Two-loop kite master integral for a correlator of two composite vertices
We consider the most general two-loop massless correlator of two composite vertices with the Bjorken fractions and for arbitrary indices and space-time dimension ; this correlator is represented by a “kite” diagram. The correlator is the generating function for any scalar Feynman integrals related to this kind of diagrams. We calculate and its Mellin moments in a direct way by evaluating hypergeometric integrals in the representation. The result for is given in terms of a double hypergeometric series—the Kampé de Férriet function. In some particular but still quite general cases it reduces to a sum of generalized hypergeometric functions . The Mellin moments can be expressed through generalized Lauricella functions, which reduce to the Kampé de Férriet functions in several physically interesting situations. A number of Feynman integrals involved and relations for them are obtained.
Keywords:Feynman integrals, hypergeometric functions
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 triangle diagram for the kinematics with hard momentum transfer , , where and 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 , where is a neutral pion and ’s are virtual photons.
Here we consider the calculation of a more general object than the one just mentioned—the two-point massless correlator of two composite vertices, which is the normalized Fourier transform of the correlator of two composite fermion currents and , see Mikhailov:1988nz ().
As it happens, a more general quantity can be evaluated technically easier than its superficially simpler counterpart with only one single composite vertex . Our goal is the calculation of the two-loop massless “kite” scalar diagram in figure (1), taken at any values of indices of the lines, , and any space-time dimension . The “kite” diagram is one of the master integrals for the two-current correlator at two and three loops, see eq. (9). The function is the generating function for any Feynman integral related to this kind of diagrams. The integrals can be obtained by convolving the function with appropriate weights and , , where symbol means integration over the longitudinal momentum fractions or . In the coordinate representation the weight function (or ) becomes an operator that acts at the vertex point z (). To return to the correlator with one composite vertex, e.g. the one with the fraction , we should integrate over the fraction . Besides, the original two-argument correlator was used to analyse the properties of the conformal composite vertices under renormalization in Craigie:1983fb ().
The zeroth Mellin moments of in both and , , 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. Chetyrkin:1980pr (), Chetyrkin et al. showed for the first time that the integral can be expressed as a double hypergeometric series for . 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 was considered by Kazakov Kazakov1985 () and Broadhurst Broadhurst:1985vq () 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 in Kazakov1985 () and in Broadhurst:1985vq (). Another result in terms of was obtained by Kotikov who refined the technique of ref. Chetyrkin:1980pr () and found a transformation from the double series to a single one even in a more general case 1996PhLB..375..240K (). Recently, the equivalence of the representations through was proved in ref. Kotikov:2016rgs (). In refs. Barfoot:1987kg () and Broadhurst:1996ur (), several different representations in terms of and were obtained by cleverly using the symmetries of the integral 1985TMP….62..232G (); Broadhurst:1986bx (); Barfoot:1987kg (). For more details and complete lists of references we suggest reading reviews 2012IJMPA..2730018G (); 2018arXiv180505109K () and the book by Grozin Grozin:2007 (). In this work, we rely on a different approach to derive the general results—a direct calculation of hypergeometric integrals occurring in the representation as suggested in Mikhailov:1988nz () for the special case (see especially the preprint Mikhailov:1988nz-JINRrep () for technical details).
The most general case of with arbitrary indices was studied in Bierenbaum2003 (). There, the corresponding Feynman integral was calculated in the Mellin–Barnes representation and expressed in terms of double hypergeometric series (for earlier less general results obtained in the same fashion for see belokurov1984calculating6082 (); Usyukina1989 ()). Acting essentially in the same way, we obtain the twofold Mellin moment with and being a natural number and a real one, respectively. The result is a sum of the Kampé de Férriet (KdF) functions. Even more general case of the twofold Mellin moment with an arbitrary real and is worked out in terms of the generalized Lauricella functions of three variables (more precisely, the series introduced by Srivastava srivastava1967generalized (); see also section 1.5 in srivastava1985multiple ()).
Evaluating master integrals in hypergeometric series is of value for at least two reasons. First of all, it opens up one of the avenues to expose analytic properties of the integrals using the machinery that has been developed in the theory of special functions. Secondly, in many physically relevant cases we can almost immediately expand these multivariate functions in dimensional regularization parameter. To this end, one can employ the method developed by Moch, Uwer, and Weinzierl some time ago 2002JMP….43.3363M () and suitable for expansions of KdF, Lauricella, and generalized hypergeometric functions in the vicinity of an even number of space-time dimensions (C++ and FORM implementations of the method are described in 2002CoPhC.145..357W (); Moch:2005uc ()). For univariate functions there are a variety of algorithms and general theorems about the Laurent expansion of these functions near even and odd , e.g. 2006JHEP…04..056K (); 2007JHEP…11..009K (); 2007JHEP…02..040K (); Huber:2005yg (); 2008CoPhC.178..755H () (see also references therein).
The rest of the paper is organized as follows. In section 2, we introduce our notation and illustrate it on the simplest one-loop example of the two-current correlator. In section 3.1, we discuss in detail our method of evaluating the integral that results in a representation for in terms of the KdF functions of two variables in section 3.2. Section 3.3 is devoted to some important cases of reduction of these KdF functions to hypergeometric series . In section 3.4, the calculation of various moments of is discussed. We consider the case of the indices , which is important for QCD applications, in section 4. There we also derive Mellin moments of and compare them with the results obtained in the literature previously. Our conclusions are given in section 5, while the definitions of the hypergeometric functions used in this paper and some technical issues are treated in three appendices.
