Momentum space approach to crossing symmetric CFT correlators. Part II. General spacetime dimension

Our previous work [1] constructed, in three-dimensional momentum space, a manifestly crossing symmetric basis for scalar conformal four-point functions, based on the factorization property proposed by Polyakov. This work extends this construction to general dimensional conformal field theory. To facilitate the treatment of symmetric traceless tensors, we exploit techniques of spherical harmonics in general dimensions.


Introduction
The crossing symmetric basis of conformal four-point functions, pioneered by Polyakov in 1974 [2], is based on the following ansatz of the expansion of the four-point functions, 1 O is called the Polyakov block in the s-channel. Each block satisfies the consistent factorization in momentum space, just as the on-shell factorization of scattering amplitudes of ordinary field theories. The Polyakov block with external scalar operators was shown in Mellin space to be nothing but the Witten exchange diagram [3,4]. Our paper [1] showed directly in three-dimensional momentum space that the Witten exchange diagram is a natural consequence of the consistent factorization, and constructed the Polyakov block with an intermediate symmetric traceless operator of general spin.

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This paper extends the construction in [1] to general dimensions. From a technical point of view, the extension to general dimensions becomes a bit more involved due to the treatment of symmetric traceless tensors of general spins. In the three dimensional case [1], we adopted the helicity representation instead of dealing with explicit vector indices. More concretely, one first fixes one of the momenta in the correlation function by using rotational symmetry without loss of generality. One then decomposes a symmetric traceless tensor in the correlator into irreducible representations of the little group of the fixed momentum. In three-dimension, the little group is O(2) and the expansion is nothing but the Fourier expansion [1,5]. In general dimension, the Fourier expansion is replaced by the expansion in spherical harmonics on the general dimensional unit sphere. In the present paper we elucidate this point in general spacetime dimension to construct a crossing-symmetric basis of scalar four-point functions.
The outline of the rest of this paper is as follows. Section 2 is a brief review of the harmonic analysis needed for our analysis. In section 3 we find the helicity representation of two-and three-point functions with spins, and discuss their analytic properties to define the cubic vertex. In section 4 we construct the crossing symmetric basis of scalar four-point functions. The two appendices present derivations of some integral formulae in section 3.

Properties of spherical harmonics
This section reviews basic properties of spherical harmonics in general spacetime dimensions. The machinery will play a main role in expanding scalar functions of momenta and polarization vectors which appear in conformal two-and three-point functions. Readers familiar with spherical harmonics in general dimension may jump safely to section 3 after checking the Funk-Hecke formula introduced in section 2.1 SO(2) spherical harmonics. Before going into the general dimensional case, let us consider lower dimensional cases, in which the expansion is achieved easily. As the simplest case, we begin with a scalar function of two unit vectors,ŵ = (ŵ 1 ,ŵ 2 ) andẑ = (ẑ 1 ,ẑ 2 ), in two dimension. More explicitly, it is a function of the inner product of the two unit vectors, f (ŵ ·ẑ). If this function is regular atŵ ·ẑ = 0, we may expand it as where λ m are constant parameters. Notice that the functions (ẑ 1 ± iẑ 2 ) m are nothing but the SO(2) spherical harmonics. If we denote a basis of the SO(2) spherical harmonics by we may rewrite eq. (2.1) as

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where we introduced the projector Π m onto the spin m sector as The expansion is nothing but the Fourier expansion, as is manifest in polar coordinates.

SO(3) spherical harmonics.
A similar expansion applies in the three dimensional case. As we physicists are familiar with, any regular scalar function of two unit vectors,ŵ and z, in three dimension may be expanded by the Legendre polynomials P m (ŵ ·ẑ). More explicitly, we may write it in the form (2.3) with some Π m ∝ P m (ŵ ·ẑ). Since the addition theorem of the SO(3) spherical harmonics states that for any orthonormal basis Y mn , we may normalize Π m such that which is again nothing but the projector onto the spin m sector.

Spherical harmonics in general dimension and Funk-Hecke formula
We can generalize the expansion of scalar functions with two unit vectors to general dimensions in a straightforward manner. The expansion is called the Funk-Hecke formula [6][7][8]: any scalar function of two unit vectors,ŵ andẑ, in D dimension is expanded as where Y mn stands for an orthonormal basis of SO(D) spherical harmonics with total spin m and dim Y D m is the dimension of the spin m representation of SO(D). 2 The coefficients λ m are evaluated by the integral, where z µ is a general D-component vector satisfying z µ = |z|ẑ µ . Note that Ω2 is independent of |z|. The non-negative integer m is the total spin. The dimension of Y D m is given by See also the footnote in appendix B for the relation to harmonic polynomials.

