Momentum space approach to crossing symmetric CFT correlators

We construct a crossing symmetric basis for conformal four-point functions in momentum space by requiring consistent factorization. Just as scattering amplitudes factorize when the intermediate particle is on-shell, non-analytic parts of conformal correlators enjoy a similar factorization in momentum space. Based on this property, Polyakov, in his pioneering 1974 work, introduced a basis for conformal correlators which manifestly satisfies the crossing symmetry. He then initiated the bootstrap program by requiring its consistency with the operator product expansion. This approach is complementary to the ordinary bootstrap program, which is based on the conformal block and requires the crossing symmetry as a consistency condition of the theory. Even though Polyakov's original bootstrap approach has been revisited recently, the crossing symmetric basis has not been constructed explicitly in momentum space. In this paper we complete the construction of the crossing symmetric basis for scalar four-point functions with an intermediate operator with a general spin, by using new analytic expressions for three-point functions involving one tensor. Our new basis manifests the analytic properties of conformal correlators. Also the connected and disconnected correlators are manifestly separated, so that it will be useful for the study of large $N$ CFTs in particular.


Introduction
Conformal field theory (CFT) is a universal testbed for physicists and mathematicians. Starting from the scale invariance of high energy scatterings, it has experienced a vast range of applications, including critical phenomena and string theory. In particular, the outstanding development in d = 2 CFT has brought about intensive interplay between physics and mathematics in connection with the representation theory, integrable systems and hypergeometric functions. The AdS/CFT correspondence [1][2][3] has established CFT as a probe of quantum gravity and triggered recent research activities in CFT on higher JHEP07(2018)136 dimensional spacetimes (see, e.g., [4][5][6][7] for review articles and references therein). Moreover, its application reaches even cosmology, where the conformal group as the de Sitter isometry is crucial to analyze inflationary correlators [8][9][10][11][12][13][14][15][16][17][18][19][20][21]. By further developing our CFT techniques, we would like to enlarge the scope of our theoretical toolkit. The basic ingredients of CFT are three-point functions. Once the operator spectrum is specified, four-point and higher order correlators can in principle be calculated from threepoint functions by using the operator product expansion (OPE). For example, four-point functions of primary scalars with an identical conformal mass dimension ∆ are given by where the summation is over all the primary operators and the numerical constants C ijk are the OPE coefficients. The conformal block G n can be determined uniquely by conformal symmetry. In this expansion, the crossing symmetry is not manifest, hence it gives a nontrivial constraint on the theory together with unitarity or reflection positivity (if any) [22]. This conformal bootstrap approach has been very successful in d = 2 [23]. Furthermore, recent years have seen a remarkable progress in its application to CFTs in higher dimensions. Following the seminal works by Dolan and Osborn [24][25][26], various properties of conformal blocks have been clarified so far and the bootstrap program has been promoted both numerically and analytically [27][28][29][30][31][32][33][34][35][36][37][38][39][40].
While the recent development in the bootstrap program is mostly based on the ordinary conformal block, Polyakov in his pioneering 1974 work [41] employed another basis for conformal correlators as a framework for the bootstrap program. Analogously to the Källén-Lehmann spectral representation of scattering amplitudes, he introduced a crossing symmetric basis for momentum space correlators as where we introduced . . . = (2π) d δ d ( k i ) . . . and W (s) n is what we call the s-channel Polyakov block enjoying a consistent s-channel factorization as we elaborate in section 3.1. In this basis, the crossing symmetry and thus (non-)analyticity in each channel are manifest by construction. On the other hand, consistency with the OPE is obscured, hence it gives a nontrivial constraint on the theory. Polyakov demonstrated that the consistency with the OPE determines anomalous dimensions and OPE coefficients of O(N ) models in the d = 4− dimension (see [42] for a recent generalization). This bootstrap approach therefore plays a complementary role to the one based on the ordinary conformal block.
Even though Polyakov's original bootstrap approach is very elegant, this direction has not been explored very much until recently, essentially because of technical complications due to conformal symmetry in momentum space. Indeed, Polyakov did not derive an explicit form of the block W (s) n in momentum space, but rather he proposed a crossing symmetric basis in position space. Recently in [43,44], Gopakumar  To further promote the recent revival of Polyakov's original bootstrap idea, we would like to complete the construction of the crossing symmetric basis in momentum space in light of recent developments on conformal correlators in momentum space. The purpose of this paper is to explicitly construct the crossing symmetric basis in momentum space, following Polyakov's original argument based on analyticity and factorization. As a first step, in the present paper, we focus on scalar four-point functions in d = 3 Euclidean space. 1 Since our work is rather technical, we would like to outline the punchline of our results in the rest of this introduction. For our construction, it is important to first clarify the analytic properties of three-point functions. Conformal threepoint functions in momentum space were intensively studied recently in [16][17][18][19][20][46][47][48][49][50][51][52][53]. For example, three-point functions of primary scalars are given by [16,[46][47][48][49] where C 123 is the OPE coefficient and B ν is the bulk-to-boundary propagator of the (wouldbe) dual bulk scalar. Interestingly, the radial coordinate z of the AdS Poincaré patch naturally appears when one solves the conformal Ward-Takahashi identity in momentum space. We emphasize that our argument does not rely on the AdS/CFT correspondence, even though we use its terminology to make the notation intuitive. As a consequence, analytic properties of three-point functions are captured by those of the bulk-to-boundary propagators. For construction of the Polyakov block for an intermediate spinning operator, we need three-point functions of two scalars and one tensor as well. As depicted in figure 1, we find in appendix B that they are obtained by simply acting a differential operator A 123m (its definition is given in section 4.2) on an integral similar to eq. (1.3) as 2 The results in general dimensions will be presented in [45]. 2 The integral (1.4) is convergent only when |Re ν1| + |Re ν2| + |Re ν3| < s + 3 2 . Otherwise, there appears a singularity near z = 0 and we need to perform analytic continuation [48,49], which may be carried out, e.g., by introducing the Pochhammer contour. A similar remark is applicable to our integral representation for the Polyakov block. where we introduced what we call the helicity basis by analogy with helicities of massless particles, and O nm has a spin s and a helicity m [18]. As we discuss in section 4.3, the differential operator A 123m is analytic in momenta, so that all the non-analyticities of threepoint functions are captured by the bulk-to-boundary propagator B ν (k; z) in the integrand. This is a technically important observation for our construction.

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The above analytic properties of three-point functions now enable us to construct the crossing symmetric basis. As we explain in section 3.1, Polyakov's idea is to require that non-analytic parts of the Polyakov block W (s) n enjoy a consistent factorization, just as the on-shell factorization of scattering amplitudes. In section 3.2, we show that the Polyakov block for an intermediate scalar is nothing but the Witten exchange diagram: where G νn (k 12 ; z 1 , z 2 ) is the bulk-to-bulk propagator. The essential point here is that the z-ordered part of the bulk-to-bulk propagator is analytic in k 12 . As a result, the non-analytic parts associated to the intermediate on-shell state manifestly reproduce the factorization required by the Polyakov ansatz. This shows that momentum space is a natural language for this program. It is then not difficult to generalize the construction to intermediate spinning operators. The Polyakov block W (s) nm for an intermediate operator O nm with a spin s and a helicity m can be obtained again by acting the differential operators A 12nm and A 34nm on an integral similar to eq. (1.5) as (see also figure 2) where a s,νn (m) is a numerical factor associated to the normalization of O nm . This is our main result. As a side remark, we would like to point out that the connected and disconnected correlators are manifestly separated in this basis (see section 3.3). We therefore

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believe that our basis is useful not only for the Polyakov type bootstrap, but also for the study of large N CFTs, which are holographically dual to weakly coupled bulk theories. The rest of this paper is organized as follows. In section 2 we discuss analytic properties of scalar two-and three-point functions in momentum space. In particular we demonstrate that non-analytic parts of three-point functions factorize into analytic cubic vertices and two-point functions. In section 3 we introduce Polyakov's analyticity based argument and illustrate that momentum space provides a natural language for this program. By requiring consistent factorization in each channel, we show that the Polyakov blocks for intermediate scalars are nothing but the Witten exchange diagrams. In section 4 we then generalize the argument to the intermediate spinning operators and complete the construction of the crossing symmetric basis. Some technical details are collected in appendices.

Analytic properties of scalar two-and three-point functions
In this section we discuss analytic properties of two-and three-point functions in momentum space. To avoid technical complication associated with spins, we first focus on scalar correlators in this section. As we will see in section 4, however, most of analytic properties discussed in this section are carried over to correlators involving operators with spins. Throughout the paper, we work on the d = 3 Euclidean space unless otherwise stated.
In the first two subsections we review basics of two-and three-point functions in momentum space. In particular we introduce a building block of three-point functions called the triple-K integral [47][48][49], which is essentially the cubic Witten diagram. We then elaborate on its analytic properties in the last subsection. The cubic vertices introduced there are relevant when constructing a crossing symmetric basis for conformal correlators.

Two-point functions
In momentum space two-point functions of primary scalars are given by where k = |k|, ∆ is the scaling dimension of O, and ν = ∆ − 3/2. In this paper, for notational simplicity, we use . . . for correlation functions with the momentum conservation factor dropped. More explicitly, . . . = (2π) 3 δ 3 ( k i ) . . . . The corresponding two-point functions in position space are where notice that C OO is positive in unitary CFTs, but C OO is not necessarily positive. As is clear from eq. (2.1), two-point functions are not analytic in k, or equivalently in k 2 , unless ν is an integer. 3 When we analytically continue k 2 , we introduce a branch cut along k 2 < 0. The discontinuity on the branch cut is

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where the imaginary part of the two-point function on the r.h.s. is evaluated at Im k 2 = with a positive infinitesimal number as usual.

Three-point functions
In momentum space three-point functions of primary scalars are given by [46][47][48][49] where B ν is the bulk-to-boundary propagator of the (would-be) dual bulk scalar: 4 Also I ν (z) and K ν (z) are the modified Bessel functions of the first and the second kinds, respectively. We call them the Bessel I and K functions in short. The bulk-to-boundary propagator B ν (k; z) is a solution for the scalar equation of motion on AdS 4 , where m 2 = ν 2 − 9/4 is the dual scalar mass. Our normalization is chosen such that As is clear from the large z behavior of modified Bessel functions, the bulk-to-boundary propagator B ν (k; z) decays in the limit kz → ∞. Since the integral (2.4) contains three K ν (z)'s, it is called the triple-K integral. We refer the reader to the references [48,49] for its detailed properties. The origin of the triple-K integral may be understood in the following three ways: 1. First, the integral (2.4) is nothing but the Witten diagram for the cubic coupling we identify z with the radial coordinate of the Poincaré patch. It respects the AdS isometry by construction, hence it is conformally covariant.
2. Indeed, we may explicitly check that the triple-K integral satisfies the conformal Ward-Takahashi identity in momentum space. By requiring an appropriate analytic behavior, scalar three-point functions may be determined uniquely up to an overall coefficient from conformal symmetry [48]. 3. It is also possible to obtain the triple-K integral by Fourier transforming the position space three-point function,

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where ∆ ijk = ∆ i + ∆ j − ∆ k . As we compute in appendix B.4, the OPE coefficient C 123 in position space is related to the one C 123 in momentum space as

Factorization of three-point functions
We then discuss analyticity of three-point functions. First, the triple-K integral (2.4) is defined in such a way that it is regular after an appropriate analytic continuation in ν i . It is therefore analytic as long as the integrand is analytic. Non-analyticity then shows up only in the squeezed limit, where one of the three momenta is much smaller than the other two. As the soft limit expansion (2.8) of the bulk-to-boundary propagator shows, the only source of non-analyticity is the second term in the square bracket of eq. (2.6). Note that the soft limit of external momenta is nothing but the momentum space counterpart of the OPE limit in position space (see figure 3). The non-analytic component is then responsible for the intermediate on-shell state and thus the singular behavior in the OPE limit. Let us take a closer look at analyticity of three-point functions ) with respect to k 3 (with k 1 fixed). As we mentioned, the only source of non-analyticity is the bulk-to-boundary propagator for O 3 . The non-analytic part of three-point functions then factorizes into analytic cubic vertices and the discontinuity (imaginary part) of twopoint functions as

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Here we defined the cubic vertex T 12;3 by which is analytic with respect to k 3 around k 3 = 0. We use this cubic vertex in the next section when constructing the crossing symmetric basis for conformal four-point functions.
For later use, it is convenient to notice that the cubic vertex (2.13) has a singularity in the collinear limit defined by k 1 + k 2 = k 3 : as we mentioned, the Bessel K and I functions behaves in the large z region as eq. (2.9), so that the bulk-to-boundary propagator has an exponential suppression ∼ e −kz . Since the integrand of cubic vertices contains the Bessel I function, it has an exponential factor ∼ e −(k 1 +k 2 −k 3 )z in the large z region. The integral then converges only when k 1 + k 2 > k 3 . In particular, the exponential suppression disappears in the collinear limit k 1 + k 2 = k 3 and thus becomes singular and non-analytic.

Crossing symmetric basis: intermediate scalar
In this section we construct a crossing symmetric basis for momentum space four-point functions, following Polyakov's argument based on analyticity and factorization [41]. For technical simplicity, we focus on the case with an intermediate scalar in this section. Extension to the case with an intermediate spinning operator will be given in the next section.
After introducing Polyakov's argument in section 3.1, we show in section 3.2 that the crossing symmetric basis is nothing but the Witten exchange diagram. Actually, this equivalence was already shown in Mellin space by explicit computations [44]. However, we find that momentum space provides a natural language for this program. In particular we elaborate on the emergence of the bulk-to-bulk propagator by studying analytic properties in momentum space. Section 3.3 is devoted to the intermediate identity operator case, which is separately treated in our construction. We there emphasize that the connected and disconnected contributions to four-point functions are manifestly separated in this approach.

Factorization ansatzà la Polyakov
In ordinary Lorentzian field theories, scattering amplitudes are analytic off shell and their non-analytic parts, associated to on-shell particle creations, enjoy the factorization property. For example, tree-level four-point amplitudes in φ 3 theory are schematically given by where s = −(k 1 +k 2 ) 2 is the Mandelstam variable. When the exchanged particle is on-shell, the propagator becomes non-analytic and the non-analytic part factorizes as Notice that cubic vertices and t, u-channel diagrams are analytic in s since our interaction is local. Also, contact vertices, if any, are analytic as long as we consider local interactions.

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x 1 Figure 4. In the collapsed limit, an internal momentum, say k 12 , becomes soft compared to external momenta (the left figure). This limit corresponds to the OPE limit in position space, where operators are separated into two groups far away as depicted in the right figure.
Let us now move on to our CFT argument. In [41] Polyakov introduced a CFT correlator analogue of such a factorization property. As we demonstrated in section 2, non-analytic parts of three-point functions can be reformulated as Notice that factorization occurs even in three-point functions because we are considering correlators rather than scattering amplitudes. Note also that T 12;3 is analytic in k 3 around k 3 = 0. In analogy with ordinary field theories, we call T 12;3 the cubic vertex. 5 Similarly, we require that non-analytic parts of four-point functions are factorized into the form, where the intermediate operator O n runs over all the primary operators. They are described in the basis with diagonal two-point functions in particular. We also introduced k ij = k i + k j . Later, we use the Mandelstam type variables s, t, and u defined by Since the on-shell conditions are not imposed on the external momenta, these variables are independent in contrast to the scattering amplitude case. Notice here that the collapsed limit k ij → 0, where the non-analyticity shows up, is the momentum space analogue of the position space OPE limit (see figure 4). In a similar fashion, we require factorization in the t, u-channels as well.
Based on the factorization property given above, Polyakov introduced a crossing symmetric basis for conformal four-point functions as 6 [41]

6)
5 To be precise, T12;3 is the cubic vertex multiplied by two-point functions of O1 and O2. Indeed, it is non-analytic with respect to k1 and k2. We, however, call it the cubic vertex for simplicity. 6 Just as in the case of the recursion relation for scattering amplitudes [54], there may exist analytic terms which cannot be determined only from the analyticity argument, even though earlier works [41][42][43][44] on the bootstrap program did not incorporate these contributions. In our context, such an ambiguity is generically associated with bulk contact interactions. We expect that other physical criteria such as consistency with OPE (see also [44]) and locality of the dual bulk theory will constrain these contributions in a significant way. We do not go into this issue in detail, leaving it to future work.

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where the label n again runs over all the intermediate primary operators. The function W (s) n , which we call the s-channel Polyakov block in the following, is a conformally covariant function enjoying the following two properties: n reproduces the non-analytic properties associated with the s-channel factorization. More explicitly, we require where we described the intermediate operators O n in the basis with diagonal twopoint functions. Such a basis can easily be taken for scalar operators. Also, in section 4, we introduce the diagonal basis for spinning operators.

W (s)
n has no non-analyticity other than the one in the s-channel collapsed limit s = 0. In particular, it is analytic in k 13 and k 14 , so that it does not introduce t, u-channel discontinuity.
Also W (t) n and W (u) n are t, u-channel analogues of W (s) n and enjoy similar properties. In contrast to the ordinary conformal block, the crossing symmetry is manifest in this basis by construction. On the other hand, the consistency with the OPE is obscured, hence it gives a nontrivial constraint on the theory as Polyakov demonstrated [41].
While Polyakov completed the bootstrap program in workable models elegantly, he did not derive an explicit form of the block W (s) n because of technical complications due to conformal symmetry in momentum space, but rather he proposed a basis in position space. Recently in [43,44], Gopakumar et al. showed in Mellin space that the Polyakov block W (s) n is nothing but the Witten exchange diagram. In the next subsection we revisit this equivalence and construct the Polyakov block in momentum space for the intermediate scalar case. Essentially because momentum space manifests the analyticity and factorization properties (3.7), our construction turns out to be quite simple and intuitive.

Witten exchange diagram is Polyakov block
We then construct the Polyakov block. Here we assume that the intermediate operator is not the identity. The identity operator case is discussed separately in the next subsection.
Lessons from a naive trial. As a first step, it is instructive to start with a naive candidate for the Polyakov block of the form, which trivially satisfies the first requirement (3.7). However, it turns out to have undesired non-analyticity and does not satisfy the second criterion. As we mentioned in the previous section, the cubic vertex,

9)
JHEP07(2018)136 In the collinear limit, two external momenta are parallel and point the same direction.
is singular and non-analytic in the collinear limit k 1 + k 2 = k 3 , essentially because of the exponential growth, I ν (z) ∼ e z (z 1), of the Bessel I function in the large z region. This singularity leads to an undesired singularity of eq. (3.8) in the collinear limit k 1 + k 2 = k 12 for example (see figure 5). The ansatz (3.8) then does not respect the second criterion. Our task is now to handle such an exponential growth of the Bessel I function appropriately.
Witten exchange diagram. We now proceed to showing that the improvement of the naive ansatz (3.8) is nothing but the Witten exchange diagram. To see this, let us first rewrite eq. (3.8) as In the large z i region (i = 1, 2), the integrand behaves as which is the origin of undesired non-analyticities mentioned above. One possible way to remove this undesired exponential growth, without spoiling conformal covariance, is to replace the two Bessel I functions by Bessel K functions. This choice is essentially the ordinary conformal block. As we explain shortly, the ordinary conformal block has an additional s-channel non-analyticity associated to the shadow operator, so that it does not satisfy the first criterion for the Polyakov block. The only thing we can do is then to replace one of the two Bessel I functions by a Bessel K function. For example, if we replace I ν (k 12 z 1 ) by K ν (k 12 z 1 ), the new integrand behaves in the large z i region as which is free from the undesired exponential growth when z 1 > z 2 . We therefore perform this replacement for the integral region z 1 > z 2 . Similarly, we replace I ν (k 12 z 2 ) by K ν (k 12 z 2 ) in the region z 1 < z 2 . Actually, this is nothing but what we usually do when constructing the bulk-to-bulk propagator. More explicitly, the bulk-to-bulk propagator in JHEP07(2018)136 momentum space is given by 7 where the second line exactly reproduces eq. (3.10) accompanied by the bulk-to-boundary propagators and integrals over the radial coordinates z i . Note also that the third line is analytic in k. The Witten exchange diagram, therefore reproduces the s-channel non-analyticity required by the factorization ansatz.
Also, it has no other non-analyticity essentially because the bulk-to-bulk boundary propagator has an exponential suppression in the large z i region. Indeed, the z-ordered part of the propagator (3.15) was originally added to remove the undesired exponential growth and thus undesired singularities of correlators. We therefore conclude that the Witten exchange diagram satisfies the two criteria of the Polyakov block and write Why not the conformal block? Before closing this subsection, let us make a brief remark on the difference of the Polyakov block and the ordinary conformal block. As we mentioned, if our goal were only to remove the undesired exponential growth in the large z i region, we could replace the two Bessel I functions by Bessel K functions, which gives the triple-K integral squared. More precisely, conformal covariance requires the form, which is regular except for the s-channel collapsed limit k 12 → 0. Since three-point functions in the squeezed limit scales as the collapsed limit k 12 → 0 of eq. (3.18) is given by which makes the prefactor in the first line to be 1. This non-analyticity is not the one required by the factorization ansatz Disc s W (s) n ∝ k 2νn 12 , but rather the one associated to the shadow operator ∝ k −2νn 12 [55][56][57][58][59] (the shadow of O n has a scaling dimension 3 − ∆ n , which corresponds to a sign flip ν n → −ν n ). Therefore, the ordinary conformal block does not satisfy the Polyakov ansatz. In this way, the absence of the non-analyticity associated with the shadow makes the Polyakov block different from the ordinary conformal block. See also figure 6 for summary of the argument in this subsection.

Polyakov blocks for an intermediate identity operator
Finally, let us consider the intermediate identity operator. The two-point function of the identity operator is 1 in position space, so that the momentum space two-point function is (3.21) Here and in what follows, we denote the Fourier transform of the identity operator 1(x) by 1(k), even though it is simply 1(k) = (2π) 3 δ (3) (k). Three-point functions involving the identity operator are also given by Here we introduced the cubic vertex T 12;id ("id" represents that the third operator is the identity operator) as Here it is worth mentioning that the Polyakov block for the intermediate identity operator is nothing but the disconnected part of four-point functions: More explicitly, the Polyakov blocks with the intermediate identity operator reproduce the disconnected four-point functions as On the other hand, the connected correlators may be expanded as where the summation is over primary operators other than the identity operator.
To contrast this property with the ordinary conformal block, let us consider the leading order of the so-called large N CFT, whose four-point functions contain only the disconnected pieces as This four-point function can be expanded in, e.g., the s-channel conformal blocks as

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Essentially because we are introducing the Polyakov blocks for all the channels, only the identity operator shows up as the intermediate operator. 8 To summarize, in the crossing symmetric basis, the disconnected and connected fourpoint functions are manifestly separated into the Polyakov blocks for the identity and non-identity operators. This property will be a big advantage for the study of large N CFTs in particular. In large N CFTs, the connected and disconnected correlators start at different orders in the 1/N expansion. As a result, in the ordinary conformal block expansion, the connected correlators are scrambled by the subleading corrections to the disconnected correlators. We expect that our new basis provides a complementary approach to large N CFTs, which are holographically dual to weakly coupled bulk theories [60].

Extension to the case with an intermediate spinning operator
In this section we extend the construction of the crossing symmetric basis to the case with an intermediate spinning operator. The basic idea is the same as the intermediate scalar case, but there appears some technical complication due to the spin structure. To handle the spin structure, it is convenient to employ what we call the helicity basis, which is analogous to the helicity basis of massless particles. In section 4.1 we first summarize the basic properties of two-point functions in momentum space and introduce the helicity basis. In section 4.2 we provide a useful expression for three-point functions of two scalars and one tensor in the helicity basis. We then discuss their analytic properties and construct the crossing symmetric basis with an intermediate spinning operator in section 4.3. Technical details of this section are collected in appendices A and B.

Two-point functions and helicity basis
A standard technique to handle symmetric traceless tensors in CFT is to contract all vector indices of the tensor operators with a null vector called the polarization vector [61,62]. 9 More explicitly, we introduce a shorthand notation [18],   1). Hence, in the large N CFT, i.e., in the limit γ → 0, the contributions from double trace operators vanish in the crossing symmetric expansion, whereas they do not in the conformal block expansion. 9 We use the bold and ordinary fonts for vectors and their components, e.g., and µ . Here µ is the vector index.

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where s is the spin of the operator O. In momentum space, two-point functions of primary operators with general spins are given by [18] (see also appendix A.1 for the derivation) where ∆ is the scaling dimension of O, two null vectors and˜ are the polarization vectors, and P (α,β) n is the Jacobi polynomial. See appendix D.1 for the definition of the Jacobi polynomial and formulae used in this paper. Note that the corresponding two-point functions in position space take the form, where the normalization factor C OO in position space is related to the momentum space one C OO by where we set k = (0, 0, k) without loss of generality by using the rotational invariance of the correlator. Angles ψ and ψ represent rotation angles around the momenta k and −k, respectively. The -dependent factors in two-point functions are then reduced to the form, so that the correlator (4.2) may be reformulated as As we show in appendix A.2, we may rewrite it as [18,63] where m is identified with the rotation charge around the momentum of each operator, so that it is analogous to the helicity of massless particles. Based on this analogy, we call m the helicity in the following. The coefficients a s,ν (m) are defined by a s,ν (m) = Γ(s + 1/2)Γ(s − ν + 1/2)Γ(m + ν + 1/2) √ π(s − m)!(s + m)! Γ(ν + 1/2)Γ(m − ν + 1/2) , (4.9)
Note in particular that a s,ν (m) = a s,ν (−m). A detailed derivation of these results is provided in appendix A. Let us now identify each m component in the summation (4.8) with the contribution from the helicity m operator. For this purpose, we define the helicity m operator as where the polarization vector is defined through eq. (4.5) upon an appropriate rotation. In this basis, the two-point function takes a simple form, Notice in particular that the helicity basis provides a diagonal basis for spinning operators.

Three-point functions
We now proceed to three-point functions of two scalars and one symmetric traceless tensor. In position space, they are given by [41] where O i (i = 1, 2) are primary scalars of dimensions ∆ i , and O 3 is a spin s primary symmetric traceless tensor of dimension ∆ 3 . The momentum space correlators can be obtained either by Fourier transformations or by explicitly solving the conformal Ward-Takahashi identities in momentum space. In appendix B, we provide detailed derivations of them in both methods. Below, we summarize the results there.
Spin 1. Before stating general results, it is instructive to elaborate the structure of threepoint functions in the spin 1 and spin 2 cases. Since the denominator of eq. (4.12) takes a similar form to the scalar three-point functions, the three-point functions in momentum space will be expressed in terms of triple-K type integrals. The polarization-dependent factor in the numerator will give derivatives in momenta. Indeed, for s = 1, we find where the last factor is a triple-K type integral. Notice that its integrand has an additional factor z compared with scalar three-point functions. On the other hand, the prefactor inside the square brackets carries information of the polarization vector . As expected, this factor contains derivatives in the momenta k i , which are packaged into a differential operator D 123 defined by , (4.14)

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where the subscripts 123 denote the labels of the operators O i . Note that when acting the operator D 123 on the triple-K integral, we regard k 1 , k 2 and k 3 as three independent variables. More explicitly, ∂ k 1 k 3 = ∂ k 2 k 3 = 0 for example. Notice also that D 123 has an odd parity, D 123 → −D 123 , under the exchange 1 ↔ 2.
Spin 2. Similarly, the spin 2 result is given by where the integrand of the triple-K type integral is accompanied by a factor z 2 . The prefactor inside the square brackets is now a second order polynomial of the differential operator D 123 . Here one might wonder that k 1 and k 2 are not symmetric in the expression (4.15). However, it is easy to find that eq. (4.15) can be reformulated as Since D 123 has an odd parity under the exchange 1 ↔ 2, it turns out that eq. (4.15) and equivalently eq. (4.16) are consistent with the exchange symmetry of the two scalars.
General spin. Three-point functions with a general spin s accommodate similar structures as s = 1, 2 mentioned above. In appendix B.2, we derive a general expression, where the triple-K type integral has an additional z s factor. Also, the prefactor is an s-th order polynomial in D 123 . Our normalization in momentum space is related to the position space one as . (4.18) Just as we did in the spin 2 case, we may rewrite it as

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which is consistent with the exchange symmetry of the two scalars. In particular we notice that three-point functions vanish when two scalars are identical and the tensor O 3 has an odd spin, as is implied also by the position space result (4.12).
Three-point functions in helicity basis. Finally, let us introduce three-point functions in the helicity basis. Without loss of generality, we set the third momentum k 3 and the polarization vector as It is convenient to parametrize the momentum k 2 as k 2 = k 2 (cos χ sin θ, sin χ sin θ, cos θ) . The polarization-dependent factor is then written as Therefore, three-point functions involving the helicity m tensor, O 3m , are (see also figure 1) Here we introduced a differential operator A 123m for −s ≤ m ≤ s as (see appendix C for derivation) where P ,|m| (cos θ) is proportional to the associated Legendre function and normalized as eq. (C.8). In the next subsection, we clarify analytic properties of three-point functions involving one tensor based on the expression (4.23). As the appearance of triple-K type integrals implies, we find that the argument goes quite similar to the scalar case discussed in the previous section.

Construction of crossing symmetric basis
We proceed to discussing analytic properties of three-point functions (4.23) and constructing the crossing symmetric basis. First, according to the parametrizations (4.20) and (4.21), the functions k s−n 2 P s−n,|m| (cos θ) is a polynomial in k 2 . Next, the differential operator D 123 enters eq. (4.24) in the combination k 3 D 123 , which is analytic because (k 2 1 − k 2 2 )/k 3 = −(k 1 − k 2 ).k 3 /k 3 = −k 1z + k 2z (see eq. (4.14)). We then conclude that the operator A 123m (k 1 , k 2 , k 3 ; D 123 ) is a polynomial in the momenta k i and the Euler operators k i ∂ k i . Since the Euler operator does not introduce new non-analyticity, non-analytic properties of three-point functions are essentially captured by the triple-K integral.

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Cubic vertex. Let us then determine the cubic vertex. Just as the scalar three-point function case, our starting point is the relation, Notice again that the I ν 3 part of the bulk-to-boundary propagator is responsible for the non-analytic properties around k 3 = 0, whereas the I −ν 3 part is for removing the singularity in the collinear limit k 1 + k 2 = k 3 . As we mentioned, the differential operator A 123m does not produce any new non-analyticity, so that the non-analytic parts of three-point functions are obtained by acting A 123m on the r.h.s. of eq. (4.25). Moreover, the differential operator D 123 does not contain derivatives in k 3 . Therefore, the operator A 123m does not change non-analytic properties around k 3 = 0. All in all, we arrive at the factorization relation, where the cubic vertex T 12;3m is given by where W (s) nm is the Polyakov block with an intermediate operator O nm with spin s, helicity m, and scaling dimension ∆ n . T 12;nm and T 34;nm are the cubic vertices we have just introduced above. As we mentioned in section 3.1, we also require that the s-channel block has no other non-analyticity. Since A 12nm and A 34nm in the cubic vertices do not change the non-analytic properties, we may apply the argument in section 3.2 in a straightforward manner to conclude

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where the second line is the Polyakov block for an intermediate scalar with an additional factor z s 1 z s 2 in the integrand. G νn (k 12 ; z 1 , z 2 ) is the scalar bulk-to-bulk propagator defined by eq. (3.15). On top of it, we have the helicity dependent differential operators A 12nm and A 34nm (see also figure 2). It is easy to see that eq. (4.29) enjoys the s-channel factorization property (4.28) and does not have any other non-analyticities. The t-and u-channel Polyakov blocks are also be defined in a similar fashion.

Conclusion
In this paper we explicitly constructed the crossing symmetric basis for conformal fourpoint functions of primary scalars, following Polyakov's analyticity based ansatz: As a concluding remark, we would like to present several promising future directions. First, we have focused on three dimensional space in the present paper, for technical simplicity. Even though there exist technical complications associated to spins and helicities, it is conceptually straightforward to extend our argument to higher dimensions. We will present this result in a forthcoming paper [45]. It will also be interesting to construct the crossing symmetric basis for conformal correlators of (external) spinning operators such as the energy-momentum tensor. These explicit forms of the crossing symmetric basis will be useful to revisit and generalize Polyakov's original bootstrap approach. Besides, as we discussed in section 3.3, the connected and disconnected correlators are manifestly separated in our basis, in contrast to the ordinary conformal block. We believe that this is a big advantage when discussing the large N CFTs, which are holographically dual to weakly coupled bulk theories. We will use this property to revisit positivity bounds on effective interactions in the bulk [60]. Another interesting direction will be application to cosmology. Recent progress in cosmology has shown that (non-)analytic structures of higher-point correlators are useful to probe new particles coupled to the inflaton sector, as is dubbed the "cosmological collider physics" program [18,[64][65][66]. Indeed, some of the momentum space techniques used in this paper were developed in the context of cosmic inflation. It will be useful to introduce a basis for inflationary correlators by extending our construction. We hope to revisit these issues in near future.

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Note added. While this paper was being finalized, ref. [67] appeared on arXiv. There, what they called the weight shifting operator was introduced in position space, whose action on scalar Witten exchange diagrams gives those involving spinning fields (see also [68,69] for other differential operators in position space with a similar property). This seems similar to the property of our differential operator A defined in momentum space as eq. (4.23), even though these two operators have different origins and their relation is not clear. It would be interesting to explore a possible connection thereof.

A Two-point functions with spins in momentum space
In this and the next appendices, we derive two-and three-point functions with spins in momentum space. For generality, we work in general dimension d = 2h in these appendices unless otherwise stated, though we set d = 3 in the main part.

A.1 Two-point functions by Fourier transformation
Let us compute the two-point function s .O(k)˜ s .O(−k) in this appendix. This has been computed in [63] by Fourier transformation in general dimensions and in [18] by solving the Ward-Takahashi (WT) identities in three dimension. Here we review the derivation by Fourier transformation, which enables us to fix the relative overall constant between position-space and momentum-space correlators.
Our convention for Fourier transformation is where O(x) is the position space operator and O(k) is the momentum space one.
Combining this with the two-point function in position space (4.3) gives

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Here we applied the binomial expansion after replacing position coordinates in the numerator of eq. (4.3) by momentum derivatives, and then used the Fourier transformation of (x 2 ) −α given by the formula, We also defined ν = ∆ − h. The momentum derivatives can be rewritten as (c = −ν − a) where we used . =˜ .˜ = 0. Applying this and the resummation where we used the following summation formula Changing the summation index m → s − m, using the definition of the Jacobi polynomial (D.1), and applying the formula Γ(−ν)Γ(ν + 1) = −π/ sin(πν), we arrive at where the overall constant C OO is given by This result for h = 3/2 gives eq. (4.2) in the main text.

