Two-loop kite master integral for a correlator of two composite vertices

We consider the most general two-loop massless correlator $I(n_1,n_2,n_3,n_4,n_5; x,y;D)$ of two composite vertices with the Bjorken fractions $x$ and $y$ for arbitrary indices $\{n_i\}$ and space-time dimension $D$; this correlator is represented by a"kite"diagram. The correlator $I(\{n_i\};x,y;D)$ is the generating function for any scalar Feynman integrals related to this kind of diagrams. We calculate $I(\{n_i\};x,y;D)$ and its Mellin moments in a direct way by evaluating hypergeometric integrals in the $\alpha$ representation. The result for $I(\{n_i\};x,y;D)$ is given in terms of a double hypergeometric series -- the Kamp\'{e} de F\'{e}rriet function. In some particular but still quite general cases it reduces to a sum of generalized hypergeometric functions $_3F_2$. The Mellin moments can be expressed through generalized Lauricella functions, which reduce to the Kamp\'{e} de F\'{e}rriet functions in several physically interesting situations. A number of Feynman integrals involved and relations for them are obtained.


Introduction
The correlators of composite vertices appear naturally as the result of "factorization" of scales in hard processes, more precisely in the technical sense-due to contractions of the so called "hard subgraphs" of the corresponding diagrams. In particular, such a two-point correlator with one composite vertex appears at the contraction of V-V subgraphs of the V (q 1 )V (q 2 )A(p) triangle diagram for the kinematics with hard momentum transfer −q 2 1 , −q 2 2 ≫ p 2 = (q 1 + q 2 ) 2 , where V =ψγ µ ψ and A =ψγ ρ γ 5 ψ are the vector and axial fermion currents, respectively. These contractions of the triangle constitute a theoretical basis of the factorization approach for the perturbative QCD calculations of the transition form factors for the processes γ * (q 1 )γ * (q 2 ) → π 0 (p), where π 0 is a neutral pion and γ * 's are virtual photons.
Here we consider the calculation of a more general object than the one just mentionedthe two-point massless correlator I of two composite vertices, which is the normalized Fourier transform of the correlator J µ (0)J ν (z) of two composite fermion currents J µ and J ν , see [1]. As it happens, a more general quantity can be evaluated technically easier than its superficially simpler counterpart with only one single composite vertex J ν . Our goal is the calculation of the two-loop massless "kite" scalar diagram I(p; {n i }; x, y; D) in figure 1, taken at any values of indices n i of the lines, {n i } = n 1 , n 2 , n 3 , n 4 , n 5 , and any spacetime dimension D. The "kite" diagram is one of the master integrals for the two-current correlator at two and three loops, see eq. (3.1). The function I is the generating function for any Feynman integral related to this kind of diagrams. The integrals can be obtained by convolving the function I with appropriate weights ϕ and φ, ϕ(x)⊗I(p; {n i }; x, y; D)⊗φ(y), where symbol ⊗ means integration over the longitudinal momentum fractions x or y. In the coordinate representation the weight function ϕ (or φ) becomes an operator ϕ(ñ · ∂ z /(ñ · p)) that acts at the vertex point z (ñ 2 = 0). To return to the correlator with one composite vertex, e.g. the one with the fraction x, we should integrate I(p; {n i }; x, y; D) over the fraction y. Besides, the original two-argument correlator I(. . . ; x, y; D) was used to analyse the properties of the conformal composite vertices under renormalization in [2]. The zeroth Mellin moments of I(. . . ; x, y; D) in both x and y, I(. . . ; 0, 0; D), 1 give the ordinary master integral of kite topology. It has been known for relatively long time that this integral can be evaluated in closed form involving hypergeometric functions. In ref. [3], Chetyrkin et al. showed for the first time that the integral I(. . . ; 0, 0; D) can be expressed as a double hypergeometric series for n 1 = n 2 = 1. To come up with this result, they expanded the integrand (in coordinate space) over a basis of the Gegenbauer polynomials and successfully solved the remaining integrals. A special case n 1 = n 2 = n 3 = n 4 = 1 was considered by Kazakov [4] and Broadhurst [5] a few years later. They applied the methods of uniqueness and integration by parts (IBP) to derive functional equations for the integral under consideration and then solved them in terms of the function 3 F 2 (−1) 2 . Another result in terms of 3 F 2 (1) was obtained by Kotikov who refined the technique of ref. [3] and found a transformation from the double series to a single one even in a more general 1 Throughout this paper, the Mellin transform of a function f (x) is indicated by underscoring its argument, i.e. f (a) = 1 0 x a f (x) dx. 2 We use sometimes the designation pFq (x) with omitted parameters implying that the values of the parameters are irrelevant in the particular context or left generic.