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where P with the Gegenbauer polynomial, Here [x] is the integer part of x and (α) m = Γ(α + m)/Γ(α) is the shifted factorial (also known as the Pochhammer symbol). We may also write the projector Π m as 3 This is called the addition theorem of spherical harmonics. In the rest of this section we summarize the basic properties of the spherical harmonics and review the derivation of the Funk-Hecke formula.

Derivation of the Funk-Hecke formula
We begin with the fact that any function on the unit sphere S D−1 can be expanded in spherical harmonics, The basis spherical harmonics Y mn are orthogonal and normalized as Y mn , Y m n = δ mm δ nn (2. 16) with the inner product defined by Here the integration measure dσ D−1 is the standard one on the unit sphere S D−1 with the normalization, We then introduce the projection operator onto the spin m subspace Y D m as In terms of the projection operator, the spin decomposition (2.15) reads (2.20)

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Explicit form of the projector. Let us derive the expression (2.14) of the projector Π m exploiting its rotational invariance. For this purpose, we first introduce the little group O(D,â) of a unit vectorâ by . It is known that any O(D,â)-invariant spherical harmonic of spin m is proportional to the normalized Gegenbauer polynomial [8], We will often use this property. As the first example, let us prove the addition theorem (2.14). Notice first that the projector Π m (ẑ,ŵ) is O(D)-invariant because the rotated spherical harmonics Y mn (Aẑ) (A ∈ O(D)) form another orthonormal basis of spin m. In particular, we have Π m (Aẑ,ŵ) = Π m (ẑ,ŵ) for any little group transformation A ∈ O(D,ŵ). Therefore, if we think of the projector Π m (ẑ,ŵ) as a function ofẑ, it is an O(D,ŵ)-invariant spherical harmonic of spin m. Therefore, it follows from the equivalence (2.22) that for each m The Funk-Hecke formula. The Funk-Hecke formula (2.8) with (2.11) is also a consequence of the equivalence (2.22). Regarding a scalar function f (x ·ẑ) as a function ofẑ, we may use the spin decomposition (2.20) to write where we used the addition theorem (2.14). Since each summand is an O(D,x)-invariant spherical function ofẑ of spin m, we may apply the equivalence (2.22) to find where we introduced This concludes the proof of the Funk-Hecke formula (2.8) with (2.11).

Two-and three-point functions
In this section we introduce helicity representation of conformal correlators with symmetric traceless tensors in general spacetime dimension d. After elaborating on the helicity JHEP10(2019)183 decomposition of spinning operators, we utilize the Funk-Hecke formula to derive helicity representation of two-and three-point functions in momentum space. 4 We then discuss their analytic properties. The cubic vertices introduced there will be used in the next section to construct the crossing symmetric basis.

Helicity decomposition of spinning operators
A standard technique to handle a symmetric traceless tensor in CFT is to contract all vector indices in the tensor operator with a null vector called the polarization vector [45,46]. 5 More explicitly, we denote this in a shorthand notation, where s is the spin of the operator O. In momentum space it is convenient to further decompose the operator by analogy with the helicity decomposition of massless on-shell particles. Without loss of generality, let us use rotational invariance to set where 0 is the (d − 1)-component zero vector. It is then convenient to parameterize the polarization vector as whereẑ is a (d − 1)-component real unit vector. With this parameterization the contracted tensor operator s .O can be thought of as a scalar function on the unit sphere S d−2 with the coordinateẑ, so that we may decompose it into the helicity operators O mn as where {Y mn } is an orthogonal normal basis for the space of spin m spherical harmonics As explained in section 2.1, this is the decomposition with respect to the spin m of the little group O(d − 1) of k: each spin m sector gives an irreducible representation of the little group. For later use it is convenient to introduce its conjugate as Helicity operators with a general momentum are defined in a similar way by performing an appropriate rotation.

Two-point functions
In momentum space, two-point functions of primary operators with general spins read [1] 6 , (3.8) where primed correlators are defined by dropping the delta function for momentum conservation as . .
with the scaling dimension ∆. In this paper we use ∆ and ν + d 2 interchangeably to simplify equations. P (α,β) n is the Jacobi polynomial defined by Following the last subsection, let us use rotational invariance to set the momentum and the polarization vectors as which is a function ofẑ 1 ·ẑ 2 multiplied by a helicity-independent factor.
Helicity representation. We then introduce the helicity representation of the two-point functions. Since the two-point function (3.11) is a scalar function of the inner productẑ 1 ·ẑ 2 , we may apply the Funk-Hecke formula (2.8) to find where Π m (ẑ 1 ,ẑ 2 ) is the projector (2.19) onto the spin m sector Y d−1 m on the unit sphere S d−2 . The factor a ν,s (m) is given by the integral formula (2.11) as (3.13) 6 The normalization of the position space correlator is given by which is related to our momentum space normalization as COO . (3.7)

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which is computed in appendix A to find (3.14) The two-point function of helicity operators now reads

Three-point functions of two scalars and one tensor
We then move on to three-point functions involving two primary scalars and one primary tensor. In momentum space they are given by [1] where we collected a helicity-independent part into the last factor as Here C 12O is a normalization factor and B ν is defined by with K ν (z) being the modified Bessel function of the second kind. Note that B ν is nothing but the bulk-to-boundary propagator of a scalar field on AdS d+1 with mass m 2 = ν 2 −d 2 /4. Also D 12O is a differential operator defined by (k 12 := |k 1 + k 2 |) We refer the reader to ref. [1] for details of the helicity independent part (3.17). In the following we instead discuss the helicity structure of the three-point function (3.16).
Helicity representation. As in the case of two-point functions, we use rotational invariance to fix the momentum of the tensor O and parameterize the polarization vector as whereẑ is a (d − 1)-component real unit vector as before. With this parameterization the three-point function (3.16) reads

(3.22)
We then decompose the three-point function (3.21) in the spin of the little group of k 3 . Since this is a scalar function of the inner productκ 2 ·ẑ, the Funk-Hecke formula (2.8) yields the following spin decomposition: In other words the three-point function with the helicity operator O mn is given by Here the m-dependent factor V 12O (m, θ) is given through the integral formula (2.11) by

Analytic properties
At the end of this section, we discuss analytic properties of three-point functions. In particular, we demonstrate that the non-analytic part of a three-point function enjoys a factorization similarly to scattering amplitudes. Our argument here is parallel to that in ref. [1], to which we refer the reader for more detailed explanations.
To discuss analytic properties, let us first rearrange eq. (3.24) into the form,

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where we introduced a differential operator A (m) 12O as Notice that A (m) 12O depends on the helicity only through the spin m of the little group. An important observation here is that the only source of non-analyticity in the three-point function (3.27) is the integral of three bulk-to-boundary propagators B ν (k; z): first, the spherical harmonics and the normalized Gegenbauer polynomial are accompanied by an appropriate power in k 2 as (k 2 sin θ) m Y * mn (κ 2 ) and k s−m−a s−a,m (cos θ), so that they are polynomials in k 2 . Second, the differential operator D 12O always appears in the form k 3 D 12O , which is a polynomial in the momenta k i and the Euler operators k i ∂ k i because (k 2 −k 1 )·k 3 /k 3 is a component of the vector k 2 −k 1 along the k 3 direction. Since the Euler operator does not introduce any new non-analyticity, we conclude that the only source of non-analyticity is the integral in eq. (3.27).
As we mentioned, the integrand of eq. (3.27) contains three bulk-to-boundary propagators B ν (k; z). By looking at analytic properties of B ν (k; z), we find that the integral enjoys the following factorization (see section 3 of ref. [1] for details): where Disc z denotes a discontinuity on the complex z plane. Also, I ν (z) is the modified Bessel function of the first kind. Since the prefactor and the differential operator A (m) 12O in eq. (3.27) do not generate any new non-analyticity as discussed above, the non-analytic part of the three-point function (3.27) also factorizes as where we introduced what we call the cubic vertex T 12;Omn as Notice that the cubic vertex is analytic at k 3 = 0. We will use it in the next section to construct a crossing symmetric basis for scalar four-point functions.
Three-point functions with a conjugate operator. For later use, it is convenient to write down three-point functions involving the conjugate operator O mn explicitly. Let us first recall that (3.32)

Crossing symmetric basis for scalar four-point functions
We then construct the crossing symmetric basis for scalar four-point functions using the ingredients introduced in the previous section. After reviewing the strategy employed in our previous work [1], we provide an explicit construction in general dimension d.

Strategy
In the previous section we have demonstrated that the non-analytic part of three-point functions factorizes into the cubic vertex (3.32) and the two-point function as which is analogous to on-shell factorization of scattering amplitudes. 7 Its conjugate counterpart is given in eq. (3.33) with the conjugate cubic vertex (3.34). Similarly, we require that the non-analytic part of four-point functions factorizes as where the first sum is over all the intermediate primary operators and the second is over helicity components O mn of the operator O. We also introduced k ij = k i + k j and the Mandelstam type variables,

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Since the on-shell conditions are not imposed on the external momenta, these variables are independent in contrast to the scattering amplitude case. We require similar factorization in the other channels as well. Based on the factorization property, we introduce a crossing symmetric basis for conformal four-point functions as [2] where the second term stands for analytic terms which cannot be determined only from analyticity. 8   O has no non-analyticity other than the one required by s-channel factorization. In particular, it is analytic in k 13 and k 14 , so free from t, u-channel discontinuity.
Also, W O and enjoy similar properties. This basis manifests the crossing symmetry whereas the consistency with the OPE is obscured, hence the latter provides a nontrivial constraint on the theory [2,24].