2 Simple example: one-loop integral
First, we introduce a generalization of the function for the one-loop integral with composite vertices :
where a slash (beside the composite vertex) on the line with momentum means factor ; is a light-cone vector, , normalized so that . The function is dimensionless and reduces to the usual one-loop function (see appendix B of ref. Chetyrkin:1980pr (), section 1.5 in Smirnov:2006 () or section 3.1 in Grozin:2007 ()) if it is integrated over :
The function has obvious symmetry:
The integral (2) is easily calculated when the propagators and are cast into the representation and the Dirac delta function is substituted by its Fourier integral. The result for the function reads
The Mellin transform of the function is
Here and in what follows, a th Mellin moment of a function is denoted by the underlined argument of the function ; the ’s with multi-indices are defined as
The definition of the two-row function is clear from eq. (5):
A slightly more general one-loop integral with also comes in handy:
where is the Heaviside step function.
3 Kite-type correlator as a generalized hypergeometric series
3.1 The correlator in the representation. Reduction to a double integral
Let us now consider a general two-loop master integral with composite external vertices:
where is the degree of divergence of the integral and the propagator factors in the denominator are
The integral (9) is symmetric under the following interchanges of its parameters:
Note that the th Mellin moment of eq. (14) is 0th moment of the Mellin convolution of two one-loop functions that is a product of 0th moments of these functions, which is a well known property of the two-loop master integral (section 4.1 in Grozin:2007 ()).
In the cases of and , the integral (9) vanishes—it reduces to a product of one-loop integral and vacuum loop that is zero in dimensional regularization (at least for ):
where is the one-loop vacuum integral, which was considered in ref. 1985TMP….62..232G ():
To evaluate the integral (9) in the general case of all being non-zero, we borrow a trick invented in ref. Mikhailov:1988nz-JINRrep () (see appendix A therein). Firstly, we substitute the denominators by their representation and the Dirac delta functions by their Fourier integrals. As usual, this allows us to evaluate momentum integral as the Gaussian one, but also leaves a trace of two Dirac delta functions. We have
Here, , , , 2, and , , 1, 2, are the Symanzik polynomials of the parameters , see figure in eq. (9), and the fractions and :
Secondly, we apply the Borel transform to both sides of eq. (21). The definition of the Borel transform and the necessary images are as follows:
Considering both sides of eq. (21) as functions of , applying the Borel transform and rescaling the parameters by the Borel parameter , , we eventually arrive at the following expression:
Finally, enjoying the plentiful amount of the Dirac deltas, we can immediately eliminate three integrals. To this end, it is convenient to make substitutions
Then, integrating over , , and , we obtain a factorized expression
Here, is a function of and is defined as a leftover double integral:
where . Note that the expression in the square brackets in eq. (27) is a function of a single variable defined as a ratio of fractions and , conformal ratio . We borrowed the name for from ref. Braun2017 () (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.
3.2 The correlator as the Kampé de Fériet function
For arbitrary nonvanishing one of the integrations in eq. (28) can be easily performed in terms of the Appell function . Indeed, making a substitution , we have as an integral over a classic Euler-type integral representation of the first Appell function fwc:07.36.07.0001.01 ():
The second equality in the equation above can be easily obtained by applying the autotransformation properties of the Appell function fwc:07.36.17.0005.01 ().
In the general case, the remaining integral over can be evaluated in terms of the Kampé de Fériet (KdF) function
The KdF function and the Appell function are defined as hypergeometric series (62) and (A), respectively (see appendix A). With the help of eqs. (71) and (72), we can also write the above KdF function in the form of series in a variable on the interval due to the step functions in (27):
3.3 Reduction to a univariate hypergeometric series
We can isolate some cases when the complicated hypergeometric series (31) reduces to a simpler one. The simplest instances of such reduction are for and . In the former case the KdF function (31) is unity. In the latter one the KdF function (31) is a product of and , which can be expressed through Euler gamma functions. It is easy to prove that we have the same results for the functions with one vanishing index as ones previously obtained in eqs. (13)–(17).
Another useful for calculations reduction is obvious from the Eulerian integral (30)—the Appell double series is simplified to the hypergeometric function if we set ,
If is a natural number, , the series truncates at . Therefore, due to the symmetries (11), (12), and (29), the function as represented by eq. (27) is a finite sum of hypergeometric functions if two adjacent external edges of the diagram have indices that are natural numbers (e.g., , ), see figure 2 below.
3.4 Mellin moments of
One- and twofold Mellin moment of the correlator (27),
can be expressed in terms of the generalized Lauricella hypergeometric function. To prove this, we should simply make the following changes of integration variables: in the above Mellin integrals (36) and in (28). As a result, we obtain
Here, is a generalized Lauricella function defined in appendix A, eq. (60). The arrays of the parameters of the two-row functions, , , and Lauricella functions , , are given explicitly in eqs. (77)–(78).
At least in some cases the threefold hypergeometric series in eqs. (37) and (38) reduce to simpler KdF functions. To see this, we can apply the method of ref. Bierenbaum2003 () to the original two-loop integral (9) (see also refs. belokurov1984calculating6082 (); Usyukina1989 () with less general results). Firstly, we write the Mellin–Barnes representation for one of the subgraphs of the two-loop diagram:
where , are dimensionless parameters and the denominators , were defined earlier in eq. (10). Note that integrating eq. (39) over leads to the Mellin–Barnes representation for the triangle diagram given in refs. Boos:1987bg (); Boos:1990rg (); Usyukina1975 () since the Dirac delta becomes identity and
For the zeroth moment the last Dirac delta in the above expression drops out and with the help of eq. (40) we arrive at a normal looking twofold Mellin–Barnes integral that can be evaluated in terms of double series:
The same is true for twofold moments: