The Generalized Hypergeometric Structure of the Ward Identities of CFT’s in Momentum Space in d > 2

: We review the emergence of hypergeometric structures (of F 4 Appell functions) from the conformal Ward identities (CWIs) in conformal ﬁeld theories (CFTs) in dimensions d > 2. We illustrate the case of scalar 3- and 4-point functions. 3-point functions are associated to hypergeometric systems with four independent solutions. For symmetric correlators, they can be expressed in terms of a single 3K integral—functions of quadratic ratios of momenta—which is a parametric integral of three modiﬁed Bessel K functions. In the case of scalar 4-point functions, by requiring the correlator to be conformal invariant in coordinate space as well as in some dual variables (i.e., dual conformal invariant), its explicit expression is also given by a 3K integral, or as a linear combination of Appell functions which are now quartic ratios of momenta. Similar expressions have been obtained in the past in the computation of an inﬁnite class of planar ladder (Feynman) diagrams in perturbation theory, which, however, do not share the same (dual conformal/conformal) symmetry of our solutions. We then discuss some hypergeometric functions of 3 variables, which deﬁne 8 particular solutions of the CWIs and correspond to Lauricella functions. They can also be combined in terms of 4K integral and appear in an asymptotic description of the scalar 4-point function, in special kinematical limits.


Introduction
Conformal symmetry has been important in the study of critical phenomena as well as in string theory, over about fifty years [1,2]. In ordinary string theory it has played a key role in its two-dimensional version (d = 2), with the identification of an infinite-dimensional Virasoro algebra which has been crucial for the characterization of the dynamics of the theory [3].
The interest in conformal field theories (CFTs), however, has developed in parallel but also independently of string theory, and in d = 2 the presence of such enhanced symmetry has allowed, on the other hand, to come up with important predictions about the behavior of several statistical models at their critical points, accounting for many of their universality properties. This, in general, allows the computations of their critical exponents and the characterization of their correlation functions [4,5].
CFT descriptions, as just mentioned, describe systems which develop, at a critical point, long-range quantum correlations with power-like decay laws as functions of their separation (coordinate) points. The absence of any dimensionful parameter in the theory only allows correlators characterized by an algebraic-rather than exponential-decay as a function of distance. These are controlled by some exponents related to the scaling dimensions of the field operators appearing in the quantum average. A set of primary fields-together with their descendents-and their correlation functions, provide information about the quantum fluctuations of the theory. The products of two primary operators describing such fluctuations obey some operatorial relations defined by an operator product expansion (OPE) which is entirely dependent on the parameter of a specific CFT (couplings, scaling dimensions), also called "the conformal data" [6].
The OPE is at the basis of the so-called "conformal bootstrap" program [6][7][8][9][10], whose goal is to generate correlators of higher point starting from the lower point ones (2 and 3-point functions). 2-and 3-point functions are essentially fixed by the symmetry, which acts on their explicit expressions via a set of Conformal Ward identities (CWIs) [11,12]. These can be generalized to n-point functions [13]. In coordinate space-we assume them to be defined in R d -such equations are of first-order and become of second-order in momentum space.
The goal of this review is to describe the CWIs of three-point functions in momentum space, illustrating the hypergeometric character of such equations, which could be of interest from the mathematical side.
Traditionally, the critical behavior of a certain theory has been investigated using the renormalization group approach [14,15]. In this approach, starting from a certain Hamiltonian of a given system, one builds a sequence of Hamiltonians, each defined at a certain distance scale (λ n ), H n ≡ H λ n , through the process of rescaling and decimation of its degrees of freedom, in order to describe the flow of the theory as we vary the fundamental scale. One looks for the fixed points of the sequence Hn +1 = Hn (for all n >n), with Hn the fixed-point Hamiltonian. The scaling dimensions of the theory are identified by an analysis of its quantum fluctuations using the fixed-point Hamiltonian of the model.
Conformal symmetry defines an independent path compared to the previous one. Exploiting the fact that at certain critical point a given system is characterized by a dynamics which is controlled by such symmetry, one is able to describe the behavior of its correlation functions without any additional input. By the use of the OPE in a CFT and its constraints in various channels-there are three channels for a four-point function, for instance-it is possible to derive-independently of any renormalization group analysis-the critical exponents of the theory. This approach requires full knowledge of the conformal blocks (or conformal partial waves) of a certain theory, which is a topic of central relevance in the study of any CFT [8].

The Momentum Space Analysis
Most of this analysis, so far, has been developed in coordinate space.
The reason is twofold. First, CFT correlator in momentum space offers a description of a correlation function which is quite close to that provided by ordinary quantum field theories, in terms of scattering amplitudes and of S-matrix elements, in which conformal symmetry plays a significant guiding role [33][34][35]. The second is related to issues concerning the UV behavior of such theories, described when all the points of a given correlator coalesce. This induces a breaking of the classical conformal symmetry at the quantum level, with the appearance of a conformal anomaly [36][37][38].
The interest in CFT in higher dimensions has grown significantly after the formulation of the Anti-De Sitter (AdS) CFT duality [23,39,40]. For d > 2, conformal symmetry is finite-dimensional, and the dynamics of such CFTs are far less constrained. Nevertheless, the correlation functions of CFTs are constrained by a finite set of conformal Ward identities (CWIs) that we are going to discuss in the next sections.
In the case of tensor correlators, additional symmetries induce additional WI's, the canonical WIs, due to Noether symmetries which must also be respected. They are related to the Poincaré symmetry.
These are hierarchical, and connect n-point functions to n − 1-point functions, and so on. In this brief review, we are going to illustrate the key steps that take to the identification of a generalized hypergeometric structure that emerges from the equations associated with such CWIs once we turn to momentum space.
We will be focusing our attention and summarize the content of some original work on the subject for scalar [41] and tensor 3-point functions in d = 4 [42][43][44], which is relevant for the analysis of the implications of conformal symmetry in several field theory contexts. Details and derivations can be found in those works. Our goal will simply be to outline some of the main results of these analyses which may raise the interest of mathematicians.
Discussions of hypergeometric systems of two variables can be found, for instance, in [45]. A more extensive review of these developments, with a detailed description of the results that we are going to summarize here, will be presented by us elsewhere.

Conformal Ward Identities (CWIs)
For a discussion of the general features of CFTs in d ≥ 2, we refer to the several reviews which have been published in the last few years [10,46,47]. Most of them deal with the analysis of such theories in coordinate space. The momentum space approach to CFT is a more recent area of research. In d = 4 it has been investigated in [41,42] and [43,44,48,49] and in more recent work in [50,51]. The hypergeometric structure of the CWIs has been identified independently in [41] and [42], as already mentioned, in the case of 3-point functions. The identification of generalized hypergeometric solutions of the CWIs for 4-point functions, which share a structure typical of 3-point functions, and of the homogenous solutions of Lauricella type, have been discussed in [51].
The CWIs are composed of special conformal and dilatation WIs, beside the ordinary (canonical) WI's corresponding to Lorentz and translational symmetries, that we are going to specify below. We recall, for instance, that in d = 4 conformal symmetry is realized by the action of 15 generators, 10 of them corresponding to the Poincaré subgroup, 4 to the special conformal transformations and 1 to the dilatation operator. In the infinitesimal form, they are given by and they can be expressed as a local rotation where µ = 1, 2, . . . , d, and Ω(x) and R µ ν (x) are, respectively, finite position-dependent rescalings and rotations with and b µ is a constant d-vector. The transformation in (1) is composed of the parameters a µ for the translations, ω µν = −ω νµ for boosts and rotations, σ for the dilatations and b µ for the special conformal transformations. The first three terms in (1) define the Poincaré subgroup, obtained for Ω(x) = 1, which leaves invariant the infinitesimal length. For a general d, the counting of the parameters of the transformation is straightforward. We have d(d − 1)/2 ordinary rotations associated to a SO(d) symmetry in R d -with parameters ω µν -d translations (P µ ) with parameters a µ , d special conformal transformations K µ (with parameters b µ ), and one dilatation D whose corresponding parameter is σ, for a total of (d + 1)(d + 2)/2 parameters. This is exactly the number of parameters appearing in general of SO(2, d) transformation. Indeed one can embed the actions of the conformal group of d dimensions into a larger R d+2 space, where the action of the generators is linear on the coordinates x M (M = 1, 2, . . . , d + 2) of such space, using a projective representation. This is at the basis of the so-called embedding formalism. We refer to [10] for more details. By including the inversion (I) we can enlarge the conformal group to O(2, d). Special conformal transformations can be realized by considering a translation preceded and followed by an inversion. We will focus our discussion mostly on scalar primary operators of a quantum CFT, acting on an certain Hilbert space, which, under a conformal transformation will transform as with specific scaling dimensions ∆ i . We start this excursus on the implication of such symmetry on the quantum correlation functions of a CFT, by considering the simple case of a correlator of n primary In all the equations, covariant variables will be shown in boldface. 3-and 4-point functions (beside 2-point functions) in any CFT are significantly constrained in their general structures due to such CWI's. For scalar correlators, the special CWI's are given by first order differential equations being the expression of the special conformal generator in coordinate space. The corresponding dilatation WI on the same n-point function Φ is given by for scale covariant correlators. In the case of scale invariance the dilatation WI takes the form with D 0 (x i ) given by Such CWIs are sufficient to completely determine the expression of a scalar three-point function of primary operators O i of scaling dimensions ∆ i (i = 1, 2, 3) in the form where x ij = |x i − x j | and C 123 is a constant which specifies the CFT. For 4-point functions the same constraints are weaker, and the structure of a scalar correlator is identified modulo an arbitrary function of the two cross ratios The general solution, allowed by the symmetry, can be written in the form where h(u(x i ), v(x i )) remains unspecified. For the analysis of n-point function it is sometimes convenient to introduce more general notations. For instance, one may define dp ≡ dp 1 dp 2 . . . dp n , where each of the integrations dp i ≡ d d p i are performed on the d-dimensional components of the momenta . It will also be useful to introduce the total momentum P = ∑ n j=1 p j characterising a given correlator, which vanishes because of the translational symmetry of the correlator in R d .
The momentum constraint in momentum space is enforced via a delta function δ(P) in the integrand. For instance, translational invariance of Φ(x) gives In general, we recall that for an n-point function Φ(x), the condition of translational invariance generates an expression of the same correlator in momentum space which of the form (17). We can remove one of the momenta, and conventionally we do it by selecting the last one, p n , which is replaced by its "on shell" versionp n = −(p 1 + p 2 + . . . + p n−1 ) Φ(x) = dp 1 dp 2 ...dp n−1 e i(p 1 x 1 +p 2 x 2 +...p n−1 x n−1 +p n x n ) Φ(p 1 , . . . p n−1 ,p n ), the Fourier transform of the original correlator (6). A discussion of the derivations of the expressions in momentum space of the dilatation and special conformal transformations can be found in [44]. The special conformal generator in momentum space takes the form The latter corresponds to (8), and then the special CWIs are given by the equation If the primary operator O i transforms under a scaling in the form in momentum space the same scaling takes the form with In momentum space, the condition of scale covariance and invariance are respectively given by and In the case of tensor correlators the structure of the special CWI's involve also the Lorentz generators Σ µν and take the form where the indices i 1 , . . . , i n and j 1 , . . . , j n run on the representation of the Lorentz group to which the operators belong. Notice that the sum over the index r selects in each term a specific momentum p r , but the last momentum p n is not included, since the summation runs from 1 to n − 1. Therefore the differentiation respect to the last momentum p n , which has been chosen as the dependent one, is performed implicitly. At the same time, the action of the rotation (Lorentz) generators Σ

2-Point Functions
The simplest application of such equations are for two-point functions [41] of two primary fields, each of spin-1, here defined as i and j. In this case, if we consider the correlator of two primary fields each of spin-1, the equations take the form For the 2-point function G S (p) of two scalar quasi primary fields, the invariance under the Poincaré group implies that the function G S depends on the scalar invariant p 2 and then G S (p) = G S (p 2 ). Furthermore, the invariance under scale transformations implies that G S (p 2 ) is a homogeneous function of degree α = 1 2 (∆ 1 + ∆ 2 − d). It is easy to show that one of the two equations in (31) can be satisfied only if ∆ 1 = ∆ 2 . Therefore conformal symmetry fixes the structure of the scalar two-point function up to an arbitrary overall constant C as If we redefine in terms of the new integration constant c S12 , the two-point function reads as and after a Fourier transform in coordinate space takes the familiar form where

The Hypergeometric Structure from 3-Point Functions F 4
In the case of a scalar correlator of three-point functions, all the conformal WI's can be re-expressed in scalar form by taking as independent momenta the magnitude p i = |p i | = p 2 i as the three independent variables. In fact, Lorentz invariance on the correlation function implies that i.e., it is a function which depends on the magnitude of the momenta p i , i = 1, 2, 3. In this case, p 3 is taken as the dependent momentum (p 3 = −p 1 − p 2 ) by momentum conservation and defining p 3 = |p 1 + p 2 |. The original equations, in the covariant version, take the form for the special conformal and dilatation WIs. In this case, K κ (p i ) doesn't involve the spin part Σ, as illustrated in the general case in Equation (30), because of the scalar nature of this particular correlation function. For this reason, the action of K κ is purely scalar and the properties of the scalar products one can re-express the differential operator for the dilatation WI as giving the equation One can show that the special conformal transformations, summarized in (36) take the form having introduced the operators It is easy to show that Equation (41) can be split into the two independent equations having used the momentum conservation equation Equation (43) takes the form which are equivalent to a hypergeometric system of equations, with solutions given by linear combinations of Appell's functions F 4 .

Hypergeometric Systems
Appell's hypergeometric functions F 1 (x, y), F 2 (x, y), F 3 (x, y), F 4 (x, y) are defined by the hypergeometric series: and are bivariate generalizations of the Gauss hypergeometric series with the (Pochhammer) symbol (α) k given by An account of many of the properties of such functions and a discussion of the univariate cases, when the two variables coalesce, can be found in [45] and related works. They are solutions of equations generalizing Euler's hypergeometric equation whose soultion is denoted as 2 F 1 written in (50). This is classified as a Fuchsian equation with singularities at z = 0, z = 1 and z = ∞. When the two arguments x, y of Appell's functions are algebraically related, the univariate specializations satisfy Fuchsian ordinary differential equations. The proof that the CWIs of the three-point functions are hypergeometric systems of equations has been shown independently in [41] and [42]. We recall that, in the case of Appell functions of type F 4 given in (49), which are the relevant ones in all our discussion, such functions are solutions of the system of differential equations as illustrated in [52], where F(x, y) can be in the most general case a linear combinations of four independent functions F 4 , hypergeometric of two variables x and y. The univariate limits of the solutions are important, from the physical point of view, for the study of the behaviour of the corresponding correlation functions in special kinematics. An example has been discussed in [53] in the case of 4-point functions.

Scalar 3-Point Functions
To show the emergence of such a system of equations, let us focus on a 3-point function of three primary scalar fields of a generic CFT defined by the two homogeneous conformal equations combined with the scaling equation where We follow the analysis of in [41], and introduce the anspoin . One of the three momenta is treated asymmetrically and takes the role of "the pivot" in the representation of the function as a series. In this case we have chosen p 1 as a pivot, but we could have equivalently chosen any of the 3 momenta. Φ is required to be homogenous of degree ∆ − 2d under a scale transformation, according to (55), and in (56) this is taken into account by the factor p ∆−2d

1
. The use of the scale invariant variables x and y takes to the hypergeometric form of the solution. One obtains by an implicit differentiation with The treatment of equation K 31 Φ = 0 proceeds in a similar way, with the obvious exchanges (a, b, x, y) → (b, a, y, x) Equation (57) acquires a hypergeometric form if we set G/x = 0, which implies that The equation K 31 Φ = 0 generates a similar condition for b by setting G /y = 0, fixing the two remaining indices Notice that the elimination of such a singularity in the equations guarantees, from the physical perspective, the analyticity of the solutions and the absence of unphysical thresholds which are not expected in a massless theory. The four independent solutions of the CWI's will all be characterized by the same 4 pairs of indices (a i , b j ) (i, j = 1, 2). It is convenient to define the two functions then The solutions takes therefore the hypergeometric form in which the functions above take the role of parameters We will refer to α, . . . , γ as to the first, . . ., fourth parameters of F 4 . Notice that the system is of order 4, since it allows 4 independent solutions which are then all of the form x a y b F 4 , where the hypergeometric functions will take some specific values for its parameters, with a and b fixed by (61) and (62) where the sum runs over the four values . The sum over a and b needs to be made explicit and, at the same time, one has to combine the 4 independent solutions in such a way that the symmetry respect to the three external momenta is restored. For this reason if we define to be the four basic hypergeometric parameters, with the remaining ones determined by shifts respect to these values, the four fundamental solutions can be expressed in the form The symmetrization with respect to the external momenta [41], by using the formula allows to reverse the ratios of the momenta in the expansion, and reduces the four constants to just one. The solution can then be written in the final form where c i are arbitrary coefficients which may depend on the scale dimensions ∆ i and on the spacetime dimension d.
By using (72), the general symmetric solution can be expressed in the form [41] It is important to verify that the symmetric solution above does not have any unphysical singularity in the physical region, reproducing the expected behaviour in the large momentum limit p 3 p 1 [54]. Indeed, the previous expression, in the limit p 3 p 1 (expressible also as p 2 3 , p 2 2 → ∞ with p 2 2 /p 2 3 → 1 fixed), behaves as and with f (d, ∆ i ) and g(d, ∆ i ) depending only on the scaling and spacetime dimensions. In the case of a scalar three-point function, the CFT correlator is equivalent to a Feynman master integral, as one can immediately realize. The result above in (74) is in complete agreement with the direct computation performed by Davydychev [55] of such integrals, obtained by a Fourier transform of (13) and the use of the Mellin-Barnes method. An equivalent version of the solution found above can be derived as in [42], where it is written in terms of K Bessel functions as where C 123 is an undetermined constant. More details are given in Appendix A. The 3K integral is related to the hypergeometric functions valid for Re α > |Re λ| + |Re µ| + |Re ν|, The Bessel functions K ν satisfy the equations which are important in order to verify that (77) satisfies the CWI's. It will be convenient to adopt the notation introduced in [42] and define the general expression which will turn useful in the analysis of scalar 4-point functions.
As we are going to see, a similar expression of the solution is obtained for dual-conformal/ conformal solutions, where the conformal symmetry in coordinate (or, equivalently, momentum) space, is accompanied by an additional symmetry in a space of dual variables. In perturbative quantum field theory, this symmetry goes under the name of dual conformal. If we impose on a 4-point function both symmetries, then the CWIs alone are sufficient to fix the solution of the conformal constraints modulo a single overall constant, with no need of introducing any extra formalism, such as the operator product expansion, which is necessary in order to determine the structure of 4-point functions in a CFT. The method to extract such dual conformal/conformal solutions is quite technically involved and has been developed by us in [51].
Before coming to a discussion of this point, we illustrate the generalization of such systems of equations obtained in the scalar case, to tensor correlators, bringing as a nontrivial example the T J J tensor correlator, and showing how the inhomogeneous systems of equations generated by the CWI's can be solved.
We will follow the approach formulated in [44], which exploits the properties of the single function F 4 in order to solve the corresponding systems of equations. The original approach for the solution of these systems of equations, entirely based on the 3K integrals of Equation (81) has been developed in [42]. The two approaches can be mapped one to the other.

Tensor Correlators: The TJJ
The T J J is a tensor correlator involving one stress-energy tensor T and two vector currents J. The interest in this correlator is manifold since it describes the leading contribution to the coupling to gravity of an ordinary gauge theory, such as quantum electrodynamics (QED) and, in the nonabelian case, quantum chromodynamics (QCD).
Perturbative studies of this correlator have been necessary in order to uncover the manifestation of the conformal (Weyl) anomaly in conformal field theory [56], which induces the breaking of a classical symmetry at the classical level. The conformal anomaly is associated with a specific functional which involves both a gravitational part and a gauge part. More details about this topic can be found in a sequel of analysis [41,57,58], and in the review [59]. The T J J is sensitive to the gauge part of the anomaly T where β(α) is the β function of the theory which describes the running of the gauge coupling α under a change of the renormalization scale. In (82) we have performed a quantum average ( ) in the background of two classical fields, the metric g µν and the gauge field A µ , as defined in terms of a functional integral (see [43]). F αβ = ∂ α A β − ∂ β A α is the field strength of the abelian gauge field A µ . Therefore, the classical result of a vanishing trace of T (T µ µ = 0), which is expected due to conformal symmetry, is modified as in (82). More details on this point can be found in [41,44,58,60].
The T J J correlator is defined by the expression and, as we have mentioned, is affected by an anomaly. This is generated when all the points x i of the correlator coalesce, and it is for this reason that the analysis of these types of correlators requires a regularization procedure. Their analysis can be drastically simplified if, for a given CFT characterized by specific conformal data, we can find a realization of the same correlator in terms of a specific free field theory. There is one simple strategy in order to perform such mapping.
1. We need to solve all the CWIs consistently, identifying all the independent constants that appear in the solution. For instance, in the case of the T J J, there will be three undetermined constants. The solution obtained is expressed in terms of linear combinations of functions F 4 or of 3K integrals.
2. We considerthree independent conformal free field theories, defined by ordinary Lagrangians, with an arbitrary particle content, and compute the Feynman contributions corresponding to the T J J vertex. These amount to one-loop diagrams and define the perturbative solution. Such a solution will be characterized by three independent constants, which can be chosen to be the number of scalars n s , fermions n F and spin-1 (gauge) fields n V , all running in the loops.
3. The match between the two results can be explicitly checked by going to special spacetime dimensions where the general hypergeometric solutions simplify. This occurs in d = 3 and d = 5, which are sufficient to establish a direct match between the three constants of the CWI' and the free field theory realization (n S , n V , n F ) bi-univocally.
As a result of the match, the hypergeometric solutions can be re-expressed in terms of simple scalar massless Feynman integrals, corresponding to the 2-and 3-point functions. Feynman amplitudes corresponding to three-point functions are indeed described by generalized hypergeometric functions, and this is at the root of the success of the correspondence between the two approaches.
The formulation of the general method which takes to the solution of the CWI's can be found in [42]. It is based on the decomposition of a tensor correlator in terms of its transverse traceless and longitudinal sectors. A similar decomposition is performed on the CWI's. In particular, the hypergeometric structure of the equations emerges from the transverse traceless sector, in a way that is quite close, though more involved, respect to the scalar case discussed above.
The entire solution is parameterized by a certain number of form factors A i which are functions of the momentum variables p 2 i , appearing in the parameterization of the transverse traceless sector of the correlator. In the case of the T J J the equations take the form and represent a hypergeometric system of equations which generalizes the simpler result presented in the scalar case.

Solving Inhomogeneous Systems in the TJJ Case
We illustrate our method of solving the system of equations above, based on the shifts of the parameters of each function F 4 . Notice that in this case, the correlator is symmetric respect to the exchange of the two J operators.
As in the case discussed above, we take as a pivot p 2 1 , and assume the symmetry under the (P 23 ) exchange of (p 2 , ∆ 2 ) with (p 3 , ∆ 3 ) in the correlator. In the case of two photons ∆ 2 = ∆ 3 = d − 1.
We start from A 1 by solving the two equations from (84) In this case we introduce the ansatz where ∆ = ∆ 1 + ∆ 2 + ∆ 3 and derive two hypergeometric equations, which are characterized by the same indices (a i , b j ) as before in (61) and (62), but new values of the 4 defining parameters. We obtain ( ∆ ≡ (∆ 1 , ∆ 2 , ∆ 3 )), with the expression of α(a, b), β(a, b), γ(a), γ (b) as given before, with the obvious switching of the ∆ i in order to comply with the new choice of the pivot (p 2 1 ) which are P 23 symmetric and with P 23 γ(a) = γ (b). If we require that ∆ 2 = ∆ 3 , as in the T J J case, the symmetry constraints are easily implemented. The 4 indices, if we choose p 1 as a pivot, are given by and in this case a = b and γ(a) = γ(b). We recall that F 4 has the symmetry F 4 (α, β; γ, γ ; x, y) = F 4 (α, β; γ , γ; y, x), and this reflects in the Bose symmetry of A 1 if we impose the constraint Now let's turn to the solution for A 2 .
The equations for A 2 are inhomogeneous We consider an ansatz of the form which provides the correct scaling dimensions for A 2 . At this point, the action of K 21 and K 31 on A 2 can be rearranged as follows wherē and we notice that the hypergeometric function which solves the equation can be taken of the form with c (2) 1 a constant and the parameters a, b fixed at the ordinary values (a i , b j ) as in the previous cases (61) and (62). This allow to remove the 1/x and 1/y poles in the coefficients of the differential operators. The sequence of parameters in (101) will obviously solve the related equation Equation (100) can be verified by observing that the sequence of parameters (α(a, b) + 1, β(a, b) + 1γ(a) − 1) allows to define a solution of (99) set to zero, for an arbitrary γ (b), since this parameter does not play any role in the solution of the corresponding equation. It is also clear that the sequence (α(a, b) + 1, β(a, b) + 1, γ (b)), on the other hand, solves the homogeneous equations associated to K 31 (i.e., Equation (102)) for any value of the third parameter of F 4 , which in this case takes the value γ(a) − 1. We can follow the same approach for the mirror solution As previously remarked, the values of the exponents a and b remain the same for any equation involving either a K i,j or aK ij , as can be explicitly verified. This clearly shows that the fundamental solutions of the conformal equations are essentially given by the four functions of the type S 1 , . . . , S 4 , for appropriate values of their parameters.
At this point we use the property which gives (for generic parameters α, β, γ 1 , γ 2 ) Obviously, such relations are valid independently of the four parameters α, β, γ 1 , γ 2 may have on the Fuchsian exponents (a i , b j ). The actions of K 21 and K 31 on the the Φ (i) 2 's (i = 1, 2) in (103) are then given by where it is clear that the non-zero right-hand-side of both equations are proportional to the form factor A 1 given in (87). Once this particular solution is determined, Equation (87) We conclude that the general solution for A 2 in the T J J case (in which γ(a) = γ (b) ) is given by combining the solution of the homogeneous form of (87) and the particular one given by (101) and (103), by choosing the constants appropriately using (110). It is explicitly given by The approach allows us to solve for all the form factors A i if used sequentially. The derivation of the solution is discussed in [44]. A similar analysis allows us to solve for other correlators, such as the TTT [43]. Notice that a reduction of the number of constants generated from (84) is obtained by imposing some additional WIs which link special CWIs and canonical WI's. They have been called secondary WIs in [42]. The solution of these equations, which connect 3-to 2-point functions can be solved by performing some specific limits on the momentum variables.

Dual Conformal/Conformal Symmetry
Dual conformal symmetry has been originally identified in the context of perturbative quantum field theory and holds for a special class of Feynman diagrams [61]. We recall that a dual conformal integral is a Feynman integral which, once rewritten in terms of some dual coordinates, under the action of K κ is modified by factors which depend only on the coordinates of the external points. The reformulation of the ordinary momentum integral in terms of such dual coordinates can be immediately worked out by drawing the associated dual diagram.
We illustrate this point in the case of the ordinary Feynman (box) diagram shown in Figure 1 Φ Box (p 1 , p 2 , p 3 , which is a function of 6 Lorentz invariants. As usual in particle theory, we use the six scalar variables p 2 i (i = 1, 2, 3, 4) and s 2 = (p 1 + p 2 ) 2 and t 2 = (p 1 + p 3 ) 2 . We introduce dual variables y i , redefining the momenta in terms of these k = y 51 , p 1 = y 12 , p 2 = y 23 , with y ij = y i − y j , and rewrite the integral in the form The action of K κ is realized in the form I · T · I, as a sequence of inversion, translation and inversion transformations, rather than as a differential action (by K κ , as in (36)). We recall that under an inversion (I) and in order to have an expression which is invariant under special conformal transformation, it is necessary to include a pre-factor in Φ Box , in the form and then it is easy to check that under the action of I the integrand with u and v given by which can be expressed directly in terms of the original momentum invariants, due to the relations (113). Notice that, by construction u and v satisfy the first order equations in the y variables Dual conformal Feynman diagrams, such as the one discussed above, satisfy in the y variables (113) CWI's as in ordinary coordinate space. The y variables are, however, variables of momentum space, being linear combinations of the fundamental momenta p i , obtained from the Fourier transform of the original correlator in coordinate space.
It then interesting to search for solutions to the 4-point functions which are at the same time conformal in the coordinate variables (x i ), as well in the auxiliary variables y i introduced by the mapping (113).
This implies that a scalar 4-point function, once written in momentum space and re-expressed in terms of the y i variables, has necessarily to take a form similar to (118), for being conformal in such variables. The simultaneous conditions of conformal and dual conformal invariance are then satisfied if a scalar correlator takes the form (118) and, at the same time, by imposing that it satisfies the ordinary CWI in the ordinary momentum variables, once the y variables are re-expressed in terms of the momenta using (113). We have shown in [51] that the form taken by F(u, v) is unique. We are going to illustrate the construction of such solutions.

The CWIs for Scalar 4-Point Correlators
To illustrate the approach in a realistic case, let's consider a generic scalar four-point function in momentum space with the definitions of the invariants and Mandelstam variables as This correlation function, to be conformal invariant, has to verify the dilatation Ward Identity and the special conformal Ward Identities which are conditions generated by the conformal symmetry in the ordinary variables x i , parameterizing the correlator in coordinate space. By turning to momentum space and using the chain rules and similar, one derives the two equations together with a third one, that we omit, which takes a similar form. A detailed discussion of such systems of equations has been presented in [51]. Clearly, these equations do not define in general, to a hypergeometric system. However, once we require that the solutions of these equations also be invariant in the momentum variables y ij , where the y ij are treated as ordinary coordinates x i , we generate a hypergeometric system of equations also for such 4-point functions.
As we have already mentioned, the strategy to solve the equations is to start with an expression of these correlators of the form (56) and to impose on them the conditions of conformal invariance. Clearly, both these conditions are only formulated in momentum space.

DCC Solutions
To identify the dual conformal/conformal solutions, we choose the ansatz Φ(p i , s, t) = s 2 t 2 n s F(x, y), where n s is a coefficient (scaling factor of the ansatz) that we will fix below by the dilatation WI, and the variables x and y are defined by the quartic ratios Such quartic ratios are nothing else but the two variables u and v of (119) re-expressed in momentum space.
By inserting the ansatz (129) into the dilatation Ward Identities, and turning to the new variables x and y, after some manipulations we obtain from the dilatation WI the constraint which determines n s = (∆ t − 3d)/4, giving We will be using this specific form of the solution in two of three special CWIs of the form C 2 and C 13 . The functional form of F(x, y) will then be further constrained.
In order to determine the conditions on F(x, y) from (127) and (128), we re-express these two equations in terms of x and y using several identities. In particular we will use the relations together with Both relations can be worked out after some lengthy computations. We start investigating the solutions of these equations by assuming, for example, that the scaling dimensions of all the fields φ i are equal Using (133) and (134), we write the first Equation (127) associated to C 2 in the new variable x and y as and the second one (128) associated to C 13 as where we recall that n s is the scaling under dilatations, now given by By inspection, one easily verifies that (135) and (136) define a hypergeometric system of two equations whose solutions can be expressed as linear combinations of 4 Appell functions of two variables F 4 , as in the case of 3-point functions discussed before. The general solution of such system is expressed as with ∆ t = ∆(1, 1, 1, 1), if we choose the operators of equal scalings. Notice that the solution is similar to that of the 3-point functions given in the previous section.
In this case, we can reformulate such solution in terms of 3K integrals using the expression which is the close in form to the result obtained from 3-point functions, but now expressed in terms of quartic ratios of momenta. Using (80) one can derive several relations, such as which generate identities such as One can show that the I integrals satisfy the differential equations In the case ∆ 1 = ∆ 3 = ∆ x and ∆ 2 = ∆ 4 = ∆ y , the special CWI's can be written as withᾱ an arbitrary constant, in agreement with the solution found for the three-point function.

Lauricella Functions
We come to the last part of this overview, illustrating the presence of another type of functions of hypergeometric type.
Lauricella systems of equations are generated if we look for asymptotic solutions of the CWIs characterized by large s and t invariants, under the condition that p 2 i s 2 , t 2 , recalling the definition of these variables in (122). In a 2-to-2 process describe d by 4-point functions, a scattering at a fixed angle is characterized by −t/s fixed. We are allowed to take both s and t large, keeping their ratio fixed. In this limit, one can show that the CWIs satisfy an approximate factorization in which we separate the dependence of Φ on the invariants p 2 i from the pair s and t. It is also possible to show that a Lauricella system describes a particular solution of the special CWIs of the form C 2 and C 13 . The analysis has been presented in [51] and [53], to which we refer for further details.
The equations in this case take the form are the dimensionless ratios x, y and z, not to be confused with coordinate points in a three dimensional space. The ansatz for the solution can be taken of the form φ(p 1 , p 2 , p 3 , p 4 ) = (p 2 4 ) n s x a y b z c F(x, y, z), which is consistent with the corresponding dilatation WI for 4-point functions if With this ansatz, equations (152) take the form Similar constraints are obtained from the equation K 34 φ = 0 that can be written as0 In this case, as for 3-poin functions, in order to perform the reduction to the hypergeometric form of the equations, we need to set G = 0, G = 0 and G = 0, which imply that the Fuchsian points a, b, c take the values We find also that as well as A = A = A = −(α(a, b, c) + β(a, b, c) + 1), indeed and finally With this redefinition of the coefficients, the equations are then expressed in the form [51]      having redefined γ 1 = γ, γ 2 = γ and γ 3 = γ and x 1 = x, x 2 = y and x 3 = z. This system of equations allows solutions in the form of the Lauricella hypergeometric function F C of three variables, which are defined defined by the series where the Pochhammer symbol (λ) k with an arbitrary λ and k a positive integer not equal to zero, was previously defined in (51). The convergence region of this series is defined by the condition The function F C is the generalization of the Appell F 4 to the case of three variables. The system of equations (166) admits 8 independent solutions given by S 1 (α, β, γ, γ , γ , x, y, z) = F C α, β, γ, γ , γ , x, y, z , having defined The most general solution of the system is obtained by taking linear combinations of such fundamental solutions with arbitrary constants. In this case, as in the cases discussed above for 3-point functions, one needs to impose the symmetry of the solution under exchanges of the momenta and of the scaling dimensions ∆ i . This is most easily accomplished by establishing a link between the Lauricella function F C and some parametric integrals of four Bessel functions, which have been introduced in [51].

Lauricella's as 4-K Integrals
The introduction of parametric representations of the solutions of the CWI's in terms of 3K integrals (81) in [42] has as its advantage the possibility of implementing the symmetry constraints of a 3-point function quite directly. On the contrary, the use of the 4 fundamental solutions introduced in (68) and (69) requires significant manipulations (see the discussion in [41]) in order to obtain the same result. In the case of 4-point functions, the possibility of expressing the dual conformal/conformal solution in the form of a 4K integral appears to be the natural generalization of such previous approach, and allowing also to discuss the kinematical limits in which the dynamics of a 4-point function reduces to that of a 3-point one.
One can show that hypergeometrics of three-variables, which belong to the class of Lauricella functions can be related to 4K integrals.
The key identity necessary to obtain the relation between the Lauricella functions and the 4K integral is given by the expression If we rewrite the solutions of such systems in the form I α−1{ν 1 ,ν 2 ,ν 3 ,ν 4 } (a 1 , a 2 , a 3 , with the Bessel functions I ν , J ν , K ν related by the identities where we have used the properties of the Gamma functions π sin(πν) the dilatation Ward identities in this case can be written as p i ∂ ∂p i I α{β 1 ,β 2 ,β 3 ,β 4 } (p 1 , p 2 , p 3 , p 4 ) = 0.
Using some properties of 4K integrals [51] one can derive the relation (α − β t + 1 + ∆ t − 3d)I α{β 1 ,β 2 ,β 3 ,β 4 } (p 1 , p 2 , p 3 , p 4 ) = 0 (177) where ∆ t = ∑ 4 i=1 ∆ i , which is identically satisfied if the α exponent is equal toα The conformal Ward identities can be re-expressed in the form Finally, the final solution can be written as whereᾱ is a undetermined constant. Notice that given I α{β 1 ,β 2 ,β 3 ,β 4 } (p 1 , p 2 , p 3 , its first derivative with respect the mgnitudes of the momenta is given by One can derive various relations satisfied by these types of integrals, as shown in [51]. For instance, using one derives the identity 4 ∑ i=1 p 2 i I α+1{β j −δ ij } = (α − β t + 1) I α{β j } , j = 1, . . . , 4 where β t = β 1 + β 2 + β 3 + β 4 . One of the advantages of the use of the 4K integral expression of a solution is the simplified way by which the symmetry conditions can be imposed. In fact, by taking each of the 8 independent solutions identified in (169), and by rewriting them in the form of 4K integrals, the permutational symmetry of the correlators under the exchanges of the external momenta p i and scaling dimensions ∆ i becomes trivial.

Conclusions
We have reviewed recent progress on the analysis of the CWIs in momentum space for 3-and 4-point functions of ordinary CFTs in d > 2.
The momentum space approach, as mentioned in our introduction, allows to look at such theories from a very different perspective, which is simply not accessible from coordinate space. It allows us to perform direct comparisons with the ordinary analysis of scattering amplitudes expressed in terms of Feynman diagrams, providing access to a wide class of methods that have been developed in this area of perturbative quantum field theories.
The emergence of hypergeometric structures in the context of the CWI's is for sure an interesting feature of such equations which will be further explored in the near future, with interesting new results in this area. The study of the conformal phases quantum field theories is a fascinating topic that will probably receive continuing attention for its important physical applications and may shed light on several phenomena, ranging from condensed matter theory to high energy theory. Therefore, the understanding of the fundamental mathematical structures which are part of this analysis and that can help in this process is of considerable importance.
Using (A4) we can calculate for instance the integrals