Construction of Polyakov block
Let us proceed to constructing the s-channel Polyakov block. In our previous work [1] we have shown that the Polyakov block with an intermediate scalar operator is nothing but the scalar-exchange Witten diagram: 9 W (s) where we introduced the bulk-to-bulk propagator of the would-be dual bulk scalar as 8 To constrain the analytic contributions, other ingredients will be necessary such as consistency with OPE or locality of the dual bulk theory. Note that the analytic terms correspond to bulk contact terms. 9 See [3,4] for construction in the Mellin space.

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The non-analytic part of the bulk-to-bulk propagator, i.e., the first line of eq. (4.7), is responsible for the factorization so that the first criterion (4.5) of the Polyakov block is satisfied. Furthermore, the last expression of eq. (4.7) guarantees that the bulk-to-bulk propagator exponentially damps down for large z i , hence there appear no undesirable singularities, e.g., in the collinear limit k 1 + k 2 = k 12 . These criteria specify the form of the Polyakov block up to analytic terms, which correspond to bulk contact interactions. See [1] for more details.
The key observation for extending the construction for intermediate scalars to general spinning operators is that the differential operators, appearing in the cubic vertices (3.32) and (3.34), do not change the non-analytic properties.
As a result, we may easily arrive at where G ν O (k 12 ; z 1 , z 2 ) is the scalar bulk-to-bulk propagator defined by eq. (4.7). We also defined the angles θ 2,4 and the unit vectorsκ 2,4 in the frame −k 12 = (0, k 12 ) such that 10 k 2 = k 2 (κ 2 sin θ 2 , cos θ 2 ) , −k 4 = k 4 (κ 4 sin θ 4 , cos θ 4 ) . (4.10) Again, the non-analytic part of the propagator is responsible for the s-channel factorization and its large z i behavior guarantees that there are no undesirable singularities. Furthermore, since in eq. (4.9) the spherical harmonics in the first line are the only n-dependent factors, the sum over n is nothing but the addition theorem (2.14) with (2.19). The result is This concludes our construction of the s-channel Polyakov block with an intermediate operator of an arbitrary spin s. The t, u-channel blocks are defined in a similar fashion. 10 Our definition of θ4 andκ4 is motivated by the fact that T34;O mn (k3, k4; k12) is conjugate to T34;O mn (−k3, −k4; −k12).

JHEP10(2019)183 5 Conclusion
This paper generalized the construction of the crossing symmetric basis of scalar fourpoint functions in three spacetime dimension [1] to general dimensions. To deal with the complication due to spins in general spacetime dimension, we utilized techniques of spherical harmonics.
A natural direction to explore along the line of the present work and our previous one [1] is to generalize the construction to correlators involving external spinning operators such as the energy-momentum tensor and other conserved currents. A first step in this direction is the construction of three-point functions with conserved currents and one primary operator of arbitrary spin, which has been done recently in ref. [40]. There will be no conceptual obstruction to constructing the crossing symmetric basis of four-point functions with external conserved currents based on the results there.
Another interesting direction is the extension to de Sitter and inflationary correlators. Some related recent works include [37,42,43]. For example, ref. [37] constructed a crossing symmetric basis of de Sitter four-point functions with external scalars of the conformal mass in four dimensions (dual to scalar operators of conformal weight ∆ = 2 in three dimensions) by solving the conformal Ward-Takahashi identities and studying (non-)analytic properties of de Sitter correlators. The extension to four-point functions with external massless scalars was also explored there aiming at applications to inflationary physics. More recently, refs. [42,43] developed a Mellin representation of exchange diagrams on (anti-)de Sitter space in momentum space for external scalars with arbitrary mass. It would be interesting to explore the relation of these works with ours. We hope to revisit these issues in the near future.
A Derivation of eq. (3.14) We derive the analytic expression (3.14) of the factor a ν,s (m). We present the integral form of a ν,s (m) again, To evaluate this, it is convenient to notice that the Gegenbauer polynomial is a special case of the Jacobi polynomial: where we introduced the angle θ by z D = |z| cos θ.
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