Abstract
In conformal field theory in Minkowski momentum space, the 3point correlation functions of local operators are completely fixed by symmetry. Using Ward identities together with the existence of a Lorentzian operator product expansion (OPE), we show that the Wightman function of three scalar operators is a double hypergeometric series of the Appell \(F_4\) type. We extend this simple closedform expression to the case of two scalar operators and one traceless symmetric tensor with arbitrary spin. Timeordered and partiallytimeordered products are constructed in a similar fashion and their relation with the Wightman function is discussed.
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1 Introduction
Conformal field theory can be formulated algebraically in terms of a set of primary operators and of rules that define the operator product expansion (OPE). Equivalently, all correlation functions of a conformal field theory can be obtained from 2 and 3point functions, which are themselves fixed by conformal symmetry up to a small number of numerical coefficients. This statement applies both to correlation functions in position space and in momentum space, but it is by far more common in conformal field theory to use the positionspace representation. There are several good reasons why it is so:

1.
All the 2 and 3point functions in position space are known and relatively easy to evaluate. In the case of scalar operators and of operators carrying lowdimensional spin representations they have been known since a long time [1,2,3,4,5]. More recently, correlation functions involving larger spin representations have been constructed using the embeddingspace formalism [6,7,8,9,10,11,12], and this construction can be extended in an algorithmic way to arbitrary spin representations using weightshifting operators [13, 14].

2.
Higherpoint functions can be computed with the help of an operator product expansion that has a large range of convergence. This OPE applies naturally when two operators are close in space, but it actually extends over most of the possible configuration space [15,16,17,18]. This implies in particular that distinct convergent OPEs can be used to compute the same correlation function, which is the key property exploited by the conformal bootstrap [19,20,21,22,23].

3.
There is a simple connection between the correlation function in Euclidean and Minkowski position space: Wightman functions in Minkowski space are obtained from Euclidean correlators by a straightforward Wick rotation. This property relates unitary Lorentzian theories to reflectionpositive Euclidean ones and explains the reality of 3point function coefficients.
All of these properties are altered in momentum space. Let us review them in reverse order:

3.
There are branch cuts in the complexified momentum space that make the Wick rotation between Minkowski and Euclidean space nontrivial. A comprehensive discussion of this phenomenon has recently appeared in Ref. [24]. We shall see in this paper that the timeordered product of operators in Minkowski momentum space is simply related to the known Euclidean expression, but also that the Wightman functions are very different objects that do not have a Euclidean counterpart.

2.
A momentumspace OPE can be defined by the Fourier transform of the positionspace OPE. Its convergent limit is when the two operators involved have both large momenta [25]. However, this momentumspace OPE has only been formulated in Euclidean theories so far. Very little is known about the Lorentzian OPE, about its convergence properties, or even whether it converges at all.

1.
Maybe more surprisingly, our knowledge of conformal 3point functions in momentum space is quite incomplete. While they have been extensively studied in Euclidean theories [25,26,27,28,29,30,31,32,33,34], partly because of their relevance for inflation [35,36,37,38,39,40,41,42,43], it is only recently that their study in Lorentzian signature has begun [24, 44,45,46]. Moreover, even in the simplest case of 3 scalar operators, the only expression available in the literature so far is in the form of a quite complicated integral over Bessel functions [24].
In spite of these difficulties, there exist strong motivations to study conformal field theory in Minkowski momentum space. For instance, the lightlike limit of momentumspace correlators is intrinsically connected with the study of local operators integrated along a light ray [47,48,49,50], which has been instrumental in the derivation of conformal collider bounds [51,52,53,54,55,56,57,58], of the proof of the average null energy condition from causality [59, 60] or even in the study of asymptotic symmetries [61]. A limit of the momentum space 3point function also enters in the lightcone Hamiltonian truncation formalism [62, 63], and the use of momentum space makes anomaly coefficients appear explicitly in correlators [44, 45]. Moreover, when the momentumspace 3point functions are known, constructing conformal blocks out of them is simple in the sense that it does not require additional integration [44,45,46]: a recent example where this technology has been put to good use is Ref. [64].
Even though this list of motivations is far from exhaustive, it makes evident that there is an interest in closing the gaps of points (1) and (2) discussed above. The goal of this paper is precisely to improve on point (1) by providing a simple closedform expression for the Wightman 3point function in Minkowski momentum space. This goal is after all quite modest since it consists in taking the Fourier transform of a known positionspace 3point function, but we will see that its computation is not quite simple. Along the way, we will also touch upon point (2), although without discussing the delicate issue of OPE convergence.
1.1 Strategy and main result
The strategy for determining the 3point function will be to use conformal Ward identities to express it in terms of the solutions of some differential equation, in the spirit of the Euclidean derivation of Ref. [25]. What is new in our case is not so much the difference between the Euclidean and Minkowskian conformal algebras as it is the boundary condition provided by the Lorentzian OPE. With this strategy, we do not perform directly the Fourier transform of the positionspace correlation function, although we use it to verify numerically and in some limits analytically the validity of our derivation. In the case of 3 scalar operators, our result is
where \(F_{\Delta _f \Delta _0 \Delta _i}\) is an Appell \(F_4\) generalized hypergeometric function of two variables defined in Eq. (36), and \({\widetilde{\lambda }}_{f0i}\) is an OPE coefficient related to the usual one by Eq. (46). We have eliminated the \(\delta \)function demanding overall momentum conservation by use of the notation
and the \(\Theta \)functions impose conditions on the momenta \(p_i\) and \(p_f\),^{Footnote 1}
This result applies in the regime \(p_f, p_i < p_0\) where the function \(F_{\Delta _f \Delta _0 \Delta _i}\) is analytic, while the discussion of Sect. 2.4 and in particular the general expression (47) covers all kinematic configurations. This result is valid for any scaling dimensions of the operators, as well as in any spacetime dimension \(d \ge 2\).
The rest of the paper is organized as follows: Sect. 2 is devoted to explaining the steps that lead to the result (1). It also contains discussions of the special cases that are generalized free field theory and \(d = 2\) spacetime dimensions. We then generalize this result in Sect. 3, replacing one of the scalar operators with a traceless symmetric tensor of arbitrary spin. In Sect. 4 we introduce the timeordering operator in the 3point function, show how the result differ from the Wightman function, and perform consistency checks. The appendix presents the direct Fourier transform of the positionspace correlation function used to verify our results numerically.
2 The Wightman 3Point Function of Scalar Operators
We begin with a derivation of the Wightman function of 3 distinct scalar operators. The momentumspace representation of this correlation function could in principle be obtained directly by performing the Fourier transform of the Wightman function in position space, which is known and relatively simple. This is the approach followed in Ref. [24], and the result is an integral over Bessel functions. We will follow instead a different approach purely based on the symmetries of the 3point function and on the existence of an operator product expansion (OPE). Our result is a closedform expression, which provides a more practical and efficient way of evaluating the scalar 3point function at any point in momentum space.
