T × T OPE does not contain any spin-3 operators, due to selection rules and Ward identities [24,47].

Finishing the argument with conformal Regge theory
The key step in the above argument was the assumption that where O i is a local operator of spin J 1 + J 2 − 1. As discussed in section 3. Euclidean OPE. 23 More precisely, this is true under the condition J 1 + J 2 − 1 > J 0 , which comes from the fact that the Lorentzian inversion formula is only guaranteed to reproduce Euclidean OPE data for spins larger than J 0 . We return to this condition later in this section.
We are also interested in the case where f 12O † i (a) is forbidden by the selection rules of the Euclidean OPE. In this case, there is nothing that we can write in the right-hand side of (4.5) and so we would like to argue that in this case We can argue for (4.6) using conformal Regge theory and boundedness in the Regge limit. Let us first review some aspects of conformal Regge theory, using a four-point function of scalars for simplicity. We follow the presentation of [22]. One starts with a four-point function in a Euclidean partial wave expansion Here, we've split each partial wave into a piece F ∆,J (x i ) that dies at large positive J and a piece H ∆,J (x i ) that dies at large negative J. For simplicity, we only keep track of F ∆,J . The sum runs over nonnegative integer J because these are the allowed spins in the Euclidean OPE.
23 Saying that f 12O † i (a) = 0 even thought it is allowed by Euclidean OPE amounts to saying that there is no corresponding pole in O ± ∆,J,λ(a) and hence no O The key step is the Sommerfeld-Watson transform: we rewrite the sum over J as a contour integral where Γ encircles all nonnegative integers clockwise. We now deform the contour Γ → Γ towards the imaginary J axis (left panel of figure 11). When we do, we pick up any poles or branch cuts in the integrand that were not encircled by the original contour Γ. We refer to such singularities as "Regge poles". In figure 11 we show a single Regge pole at J = j(ν). To see this, imagine applying conformal Regge theory to T T O 3 O 4 . We will arrive at the generalization of (4.8), where the factor 24 1 1 + e −iπJ (4.9) will create poles for all odd J, including J = 1 and J = 3. However, since J = 1, 3 are not allowed in the Euclidean OPE, the contour Γ must not circle these poles, see right panel of figure 11. This implies that these poles will be picked up by Γ . If J 0 < 3, we must conclude that the residue of J = 3 pole vanishes, and so This straightforwardly generalizes to other situations, and we conclude that (4.5) holds provided J 1 + J 2 − 1 > J 0 . If this condition is satisfied, the arguments in the previous section This is precisely the same result as obtained in [1], where it was shown that J 1 + J 2 − 1 > J 0 is a necessary condition for the product to be well-defined and commutative.

Superconvergence in ν-space
We have seen that when J 1 + J 2 − 1 > J 0 , the commutator (4.2) vanishes. This follows from the analysis of [1], or alternatively from the arguments of sections 4.1 and 4.2 using the light-ray OPE and conformal Regge theory. 25 From (4.2), commutativity is equivalent to the statement that where we have written ∆ = d 2 + iν, and the above conditions hold for all ν ∈ R. Using (3.40), we can also write this as What constraints do these conditions imply on CFT data? For simplicity, let us specialize to the case where O 1 = O 2 = O 3 = O 4 = φ are identical scalars (so that λ = • and the labels a, b are trivial). Recall that C ± (∆, J) is computed by plugging the physical four-point function g(z, z) into the Lorentzian inversion formula (3.26) and performing the integral. The four-point function has an expansion in t-channel conformal blocks that converges exponentially inside the square z, z ∈ (0, 1) [48]: into the t-channel block expansion, the t-channel block expansion for dDisc[g] converges exponentially inside the square as well.
Inserting (4.13) into the Lorentzian inversion formula, we obtain an expression for C ± (∆, J) as a sum where B ± (∆, J; ∆ , J ) is the Lorentzian inversion of a single t-channel block. The functions B ± (∆, J; ∆ , J ) were computed in d = 2 and d = 4 dimensions in [49]. We expect the sum (4.14) to converge whenever ∆ = d 2 + iν is on the principal series and J > J 0 is larger than the Regge intercept. We argue for this using the Fubini-Tonelli theorem in appendix E.
Plugging (4.14) into (4.12), we obtain an infinite set of sum rules 26 As we will see in section 5.4, these are precisely the superconvergence sum rules of [1], written as a function of a different variable ν. In ν-space, we have a clear argument that the sum is convergent. More generally, for spinning light-ray operators, we find where B ± ab;cd is the spinning analog of B ± , including three-point structure labels a, b, c, d. Equation (4.16) may be a good starting point for analyzing contributions of stringy states to superconvergence sum rules in holographic theories.
We give more details on the relationship between (4.16) and the sum rules from [1] in section 5.4.

Celestial blocks
For the purpose of computing event shapes, we would like to apply the light-ray OPE inside momentum eigenstates. Matrix elements of individual light-ray operators O ∆,J in momentum eigenstates are proportional to the one-point event shape (2.37). To apply the OPE (3.88), we must understand how to apply the differential operator C δ,0 (z 1 , z 2 , ∂ z 2 ) to these matrix elements: We call the resulting objects "celestial blocks" because they capture the full contribution of a light-ray operator and its z-derivatives to an event shape. The right-hand side of (5.1) is fixed by Lorentz-invariance and homogeneity to have the form 2) 26 We expect that J0 < −1 is not true in most interesting theories. Here, we have this condition because we specialized to scalar operators for simplicity.
where the cross-ratio ζ is given by Furthermore, it is an eigenvector of the quadratic Casimir of the Lorentz group acting simultaneously on z 1 , z 2 , or equivalently acting on p. Specifically, it is killed by the differential operator This gives the Casimir differential equation Meanwhile, from the definition of C δ,0 , we see that where ". . . " represent higher-order terms in the separation between z 1 and z 2 on the celestial sphere.
In terms of f (ζ), this becomes The solution to the Casimir equation with boundary condition (5.7) is where we have written δ i = ∆ i − 1 for future convenience. Essentially the same function has appeared previously in the literature as the conformal block for a two-point function of local operators in the presence of a spherical codimension-1 boundary [25,26]. The reason is that the momentum p breaks SO(d − 1, 1) in a similar way to a boundary in a d − 2 dimensions. To see this, consider an embedding space coordinate X ∈ R d−1,1 for a d − 2-dimensional CFT. A spherical codimension-1 boundary is specified by P · X = 0, for some spacelike P ∈ R d−1,1 [50]. The vector P breaks the symmetry from SO(d−1, 1) to SO(d−2, 1). In our case, we have a timelike vector p that breaks the symmetry from SO(d − 1, 1) to SO(d − 1). However, the Casimir equation is the same in both cases, and the only difference is a minus sign in our definition of the cross-ratio ζ. Now, we can finally write the light-ray OPE for a two-point event shape. For simplicity, we consider the case where the sink, source, and detectors are all scalars. From the OPE (3.88), we have where 0|φ 4 Oφ 3 |0 is the standard Wightman structure (3.89) with 2 → 4 and 1 → 3. Plugging in the expression (2.37) for the light transform and Fourier transform (with appropriate relabelings), and using (5.2) we find In the special case where the sink and source are the same φ 3 = φ 4 = φ, it is natural to define an expectation value by dividing by a zero-point event shape:

Contour deformation in ∆ and spurious poles
In (5.11), the celestial block expansion of G φ 1 φ 2 (ζ) takes the form of an integral over the principal series ∆ ∈ d 2 + iR. When ζ < 1, the celestial block f ∆ 1 ,∆ 2 ∆ (ζ) is exponentially damped in the right-half ∆-plane, so we can deform the contour into this region and pick up poles in the integrand.
The coefficient function C ± (∆, J) contains poles of the form where p ± i (J) are products of OPE coefficients analytically-continued in J, and ∆ ± i (J) are dimensions analytically-continued in J. 27 When we deform the ∆-contour, we pick up contributions from these poles. They are interpreted as light-ray operators in the light-ray OPE.
In general, C ± (∆, J) can also contain "spurious" poles at ∆ = d + J + n for integer n, originating from poles in the conformal block G J+d−1,∆−d+1 (z, z) in the Lorentzian inversion formula (3.26). In the usual conformal block expansion, these spurious poles cancel with poles in G ∆,J (z, z) that are encountered when deforming the ∆-contour from the principal series to the positive real axis [23,44,51,52]. However, the celestial block f ∆ 1 ,∆ 2 ∆ (ζ) does not have poles in ∆ to the right of the principal series. 28 Thus, it is not clear how spurious poles in C ± (∆, J) could cancel.
Remarkably, it turns out that when we set J = −1, spurious poles in C ± (∆, J) are absent. This can be seen as follows. Note that the following combination of conformal blocks is free of poles to the right of ∆ = d 2 [23]: .
so that G J+d−1,∆−d+1 | J=−1 is again free of poles to the right of ∆ = d 2 . 29 Let us also comment on the case d = 3. There, the Lorentz group is SL(2, R), whose harmonic analysis is slightly different than for the higher-dimensional Lorentz groups. In particular, the Plancherel measure for SL(2, R) has support on discrete series representations in addition to principal series representations. In this case, we expect the contribution of discrete series representations to be cancelled by poles in C ± (∆, J), in the same way as occurs in the four-point function of fermions in the SYK model [55].
The end result is that spurious poles and discrete state contributions are absent in the celestial block expansion for all d > 2. Deforming the ∆-contour, we obtain G φ 1 φ 2 (ζ) as a 27 We comment on the possibility of non-simple poles or branch-cuts in ∆ below. 28 Assuming |∆1 − ∆2| is not too large. See [22,46] for examples of how to treat the case where |∆1 − ∆2| is large. 29 Note that the case d = 2 is not relevant to our discussion, since there is no transverse space R d−2 in which to consider the light-ray OPE.
sum of contributions from light-ray operators when ζ < 1 Here, i labels Regge trajectories and we have abbreviated ∆ i = ∆ i (J = −1) and p ± ∆ i = p ± i (J = −1). When ζ = 1, the celestial block f ∆ 1 ,∆ 2 ∆ (ζ) is no longer exponentially damped at large positive ∆, so (5.18) does not apply. We will see examples of how to treat the case ζ = 1 in section 7.3.
We expect that the above analysis extends to the more general light-ray OPE where O 1 and O 2 have general spins J 1 and J 2 . In this case, the contour integral over ∆ in (3.98) should become (in schematic notation) Let us return to the assumption that C ± (∆, J) (more generally O ± ∆,J,λ(a) ) has only simple poles in ∆. This is known to be true when the signature and spin are such that C ± (∆, J) describes light-transforms of local operators, i.e. when J ∈ Z ≥0 and ±1 = (−1) J . However, for more general values of J, the singularity structure of C ± (∆, J) as a function of ∆ is not known. In the presence of other types of singularities like higher poles and branch cuts, one can define light-ray operators O ± i,J in terms of discontinuities across those singularities, and then suitable generalizations of (5.18) and (5.19) apply.

Consider an event shape of identical scalars φ(p)|L[φ]L[φ]|φ(p) .
(5.20) The four-point function of φ's can be split into connected and disconnected pieces After taking the light-transforms to compute (5.20), the disconnected terms in (5.21) must drop out. The reason is that the light-transform of a Wightman two-point function vanishes, since the light-transformed operator annihilates the vacuum.
Despite the simplicity of this argument, vanishing of disconnected contributions in the celestial block expansion is slightly nontrivial. The mechanism is similar to the vanishing of spurious poles discussed in section 5.2. Note that the contribution of disconnected terms to C + (∆, J) is given by the OPE coefficient function of Mean Field Theory (MFT). This is [45,46] C MFT (∆, J) Due to the factor Γ(J + 1) −1 , this function vanishes at J = −1. Thus, we have where the subscript c indicates the contribution of the connected term alone. Consequently, disconnected terms don't contribute to the celestial block expansion (5.11), as expected.

Relationship to t-channel blocks and superconvergence
In [1], we introduced an alternative expansion for event shapes in terms of t-channel eventshape blocks G t ∆ ,J (p, z 1 , z 2 ). We computed G t ∆ ,J (p, z 1 , z 2 ) by inserting a projector onto an individual conformal multiplet O ∆ ,J between L[O 1 ] and L [O 2 ]. An alternative way to obtain it is to first find the contribution of the t-channel four-point block G ∆ ,J (1 − z, 1 − z) in the Lorentzian inversion formula and then plug this into the celestial block expansion (5.11).
For example, in the case of scalars O i = φ i with dimensions ∆ i , we claim that Here B(∆, J; ∆ , J ) is the Lorentzian inversion of a single t-channel block (defined near (4.14)) and G t ∆ ,J (p, z 1 , z 2 ) are the functions defined in (5.160) in [1]. We have verified this identity numerically for some special cases in d = 2 and d = 4 using formulas for B ± from [49].
One property of event-shape t-channel blocks is that they are regular in the limit z 1 → z 2 . This is consistent with (5.25) because the Lorentzian inversion of a single t-channel block contains double and single poles at double-trace values of ∆, and no other singularities in ∆ [23,49]. Thus, when we deform the ∆-contour in (5.25) to pick up poles, we obtain only double-trace celestial blocks, which are indeed regular near ζ = 0. Equation (5.25) lets us clarify the relationship between (4.16) and the superconvergence sum rules written in [1]. Equation (4.16) is a superconvergence sum rule written in ν-space, obtained by decomposing the commutator (4.2) into celestial conformal partial waves. By contrast, the sum rules of [1] are obtained by decomposing the commutator into t-channel conformal multiplets (each of which is a finite sum of spherical harmonics on the celestial sphere). To go from (4.16) to the formulas of [1], we can integrate (4.16) against celestial blocks.

Contact terms
In addition to giving a convergent expansion for the product for z 1 ∝ z 2 , the OPE expansion (3.98) can also capture contact terms in the limit z 1 → z 2 , such as those studied in [10]. A complete description of possible contact terms is beyond the scope of this work. Instead, in this section, we will study two specific examples and explain how (3.98), suitably interpreted, can be used to determine contact terms at z 1 ∝ z 2 . The contact terms in both examples ultimately arise for the same reason: we must be careful about the distributional interpretation of the integrand in (3.98). In particular, we must ensure that this distribution is analytic in ∆.

Charge detector commutator
Our first example concerns contact terms in the OPE of charge detectors, 30 where J a is a current for a global symmetry group G, and a is an adjoint index for G. From [10], the commutator should contain a contact term where f abc are the structure constants of G, and δ d−2 (z 1 , z 2 ) is a delta-function on the null cone. To see this, note that

4)
30 Note that a sufficient condition for the charge-charge correlator to exist is J0 < 1. Therefore, we expect that we encounter divergences in gauge theories both in the weak and strong coupling perturbative expansion.
On the other hand, we expect that it exists in the critical O(N ) model. and we should have ](x, z 2 )] vanishes for z 1 ∝ z 2 we arrive at (6.3). Vanishing of this commutator for z 1 ∝ z 2 was justified in [1] if J 0 < 1. This also follows from the arguments of section 4.1.
We would now like to argue for (6.3) using the light-ray OPE. Recall that the commutator is a sum of light-transforms of local operators with spin J 1 + J 2 − 1 = 1. Thus, we must understand three-point structures where O c ∆ is a local spin-1 operator in the adjoint representation of G, with dimension ∆. There exist two tensor structures .

(6.8)
Here we used the convention V 1 = V 1,23 and its cyclic permutations, and H ij , V i,jk , X ij are defined in appendix D, see also [40]. The second structure cannot appear in the local threepoint function (6.6) for generic ∆ because of the term involving V −1 3 . However, when O = J and ∆ = d − 1, the term with V −1 3 vanishes, and this structure is allowed. 31 Moreover, at ∆ = d − 1 Ward identities fix the coefficient λ 2 of the second structure as where C J is defined by (6.10) We will now argue that the second structure survives the map to celestial structures even at ∆ = d − 1 as a contact term. According to the results of section 3.4.4, naïvely, when ∆ = d − 1 the structure (6.8) does not survive the map to celestial structures because it does not contain factors of V −1 3 .
However, this is only true for z 1 ∝ z 2 . When z 1 ∝ z 2 , this claim must be modified. It should be possible to see this directly by performing a more careful analysis of the map to celestial structures. However, we can also use the following indirect argument. According to the results of section 3.4.4, for generic ∆ the structure (6.8) gets mapped to the following OPE contribution This result is obtained using (3.98) and (3.100). Here, we put x 0 at past null infinity and used transverse coordinates y i to parametrize the detectors. The factor (∆ − d + 1) appears because only the term with V −1 3 in (6.8) contributes. We can now take the limit ∆ → d − 1 in this expression, taking into account that while the higher-order terms in the parenthesis in (6.11) are less singular and go to zero. We then find It follows from the discussion in 4.1 that this is the only term that survives after taking the commutator, 32 and so we find as expected. We expect that it should be possible to generalize this discussion to other commutators considered in [10]. The main difficulty in this generalization is that the operators considered in [10] are descendants of light transforms [1]. We expect that the light-ray OPE can be generalized to OPE of these descendants; we briefly discuss this direction in section 8.

Contact terms in energy correlators in N = 4 SYM
Our second example concerns the celestial block expansion (5.11). For simplicity, we will specialize to ∆ i = 2, which is relevant for the case of energy-energy correlator in N = 4 SYM studied in the next section, see (7.11) and (7.12).
We will focus on the function that multiplies C + (∆, −1) under the integral in (7.12). Naïvely, this function vanishes at ∆ = 3 + 2n due to the Γ-function in the denominator. However, at the same time the factor ζ ∆−7 2 becomes singular as ζ n−2 if n = 0, 1. To interpret (7.12) in a distributional sense and simultaneously treat it as the integral of an analytic function, we must ensure that we make sense of f ∆ (ζ) as a distribution that is analytic in ∆. This distribution must be defined for ζ ∈ [0, 1].
For Re ∆ > 5, f ∆ (z) is integrable near ζ = 0 and thus uniquely defines a distribution analytic in ∆. Therefore, for all other ∆ the distribution f ∆ (z) must be defined by analytic continuation in ∆. For example, and similarly The other values of ∆ that give negative integer powers of ζ are ∆ = 1−2n for n ≥ 0. In these cases, we find f ∆ (ζ) = 0, due to higher-order zeros coming from Γ( ∆−1 2 ) 3 in the denominator. For other values of ∆, the exponent of ζ, even if large and negative, is non-integer, and analytic continuation in ∆ defines a distribution even though there is no zero coming from the Γ-functions.
As we will see in the next section, the function relevant for scalar event shapes in N = 4 SYM is ζ 2 f ∆ (ζ). Since we only a found delta function and its first derivative in f ∆ (ζ), this means that there are no contact terms in the scalar event shapes. Alternatively, by repeating the above analysis for ζ 2 f ∆ (ζ) we find that it stops being integrable at ∆ = 1, at which point the Γ( ∆−1 2 ) 3 factor in denominator kicks in, and we do not get interesting distributions. We will also need a slight refinement of the result for f ∆ (ζ) near ∆ = 5. Near this point, the only term non-integrable in ζ comes from the leading term of the 2 F 1 , so we can write and ζ ∆−7 2 Here the distribution [ζ −1 ] 0 is in principle defined by the Laurent expansion in which it appears. Otherwise, one can define it as the unique distribution which agrees with ζ −1 on test functions which vanish at ζ = 0 and for which Similar comments apply to [ζ −1 log ζ] 0 . It is straightforward to obtain subleading terms in (6.21). In section 7 we will see that the contact terms we just described are necessary to satisfy the Ward identities for the energy-energy correlator.

Event shapes in N = 4 SYM
In this section, we apply the machinery derived above to scalar half-BPS operators in N = 4 SYM. We re-derive some previous results both at weak and strong coupling and make further predictions. The basic operators of interest are which transform as traceless symmetric tensors of SO(6), i.e. in the 20 representation. These operators are part of a supermultiplet that also contains R-symmetry conserved currents, supersymmetric currents, and the stress tensor, among other operators. We will study a scalar event shape, where the detectors, source, and sink are all built from O IJ 's. Superconformal Ward identities relate the four-point function of 20 scalars to four-point functions of other operators in the stress tensor multiplet [56,57]. These relations were explicitly worked out in [27,28]. In particular they imply a simple relation between scalar event shapes and energy-energy correlators which we review below.
The structure of the section is as follows. We first review basic properties of the fourpoint function of 20 operators and define the scalar event shape of interest. We then explain its relation to the energy-energy correlator which is the main subject of our interest. In sections 7.3, 7.4, 7.5, we apply the light-ray OPE at weak coupling at tree-level, 1-loop, and 2-loops (at leading and subleading twist), finding agreement with previous results and completing them with contact term contributions. In section 7.7, we use known OPE data to make a new prediction for the the small-angle limit at 3 and 4-loops. In section 7.8, we apply the OPE at strong coupling, again finding agreement with previous results.

Review: event shapes in N = 4 SYM
The scalar event shape of interest is built from O IJ 's, where the R-symmetry indices are contracted with particular polarizations. Following the conventions of [6], we treat the inand out-states differently from the detectors. For the in-and out-states, we contract O IJ with null polarization vectors Y I ∈ C 6 , The two-point function of O(x, Y ) is given by  , z), where z is a future-pointing null vector. 33 Our scalar event shape is defined by This event shape is sometimes called "scalar flow," by analogy with energy flow observables that measure the flow of energy at null infinity. Following [6], let us choose the R-symmetry structures  33 The factor of 1 2 is for consistency with the definitions of [6].
(The energy correlators are independent of Y .) In [27], this relation was derived while ignoring contact terms at z 1 ∝ z 2 . We will find that consistency with the OPE requires correcting this relation by contact terms. We expect that these contact terms come from the protected part of the 20 four-point function. We discuss them in more detail below. Using (5.13), the scalar event shape can be written where the factor ( 1 2 ) 2 in (7.9) comes from O(z) z). The function G OO (ζ) has a celestial block expansion given by (5.11): Here, C + (∆, −1) encodes the OPE data of the OOOO four-point function, analytically continued to spin J = −1. We discuss this four-point function in section 7.2. Since the 105 representation appears in the symmetrized tensor square of the 20 representation, the OPE contains only even spin operators. This is the reason for the absence of C − (∆, −1) in (7.10). The superconformal Ward identity (7.8) lets us obtain a celestial block expansion for the energy-energy correlator in terms of OPE data for the scalar four-point function. Let us define the function F E (ζ) by Here we include the factor 4 vol S 2 = 16π because it simplifies the Ward identities discussed below. The relation (7.8) implies that F E has the celestial block expansion ). (7.14) Here, C + (∆, −1) is the same function that enters (7.10). The function ξ(ζ) represents the protected coupling-independent contact terms mentioned in (7.8). Below, we fix ξ(ζ) by requiring consistency with Ward identities and check that it is indeed independent of the coupling (at one and two loops, and at strong coupling). Its effect is to remove the contribution of short multiplets from C + (∆, −1) in (7.12). It would be interesting to derive the presence of ξ(ζ) from first principles along the lines of [27].
For 0 < ζ ≤ 1, the superconformal Ward identity relating scalar flow and the energyenergy correlator takes the simple form However, the celestial block expansion (7.12) also captures contact terms at ζ = 0 that are not captured by (7.15). When evaluating the celestial block expansion for ζ < 1, we will find it convenient to close the ∆-contour to the right as discussed in section 5.2 and write the event shape as a sum over Regge trajectories, see (5.18). For example, we have Before computing F E (ζ), let us comment on some of its properties. First, F E (ζ) is constrained by Ward identities. By integrating E(z 1 ) over the celestial sphere with the appropriate weight, we can produce different components of the translation generators P µ . In the energy correlator (7.11), these must evaluate to p µ , which leads to the Ward identities Since (7.17), (7.18) are sensitive to the values of F E (ζ) at arbitrary angle ζ they can be used as a nontrivial consistency check on the computations of F E (ζ). Finally, note that F E (ζ) has a weak-coupling expansion F E (ζ) is explicitly known up to two-loop order [32], and as a two-fold integral at three loops [33]. 34 It is also easily computable at strong coupling, reproducing the result of Hofman and Maldacena [5].

Review: four-point function of 20 operators
The main ingredient in computing F E (ζ) is the four-point function of 20 operators that enters in the definition of the scalar event shape (7.5), specialized to the R-symmetry structures (7.7). This is (7.20) 34 The quantity EEC(ζ) computed in [32,33] is equal to our FE (ζ).
It will be convenient to write G (105) (u, v) in two different ways. Firstly, we have where the central charge c is given by v encodes the dependence of the correlator on the coupling a (it is zero for a = 0). It is known explicitly up to three loops [58]. The integrand for G (105) (u, v) is known up to ten loops in the planar limit [59,60].
The other way of writing G (105) (u, v) is to organize it into the contribution of short and long supermultiplets in the superconformal block expansion, where G (short) (u, v) encodes the contribution from protected operators and was computed in [61]. H(u, v) encodes the contribution of long multiplets and can be written in terms of superconformal blocks as follows where g ∆,J (u, v) are the usual conformal blocks and a ∆,J is the three-point coupling to a given superconformal primary, see e.g. [61]. 35 We will use p ∆,J to denote the three-point coupling to a given conformal primary. Note that only even spin operators enter in the OPE decomposition of G (105) (u, v). Because of the factor Γ( 3−∆ 2 ) −1 in (7.16), most protected operators from G (short) (u, v) will not contribute to F E (ζ). However, operators with dimensions ∆ = 3 and ∆ = 5 can contribute contact terms at ζ = 0, in accordance with the discussion in section 6.2.

Tree level
To get the tree-level correlator we set Φ(u, v) = 0 in (7.20). Recall from section 5.3 that where C c (∆, −1) corresponds to the connected part of the four-point function. Written in terms of cross-ratios, the connected tree-level correlator takes the form (7.26) 35 Note that [62] used a different conformal block normalization.
Plugging into the inversion formula, we have (7.27) where the factor of 2 in front comes from the fact that the t-and u-channel terms in the inversion formula are equal. dDisc 1 1−z is delta-function supported near z = 1. To regulate it, we will replace Recall that [53,54] Doing the integral and removing the regulator δ → 0 leads to the result We can now compute the energy-energy correlator by plugging (7.30) at J = −1 into (7.12).
The result is . (7.32) This expression is free of poles to the right of the principal series, so by closing the contour in (7.12) to the right we conclude that F E (ζ) = 0 for 0 < ζ < 1. This ignores the possibility of contact terms discussed in section 6.2, which we now address. Let us start by studying contact terms at ζ = 0. As explained in section 6.2, apart from the protected contact term ξ(ζ) in (7.14), the energy correlator F E (x) may receive contact terms from the integral (7.12). Indeed, when ζ = 0, the distribution f ∆ (ζ) does not vanish at ∆ = 3, 5, and we in fact have Let us now analyze contact terms at ζ = 1. When ζ = 1, we should worry about the convergence of the integral when closing the contour, since there is no suppression coming from ζ ∆− 7 2 in the celestial block. To probe possible delta-function terms localized at ζ = 1 let us consider moments of the energy flow where we deformed the integration contour to Re[∆] = C 0 > 5 so that the integral 1 0 dζ ζ N f 4,4 ∆ (ζ) converges for N ≥ 0. We find that at large |∆| 1 the integrand behaves as where we evaluated the ∆ integral using the principal value prescription. If we subtract off this leading behavior, then the contour deformation in ∆ becomes legitimate and we get 0 for the remainder. This implies that F (0) , in agreement with the straightforward scattering amplitude evaluation, see e.g. [6]. More generally, we see that distributional terms supported at ζ = 1 are encoded in the large-∆ behavior of C + (∆, J = −1).
To summarize, the energy-energy correlator at tree-level is given by Note that this is the unique expression with delta functions at ζ = 0 and ζ = 1 that satisfies both Ward identities (7.17) and (7.18).

One loop
To study perturbative corrections, let us briefly discuss how they are encoded in C + (∆, J). Non-perturbatively, we have poles of the form where a i (a) and ∆ i (a) are, respectively, the product of OPE coefficients and scaling dimension of an exchanged operator. We furthermore have expansions i + · · · , (7.38) i + · · · , (7.39) and thus where we introduced the notation · · · (used extensively below) representing the total contribution of operators that are degenerate at tree level. Below, the subscript * will be replaced by a label referring to the degenerate group of operators. The contribution of these poles to (7.12) now becomes In this section, we will not compute C + (∆, −1), but rather use the known OPE data a (1) * and a (0) * γ (1) * , analytically continued to J = −1. The complete OPE data for the one-loop correlator was written down in [63]. Recall from section 7.2 that the contribution of long multiplets, which are the ones that receive loop corrections, is given by At tree level, this can be decomposed into superconformal blocks (7.24) as follows, see (2.21) in [63], where we used twist τ = ∆ − J and even spin J ≥ 0 to label the operators. Note that for τ > 2 there are degeneracies in the spectrum, so · · · notation is necessary. One can check that (7.44) indeed correctly reproduces (7.43) upon setting Φ(u, v) to zero.
In perturbation theory, we write (7.45) and similarly for H(u, v). At one loop we have Analogously to (7.42), the OPE data enters as where for convenience we labeled the superconformal primaries by twist τ = ∆ − J instead of the dimension (as we did in (7.24)).
The result of the one-loop decomposition for anomalous dimensions is, see (A.24-A.25) in [63], where following [63] we introduced η = (−1) τ /2 and We can concisely write the OPE coefficients at one loop by defining The coefficients α τ,J are Note that for superconformal primaries of twist τ and spin J we should set ∆ (0) * = 4+τ +J in (7.42). Here the shift by 4 is due to the form of the superconformal block in (7.24). This means that for twist τ = 2n, n ≥ 1, and spin J = −1 we have to use ∆ (0) * = 3 + 2n. In this case, the first term in (7.42) vanishes for ζ = 0 due to the factor Γ( ∆−3 2 ) −1 . Thus, the only relevant term is the one proportional to a (0) (7.52) From this we conclude that for 0 < ζ < 1 Again our results are in perfect agreement with the direct evaluation performed in [6].
Let us now analyze the contact terms at ζ = 0 and ζ = 1 in F E (ζ). First, let us fix these contact terms using the result for 0 < ζ < 1 together with Ward identities. We will then check that we reproduce the same contact terms at ζ = 0 using the light-ray OPE. We can rewrite (7.55) as where F (1),reg E (ζ) is integrable near 0 and 1, and so has an unambiguous distributional interpretation. We then only need to interpret the first two terms. The most general expression we can write is 36 where [· · · ] 0 is defined near (6.22), and the definition of [· · · ] 1 is analogous with ζ → 1 − ζ.
from which we find We would now like to reproduce the distributional piece near ζ = 0 from the OPE. From the discussion in section 6.2 together with (7.42), this piece is given by where we used (6.21). This is precisely the expected result.
To summarize, the full one-loop energy-energy correlator takes the form where F (1),reg E (ζ) is defined via (7.56). The distributional part at ζ = 1 is in agreement with the one obtained in [6]. We also derive this ζ = 1 contact term from a different point of view in appendix F.

Two loops
Next, we would like to perform a similar analysis for the two-loop result [64,65]. In this case, we must expand both the three-point coefficients and the anomalous dimensions up to second order. We have and a similar extension of (7.42) for the celestial block expansion (7.16 A complete OPE expansion of this result is not available in the literature (as far as we know). Otherwise, we could simply evaluate the OPE data at J = −1, plug into the celestial OPE formula, and read off the answer for the energy-energy correlator. Some parts of the OPE expansion were obtained in [66], whose results we use below. For simplicity we focus on the term that involves a (1) τ,J (γ (1) τ,J ) 2 , which on the celestial sphere maps to terms containing log 2 ζ.
Below, it will be useful to explicitly write the small-z expansion of ∂ 2 τ G τ +4,J , which takes the form where we only kept the terms containing log 2 z.

Leading twist
The leading-twist contribution to H (2) takes the form (zz) 3 log 2 zf 3 (z), where Since there is no tree-level degeneracy for twist-two operators, this is equal to Indeed one can check that (7.67) reproduces (7.66) .

Subleading twist
Knowing a (1) τ,J (γ (1) τ,J ) 2 at τ = 2 allows us to compute the ζ 2 log ζ piece in F (2) (ζ). A really nontrivial check would be to reproduce the ζ 3 log ζ term. Indeed the two-loop result of [32] contains both rational and transcendental pieces (π 2 ) at this order. The latter should come from the analytically continued γ 2 = γ 2 , due to the degeneracy of twist 4 operators.
We can compute the required OPE data from the piece (zz) 4 log 2 zf 4 (z) ∈ H (2) , where This receives contributions from descendants of twist-two operators as well as from the subleading twist-four Regge trajectory. The subleading trajectory has tree-level degeneracies that we have not resolved, and therefore we cannot simply compute the result using our one-loop analysis.
The function ( and (7.65) it is easy to compute the contribution of descendants of twist 2 operators. After that we are left with the contribution of twist-four primaries which admits the decomposition (7.69) with the second term only. From this we find 37 Note the appearance of the transcendental quantity π 2 which is absent for even integer J.

Two-loop energy correlator
Expanding (7.16) to the second order we get Here, we used the fact that corrections to three-point coefficients alone do not contribute to scalar flow, due to the vanishing of the prefactor in (7.16) at tree-level twists.
Since we do not have degeneracies at twist two, we can fully predict the n = 1 term in (7.74). For n = 2, corresponding to twist-four operators, we only computed the term 37 To solve this decomposition problem, one can use the methods of [66].
where we used standard methods [67] to perform the analytic continuation. Plugging everything back, we get the following prediction for the small-angle expansion of the scalar flow observable at two loops This coincides with the expansion of the result in [32]. In principle, by performing the OPE decomposition of the small z expansion of the two-loop result (7.64) further and evaluating it at J = −1, we can predict higher order terms in the small-angle (small ζ) expansion of the scalar event shape.

Contact terms
Let us also check that we reproduce the correct ζ = 0 contact terms in F E (ζ). Firstly, as in the one-loop example, we can use the Ward identities to fix the contact terms in the two-loop result of [32]. We have where F (2),reg E (ζ) is integrable both at ζ = 0 and ζ = 1. We show the plot of F (2),reg E (ζ) in figure 12. It only has integrable log k -type singularities at the endpoints. To demonstrate this, we show also the ratio F 3 . This ratio is finite, but approaches its limits near ζ = 0, 1 in a non-analytic way due to 1/ log k type non-analyticities.
As before, we make an ansatz for the distribution by writing where [ζ −1 log k ζ] 0 is defined by the Taylor expansion of ζ −1+ in to the appropriate order, and similarly for [(1 − ζ) −1 log k (1 − ζ)] 1 . The Ward identities (7.17) and (7.18) require that 0 =c The explicit expression for F (2),reg E (ζ) follows easily from the definition and the results of [32]. Due to its complexity, we computed the above integrals numerically, Using Mathematica's FindIntegerNullVector we found that to the available precision these numbers are given by To summarize, the distributional piece of F log ζ ζ 0 + · · · . (7.86) As at one loop, from the OPE point of view these pieces are determined completely by twist-two OPE data. In particular, we have All OPE data in this equation except for a (2) τ =2,−1 has been described above. We give a (2) τ =2,−1 in the next section in equation (7.99). Using these results and (6.21) we precisely reproduce (7.86). A calculation in appendix F also reproduces the value of c (2) 1 in (7.85). Note that this is non-trivial consistency check of the result [32], since in order to fix the contact terms we used Ward identities which involve integrals of the even shape over ζ, not just the ζ → 0 and ζ → 1 limits.
To summarize, the full two loop energy-energy correlator is given by (7.79), where c 0 and c (2) 1 are given by (7.85). This completes the 0 < ζ < 1 result of [32]. We checked numerically that the complete two-loop energy-energy correlator satisfies Ward identities (7.17) and (7.18). This check was also performed in [29].

Three loops
Recently the three loop the energy-energy correlator have been computed in [33]. The authors have verified that the leading ζ asymptotic of their result agrees with our prediction (see section 7.7). 38 In this section we extend this check to contact terms at ζ = 0, similarly to what we did at the two-loop level above. Namely, we will use the results of [33] and Ward identities to fix the contact terms at ζ = 0 and ζ = 1, and then compare to the ζ = 0 contact terms predicted by the light-ray OPE. This provides a highly non-trivial consistency check of the results of [33], since the Ward identities involve integrals of F We proceed as before, by writing order to perform numerical integration of these singularities we change the variable from ζ ∈ [0, 1] to p ∈ [0, 1] defined as This change of variables is designed so that the Jacobian ∂ζ ∂p has appropriate 1/ log k behavior to cancel log k singularities of F (ζ) in the right panel of figure 13. The singular part, except from the delta functions (and distributional interpretation of other pieces), can be obtained from the results of [33]. We can fix the coefficients c Integrating the result of [33] numerically we find In [33], F E (ζ) contains a piece expressed as a double integral, and the integrals above are therefore effectively triple integrals. Because of this, it is non-trivial to control the numerical errors, and we have not attempted to get an a priori error estimate for (7.92). Based on the agreement with the light-ray OPE below, we expect that the errors in the numbers above are in the last digit.
Using this data we find 1 ≈ −5.32939. (7.93) Using the same methods as above, and the OPE data described in section 7.7, we find the light-ray OPE prediction for c This agrees well with (7.93), and based on the accuracy of the agreement, we expect for c We show in appendix F that c 1 is given by which precisely agrees with (7.95). 39 This numerical check was also done in [29].
To summarize, the complete three-loop energy-energy correlator, including contact terms, is given by (7.88), where c  E (ζ) follows from its definition and results of [33]. We checked numerically that the complete three-loop energy-energy correlator satisfies Ward identities (7.17) and (7.18).

Four loops in the planar limit and finite coupling
Using known results for the OPE data of twist-2 operators, we can make new predictions for the leading small-angle asymptotics of the energy-energy correlator. At finite coupling the contribution of twist-two operators takes the form where by (+) we indicate analytic continuation from even spin. Note that γ (+) 2,−1 can be computed at any 't Hooft coupling using integrability methods [68,69]. At small angles we Therefore, at weak coupling (7.97) controls the small angle ζ → 0 expansion of the EEC. When the coupling becomes large, operators with twist two at tree level become very heavy and the leading small-angle asymptotic is controlled by the 39 In deriving (7.96) we used the three-loop result for the so-called coefficient function H(a) [29].
approximately twist-four double trace operators. This transition happens at a ≈ 2.645, see figure 14.
At finite coupling there is no contact term coming from (7.97), since the anomalous dimension of twist-two operators is finite. The term ξ(ζ) in (7.16) is completely canceled by a contribution of a protected operator. This cancellation is the same as at strong coupling and is described in the next section. In summary, the event shape at finite coupling is integrable near ζ = 0 and the contact terms only appear at weak coupling through the expansion (6.21). Using (7.97) we can easily make a planar four-loop prediction for the leading asymptotic of F(ζ). 40 The relevant OPE data takes the form where for our normalization of the four-point function the tree-level three-point function is a (0) 2,−1 = 1 32π 4 . Up to three loops, the results can be found in [71], where the three-loop correction to the structure constant was first explicitly computed. 41 For the four-loop anomalous dimensions, we combined the results of [73] and [74]. To analytically continue in spin, we used the HPL package [75] together with the supplement developed in [76]. 42 Plugging these results into (7.97), we easily obtain the leading small-angle expansion of the energy-energy correlator up to four loops. Due to the factor only the three-loop correction to three-point coefficients is needed to compute the four-loop result for 0 < ζ < 1. At ζ = 0, ζ = 1, there are contact terms that depend on additional 40 Starting from the four loops there are non-planar corrections to the correlator [70]. 41 The currently available online version (arXiv v1) of [71] contains a typo. The corrected version of the formula can be found for example in [72] which we used in our computation. 42 In the papers cited above, the anomalous dimension and three-point coupling of Tr[ZD J Z] operator are computed. These operators transform in the 20 representation and their dimensions and couplings are related to the anomalous dimension γτ=2,J and aτ=2,J of the superconformal primaries that appear in (7.24) by a spin shift J → J + 2, see e.g. [77]. Therefore, the formulas of [71] should be evaluated at J = 1 for our purposes. 2,−1 , the red dashed line corresponds to the first four terms at the strong coupling expansion [69]. The solid line was obtained using the quantum spectral curve technique [69]. The curve intersects γ data at four loops (discussed below). The first two terms in the expansion in the coupling reproduce the two-loop computation of [32]. The three-and four-loop predictions are new. Our three-loop prediction was recently independently confirmed in [33].
In more detail, we can write the following expression for the planar four-loop energy-energy correlator 43 where F (4),reg E (ζ) is integrable at ζ = 0, 1. We also included the leading terms in the ζ → 1 limit, which we obtained using results of [29,32,78] as described in appendix F (recall y = 1 − ζ). 44 The contact term coefficients c (4) 0 and c (4) 1 are equal to   44 Here we again made use of the three-loop result for H(a) [29]. use (7.103) to predict a 4 and H 4 . These values can then be used to predict leading five-loop asymptotics at ζ → 0 and ζ → 1.

Strong coupling in the planar limit
The four-point function at strong-coupling is simple enough that we can directly compute C +,sugra (∆, J) and use the celestial block expansion to obtain the full scalar flow observable as a function of ζ. The four-point function is [79] Φ (sugra) (u, v) = uvD 2422 (u, v). (7.104) For a review of D-functions see e.g. [77]. As explained in [62], remarkably the tree-level supergravity answer is fixed by the protected half-BPS data and is given by where f (z, z) is regular at z, z = 1 and is symmetric under permutations of z and z. The relation to the G(z, z) used in [62] is G (105) where we have regulated the integral by introducing δ in the same way as we did in section 7.3.
To isolate the contribution that survives as δ → 0, we rewrite z − z = (1 − z) − (1 − z). By the symmetry of the integral under the exchange of z and z, each of the terms produces an identical contribution, giving a factor of 2. We can rewrite the integral as where we set z = 1 in f (z, z) since it does not affect the δ = 0 result, and used f (z, 1) = [62]. We have also introduced the differential operator which is the Casimir operator of which k β (z) is an eigenfunction with eigenvalue β(β−2)

4
. Doing the integrals, we get where the second term in the second line comes from boundary terms when we integrate by parts. Its contribution to C +,sugra (∆, −1) is equal to zero. Specializing to J = −1, we find (7.111) This provides the data needed to compute F (sugra) using the celestial block expansion. Formula (7.12) gives an integral which we can evaluate by residues when 0 < ζ < 1, where r h was defined in (7.54). This answer coincides with the one obtained in [5]. Alternatively, we could have directly continued the known OPE decomposition of the correlation function to J = −1. Indeed, in the one-loop example above the sum above is equal to . After an appropriate overall rescaling related to the normalization of the conformal blocks the coefficients in the celestial block expansion (7.112) and (7.113) coincide with the analytic continuation of the OPE data worked out in [62] to J = −1.
The result (7.112) already satisfies Ward identities (7.17) and (7.18), so we do not need to add any distributional terms at ζ = 0 or ζ = 1. Let us now check this using the light-ray OPE. Using (7.111) and formulas from 6.2 we find for the distributional terms at ζ = 0 Similarly, to probe distributional terms at ζ = 1 we consider 1 0 dζ ζ N F (sugra) (ζ) and evaluate the integral over ∆. The result is that distributional terms are absent.
To summarize, the complete strong coupling result takes the form

Comments on supergravity at one loop
Recently, the function G (105) (u, v) was also computed at strong coupling to the 1 N 4 order [80], see also [62,81,82]. It corresponds to a one-loop computation in supergravity. It is therefore natural to ask if we can use it to compute the corresponding correction to the two-point energy correlator. As discussed in [1] the existence of the two-point energy correlator is guaranteed in the non-perturbative theory as well as in the planar theory. This, however, does not have to be the case in 1 N 2 perturbation theory. Indeed, in this case the Regge behavior of the correlation function becomes more and more singular and the condition for the existence of the energy correlator J 0 < 3 can be violated (here J 0 is the Regge intercept of the correlator).
At infinite 't Hooft coupling and order 1 N 4 we have J 0 = 3 and thus the energy correlator becomes ill-defined. In other words, to compute it we have to first re-sum 1 N 2 corrections before doing the light transforms and taking the coincident limit, see [1]. It is very easy to see the manifestation of the problem at the level of the OPE as well. If we are to try to evaluate corrections to the spectrum at J = −1 as we did above in section (7.5) we find a pole in a τ,−1 [γ τ,−1 ] 2 , see e.g. (3.15) in [62]. It is an interesting question how to compute subleading large N corrections to the energy correlator. We leave this question for the future.

Multi-point event shapes
It is also interesting to consider higher-point event shapes. To our knowledge, the only higherpoint event shapes available in the literature are the ones due to Hofman and Maldacena [5] for planar N = 4 SYM at strong coupling. In principle, higher-point event shapes can be computed via repeated light-ray OPEs, in the same way that correlation functions of local operators can be computed by repeated local OPEs. (Alternatively, we can use the t-channel block decomposition introduced in [1].) Although we have not developed the formalism for taking OPEs of completely general light-ray operators in this work, it is reasonable to conjecture that the light-ray OPE closes on the light-ray operators of [22]. This is already enough information to make nontrivial predictions about the small-angle limit of multi-point event shapes.
As a simplest nontrivial example, consider a three-point event shape of null-integrated scalars. We assume that the Regge behavior of the theory is such that the event shape exists, and the null-integrated scalars commute. By taking consecutive OPEs, we have where for simplicity we have ignored transverse spins in the second OPE and we are dropping overall constants. We have also abused notation and written the light-ray operators as a function of the transverse position y, as opposed to x, z used in most of this work. Inserting the above expression into an event shape, we obtain a sum of multi-point celestial blocks (which would be interesting to compute explicitly). In the limit | y 12 | | y 23 | 1, the product of operators is dominated by the lightest-dimension terms in each OPE lim (7.117) where ∆ + −1 and ∆ + −2 represent the lightest dimensions at spin −1 and −2. Similarly, we can take repeated OPE limits of an arbitrary number of scalar light-ray operators (assuming their products exist). This leads to a very simple formula for the multicollinear limit of scalar event shapes where we have suppressed subleading terms and an overall proportionality constant that does not depend on relative angles. Of course, a more physically interesting case is to consider multi-point energy correlators. A difference compared to the scalar case is that the OPE of ANEC operators contains lightray operators transforming nontrivially under SO(d − 2) (except for d = 3), see (3.98) and [5]. Let us ignore this for the moment. Repeated OPEs give Here τ + J represents the leading twist at spin J. When operators transform non-trivially under SO(d − 2), the overall scaling with respect to the corresponding small angle will not change -it will still be controlled by the minimal twist [5].
A fascinating property of repeated ANEC OPEs is that alternating steps are controlled by local operators. Specifically, after a single OPE, we obtain light-ray operators with even signature and spin 3. After taking an additional OPE with an ANEC operator, we obtain light-ray operators with even signature and spin 4. These are the quantum numbers of a lighttransformed local operator. We expect that arguments like the ones in sections 4.1 and 4.2 establish that the resulting operator is indeed the light-transform of a local spin-4 operator. Thus, the structure of the light-ray OPE is 45 (7.120) We have already determined the form of the first line above. To understand OPEs for multipoint event shapes, it suffices to understand the second line.
8 Discussion and future directions

Generalizations
In this work, we derived an OPE for a product of null-integrated operators on the same null plane. There are several possible generalizations that would be interesting to consider. One possibility is to derive OPEs of more general continuous-spin light-ray operators [22]. Such an OPE would enable repeated OPEs in multi-point event shapes. For example, a three-point energy correlator could be computed by applying the OPE in this paper to merge two ANEC operators into spin-3 light-ray operators, followed by a generalized OPE with the remaining ANEC operator to produce spin-4 light-ray operators. From symmetries, a multi-point OPE of n ANEC operators will produce light-ray operators with spin n + 1. The average null energy condition implies positivity of the leading light-ray operator in this product, which is presumably the lowest-twist light-ray operator with spin n + 1. 46 This gives an alternative derivation of the higher-even-spin ANEC [12] that additionally includes the case of odd spins, but is not as general as the continuous spin version in [22].
A possible application of repeated OPEs for multi-point event shapes is to set up a bootstrap program for event shapes similar to the bootstrap program for four-point functions of local operators [18,19]. 47 Specifically, one could demand that the light-ray OPE is associative and use this condition to study the space of possible event shapes abstractly. One can also consider mixed light-ray and t-channel OPEs of the type discussed in [1]. With sufficient positivity conditions, perhaps one could apply numerical bootstrap techniques [83][84][85]. Even without deriving the details of the generalized light-ray OPE, it is reasonable to conjecture that it closes on the light-ray operators of [22], and thus multi-point event shapes should admit an expansion in multi-point celestial blocks (which would be interesting to compute).
A surprising property of the light-ray OPE is boundedness in transverse spin, i.e. in spin on the celestial sphere. This is a vast simplification compared to the naïve expectation that a product of point-like objects on the celestial sphere might result in arbitrarily high spin on the celestial sphere. Boundedness in transverse spin is a strong constraint on event shapes that would be interesting to test either analytically or experimentally. It also might have implications for the multi-point event-shape bootstrap. In the bootstrap of local operators, the presence of unbounded spin is important for associativity of the operator algebra [86][87][88][89][90][91][92]. It would be interesting to understand how this works for the light-ray OPE.
It would also be interesting to study OPEs of other types of null-integrated operators, such as those studied in [8,10]. As explained in [1], these can be viewed as descendants of light-transformed operators L [O]. Consider two such descendants inserted at the same point, say x = 0, where we denoted the descendants schematically by P k i L[O i ] and suppressed polarizations associated to P . Acting on this with K k 1 +k 2 +1 we get 0, and so we must conclude that this product has an expansion in terms of descendants of light-ray operators at level at most k 1 + k 2 . A conformally-invariant way to think about descendants P k i L[O i ] is in terms of weight-shifting operators [1,93]. It is likely that the derivation of the light-ray OPE in this paper can be dressed appropriately with weight-shifting operators using methods described in [22,93].
Another generalization is to allow null-integrated operators to be on different null planes that approach each other. It should still be possible to relate matrix elements of such a product to the Lorentzian inversion formula. We expect that light-ray operators with spin other than J 1 + J 2 − 1 would appear.
In [1], we introduced shock amplitudes, which describe the flat-space limit of the bulk dual of a null-integrated operator. In theories with bounded Regge growth, it should be possible to analytically continue shock amplitudes in spin, giving a vast generalization of the amplitudes usually considered. This work suggests a simple way to partially achieve this generalization: one can take coincident limits of shock particles to produce other types of shocks with different (integer) spin. For example, a coincident limit of shock gravitons produces a spin-3 "stringy" shock, as studied by Hofman and Maldacena [5].
A more speculative possible direction is to derive a nonperturbative OPE for amplitudes, describing a convergent expansion around the collinear limit. Such an OPE expansion exists in planar N = 4 [94][95][96][97][98][99], relying on special properties of the theory like amplitude-Wilsonloop duality and integrability; it would be nice to generalize to a generic CFT. (Presumably, this would also require finding a good nonperturbative definition of an amplitude in a generic CFT.) Perhaps the conformal basis [100,101] could be helpful for this. The soft limit of an external particle should correspond to the insertion of a null-integrated operator, so perhaps the hypothetical amplitudes OPE would be related to the light-ray OPE in this limit.