A.2 Helicity basis
The expression (4.2) is compact, but to understand the helicity dependence, it is more useful to rewrite it in such a form that the angles among momenta and helicity vectors are manifest [18,63]. In this subsection we concentrate only on the three-dimensional case. A general dimensional version will be presented in a forthcoming paper [45].

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We can easily see that the Jacobi polynomial is defined on the unit circle and only depends on the angle ψ +ψ . Furthermore, P (ν−s,−1/2) (t) is a polynomial of degree s, and any power cos m (ψ + ψ ) can be expanded by {cos n(ψ + ψ )} with n = 0, 1, · · · , m. Therefore, the Jacobi polynomial P (ν−s,−1/2) s (− cos(ψ + ψ )) can be expanded as where the coefficients α m are given by The integration (A.11) was carried out using the integration formula (D.4) after rewriting cos mθ as a Chebyshev polynomial defined in eq. (D.5). Substituting these results into the two-point function (A.9), we find where we introduced a s,ν (m) = (−) s α m for m ≥ 0 and a s,ν (m) = a s,ν (−m) for m < 0.

B Three-point functions of two scalars and one tensor
In this appendix we derive two types of closed forms for three-point functions involving two scalars and one tensor. The first expression is the one we used in the main text: 11 where D 123 is a differential operator in k i 's, defined by eq. (B.29). As we discussed in section 4, this expression manifests analytic properties of three-point functions, so that it is useful for our construction of the crossing symmetric basis. We also derive the other form,

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where c 123 and d 123 are numerical coefficients given by eq. (B.51) and eq. (B.52), respectively, and I s is a triple-K type integral, As we discuss in appendix B.4, this expression naturally arises when we perform Fourier transformation of the position space correlators. Even though the derivation itself is a bit technical, the two closed forms are a part of the main results of this paper. The rest of this appendix goes along the line of the derivation of the first expression (B.1). Following the argument in appendix C of [18], we begin with a general ansatz, 12 where ∆ t = ∆ 1 + ∆ 2 + ∆ 3 and we introduced Notice that we already used the dilatation symmetry to fix the exponent of k 3 . Also it is easy to show that eq. (B.4) provides the most general ansatz consistent with the dilatation symmetry and the polarization vector dependence. Our task is now to determine the functions a n (κ 1 , κ 2 ). In appendix B.1, we first derive a recursion relation for a n by using a subset of special conformal Ward-Takahashi (WT) identities. We then solve the recursion relation explicitly in appendix B.2. The initial condition for this program will be given in appendix B.3. The relation between the momentum space correlator and the position space one is discussed in appendix B.4 by performing Fourier transformation.

B.1 Recursion relations for a n
In momentum space, the special conformal WT identity is stated as where an explicit form of the special conformal generator K is given by For later convenience, we contracted the generator with an arbitrary vector b. Also, we introduced a shorthand notation, (B.8) 12 Our definition of an is slightly different from the one in [18]: a

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A useful observation is that the second line of eq. (B.7) trivially acts on k 3 -independent functions. We therefore write the three-point functions as functions of k 1 and k 2 . In particular, k 3 should be understood as k 2 3 = (k 1 + k 2 ) 2 . Another observation is that the last term in the first line of eq. (B.7) vanishes when b.k 1 = b.k 2 = 0. For this choice of b (we also assume .b = 0 for later convenience), the special conformal WT identity is simply [ Notice that this is a subset of the special conformal WT identity, hence there still exist two independent conditions associated with b which does not satisfy b.k 1 = b.k 2 = 0. These two conditions may be used to constrain a 0 as we discuss in appendix B.3. We then apply the condition (B.9) to the ansatz (B.4). By reorganizing the differential operator in terms of b.∂ 1 ± b.∂ 2 , eq. (B.9) can be rephrased as where we used the fact that three-point functions have a scaling dimension ∆ t − 2d (after dropping the delta function for momentum conservation). The next step is to rewrite the momentum derivatives as those in γ, δ, κ 1 , κ 2 . For this, we use the following relations: b.∂ i f (k 1 , k 2 , k 3 ) = 0 (i = 1, 2) , An important point is that the operator b.∂ i trivially acts on a n since this is a function of k 1 , k 2 , and k 3 . We then arrive at γ s−n δ n a n (κ 1 , κ 2 ) . (B.12) Here we used .b = 0 and introduced a differential operator, which contains derivatives in κ 1 and κ 2 only. Also note that we regard γ, δ, κ 1 and κ 2 as independent variables in the expressions (B.12)-(B.13). For example, we have ∂ κ i γ = 0 and [D, γ] = 0, which let us solve the recursion relation as we discuss in the next subsection.

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Finally, by rewriting eq. (B.12) into the form, s n=0 − (s − n)γ s−n−1 δ n D + n(∆ 3 − 1 + s − n)γ s−n δ n−1 + (s − n)(s − n − 1)γ s−n−2 δ n+1 a n = 0 , (B.14) we may pick up the desired recursion relation for a n from the O(δ n ) terms as Da n − (s − n + 1)a n−1 , where we define a −1 = 0 for convenience. Also note that this recursion relation is consistent with a s+1 = 0. Once a concrete form of a 0 is given, we may compute a n at least recursively. We can easily see that a n is schematically of the form, a n ∼ where we wrote numerical coefficients as α n,k . Note that α 0,0 = 1 in particular.