2.1 Momentum eigenstates and support
Before we begin with the derivation, it is useful to recall some properties of the momentumspace representation. The Hilbert space of a conformal field theory can be constructed in terms of a (infinite) set of primary states \(\left {\mathcal {O}} \right\rangle \) and of their descendants obtained by acting repeatedly with the generator of translations \(P^\mu \),
An equivalent representation of this Hilbert space is in terms of distributions over flat Minkowski space,
The state/operator correspondence associates to each such state a local operator \({\mathcal {O}}(x)\) such that
Taking \(P^0\) as the Hamiltonian of the theory, one should include in the definition (5) the prescription \(x^0 \rightarrow x^0 + i \epsilon \) with positive \(\epsilon \) so that the norm of the state \(\left {\mathcal {O}}(x) \right\rangle \) is welldefined when the Hamiltonian is bounded from below.^{Footnote 2} For a scalar state \(\left \phi (x) \right\rangle \), this norm, or equivalently the Wightman 2point function of the operator \(\phi (x)\), is given by
where \(\Delta \) is the scaling dimension of the operator \(\phi \) and the normalization is conventionally chosen.
There exist yet another equivalent representation of the Hilbert space given by the states
which are eigenstates of the generator of translations, \(P^\mu \left {\mathcal {O}}(p) \right\rangle = p^\mu \left {\mathcal {O}}(p) \right\rangle \). As before, these states are in onetoone correspondence with the set of operators
The advantage of this basis is that the states are orthogonal in a distributional sense: their norm satisfies^{Footnote 3}
and hence vanishes if \(p_f \ne  p_i\). The function \(\Theta \) is defined in Eq. (3): it indicates that the norm only has support when \(p_i\) (and thus \(p_f\)) is timelike and has positive energy. When this condition is not satisfied, the state must be null:
Note that this property is specific to the Lorentzian theory: momentum eigenstates can be constructed in an Euclidean theory but they have different characteristics. In the notation of Eq. (2), the Wightman 2point function of a scalar operator is therefore
These general considerations are also important for the Wightman 3point function since it can be written as the expectation value of an operator between two momentum eigenstates,
where we have used the labels i for “initial” and f for “final” states. Because of the condition (11) on the states, this 3point function only has support when both momenta \(p_i\) and \(p_f\) are timelike with positive energies, i.e.
By translation invariance, correlation functions in momentum space are always proportional to a \(\delta \)function, in this case enforcing \(p_f + p_0 + p_i = 0\). Nevertheless, the constraint (14) does not restrict the intermediate momentum \(p_0\), which can be either spacelike or timelike, with positive or negative energy. Two possible configurations of momenta are shown in Fig. 1.
There are additional constraints coming from conformal symmetry: using Lorentz symmetry, we can choose to parametrize the 3point function in terms of the three invariant quantities \(p_f^2\), \(p_0^2\) and \(p_i^2\). This choice is not unique, but it will turn out to be the most convenient in the next sections. The scale symmetry determines the overall scaling dimension of the 3point function. Taking \(p_0^2\) as the reference scale, we can write
where F is a function of two dimensionless arguments. This might seem a curious choice of reference scale since both \(p_f^2\) and \(p_i^2\) are positive over the region of support whereas \(p_0^2\) can potentially change sign. We will see in the next section that this choice is motivated by the various OPE limits of the 3point function. Moreover, note that the scalar 3point function enjoys the conjugation symmetry
which means that the choice (15) makes F a symmetric function under the simultaneous exchange of its two arguments and of the scaling dimensions \(\Delta _f\) and \(\Delta _i\). Finally, there are constraints coming from the special conformal symmetry that will completely restrict the form of F. Since these constraints are much more involved, we dedicate Sect. 2.3 to their study. But before proceeding with them, we discuss the role played by the operator product expansion.
2.2 OPE limits in momentum space
Besides the Hilbert space construction discussed in the previous section, the other key property of conformal field theory is the existence of an operator product expansion. The OPE expresses how a local operator acts on the Hilbert space of the theory: in the positionspace representation,
where the operator \(C_{{\mathcal {O}}12}(x, P)\) is completely fixed by conformal symmetry, while the OPE coefficients \(\lambda _{{\mathcal {O}}12}\) encode the dynamical content of the theory. \(C_{{\mathcal {O}}12}\) is a series expansion in the generator \(P^\mu \), for instance in the case where the operators \(\phi _1\), \(\phi _2\) and \({\mathcal {O}}\) are scalars
In an Euclidean CFT x would be the Euclidean norm; in a Lorentzian CFT, it is its analytic continuation \(x^2 = (x^0  i\epsilon )^2 + \mathbf {x}^2\).
Since the OPE ultimately expresses the completeness of the Hilbert space, there must exist a similar statement in the momentumspace representation. Taking the Fourier transform of Eq. (17) with respect to both \(x_1\) and \(x_2\), one can write
where we have defined
We have used the fact that \(\left {\mathcal {O}}(p_1 + p_2) \right\rangle \) is a momentum eigenstate to replace the generator \(P^\mu \) by its eigenvalue. For this reason, \({\widetilde{C}}_{{\mathcal {O}}12}(p, q)\) is not anymore a derivative operator acting on the primary \({\mathcal {O}}\) but just a number. This is a consequence of the orthogonality of momentum eigenstates. One should realize however that this definition of the momentumspace OPE is purely formal so far, and it faces two major problems. First, we have not established whether the Fourier transform commutes with the sum over conformal primaries. It is known that the OPE for Wightman functions converges in the sense of distributions [65], but the series might not converge at every given configuration of momenta (see Ref. [66] for a discussion and examples in \(d = 2\) dimensions). In any case, this problem is absent when the OPE applies to a 3point function since the sum is given by a single term. The second problem is a practical one: using the expansion (18), one can formally write
and recognize in the integral on the righthand side the Wightman 2point function of a fictitious operator with scaling dimension \((\Delta _1 + \Delta _2  \Delta _{\mathcal {O}})/2\). This integral is discontinuous at \(p^2 = 0\) and the dependence of \({\widetilde{C}}_{12{\mathcal {O}}}(p, q)\) on p is difficult to establish, which means that this formal definition of the OPE in impractical for computations, but it establishes a property that will be crucial in the next section: by definition, \({\widetilde{C}}_{12{\mathcal {O}}}(p, q)\) is an analytic function in q around \(q = 0\).
Applying this momentumspace OPE to the Wightman 3point function, one gets
where the line above the 3point function indicates that the OPE is taken between \(\phi _0\) and \(\phi _i\). In the limit \(p_f \rightarrow 0\), the series (21) for \({\widetilde{C}}_{f0i}( p_0, p_f)\) is dominated by its first term, and since the integral is a Wightman 2point function for an operator with scaling dimension \((\Delta _i + \Delta _0  \Delta _f)/2\), we can use Eq. (12) to get
Note that \(p_0\) is necessarily timelike in this limit since it approaches \(p_i\), as illustrated in Fig. 2a. We obtain therefore the limit
The limit \(p_i \rightarrow 0\) can established in a similar fashion starting from the OPE
The result corresponds to exchanging the labels f and i in Eq. (24).