More applications to event shapes
It would be interesting to understand whether the light-ray OPE can be applied to asymptoticallyfree theories like QCD. The small angle behavior of the EEC in QCD was analyzed in [102]. A more general factorization formula describing the collinear limit ζ → 0 and applicable to any weakly coupled gauge theory was derived in [30]. The energy-energy correlator (EEC) in QCD was recently computed at 2 loops (NLO) for arbitrary ζ [103,104]. The light-ray OPE gives a way to resum large logarithms using symmetries as opposed to RG equations. The celestial block expansion is ultimately a consequence of Lorentz symmetry, which is still present when conformal symmetry is broken. Thus, event shapes in any theory should admit a celestial block expansion. However, when dilatation symmetry is broken, the selection rule J = J 1 + J 2 − 1 will no longer hold. Thus, we expect the celestial block expansion in asymptotically-free theories to involve light-ray operators with spin other than 3. 48 In [5], it was shown how to relate the EEC to spin-3 moments of PDFs. Because these spin-3 moments compute matrix elements of spin-3 light-ray operators, it is natural to guess that spin-J moments of PDFs for general J ∈ C compute matrix elements of general spin-J light-ray operators. 49 It would be interesting to derive this connection directly.
The celestial block expansion suggests a way of "perturbatively bootstrapping" the EEC in the same sense as the perturbative bootstrap for amplitudes and Wilson loops in N = 4 SYM [105][106][107][108][109][110][111]. The idea of the perturbative bootstrap is to guess a basis of functions for the answer at some loop order (for example, by guessing the symbol alphabet). One then imposes consistency conditions to fix the coefficients in this basis. In the case of amplitudes in N = 4, this program has been wildly successful, for example resulting in expressions for the 6-point gluon amplitude up to 7 loops [112]. There, consistency with the OPE for amplitudes [94][95][96][97][98][99] and data from integrability provide powerful constraints. The celestial block expansion can provide analogous constraints for the EEC. Furthermore, in [1], we gave a different expansion for the EEC in terms of "t-channel blocks." OPE data from integrability can be used in either channel to make predictions that could help bootstrap the EEC.
An important ingredient in the perturbative bootstrap is the presence of contact terms in perturbative event shapes at ζ = 0 and ζ = 1. Because of Ward identities, the coefficients of contact terms serve as a check on the entire event shape. The light-ray OPE gives a systematic way to compute contact terms at ζ = 0. Furthermore, it provides a connection between the ζ = 0 contact term at L loops and the leading non-contact term as ζ → 0 at L + 1 loops. 50 It would also be interesting to understand event shapes in N = 4 SYM in a systematic expansion in 1/λ and 1/N . The leading 1/λ corrections to energy-energy correlators were computed in [5], see also [113]. They take the form of a finite sum of the t-channel eventshape blocks defined in [1]. This suggests that t-channel blocks could be simple ingredients for setting up a perturbative expansion in 1/λ. One advantage of the t-channel expansion is the absence of contributions from double-trace operators in the planar limit. (By contrast, the light-ray OPE discussed in this paper gets contributions from both single-and double-trace operators.) The extreme simplicity of the 1/λ corrections in [5] stems from the fact that the string shockwave S-matrix, expanded to leading order in α , only mixes adjacent levels on the string worldsheet, see e.g. [1]. 48 We thank Ian Moult for discussions on this point. 49 We thank Juan Maldacena and Aneesh Manohar for making this suggestion, and Ian Moult and Cyuan Han Chang for discussions. 50 Meanwhile, the back-to-back expansion (F.1) provides a description of contact terms and leading noncontact terms at ζ = 1, given knowledge of the hard function H(a) and cusp/collinear anomalous dimensions. Given this, one could imagine a poor-man's version of the perturbative bootstrap, where one uses contact terms at L loops to predict leading non-contact terms at L + 1 loops, fits the leading non-contact terms to a simple ansatz, integrates the ansatz to obtain contact terms at L + 1 loops, and repeats.
The problem of developing a 1/N expansion at large λ is conceptually interesting because the condition J 0 < 3 for the event shape to be well-defined is violated in naïve 1/N perturbation theory. To study 1/N corrections, it will be necessary to re-sum the four-point function in the Regge regime.

Other applications and future directions
Null-integrated operators arise naturally in information-theoretic quantities in quantum field theory. For example, the full modular Hamiltonian in the vacuum state of a region bounded by a cut v = f ( y) of the null plane u = 0 is [8] H f = 2π(K − P f ), where K is the generator of a boost in the u-v plane. Here, we have abused notation and written L[T ] as a function of the transverse position y, instead of the usual arguments x, z.
The vacuum modular flow operator is U f (s) = e −isH f . It is interesting to ask how U f changes as we deform the cut f ( y) → f ( y)+δf ( y). Because the ANEC operator L[T ] appears in the modular Hamiltonian, we can use the algebra of K and P f together with the light-ray OPE to do perturbation theory in δf ( y): where t = e 2πs − 1. Similarly, at n-th order in δf , light-ray operators with spin J = n + 1 will appear. The expression (8.3) gives a direct connection between the spectrum of a CFT and the shape dependence of the vacuum modular flow operator. It may be useful for understanding aspects of the quantum null energy condition (QNEC) [9,[114][115][116]. Furthermore, it would be interesting to see whether it (or other manifestations of the light-ray OPE) has implications for bulk locality in holographic theories. It would also be interesting to study the light-ray OPE for strongly-coupled theories like the 3d Ising model. With enough CFT data, it may be possible to compute event shapes and study modular flow quantitatively in this theory.
Particle colliders like the LHC have given us a wealth of data on event shapes in the Standard Model. In principle, it should be possible to measure event shapes in condensed matter systems using a tabletop collider. One must prepare a material in a state described by a QFT, excite it at a point, and measure the pattern of excitations on the boundary of the material. Several quantum critical points have both Euclidean and Lorentzian avatars in the laboratory. Traditionally, the most precise measurements are available for the Euclidean avatars, in the form of scaling dimensions of low-dimension operators. Event shapes for these systems could reveal intrinsically Lorentzian dynamics that would otherwise remain deeply hidden in the Euclidean measurements.
Finally, it could be interesting to study event shapes in gravitational theories in an asymptotically flat spacetime, see e.g. [117] and references therein. 51 In this case, physical measurements are performed at the future null infinity I + . As in a particle collider experiment, one can measure energy flux through the celestial sphere created in a gravitational collision. In addition to energy carried away by matter fields, there is a contribution due to gravity waves E( n) ∼ I + News 2 which is quadratic in the so-called news tensor. In a gravitational theory, however, it is also natural to consider light-ray operators that are linear in the metric, similar to the ones measured in the current gravitational wave experiments. One such example is a memory light-ray operator M( n) ∼ I + News which measures the memory effect on the celestial sphere. As in the main body of the paper, we can consider multi-point gravitational event shapes and possibly study the corresponding light-ray OPE. One appealing feature of these observables is that they are IR safe -in other words all IR divergencies that arise in the computations of scattering amplitudes should cancel in the event shapes. BMS symmetry [118] and familiar soft theorems [119] should become statements that relate different gravitational event shapes. 52

A Notation
In this appendix, we summarize some of our notation. Many of our conventions are taken from [22].
It is useful to distinguish between physical correlation functions and conformally invariant structures. A correlation function in the state |Ω represents a physical correlation function in a CFT. For example, is a Wightman n-point function in a physical theory, and is a time-ordered n-point function in a physical theory. Two-or three-point functions in the fictitious state |0 represent conformally-invariant functions that are fixed by conformal invariance. If conformal symmetry allows a finite set of possible tensor structures, then we index the possibilities by a label (a), (b), etc.. For example, represents a conformally-invariant tensor structure for the representations of O 1 , O 2 , O 3 , and a runs over the possible solutions to the conformal Ward identities. The above structure has an i prescription appropriate for a Wightman function. Meanwhile, represents a conformally-invariant structure with the i prescription of a time-ordered correlator. Primary operators are labeled by weights (∆, ρ) with respect to the conformal group SO(d, 2). Here, ∆ ∈ C and ρ is an irreducible representation of SO(d − 1, 1). (∆ is constrained to be real and sufficiently positive for local operators in unitary theories.) The weights of ρ can be futher decomposed into ρ = (J, λ), where J is a positive integer for local operators, but in general J ∈ C can be continuous in Lorentzian signature. Here, λ is a finite-dimensional representation of SO(d − 2). We can think of J as the length of the first row of the Young diagram of ρ, while λ encodes the remaining rows. Altogether, we specify a conformal representation by the triplet (∆, J, λ).
We often use the symbol O to stand for the conformal representation with quantum numbers (∆, J, λ). We use φ to represent a scalar operator with quantum numbers (∆ φ , 0, •), where • is the trivial representation. (An exception is in section 7, where O IJ refers to a 20 operator in N = 4 SYM.) If O is a local operator, then ρ is a finite-dimensional representation. In this case, we define shadow and Hermitian conjugate representations as follows where ρ R denotes the reflection of ρ and (ρ R ) * is the dual of ρ R . For continuous-spin operators, ρ = (J, λ) is infinite-dimensional. The light transform turns a local operator into a continuous spin operator To define a conformally-invariant pairing for continuous spin operators we define Similarly, we define O F as an operator that can be paired with L[O] (upon Hermitian conjugation) To describe the causal relation between two points we use the following symbols: • x ≈ y if x and y are space-like; • x > y (x < y) if x lies in the future (past) light-cone of y; • x y (x y) if x is on the future (past) null cone of y.
In section 3, we extensively use Euclidean and Lorentzian pairings between the 2-, 3-and 4-point functions. These are described in detail in appendix C and D of [22] correspondingly.