B.2 A closed form of 3pt functions
It might seem difficult to find a closed form of α n,k , but it is actually possible to find a closed form of (B.17) by solving the special conformal WT identity (B.12). The special conformal WT identity in terms of (B.17) reads It is convenient to further rewrite it as for γ and δ. By setting z = γ/δ in the above, we can reduce it into Notice that the function h D (z), by construction (B.20), is a polynomial in z of order s. We can therefore find its unique solution, which is a hypergeometric function, This is indeed a polynomial of order s because the hypergeometric function has negative integer −s in its first slot. 13 The coefficient of proportionality may depend on D, but we can fix it by requiring h D (z) = z s + · · · , which follows from α 0,0 = 1 mentioned in the previous subsection. We then arrive at By substituting this back into (B.17), we obtain where we performed a change n → s − n at the second equality. Our final task is now to find the initial condition a 0 for the recursion relation. We work on this problem in the next subsection and find where the OPE coefficient C 123 in momentum space is related to the position space one as eq. (B.59). Also its normalization is chosen in such a way that the final expression is simple. By substituting this explicit form into eq. (B.26) and rescaling the integration variable in the triple-K integral as z → k 3 z, we conclude that 13 The concrete relation is where we introduced the rescaled differential operator D 123 as 14 This is the first closed form (B.1). Here k 1 , k 2 and k 3 should be understood as independent variables when D 123 acts on the triple-K integral. More concretely, ∂ k i k 3 = 0 (i = 1, 2) for example. Note also that the final form (B.28) depends on the space dimension d only through the differential operator D 123 and the triple-K integral. Before discussing the initial condition problem, let us take a quick look at the symmetry under the exchange 1 ↔ 2 of the two scalars. Since eq. (B.17) implies the invariance of three-point functions under (δ, D) → (−δ, −D), we may reformulate eq. (B.26) as This yields another expression for three-point functions: By comparing this expression with eq. (B.28), we see in particular that the three-point function vanishes when two scalars are identical and the tensor has an odd spin s.

B.3 Initial condition a 0
So far, we have used a subset of the special conformal WT identity (B.6) with b satisfying b.k 1 = b.k 2 = 0. In this subsection, we determine the initial condition a 0 by solving the residual WT identities for b.k 1 = 0 and b.k 2 = 0. To simplify the calculation, we also set b. = 0 [18]. For later convenience, let us first modify the ansatz (B.4) as where we rescaled a n as a n (k 1 , k 2 , k 3 ) = k ∆t−2d−s 3 a n (k 1 /k 3 , k 2 /k 3 ). Also, it follows from the dilatation WT identity that a n 's are (∆ t − 2d − s)-th order homogeneous polynomials. The special conformal WT identity for b.k 1 = 0, b.k 2 = 0 and b. = 0 is then given by b.K s n=0 γ s−n δ n a n (k 1 , k 2 , k 3 ) = 0 , (B.33)

B.4 Three-point functions by Fourier transformation
Finally, let us discuss the relation between the momentum space correlator and the position space one. The Fourier transformation of the position space correlator (4.12) is given by where we replaced the position coordinates in the numerator of eq. (4.12) by momentum derivatives, after the binomial expansion. The integral here is essentially the same as the scalar three-point function, hence it gives a triple-K integral. A more precise relation is given through a formula (see, e.g., [47] for derivation), where we defined a t = a 1 + a 2 + a 3 . Substituting (B.42) back into (B.41), we find the following derivatives: where we introduced m 3 = 0 for notational simplicity. Since the triple-K integral depends only on the magnitudes k i of the momenta k i , we can replace the derivatives in k i by those in k i . More explicitly, for i = 1, 2 we have where each function f depends only on the magnitudes of the momenta. Also, we used . = 0. Combining them with the generalized Leibniz rule (i = 1, 2), we have

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Eq. (B.43) may then be reformulated as (B.47) Here and in the rest of this section, we use k 3 and |k 1 + k 2 | interchangeably because there are no more derivatives. Also we used a relation following from eq. (D.10), which induced the shift of the arguments in the triple-K integrals. By applying this result to (B.43), we arrive at the second expression for momentum space correlators, where we wrote the triple-K integral in terms of the bulk-to-boundary propagator B ν as We also introduced a new overall factor, , (B.51) and a numerical coefficient,  . (B.59)

C Explicit form of A 123m
In this appendix we derive the analytic form where we performed a change of the integration variable ψ → ψ + χ at the second equality. 16 One might wonder that the exchange symmetry of two scalars is not manifest in the second expression.
However, we may also write

JHEP07(2018)136 D Special functions
We summarize the definition and properties of special functions used in this paper.

D.1 Jacobi polynomials
The Jacobi polynomial is defined as P (α,β) n (t) = Γ(α + n + 1) n! Γ(α + β + n + 1) In general, the following Rodrigues' formula plays a central role: The Jacobi polynomials are orthogonal in the sense that = 2 α+β+1 α + β + 2n + 1 Γ(α + n + 1)Γ(β + n + 1) n! Γ(α + β + n + 1) δ mn . (D. 3) The orthogonality for m = n follows from the Rodrigues' formula (D.2). The normalization factor can be found as a special case α = γ of the integral formula (7.391.10, p.807, [71]) We obtain eq. (D.3) for m = n by first setting m = n to cancel Γ(γ − α) and then setting α = γ, whereas m = n can be obtained by setting α = γ first. In this paper we use the Chebyshev and Gegenbauer polynomials, which are defined as special cases of the Jacobi polynomial. The Chebyshev polynomial is defined as T n (t) = n! √ π Γ(n + 1 2 ) which has the property T n (cos θ) = cos nθ and T 0 (t) = 1. The Gegenbauer polynomial is defined as C (α) n (t) = (2α) n (α + 1 2 ) n P (α− 1 2 ,α− We call these functions the Bessel I and K functions in short. Also, the Bessel differential equation, the defining differential equation of these functions, is given by which is useful to derive the bulk-to-boundary propagator B ν (k; z) from the equation (2.7). A useful relation of the modified Bessel functions is In the main text, we also use the asymptotic behavior around z ∼ ∞, which is given by Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.