Simple example: one-loop integral
First, we introduce a generalization of the G function for the one-loop integral with composite vertices ⊗: where a slash (beside the composite vertex) on the line with momentum k means factor δ(x −ñk);ñ µ is a light-cone vector,ñ 2 = 0, normalized so that (ñp) = 1. The function G(n 1 , n 2 ; x; D) is dimensionless and reduces to the usual one-loop G function (see appendix B of ref. [3], section 1.5 in [14] or section 3.1 in [29]) if it is integrated over x: (2. 2) The function G(n 1 , n 2 ; x; D) has obvious symmetry: The integral (2.1) is easily calculated when the propagators 1/[k 2 ] n 1 and 1/[(k − p) 2 ] n 2 are cast into the α representation and the Dirac delta function is substituted by its Fourier integral. The result for the G function reads The Mellin transform of the function G is (2.5) Here and in what follows, a νth Mellin moment of a function f (x) is denoted by the underlined argument of the function f (ν); the n's with multi-indices are defined as The definition of the two-row Γ function is clear from eq. (2.5): . (2.7) , (2.8) where Θ(x) is the Heaviside step function.

20)
F (n 1 , n 2 , n 3 , n 4 , n 5 ; z; D) = F (n 4 , n 3 , n 2 , n 1 , n 5 ; z; D), (3.21) where λ = D/2−1. Note that the expression in the square brackets in eq. (3.19) is a function of a single variable defined as a ratio of fractions x and y, conformal ratio z = (yx)/(xȳ). We borrowed the name for z from ref. [30] (see also references therein), since the form of the ratio closely resembles that appearing in the evolution kernel for light-ray operator as a consequence of the conformal group.

The correlator as the Kampé de Fériet function
For arbitrary nonvanishing n r one of the integrations in eq. (3.20) can be easily performed in terms of the Appell function F 1 . Indeed, making a substitution b = aw, we have as an integral over w a classic Euler-type integral representation of the first Appell function (A.9): The second equality in the equation above can be easily obtained by applying the autotransformation properties of the Appell function (A.10).
In the general case, the remaining integral over a can be evaluated in terms of the Kampé de Fériet (KdF) function f

Correlator
If all indices of the external edges of the diagram are equal to one (see figure 3), the corresponding integral splits into two univariate functions-a hypergeometric function of the conformal ratio z and a power of the difference x − y: G(1, 1, 1, 1, n; x, y; D) =Ŝ f (n; z; D) where z = (xy)/(xȳ) is the conformal ratio defined earlier, i.e.Ŝf (z) = f (z)+f (1/z). In eq. (4.2) we introduce renormalized hypergeometric functions for the sake of brevity (see appendix A): where Iz(a, b) is an incomplete Euler B function normalized by the complete one:

Mellin moments of the correlator as the Kampé de Fériet functions
Both one-and twofold moments of the function (4.1) can be written in terms of the KdF functions for arbitrary real orders a and b of the moments: (4.8)

Reduction to hypergeometric series in one variable
At least some of KdF functions in eqs. (4.7) and (4.8) reduce to onefold hypergeometric series: The proofs of the above reductions are reserved for the appendix B. All other KdF functions in eqs. (4.7) and (4.8) reduce to generalized hypergeometric functions for b = 0: (4.12) An immediate consequence of the variable splitting in eq. (4.1) is that a moment (x, N + b) shifted by a natural number N with respect to a lower moment (x, b) can be obtained by differentiating the latter one: 1, 1, 1, n; x, N + b; D) 1, 1, 1, n; x, b; D) . This implies that any moment G (1, 1, 1, 1, n; x, N ; D) for a natural N is a finite sum of functions 3 F 2 (x), 3 F 2 (x), and simpler functions.
Setting N to 1 and integrating (4.13) with x a by parts, we obtain a simple recurrence relation for twofold moments: (ω/2 + a + b + 2)G(1, 1, 1, 1, n; a, b; D) = (ω/2 + a + 2)G(1, 1, 1, 1, n; a + 1, b; D) + (ω/2 + b + 2)G (1, 1, 1, 1, n; a, b + 1; D). (4.14) Deriving the above relation, we assumed that the limits of x 1+ax G (1, 1, 1, 1, n; x, b; D) at the endpoints are zero. The recurrence relation allows us to express all moments (a + k, b + l) for any natural numbers k and l through a set of independent moments chosen, for instance, as (a + 2k, b), k = 1, 2, . . . . Evaluating a-moments of eq. (4.12), we arrive at This and the recurrence relation (4.14) give also all twofold moments G(1, 1, 1, 1, n; a, N ; D) for any natural N . The moments G(1, 1, 1, 1, n; a, N ; D) can always be represented as a sum of 4 f 3 (1) and simpler functions. Finally, to compare eq. (4.15) with earlier results in the literature, we can write the following expression for the special case a = b = 0 that is valid for arbitrary D and n: (4.16) The above equation reproduces the already known results [4,6,9] in its different domain of applicability. Here, we have taken into account the fact that the function 3 f 2 (1) converges if and only if its parametric excess-the difference between the sums of all lower and all upper parameters-is positive. The Heaviside step functions "switch off" the corresponding the step function of 1 − 2λ + n is exactly the expression found by Kotikov [6], while the other half of eq. (4.16) can be easily transformed to one of the representations suggested by Broadhurst et al. [9] (see also eq. (6.11) in ref. [12]). 3