These OPE limits are important, but in practice they will not be convenient to determine the Wightman 3point function completely. Instead, there is another case that can be resolved with the help of the OPE: the lightcone limit \(p_f^2 \rightarrow 0_\) with \(p_f \ne 0\). To understand this limit, consider that the coefficient \({\widetilde{C}}_{f0i}(p_0, p_f)\) is invariant under Lorentz transformations. It can therefore be written in terms of the invariant quantities \(p_f^2\), \(p_0^2\) and \(p_i^2\). The analyticity in \(p_f\) implies that \({\widetilde{C}}_{f0i}(p_0, p_f)\) is also analytic in \(p_f^2\). When applied to Eq. (22), this means that the 3point function is equal to an analytic function of \(p_f^2\) multiplying the 2point function, and therefore
where f is an unknown function. The same argument can be applied in the limit \(p_i^2 \rightarrow 0\) to the OPE (25) to establish that
for a different function \(f'\). Taking both limits \(p_f^2 \rightarrow 0_\) and \(p_i^2 \rightarrow 0_\) simultaneously, in a configuration of momenta similar to Fig. 2b in which \(p_0\) is necessarily spacelike, one must have
This form is consistent with the ansatz (15) for the Wightman 3point function, and it establishes that the function F has the asymptotic limit
When combined with the constraints from conformal Ward identities, this will completely fix the scalar Wightman function up to an overall coefficient, which in turn will be determined by the limit (24). This is the topic of the next section.
2.3 Conformal Ward identities
The form (15) of the Wightman 3point function already takes into account all the information from Poincaré and scale symmetry. Only the symmetry under special conformal transformation remains to be imposed. To do so, we follow the approach pioneered in Ref. [25]. It consists in writing down a system of differential equations for the unknown function F.
The infinitesimal transformations of the momentumspace operators under the conformal group are given in Appendix A. In particular, the action (102) of the generator of special conformal transformation is a second order differential operator. When applied to the scalar 3point function, written in this case as a function of the two momenta \(p_f\) and \(p_i\) only, it gives the Ward identity
where
This equation is a Lorentz vector with d components. However, its projection along a direction perpendicular to both \(p_i\) and \(p_f\) is trivial. There are thus only 2 scalar equations that are generated by \(p_f \cdot {\widehat{K}}\) and \(p_i \cdot {\widehat{K}}\). Using the ansatz (15), these two equations become partial differential equations for the function F,
with
This system of equation is of the type satisfied by Appell’s \(F_4\) generalized hypergeometric function of two variables [67, eq. 16.14.4]. The \(F_4\) function is defined around \((z_f, z_i) = (0,0)\) by the double infinite series
The most general solution to the system (32) is a linear combination of the four functions
where we have introduced a shorthand notation for the Appell \(F_4\) function
and denoted \({\widetilde{\Delta }}= d  \Delta \).^{Footnote 4}
Of the four solutions (35), only the first one is consistent with the asymptotic behavior (29) for generic values of the scaling dimensions \(\Delta _f\) and \(\Delta _i\). We conclude that
with an unknown coefficient \({\widetilde{\lambda }}_{f0i}\). This is the equation quoted in the introduction and can be considered the main result of this work. Its simplicity is striking when compared with the integral representation in terms of Bessel functions of Ref. [24]. In the special case where \(\Delta _i, \Delta _f = \frac{d}{2} + n\) with \(n \in {\mathbb {N}}\), the OPE limit (28) does not unambiguously select a unique solution. This situation can however be understood by analytic continuation in \(\Delta _i\) and \(\Delta _f\) of the general case, since a careful analysis of the Fourier transform shows that Wightman functions do not have nonanalyticities when \(\Delta = \frac{d}{2} + n\) [24]. The result (37) is indeed analytic in \(\Delta _f\), \(\Delta _0\), \(\Delta _i\) and d as long as the unitarity bound is satisfied (\(\Delta \ge \frac{d2}{2}\)), and it applies therefore in all generality.
This result is not complete, however, because some kinematicallyallowed range of the arguments \(p_f^2/p_0^2\) and \(p_i^2/p_0^2\) fall outside the radius of convergence of the Appell \(F_4\) series. In particular, there is a singularity as \(p_0^2 \rightarrow 0\) that need to be resolved. This will be done in the next section with the help of the other momentumspace OPE limit.
2.4 Analytic continuation and normalization
At fixed \(z_f\), the radius of convergence of the double hypergeometric series (34) in \(z_i\) is \(( 1  \sqrt{z_f} )^2\). However, the first singularity in \(z_i\) appears on the positive real axis, and it turns out that the Appell \(F_4\) function is analytic over the full negative real axis \(z_i \in (\infty , 0]\). This is made manifest by the existence of a transformation formula stating that [67, eq. 16.16.10]
where
For the Wightman 3point function, this transformation formula and its conjugate taking \((z_f, z_i) \rightarrow (1/z_f, z_i/z_f)\) show that the result (37) applies over the whole kinematic range in which \(p_0\) is spacelike. When \(p_0\) approaches the light cone, there is a branch point singularity: applying the transformation (38) to the 3point function,
The first term on the righthand side is analytic at \(p_0^2 = 0\), and the nonanalyticity only arises from the factor \((p_0^2)^{\Delta _0  d/2}\) in the second term.^{Footnote 5} This representation suggests that the 3point function can be continued past the light cone \(p_0^2 = 0\), but the continuation is ambiguous.
In fact, the structure of Eq. (40) is not surprising: when expressing the conformal Ward identities in terms of the variables \(p_f^2/p_i^2\) and \(p_0^2/p_i^2\), they still admit four solutions, of which only two are consistent with the asymptotic limit (26). The most general form of the 3point function at timelike \(p_0\) is therefore
The two unknown coefficients \({\widetilde{\lambda }}_{f0i}^{(a)}\) and \({\widetilde{\lambda }}_{f0i}^{(b)}\) can be fixed with the help of the OPE limit \(p_f \rightarrow 0\). This limit is subtle, however, since the second argument of the Appell functions in Eq. (41) approaches their radius of convergence, \(p_0^2 / p_i^2 \rightarrow 1\), and it is not possible to evaluate them using the hypergeometric series definition (34). Instead, one can study the OPE limit by taking \(p_f^2 \rightarrow 0_\) first, as shown in Fig. (2)a. In this case, the Appell functions turn into ordinary hypergeometric functions
with a wellknown behavior at argument \(z = 1\), where we have the asymptotic limit
The nonanalytic term in \(1z\) in the second line is in contradiction with the existence of the limit \(p_f \rightarrow 0\). Therefore, the nonanalytic terms coming from the functions \(F_{\Delta _f \Delta _i \Delta _0}\) and \(F_{\Delta _f \Delta _i {\widetilde{\Delta }}_0}\) in Eq. (41) must cancel exactly, and the analytic terms must add up to the limit determined in Eq. (24). This gives a linear system of equations for the coefficients \({\widetilde{\lambda }}_{f0i}^{(a)}\) and \({\widetilde{\lambda }}_{f0i}^{(b)}\), whose unique solution is
where \({\mathcal {N}}\) is related to the OPE coefficient \(\lambda _{f0i}\) by
This result is valid for any configuration of momenta as long as \(p_0\) is spacelike and \(p_f^2 < p_i^2\). There exists a similar expression covering the case \(p_f^2 > p_i^2\), which by the conjugation symmetry (16) can be obtained from the simultaneous exchange \(p_i^2 \leftrightarrow p_f^2\) and \(\Delta _i \leftrightarrow \Delta _f\) in Eq. (41). The special case \(p_i^2 = p_f^2\) is covered by Eq. (37) since it necessarily implies that \(p_0\) is spacelike.