B Representations of orthogonal groups
B.1 General index-free notation A finite-dimensional representation of SO(d) is labeled by a sequence m d = (m d,1 , . . . , m d,n ) such that The m d,i are either all integers (in the case of tensor representations) or all half-integers. When they are integers, we can think of them as lengths of rows of a Young diagram. See [121] for a recent review.
A spin-J traceless symmetric tensor has labels m d = (J, 0, . . . , 0), corresponding to a single-row Young diagram with length J. More generally, an object in the representation m d is a tensor with indices For a given Young diagram, we can choose to make either symmetry of the rows manifest or antisymmetry of the columns manifest. We choose to make symmetry of the rows manifest. Thus, f is symmetric in each of its n groups of indices Furthermore, it is traceless in all pairs of indices. Antisymmetrization of columns of the Young diagram is reflected in the fact that if we try to symmetrize too many indices, we get zero. For example, Finally, in even dimensions, the tensor f can satisfy µ 1 ···ρ 1 µ 0 ···ρ 0 f µ 0 µ 2 ···µm d,1 ··· ρ 0 ρ 2 ···ρm d,n z 1µ 1 · · · z 1µm d,1 z 2ν 1 · · · z 2νm d,2 · · · z nρ 1 · · · z nρm d,n = ±p n f (z 1 , . . . , z n ) (B.11) where p n is a constant depending only on n. This is equivalent to imposing an (anti-)selfduality condition on the polarization vectors µ 1 ···ρ 1 µ 0 ···ρ 0 z 1µ 1 · · · z nρ 1 = ±p d n!z [1µ 0 · · · z nρ 0 ] . (B.12) To summarize, the representation m d is equivalent to the space of homogeneous polynomials of polarization vectors z 1 , . . . , z n ∈ C d with degrees m d,1 , . . . , m d,n , satisfying the orthogonality conditions (B.8), duality condition (B.12) in even dimensions, and subject to gauge-redundancy (B.10).
We have essentially arrived at the Borel-Weil theorem, specialized to orthogonal groups. The theorem states that each irreducible finite-dimensional representation of a reductive Lie group G is equivalent to the space of global sections of a holomorphic line bundle on the flag manifold G/B, where B ⊂ G is a Borel subgroup. In the case G = SO(d), the flag manifold G/B is the projectivization of the space of vectors z 1 , . . . , z n satisfying the above conditions and gauge-redundancies. A section of a line bundle on this space is a homogeneous polynomial of the polarization vectors.

B.3 Application to CFT
Most of the above constructions still work when some of the weights m d,i become continuous. We can now no longer demand that f is a polynomial in the polarization vectors with continuous weights. However, we can still demand that f is a homogeneous function. Such homogeneous functions yield infinite-dimensional representations of SO(d). 53 We are interested in studying infinite-dimensional representations of SO(d, 2), corresponding to operators in CFT. These are labeled by a weight m d+2 = (−∆, m d,1 , . . . , m d,n ), where ∆ is not necessarily a negative integer. To describe light-ray operators, we must additionally allow m d,1 = J to be non-integer. We often use the notation The elements of the representation with weights m d+2 are homogeneous functions of the kind described in section B.1. Here, we simply introduce some specialized notation for the case at hand. The functions are O(X, Z, W 1 , . . . , W n−1 ), where the vectors X, Z, W i are null and mutually orthogonal. Furthermore, they satisfy gauge redundancies Z ∼ Z + #X W 1 ∼ W 1 + #Z + #X . . . Here, z, w i are mutually orthogonal null vectors, subject to the gauge redundancies w 1 ∼ w 1 + #z w 2 ∼ w 2 + #w 1 + #z . . . where z 0 = (1, 1, 0, . . . , 0) is a null vector in the v direction. We almost always abuse notation and drop the ↓ superscripts, relying on the arguments of O to distinguish between the embedding-space function and its restrictions to Poincare patches. We also often use mixed index-free notation, where we strip off the w i 's to obtain a tensor operator O(x, z, w 1 , . . . , w n−1 ) = O µ 1 ···µm d,2 ··· ν 1 ···νm d,n (x, z)w 1µ 1 · · · w 1µm d,2 · · · w n−1ν 1 · · · w n−1νm d,n . (B.33) The tensor O µ 1 ···µm d,2 ··· ν 1 ···νm d,n (x, z) has indices symmetrized using the Young tableau λ = (m d,2 , . . . , m d,n ), and furthermore all its indices are transverse to z. Finally, we often suppress tensor indices and simply write O(x, z), where it is understood that O can carry indices transverse to z.
All of these different formalisms for representing O are equivalent, and they are convenient for different purposes. For example, to define the celestial map in section 3.4.4, it is convenient to use embedding-space operators O(X, Z, W 1 , . . . , W n−1 ). To define the Lorentzian pairings (3.44) and (3.46), it is convenient to use the object O µ 1 ···µm d,2 ··· ν 1 ···νm d,n (x, z) which caries a finite set of indices transverse to z. We move freely between the different formalisms as needed.

C More on analytic continuation and even/odd spin
In this section, we give more detail on the relationship between CRT and the generalized Lorentzian inversion formula. In particular, we explain how to go from the formula in [22] to the formula (3.41) in the main text.
The formula derived in [22] is . 54 Time-ordered structures only make sense for integer J (see appendix A of [22]), so we must give a prescription for how to analytically continue (C.1) in J. Such a prescription was described in [22]. 55 However, for our purposes, it will be helpful to phrase it in a different way. In particular, this requires clarifying the role of the ± sign in the definition of O ± ∆,J,λ(a) . Note that there are two terms in the Lorentzian inversion formula. The t-channel term written explicitly in (C.1) depends on On the right, we indicate the causal relationship between points for which the structure is needed. We also give light-transformed Wightman structures that equal the light-transformed time-ordered structures when those causal relationships hold. Meanwhile, the u-channel term (1 ↔ 2) depends on instead of (C.2). We see from (C.2) and (C.4) that the Lorentzian inversion formula actually depends on a pair of Wightman structures (C.5) 54 By a "time-ordered structure," we mean a conformally-invariant function of positions, with the i prescription appropriate for a time-ordered correlator. By a "Wightman structure," we mean a conformally-invariant function of positions, with the i prescription appropriate for a Wightman function with the given ordering. 55 It is as follows: we should first compute O1O2L[O † ] (a) for general nonnegative integer J (where J is the spin of O). The result is no longer a time-ordered structure (e.g. it has θ-functions of positions). It can then analytically continued from even or odd J, depending on whether we are computing C + ab (∆, J, λ) or C − ab (∆, J, λ). The analytic continuations are fixed by demanding that they are well-behaved in the right-half J-plane.
It is easy to separately analytically continue each Wightman structure in spin. However, we should take care to preserve the correct relationship between the structures. Let us describe this relationship when J is an integer, and then generalize to non-integer J.
The simplest way to relate the structures (C.5) for integer J is to demand that they are equal when all operators are spacelike separated. Unfortunately, this type of relationship does not generalize to non-integer J due to branch cuts in the spacelike region [22].
A different way to state the relationship between the structures (C.5) for integer J is to say how they transform under a combination of CRT and Hermitian conjugation. Recall that CRT is an anti-unitary symmetry that takes x = (x 0 , x 1 , x 2 , . . . , x d−1 ) to its Rindler reflection x = (−x 0 , −x 1 , x 2 , . . . , x d−1 ). Its action on a local operator is given by Note that Rindler conjugation preserves operator ordering, since it is simply conjugation by a symmetry. If we combine Rindler conjugation with Hermitian conjugation, we obtain a linear map that reverses operator ordering (One way to understand why this reverses operator ordering is that such a rotation reverses all the i 's.) However, for non-local operators, (C.8) cannot be described in terms of a Euclidean rotation. We call the eigenvalue of an operator under (C.9) its "signature." Let us understand how signature is encoded in the inversion formula. Since (C.8) acts as a complexified Lorentz transformation (C.9) on local operators, it is an operator-orderreversing "symmetry" of three-point functions of local operators. Let O 1 , O 2 be any local operators. We have Recall that X ij ≡ −2X i · X j . The building blocks for the structures are [40] X ij ≡ −2X i · X j , (D.4) For brevity, we define V i ≡ V i,jk for {i, j, k} in cyclic order. We have shown in [1] that The new structure 0|O 2 φO 1 |0 (a ) is the unique one that has    at ζ = 1 if we carefully expand (F.1) in powers of a. Naïve expansion yields terms of the form y −1 log k y. In our conventions for the distributional part of F E (ζ) we interpret these terms as [y −1 log y] 1 , which satisfy This can be expanded in powers of a, with b-dependence entering as powers log b b 0 . Note that naïvely this function has an expansion in powers of √ a. However, all non-integer powers of a will go away after performing b-integral.
We now want to perform the a-expansion of the integral can be expanded in a with coefficients polynomial in log b b 0 . This is legal since the integral still converges after the expansion. This means that it suffices to compute the integrals where we treat Γ cusp as arbitrary parameter. It suffices only to compute this in the case k = 0, 1 since to get higher k we can simply take derivatives with respect to Γ cusp . Let us consider the case k = 0; k = 1 is completely analogous. We first integrate by parts, Now the integral converges even for Γ cusp = 0, so we can expand the exponential since Γ cusp ∈ O(a). This way, we reduce to integrals . (F.11)