Conclusion
We Considering the integral I(p; {n i }; x, y; D) in the α representation, we have evaluated the integral in terms of the hypergeometric Kampé de Fériet (KdF) function of two variables, f(−z/z, 1). In some important cases with two natural indices (figure 2), the KdF function reduces to a sum of univariate hypergeometric functions 3 F 2 (z). The hypergeometric part of the integral depends on only one combination of the Bjorken variables, the conformal ratio z = (xy)/(xȳ), which could be a manifestation of the conformal symmetry. We have also calculated one-and twofold Mellin moments I(p; {n i }; x, y; D). In the general case they are expressed through the generalized Lauricella functions. For natural moments, however, the Lauricella functions reduce to simpler KdF functions, which is proved most easily in their Mellin-Barnes representation.
We paid close attention to the special cases of {n i } = {1, 1, 1, 1, n}, which appear in calculating two-and three-loop quark correlator (more precisely, the part of the correlator proportional to β 0 ). These master integrals and the Mellin moments thereof can always be expressed through the KdF functions reducing to 4 F 3 and simpler functions at least in most of the practically important situations. Taking the double zeroth Mellin moment of I(p; 1, 1, 1, 1, n; x, y; D)-a twofold integral over x and y-yielded the well known expression for the two-loop integral I(p; 1, 1, 1, 1, n; D) in terms of 3 F 2 . This can be viewed as a curious methodological byproduct of our consideration-evaluating directly hypergeometric Eulerian integrals occurring in the α representation provides us with the third alternative approach to derive the result for I(p; 1, 1, 1, 1, n; D), the first two being: solving functional equations coming from the star-triangle and integration-by-parts relations [4,5,9], and integrating with the help of expansion in the basis of Gegenbauer polynomials [6]. Also, we have derived the recurrence relations allowing us to express higher moments shifted by some natural numbers through a basis of lower moments. It should be noted that these recurrence relations in our approach are not a consequence of integration-by-parts relations as it is usually the case but of variable splitting in the integral, which is a product of function of the conformal ratio z and function of the difference x − y.
Finally, for the sake of completeness of the paper, we list relations referred to in the text: The purpose of this appendix is to encapsulate some properties of the KdF function f 1:2;1 1:1;0 (x, y) that has occurred in eqs. (3.23) and (4.7). It admits three representations as the integrals of the Gauss and Appell functions, and 3 f 2 : where Re b > Re a > 0 and, for (B.3), Re b > Re c > 0; f 1 (x, y) = f 1:1;1 1:0;0 (x, y) in eq. (A.4). The equivalence of the first two representations can be easily proved by using the Euler integrals for 2 f 1 (A.13) and f 1 (A.9). The third representation can be obtained from the second one with the help of the following representation for f 1 [47]: Eqs. (B.1)-(B.3) allows us to obtain (4.7) readily by integrating (4.1) with y b over y (one should simply change the integration variable to z).
Making the substitution z →z in eq. (B.1), setting y = 1, and using the Kummer transformation (A.11), we obtain the following autotransformation property for f 1:1;2 1:0;1 (x, 1): x c f 1:1;2 Also, we can make the substitution z →z in eq. (B.1), expand the denominator in the integrand, and evaluate the integral of the series term by term with the help of the integral representation of 3 F 2 (A.12). This gives In a number of cases f 1:1;2 1:0;1 (x, 1) reduces to simpler hypergeometric functions. One of them follows from eq. (B.1)-the integral over z can be easily done in terms of 3 f 2 if a is equal to g [48]. Indeed, expanding the denominator of the integrand as a series in −x/x, evaluating the integral over z, and using the well-known summation formula for 2 f 1 (1) (A.17), we get x c f 1:1;2 Another possibility of reduction important for us has been pointed out in our sketching the proof of eq. (3.26). It is related to the reduction (A.18) of f 1 to 2 f 1 in the integral representation (B.2) if b = c + d. We will not develop the consideration of section 3.3 here. It should be noted, however, that the reduction formula (4.9) can be derived (rather cumbersomely) in the same way-use the representation (B.2), write the Appell function as 2 f 1 (see (A.18)), transform 2 f 1 with the help of (A.11), evaluate the resulting simple integrals, and apply the identity (A.20) to 3 f 2 (1/x). Now, let us prove the reduction (4.10). To this end, we represent the KdF function as a series with 3 f 2 (1) in coefficients: Then, we transform 3 f 2 (1) with the help of the identity (A.15) 3 f 2 1 + r,ṅ + 1, n 1 + λ + r, 2(ṅ + 1) 1 = Γ 1 + r, −ṅ, λ −ṅ − 1, r − 2ṅ,ṅ + 1, n λ, −2ṅ − 1, r −ṅ + 1, r + λ −ṅ, 2(ṅ + 1) This results in the following expression: The last term in the above equation can be simplified by virtue of the reduction (B.7), which leads us immediately to eq. (4.10). Finally, to derive the reduction of the third term in eq. (4.7) for b = 0, we can make use of the autotransformation property (B.5), which gives two terms-the Appell function f 1 (−x/x, 1) and a new f 1:1;2 1:0;1 (−x/x, 1). Both terms can be reduced to series in one variable due to the reduction formulas (B.7) and (A. 19). In the case of b = 0, the fourth term in eq. (4.7) is equal to the third one up to replacing x byx.