The result (41) also turns out to be a straightforward analytic continuation of Eq. (40), sharing the same structure with a branch point singularity at \(p_0^2 = 0\). The parts that are analytic in \(p_0^2\) match provided that one makes the identification
\({\widetilde{\lambda }}_{f0i}\) is real and analytic in the scaling dimensions \(\Delta _f\), \(\Delta _0\) and \(\Delta _i\) as well as in the spacetime dimension d. It has zeroes when \(\Delta _f = \Delta _i + \Delta _0 + 2n\) and when \(\Delta _i = \Delta _f + \Delta _0 + 2n\) with \(n \in {\mathbb {N}}\). This is the situation of generalized free field theory discussed in more detail in Sect. 2.5. In contrast, the 3point function (41) at timelike \(p_0\) does not vanish in generalized free field theory, but it it still analytic in the scaling dimensions, even though the coefficients \({\widetilde{\lambda }}_{f0i}^{(a)}\) and \({\widetilde{\lambda }}_{f0i}^{(b)}\) are not. This is best seen from the following compact expression for the Wightman 3point function that covers both the spacelike and the timelike regions in \(p_0\):
This representation somehow obscures the facts that the 3point function is real and that it has a branch point at \(p_0^2 = 0\), but it makes the analyticity in \(\Delta _f\), \(\Delta _0\), \(\Delta _i\) and d manifest over its whole region of support.
This is our final result for the Wightman 3point function of scalar operators. It should be noted that this result has been successfully compared with a direct evaluation of the Fourier transform of the positionspace 3point function, analytically in the OPE limits of Sect. 2.2 and numerically for generic kinematics. Some details about the direct computation of the Fourier transform are presented in Appendix B. Before moving on to the study of other correlation functions, we will discuss some interesting features of the Wightman function.
2.5 Generalized free field theory
As already mentioned, something special happens to the 3point function when one scaling dimension equals the sum of the other two. Let us assume that \(\Delta _0 = \Delta _f + \Delta _i\) first. In this case we will interpret \(\phi _0\) as the composite operator \([\phi _f \phi _i]\). Of the three terms in Eq. (47), only the first one remains because \({\widetilde{\lambda }}_{fi0} = {\widetilde{\lambda }}_{0fi} = 0\). Moreover, the Appell function \(F_{\Delta _f\Delta _0\Delta _i}\) is trivially equal to one in this case. We obtain therefore
The dependence on the momenta \(p_i\) and \(p_f\) factorizes, and we see a similar factorization in the OPE coefficient,
This means that we can write
This result is expected from a generalized free field theory, and since the only dynamical data in a conformal 3point function is encoded in the OPE coefficient, the kinematics must match that of the generalized free theory whenever the scaling dimensions obey such relations.
A similar study of the case \(\Delta _i = \Delta _f + \Delta _0\) shows that
If instead we take \(\Delta _0 = \Delta _f + \Delta _i + 2n\) where n is a positive integer, the righthand side of Eq. (50) gets multiplied by a homogeneous polynomial of degree n in \(p_f^2\), \(p_i^2\) and \(p_f \cdot p_i\), because the hypergeometric series that defines \(F_{\Delta _f \Delta _0 \Delta _i}\) terminates at order n. This provides a way of resolving the exact structure of the doubletrace operator \([ \phi _f \square ^n \phi _i]\).
The factorization of 3point functions into 2point functions is actually a trivial statement in the positionspace representation, and it is easy to take their Fourier transform directly and reproduce expressions like (50) and (51). Nevertheless, it is important to see that our general result (47) covers these special cases in a quite nontrivial manner.
2.6 Holomorphic factorization in two dimensions
Another curiosity occurs in two spacetime dimension. Using lightcone coordinates \(p^2 = p^+ p^\) together with a special identity of the Appell \(F_4\) function that only applies when \(d = 2\) [67, eq. 16.16.6], we can write
This allows to write the 3point function in the fully factorized form
where we have defined
\(\Theta \) here is the ordinary Heaviside step function. This is consistent with the fact that the Wightman 3point function in position space can be factorized into holomorphic and antiholomorphic pieces, or equivalently into left and rightmovers. This result is also found to match the direct Fourier transform of the positionspace correlator, which can be easily performed in this case. The interesting way in which this factorization arises from the general expression (47) is another verification of its validity.
3 Adding Spin: Traceless Symmetric Tensor
We will now discuss how to incorporate an operator that is not a scalar in the analysis of the previous section. Our approach is not meant to be systematic, but instead focuses on the simplest case as an example.
3.1 Poincaré and scale symmetry
Starting with the Wightman 3point function (1), we choose to keep the operators \(\phi _0\) and \(\phi _i\) scalar and replace \(\phi _f\) by an operator \({\mathcal {O}}_f\) carrying spin. The only type of spin representations allowed by conformal symmetry are are traceless symmetric tensors. In this case it is convenient to introduce a null polarization vector \(\zeta ^2 = 0\) and define the momentumspace operator with spin \(\ell \) by [7, 69, 70]
Both the symmetry and the tracelessness of the operator are automatically encoded in this definition. As in the scalar case, this operator is in onetoone correspondence with a momentum eigenstate \(\left {\mathcal {O}}^{(\ell )}(p, \zeta ) \right\rangle \equiv {\mathcal {O}}^{(\ell )}(p, \zeta ) \left 0 \right\rangle \). The only novelty is that not all such state are linearly independent, since states related by a little group transformation on \(\zeta \) are equivalent, and some states are even null in the case of a conserved operator \(\partial _\mu {\mathcal {O}}^{\mu \nu \ldots } = 0\). But these considerations do not affect the construction of Sect. 2.1. We can still construct the most general ansatz consistent with Poincaré and scale symmetry, the only new constraint being that it must be a polynomial of degree \(\ell \) in the polarization vector \(\zeta \). Therefore we can write
where the \(F^{(\ell )}_n\) are \(\ell \) distinct functions to be determined. This ansatz is valid as long as \(p_0\) is spacelike as in Fig. 1a, and the general case will again be obtained by analytic continuation.
Note that this treatment of the spin does not apply in \(d = 2\) spacetime dimensions: In that case the ansatz (56) is redundant because the polarization vector \(\zeta \) can be expressed as a linear combination of \(p_f\) and \(p_i\). In \(d = 2\), all operators can be viewed as scalars with different conformal weights for the holomorphic and antiholomorphic pieces, and it is easy to generalize the results of Sect. 2.6 in that case.
3.2 Conformal Ward identities
Further restrictions on the functions \(F^{(\ell )}_n\) in Eq. (56) are provided by the Ward identity for special conformal transformations
where now instead of Eq. (31) the differential operator is
The ansatz (56) contains more freedom than its scalar counterpart Eq. (15) as it is written in terms of \(\ell \) distinct unknown functions, but it should be noted that the Ward identity (57) is also more constraining than Eq. (30): it does not only have components in the plane spanned by \(p_i\) and \(p_f\), but also along the orthogonal direction. If we denote by \(p_\perp \) a vector such that \(p_\perp \cdot p_i = p_\perp \cdot p_f = 0\), then the Ward identity generated by the operator \(p_\perp \cdot {\widehat{K}}\) takes the form
This recursion relation determines all the functions \(F^{(\ell )}_n(z_f, z_i)\) in terms of \(F^{(\ell )}_0(z_f, z_i)\). Moreover the projections of the differential operator (58) along the direction of \(p_f\) and \(p_i\) are such that they never raise the power of \(p_i \cdot \zeta \), which means that we get a closed system of differential equations for \(F^{(\ell )}_0(z_f, z_i)\), which reads
with
This system is identical to that of Eq. (32) but with different parameters \(\alpha \) and \(\beta \). Hence its most general solution is also a linear combinations of four Appell \(F_4\) hypergeometric functions,
where now
Note that \(F^{(\ell )}_{\Delta _f \Delta _0 \Delta _i}(z_f, z_i)\) can be obtained from the scalar function \(F_{\Delta _f \Delta _0 \Delta _i}(z_f, z_i)\) by a shift of all scaling dimensions \(\Delta \rightarrow \Delta + \ell \) accompanied by a shift \(d \rightarrow d + 2 \ell \) in the spacetime dimension, under which the combination \(\Delta  \frac{d}{2}\) is invariant.
Without going into the details of it, a logic similar to that of Sect. 2.2 can be used to argue that among the four solutions (62), only the first one is consistent with the OPE. Thus we arrive at the result
and the other functions \(F^{(\ell )}_n(z_f, z_i)\) are defined recursively by Eq. (59).^{Footnote 6}
3.3 Analytic continuation and normalization
The analytic continuation of this result to the regions of timelike \(p_0\) proceeds as in Sect. 2.4, where we had seen that it is uniquely determined by the existence of the OPE limits \(p_f \rightarrow 0\) and \(p_i \rightarrow 0\). We do not provide the details of all such analytic continuations here as the result is quite complicated, but focus instead on the simplest case that allows to determine the coefficient \({\widetilde{\lambda }}_{f0i}^{(\ell )}\) in Eq. (64).
Let us study the limit \(p_f \rightarrow 0\) of the 3point function. In order to achieve this, we apply the transformation (38) to the function \(F^{(\ell )}_0\) of Eq. (64), and then continue the noninteger power of \(p_0^2\) in such a way that the resulting contribution to the 3point function is analytic around the point \(p_f = 0\). When this procedure is complete, we are left with the asymptotic limit
The ellipsis indicate terms of order \((\zeta \cdot p_f) / p_f\), which arise from the analytic continuation of the functions \(F^{(\ell )}_n\) with \(n > 0\). It is important to realize that the term \(n = 0\) in the ansatz (56) is not the only contributor in the limit \(p_f \rightarrow 0\) since the functions \(F^{(\ell )}_n\) contain increasingly divergent powers of \(p_f\).
This result can be matched with the positionspace OPE
where we ignored the contribution of all other operators besides the traceless symmetric tensor \({\mathcal {O}}^{\mu _1 \ldots \mu _\ell }\). The operator \(C_{{\mathcal {O}}12}^{\mu _1 \ldots \mu _\ell }(x, P)\) admits a series expansion in \(P^\mu \), given at lowest order by
Taking the Fourier transform of this OPE as in Sect. 2.2, we get
where \({\widetilde{C}}^{\mu _1 \ldots \mu _\ell }_{{\mathcal {O}}12}(p, q)\) is the Fourier transform of \(C_{{\mathcal {O}}12}^{\mu _1 \ldots \mu _\ell }(x, q)\), given at lowest order in q by
We did not bother to write down the trace terms involving the metric \(\eta ^{\mu _i \mu _j}\) explicitly as these vanish when contracted with the traceless symmetric tensor \({\mathcal {O}}^{\mu _1 \ldots \mu _\ell }\). To use this OPE in the 3point function, we also need the momentumspace 2point function of traceless symmetric tensor operators. It was computed for instance in Ref. [46], and can be written in a compact form as
The ellipsis indicate that we have omitted terms proportional to \(\zeta \cdot p\). The structure of these terms is quite complicated, but we do not need them to perform the comparison with Eq. (65), where they are neglected as well. We obtain finally
Like the coefficient \({\widetilde{\lambda }}_{f0i}\) of Eq. (46), \({\widetilde{\lambda }}^{(\ell )}_{f0i}\) is analytic in all the scaling dimensions and in d, and it has zeroes at the dimensions of doubletrace operators, in this case when \(\Delta _f = \Delta _i + \Delta _0 + \ell + 2n\) and \(\Delta _i = \Delta _f + \Delta _0 + \ell + 2n\) with \(n \in {\mathbb {N}}\). These zeroes are consistent with the vanishing of the 3point function in generalized free field theory when \(p_0\) is spacelike. The analytic continuation to timelike \(p_0\) comes with a pole that cancels either one of these zeroes, and in that case the Appell \(F_4\) function can be expressed as a finite hypergeometric sum. As in the scalar case this can be used to resolve the exact structure of the doubletrace operators.
This concludes our study of Wightman functions involving a traceless symmetric tensor. The problem of generalizing our findings to arbitrary spin representations for each of the three operators is left for future work.
4 TimeOrdered Products
In this section we consider correlation function involving timeordered products of operators and show to what extent the method of Sect. 2 can be used. The results also illustrate how different timeordered correlation function are from the Wightman function in momentum space.
4.1 Partial timeordering
We consider first the case in which two out of the three operators in the correlation function are timeordered, as in
The timeordering operator is defined in position space by
Our notation for the operators differs from Sect. 2 because of the different physical interpretation of this correlation function. If \(\left\langle \phi _f(p_f) \right \) still defines a final state created by a single operator, there is no notion of an initial state created by a local operator in the correlator (72). As a consequence, the momenta \(p_1\) and \(p_2\) might be timelike as well as spacelike. The only requirement is that they add up to \(p_f\) that is timelike and has positive energy, by the condition (11) on the final state. The correlation function (72) is also obviously symmetric under the exchange of the operator \(\phi _1\) and \(\phi _2\), which means that it must be represented by a function that is symmetric under the simultaneous exchange of the momenta \(p_1 \leftrightarrow p_2\) and of the scaling dimensions \(\Delta _1 \leftrightarrow \Delta _2\). This suggests the ansatz
in which the function \(F_{12}\) enjoys the aforementioned symmetry. Since the partiallytimeordered 3point function obeys the same conformal Ward identities as the Wightman function, it is possible to express the function \(F_{12}\) as a linear combination of Appell \(F_4\) functions of the type of Eq. (35). In other words, the two correlators are different solutions to the same system of partial differential equations, but with a different boundary condition.
As before, this boundary condition is provided by an OPE limit. Since the basis of momentum eigenstates created by single local operator insertion is complete, it must be possible to write
for some function \({\widetilde{C}}_{{\mathcal {O}}12}^{{{\,\mathrm{T}\,}}}(p, q)\) that differs from \({\widetilde{C}}_{{\mathcal {O}}12}(p, q)\) of Eq. (19). This function admits the formal expansion
where the norm \((x^2 + i \epsilon )\) is the timeordered analog of the norm appearing in Eq. (21).^{Footnote 7} It corresponds to the timeordered 2point function of a fictitious operator with scaling dimension \((\Delta _1 + \Delta _2  \Delta _{\mathcal {O}})/2\). Therefore, the integral on the righthandside of Eq. (76) can be written as a momentumspace 2point function,
This gives immediately the asymptotic limit \(p_f \rightarrow 0\) of the correlation function (72):
The symmetry \(\phi _1 \leftrightarrow \phi _2\) is obvious on the righthand side since \(p_1^2 = p_2^2\) in this limit. Unlike the Wightman function, the partially timeordered 3point function do not admit any other Lorentzian OPE limit.
It is not necessarily obvious how to reconcile this OPE limit with the symmetric ansatz (74) for the 3point function. That ansatz gives a good description of the correlation function in a neighborhood of \(p_1^2 = p_2^2 = 0\), which corresponds to a configuration of momenta as in Fig. 3a. Both \(p_1\) and \(p_2\) lie close to the lightcone and have positive energy in that case, while the limit \(p_f \rightarrow 0\) requires \(p_2 \rightarrow p_1\). It is therefore useful to introduce the different ansatz
covering in particular configurations like Fig. 3b in which both \(p_f^2\) and \(p_2^2\) are small compared to \(p_1^2\). We will assume for now that both \(p_1\) and \(p_2\) spacelike as in the figure and discuss later what happens when either one of them crosses a lightcone. With this ansatz, the Ward identity (30) for special conformal transformations implies that \(F_{f2}(z_f, z_2)\) is a linear combination of the four functions
where \(F_{\Delta _f \Delta _1 \Delta _2}\) is the Appell \(F_4\) function given in Eq. (36). Besides providing an explicit value for the limit \(p_f \rightarrow 0\), the OPE (76) also implies that the 3point functions must scale like \((p_f^2)^{\Delta _f  d/2}\) in the limit \(p_f^2 \rightarrow 0_\). Among the 4 functions in Eq. (80), only the first two follow this asymptotic behavior. We must therefore have
for some coefficients A and B. This form is readily compatible with the limit \(p_f \rightarrow 0\), which corresponds to \(z_f \rightarrow 0_\) and \(z_2 \rightarrow 1_\). The limit \(z_f \rightarrow 0\) should be taken first, and one can then use Eq. (43) to obtain the limit \(z_2 \rightarrow 1_\). For generic A and B there is a nonanalytic piece proportional to \((1  z_2)^{1  \Delta _f}\) in this limit. Requiring that this term vanishes and that the limit reproduces Eq. (78), one obtains
The symmetry \(\phi _1 \leftrightarrow \phi _2\) is not at all obvious in this result. But one can now make use of the transformation property (38) of the Appell \(F_4\) function to bring the 3point function in the form of the ansatz (74) where the symmetry becomes evident. We find
where we have denoted
This is our result for the partiallytimeordered 3point function. The symmetry \(\phi _1 \leftrightarrow \phi _2\) follows from the property \(F_{\Delta _1 \Delta _f \Delta _2}(z_1, z_2) = F_{\Delta _2 \Delta _f \Delta _1}(z_2, z_1)\) of the Appell \(F_4\) function. The results of Refs. [44, 46] are special cases of this expression. They correspond to the limit \(p_1^2, p_2^2 \rightarrow 0_+\) which is finite under the assumption that \(\Delta _1, \Delta _2 > \frac{d}{2}\).
Unlike the Wightman function, this correlation function is not analytic in the scaling dimensions. It has poles when \(\Delta _1, \Delta _2 = \frac{d}{2} + n\) with \(n \in {\mathbb {N}}\). This is a wellknown feature of the timeordered correlation function, which is also present in the twopoint function (77): correlation functions involving operators with these special dimensions have anomalies and must be renormalized. In momentum space, this renormalization leads to the appearance of logarithms [44]. The result (83) also shows that the 3point function diverges when \(\Delta _f = \Delta _1 + \Delta _2  d  2n\). We do not have an explanation for the presence of these poles.
4.2 Relationship with the Wightman function
The relationship between timeordered and Wightman functions in momentum space is complicated. There is no simple way to go from one to the other without invoking the positionspace representation. Nevertheless, there exists a link between the two given by the operator identity
where \({{\,\mathrm{{\overline{T}}}\,}}\) indicates the reverse timeordering operation. When applied to the 2point function, it implies that
where have used the fact that the antitimeordered 2point function is the complex conjugate of the timeordered one. This equality is satisfied by the 2point functions given in Eqs. (12) and (77) for all values of the momentum p: when p is spacelike both Wightman functions vanish and the timeordered function is purely imaginary; when p is timelike, the nontrivial phase in the timeordered function is precisely matched by the Wightman function.
The identity (85) become more interesting when we apply it to the 3point function, as it provides an interesting verification of our result. We have
In Sect. 4.1 we focused on the regime where \(p_1\) and \(p_2\) are both spacelike. In this case the Wightman functions on the righthand side vanish, and we found indeed that the partiallytimeordered 3point function (83) is purely imaginary. When either \(p_1\) or \(p_2\) crosses the lightcone, necessarily with positive energy, then the corresponding Wightman function on the righthand side becomes nonzero and equal to Eq. (37). Thus the partiallytimeordered function must acquire a nontrivial phase. The ansatz (79) is wellsuited to study the case where \(p_2\) crosses the lightcone. Consistency with the lightcone limit (78) actually indicates that one should replace \(p_2^2 \rightarrow p_2^2  i \epsilon \) in that case, and one obtains therefore the relation
It can be verified from the definitions (82) for A and (46) for \({\widetilde{\lambda }}_{f12}\) that this is indeed satisfied.
Finally, it would be interesting to check the identity (87) in the case where both momenta \(p_1\) and \(p_2\) are timelike. However, the analytic continuation of the partiallytimeordered function is ambiguous, and the information provided by the OPE is not sufficient to resolve it. A naive guess would be to replace \(z_1 \rightarrow z_1 + i \epsilon \) and \(z_2 \rightarrow z_2 + i \epsilon \) in the expression (83). However, it can be verified that this guess does not satisfy (87): at least one additional term proportional to \(z_1^{\Delta _1  d/2} z_2^{\Delta _2  d/2} F_{\Delta _1 \Delta _f \Delta _2}(z_1, z_2)\) must be present when both \(p_1\) and \(p_2\) are timelike.^{Footnote 8} Note that the presence of this additional term is consistent with an observation that can be made using generalized free field theory: when the scaling dimensions satisfy \(\Delta _f = \Delta _1 + \Delta _2\), the 3point function is expected to factorize as
The righthand side is nonzero when \(p_1\) and \(p_2\) are both timelike with positive energies. However, the function \(F_{12}\) in Eq. (83) vanishes identically when \(\Delta _f = \Delta _1 + \Delta _2\). Therefore it cannot be the complete answer when both \(p_1\) and \(p_2\) are timelike. On the other hand, the other relation obtained from a generalized free field theory correlation function,
is satisfied by Eq. (81) even when \(p_2\) is timelike, provided that one makes the substitution \(p_2^2 \rightarrow p_2^2  i \epsilon \). In summary, the computation of the partiallytimeordered 3point function in the regime of timelike \(p_1\) and \(p_2\) remains an interesting open problem that cannot be directly addressed with our method.
4.3 The fully timeordered 3point function
The last momentumspace 3point function that one can consider is the fully timeordered product
It cannot be understood as the overlap of momentum eigenstates, and so the OPE analysis of the previous sections do not apply. But the permutation symmetry \(\phi _1 \leftrightarrow \phi _2 \leftrightarrow \phi _3\) between the 3 operators is actually sufficient to determine the 3point function uniquely up to an overall coefficient: when all three momenta are spacelike, the Euclidean result of Ref. [25] applies readily,^{Footnote 9}
where
and the coefficients \(g_{\Delta _1 \Delta _3 \Delta _2}\) are defined by
Once again the 3point function is expressed in terms of the solutions to the conformal Ward identities (30). This time all four solutions appear in the 3point function, no matter which of the momenta is taken as the reference momentum.
This fullytimeordered 3point function can be related to the partiallytimeordered correlator using an identity similar to Eq. (85):
We have omitted to write the argument of the operators because this identity is purely combinatoric and applies in position space as well as in momentum space. In position space it can be verified by expanding the timeordered product according to its definition. In momentum space, the content of the identity is more interesting as it provides an interesting relation between Eq. (92) and the results of the previous sections. When all three momenta are spacelike, each of the 3point functions on the righthand side vanishes individually as it involves a null state. To see that the lefthand side vanishes as well, it is sufficient to note that the timeordered and antitimeordered products are related by complex conjugation,^{Footnote 10} so that the identity becomes
The function \(F_T\) in Eq. (93) is indeed real. This means that there is no distinction between the timeordered and antitimeordered products when all momenta are spacelike.
If one of the momenta is instead timelike (say \(p_1\)), one term on the righthand side of the identity (95) is nonzero. In the timeordered 3point function, this configuration can be reached by analytic continuation in \(p_1^2\). There are only two possible analytic continuations of the function \(F_T\) that preserve the permutation symmetry of the operators: one of them consist in taking \(p_1^2 \rightarrow p_1^2  i \epsilon \) and the other \(p_1^2 \rightarrow p_1^2 + i \epsilon \). From the representation of the 3point function as the Fourier transform of the positionspace correlator, it is easy to see that the first of these analytic continuations corresponds to the timeordered product and the second to the antitimeordered product. They are the complex conjugate of each other, and thus one can write
When \(p_1\) has positive energy, the righthand side is given by Eq. (74), or equivalently Eq. (79). It can be verified that these expressions precisely match the imaginary part of Eq. (92) with \(p_1^2 \rightarrow p_1^2  i \epsilon \). This provides a simple and yet nontrivial verification of the results of Sect. 4.1.
5 Discussion
In this paper, we have provided a simple closedform^{Footnote 11} expression for the momentumspace Wightman function of 3 scalar operators, as well as two scalars and one traceless symmetric tensor with arbitrary spin. Besides the explicit results, we have given a detailed explanation of the logic underlying this approach so that the interested readers might themselves proceed to the derivation of correlation functions not given here. This method should be particularly suited to study the Wightman correlation functions of operators such as conserved currents or the energymomentum tensor. For other spin representations, one might want to develop the theory of weightshifting operators in momentum space.
In addition, studying the (partially)timeordered correlation function in the case of scalar operators, we have found that they take generically a more complicated form, which we were not able to fix completely by analytic continuation away from the OPE limits. One observes also that the timeordered correlation functions can be expressed as a sum of the Wightman function and of some its shadow transforms. This supports the idea that the Wightman function are really the building blocks in a Lorentzian conformal field theory.
Finally, we would like to emphasize that the computation of the Wightman 3point function opens the door to the computation of higherpoint functions through the momentumspace OPE: one way of interpreting our results is in terms of the relation
which we have formally established for any operator \({\mathcal {O}}\) and in every kinematic configuration of the momenta \(p_1\) and \(p_2\).^{Footnote 12} Many questions about the convergence of this OPE in a 4point function remain to be answered: it is guaranteed to converge in a distributional sense only [65], but not necessarily at every single point in momentum space. The problem has been addressed in \(d = 2\) [66], but it remains open in higher dimensions.
Notes
Note that this definition is equivalent to \(\Theta (p) = \Theta (p^0  \mathbf {p})\) used in Ref. [24], but we write it as a function of \(p^2\) and \(p^0\) to emphasize the fact that it is a Lorentzinvariant object.
The Minkowski metric is taken in the “mostly \(+\)” convention, i.e. \(\eta = \text {diag}(1, 1, \ldots , 1)\) and Lorentz indices run from 0 to \(d  1\).
Our definition of the conjugate state is \(\left\langle {\mathcal {O}}(p) \right = \left\langle 0 \right {\mathcal {O}}(p)\) and therefore \(\left {\mathcal {O}}(p) \right\rangle ^\dag = \left\langle {\mathcal {O}}(p) \right \).
This notation is of course reminiscent of the scaling dimension of “shadow” operators (see e.g. Ref. [68] for a modern discussion). The fact that the Ward identity has four solutions is actually related to the existence of a shadow transform that can be applied either to the initial or to the final state, or to both. The 3 discarded solutions correspond then to the correlation function \(\langle \!\langle {\widetilde{\phi }}_f \phi _0 \phi _i \rangle \!\rangle \), \(\langle \!\langle \phi _f \phi _0 {\widetilde{\phi }}_i \rangle \!\rangle \) and \(\langle \!\langle {\widetilde{\phi }}_f \phi _0 {\widetilde{\phi }}_i \rangle \!\rangle \). Note that in a Wightman function it is not possible to define the shadow transform of the middle operator \(\phi _0\). This is reflected in the fact that the solutions are invariant under \(\Delta _0 \leftrightarrow d  \Delta _0\), i.e. \(F_{\Delta _f \Delta _0 \Delta _i}(z_f, z_i) = F_{\Delta _f {\widetilde{\Delta }}_0 \Delta _i}(z_f, z_i)\).
The divergence present when \(\Delta _0 < \frac{d}{2}\) is actually integrable by the unitarity bound \(\Delta _0 > \frac{d2}{2}\). This suggests that it should be possible to take the inverse Fourier transform of this expression and recover the positionspace 3point function.
Derivatives of Appell \(F_4\) functions can be again expressed in terms of Appell \(F_4\) functions with parameters shifted by integers, but we did not find a form simple enough for the generic function \(F^{(\ell )}_n(z_f, z_i)\) to be reproduced here.
One can write a similar OPE for the antitimeordered product of operators in which the sign of the \(i \epsilon \) prescription is opposite.
Consider for instance that one can add to the function \(F_{12}\) in Eq. (81) a term proportional to
$$\begin{aligned} \left[ (z_1  i \epsilon )^{\Delta _1  d/2}  (z_1 + i \epsilon )^{\Delta _1  d/2} \right] \left[ (z_2  i \epsilon )^{\Delta _2  d/2}  (z_2 + i \epsilon )^{\Delta _2  d/2} \right] F_{\Delta _1 \Delta _f \Delta _2}(z_1, z_2). \end{aligned}$$This term satisfies the conformal Ward identity, preserves the symmetry \(\phi _1 \leftrightarrow \phi _2\) and vanishes whenever \(p_1\) or \(p_2\) is spacelike. It might actually be the unique term with these properties.
This can be seen in position space, where the timeordered and antitimeordered correlation functions only differ by the sign of the Feynman \(i \epsilon \) prescription. Since both functions are symmetric under \(x \rightarrow x\), the Fourier transform preserves this property.
The terminology of “closedform” is used because the Appell \(F_4\) function is a recognized function whose properties are known and welldocumented, but really it does not carry much meaning: to evaluate it, one must either use one of its integral representations or the doubleinfinite hypergeometric series (34). Nevertheless, we have provided for every possible momentum configurations at least one representation in terms of which the double hypergeometric series converges, and thus the function can be approximated to arbitrary precision with a truncated series.
In practice determining \({\widetilde{C}}_{{\mathcal {O}}12}(p_1, p_1 + p_2)\) from the 3point function still requires multiplication with the inverse of the 2point function of the operator \({\mathcal {O}}\), which for operators of large spin can be a bit tedious.
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Acknowledgements
Open access funding provided by Scuola Internazionale Superiore di Studi Avanzati  SISSA within the CRUICARE Agreement. The author would like to thank Brian Henning, Xiaochuan Lu, Markus Luty, Guram Mikaberidze, Marco Meineri, and João Penedones for discussion. The work of the author at EPFL was supported by the Swiss National Science Foundation through the NCCR SwissMAP.
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Appendices
Conformal Algebra
In this appendix we describe our conventions for the generators of the Lorentzian conformal group \(\text {SO}(d,2)\) and the infinitesimal transformation of the operators in the momentumspace representation. There are \(d (d+1) / 2\) generators of \(\text {SO}(d,2)\), denoted by the antisymmetric tensors \(J^{AB}\) with indices \(A,B = 0, \ldots , d+1\). They are hermitian, \((J^{AB})^\dag = J^{AB}\), and obey the algebra
with metric \(\eta ^{AB} = \text {diag}( 1, +1, \ldots , +1, 1)\). We take the Lorentz indices in the range \(0, \ldots , d1\) and identify
This gives the commutation relations^{Footnote 13}
and all other commutators vanish. The transformation rules for a primary operator \({\mathcal {O}}(p)\) with scaling dimension \(\Delta \) are
where \(\Sigma ^{\mu \nu }\) is the spin matrix acting on the indices of the operators \({\mathcal {O}}\) which are implicit here. These transformations follow from the definition (9) of operators in momentum space, together with the decomposition of the Hilbert space into irreducible representations of the Lorentz group and dilatations,
Direct Fourier Transform
Consider the positionspace Wightman 3point function of scalar operators
where we have denoted \(x_{ab}^2 = (x_a^0  x_b^0  i \epsilon )^2 + (\mathbf {x}_a  \mathbf {x}_b)^2\). The goal of this appendix is to bring the Fourier transform of this expression in a form where it can easily be evaluated numerically, and to derive results analytically in limits where the integrals are tractable.
Since Eq. (104) is the product of three Wightman function, its Fourier transform can be written
where
and we have defined
We will denote the integral in (105) with \(W_{\alpha \beta \gamma }(p_f, p_i)\). Using the formula (12) for \(W_\alpha (k)\),
This integral is free of ultraviolet divergences since the region of integration in k is bounded by the presence of the \(\Theta \) functions defined in Eq. (3). It can however have infrared divergences depending on the value of the parameters \(\alpha \), \(\beta \) and \(\gamma \) and on the kinematics. To avoid this situation we will assume that the momenta \(p_i\) and \(p_f\) are noncolinear and that
Working in \(d > 2\) spacetime dimensions, it is convenient to introduce lightcone coordinates \(k = (k^+, k^, \mathbf {k}^\perp )\), such that the scalar product of two vectors is \(k_1 \cdot k_2 =  \frac{1}{2} \left( k_1^+ k_2^ + k_1^ k_2^+ \right) + \mathbf {k}_1^\perp \cdot \mathbf {k}_2^\perp \) and the integration measure \(d^dk = \frac{1}{2} dk^+ dk^ d^{d2}\mathbf {k}^\perp \). This corresponds to choosing \(k^0 = \frac{1}{2} \left( k^+ + k^ \right) \) and \(k^{\Vert } = \frac{1}{2} \left( k^+  k^ \right) \), where \(k^{\Vert }\) indicates some preferred space direction. We can take in particular this direction in the plane of \(p_f\) and \(p_i\), so that \(p_f = (p_f^+, p_f^, 0)\) and \(p_i = (p_i^+, p_i^, 0)\). Now the \(\Theta \) functions give the conditions
and
It is immediately obvious that the integral is nonzero if \(p_f^\pm < 0\) and \(p_i^\pm > 0\) only, i.e. the result will be proportional to \(\Theta (p_f) \Theta (p_i)\) as expected. Using spherical coordinates for \(k^\perp \), we have
where we have now defined
In this form, the integral is easy to evaluate numerically, but still hard to handle analytically. Besides numerical checks of the results of Sect. 2 that have been performed, we consider two kinematic limits in which it can be evaluated explicitly.
The first limit is
In this case \(p_0 = p_f  p_i\) is spacelike since \(p_0^2 \approx  p_f^+ p_i^ > 0\). We are therefore in the situation of Eq. (28),
The integral (113) simplifies to
With the change of variable \(k^+ = p_i^+ u\), \(k^ = p_f^ v\), \(k^\perp = (p_f^ p_i^+ w)^{1/2}\), the dependence on the momenta can be factored out,
After rescaling \(w \rightarrow u v w\), the three integrals factorize and one arrives at
or for the complete integral (108),
This is in agreement with Eq. (37) and the definition (46) of \({\widetilde{\lambda }}_{f0i}\). Note that the singularity in \(d = 2\) that appears at intermediate steps of the computation is absent in the final limit; one can verify that a derivation using lightcone coordinates in \(d = 2\) (i.e. without the orthogonal component \(k^\perp \)) gives an identical result. Similarly, this limit is completely analytic in \(\alpha \), \(\beta \) and \(\gamma \) so that the assumption (109) can be relaxed.
The second limit that can be taken analytically is
It corresponds to
In this case the integral (113) can be approximated with
which after the change of variables \(k^+ = p_f^+ u\), \(k^ = p_f^ v\), \(k^\perp = (p_f^+ p_f^ w)^{1/2}\) becomes
To evaluate the remaining integral, one performs the change of variable
in terms of which
This gives again three independent integrals that can be expressed as ratios of \(\Gamma \)functions, and we find
This gives finally for the integral (108)
Again, we find perfect agreement with the OPE limit (24).
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Gillioz, M. Conformal 3Point Functions and the Lorentzian OPE in Momentum Space. Commun. Math. Phys. 379, 227–259 (2020). https://doi.org/10.1007/s00220020038368
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DOI: https://doi.org/10.1007/s00220020038368